Model based control of a flight simulator motion system - DCSC
Model based control of a flight simulator motion system - DCSC
Model based control of a flight simulator motion system - DCSC
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<strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong><br />
<strong>system</strong>
<strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong><br />
<strong>system</strong><br />
PROEFSCHRIFT<br />
ter verkrijging van de graad van doctor<br />
aan de Technische Universiteit Delft,<br />
op gezag van de Rector Magnificus pr<strong>of</strong>. ir K.F. Wakker<br />
voorzitter van het College voor Promoties,<br />
in het openbaar te verdedigen op maandag 10 december 2001 om 16.00 uur<br />
door<br />
Sjirk Holger KOEKEBAKKER<br />
werktuigkundig ingenieur<br />
geboren te Ermelo
Dit proefschrift is goedgekeurd door de promotor:<br />
Pr<strong>of</strong>. ir O.H. Bosgra<br />
Samenstelling promotiecommissie:<br />
Rector Magnificus voorzitter<br />
Pr<strong>of</strong>. ir O.H. Bosgra Technische Universiteit Delft, promotor<br />
Dr ir A.J.J. Van der Weiden Technische Universiteit Delft, toegevoegd promotor<br />
Pr<strong>of</strong>. dr ir M. Steinbuch Technische Universiteit Eindhoven<br />
Pr<strong>of</strong>. dr ir J.A. Mulder Technische Universiteit Delft<br />
Pr<strong>of</strong>. dr ir P.M.J. van den H<strong>of</strong> Technische Universiteit Delft<br />
Pr<strong>of</strong>. dr ir J.H. de Leeuw University <strong>of</strong> Toronto<br />
Ir P.C. Teerhuis Technische Universiteit Delft<br />
Ir P.C. Teerhuis heeft als begeleider in belangrijke mate aan de<br />
totstandkoming van het proefschrift bijgedragen.<br />
Keywords: model <strong>based</strong> <strong>control</strong>, <strong>flight</strong> simulation, parallel <strong>motion</strong> <strong>system</strong>s<br />
ISBN 90-370-0194-7<br />
Copyright c 2001 by S.H. Koekebakker<br />
All rights reserved. No part <strong>of</strong> the material protected by this copyright notice may be reproduced or<br />
utilized in any form or by any means, electronic or mechanical, including photocopying, recording or<br />
by any information storage and retrieval <strong>system</strong>, without written permission from the copyright owner.<br />
Printed in the Netherlands by Ponsen & Looijen b.v.
To faith<br />
We’re on a road to nowhere<br />
Come on inside<br />
Takin’ that ride to nowhere<br />
We’ll take that ride<br />
Talking Heads, 1985<br />
Running is no sport<br />
but a way <strong>of</strong> travelling<br />
displacing body and mind<br />
Vrij naar Jan Knippenberg
Voorwoord<br />
In de ruwweg zeven jaren, maanden, dagen en uren tussen de start van dit onderzoek en het<br />
moment waarop het verdedigd zal worden is veel gebeurd. Beginnend vanuit een kamertje<br />
in een studentenhuis in Delft kon ik me weinig voorstellen bij de grote, relatief eenzaam<br />
verrichte inspanningen waar kond van werd gedaan in het voorwoord van de verschillende<br />
proefschriften. Onderzoek doen is leuk en zeker in het kader van een multidisciplinair<br />
project werk je daarbij met veel mensen samen. Pas nadat het onderzoek na vier jaar<br />
afgerond maar nog niet helemaal opgeschreven was en naast een aantrekkelijke voltijds<br />
baan en een fantastisch gezin met inmiddels twee dochters vanuit Venlo het werk in groter<br />
geheel vastgelegd moest worden, kon ik mij vinden in vele van die eerder gelezen frasen.<br />
Alhoewel een groot aantal mensen je op een <strong>of</strong> andere manier bijstaan tijdens de gehele periode,<br />
moet het schrijfwerk toch grotendeels alleen werkend achter de computer gebeuren.<br />
Hier een punt achter zettend is eindelijk de tijd ook gekomen om al deze mensen uitdrukkelijk<br />
te bedanken voor de hulp die ze me op verschillende wijze hebben geboden. Een<br />
aantal mensen zal ik alleen als groep noemen gezien de vele veranderingen die zich over<br />
deze lange periode hebben voorgedaan en het risico daarbij individuele namen over te slaan.<br />
Ten eerste wil ik mijn promotor Okko Bosgra bedanken die me in staat heeft gesteld<br />
mijn promotieonderzoek te verrichten binnen zijn vakgroep. Vervolgens bedank ik mijn<br />
directe begeleiders, Ton van der Weiden met zijn directheid en politiek gevoel voor tact<br />
waar nodig in dit project en Piet Teerhuis met zijn aanstekelijke liefde voor de techniek en<br />
in het bijzonder de hydraulische <strong>system</strong>en.<br />
Zoals in de referenties terug te vinden is, heeft een grote groep studenten het laatste (anderhalf<br />
<strong>of</strong> meer) jaar van de ingenieursopleiding een afstudeerproject in het kader van mijn<br />
onderzoek kunnen verrichten. Dit heb ik als zeer prettig ervaren. Zeker op het moment dat<br />
er gelijktijdig vijf, zes enthousiaste mensen de ’Simona <strong>motion</strong> <strong>control</strong> room’ bij werktuigbouwkunde<br />
bevolkten en we door slim tactisch spel een (audiovisio)hardloopwedstrijd konden<br />
winnen zonder als eerste aan te komen. Boris Rijnten, Vasken der Kevorkian, Sander<br />
Bettendorf, Philippe Piatkiewitz, Jan van Hulzen, Maris Franken, Riad Al-Saidi, Etienne<br />
van Zuijlen en Arne Scheffer bedankt voor jullie mede- en vooral samenwerking.<br />
Alle groeps- en ex-groepsleden van de vakgroep Systeem- en regeltechniek wil ik bedanken<br />
voor de collegialiteit, ook in de periode dat ik al uit Delft weg was. Met name door<br />
de combinatie van een grote groep promovendi, de ervaren staf en de rechtdoorzee support<br />
groep was er sprake van een levendige en productieve sectie. Marco Dettori bracht daar<br />
als langstzittende kamergenoot nog wat Italiaanse cultuur in die gecombineerd met Carsten<br />
Scherers internationale en vooral ook wiskundige achtergrond tot prettig pittig smakende<br />
lunchdiscussies leidden. Thomas de Hoog wil ik ook speciaal noemen vanwege het feit dat<br />
vii
viii Voorwoord<br />
ik de template van zijn proefschrift en allerhande LateX-advies goed heb kunnen gebruiken.<br />
Door het interfacultaire karakter van het <strong>simulator</strong>project is er interactie met veel mensen<br />
die ik als zeer leerzaam heb ervaren. Ik bedank alle mensen betrokken bij Simona, vooral<br />
ook door de eigenlijk nooit ontbrekende en ook zeker noodzakelijke passie voor het maken<br />
van een technisch complex apparaat waarbij altijd vele hobbels genomen moeten worden.<br />
Ik wil een aantal mensen hiervan met name noemen waarmee ik door hun directe betrokkenheid<br />
bij mijn onderzoek nauw samengewerkt heb. Sunjoo Advani, die met zijn enthousiasme<br />
het Simonaproject als projectleider lange tijd over pieken en dalen getrokken<br />
heeft en ook tot nu nog grote betrokkenheid bij dit onderzoek heeft getoond. Gert van<br />
Schothorst, die, twee jaar eerder als werktuigkundig promovendus bij het <strong>simulator</strong>project<br />
begonnen, veel werk op het gebied van de modelvorming rond hydraulische <strong>system</strong>en heeft<br />
verricht waarop ik met mechanica en regeling verder kon bouwen. Met Peter Valk kon ik,<br />
gegeven ook de gelijkgestemdheid op sportief gebied, eindeloze discussies voeren. Rens<br />
de Keijzer, Kees Slinkman, Rolf van Overbeek, Fred den Hoedt, John Dukker en Ad van<br />
der Geest brachten allen vanuit onze faculteit een praktisch technische insteek waar ik nog<br />
steeds niet aan kan tippen. Ook de wizardous meet- en communicatiekastjes van de prettig<br />
nuchtere Henk Huisman moet ik niet vergeten.<br />
Buiten het werk brachten in de eerste jaren vooral de gezamenlijk gelopen ettelijk<br />
honderden trainingskilometers, tientallen etentjes en (estafette)wedstrijden onder andere in<br />
Londen, Parijs, Barcelona en een tweede keer ook weer tactisch winnend rond het IJsselmeer<br />
met het losse en toch ook weer vaste groepje jonge oude Delvers de nodige afleiding<br />
en zeker ook vriendschap. Daarnaast leefden ook veel andere vrienden, kennissen en<br />
familie mee. Allen bedankt!<br />
Alhoewel de frequentie verlaagd is houdt nu een groep enthousiaste collega’s de sportiviteit<br />
en de felle en gelukkig lekker oeverloze en daarmee gevarieerde discussies tijdens de<br />
gezamenlijke etentjes erin. Ik wil graag hierbij ook mijn ho<strong>of</strong>d bij de researchafdeling van<br />
Océ, Jo Geraedts, bedanken voor zijn steun ten aanzien van mijn werk voor dit onderzoek<br />
en dan met name zijn motivatie om er vooral op een goede manier een punt achter te zetten.<br />
Tenslotte kan ik van hieruit nooit mijn lieve Els, Hanneke en Ingrid voldoende bedanken<br />
voor het geduld dat ze met me en het vertrouwen dat ze in me hebben gehad. Ik zal van<br />
’pappa z’n werk’ met vier PC’s en meer dan twintig meter stapels papieren archief op twaalf<br />
vierkante meter maar (g?)een museum maken.<br />
Sjirk Koekebakker<br />
Venlo, oktober 2001.
Summary<br />
The extremely high safety requirements in air transportation require a proper understanding<br />
<strong>of</strong> human pilot behaviour under extreme weather conditions. Advanced research in this<br />
area must utilize <strong>flight</strong> <strong>simulator</strong> equipment that is able to reproduce critical <strong>flight</strong> conditions<br />
with great fidelity. The turbulence and wind shear phenomena in the lower part <strong>of</strong><br />
the atmosphere contain a velocity spectrum that induces large forces on the aircraft over a<br />
wide frequency range. Thus, <strong>motion</strong> reproduction and generation <strong>of</strong> desired accelerations<br />
in a high quality <strong>flight</strong> <strong>simulator</strong> constitutes a complicated <strong>control</strong> task that must exploit<br />
the capabilities <strong>of</strong> an available actuation <strong>system</strong> to its limits. Predominately, these <strong>system</strong>s<br />
are comprised <strong>of</strong> six hydraulically actuated linear servo motors, similar to many robotic<br />
manipulators. The <strong>control</strong> <strong>of</strong> modern robotic devices employs advanced model <strong>based</strong> <strong>control</strong><br />
strategies that allow higher performance to be achieved than less structured approaches,<br />
and can provide more insight into the <strong>system</strong> limitations. In <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>control</strong>,<br />
application <strong>of</strong> such methods has been almost absent. One <strong>of</strong> the reasons might be that<br />
the <strong>motion</strong> <strong>system</strong>s have a degree <strong>of</strong> complexity, which does not allow exact modelling or<br />
detailed models as part <strong>of</strong> the <strong>control</strong>ler. Application has required an intermediate step <strong>of</strong><br />
extracting a reasonably detailed model, which described the most relevant dynamic characteristics.<br />
The central problem <strong>of</strong> this research was to investigate which relevant <strong>system</strong> knowledge<br />
should be used in a model <strong>based</strong> <strong>control</strong> strategy, and to quantify the extent to which this improves<br />
the <strong>control</strong>led dynamics <strong>of</strong> the <strong>simulator</strong>. First, the full design process was defined,<br />
structured and evaluated. This began with modelling and analysis, followed by <strong>control</strong> synthesis<br />
and, finally, testing the <strong>system</strong>. Many solutions and procedures <strong>of</strong> the steps taken<br />
were given in literature, but an integral approach, having the specific properties and requirements<br />
<strong>of</strong> <strong>flight</strong> simulation <strong>motion</strong> generation in mind, was still lacking. In this research,<br />
integratibility and applicability were the main arguments in choosing the most suitable alternative<br />
or, if necessary, newly proposed variant, <strong>of</strong> each step in the <strong>system</strong>/<strong>control</strong> design<br />
procedure.<br />
First, a relevant model structure for analysis and <strong>control</strong> through physical modelling was<br />
derived. Next, the structure was evaluated by experiments. <strong>Model</strong> parameters were identified<br />
and boundaries were provided to the extent for which the model validity holds. Then,<br />
the most appropriate model <strong>based</strong> robot <strong>control</strong> strategy was chosen and, given the particular<br />
<strong>system</strong> properties and performance requirements, modified for this application. The next<br />
step was actual application within the <strong>simulator</strong> environment, after which the testing procedure<br />
was defined in order to quantify the performance with respect to the requirements, and<br />
to conventional <strong>system</strong>s.<br />
ix
x Summary<br />
Structural insight into the <strong>system</strong> kinematics and dynamics was required in order to<br />
maintain at least the level <strong>of</strong> robustness <strong>of</strong> conventional <strong>control</strong> strategies, and simultaneously<br />
increase <strong>motion</strong> performance and predictability. The physical modelling began with<br />
an analysis <strong>of</strong> the <strong>system</strong> kinematics and, as a contribution <strong>of</strong> this thesis, a method was proposed<br />
to exclude singular points from the operational workspace and, subsequently, safely<br />
and efficiently determine platform pose from the actuator positions in real time. Calibration<br />
<strong>of</strong> the physical <strong>system</strong> improved further its positioning accuracy by an order <strong>of</strong> magnitude.<br />
The dynamics <strong>of</strong> the Stewart Platform <strong>of</strong> this <strong>simulator</strong> were expressed by a set <strong>of</strong> explicit<br />
differential equations using platform coordinates. It was shown in theory and practice<br />
that the rigid body modes resulting from interaction <strong>of</strong> hydraulics and mechanics form the<br />
most relevant <strong>motion</strong> <strong>system</strong> dynamics in <strong>simulator</strong> applications. The platform to actuator<br />
coordinate jacobian is a central operator here. The research showed also how the parallel<br />
actuation can be viewed from a new set <strong>of</strong> coordinates as six nearly independent hydraulic<br />
<strong>system</strong>s driving a single mass.<br />
The applied model <strong>based</strong> <strong>control</strong>ler attained several goals, which could not be implemented<br />
conventionally. First, it directs the required forces for the desired accelerations<br />
along the appropriate vectors experiencing largely different masses. Then, the feedback<br />
paths <strong>of</strong> the positional errors also use these decoupled directions. Finally, the interaction<br />
between hydraulics and mechanics is minimised over the feed forward path primarily by<br />
applying compensation <strong>of</strong> the required oil flow given the desired velocity. Implementation<br />
in practice showed an even higher bandwidth, less peaking, and a more equalised response<br />
<strong>of</strong> the model <strong>based</strong> <strong>control</strong>ler over each degree <strong>of</strong> freedom compared with a conventional<br />
strategy. Even more can be gained since model <strong>based</strong> <strong>control</strong> can more effectively use<br />
predictive information from the vehicle simulation model.<br />
Performance limitations to this technique can be found at frequencies where parasitic<br />
effects such as structural flexibility, and hydraulic valve and transmission line dynamics<br />
become significant, or when the <strong>system</strong> no longer behaves as being rigidly attached to the<br />
floor. However, through the application <strong>of</strong> a light weight and rigid design <strong>of</strong> the moving<br />
platform, and with the <strong>control</strong> strategy proposed in this thesis, the <strong>system</strong> bandwidth can<br />
be raised to a sufficiently high level, thereby enabling a wider range <strong>of</strong> <strong>motion</strong> <strong>based</strong> <strong>flight</strong><br />
simulation research to be executed than heret<strong>of</strong>ore.
Contents<br />
Voorwoord vii<br />
Summary ix<br />
1 Introduction 1<br />
1.1 Flight simulation . .............................. 1<br />
1.1.1 The art <strong>of</strong> simulation . . ....................... 2<br />
1.1.2 Mechanism <strong>of</strong> <strong>motion</strong> perception . . . ............... 4<br />
1.1.3 Historical perspective . . ....................... 8<br />
1.1.4 Motion simulation . . . ....................... 9<br />
1.1.5 SIMONA project . . . . ....................... 12<br />
1.2 Motion <strong>control</strong> . . .............................. 17<br />
1.2.1 Motion <strong>control</strong> structure ....................... 17<br />
1.2.2 Motion <strong>control</strong> specifications . . ................... 19<br />
1.2.3 Conventional <strong>simulator</strong> <strong>motion</strong> <strong>control</strong> ............... 21<br />
1.2.4 Robot <strong>control</strong> . . . . . . ....................... 22<br />
1.3 Problem statement .............................. 25<br />
1.3.1 General problem statement . . . ................... 25<br />
1.3.2 Structuring the problem statement;<br />
approach in research . . ....................... 25<br />
1.4 Outline <strong>of</strong> the thesis .............................. 28<br />
2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s 31<br />
2.1 Parallel <strong>motion</strong> <strong>system</strong>s . . . . . ....................... 32<br />
2.2 Kinematics .................................. 33<br />
2.2.1 Notation and definitions ....................... 33<br />
2.2.2 Calculating velocity and acceleration by differentiation . . . .... 34<br />
2.2.3 Parametrising orientation by euler angles . . . . . . ........ 36<br />
2.2.4 Parametrising orientation by euler parameters . . . . ........ 38<br />
2.2.5 Jacobian matrices . . . . ....................... 40<br />
2.2.6 Parallel, serial and singular configurations . . . . . . ........ 41<br />
2.2.7 Stewart platform definitions and assumptions . . . . ........ 43<br />
2.2.8 Stewart platform kinematics . . ................... 44<br />
xi
xii Contents<br />
2.2.9 Interpretation and use <strong>of</strong> the jacobian matrix, J l�x .......... 45<br />
2.2.10 Velocity and acceleration <strong>of</strong> the actuator joints . . . ........ 47<br />
2.2.11 The Simona <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> kinematics . . . .... 49<br />
2.3 Analysis <strong>of</strong> the Stewart platform kinematics . . ............... 50<br />
2.3.1 Convergence NR-iteration . . . ................... 53<br />
2.3.2 NR-convergence Stewart platform . . . ............... 54<br />
2.3.3 Lipschitz condition on jacobian ................... 56<br />
2.3.4 Exclusion <strong>of</strong> singular points . . ................... 58<br />
2.3.5 Sufficient update frequency . . ................... 59<br />
2.4 <strong>Model</strong>ling the mechanical part <strong>of</strong> the <strong>system</strong> dynamics . . . ........ 60<br />
2.4.1 General theory in modelling the dynamics <strong>of</strong> mechanical <strong>system</strong>s . 60<br />
2.4.2 Example in modelling using Kane’s method . . . . . ........ 64<br />
2.4.3 Stewart platform dynamics . . . ................... 66<br />
2.4.4 Basic Stewart platform mechanical model dynamics ........ 66<br />
2.4.5 Influence <strong>of</strong> the actuator inertial forces ............... 67<br />
2.4.6 The Stewart platform model including actuator inertia . . . .... 69<br />
2.4.7 Parasitic mechanical aspects: modelled as linear flexibility . .... 70<br />
2.5 Chapter Resume . . .............................. 73<br />
3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s 75<br />
3.1 The basic structure <strong>of</strong> hydraulic actuators . . . ............... 76<br />
3.1.1 Leakage . . .............................. 78<br />
3.1.2 Basic hydraulically driven mechanical <strong>system</strong> model ........ 79<br />
3.2 Extensions . .................................. 80<br />
3.2.1 Servo valve .............................. 80<br />
3.2.2 Transmission lines . . . ....................... 85<br />
3.3 Integrated <strong>system</strong> . .............................. 89<br />
3.3.1 Passive input/output pairs in <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s .... 90<br />
3.3.2 The 1-d.o.f. hydraulically driven mechanical <strong>system</strong> ........ 94<br />
3.3.3 Additional mechanical modes . ................... 97<br />
3.4 Hydraulically driven Stewart platform . ................... 101<br />
3.4.1 Basic model structure hydraulically driven <strong>system</strong>s . ........ 102<br />
3.4.2 Dynamical and kinematical properties <strong>of</strong> the SRS-<strong>motion</strong> <strong>system</strong>;<br />
Design aspects . . . . . ....................... 104<br />
3.4.3 Actuator inertial effects . ....................... 110<br />
3.4.4 Analyzing the SRS hydraulically driven <strong>motion</strong> <strong>system</strong> model . . . 112<br />
3.4.5 Connection <strong>of</strong> a flexible foundation to the SRS <strong>system</strong> model . . . 116<br />
3.5 Chapter Resume . . .............................. 118<br />
4 Parameter identification and model validation 119<br />
4.1 Calibration .................................. 120<br />
4.1.1 Calibrating the Stewart platform ................... 121
Contents xiii<br />
4.1.2 Redundant measurement <strong>of</strong> the platform pose . . . . ........ 123<br />
4.1.3 Stewart platform pose measurement in practice . . . ........ 126<br />
4.1.4 Identification <strong>of</strong> the kinematical parameters . . . . . ........ 128<br />
4.1.5 Results in calibrating the SRS <strong>motion</strong> <strong>system</strong> . . . . ........ 131<br />
4.2 Stewart platform model parameter identification ............... 133<br />
4.2.1 Gravitational force determination . . . ............... 133<br />
4.2.2 Identification <strong>of</strong> the inertial properties . ............... 136<br />
4.3 Frequency response model validation . . ................... 141<br />
4.3.1 Frequency response dummy platform . ............... 142<br />
4.3.2 Additional dynamics into the higher frequency area . ........ 147<br />
4.4 Chapter Resume . . .............................. 161<br />
5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> 163<br />
5.1 Introduction .................................. 163<br />
5.2 Another look at the <strong>control</strong> problem . . ................... 164<br />
5.3 Control Strategy . . .............................. 166<br />
5.4 Inner loop pressure <strong>control</strong> . . . ....................... 168<br />
5.5 Multivariable feedback linearisation . . ................... 173<br />
5.5.1 Feedback linearising robotic manipulators . . . . . . ........ 173<br />
5.5.2 Feedback linearisation <strong>of</strong> a Stewart platform . . . . ........ 174<br />
5.5.3 Implicit state measurement requirements . . . . . . ........ 176<br />
5.5.4 Outer loop <strong>control</strong> . . . ....................... 177<br />
5.6 Reference model <strong>based</strong> feed forward . . ................... 178<br />
5.6.1 Reference model <strong>based</strong> predictors . . . ............... 179<br />
5.6.2 Construction <strong>of</strong> the reference model <strong>based</strong> <strong>control</strong>ler ........ 181<br />
5.7 Implementational issues . . . . . ....................... 185<br />
5.8 Performance quantification . . . ....................... 185<br />
5.8.1 Motion <strong>system</strong> evaluation methods and requirements ........ 187<br />
5.8.2 New test procedure . . . ....................... 189<br />
5.9 Experimentally evaluating performance ................... 191<br />
5.9.1 Experimental set up . . . ....................... 191<br />
5.9.2 Characteristics <strong>of</strong> the Simona <strong>motion</strong> <strong>system</strong> . . . . ........ 193<br />
5.10 Chapter Resume . . .............................. 200<br />
6 Review and discussion on the results 203<br />
6.1 Flight simulation . .............................. 203<br />
6.2 Motion <strong>system</strong> specifications . . ....................... 204<br />
6.3 Theoretical modelling . . . . . . ....................... 205<br />
6.4 Experimental modelling and validation . ................... 206<br />
6.5 Control strategy and evaluation . ....................... 208<br />
7 Conclusion and recommendations 211
xiv Contents<br />
7.1 Conclusion .................................. 211<br />
7.2 Recommendations . .............................. 213<br />
A Frequency domain measurements 215<br />
B Derivation <strong>of</strong> actuator inertial properties 227<br />
Bibliography 231<br />
Glossary <strong>of</strong> symbols 241<br />
Samenvatting en CV 247
Chapter 1<br />
Introduction<br />
”As man travels, he vibrates. For as he travels, he generally experiences accelerations<br />
not perfectly uniform in magnitude nor direction, and these he feels as varying forces or<br />
vibrations. Travelling faster, these vibrations become more severe, until he learns new<br />
ways to reduce them. But the pioneers will vibrate considerably.” [23] These phrases by<br />
Clark, originally used to motivate research into the tolerable limits <strong>of</strong> <strong>motion</strong> cues for human<br />
beings in the early development <strong>of</strong> the U.S. space program, exactly point at the aspect <strong>of</strong><br />
piloted simulation, which is the subject <strong>of</strong> this research. The <strong>motion</strong> cues, which are a<br />
consequence <strong>of</strong> travelling, can be stimulated without actually travelling, thereby enhancing<br />
perceived realism in a <strong>simulator</strong>. Further, since proper <strong>control</strong> <strong>of</strong> a vehicle requires sufficient<br />
reduction <strong>of</strong> vibration, the actual sensation <strong>of</strong> these resonances forms an essential part <strong>of</strong><br />
basic skill training.<br />
The tool to achieve this, is a <strong>motion</strong> <strong>system</strong>. By automatic <strong>control</strong>, the dynamics <strong>of</strong><br />
the <strong>motion</strong> <strong>system</strong> will have to be transformed into the dynamics <strong>of</strong> the vehicle with its<br />
environment. This requires knowledge <strong>of</strong> both <strong>system</strong>s. With the developments in computer<br />
technology, more and more information can be directly incorporated in the <strong>control</strong>ler. Given<br />
feasible trajectories generated by a reference generating <strong>system</strong> <strong>of</strong> e.g. an aircraft model in<br />
our case, this research considers the use <strong>of</strong> <strong>motion</strong> <strong>system</strong> knowledge in a model <strong>based</strong><br />
<strong>control</strong>ler.<br />
In this chapter, first an overview <strong>of</strong> the most important aspects <strong>of</strong> <strong>flight</strong> simulation will be<br />
given in Section 1.1. Then, the specific requirements and approaches towards <strong>motion</strong> <strong>control</strong><br />
will be discussed in Section 1.2. This will lead to the problem statement and approach <strong>of</strong><br />
this research in Section 1.3. Finally, the outline <strong>of</strong> this thesis will be given in Section 1.4.<br />
1.1 Flight simulation<br />
First, the different aspects <strong>of</strong> the complex <strong>system</strong> <strong>of</strong> <strong>flight</strong> simulation will be discussed.<br />
With an emphasis on the <strong>motion</strong> aspect, the evolution <strong>of</strong> these <strong>system</strong>s over the years will<br />
be sketched. Then, the Simona project centred around the development <strong>of</strong> a modern research<br />
<strong>simulator</strong>, in which this investigation was embedded, will be introduced. Finally, the specific<br />
features <strong>of</strong> the six degrees-<strong>of</strong>-freedom Stewart-Gough Platform <strong>motion</strong> <strong>system</strong>, nowadays<br />
generally used in <strong>flight</strong> simulation, will be stated.<br />
1
2 1 Introduction<br />
1.1.1 The art <strong>of</strong> simulation<br />
The fact that simulation is <strong>of</strong>ten still considered an art instead <strong>of</strong> a science [25],[131], reflects<br />
the fact that this complex mechanism <strong>of</strong> creating a virtual environment is not fully<br />
understood yet. The <strong>flight</strong> <strong>simulator</strong> or more specifically piloted simulation is, however,<br />
widely used nowadays, especially in the area <strong>of</strong> aerospace and astronautics. Following Advani<br />
[7], several areas <strong>of</strong> application can be recognised.<br />
Area <strong>of</strong> application<br />
Engineering <strong>simulator</strong>s are used to evaluate the characteristics <strong>of</strong> a vehicle. In the development<br />
<strong>of</strong> a new aeroplane, the <strong>simulator</strong> is currently intensively used during the whole<br />
process <strong>of</strong> design, due to their ability to predict problems and support <strong>flight</strong> clearance. With<br />
simulation, large reduction in <strong>flight</strong> testing could be achieved, which saved costs and allowed<br />
earlier certification [131]. Most engineering <strong>simulator</strong>s are fixed base i.e. without<br />
inertial <strong>motion</strong> since <strong>motion</strong> cues are not always important and necessary.<br />
However, the <strong>motion</strong> <strong>system</strong> is essential in predicting handling qualities, e.g. a pilot<br />
induced oscillation (PIO) due the dynamics <strong>of</strong> the vehicle, in a proper way [121]. Low<br />
frequent modes <strong>of</strong> the aircraft can be damped more appropriately with <strong>motion</strong>, while vibrations<br />
at higher frequencies amplified by the pilot in the loop, e.g. <strong>of</strong>ten modes due to<br />
structural aeroelasticity, can usually only be detected if the <strong>control</strong>led <strong>motion</strong> <strong>system</strong> leaves<br />
these modes unaltered.<br />
Training <strong>simulator</strong>s are used to train pilots both in the procedural (e.g. <strong>flight</strong> management)<br />
as well as the basic skill tasks (e.g. manual <strong>control</strong>). In military applications the procedural<br />
training is <strong>of</strong>ten done in fixed base <strong>simulator</strong>s. The basic skill training is mostly performed<br />
in-<strong>flight</strong> since the discrepancy between the <strong>simulator</strong> and aircraft is considered too large in<br />
extreme manoeuvres and <strong>of</strong> course actual <strong>flight</strong> training is much more exciting. In commercial<br />
airlines, the economic attractiveness <strong>of</strong> <strong>motion</strong> base <strong>simulator</strong>s is preferred over<br />
actual <strong>flight</strong> time and many hours are spent in training <strong>simulator</strong>s. Pr<strong>of</strong>iciency checks and<br />
re-currency training i.e. skill examinations and transfer to another type <strong>of</strong> plane, are allowed<br />
by the regulating authorities without actually flying for pilots with over 500 hours experience<br />
on an aircraft <strong>of</strong> the same group or 1500 (Europe) to 2500 hours (USA) grand total<br />
[25]. There is a tendency to further reduce this prescribed number <strong>of</strong> hours [143], which<br />
allow this so-called Zero-Flight-Time (ZFT) training, but exact knowledge <strong>of</strong> the required<br />
simulation fidelity is still lacking.<br />
Research <strong>simulator</strong>s are <strong>of</strong>ten applied for fundamental investigation into pilot/vehicle interaction<br />
and human perception research. These kind <strong>of</strong> <strong>simulator</strong>s most <strong>of</strong>ten require the<br />
highest level <strong>of</strong> performance w.r.t. the <strong>motion</strong> <strong>system</strong> as the resulting sensory input is essential<br />
in the manual <strong>control</strong> task [47].<br />
Apart from this fundamental scientific work, the <strong>simulator</strong> has also proved valuable<br />
for accident investigations [131]. Research <strong>simulator</strong>s have been used to study the circum-
1.1 Flight simulation 3<br />
stances <strong>of</strong> incidents due to clear air turbulence and other types <strong>of</strong> severe turbulence like gust,<br />
and come up with more beneficial strategies in <strong>control</strong> and better operational procedures.<br />
In these experiments, the <strong>motion</strong> <strong>system</strong> had to be capable <strong>of</strong> producing vibrations which<br />
made the pilot unable to read the instruments.<br />
Advantages <strong>of</strong> <strong>simulator</strong>s<br />
Reviewing the area <strong>of</strong> application reveals the various specific qualities <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong>.<br />
Over the years it has proved to have several advantages as compared to flying an actual<br />
aircraft.<br />
-Cost effectiveness. Both with respect to initial investment as to operational costs, the <strong>flight</strong><br />
<strong>simulator</strong>, though still requiring a considerable amount <strong>of</strong> money (<strong>of</strong>ten over 10M$<br />
and about 1k$/h), is still an order <strong>of</strong> a magnitude less expensive than most public<br />
transportation aircrafts. Further, the effectiveness <strong>of</strong> training and testing is in many<br />
cases even better.<br />
-Safety requirements. The <strong>simulator</strong> is a safe tool to test and train in conditions, which<br />
are dangerous in actual <strong>flight</strong>. Training <strong>simulator</strong>s can safely be used to show pilots<br />
the best operating procedures in unusual circumstances. Inexperienced pilots can<br />
be taught the bulk <strong>of</strong> basic skills and procedures avoiding the risk <strong>of</strong> putting them<br />
into the air. Many potential <strong>flight</strong> design errors can be detected in advance. Further,<br />
the fact that accidents can be analysed in detail with a <strong>simulator</strong>, enhances <strong>flight</strong><br />
safety. An indirect safety aspect is the cost effectiveness enabling a thorough and still<br />
economically feasible training program.<br />
-Environmental considerations. Every hour spent in a <strong>simulator</strong> instead <strong>of</strong> an aircraft,<br />
directly saves a considerable amount <strong>of</strong> kerosine burnt in the air. Indirectly, environmentally<br />
less harmful <strong>flight</strong> operation can be tested and trained in a <strong>simulator</strong>.<br />
Since the objective is not to duplicate, but to simulate an environment which creates the<br />
best and most effective training and testing results [25],[117], the <strong>flight</strong> <strong>simulator</strong> should<br />
also be considered as a training or testing tool in its own right, rather than as a substitute for<br />
the aircraft [143]. In this respect the <strong>simulator</strong> is <strong>of</strong>ten more effective.<br />
Training can be split up in parts and these modules can be taught separately, in a step by<br />
step approach. The <strong>simulator</strong> provides an objective tool to evaluate different subjects under<br />
equivalent conditions as the repeatability <strong>of</strong> the <strong>system</strong> is much higher. A <strong>simulator</strong> is very<br />
flexible in performing specific tests e.g. under desired weather conditions or at a certain<br />
airport.<br />
Flight <strong>simulator</strong> components<br />
In piloted simulation, a complex environment is created, which has to reflect the impression<br />
<strong>of</strong> travelling in the actual vehicle with an emphasis on the perceived realism <strong>of</strong> the task to<br />
be investigated or learned. This environment can be divided into several sub<strong>system</strong>s:<br />
-Interior. First <strong>of</strong> all the static environment <strong>of</strong> the pilot has preferably the look <strong>of</strong> the interior<br />
<strong>of</strong> the vehicle cockpit.
4 1 Introduction<br />
-Visual <strong>system</strong>. The exterior out-<strong>of</strong>-the-window sight is simulated with an image generating<br />
<strong>system</strong> in case <strong>of</strong> civil aircraft simulation <strong>of</strong>ten projected via a wide angle mirror<br />
attached in front <strong>of</strong> the <strong>simulator</strong> cockpit.<br />
-Instruments. The instruments in the cockpit give information about the state <strong>of</strong> the simulated<br />
vehicle and should actively be changed accordingly.<br />
-Motion <strong>system</strong>. The required inertial <strong>motion</strong>, the actual accelerations <strong>of</strong> the <strong>system</strong>, can<br />
be generated to some extent by actively moving the <strong>system</strong>.<br />
-Control loading. The pilot <strong>control</strong>s to the vehicle <strong>of</strong>ten responds to the given input by<br />
feedback <strong>of</strong> forces which reflect some <strong>of</strong> the external forces on the vehicle e.g. the<br />
aerodynamic force on the rudder, etc. These forces have to be generated artificially<br />
and in this case called <strong>control</strong> loading in a <strong>simulator</strong> and vehicles, where this has been<br />
decoupled mechanically (fly-by-wire).<br />
-Audio <strong>system</strong>. The generation <strong>of</strong> the sound a vehicle together with its environment produces<br />
adds to the perceived realism in the <strong>simulator</strong> and, above this, is used in the<br />
<strong>control</strong> task (gas throttle). In some cases, the audio <strong>system</strong> can also mask the parasitic<br />
noise produced by other sub <strong>system</strong>s [10].<br />
Apart from the static interior, all these sub<strong>system</strong>s add to the <strong>motion</strong> awareness and all cues<br />
(noticeable stimuli), which can be generated by these sub<strong>system</strong>s, have to be considered in<br />
evaluating a manual <strong>control</strong> task.<br />
The main unknown remains to what extent the <strong>simulator</strong> environment reflects the inthe-air<br />
conditions and to what extent this is required to achieve a certain training or test<br />
objective. First, the <strong>simulator</strong> is only as good as the data it contains i.e. the quality <strong>of</strong><br />
the model representing the aircraft and the <strong>flight</strong> environment to be simulated. Secondly,<br />
inherent limitations <strong>of</strong> the <strong>system</strong> used to simulate can be the cause <strong>of</strong> mismatches. The<br />
<strong>motion</strong> <strong>system</strong> <strong>of</strong> a <strong>simulator</strong> does posses one <strong>of</strong> the most obvious constraints with its very<br />
limited amount <strong>of</strong> travel. Fortunately, the pilot can be made to perceive the <strong>motion</strong> desired<br />
if the main cues are correctly simulated, allowing for some deviation in the others.<br />
1.1.2 Mechanism <strong>of</strong> <strong>motion</strong> perception<br />
Perception <strong>of</strong> <strong>motion</strong> has been studied for more than hundred years [88]. At that time,<br />
mainly experimental data was obtained. In the late 60’s <strong>of</strong> the past century models were<br />
proposed, which described the human behaviour in a <strong>motion</strong> <strong>control</strong> loop [69],[94]. With<br />
the crossover model <strong>of</strong> McRuer et al. [94], the pilot acts according to a describing function,<br />
H(j!), over the frequency, !,<br />
H(j!)=Kh 1j! +1<br />
2j! +1 e; dj! : (1.1)<br />
This model assumes only one main cueing input and considered additional dynamics <strong>of</strong> the<br />
sensory and response <strong>system</strong> as a simple time delay, e ; dj! . The parameters, gain, Kh, lead<br />
time constant, 1, and lag time constant, 2, are adapted such that a slope <strong>of</strong> -20 dB/decade
1.1 Flight simulation 5<br />
in the describing function frequency plot around the crossover frequency, ! c is attained <strong>of</strong><br />
the open loop <strong>system</strong> <strong>of</strong> human and the <strong>system</strong> to be <strong>control</strong>led, which usually results in<br />
a sufficient stability margin in the closed loop. This frequency is approximately 2 rad/s to<br />
5 rad/s in case <strong>of</strong> a <strong>flight</strong> <strong>control</strong> task (involving tracking and disturbance rejection) [53, 94,<br />
149]. With aircraft vehicle <strong>control</strong>, on which this model was experimentally validated, the<br />
open loop transfer function with a double integrator, 1=(j!) 2 , can be <strong>control</strong>led with PD,<br />
proportional/differential action in which 2 < 1=(!c) < 1.<br />
The optimal <strong>control</strong> model <strong>of</strong> Kleinman et al.[69] assumes that the pilot operates in an<br />
optimal way, given all the perceived cues with their respective limitations. This motivated<br />
the research into the mechanism <strong>of</strong> <strong>motion</strong> perception ([159] and the references therein) with<br />
a model <strong>based</strong> approach in which the interaction between the different stimuli, e.g. inertial<br />
<strong>motion</strong> and visual inputs, is taken into account. These models are still being refined [53]<br />
as the <strong>system</strong> is incompletely understood. Though the quantitative values are seen to be<br />
dependent on the subject, task and environment involved, the qualitative <strong>control</strong> behaviour<br />
<strong>of</strong> a well trained subject is reasonable consistent and reproducible.<br />
Motion results in stimuli inputs to the sensory organs. Above some sensory threshold,<br />
the sensory organs start to fire pulses to the central nervous <strong>system</strong>. The subject perceives<br />
<strong>motion</strong> from the set <strong>of</strong> cues generated in this way. These cues have to be reasonably consistent<br />
relative to each other. Otherwise disorientation and <strong>motion</strong> sickness can occur. Some<br />
deviations can be tolerated as long as the coherence functions <strong>of</strong> the subject from each cue to<br />
the corresponding movement have sufficient overlap [148]. Advantage from these tolerable<br />
differences is taken in simulation, which allows perceived <strong>motion</strong> to be simulated without<br />
the need for very large <strong>simulator</strong> excursions.<br />
The main sensory mechanisms to detect <strong>motion</strong> as given in the AGARD Advisory report<br />
159, [2] are:<br />
- Semicircular canals and Otoliths are called the vestibular organs. Both reside in the<br />
inner ear. The function <strong>of</strong> the semicircular canals is roughly that <strong>of</strong> rate gyros, from<br />
which angular velocity can be measured. The otoliths sense the specific force which<br />
is the combination <strong>of</strong> inertial acceleration and gravity.<br />
- Tactile or somatosensory receptors permit sensing a change <strong>of</strong> force on the body, e.g. due<br />
to a change in orientation. These sensors have a high-pass characteristic, which causes<br />
sustained uniform pressure to be faded to a certain reference level. Especially the fingertips<br />
are very capable detecting high frequency vibrations.<br />
- Proprioceptive and kinesthetic senses lead to information about the relative positions <strong>of</strong><br />
the parts <strong>of</strong> the body as well as their movements i.e. the body acceleration. This is<br />
attained through the double sided muscular coupling <strong>of</strong> force and velocity/position<br />
and an internal model <strong>of</strong> the body<br />
- The eyes primarily detect <strong>motion</strong> through the peripheral retina. With uniform <strong>motion</strong> <strong>of</strong><br />
a wide visual field, it is possible to create a so-called self <strong>motion</strong> sensation, which<br />
is called vection. The foveal area <strong>of</strong> the retina or central part is primarily associated<br />
with scanning and recognition i.e. the cognitive sense <strong>of</strong> self-<strong>motion</strong>.
6 1 Introduction<br />
Advani [7] also mentions the auditory <strong>system</strong> in the ears, from which through sound <strong>motion</strong><br />
can be detected.<br />
With the visual and auditory <strong>system</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong>, the pilot can be isolated from<br />
detecting inertial <strong>motion</strong> through sound or sight. With the other organs, inertial <strong>motion</strong> cues<br />
will be perceived as <strong>motion</strong>. In case <strong>of</strong> required high forces, the proprioceptive and tactile<br />
senses can also be stimulated with a so-called g-suit, mainly used in military applications.<br />
Control forces and vibrations provided by the <strong>control</strong> loading <strong>system</strong> will also influence the<br />
tactile senses.<br />
The vestibular <strong>system</strong> is considered to generate the main cues in the frequency area<br />
around the crossover frequency (:3 to 1Hz), essential to learn skill <strong>based</strong> <strong>control</strong> behaviour.<br />
This is the central issue in <strong>motion</strong>-<strong>based</strong> <strong>flight</strong> simulation. In order to recreate the skill<br />
<strong>based</strong> behaviour, the pilot in the <strong>simulator</strong> must be provided with well coordinated visual<br />
and vestibular cues. Motion cues around the crossover frequency frequency area can only<br />
be stimulated with actual inertial <strong>motion</strong> and this is the main reason a <strong>motion</strong> <strong>system</strong> is<br />
essential in case <strong>of</strong> piloted <strong>control</strong> tasks in <strong>flight</strong> simulation.<br />
In the low frequency area ( :1 Hz), <strong>motion</strong> is much easier to detect through the visual<br />
information <strong>of</strong> position and orientation at the instrument displays and exterior screen. Some<br />
mismatch with inertial <strong>motion</strong>, with the already low accelerations related to these frequencies,<br />
is allowed, even slightly superthreshold, without the pilot noticing discrepancies.<br />
Schroeder [126] notes that the review <strong>of</strong> different studies resulted in a wide range <strong>of</strong><br />
threshold values. This is confirmed by the study reported in [54] in which it is shown that a<br />
threshold depends on several factors such as the frequency contents <strong>of</strong> the stimulus and the<br />
task the subject has to fulfil i.e. mental load. Values range for the translational accelerations<br />
from :08 m=s 2 in heave, no task involved to :7 m=s 2 in a very busy situation experiencing<br />
surge (longitudinal direction). Rotational thresholds are found to be in a closer range <strong>of</strong><br />
:8 deg =s 2 to 1:5 deg =s 2 for pitch and roll, though Schroeder [126] reports a range <strong>of</strong><br />
:0035 deg =s 2 to 2 deg =s 2 for yaw <strong>motion</strong> (vertical axis rotation).<br />
Hosman [53] provides some transfer functions for the vestibular <strong>system</strong>. Both the semicircular<br />
channels as the otholits are modelled according to a function specified in the frequency<br />
domain:<br />
xout<br />
=<br />
xin<br />
K( Lj! +1)<br />
� (1.2)<br />
( 1j! + 1)( 2j! +1)<br />
where the ratio between (actual) input acceleration, x in, and noticed acceleration (xout in<br />
impulses per second, ips), is given by a second order transfer function with gain, K, two<br />
lag time constants, 1 and 2, and a lead time constant, L.<br />
Although the models <strong>of</strong> the two vestibular organs, which give the sensitivity over the frequency,<br />
have the same structure, the characteristic is different. This is due to the parameters,<br />
which are given by Hosman [53] as in Table 1.1.<br />
This means the sensitivity <strong>of</strong> the semicircular channels drops over the range from 0.025 Hz<br />
to 1.4 Hz after which it starts to drop again at 30 Hz, while the otholits become more sensitive<br />
from .16 Hz to .3 Hz, after which it starts to drop again at 10 Hz. This opposite effect<br />
in the relative important frequency range could maybe explain the fact that different conclusions<br />
can be drawn towards the question whether translational accelerations or angular
1.1 Flight simulation 7<br />
Semicircular Channels Otholits<br />
K 2 ips=( =s 2 ) 3:4 ips=(m=s 2 )<br />
L 0:11 s 1 s<br />
1 5:9 s 0:5 s<br />
2 0:005 s 0:016 s<br />
Table 1.1: The parameters <strong>of</strong> the models <strong>of</strong> the vestibular organs<br />
accelerations are more important. In case <strong>of</strong> roll and lateral <strong>motion</strong> in large aircrafts and yaw<br />
and lateral <strong>motion</strong> in helicopters, lateral <strong>motion</strong> cues are shown to be more important [126].<br />
In other cases reported in [54], [126] rotational cues are considered more relevant. Chung<br />
et al. [22] conclude that the translational and rotational <strong>motion</strong> relative to each other should<br />
not differ in phase nor amplitude since these types <strong>of</strong> <strong>motion</strong> are coupled as a function <strong>of</strong><br />
place.<br />
The <strong>motion</strong> cues provided through the vestibular <strong>system</strong> with inertial <strong>motion</strong>, change<br />
the describing function <strong>of</strong> the pilot [53], [149]. With the introduction <strong>of</strong> <strong>motion</strong>, the time<br />
delay in the describing function is seen to drop from 0.23 s to 0.12 s in case <strong>of</strong> a disturbance<br />
rejection task. The crossover frequency a pilot can achieve is lower in the absence <strong>of</strong> inertial<br />
<strong>motion</strong> and, as a result, also the performance is. Hall [47] argues and confirms by an<br />
example that, even with equivalent performance, the pilot can develop a different and false<br />
strategy if the cues in a <strong>simulator</strong> deviate from those in the aircraft. Especially in case <strong>of</strong><br />
relatively inexperienced pilots, all cues are important in the learning process <strong>of</strong> picking the<br />
most relevant [143].<br />
There is no doubt inertial <strong>motion</strong> is necessary in the appropriate perception <strong>of</strong> <strong>motion</strong>.<br />
Not much is known to what extent the inertial <strong>motion</strong> <strong>of</strong> the <strong>simulator</strong> is allowed to differ<br />
from the aircraft. In the low frequency area it is concluded that high pass filtering at .08 Hz is<br />
acceptable [149]. The crossover region between .1 Hz and 1 Hz is likely to be most sensitive<br />
to deviations. At higher frequencies, the situation is less clear. Pilots are seen to provide<br />
lead well over 3 Hz and consequently still respond to <strong>motion</strong> with increasing gain in this<br />
frequency area. Further, pilots respond much earlier when inertial <strong>motion</strong> is present. In the<br />
more general requirements <strong>of</strong> the international standard ISO 2631 [138], which evaluates<br />
human exposure to whole-body vibration, it is noted that humans are most sensitive to heave<br />
accelerations in the frequency area between 4 Hz and 8 Hz. The bandwidth <strong>of</strong> many <strong>motion</strong><br />
<strong>system</strong>s in this frequency area makes these <strong>system</strong>s prone to errors exactly in the area where<br />
the level <strong>of</strong> comfort is most sensitive.<br />
Over the years, many discrepancies in the <strong>motion</strong> <strong>system</strong>s <strong>of</strong> <strong>simulator</strong>s have been considered<br />
acceptable. There was no economical or technical feasible alternative. Other sub<strong>system</strong>s<br />
were also unable to represent reality in a satisfactory manner. A pilot was required<br />
to fly an aircraft on the basis <strong>of</strong> instrument displays and external sight without <strong>motion</strong> cues.<br />
Being able to fly a <strong>simulator</strong> without appropriate inertial <strong>motion</strong> qualified since it is more<br />
difficult than with an aircraft. In-<strong>flight</strong> training still formed a considerable part <strong>of</strong> the learning<br />
process. This point <strong>of</strong> view is changing as can be seen by putting the current state <strong>of</strong> the<br />
art in a historical perspective.
8 1 Introduction<br />
1.1.3 Historical perspective<br />
In [4], Adorian et al. provide a nice overview <strong>of</strong> the development <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong>s over<br />
the years. Almost directly with the introduction <strong>of</strong> the first aeroplanes, the first <strong>simulator</strong>s<br />
were built. Already in those days, a <strong>motion</strong> <strong>system</strong> was part <strong>of</strong> the <strong>simulator</strong>. Around 1910,<br />
these <strong>motion</strong> <strong>system</strong>s had to be driven by hand by the instructor, or wind, and mainly had<br />
to give the experience <strong>of</strong> a changing orientation around one to three axes. Soon the inertial<br />
<strong>motion</strong> was provided automatically. The Ruggles Orientator, developed in 1917, had full rotation<br />
around three axes and additionally vertical movement, all driven by electrical motors.<br />
The most successful device, the Link trainer, developed in 1927-29, was used well into the<br />
50’s. It was driven pneumatically and operated by the stick and rudder in the cockpit. The<br />
simulated effects were adjusted by trial and error independently for ailerons, elevators and<br />
rudder. In this way, the aircraft’s coordinated behaviour could not be represented properly.<br />
After 1930, for thirty years, the main developments concerned other sub<strong>system</strong>s than the<br />
<strong>motion</strong> <strong>system</strong> <strong>of</strong> the <strong>simulator</strong>. Early in the Second World Ward, hundreds <strong>of</strong> the so-called<br />
Celestrial Navigation Trainers, massive bomber crew trainers, were built. A moving dome<br />
provided for the ’correct’ out-<strong>of</strong>-the-window sight in case <strong>of</strong> a nightly atlantic ocean crossing.<br />
During the war, the visual <strong>system</strong> was developed further into actual image projection<br />
<strong>system</strong>s.<br />
With the instruments in the cockpit, all basic flying behaviour, all engine, electrical and<br />
hydraulic <strong>system</strong>s could be simulated. The computation, at first pneumatically, could solve<br />
the equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> an aircraft with the introduction <strong>of</strong> the analog computer.<br />
The first full aircraft <strong>simulator</strong> operated by an airline was installed in 1948 (a Boeing<br />
377 by PanAm). Most full aircraft <strong>simulator</strong>s did not have a <strong>motion</strong> <strong>system</strong> at that time,<br />
which was justified by the statement that ”modern pilots should not fly by the seat <strong>of</strong> their<br />
pants.” Partly, the <strong>control</strong> loading <strong>system</strong> was thought to compensate for the lack <strong>of</strong> <strong>motion</strong><br />
and provide for a realistic feel.<br />
With the improvement <strong>of</strong> <strong>flight</strong> test data and the increasing complexity <strong>of</strong> the aircrafts,<br />
the analog computer became the bottle-neck in the <strong>simulator</strong>. The demand for increased<br />
fidelity and reliability motivated the introduction <strong>of</strong> the digital computer in the <strong>flight</strong> <strong>simulator</strong><br />
<strong>system</strong>.<br />
However, correlation was poor with <strong>flight</strong> tests concerning low damping, stability or<br />
characteristics <strong>of</strong> pilot induced oscillations. Further, strong visual cues supplied by the<br />
improved wide field <strong>of</strong> view visual <strong>system</strong> were disorienting in the absence <strong>of</strong> <strong>motion</strong> cues.<br />
In 1959, the first <strong>motion</strong> research <strong>simulator</strong> was built at NASA, which also indicated the<br />
importance <strong>of</strong> <strong>motion</strong> cues [10]. Analysis <strong>of</strong> loss <strong>of</strong> <strong>control</strong> in extreme operations motivated<br />
the research with centrifuge <strong>simulator</strong>s [23]. The many <strong>motion</strong> deficiencies, due to the<br />
very limited vertical stroke, the vibration noise and incorrect centrifugal forces, were so<br />
distracting that most <strong>of</strong> the research was still performed with fixed base <strong>simulator</strong>s.<br />
Motion <strong>system</strong>s with alternative construction were designed. The introduction <strong>of</strong> the<br />
wide body transport aircraft, such as the B747, required lateral acceleration and led to the<br />
four and six degrees-<strong>of</strong>-freedom (d.o.f.) <strong>motion</strong> <strong>system</strong>s. The first six-degrees-<strong>of</strong>-freedom<br />
research <strong>simulator</strong> at NASA Ames was put into operation in 1964. These <strong>system</strong>s were <strong>of</strong>ten<br />
given too much stroke (up to 30 m), which was not necessary given the limited velocity and<br />
resulted in inferior dynamics <strong>of</strong> lengthy cables (low eigenfrequency). Also the turn-around-
1.1 Flight simulation 9<br />
bump due to coulomb friction prohibited unnoticeable fade to a central position.<br />
In the late sixties, the synergetic, fully parallel, hydraulically driven <strong>motion</strong> <strong>system</strong> came<br />
into operation. The most commonly used is the Stewart-Gough platform [140]. Through<br />
the hydrostatic bearings introduced by Viersma [153] in 1969 in the first (3-d.o.f.) <strong>flight</strong><br />
<strong>simulator</strong> <strong>motion</strong> <strong>system</strong> at Delft University <strong>of</strong> Technology [12], these <strong>system</strong>s had virtually<br />
no coulomb friction. The first commercially available 6-d.o.f. <strong>motion</strong> <strong>system</strong> in 1977 with<br />
hydrostatic bearings was reported on by Baret in [13]. Nowadays, almost every <strong>simulator</strong><br />
<strong>motion</strong> <strong>system</strong> operates in this manner.<br />
In the late 70’s, an attempt was made to classify the characteristics <strong>of</strong> <strong>motion</strong> <strong>system</strong>s in<br />
<strong>flight</strong> <strong>simulator</strong>s [1], [67]. Travelling limits, response time (latency), roughness and noise<br />
were considered most important. The regulating authorities such as the FAA (USA) and<br />
JAR (Europe), given confidence in the use <strong>of</strong> simulation, came up with required <strong>motion</strong><br />
<strong>system</strong> specifications, which reflect the abilities <strong>of</strong> <strong>motion</strong> <strong>system</strong>s at that time [3], [25],<br />
[118], [139]. It is argued that the fact that these standards have been basically unchanged<br />
for the past 15 years, has stopped virtually all fundamental development in <strong>flight</strong> <strong>simulator</strong>s<br />
[143]. Further, since there are still no objective tests, no prescribed manoeuvres and no<br />
criteria for accurate <strong>motion</strong> cueing, makes it difficult to establish new standards. There is,<br />
however, a market demand from the airlines to come up with additional knowledge to enable<br />
a decrease in required <strong>flight</strong> experience for zero <strong>flight</strong> time training.<br />
In the 80’s and 90’s, the main developments concerned the rapid growing abilities in<br />
the visual and computational <strong>system</strong>s. As the main selling argument in the 60’s came from<br />
the all but perfect though impressive <strong>motion</strong> <strong>system</strong>s, now most interest stems from the fast<br />
improving, marketable, visual <strong>system</strong>. As in the past, higher standards for other sub<strong>system</strong>s,<br />
such as the quick response <strong>of</strong> the image generating <strong>system</strong>s now available [100], the<br />
increased computational power enabling to include aer<strong>of</strong>lexible modes in the aircraft model<br />
and more advance <strong>control</strong> strategies for <strong>motion</strong> <strong>control</strong>, will be a drive for higher fidelity<br />
with respect to <strong>motion</strong>.<br />
In [10] it is not expected that any major changes in <strong>motion</strong> travel will be made in <strong>flight</strong><br />
simulation <strong>motion</strong> <strong>system</strong>s. However, within the current structure improvements can be attained<br />
[7] and more <strong>of</strong>ten multi-stage, cascaded, <strong>motion</strong> <strong>system</strong>s can be observed, in which<br />
the need for <strong>motion</strong> cues (with the current 6-d.o.f. structure) and travel (additional for lateral<br />
and longitudinal travel over rails as in the National Advanced Driving Simulator [20], [92])<br />
have been decoupled but still have considerable interaction in the dynamics. Reduction <strong>of</strong><br />
time delays in <strong>motion</strong> <strong>system</strong> response and interaction is needed for solving more complex<br />
problems with visual/<strong>motion</strong>/model mismatches [10]. At this moment, high frequency <strong>motion</strong><br />
cues ( 4 Hz) require special effects included in the <strong>motion</strong> <strong>system</strong> <strong>control</strong>ler and<br />
phase lag in the crossover region is compensated for by additional lead (differential action)<br />
on the reference signals [10], [24], [143]. A more structural model <strong>based</strong> <strong>based</strong> approach<br />
will be strived for in this research.<br />
1.1.4 Motion simulation<br />
As already mentioned, in <strong>flight</strong> simulation, the pilot relies on the perception <strong>of</strong> self-<strong>motion</strong><br />
through several stimuli, and uses this perception to exercise <strong>control</strong> over the aircraft. From
10 1 Introduction<br />
Experimentator<br />
Situation<br />
Aircraft<br />
<strong>Model</strong><br />
Wash-Out<br />
Filter<br />
Pilot<br />
Motion<br />
Controller<br />
Simulation computer Motion Computer<br />
Motion<br />
System<br />
Continuous World<br />
Fig. 1.1: A block diagram representation <strong>of</strong> <strong>motion</strong> simulation.<br />
the pilots point <strong>of</strong> view, the <strong>simulator</strong> should be able to replace the aircraft.<br />
In the <strong>control</strong> task, the pilot uses his visual perception to make a good estimation <strong>of</strong><br />
the aircraft’s long term attitude and velocity. The approximate frequency response <strong>of</strong> visual<br />
perception can be modelled as a first order low-pass filter with a break frequency <strong>of</strong> 0.1 Hz<br />
[92]. Fortunately, the inertial <strong>motion</strong> <strong>of</strong> the <strong>simulator</strong> has a minor role below this frequency.<br />
There is a potential pay<strong>of</strong>f in tuning the <strong>motion</strong> cues to match the human physiological<br />
sensors’ responses, rather than attempting to match the aircraft <strong>motion</strong> themselves. If done<br />
properly, the effective provision <strong>of</strong> <strong>motion</strong> can be obtained in an envelope substantially<br />
reduced from that <strong>of</strong> the aircraft <strong>motion</strong>.<br />
The stimulation through actual inertial <strong>motion</strong> with a <strong>simulator</strong> can usually be represented<br />
by the block scheme given in Fig. 1.1. The pilot responds to <strong>simulator</strong> cues and<br />
the task at hand provided through the experiment leader. With a model <strong>of</strong> the aircraft, it is<br />
calculated what the actual aircraft <strong>motion</strong> would have done due to the action <strong>of</strong> the pilot,<br />
the situation (on the ground the aircraft has different dynamics in interacting with the track<br />
than in the air) and the situation dependent disturbances such as turbulence.<br />
With washout, the calculated aircraft <strong>motion</strong> has to be translated into feasible <strong>motion</strong><br />
<strong>system</strong> trajectories regarding the limited stroke, velocity and accelerations. Washout must<br />
provide some form <strong>of</strong> high pass filtering to limit the <strong>simulator</strong> excursions given the unlimited<br />
aircraft travel. Washout does usually separate the <strong>motion</strong> cues into high (”onset”)<br />
and low (”sustained”) frequency components. Further, by also introducing a scaling factor<br />
<strong>of</strong> <strong>simulator</strong> <strong>motion</strong>, all onsets can be performed by the <strong>simulator</strong> without attaining the<br />
inherent limitations <strong>of</strong> the <strong>motion</strong> <strong>system</strong>.<br />
Most sustained cues can not be represented through inertial <strong>motion</strong> <strong>of</strong> the <strong>simulator</strong> and<br />
have to be ”washed out”. Sustained forward or lateral acceleration is sometimes represented<br />
through a trick already mentioned by Johnson in 1931 [4]. Through the help <strong>of</strong> the gravity<br />
force, a backward tilt gives the same sensation as a forward linear acceleration. Problem
1.1 Flight simulation 11<br />
Fig. 1.2: A typical modern <strong>flight</strong> <strong>simulator</strong> equipped with a synergistic (hexapod) <strong>motion</strong><br />
<strong>system</strong>. 1<br />
with this so-called tilt coordination is to attain the tilted pose without the pilot noticing<br />
i.e. pitching or rolling with subthreshold angular <strong>motion</strong>. In a critical survey, Advani [7]<br />
notices that the mostly used classical linear washout leads to considerable false cues in the<br />
important frequency area between 0.1 Hz and 1 Hz. In an overview <strong>of</strong> wash-out techniques<br />
in [92] by Martin, also non-linear wash-out is mentioned, which however has not led to<br />
completely acceptable solutions so far.<br />
The feasible washed-out <strong>motion</strong> trajectories are the reference input to the <strong>motion</strong> <strong>control</strong>ler.<br />
The task <strong>of</strong> the <strong>motion</strong> <strong>control</strong>ler here is to attain a one-to-one correspondence<br />
between the actual <strong>motion</strong> <strong>system</strong> trajectories and the desired feasible washed-out <strong>motion</strong><br />
trajectories. In the usual configuration, the references contain positional information, which<br />
is also measured and can be used in feedback. Further, the desired acceleration pr<strong>of</strong>iles are<br />
provided for, which can be applied in generating an appropriate feed forward.<br />
As it is physically impossible to generate immediate acceleration with the <strong>motion</strong> <strong>system</strong>,<br />
imperfect cues are generated, which lag the desired cues. Enlarging the bandwidth<br />
decreases the lag but has to be compromised with sufficient attenuation <strong>of</strong> false cues resulting<br />
from noise and excitation <strong>of</strong> parasitic dynamics, which mainly reside in the high<br />
frequency area. Considering a more general setting than the one given in Fig. 1.1, can possibly<br />
lead to a less severe compromise by e.g. allowing available predictive information in<br />
the simulation model to be sent to the <strong>motion</strong> <strong>control</strong>ler.<br />
The <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s, nowadays, have almost invariably a so-called 6d.o.f.<br />
synergetic construction. A <strong>motion</strong> <strong>system</strong> <strong>of</strong> this type is shown in Fig. 1.2. 1 Six<br />
1 Picture courtesy <strong>of</strong> Rexroth-Hydraudyne.
12 1 Introduction<br />
actuators are connected in parallel to a base frame at the foundation with one gimbal, a<br />
freely rotating joint, and at the other end to the <strong>simulator</strong> with another gimbal. Kinematically,<br />
the six lengths <strong>of</strong> the actuators can be driven independently. However, dynamically,<br />
this <strong>system</strong> is highly interactive i.e. an elongation force <strong>of</strong> one actuator generally induces reaction<br />
forces in all others. Further, prescribed platform <strong>motion</strong> requires coordinated <strong>motion</strong><br />
<strong>of</strong> all actuators, which varies with the platform position.<br />
The main advantages <strong>of</strong> this construction are the parallel connections resulting in a <strong>system</strong><br />
with higher stiffness given a specific actuator and its simplicity. Six identical <strong>system</strong><br />
are easy to build and maintain. The stiffness is due to the fact that springs in parallel result<br />
in higher stiffness while springs in series will result in a decrease. This is important in <strong>flight</strong><br />
<strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s since the payload to be accelerated is usually relatively high. With<br />
the synergetic construction the static load per actuator will be lower and the eigenfrequencies<br />
will shift to higher values. The forces to be applied with the actuators are usually still<br />
considerably high. Most <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s are therefore driven by hydraulic<br />
servo actuators, since these <strong>system</strong>s provide superior performance in generating high power<br />
long stroke linear <strong>motion</strong>s [124]. Further, the use <strong>of</strong> hydrostatic bearings results in very low<br />
coulomb friction forces, which minimises the velocity reversal bump. As <strong>simulator</strong> wash<br />
out <strong>motion</strong> reversal and aircraft <strong>motion</strong> do not coincide, an unnoticeable change <strong>of</strong> velocity<br />
sign is very important in <strong>flight</strong> simulation.<br />
With the introduction <strong>of</strong> improved linear electrical drives, there is a trend to use electrically<br />
driven <strong>motion</strong> <strong>system</strong> if a <strong>simulator</strong> has relatively low payload up to 2500 kg<br />
[24].<br />
1.1.5 SIMONA project<br />
Through a preliminary investigation for an upgrade programme <strong>of</strong> the older DUT 3-d.o.f.<br />
<strong>simulator</strong>, a new international centre for research into <strong>flight</strong> simulation techniques, Simona,<br />
was initiated in 1992. Simona stands for Simulation, Motion and Navigation, which reflects<br />
the areas <strong>of</strong> interest in the research programme <strong>of</strong> the three groups, who initiated this centre.<br />
Instead <strong>of</strong> an upgrade <strong>of</strong> the older <strong>simulator</strong>, which has been dismantled, the core <strong>of</strong> Simona<br />
will be a full scale 6-d.o.f. <strong>flight</strong> <strong>simulator</strong>, the Simona Research Simulator (SRS) [6]. The<br />
development <strong>of</strong> the SRS is also performed within Simona. The actual fabrication <strong>of</strong> the<br />
<strong>simulator</strong> was started in 1994, after a large donation <strong>of</strong> the Dutch Government to enable a<br />
multi-discipline international centre with a high fidelity <strong>simulator</strong>.<br />
The three groups involved, each have their specific research interest and contribution to<br />
the project.<br />
The Simulation and Control Group <strong>of</strong> the Faculty <strong>of</strong> Aerospace Engineering performs<br />
studies into <strong>flight</strong> <strong>control</strong>, the man-machine interaction <strong>of</strong> the pilot in the aircraft<br />
and modelling and real-time simulation <strong>of</strong> the aircraft and its <strong>system</strong>s. In the, now<br />
ongoing, realisation phase <strong>of</strong> the <strong>simulator</strong> the group takes part in the development<br />
<strong>of</strong> simulation s<strong>of</strong>tware, the interior and the visual <strong>system</strong>. The Group <strong>of</strong> Production<br />
Technology within Aerospace Engineering did the design <strong>of</strong> the integrated platformcockpit<br />
shuttle and planned connections <strong>of</strong> projection platform and image projection<br />
screens, which will be connected in future.
1.1 Flight simulation 13<br />
Fig. 1.3: Artist’s impression <strong>of</strong> what will be the full operational Simona <strong>flight</strong> <strong>simulator</strong>
14 1 Introduction<br />
Fig. 1.4: Simona <strong>motion</strong> <strong>system</strong> with empty shuttle on top, configuration B.
1.1 Flight simulation 15<br />
The Mechanical Engineering Systems and Control Group <strong>of</strong> the Faculty <strong>of</strong> Design,<br />
Engineering and Production performs research into the design, development, modelling<br />
and <strong>control</strong> <strong>of</strong> hydraulic servo and <strong>motion</strong> <strong>system</strong>s. The 6-d.o.f. hydraulically<br />
driven <strong>motion</strong> <strong>system</strong> has been realised and tested and so have the first model <strong>based</strong><br />
<strong>control</strong>lers as part <strong>of</strong> this research. In future, alternative <strong>control</strong> concepts will have to<br />
be developed. For one, to interconnect simulation models and <strong>motion</strong> <strong>control</strong> in an<br />
optimal way. Further, to adapt to the changing characteristics <strong>of</strong> the <strong>motion</strong> <strong>system</strong><br />
with e.g. a, to some extent flexibly attached, visual <strong>system</strong>.<br />
The Telecommunications and Traffic-Control Systems and Services Group <strong>of</strong> the Faculty<br />
<strong>of</strong> Information Technology and Systems does research into navigation technology<br />
and advanced cockpit display <strong>system</strong>s. The operational SRS can serve as an experimental<br />
set-up to test new navigation techniques and approaches to simulate these<br />
techniques appropriately. The group has been involved in the development <strong>of</strong> a realtime<br />
simulation platform.<br />
An impression <strong>of</strong> what will be the full operational <strong>flight</strong> <strong>simulator</strong> is given in Fig. 1.3.<br />
The <strong>motion</strong> <strong>system</strong> consists <strong>of</strong> six hydraulic servo actuators which are connected in parallel<br />
at the base frame and <strong>simulator</strong>. The actuators together can drive the <strong>simulator</strong> in all three<br />
translational and rotational degrees <strong>of</strong> freedom. At the <strong>simulator</strong> side, the actuators are connected<br />
to the bottom <strong>of</strong> the shuttle, an integrated platform/cockpit made <strong>of</strong> the light weight<br />
material TWARON/carbon. In front <strong>of</strong> the shuttle, a wide angle visual mirror projection<br />
screen will be attached. With projectors on top <strong>of</strong> the shuttle and a back projection screen<br />
in between, the pilot in the cockpit can be provided with a out-<strong>of</strong>-the-window view via the<br />
mirror in front <strong>of</strong> the shuttle.<br />
The new design <strong>of</strong> the integrated platform/cockpit and the use <strong>of</strong> light weight materials<br />
will lead to a relative low centre <strong>of</strong> gravity and considerably less mass to be accelerated<br />
as compared to conventional <strong>simulator</strong>s. These properties strived for in the design <strong>of</strong> the<br />
construction, will have to result in favourable <strong>motion</strong> characteristics. In what sense the<br />
expectedly 4 tons <strong>of</strong> the SRS instead <strong>of</strong> the 12 to 15 tons <strong>of</strong>ten reported for other <strong>system</strong>s<br />
[24], [92] can be taken advantage <strong>of</strong>, is one <strong>of</strong> the issues to be discussed in this research.<br />
After separate testing <strong>of</strong> all six hydraulic actuators in a special experimental set-up in<br />
1994 [124], the integration <strong>of</strong> the <strong>motion</strong> <strong>system</strong> was complete early in 1997, and the first<br />
free-run tests <strong>of</strong> the integrated Simona <strong>motion</strong> platform were performed. The <strong>motion</strong> <strong>system</strong><br />
was tested in three configurations in the Central Workshop at the faculty <strong>of</strong> Mechanical<br />
Engineering.<br />
A. The first tests were performed with a steel platform, the dummy platform <strong>of</strong> 2200 kg.<br />
B. In the autumn <strong>of</strong> 1997 the empty (only chairs) shuttle <strong>of</strong> 1700 kg was put on top <strong>of</strong> the<br />
<strong>motion</strong> <strong>system</strong>, as shown in Fig. 1.4.<br />
C. Finally, tests were performed with the estimated final load <strong>of</strong> 4000 kg with the dummy<br />
platform and additional masses (6 times 250 kg) on top, shown in Fig. 1.5, in the<br />
months <strong>of</strong> May and June, 1998.<br />
In the early spring <strong>of</strong> 1999, the <strong>motion</strong> <strong>system</strong> was to be operated at the building especially<br />
made to house the Simona centre, eventually with the full operational <strong>simulator</strong>. Main
16 1 Introduction<br />
Fig. 1.5: Simona <strong>motion</strong> <strong>system</strong> with dummy platform and additional masses, configuration<br />
C.
1.2 Motion <strong>control</strong> 17<br />
w z<br />
- -<br />
- Psp u<br />
K<br />
Fig. 1.6: A standard <strong>control</strong> design structure<br />
efforts are now in progress to complete the cockpit interior, the real-time simulation environment<br />
and the visual <strong>system</strong>.<br />
1.2 Motion <strong>control</strong><br />
Having sketched the <strong>flight</strong> simulation process and the place <strong>motion</strong> generation has in this<br />
environment, the <strong>motion</strong> <strong>control</strong> problem can be defined. First the process <strong>of</strong> <strong>motion</strong> generation<br />
is put into a generalised framework. From this perspective the <strong>motion</strong> <strong>control</strong> specifications<br />
are stated and the conventional approach towards the design <strong>of</strong> <strong>motion</strong> <strong>control</strong> in<br />
<strong>flight</strong> <strong>simulator</strong>s is discussed. Then realising that the <strong>motion</strong> <strong>system</strong> <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> can<br />
be considered a robotic <strong>system</strong>, <strong>of</strong> course with its specific demands and properties, some <strong>of</strong><br />
the modern <strong>control</strong> strategies developed in robot <strong>control</strong> are put forward, since these could,<br />
with some modification, possibly be <strong>of</strong> benefitin<strong>flight</strong> simulation.<br />
1.2.1 Motion <strong>control</strong> structure<br />
A standard <strong>control</strong> framework, adopted in many textbooks on modern <strong>control</strong> [17], [89],<br />
[160], is given in Fig. 1.6. A <strong>control</strong>ler, K, is provided with measurement signals, y, and<br />
has to stabilise a plant, Psp, with input signals, u, such that the cost variables, z, are minimal<br />
in some sense, despite the disturbance signals, w.<br />
The plant, Psp, is <strong>of</strong>ten called the generalised plant or standard plant since it usually<br />
does not only consists <strong>of</strong> the plant to be <strong>control</strong>led, the <strong>motion</strong> <strong>system</strong> in <strong>flight</strong> simulation<br />
<strong>motion</strong> <strong>control</strong>, but can also contain weightings e.g. sensory thresholds weighting on the<br />
cost variable specific force or a reference model e.g. the aircraft to be simulated. Also the<br />
other entities can be viewed in a generalised way, e.g. reference signals can be incorporated<br />
as disturbances w and additional feedback paths can be taken in case <strong>of</strong> a robust <strong>control</strong><br />
problem to describe a set <strong>of</strong> <strong>system</strong>s i.e. uncertainty.<br />
In a general simulation <strong>motion</strong> <strong>control</strong> problem, as shown in Fig. 1.7, the performance<br />
cost variables, zp, will be the difference <strong>of</strong> <strong>motion</strong> in the aircraft and in the <strong>simulator</strong> with<br />
the pilot in-the-loop weighted with the ability <strong>of</strong> the pilot to perceive these differences,<br />
Wp. Further, <strong>motion</strong> <strong>system</strong> limitations should be penalised by additional cost variables,<br />
zc. So the standard plant will consist <strong>of</strong> the <strong>motion</strong> <strong>system</strong>, the simulation model (Sim),<br />
the aircraft, the pilot and weighting functions. The generalised disturbances are both the<br />
y
18 1 Introduction<br />
wda<br />
wr<br />
wdm<br />
u<br />
-<br />
-<br />
-<br />
-<br />
-<br />
Aircraft<br />
Pilot<br />
Pilot<br />
Motion System<br />
Sim - Ww<br />
Motion Control<br />
e? - Wp -<br />
6<br />
Fig. 1.7: The <strong>flight</strong> simulation <strong>motion</strong> <strong>control</strong> problem put in the generalised plant structure<br />
tasks to be tracked, wr and the external disturbances to be rejected. There will be both<br />
external disturbances on the aircraft, wda, such as forces due to turbulence, which should<br />
be simulated, and external disturbances on the <strong>motion</strong> <strong>system</strong>, w dm, such as friction forces,<br />
which are to be rejected.<br />
The <strong>control</strong>ler receives information from both the <strong>motion</strong> <strong>system</strong>, y m, which has to<br />
be stabilised and the other sub<strong>system</strong>s <strong>of</strong> the generalised plant to construct an appropriate<br />
reference to be tracked, yr. In case <strong>of</strong> the <strong>motion</strong> <strong>control</strong>ler in the conventional setting as<br />
given in Fig. 1.1 this will be the desired <strong>motion</strong> reconstructed by the wash-out filters, W w.<br />
If the reconstruction <strong>of</strong> a feasible trajectory is taken as a part <strong>of</strong> the <strong>control</strong> problem, as is<br />
done by Idan et al. in [58], the aircraft <strong>motion</strong> can be taken as the reference and the wash<br />
out model becomes part <strong>of</strong> the <strong>motion</strong> <strong>control</strong>ler to be designed. When the whole inertial<br />
<strong>motion</strong> simulation is considered, the ’simulation <strong>control</strong>ler’ is fed by actions <strong>of</strong> the pilot and<br />
the experiment leader. The <strong>control</strong>ler can act through the actuators <strong>of</strong> the <strong>motion</strong> <strong>system</strong>, u.<br />
To calculate an appropriate <strong>control</strong>ler, a mathematical model <strong>of</strong> the sub<strong>system</strong>s in the<br />
standard plant can be derived. If these models are sufficiently simple, e.g. linear time-<br />
+<br />
-<br />
yr<br />
ym<br />
zp<br />
zc<br />
-
1.2 Motion <strong>control</strong> 19<br />
invariant, an optimal <strong>control</strong>ler can be synthesised, which for instance minimises some<br />
maximal amplification on some norm on the cost, z, given a bound on the norm <strong>of</strong> the<br />
disturbances, w [160]. The <strong>flight</strong> simulation <strong>motion</strong> <strong>control</strong> problem is yet too complex to<br />
be solved in this way, but some observations can be made from the structure <strong>of</strong> the standard<br />
plant given in Fig. 1.7.<br />
The task <strong>of</strong> tracking a reference model is a particular element in <strong>flight</strong> simulation,<br />
which requires special consideration. Several techniques, like model following [145], model<br />
matching [156], the servo mechanism [33], etc., exist to achieve such tasks. Tracking results<br />
through the servo-mechanism with linear time-invariant <strong>system</strong>s if the internal model<br />
principle [40] is satisfied, which states that the (unstable) parts <strong>of</strong> the <strong>system</strong> to be tracked<br />
have to be part <strong>of</strong> the <strong>control</strong>ler. Also with the other techniques, knowledge <strong>of</strong> the <strong>system</strong><br />
to be matched or followed is incorporated in the <strong>control</strong>ler (design). Although the reference<br />
aircraft model is available in <strong>flight</strong> simulation, the specific properties <strong>of</strong> these models are<br />
not yet taken into account in the <strong>motion</strong> <strong>control</strong>ler design.<br />
The reference tracking problem typically leads to a two degree <strong>of</strong> freedom <strong>control</strong>ler<br />
design problem [152], as given in Fig. 1.8, in which a feedforward, C 1, and a feedback part,<br />
C2, can be distinguished. With feedforward, ideally, the input, u, required to achieve the<br />
desired reference, r, should be generated, i.e. an inverse model <strong>of</strong> the <strong>system</strong>, G. Usually,<br />
however, the <strong>system</strong> does not have a causal and/or stable inverse and therefore an inverse<br />
model cannot be used. Further, the feedforward should not unnecessarily excite unmodelled<br />
dynamics in the <strong>system</strong>. Typically, this dynamics resides in the high frequency area, where<br />
an inverse model would, moreover, usually result in high amplification.<br />
The feedback part should take care <strong>of</strong> uncertainties such as unmodelled dynamics and<br />
other external disturbances and, if necessary, stabilisation <strong>of</strong> the plant using the error, e, the<br />
difference between the measured output, y, and the reference, r. Where feedforward can<br />
be seen as specific shaping <strong>of</strong> the plant w.r.t. reference signals only, the feedback shapes<br />
the sensitivity <strong>of</strong> the plant towards all disturbances. Feedback usually results in a frequency<br />
area (<strong>of</strong>ten the lower frequencies) where the <strong>control</strong>led plant becomes less sensitive, an area<br />
where sensitivity is higher (around the bandwidth) and an area with unchanged characteristics<br />
(high frequency area) [89].<br />
De Roover argues in [30] that a third degree <strong>of</strong> freedom in the design <strong>of</strong> the <strong>control</strong>ler<br />
should be considered. This should take care <strong>of</strong> the generation <strong>of</strong> an appropriate reference<br />
signal. In case <strong>of</strong> the <strong>flight</strong> simulation <strong>motion</strong> <strong>control</strong>, the causality problem in the mechanics<br />
is <strong>of</strong>ten overcome by generating an acceleration reference signal instead <strong>of</strong> position<br />
only. Calculating the forces required given a position reference would lead to a non-causal<br />
transfer function. This is not the case with acceleration. It is questionable, whether an acceleration<br />
reference signal is sufficient to fully overcome the causality problem since this<br />
would imply that immediate force generation is possible.<br />
1.2.2 Motion <strong>control</strong> specifications<br />
The main task to be accomplished after wash-out, tracking <strong>of</strong> the reference acceleration<br />
signals should not only be accomplished w.r.t. the asymptotic behaviour at infinitely long<br />
time spans but especially the transient behaviour, the generation <strong>of</strong> appropriate onset mo-
20 1 Introduction<br />
-<br />
C1<br />
r + e +<br />
- e - C2 - e?<br />
u<br />
y<br />
- G<br />
-<br />
+<br />
6<br />
Fig. 1.8: Two degree <strong>of</strong> freedom <strong>control</strong> structure<br />
tion, is most relevant. By using (model) knowledge <strong>of</strong> the reference generating <strong>system</strong>, the<br />
disturbed aircraft, the pilot and the plant, the <strong>motion</strong> <strong>system</strong>, use can be made <strong>of</strong> the two<br />
additional degrees <strong>of</strong> freedom in the <strong>control</strong>ler to enhance performance [32].<br />
As acceleration is the perceived unit, the performance <strong>of</strong> such <strong>motion</strong> <strong>control</strong> is much<br />
more sensitive as (disturbance) signals become faster, as compared to <strong>system</strong>s with position<br />
as cost variable. Attaining higher bandwidths than required can make the <strong>system</strong><br />
unnecessarily sensitive. Though it is known that below approximately 0.1 Hz <strong>motion</strong> is not<br />
perceived through inertial movement, the required bandwidth is yet unknown. Of course,<br />
the crossover frequency area <strong>of</strong> the pilot/aircraft loop around 1 Hz should be simulated with<br />
close correspondence but the question <strong>of</strong> whether the vestibular organs, which are sensitive<br />
up to 30 Hz, should be stimulated over this whole frequency area is not seen to be<br />
answered in literature. It can be expected that striving for a bandwidth <strong>of</strong> 30 Hz will lead<br />
to compromising false cues due to parasitic dynamics <strong>of</strong> the <strong>motion</strong> <strong>system</strong> with required<br />
cues.<br />
In summary, inertial <strong>motion</strong> <strong>control</strong> <strong>of</strong> the cockpit in <strong>flight</strong> <strong>simulator</strong>s is directed at<br />
providing appropriate cues in the following areas:<br />
- The onset <strong>of</strong> the aircraft response to the actions <strong>of</strong> the pilot. Both shape (amplitude) and<br />
timing (phase) <strong>of</strong> the response is important here. As the bandwidth <strong>of</strong> the pilot usually<br />
does not exceed 1 Hz, in a first attempt, a bandwidth <strong>of</strong> an order <strong>of</strong> a magnitude higher<br />
can be strived for in this respect.<br />
- A realistic generation <strong>of</strong> disturbances resulting from e.g. turbulence. At higher frequencies,<br />
above 2-3 Hz, the shape <strong>of</strong> these disturbances becomes most important while<br />
(linear) phase lags, small time delays, are not so relevant.<br />
- The dynamics related to the aircraft resulting in vibrations. This can be both due to the<br />
pilot in the loop as the disturbances. This dynamics can change drastically with the<br />
situation e.g. on the ground or airborne. This also means sufficient suppression or at<br />
least minimal excitation <strong>of</strong> <strong>simulator</strong> related dynamics i.e. parasitic dynamics, which<br />
do not correspond with the pilot in-the-loop with the actual aircraft.<br />
- Given feasible reference trajectories, the position <strong>control</strong> should be such that errors in
1.2 Motion <strong>control</strong> 21<br />
the actuator position do not result in running the actuators out-<strong>of</strong>-stroke. The corrective<br />
action should not lead to false cues and should therefore typically be <strong>of</strong> a low<br />
frequency nature and thus have very limited bandwidth.<br />
Not only the quality <strong>of</strong> the <strong>motion</strong> <strong>control</strong> is important, but also the predictability. In doing<br />
pilot/aircraft interaction experiments or in design <strong>of</strong> a training programm, it is relevant<br />
to know to what degree the simulated environment corresponds with the actual situation.<br />
Finally, the main requirement in doing piloted simulation experiments is safety, which demands<br />
stability <strong>of</strong> the <strong>motion</strong> <strong>control</strong> loop at all time.<br />
1.2.3 Conventional <strong>simulator</strong> <strong>motion</strong> <strong>control</strong><br />
Most <strong>motion</strong> <strong>control</strong> schemes, reported on in <strong>flight</strong> simulation, focus on the kinematics <strong>of</strong><br />
the mechanical <strong>system</strong> and the dynamics <strong>of</strong> independent hydraulically driven masses [12,<br />
24, 46, 51, 82]. Given a desired <strong>simulator</strong> trajectory, the required actuator trajectories can be<br />
calculated with the inverse kinematics (end-effector to actuator coordinate transformation)<br />
and with a fully parallel driven robot these relations can be given explicitly. By neglecting<br />
the interaction between the actuators, each feedback loop can be designed given the transfer<br />
function <strong>of</strong> a hydraulic actuator driving a mass. The linearised version from valve input,<br />
i, to position, q, as in [99, 153] consists <strong>of</strong> a lightly damped second order <strong>system</strong> in series<br />
with an integrator. This is described in the frequency domain as<br />
q<br />
i =<br />
K<br />
s(s2 =! 2 o +2 s=!o<br />
� (1.3)<br />
+1)<br />
considering s = j!.<br />
So at low frequencies the <strong>system</strong> acts as a velocity generator and at higher values the<br />
resonant p characteristics <strong>of</strong> second order <strong>system</strong> plays a role with the eigenfrequency, ! o =<br />
c=m, which depends on the driven mass, m, and the limited oil stiffness, c, <strong>of</strong>ten with<br />
poor damping,
22 1 Introduction<br />
the characteristics as described for the one degree-<strong>of</strong>-freedom <strong>system</strong>. Pressure feedback,<br />
therefore, still results in damping only. As the eigenfrequencies can vary considerably for<br />
each direction and <strong>simulator</strong> position, the relative damping provided by pressure feedback<br />
will differ. The positional feedback will have to be tuned according to the lowest occurring<br />
<strong>system</strong> eigenfrequency. The resulting <strong>control</strong>led <strong>system</strong> has a bandwidth corresponding to<br />
this frequency and is <strong>of</strong>ten somewhat lower since sufficient damping <strong>of</strong> the faster modes can<br />
require overdamped characteristics <strong>of</strong> the slowest.<br />
By using the acceleration reference signals as lead feedforward tuned according to a<br />
neutral <strong>simulator</strong> position 2 [24, 46], the bandwidth up to which the <strong>system</strong> reacts properly<br />
to these references can be extended. In principle, the second degree <strong>of</strong> freedom in the <strong>control</strong>ler<br />
is used to create a local inverse <strong>of</strong> the <strong>control</strong>led <strong>system</strong> up to a frequency somewhat<br />
higher than the bandwidth attained with feedback.<br />
Although the parallel structure used in <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s requires some special<br />
consideration, the <strong>system</strong> can be viewed as a robotic manipulator for which a wide field<br />
<strong>of</strong> literature exists on <strong>control</strong> strategies, which make more extensive use <strong>of</strong> <strong>system</strong>/model<br />
knowledge. Not until very recently, this is seen to be applied in <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s<br />
[20, 58, 82]. It seems worthwhile to review the robot <strong>control</strong> strategies, which can be <strong>of</strong> use.<br />
1.2.4 Robot <strong>control</strong><br />
Although the dynamics <strong>of</strong> robot manipulators is highly nonlinear, a very specific structure<br />
can be recognised. Most robot <strong>control</strong> strategies take advantage <strong>of</strong> this structure. In [136], a<br />
selection <strong>of</strong> articles provides a state <strong>of</strong> the art and an overview is given <strong>of</strong> the developments<br />
in robot <strong>control</strong> over the years. There are also numerous textbooks on the subject. In recent<br />
works like [11, 31, 81, 102, 113, 115], modern issues such as passivity, robustness, tracking,<br />
elasticity and flexibility, are treated.<br />
Computed torque<br />
One <strong>of</strong> the earliest applications <strong>of</strong> nonlinear <strong>control</strong> <strong>of</strong> robot manipulators was the method <strong>of</strong><br />
computed torque, which started in the early seventies. Using the Euler-Lagrange equations<br />
<strong>of</strong> a rigid robot manipulator with n input torques, and output positions, q given by<br />
M (q)q + C(q� _q)_q +g(q) = � (1.5)<br />
the required input torque can be calculated, compensating the coriolis/centripetal terms, C,<br />
and gravity, g, measuring the positions, q, and velocities, _q, and filling in a new input vector,<br />
a, instead <strong>of</strong> q, to end up with<br />
M (q)q = M (q)a: (1.6)<br />
2This position usually corresponds with all the actuators half stroke and is the position to which the wash-out<br />
fades
1.2 Motion <strong>control</strong> 23<br />
Since the inertia matrix, M, can be inverted, linear, parallel (decoupled) double integrator<br />
<strong>system</strong>s result from a to q,<br />
q = 1<br />
s 2 In n a (1.7)<br />
The double integrator <strong>system</strong>s can easily be <strong>control</strong>led with any desired bandwidth by a<br />
simple proportional/derivative (PD) <strong>control</strong>ler.<br />
a = Kpe + Kd _e + q d<br />
(1.8)<br />
with the positional error, e =(qd ; q), the desired positions, qd, can be tracked with step<br />
and ramp inputs.<br />
In [77], several <strong>control</strong> strategies like the computed torque method, feedback linearisation,<br />
resolved acceleration method and inverse dynamics are shown to be very similar and<br />
unifiable. In all cases, exact linearisation and decoupling is strived for. This can be done<br />
both in end effector or joint coordinates.<br />
The main disadvantages <strong>of</strong> the computed torque method are its robustness properties.<br />
Both with inexact cancellation due to parameter uncertainties (e.g. an inexactly known inertia<br />
matrix) [9] and unmodelled dynamics, such as joint elasticity [85], this <strong>control</strong>ler can<br />
go unstable.<br />
Robust robot <strong>control</strong><br />
Robustness can be regained by adding an outer loop such as in the sliding mode <strong>control</strong><br />
<strong>of</strong> Slotine and Li [133] or the saturating <strong>control</strong>ler applied by Spong [135]. Both methods<br />
employ local high gain, which is sufficiently high to track in the presence <strong>of</strong> uncertainties<br />
but on the other hand is bounded. With the discontinuous sliding mode feedback method,<br />
high frequency unmodelled dynamics is easily excited and chattering is <strong>of</strong>ten reported in<br />
practice. Transient response in saturating <strong>control</strong> is worse. Both properties do not seem<br />
most suitable for <strong>flight</strong> simulation <strong>motion</strong> <strong>control</strong>.<br />
Yet another robustifying approach is to compensate for the main nonlinearities and consider<br />
the <strong>system</strong> linear with some, possibly varying, uncertainty as was done by Van de Linden<br />
[146]. Robust <strong>control</strong> techniques developed for linear <strong>system</strong>s as described in textbooks<br />
as [132, 160] can then be used. Main problem is to arrive at a nonconservative description<br />
<strong>of</strong> an uncertainty in the nonlinear plant at the level <strong>of</strong> the outer loop.<br />
Passive robot <strong>control</strong><br />
Alternatively, use can be made <strong>of</strong> the passivity (’no energy generating’) property <strong>of</strong> a robotic<br />
<strong>system</strong> [11, 14]. Feedback with a passive <strong>control</strong>ler results in a passive closed loop <strong>system</strong>,<br />
which under some assumptions has specific stability properties. The feedback <strong>control</strong>ler can<br />
e.g. have a PD structure with possibly time varying gains (dependent on estimated mass matrix).<br />
W.r.t. feedforward, the computed torque structure can still be employed. This strategy<br />
even generalises to a much larger class <strong>of</strong> <strong>system</strong>s [147] among which are the mechanical<br />
<strong>system</strong>s with possibly flexible structures [63, 66].<br />
Most textbooks on robotic <strong>system</strong>s deal with rigid body structures only. The complexity<br />
<strong>of</strong> the <strong>system</strong> increases drastically in including deformations. The flexible structure has
24 1 Introduction<br />
more degrees <strong>of</strong> freedom than <strong>control</strong> inputs. Trajectory tracking <strong>of</strong> a robot end-effector becomes<br />
very difficult and requires accurate models [31]. The eigenfrequency <strong>of</strong> the flexible<br />
mode is usually a hard constraint for the attainable performance such as bandwidth. With<br />
passive <strong>control</strong>, though stable, performance is <strong>of</strong>ten hard to specify.<br />
Actuator dynamics<br />
Also passivity (<strong>control</strong>) <strong>of</strong> robot manipulators with dynamics in the actuators is addressed<br />
as by Arimoto [11]. The DC-actuator employed there is equivalent with a linearised version<br />
<strong>of</strong> a hydraulic actuator [137]. Although the structure <strong>of</strong> the (voltage driven) DC-motor and<br />
the hydraulic actuator may be equivalent, the parameters usually vary largely. Therefore,<br />
the DC-motor can <strong>of</strong>ten be considered relatively fast in comparison with the robot mechanics.<br />
On the contrary, the (finite oil stiffness <strong>of</strong> the) hydraulic actuator forms a relevant part<br />
<strong>of</strong> the dynamics in e.g. a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> around the achievable bandwidth.<br />
The influence <strong>of</strong> the mechanics can nominally be decoupled in the hydraulics by local<br />
velocity compensation [127]. By an inner feedback loop the hydraulic force generation<br />
becomes relatively fast compared to the mechanics. This cascaded approach was successfully<br />
implemented by Heintze and Van Schothorst [48, 124]. In another cascaded approach<br />
suggested by Qu and Dawson [113], actuator-level robust <strong>control</strong> is achieved through a<br />
recursive or backstepping design [78]. Though this method is going through rapid development<br />
[128], not much practical applications have seen to be reported yet.<br />
Parallel <strong>system</strong>s<br />
Parallel driven <strong>system</strong>s, such as the <strong>flight</strong> <strong>simulator</strong> Stewart platform <strong>motion</strong> <strong>system</strong> in<br />
Fig. 1.2, are usually not discussed in considering robotic <strong>system</strong>s. Only recently, in the late<br />
eighties, the advantages <strong>of</strong> parallel driven robotic manipulators were recognised and now<br />
become more widely applied in industry [5, 50, 96]. Such <strong>system</strong>s are inherently stiffer<br />
and due to the parallel actuation, less actuator weight has to be moved, which makes them<br />
suitable for fast assembly lines.<br />
The computed torque <strong>control</strong> laws for Stewart platforms presented by Liu et al. in [83],<br />
reveal some <strong>of</strong> the additional problems with application <strong>of</strong> model <strong>based</strong> <strong>control</strong>ler for the<br />
parallel <strong>system</strong>s. The linearising <strong>control</strong> in end-effector coordinates requires matrix inversion<br />
(<strong>of</strong> the jacobian) and the forward kinematical problem has to be solved on-line.<br />
This means computing the platform position <strong>based</strong> on joint measurements, which is easy<br />
in serial robots. No closed form explicit solution, however, has seen to be found for this<br />
problem for Stewart platforms. Choice <strong>of</strong> linearising in joint coordinates does not solve<br />
any <strong>of</strong> these problems but actually adds the requirement <strong>of</strong> calculating derivative jacobian<br />
matrices (solved in [38] by Dutre et al.).<br />
Considering general forms <strong>of</strong> parallel or closed chain mechanisms, the situation becomes<br />
even more difficult as discussed by Ghorbel in [44], where several parallel mechanisms<br />
and <strong>control</strong> strategies are classified. In some cases, it is not possible to describe such<br />
a <strong>system</strong> with a minimal number <strong>of</strong> coordinates globally and due to this fact and the specific<br />
singularities, which only appear in parallel <strong>system</strong>s, global stability results are generally<br />
hard to derive, especially for tracking problems.<br />
Many <strong>control</strong> strategies presented for parallel <strong>system</strong>s are designed for appropriate in-
1.3 Problem statement 25<br />
teraction with an environment. Force or hybrid position/force <strong>control</strong> and creating a passive<br />
compliance are <strong>of</strong>ten strived for [96, 103, 110, 116, 142]. Another field <strong>of</strong> application is the<br />
area <strong>of</strong> active vibration <strong>control</strong>, mostly in space structure applications [42, 108, 93]. This<br />
requires high bandwidth <strong>control</strong>lers, which, however, have very limited stroke and thus do<br />
not have to include typical position dependent nonlinearities.<br />
Summarising, in principle, most <strong>of</strong> the robot <strong>control</strong> strategies take advantage <strong>of</strong> the<br />
specific structure in the dynamics <strong>of</strong> mechanical <strong>system</strong>s. Though not <strong>of</strong>ten seen, the model<br />
<strong>based</strong> schemes are applicable to the parallel driven <strong>flight</strong> simulation <strong>motion</strong> <strong>system</strong>s, but<br />
all require some modifications in analysis and design. The main mechanical nonlinearities<br />
can be compensated for by a computed torque strategy, possible feedforwarded. Further,<br />
several alternatives exist to robustify the feedback loop. An important source <strong>of</strong> additional<br />
relevant dynamics can be expected from the hydraulic servo actuators, which are <strong>of</strong> main<br />
consideration in the conventional <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>control</strong> and usually not taken into<br />
account in robot <strong>control</strong>.<br />
1.3 Problem statement<br />
Due to the rapid progress in all kinds <strong>of</strong> <strong>flight</strong> <strong>simulator</strong> sub<strong>system</strong>s such as simulation<br />
computing power, image generating <strong>system</strong>s, etc., also enhanced quality <strong>of</strong> <strong>flight</strong> <strong>simulator</strong><br />
inertial <strong>motion</strong> generation is desired. At least, high fidelity <strong>motion</strong> generation would enable<br />
research into the specific requirements <strong>of</strong> <strong>motion</strong> simulation, which are still not exactly<br />
known. In similar <strong>system</strong>s, such as robotic manipulators, advanced model <strong>based</strong> <strong>control</strong><br />
strategies exist, which can attain higher performance than less structured methods, provide<br />
more insight into the <strong>system</strong>s limitations and possibly point at constructional properties,<br />
which can be taken advantage <strong>of</strong>. In <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>control</strong>, application <strong>of</strong> such<br />
methods has been almost absent. One <strong>of</strong> the reasons might be that the <strong>motion</strong> <strong>system</strong>s have<br />
a degree <strong>of</strong> complexity, which does not allow exact modelling or detailed models as part <strong>of</strong><br />
the <strong>control</strong>ler. Application requires an intermediate step <strong>of</strong> extracting a reasonably detailed<br />
model, which describes the most relevant dynamic characteristics.<br />
1.3.1 General problem statement<br />
The foregoing leads to the following problem statement for this research:<br />
Investigate what relevant <strong>system</strong> knowledge should be used in a model <strong>based</strong> <strong>control</strong><br />
strategy and to what extent does this improve the <strong>control</strong>led dynamics <strong>of</strong> a <strong>flight</strong><br />
<strong>simulator</strong> <strong>motion</strong> <strong>system</strong>.<br />
1.3.2 Structuring the problem statement;<br />
approach in research<br />
To solve this problem, the full design process has to be defined, structured and evaluated.<br />
This process starts with modelling and analysis, proceeds with <strong>control</strong> synthesis and finally<br />
testing <strong>of</strong> the <strong>system</strong>. Many solutions and procedures <strong>of</strong> the steps to be taken are given
26 1 Introduction<br />
in literature, but an integral approach, having the specific properties and requirements <strong>of</strong><br />
<strong>flight</strong> simulation <strong>motion</strong> generation in mind, still lacks. In this research, integratibility and<br />
applicability will be the main arguments in choosing the most suitable alternative or, if<br />
necessary, newly proposed variant, <strong>of</strong> each step in the <strong>system</strong>/<strong>control</strong> design procedure.<br />
Since the overall investment is high and the fidelity <strong>of</strong> the <strong>motion</strong> <strong>system</strong> is considered<br />
important, the full <strong>motion</strong> <strong>control</strong> design process <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> is allowed a fairly<br />
elaborate procedure optimised towards the specific properties <strong>of</strong> the <strong>system</strong> at hand, as<br />
compared to the industrial robot <strong>control</strong> design. Only, <strong>of</strong> course, if this leads to enhanced<br />
quality <strong>of</strong> the resulting <strong>system</strong>.<br />
In the <strong>control</strong> design procedure employed in this research, the following fairly standard<br />
subproblems in application directed <strong>system</strong>s and <strong>control</strong>, will be distinguished:<br />
- Obtain a relevant model structure for analysis and <strong>control</strong> through physical modelling.<br />
- Evaluate the structure by experiments, identify model parameters and provide boundaries<br />
for the extent to which the model validity holds.<br />
- Choose and, if necessary, modify the most appropriate model <strong>based</strong> robot <strong>control</strong> strategy,<br />
given the <strong>system</strong>s properties and requirements. Design and implement this <strong>control</strong>ler<br />
on the <strong>motion</strong> <strong>system</strong> <strong>of</strong> the Simona <strong>flight</strong> <strong>simulator</strong>.<br />
- Define a test to quantify performance <strong>of</strong> the <strong>control</strong>led <strong>system</strong> and use this test to compare<br />
with the usual <strong>control</strong> design approach.<br />
Related to the <strong>flight</strong> simulation application and the properties <strong>of</strong> the <strong>motion</strong> <strong>system</strong>s used,<br />
specific elements have to be emphasised in each <strong>of</strong> these steps.<br />
Physical modelling<br />
<strong>Model</strong>ling according to physical relations such as balance relations i.e. force balance equations<br />
in mechanics and flow balance equations in hydraulics, has two important advantages<br />
as compared to the so-called black box experimental identification. First, as the structure<br />
<strong>of</strong> the model points at the underlying physical properties <strong>of</strong> the <strong>system</strong>, analysis and (constructional)<br />
design considerations are performed much easier. Secondly, modelling <strong>of</strong> a<br />
nonlinear <strong>system</strong> such as the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> platform, does not impose additional<br />
difficulties.<br />
As already mentioned, the mechanics and hydraulics are considered the main sub<strong>system</strong>s<br />
to be taken into account. These sub<strong>system</strong>s specifically, and the integration <strong>of</strong> the<br />
two, will form the basis <strong>of</strong> the models to be derived. The mechanical <strong>system</strong> is, even if<br />
the parts are considered rigid, relatively complex. This complexity mainly stems from the<br />
parallel kinematical structure, which, however, has its influence on the <strong>system</strong>s dynamics.<br />
The kinematics <strong>of</strong> parallel <strong>system</strong>s, and <strong>of</strong> the Stewart platform specifically, will have to be<br />
analysed.<br />
Nowadays, the equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> complex rigid body <strong>system</strong>s can <strong>of</strong>ten be generated<br />
automatically in symbolic form. This way <strong>of</strong> working will be evaluated as opposed to<br />
the derivation ’by hand’ for this <strong>system</strong>.<br />
The models describing the dynamics <strong>of</strong> the mechanical <strong>system</strong> will have to be evaluated<br />
for use in a real-time model <strong>based</strong> <strong>control</strong> structure. Further, it will have to be analysed, how
1.3 Problem statement 27<br />
choices in the design <strong>of</strong> the construction resulting in the specific kinematical and inertial<br />
(mass) properties, influence the dynamics.<br />
The hydraulic servo actuators <strong>of</strong> the Simona <strong>motion</strong> <strong>system</strong> have been modelled and<br />
evaluated in detail in [124] by Van Schothorst. In this research, only models will be discussed,<br />
which describe the most relevant dynamics for <strong>control</strong>. Finally, the properties <strong>of</strong> the<br />
integrated hydraulically driven mechanical <strong>system</strong> will have to be analysed.<br />
Experimental validation<br />
Since the kinematical structure <strong>of</strong> the <strong>system</strong> forms the basis <strong>of</strong> the dynamical properties <strong>of</strong><br />
this <strong>motion</strong> <strong>system</strong>, some attention will have to be paid to the validity <strong>of</strong> the structure and the<br />
identification <strong>of</strong> the parameters to be used. An, ’after construction’, calibration procedure<br />
will be evaluated as opposed to setting tight (and <strong>of</strong>ten expensive) tolerance specifications<br />
in fabrication.<br />
Experimental identification procedures will have to be developed and applied to reconstruct<br />
the parameters such as dissipative terms, which are difficult to predict in advance,<br />
and evaluate others such as the inertial properties. The models will be evaluated through<br />
comparison with the experimental data.<br />
The area <strong>of</strong> validity <strong>of</strong> the models is also an important aspect, which has to be checked.<br />
The cause and severity <strong>of</strong> possible parasitic effects will be investigated. Especially flexibility<br />
is known to constrain the achievable <strong>control</strong> performance and difficult to predict.<br />
Control strategy<br />
To maintain structure in a model <strong>based</strong> <strong>control</strong>ler for a complex <strong>system</strong> such as the <strong>motion</strong><br />
<strong>system</strong> under consideration and to let a <strong>control</strong>ler result, which can be implemented on a<br />
real-time application, modularization <strong>of</strong> the tasks is desired. The physical structure <strong>of</strong> the<br />
<strong>system</strong> provides a reasonable way to achieve this with a multi level <strong>control</strong>ler in which for<br />
each level specific tasks are defined in close relation with the other levels.<br />
It should be possible to turn the hydraulic actuators into fast and smooth force generators<br />
by local inner loop pressure <strong>control</strong> as done by Heintze in [48]. The applicability <strong>of</strong> this<br />
design step, which e.g. assumes accurate velocity measurement, should be evaluated.<br />
Any standard robot <strong>control</strong> strategy, which assumes direct torque, can be put on top <strong>of</strong><br />
this lower level. As first basic step, the computed torque schemes will have to be made fit for<br />
use in parallel <strong>system</strong>s. This mainly concerns the choice <strong>of</strong> an appropriate set <strong>of</strong> coordinates<br />
and evaluation <strong>of</strong> safe (convergent and stable) and sufficiently fast reconstruction <strong>of</strong> these<br />
coordinates.<br />
In the outer level, the two degrees <strong>of</strong> freedom, smooth reference feedforward, possibly<br />
taking reference model knowledge into account, and position feedback for stabilisation and<br />
preventing the <strong>system</strong> from running out <strong>of</strong> stroke, should not interfere with each other in<br />
performing their basic task.<br />
Performance evaluation<br />
In evaluation <strong>of</strong> the model <strong>based</strong> <strong>control</strong> strategy the feedback oriented conventional decentralised<br />
<strong>control</strong>ler will be taken as a reference. It is understood that, in practise, this<br />
conventional <strong>control</strong>ler can be improved by design <strong>of</strong> an additional feed forward path, but
28 1 Introduction<br />
this usually leads to local improvement only and is either an ad hoc approach or takes a<br />
similar strategy as in the more structured model <strong>based</strong> <strong>control</strong> design.<br />
Evaluation test procedures exist [1], but are outdated and should be modified to current<br />
standards <strong>of</strong> <strong>control</strong> performance quantification. In this research, the following approach<br />
will be taken. By assuming the high fidelity <strong>control</strong>ler being capable <strong>of</strong> achieving a reasonably<br />
linear <strong>system</strong> response, the frequency response measurements are considered most<br />
important. A set <strong>of</strong> other tests is mainly developed to evaluate the assumption <strong>of</strong> closed<br />
loop linear <strong>system</strong> response.<br />
Further, it will be observed that the existing evaluation procedures have two main limitations.<br />
First, they have a local character and consequently do not show whether the <strong>control</strong>ler<br />
does adequately respond to the nonlinear characteristics <strong>of</strong> the <strong>system</strong> over the workspace.<br />
Secondly, the <strong>system</strong> is not evaluated performing its actual task, generating inertial <strong>motion</strong><br />
cues in <strong>flight</strong> simulation manoeuvres. A modified test will, as well as possible, have to<br />
overcome these limitations.<br />
1.4 Outline <strong>of</strong> the thesis<br />
The thesis roughly follows the line <strong>of</strong> the reasoning for the approach sketched in the preceding<br />
section, to go through the full <strong>control</strong>ler design process. Subsequently the subproblems<br />
in modelling, identification, <strong>control</strong> and evaluation <strong>of</strong> the <strong>motion</strong> <strong>system</strong>/<strong>control</strong> design with<br />
the <strong>flight</strong> simulation application in mind, will be treated.<br />
Physical modelling<br />
By considering the laws <strong>of</strong> physics, dynamical models <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
will be obtained. In Chapter 2, the mechanical part will be discussed. Specifically, attention<br />
will be paid to the kinematics and dynamics <strong>of</strong> a parallel rigid body <strong>system</strong>. In Chapter 3,<br />
the basic structure <strong>of</strong> the hydraulic servo actuators and extensions to this model will first be<br />
discussed. The integration <strong>of</strong> the hydraulic mechanical <strong>system</strong>, the full model <strong>of</strong> the <strong>motion</strong><br />
<strong>system</strong> and its properties are the subject <strong>of</strong> Section 3.3 and its specifics in a Stewart platform<br />
construction are pointed at in Section 3.4.<br />
’Open loop’ experimental verification<br />
Chapter 4 deals with the experimental validation <strong>of</strong> the physical models, the identification<br />
<strong>of</strong> model parameters and the parasitic ’additional’ dynamics. The identification <strong>of</strong> the<br />
static parameters in the kinematical structure, calibration through redundant measurements<br />
is discussed in Section 4.1. Dynamic (inertial) parameter identification by modal analysis<br />
techniques is the subject <strong>of</strong> Section 4.2. Frequency response measurements are the main<br />
ingredients in the model validation considered in Section 4.3. This includes getting track<br />
<strong>of</strong> the parasitic dynamics caused by flexibility in the <strong>system</strong> and its environment in Section<br />
4.3.2, which concludes the discussion on the ’open loop’ experiments taken in this<br />
chapter.<br />
Control strategy<br />
In Chapter 5, the <strong>control</strong> strategy is built up by defining the general approach for multi level
1.4 Outline <strong>of</strong> the thesis 29<br />
<strong>control</strong> in Section 5.3 after a short introduction to the main <strong>control</strong> aspects in Section 5.1<br />
and Section 5.2. Then, each part <strong>of</strong> this <strong>control</strong>ler will be treated. In Section 5.4, the local<br />
pressure <strong>control</strong> <strong>of</strong> the hydraulic servo <strong>system</strong>s is outlined. Section 5.5 discusses the computed<br />
torque <strong>control</strong> <strong>control</strong> level to compensate for the multivariable, nonlinear mechanics.<br />
This requires an iterative coordinate reconstruction and is followed by the feedback path defined<br />
by the problem <strong>of</strong> position stabilisation. Interconnection with the trajectory generating<br />
<strong>system</strong> with reference model <strong>based</strong> <strong>control</strong>, treated in Section 5.6, defines the forward path<br />
at the outer level.<br />
Closed loop experimental evaluation<br />
The evaluation <strong>of</strong> the <strong>control</strong>ler on the actual experimental set-up is the subject <strong>of</strong> the second<br />
part <strong>of</strong> Chapter 5 starting with discussing the implementational issues in Section 5.7. First,<br />
the procedure to test the performance <strong>of</strong> the <strong>control</strong>led <strong>system</strong> is defined in Section 5.8.<br />
Then, this procedure is applied to the Simona <strong>motion</strong> <strong>system</strong> with the newly proposed <strong>control</strong>ler,<br />
which is discussed in Section 5.9. As a reference, comparing with a conventional<br />
<strong>control</strong>ler strategy.<br />
Final considerations<br />
The main aspects in the design <strong>of</strong> <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> <strong>control</strong>lers evaluating the<br />
experience <strong>of</strong> going through the full <strong>system</strong>/<strong>control</strong> design process defined are treated in<br />
Chapter 6 giving a review and discussion on the results. Directed at the problem statement,<br />
defined in Section 1.3, the final Chapter 7 states conclusions towards the opportunities given<br />
in this research by the application <strong>of</strong> a model <strong>based</strong> <strong>control</strong>ler in <strong>flight</strong> simulation <strong>motion</strong><br />
<strong>control</strong>. Also recommendations are provided.
Chapter 2<br />
Mechanics <strong>of</strong> parallel driven<br />
<strong>motion</strong> <strong>system</strong>s<br />
The, fully parallel, hydraulically driven construction <strong>of</strong> a Stewart platform is <strong>of</strong>ten applied<br />
for <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s. In this research, the use <strong>of</strong> model <strong>based</strong> <strong>control</strong> for such<br />
<strong>system</strong>s is investigated. In this chapter, the first step in modelling and analysis is performed<br />
by considering the mechanics.<br />
In general, models <strong>of</strong> parallel robots result in combined algebraic and differential equations,<br />
which causes difficulties with simulation, analysis and model <strong>based</strong> <strong>control</strong>. This<br />
problem will be stated in discussing some literature on parallel manipulators and modelling<br />
<strong>of</strong> mechanical <strong>system</strong>s in Section 2.1.<br />
In Section 2.2, first a short introduction into basic kinematics will be given. Then, the<br />
structure <strong>of</strong> kinematic parallel and serial singularities will be treated. The, less standard,<br />
parallel singularity causes local loss <strong>of</strong> <strong>control</strong>lability and observability <strong>of</strong> the mechanical<br />
<strong>system</strong> and should be avoided. In Section 2.3, a new method for the Stewart platform is<br />
introduced to ensure exclusion <strong>of</strong> these points from the working area <strong>of</strong> the manipulator.<br />
Moreover, this is also a necessary condition for guaranteed sufficiently fast convergence <strong>of</strong><br />
an iterative scheme in reconstructing the platform coordinates. For the <strong>motion</strong> <strong>system</strong> at<br />
hand, sufficient accurate real time convergence is proven, which enables safe use <strong>of</strong> this<br />
scheme in model <strong>based</strong> <strong>control</strong> later on in Chapter 5.<br />
The platform coordinates are a proper choice in parametrization <strong>of</strong> the Stewart platform,<br />
thereby circumventing difficulties with combined algebraic and differential equations. In<br />
Section 2.4, which deals with the dynamics <strong>of</strong> the mechanical part <strong>of</strong> the <strong>motion</strong> <strong>system</strong>,<br />
it is shown that an explicit differential model results from this choice. Thereby, the use <strong>of</strong><br />
a suitable projection method, following Kane [64], is exploited to arrive at the equations<br />
<strong>of</strong> <strong>motion</strong>. In this way, a proper starting point for simulation, analysis, model reduction<br />
and model <strong>based</strong> <strong>control</strong> is provided for. Some extensions, e.g. parasitic modes <strong>of</strong> the<br />
foundation, are also included.<br />
31
32 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
2.1 Parallel <strong>motion</strong> <strong>system</strong>s<br />
The use <strong>of</strong> robot manipulators is widely spread in industry nowadays. Most <strong>of</strong> these manipulators<br />
are constructed as a series connection <strong>of</strong> joints and links. The dual form <strong>of</strong> these<br />
robots, the parallel manipulator, is less <strong>of</strong>ten seen to be applied. In (<strong>flight</strong>)simulation <strong>motion</strong><br />
<strong>system</strong>s however, the parallel construction is almost invariably in use. The Stewart platform<br />
is a 6 degrees-<strong>of</strong>-freedom (d.o.f.) parallel manipulator, which is applied in most <strong>of</strong> the current<br />
high fidelity <strong>flight</strong> <strong>simulator</strong>s. It was named after Stewart [140] who illustrated the use<br />
<strong>of</strong> such a parallel structure for <strong>flight</strong> simulation in 1965. Nowadays, it is also referred to<br />
as Gough-platform, since it was Gough who presented the practical use <strong>of</strong> such a <strong>system</strong><br />
somewhat earlier in 1949 rediscovering the structure already described around 1800 by the<br />
mathematician Cauchy [95]. As a manufacturing and fabrication robot it is also named a<br />
hexapod.<br />
There are several advantages in applying a parallel construction. This kind <strong>of</strong> manipulators<br />
have higher rigidity and accuracy due to the parallel force path and averaged link to<br />
end-effector error. The inverse kinematics (from end-effector to link coordinates) which is<br />
a problem in path generation <strong>of</strong> serial manipulators is easily solved in parallel robots. There<br />
are also disadvantages. The dual forward kinematics is a complex algebraic problem and<br />
has in general more than one solution [114]. <strong>Model</strong>ling the dynamics is also less straight<br />
forward.<br />
For several reasons, feedback <strong>control</strong> <strong>of</strong> these <strong>motion</strong> <strong>system</strong>s is still decentralized<br />
i.e. per actuator without taking mechanical coupling into account. Setting higher standards<br />
<strong>of</strong> <strong>motion</strong> realism in simulation will involve modern <strong>control</strong> strategies in order to fully benefit<br />
from recent constructional and computational improvements in <strong>flight</strong> <strong>simulator</strong>s e.g. light<br />
weight, low centre-<strong>of</strong>-gravity (c.o.g.) platforms, high frequency airplane dynamics simulation<br />
[6]. Also use <strong>of</strong> a Stewart platform as a more general robot, as presented by Nguyen<br />
et al. [104], will be enhanced with high performance <strong>control</strong> <strong>of</strong> <strong>motion</strong>, as these <strong>system</strong>s<br />
usually require minimal time tasks.<br />
By incorporating more structural <strong>system</strong> information i.e. a model into the <strong>control</strong>ler,<br />
it is possible to achieve higher performance. Most modern <strong>control</strong> strategies are therefore<br />
model <strong>based</strong> in some way (directly, in design or evaluation). In this case the quality <strong>of</strong><br />
<strong>motion</strong> depends on the fidelity <strong>of</strong> the model. Deriving a model <strong>of</strong> the mechanics <strong>of</strong> the<br />
Stewart platform manipulator for analysis, design and <strong>control</strong> will be the subject <strong>of</strong> this<br />
chapter. To arrive at a model with structure from which insight can be gained, modelling<br />
will be done <strong>based</strong> on physical laws.<br />
<strong>Model</strong>ling the dynamics <strong>of</strong> a Stewart platform as a multibody <strong>system</strong> has been done<br />
by Lee et al. [80] who claim to be the first to present a complete model and with more<br />
simplifications by Do et al. [36] and Liu et al. [83]. <strong>Model</strong>ling the mechanics <strong>of</strong> this platform<br />
can be done in several ways and with various objectives in mind. The equations <strong>of</strong> <strong>motion</strong><br />
can be derived by using the classical approach <strong>of</strong> Lagrange [80] or Newton-euler [36].<br />
In general, deriving the equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> a parallel manipulator results in combined<br />
differential and algebraic (constraint) equations (see e.g. Roberson and Schwertassek<br />
[120]). In simulation and <strong>control</strong> this formulation can cause difficulties, usually referred to<br />
as index problems [19]. Here it is shown that an explicit differential model for the Stewart
2.2 Kinematics 33<br />
platform results if one makes the right choices in parametrization. Dependent variables are<br />
explicit functions <strong>of</strong> the integrable differential equations. In this way index problems, etc.<br />
are circumvented.<br />
By using a modern projection or dual basis method [79] like Kane’s [64], which have<br />
the advantages <strong>of</strong> both the Newton-euler and Lagrange formulation but without the corresponding<br />
disadvantages [56], it will be shown that applying this approach can result in a<br />
model from which more insight can be gained.<br />
Together with the alternative parametrization, this is advantageous over the models earlier<br />
presented in literature, if one wants to apply a model for both analysis, simulation and<br />
<strong>control</strong>. <strong>Model</strong> <strong>based</strong> feedback, however, is still more complex for parallel manipulators,<br />
since the dynamics are only described in end-effector coordinates and the measured signals<br />
are link related. Next to the fairly low performance requirements <strong>of</strong> <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong><br />
<strong>system</strong>s in the past, this will probably be the reason that in most <strong>of</strong> these <strong>motion</strong> <strong>system</strong>s,<br />
feedback <strong>control</strong>lers are decentralized (one SISO loop per parallel link i.e. actuator).<br />
Through analysis <strong>of</strong> the derived model, some <strong>of</strong> the disadvantages <strong>of</strong> decentralized <strong>control</strong><br />
can be revealed. To apply a simple, but accurate model <strong>based</strong> <strong>control</strong>ler in practise, one<br />
would like to quantify the errors made by undermodelling, to be able to do robust analysis<br />
<strong>of</strong> the <strong>control</strong> scheme. The modelling approach taken here aims at a model from which<br />
the influences <strong>of</strong> different <strong>system</strong> parameters, like masses, inertias, velocity, gravity can be<br />
clearly separated.<br />
2.2 Kinematics<br />
After stating some notational issues, the fundamental formulas <strong>of</strong> mechanics to describe the<br />
kinematics will be introduced. Then the Stewart platform will be defined in order to derive<br />
a model <strong>of</strong> its specific kinematics. After some model analysis with the <strong>control</strong> objective in<br />
mind, this part will be concluded by a resume.<br />
2.2.1 Notation and definitions<br />
Capital symbols, X are used for matrices, x for vectors, x for scalars. With some scalar<br />
(energy) functions X is used. x y denotes the vector product which can also be written<br />
as ~ Xy = ( ~ Y ) T x where the vector product matrix, ~ X, is a skew symmetric (X = ;X T )<br />
matrix parametrized by the vector, x T = x1 x2 x3 , such that the result is the vector<br />
product.<br />
~X =<br />
2<br />
4 0 ;x3 x2<br />
x3 0 ;x1<br />
;x2 x1 0<br />
3<br />
5 (2.1)<br />
X Y denotes vector wise product <strong>of</strong> the columns stacked in the matrices.<br />
p<br />
The index xn is used for the normalising operation xn = x= j x j with j x j= xT x.<br />
Pxn denotes the projector to the (hyper)plane with normal vector x n and can be constructed
34 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 2.1: Projection <strong>of</strong> arbitrary vector, s onto the plane with normalised normal vector, x n.<br />
The projection matrix, Pxn, can be composed from the vector product (matrices,<br />
~Xn).<br />
from the vector product matrix<br />
Pxn =(I ; xnx T n )=(~ Xn) 4 = ~ Xn ~ X T n = ;( ~ Xn) 2 : (2.2)<br />
Projection matrices have some properties, which can be taken advantage <strong>of</strong>, like<br />
Pxn = P T xn = P k xn� (2.3)<br />
with k>0 and integer. In some cases it will be appropriate to use such a projection plane<br />
for construction <strong>of</strong> a frame as done in Fig. 2.1.<br />
Motion can be described w.r.t. various frames. A matrix or vector described in some<br />
frame can have a superscript referring to this frame. For the inertial frame, G, or ground<br />
coordinates, the index x g will be used. As a function <strong>of</strong> the moving end-effector or platform<br />
frame, M, vectors will be denoted x m . If a (rotation) matrix maps a vector into another<br />
frame it will be denoted as g R m if R maps from M to G.<br />
The subscript index like ai will be used to refer to the i th -actuator if also non actuator<br />
dependent variables appear in the equation.<br />
2.2.2 Calculating velocity and acceleration by differentiation<br />
An important part <strong>of</strong> kinematics is the ability to derive velocity and acceleration vectors <strong>of</strong><br />
any part <strong>of</strong> a (partly) moving construction. The basic formulas to do this will be derived in<br />
this section.<br />
The <strong>motion</strong> <strong>of</strong> a point (mass particle, joint, etc.) is usually most conveniently and invariantly<br />
defined w.r.t. the body frame to whom it’s connected. The <strong>motion</strong> <strong>of</strong> a frame put<br />
in another frame generally consists <strong>of</strong> a translation, which can be described by a vector, t,<br />
and a rotation for which a matrix, R, can be used. The orientation <strong>of</strong> a whole frame can<br />
be described by a rotation matrix. A rotation matrix consists <strong>of</strong> perpendicular unit vectors<br />
which describe the basis <strong>of</strong> the frame into the other frame. As a result a rotation matrix, R
2.2 Kinematics 35<br />
has the following property:<br />
R T R = I (2.4)<br />
Any real 3 3-matrix with this property and the property det(R) = 1 is a rotation matrix,<br />
which set is <strong>of</strong>ten referred as the SO(3)-group [77]. SO(3) stands for the Special<br />
(det(R) =1) Orthonormal (RT R = I) group <strong>of</strong> matrices <strong>of</strong> size 3x3. With det(R) =;1<br />
also the mirror operation is included (transformation <strong>of</strong> right hand frames to left hand frames<br />
and vice versa). The position <strong>of</strong> a point pm in frame M can now be described in frame G<br />
by:<br />
p g = t g + g R m p m<br />
(2.5)<br />
To describe the velocity <strong>of</strong> this point in the other frame one can simply differentiate this<br />
equation. Some properties <strong>of</strong> the time derivative <strong>of</strong> the rotation matrix can be derived by<br />
differentiating (2.4). This results in skew symmetric matrices which can be parametrized by<br />
the vector product matrix <strong>of</strong> the (thereby defined) angular velocity ! m .<br />
with<br />
~ m =<br />
R T _ R = ; _ R T R = ~ m<br />
2<br />
4 0 ;!3 !2<br />
!3 0 ;!1<br />
;!2 !1 0<br />
(2.6)<br />
3<br />
5 (2.7)<br />
Now, for the vector, which is rigidly attached to the frame M, _p m = 0. Hence, by differentiating<br />
(2.5) and using (2.6), one obtains<br />
_p g = _ t g + g R m ~ m p m = _ t g + ~ g p g = _ t g +! g<br />
p g<br />
(2.8)<br />
where the change <strong>of</strong> frame for the matrix ~ ,isgivenby ~ g = g R m ~ mm R g and using<br />
the inverse rotation given by m R g =( g R m ) T with (2.4).<br />
By differentiating (2.8) the acceleration <strong>of</strong> the point, still rigidly attached to the frame<br />
M(_p m = 0), can be calculated, using ~ P m ! m = ~ m p m :<br />
p g = t g + g R m ( ~ P m ) T _! m + g R m ( ~ m ) 2 p m<br />
= t g + ( ~ P g ) T _! g +( ~ g ) 2 p g<br />
= t g ; p g _! g +! g<br />
(! g<br />
p g ) (2.9)<br />
In other cases, the point considered can already be moving in the frame M (p m ,v m m =<br />
_p m ,a m m = pm ). Here, by differentiation <strong>of</strong><br />
_p g = _ t g + g R m ( ~ ) m p m + g R m v m p<br />
the coriolis acceleration appears as the third term in<br />
p g = a g p + g R m a m m +2g R m ~ m v m m<br />
= a g p +ag m +2~ g v g m =a g p +ag m +2(!g<br />
(2.10)<br />
v g m ): (2.11)<br />
Where p is a point connected to M momentarily at the same position as p. Its acceleration,<br />
a g p , is given by (2.9).
36 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 2.2: Physical structure <strong>of</strong> the euler angle representation.<br />
2.2.3 Parametrising orientation by euler angles<br />
In stating the equations <strong>of</strong> <strong>motion</strong>, the state which describes the orientation, usually only<br />
appears in the rotation matrix. It is possible to parametrise the rotation matrix in several<br />
ways.<br />
Although the rotation matrix consists <strong>of</strong> nine entries, its properties, given by (2.4), put<br />
constraints on these entries. The orientation, i.e. the SO(3)-group, can locally be represented<br />
by 3 parameters. It can, however, not be covered by a single coordinate chart <strong>of</strong> this<br />
size [77]. Several parametrizations are used in literature. The euler angle description consists<br />
<strong>of</strong> three subsequent simple planar rotations around momentary axes. The euler angles<br />
are not convenient for use in modelling the Stewart platform but are merely discussed for<br />
reference since it is a popular description in the aerospace community.<br />
An <strong>of</strong>ten used example <strong>of</strong> euler angle rotation is physically represented in Fig. 2.2. Both<br />
three unit vectors nG�M <strong>of</strong> the ground frame, G, and moving frame, M, are drawn. A simple<br />
x�y�z<br />
planar rotation around the x-axis has the following structure<br />
Tx =<br />
2<br />
4<br />
1 0 0<br />
0 cos( ) ; sin( )<br />
0 sin( ) cos( )<br />
A vector, x m , quantified in the axes <strong>of</strong> M can be represented in G by<br />
x g = Rx m = Tz�Ty�Tx�xm 3<br />
5 (2.12)<br />
(2.13)<br />
Since the subsequent rotations are not commutative, it is important to define the order <strong>of</strong><br />
rotation. The order defined in (2.13) is the usual order applied in the aerospace community.
2.2 Kinematics 37<br />
This means that the airplane, at first positioned so that the body axis is parallel to the fixed<br />
inertial frame, G, has to be rotated as follows:<br />
- First rotate over an angle , the yaw angle, around the inertial n g z-axis. In taxiing, this<br />
typically will be the main angle <strong>of</strong> rotation.<br />
- Then pitch over an angle around the intermediate y � -axis. This will usually be the main<br />
angle to be changed in take-<strong>of</strong>f. Note, e.g. from Fig. 2.2 that this axis is in general not<br />
an axis <strong>of</strong> the inertial or the moving frame.<br />
- Finally roll around the airplanes’ longitudinal axis, n m x , over an angle to obtain its actual<br />
orientation.<br />
The fully parametrized rotation matrix is then as follows:<br />
R =<br />
2<br />
4<br />
c c c s s ; s c s s + c s c<br />
s c c c + s s s s s c ; c s<br />
;s c s c c<br />
3<br />
5 (2.14)<br />
In this equation s = sin( ), c = cos( ) and so on. The three euler angles have the<br />
disadvantage <strong>of</strong> a highly non-linear appearance in both the rotation matrix and the euler<br />
angle velocity to angular velocity transformation. The latter can even become singular.<br />
Since, although any rotation matrix can be described by this parametrization, in some states<br />
(consider = =2 in Fig. 2.2) this chart does not allow three independent changes <strong>of</strong><br />
orientation, i.e. the kinematical relationship between =[ ] T and the angular velocity<br />
in the body frame, ! m , given by<br />
! m =<br />
2<br />
4 p<br />
q<br />
r<br />
2<br />
4 1<br />
0<br />
0<br />
3<br />
5 = ( ) _ =<br />
3 2<br />
5 _ T<br />
+ Tx0 2<br />
4<br />
4 0<br />
1<br />
0<br />
1 0 ;s<br />
0 c c s<br />
0 ;s c c<br />
3<br />
5 _ T<br />
+ Tx0T T y0 2<br />
4 0<br />
0<br />
1<br />
32<br />
5<br />
4 _<br />
_<br />
_<br />
3<br />
5 =<br />
3<br />
5 _ (2.15)<br />
becomes singular. p, q and r are the components <strong>of</strong> the angular velocity, ! m , measured in<br />
moving body frame. In an airplane this will usually be the output <strong>of</strong> the gyros.<br />
In some experiments, a rotation matrix can be identified. With ; =2 < � � < =2,<br />
the euler angles can be calculated from such rotation matrix as follows:<br />
sin( ) = ;R(3� 1) (2.16)<br />
sin( ) = R(2� 1)= cos( )<br />
sin( ) = R(3� 2)= cos( )<br />
As already said, the euler angle description <strong>of</strong> rotation is not convenient in describing the<br />
model <strong>of</strong> the Stewart platform. This is mainly because evaluation <strong>of</strong> the high number <strong>of</strong> sine<br />
and cosine functions is both analytically and numerically relatively troublesome and this can<br />
fully be avoided in modelling the <strong>motion</strong> <strong>system</strong> mechanics by using euler parameters.
38 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
G<br />
nµ<br />
n<br />
g<br />
z<br />
n<br />
n<br />
g<br />
x<br />
g<br />
y<br />
s m<br />
µ<br />
s g<br />
nµ x sm<br />
µ<br />
s<br />
s<br />
g<br />
m<br />
2sin(1/2 µ )<br />
( nµ x sm)xnµ<br />
Fig. 2.3: Visualising the euler parameter representation by a rotation around the unit vector,<br />
n , over an angle, . The lower right schematic drawing is the plane orthogonal<br />
to n through the points sg and sm.<br />
2.2.4 Parametrising orientation by euler parameters<br />
An attractive alternative <strong>of</strong> euler angles are the unit quaternions [130] called euler parameters.<br />
No singularities and, even more important, numerically convenient relationships with<br />
both the rotation matrix and the angular velocity and as with the euler angles, an interpretation<br />
which can be visualised. This at the cost <strong>of</strong> an additional fourth parameter and<br />
consequently an extra constraint equation.<br />
The orientation <strong>of</strong> one frame w.r.t. another frame can always be described by a single<br />
rotation over an angle , about a unique axis with direction n . The four euler parameters,<br />
T T<br />
, parametrise and n as follows:<br />
e = 0<br />
The constraint equation,<br />
= 1 2 3<br />
0 = cos(1=2 ) (2.17)<br />
T = sin(1=2 )n (2.18)<br />
T<br />
e e =1� (2.19)<br />
is a direct result from this definition. By not using the angle <strong>of</strong> rotation itself as a coordinate<br />
but the sine and cosine instead, these geometric functions do not have to be evaluated anymore.<br />
And as a result we will see in the subsequent sections that in a full mechanical model<br />
<strong>of</strong> the Stewart platform not one geometric function will appear.
2.2 Kinematics 39<br />
With Fig. 2.3 the rotation <strong>of</strong> a vector, sm, in the body frame, M, to its direction in the<br />
inertial frame, G, given as sg, can be described using the unit vector, n , along the axis <strong>of</strong><br />
rotation and the angle <strong>of</strong> rotation, .<br />
The following derivation starts with constructing s g from sm adding the two component<br />
vectors in the plane given in Fig. 2.3 and is directed towards the specification <strong>of</strong> the rotation<br />
matrix.<br />
sg = sm ; (1 ; cos( ))(n sm) n + sin( )(n sm)<br />
= sm + 2 sin 2 (1=2 )n (n sm) +2cos(1=2 )sin(1=2 )(n sm)<br />
= I +2~~ +2 0~ sm<br />
= 2<br />
=<br />
2<br />
2<br />
4 2<br />
0<br />
2<br />
+ 1 ; 1=2 1 2 ; 0 3 1 3 + 0 2<br />
2<br />
1 2 + 0 3 0 + 2 2 ; 1=2 2 3 ; 0 1<br />
1 3 ; 0 2 2 3 + 0 1 ; 1=2<br />
4 ; 1 0 ; 3 2<br />
; 2 3 0 ; 1<br />
; 3 ; 2 1 0<br />
3 2<br />
5<br />
2<br />
0<br />
+ 2<br />
3<br />
3<br />
5 sm<br />
4 ; 1 0 3 ; 2<br />
; 2 ; 3 0 1<br />
; 3 2 ; 1 0<br />
3T<br />
5 sm<br />
= G( e)L T ( e)sm = R( e)sm (2.20)<br />
So, rotation appears to be equal to two simple subsequent transformations, G and L,<br />
which are linear in the euler parameters. By Nikravesh et al. [107] it is shown that this<br />
parametrization and using _ RR T = ~! g leads to the following relation to calculate _ e or _<br />
from ! and e,<br />
_e = 1<br />
2 GT ( e)! g<br />
In only constructing the variable, _, a reduced version is given by<br />
(2.21)<br />
_ = 1<br />
2 GT s ( e)! g � (2.22)<br />
where the matrix, Gs, is given by the last three columns <strong>of</strong> G. This equation can be used to<br />
get from the angular velocity to the rotation matrix with help <strong>of</strong> an integration routine and<br />
initial conditions on . Further, the fairly simple relations like<br />
! g =2G( e)_ e� (2.23)<br />
! m =2L_ e and e =1=2G T _! g can be derived.<br />
With angles ; < < , can be used as the (orientation) state from which 0 =<br />
p 1 ; T is solved. In that case, a three parameter setting is used at the cost <strong>of</strong> possible<br />
singularities. The nonsingular envelope is, however, twice as large as in case <strong>of</strong> using euler<br />
angles.
40 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 2.4: A 2-d.o.f. serial <strong>system</strong> in a singular configuration<br />
It is possible to calculate euler parameters from any rotation matrix where j j< .<br />
0 = 1p<br />
(1 + tr(R))<br />
2<br />
(2.24)<br />
1 = (R(3� 2) ; R(2� 3))=(4 0) (2.25)<br />
2 = (R(1� 3) ; R(3� 1))=(4 0) (2.26)<br />
3 = (R(2� 1) ; R(1� 2))=(4 0) (2.27)<br />
where tr(R) = P i=3<br />
i=1 R(i� i) is the trace <strong>of</strong> the rotation matrix. Only in case <strong>of</strong> 0 = 0<br />
another solution strategy has to be applied. With the earlier given relationships, (2.14) and<br />
(2.16) between R and and vice versa, euler parameters can be calculated from euler angles<br />
and vice versa.<br />
2.2.5 Jacobian matrices<br />
Connected parts <strong>of</strong> a mechanical <strong>system</strong> will result in less freedom <strong>of</strong> <strong>motion</strong> reflected by<br />
constraint equations which can be explicit as in y = f(x) or implicit as in g(x� y) = 0.<br />
Such equations also constrain the time derivatives <strong>of</strong> these coordinates, i.e. the velocities<br />
and accelerations. In describing a mechanical <strong>system</strong>, e.g. stating its equations <strong>of</strong> <strong>motion</strong>,<br />
it is <strong>of</strong>ten convenient to state the <strong>motion</strong> <strong>of</strong> the <strong>system</strong> as a function <strong>of</strong> a limited number <strong>of</strong><br />
(generalised) variables (coordinates and/or velocities).<br />
If some variations or velocities can be described as a product <strong>of</strong> a (position-dependent)<br />
matrix and a vector <strong>of</strong> other variations this matrix will be called a (semi-)Jacobian matrix.<br />
The jacobian matrix between two sets <strong>of</strong> variables usually comes out naturally by a time<br />
differentiated version <strong>of</strong> an equation in which one <strong>of</strong> the sets is explicitly stated as in y(t) =
2.2 Kinematics 41<br />
f(x(t)),<br />
_y(t) = @f<br />
@x (x(t)) _x(t) =Jy�x(x(t)) _x(t) (2.28)<br />
In mechanical <strong>system</strong>s such mappings <strong>of</strong>ten result in an indirect way from differentiation<br />
<strong>of</strong> the rotation matrix and using ! as velocity term. The angular velocity, !, however, is no<br />
time derivative <strong>of</strong> any representation <strong>of</strong> attitude. E.g. given (2.8),<br />
_p g = I g R m ( ~ P m ) T<br />
_t g<br />
! m = Jp g �x _x (2.29)<br />
Also matrices like Jp g �x will be termed jacobian, although this is not precise i.e. not a result<br />
<strong>of</strong> (2.28), but just states one set <strong>of</strong> velocities as a function <strong>of</strong> another set.<br />
2.2.6 Parallel, serial and singular configurations<br />
In kinematics, jacobian matrices <strong>of</strong>ten play an important role in relating input/actuator and<br />
output/ end-effector coordinates. With parallel manipulators, it will be shown using some<br />
examples, that these relations differ from those constructed from a serial manipulator. In<br />
Fig. 2.4, a 2 degree-<strong>of</strong>-freedom serial joint robot is depicted. The end-effector coordinates<br />
x and y can in the serial configuration explicitly be stated as functions <strong>of</strong> actuated rotating<br />
joints angles q1 and q2 as<br />
And by differentiation<br />
x = l1 cos(q1) +l2 cos(q1 + q2) (2.30)<br />
y = l1 sin(q1) +l2 sin(q1 + q2) (2.31)<br />
_x<br />
_y = ;l1 sin(q1) ; l2 sin(q1 + q2) ;l2 sin(q1 + q2)<br />
l1 cos(q1) +l2 cos(q1 + q2) l2 cos(q1 + q2)<br />
_q1<br />
_q2<br />
= Jx�qq (2.32)<br />
At q2 =0 , the jacobian <strong>of</strong> this configuration becomes singular. At such configurations,<br />
physically, the actuators can only generate one d.o.f. <strong>motion</strong>/velocity <strong>of</strong> the end-effector<br />
locally. Further, it can be noted that, for each nonsingular configuration, two actuator positions<br />
generate the same end-effector state.<br />
The dual parallel configuration <strong>of</strong> a 2 d.o.f. robot is given in Fig. 2.5. In fully parallel<br />
<strong>system</strong>s, the sliding actuator lengths can explicitly be stated as functions <strong>of</strong> the end-effector<br />
coordinates. With xT =[xy] and li(x) =x ; bi<br />
q<br />
klik = lT i li<br />
(2.33)<br />
and by differentiation<br />
k _ lik = l T n�i _x (2.34)<br />
Thus J T l12�x = [ln�1 ln�2]. If the unit actuator direction vectors, ln�i are in parallel, the<br />
configuration becomes singular. In this case, however, the end-effector can still move but
42 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 2.5: A 2-d.o.f. parallel <strong>system</strong> in a singular configuration<br />
Fig. 2.6: A 3-d.o.f. <strong>system</strong> with serial and parallel singular configurations<br />
not be forced in 2 d.o.f. and from the speed <strong>of</strong> the actuators, the end-effector velocity can<br />
not be determined. In this <strong>system</strong>, in a nonsingular configuration, there are two end-effector<br />
states for each actuator position.<br />
In general, a rigid body mechanical <strong>system</strong>, will have combinations <strong>of</strong> serial and parallel<br />
connections as in Fig. 2.6. In this example, the actuator coordinates, q, are no explicit functions<br />
<strong>of</strong> the end-effector coordinates, x and v.v. Also both kind <strong>of</strong> singular configurations<br />
can usually occur. An important observation in the following sections will be the fact that<br />
the Stewart platform <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> is a purely parallel <strong>system</strong>.<br />
The parallel manipulator construction <strong>of</strong> the Stewart platform is first defined. To derive<br />
the equations <strong>of</strong> <strong>motion</strong>, the velocity and accelerations should be described w.r.t. a<br />
limited number <strong>of</strong> generalized variations. This defines the kinematics after which the semiequilibrium<br />
equations <strong>of</strong> the active and inertial forces can be stated.<br />
φ
2.2 Kinematics 43<br />
2.2.7 Stewart platform definitions and assumptions<br />
The Stewart platform (Fig. 2.7) consists <strong>of</strong> an end-effector body with mass, m c, and 3x3<br />
inertia matrix I m c w.r.t. the end-effector frame, M, connected to the centre <strong>of</strong> gravity (c.o.g.)<br />
<strong>of</strong> the body which has varying coordinates, c, in the inertial frame, G. The end-effector body<br />
or platform is connected by six parallel actuators at the moving upper gimbal points with<br />
fixed coordinates, ai, in the moving frame, M, tobi in the inertial frame, G. The length <strong>of</strong><br />
the six actuators can be varied. In describing a specific actuator the subscript i for the i th<br />
actuator will be left away.<br />
The platform position is defined as<br />
sx =<br />
c<br />
s<br />
(2.35)<br />
With the last three euler parameters, ,todefine the rotation matrix, T ( ), asR in (2.20),<br />
from the moving (platform) frame, M, to inertial (ground) frame, G and a scaling factor<br />
s = 2kakmax equal to the maximum distance <strong>of</strong> the c.o.g. to any <strong>of</strong> the upper gimbal<br />
points. The platform speed is not defined as the derivative <strong>of</strong> platform position but as<br />
_x =<br />
_c<br />
! � (2.36)<br />
where ! is the angular velocity <strong>of</strong> the platform. The platform (generalised) speed and<br />
position are related through the jacobian, J sx�x, using (2.22)<br />
sx _ =<br />
_c<br />
s_<br />
= I 0<br />
0 sG T s =2<br />
_c<br />
! = Jsx�x _x (2.37)<br />
An actuator (Fig. 2.8) will be modelled as 2 bodies, the rotating cylinder and the moving<br />
piston. The rotating body, with mass mb and a constant distance <strong>of</strong> rb <strong>of</strong> the c.o.g., bc, to<br />
the connection <strong>of</strong> a 2-d.o.f.-rotational gimbal joint to the inertial frame at b. The moving<br />
actuator body, i.e. the piston, with mass ma with a constant distance <strong>of</strong> ra <strong>of</strong> the c.o.g.,<br />
ac, is connected with a 3-d.o.f.-rotational gimbal joint to the platform at a. Witha1d.o.f.<br />
<strong>control</strong>led sliding joint between these two bodies the length <strong>of</strong> the actuator can be<br />
varied.<br />
It is assumed that the inertia <strong>of</strong> the actuator bodies can be neglected around the actuator<br />
axis and to be uniform perpendicular to this axis. i a is the inertia <strong>of</strong> the moving actuator<br />
body at ac and any axis perpendicular to the actuator. i b is the inertia <strong>of</strong> the rotating part<br />
<strong>of</strong> the actuator along any axis perpendicular to the actuator w.r.t. the connection point to the<br />
inertial frame (b) .<br />
With these assumptions also the case, <strong>of</strong>ten seen in practise, in which the moving part <strong>of</strong><br />
the actuator not only slides at the connection with the rotating part but also rotates around<br />
the l-axis , and has only a 2-d.o.f. rotation gimbal in connection to the platform, results in<br />
the same dynamics. Finally, the c.o.g.’s <strong>of</strong> the actuator parts are assumed to lie on the line<br />
connecting the upper and lower gimbal point.
44 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 2.7: A schematic view <strong>of</strong> the Stewart platform<br />
2.2.8 Stewart platform kinematics<br />
The kinematics <strong>of</strong> the Stewart platform will be described by first defining the transformation<br />
<strong>of</strong> the platform to actuator coordinates. Then by differentiation also velocity and acceleration<br />
<strong>of</strong> all relevant points can be calculated as a function <strong>of</strong> the platform <strong>motion</strong>, whose<br />
velocities will be taken as the generalized speeds.<br />
Almost all vectors can be conveniently described in the inertial frame. Apart from a m i<br />
whose time derivative in the moving frame is 0.<br />
As shown in Fig. 2.7, the vector, li, between the two attachment points <strong>of</strong> an actuator<br />
can be described by<br />
li =c + T a m i ; bi (2.38)<br />
Now the squared length <strong>of</strong> the actuator, j li j 2 = l T i li, and the unit vector in direction <strong>of</strong><br />
the actuator, ln�i = li<br />
jlij can be calculated from the platform variables c g and the orientation<br />
matrix T = g R m which will be the only rotation matrix used.<br />
The velocity <strong>of</strong> the length <strong>of</strong> the actuators can be calculated by projection <strong>of</strong> the velocity<br />
<strong>of</strong> the upper gimbal attachment point, v a, in the direction <strong>of</strong> the actuator, since<br />
d<br />
dt j li j= d<br />
dt<br />
q l T i li = lT i va<br />
j li j = lT n va: (2.39)
2.2 Kinematics 45<br />
Using (2.8), the velocity <strong>of</strong> the upper gimbal points is given by<br />
vai = _c +! T a m i � (2.40)<br />
where ! is the angular velocity <strong>of</strong> the moving frame in the inertial frame. By projection <strong>of</strong><br />
this velocity with (2.39) and filling in (2.40), the velocity <strong>of</strong> the actuator is described as a<br />
function <strong>of</strong> platform velocities and poses.<br />
_li = l T n�ivai = lT n�i _c + l T n�i (! T ami ) (2.41)<br />
With some reordering and written as matrix equation (ln�i, am i and vai stacked in Ln, Am and Va) for all the actuators the jacobian between the actuator and platform velocities is<br />
defined.<br />
_<br />
l = L T n _c +(TA m Ln) T ! = Jl�x _x = L T n Va (2.42)<br />
The semi-jacobian matrix, Jl�x, is one <strong>of</strong> the most important variables in the Stewart platform,<br />
relating the platform coordinates to be <strong>control</strong>led and used as basic model coordinates,<br />
and the actuator lengths, which can be measured. Further, in transposed form, J l�x relates<br />
the actuator input forces with the platform forces as will be shown in deriving the platform<br />
dynamics. Note that with defining _x T =[_c T ! T ], x does not exist.<br />
2.2.9 Interpretation and use <strong>of</strong> the jacobian matrix, Jl�x<br />
If the <strong>system</strong> has to be <strong>control</strong>led by the actuators, the jacobian specifies how the <strong>control</strong><br />
inputs, the actuator forces, influence the platform (accelerations), which are, especially in<br />
<strong>flight</strong> simulation applications, <strong>of</strong>ten the variables to be <strong>control</strong>led. Further, the measurable<br />
outputs are <strong>of</strong>ten only the actuator lengths. The derivatives (actuator speed) <strong>of</strong> these outputs<br />
are given by the product <strong>of</strong> the jacobian and the platform speed.<br />
There are two interpretations to the jacobian. In the force interpretation the rows <strong>of</strong> J l�x<br />
give the (generalized) forces in the platform coordinates given a unit force in an actuator. In<br />
the velocity interpretation the columns <strong>of</strong> Jl�x specify the velocity <strong>of</strong> the actuators required<br />
to have unit velocity <strong>of</strong> the platform.<br />
In model <strong>based</strong> <strong>control</strong>, the inverse information is <strong>of</strong> interest. The measured variations<br />
<strong>of</strong> the actuator have to be put in platform variations to calculate corrections in a model<br />
specified in platform coordinates. Each column <strong>of</strong> the inverse jacobian, J ;1<br />
l�x , specifies what<br />
velocity (angular velocity included) <strong>of</strong> the platform is necessary to have elongation <strong>of</strong> just<br />
one actuator while the others only rotate. Given measurable actuator velocities, the platform<br />
velocities can be calculated.<br />
The correction forces in a model <strong>based</strong> <strong>control</strong>ler are also most conveniently calculated<br />
as a function <strong>of</strong> platform coordinates. Each row <strong>of</strong> J ;1<br />
l�x specifies the forces necessary in the<br />
actuators to have unit force correction in platform coordinates.<br />
The inverse jacobian appears in feedback linearising structures (like computed torque,<br />
etc.), which will be dealt with in the forthcoming chapters.<br />
Another problem <strong>of</strong> a parallel manipulator, with only the link position measured, are<br />
the forward kinematics. It is not known how to analytically calculate the platform position<br />
(without decision making about set <strong>of</strong> possible solutions) from link measurements.
46 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
r b<br />
l<br />
bc<br />
ac<br />
b<br />
Lower Gimbal<br />
ra<br />
Upper Gimbal<br />
Fig. 2.8: Stewart platform actuator link construction<br />
The jacobian Jl�x(sx) is an important element in the jacobian Jsx�l(sx) from actuator<br />
variation to the platform position variation.<br />
Jsx�l(sx) =J ;1<br />
;1<br />
l�sx (sx) =Jsx�x(sx)J l�x<br />
(sx)� (2.43)<br />
where Jsx�x is given by (2.37).<br />
This jacobian provides a way to apply a Newton-Raphson iteration to calculate the solution<br />
provided one starts in a point sufficiently close to the solution and away from jacobian<br />
singularities.<br />
sxj+1 = sxj + J ;1<br />
l�sxj (lmeasured ; lj) (2.44)<br />
The condition number <strong>of</strong> Jl�x also provides a measure for the <strong>control</strong>lability <strong>of</strong> the platform<br />
from the actuators which becomes un<strong>control</strong>lable at singularities <strong>of</strong> this matrix.<br />
Further, most <strong>of</strong> the constraints <strong>of</strong> the platform are caused by the characteristics <strong>of</strong> the<br />
actuators like limited stroke, maximum speed and force. The jacobian plays an important<br />
role in translating these limitations into platform coordinates. E.g. given a maximum velocity,<br />
j v jmax, in extending an actuator either direction, the maximum velocity in the j’th<br />
platform coordinate is given by the 1-norm <strong>of</strong> the respective row <strong>of</strong> J ;1<br />
l�x times the maximum<br />
actuator velocity.<br />
_xi�max = kJ ;1<br />
l�x (i� )k1 j v jmax=<br />
6X<br />
j=1<br />
j J ;1<br />
l�x (i� j) jj v jmax (2.45)<br />
a
2.2 Kinematics 47<br />
So, the maximal platform velocity along a specific coordinate can be calculated by adding<br />
all absolute values <strong>of</strong> the specific row <strong>of</strong> the inverse jacobian and multiplying with the<br />
maximum actuator velocity. In this case, every other platform coordinate is left unspecified.<br />
In the more interesting case, with the other platform coordinates constraint to zero<br />
_xi�max =j v jmax =kJl�x(i� )k1 =j v jmax = max(j<br />
Jl�x(i� j) j)� (2.46)<br />
j<br />
e.g. the maximum pure surge speed can be calculated by dividing the maximum actuator<br />
velocity by the largest number in the first column <strong>of</strong> the Jacobian. For position and acceleration,<br />
these formulas can be used as an approximation.<br />
2.2.10 Velocity and acceleration <strong>of</strong> the actuator joints<br />
If the inertia <strong>of</strong> the actuator joints is important or gimbal forces have to be calculated, the<br />
kinematical formulas have to be expanded with those to derive the velocities and accelerations<br />
<strong>of</strong> these joints.<br />
The jacobian between the platform and the upper gimbal point velocity is defined by<br />
vai = I T ( ~ A m i )T T T _x = Jai�x _x (2.47)<br />
To determine the inertial forces <strong>of</strong> the actuators, the jacobians, from gimbal point to the<br />
c.o.g.’s <strong>of</strong> the actuators, are also important. The angular velocity <strong>of</strong> the actuator perpendicular<br />
to the actuator, !a,isdefined by<br />
va<br />
!a = ln<br />
(2.48)<br />
j l j<br />
Now the velocities <strong>of</strong> the c.o.g.’s <strong>of</strong> the actuator bodies v ac and vbc can be stated as a function<br />
<strong>of</strong> va:<br />
and<br />
vac =va +!a<br />
(;raln) =(I ; ra<br />
j l j Pln )va = Jac�ava� (2.49)<br />
vbc =!a rbln = rb<br />
j l j Plnva = Jbc�ava� (2.50)<br />
as becomes more clear by looking at Fig. 2.9. The acceleration <strong>of</strong> the actuators can be<br />
calculated by differentiating (2.42),<br />
l = Jl�xx + _<br />
Jl�x _x = L T n _ Va + _ L T n Va: (2.51)<br />
The derivative <strong>of</strong> the unit vectors, ln�i, <strong>of</strong> each actuator, i, can be calculated with:<br />
_<br />
ln = d l<br />
dt j l j =<br />
l<br />
_ j l j;ld dt j l j<br />
j l j2 = (I ; lnlT n )<br />
va =<br />
j l j<br />
1<br />
j l j Plnva<br />
(2.52)
48 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
rb ln<br />
vbc<br />
ω a<br />
bc<br />
ω a<br />
ac<br />
Lower Gimbal<br />
va<br />
ω a<br />
vac<br />
ra ln ra l n<br />
x<br />
ω a<br />
ln .<br />
ln<br />
l<br />
Upper Gimbal<br />
Fig. 2.9: Stewart platform actuator construction <strong>of</strong> link velocities. With the upper gimbal<br />
velocity, va, which can be derived from the platform velocity by (2.40), all other<br />
actuator velocities can be constructed. The actuator speed, _ l, is the projection <strong>of</strong><br />
va along the actuator ln (2.41). The figure is best read by considering v a in the<br />
plane <strong>of</strong> the picture. Then, only the actuator angular velocity, ! a, given by (2.48),<br />
is out <strong>of</strong> the plane. In fact, it is orthogonal to the plane.
2.2 Kinematics 49<br />
The acceleration <strong>of</strong> the actuator length consists <strong>of</strong> a term which is the projection <strong>of</strong> the<br />
acceleration <strong>of</strong> the upper gimbal in the direction <strong>of</strong> the actuator and a positive quadratic<br />
term, which is the centripetal acceleration <strong>of</strong> the actuator.<br />
l = l T n _va +v T 1<br />
a ( Pln )va<br />
(2.53)<br />
j l j<br />
So the acceleration <strong>of</strong> the actuator length is always positive if the platform is moving with<br />
constant translational speed in any direction and constant orientation since _va =0in that<br />
case. The acceleration <strong>of</strong> the upper gimbal can be derived directly with (2.9) and (2.40)<br />
_vai = c + _! ai +! (! ai) =Jai�xx; j! j 2 P!ai� (2.54)<br />
where the projection matrix, P! =(I +!n! T n ), projects the upper gimbal vector, ai, on the<br />
plane orthogonal to the normalised angular velocity direction <strong>of</strong> the platform, ! n.<br />
The acceleration <strong>of</strong> the c.o.g. <strong>of</strong> the moving actuator part also generates inertial forces<br />
and can be written as a function <strong>of</strong> platform <strong>motion</strong>.<br />
_vac = d<br />
dt vac = d<br />
dt (Jac�ava) =Jac�a _va + _ Jac�ava<br />
So this jacobian needs to be differentiated. With (2.49) and (2.52),<br />
Now with j Plnva j 2 =v T a Plnva,<br />
Jac�a<br />
_ = d ra<br />
(I ;<br />
dt j l j<br />
=<br />
ra<br />
j l j2 vT ra<br />
a lnPln +<br />
j l j2 (Plnval T n + lnv T a Pln )<br />
(2.55)<br />
Pln ) (2.56)<br />
Jac�ava<br />
_ = ra<br />
j l j2 (j Plnva j 2 ln +2(v T a ln)Plnva)� (2.57)<br />
which shows a quadratic centripetal term in actuator direction and a coriolis term orthogonal<br />
to the actuator. Clearly, the piston acceleration, _vac, only depends on the velocity, va and<br />
acceleration, _va, <strong>of</strong> the upper gimbal, given the positional coordinates.<br />
The <strong>motion</strong> <strong>of</strong> the actuators has now been explicitly stated as a function <strong>of</strong> the platform<br />
coordinates and its derivatives. The kinematic relations thus provide means to state the<br />
<strong>motion</strong> <strong>of</strong> the <strong>system</strong> as a function <strong>of</strong> a limited number <strong>of</strong> variables. Together with the<br />
dynamics i.e. semi-equilibria <strong>of</strong> active and inertial forces stated in Section 2.4, the equations<br />
<strong>of</strong> <strong>motion</strong> result.<br />
2.2.11 The Simona <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> kinematics<br />
In this thesis, any <strong>system</strong> connected to the environment through six parallel joints with five<br />
(passive) rotational and one (active) translational d.o.f. is considered a Stewart platform.<br />
The structure <strong>of</strong> the Simona <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> design is much more specific. It<br />
is schematically depicted from above in Fig. 2.10 and the measures are given in Table 2.1.
50 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 2.10: The Simona <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> kinematics represented schematically<br />
in a top view in an ’all the actuators down’ position<br />
Both the upper as the lower gimbal points virtually form a platform, which looks like two<br />
blunt equilateral triangles rotated 180 degrees with respect to each other.<br />
All the upper gimbal points as well as the lower lie on a circle. The circles lie in parallel<br />
planes if the actuator lengths are equal. All the minimal and maximal actuator lengths are<br />
equal. The actuators and gimbal points, which are numbered 1 to 6, are pairwise distributed<br />
clockwise over the circle. (6,1), (2,3) and (4,5) for the moving upper and (1,2), (3,4) and<br />
(5,6) for the inertial lower platform. The distance between the two elements <strong>of</strong> the pairs are<br />
also specified in the table. The upper pair (6,1) is connected on the positive x-side <strong>of</strong> the<br />
<strong>simulator</strong>. The lower pair (3,4) at the negative x-side <strong>of</strong> the ground.<br />
In most <strong>of</strong> the derivations, the <strong>motion</strong> <strong>system</strong> kinematics is assumed to belong to the<br />
most general class <strong>of</strong> Stewart platforms. This leaves room to adjust easily to a structure<br />
which differs from its design.<br />
This concludes the kinematical modelling, which is used to derive the equations <strong>of</strong> <strong>motion</strong><br />
in modelling the mechanical <strong>system</strong> in Section 2.4. The nest section finishes the treatment<br />
<strong>of</strong> the kinematics with two issues, which will be specifically important for the model<br />
<strong>based</strong> <strong>control</strong> structure to be used in Chapter 5.<br />
2.3 Analysis <strong>of</strong> the Stewart platform kinematics<br />
A vast majority <strong>of</strong> the literature on parallel robotic <strong>system</strong>s is devoted to their kinematics.<br />
This is an important subject because in the design <strong>of</strong> the kinematics <strong>of</strong> a robotic <strong>system</strong>,
2.3 Analysis <strong>of</strong> the Stewart platform kinematics 51<br />
Table 2.1: The Simona <strong>system</strong> parameters<br />
variable value description<br />
ra 1:60 m upper gimbal radius<br />
rb 1:65 m lower gimbal radius<br />
du 0:20 m upper gimbal spacing<br />
dl 0:60 m lower gimbal spacing<br />
lmin 2:08 m minimal actuator length<br />
qmax 1:25 m actuator stroke<br />
qop 1:15 m operational actuator stroke<br />
_lmax 1 m=s maximum actuator speed<br />
its manipulability, its workspace, etc. is to be fixed. This problem in combination with the<br />
application to <strong>flight</strong> simulation has been studied by Advani [7].<br />
In this section, two new solutions towards kinematical problems will be presented,<br />
which have a direct relation with the ability to perform real time model <strong>based</strong> feedback<br />
safely on a specific kinematic design <strong>of</strong> a Stewart platform <strong>motion</strong> <strong>system</strong>.<br />
- Given a specific design <strong>of</strong> the manipulator, one has to ensure the <strong>system</strong> can not be driven<br />
into a singular configuration.<br />
- Further, model coordinates have to be calculated from the measurements taken in real<br />
time (e.g. a finite number <strong>of</strong> calculation steps) without the chance to have divergence<br />
between actual and calculated coordinates.<br />
In parallel robots such as most <strong>flight</strong> simulation <strong>motion</strong> platforms, the position <strong>of</strong> the<br />
<strong>system</strong> is usually indirectly measured by the length <strong>of</strong> the actuators. The forward kinematical<br />
problem <strong>of</strong> calculating the platform coordinates given the actuator lengths <strong>of</strong> the Stewart<br />
platform is seen to be solved in roughly two ways in literature.<br />
Using analytic techniques, the problem can be transformed to a set <strong>of</strong> combined polynomial<br />
equations whose roots have to be found to solve the forward kinematics e.g. [57].<br />
Although these equations can provide insight into the structure <strong>of</strong> the problem, closed form<br />
solutions are only seen to be presented for special classes <strong>of</strong> platforms e.g. Merlet [97],<br />
which still is an active area <strong>of</strong> research [68]. Solving the roots <strong>of</strong> the equations, still leaves<br />
the problem <strong>of</strong> choosing the actual pose. The complexity <strong>of</strong> this problem is reflected by the<br />
fact that it has been shown that the general version <strong>of</strong> the Stewart platform can have up to<br />
40 real solutions [34].<br />
Secondly, for long, the forward kinematics has been tackled numerically by performing<br />
the Newton-Raphson (NR) iteration scheme [35] already given in (2.44).<br />
sxk+1 = sxk + J ;1<br />
l�sx (sxk)(lmeasured ; lk) (2.58)
52 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
As the actuator lengths, l, are explicit functions <strong>of</strong> the platform coordinates, sx, the<br />
jacobian, Jl�sx, is a function <strong>of</strong> platform coordinates.<br />
Jl�sx(x)(i� j) = @li(sx)<br />
@sxj<br />
(2.59)<br />
This latter method is preferred here as it is less involved to be implemented in a realtime<br />
model <strong>based</strong> <strong>control</strong>ler where inversion <strong>of</strong> the jacobian is already part <strong>of</strong> the <strong>control</strong><br />
structure. Convergence or convergence to the actual physical pose in this scheme is not<br />
guaranteed in general. As the forward kinematics <strong>of</strong> a Stewart platform have more than one<br />
solution [57], and since singular points <strong>of</strong> the jacobian for unconstrained actuator lengths<br />
exist [87], the iteration scheme does not converge globally to the right solution. This problem<br />
has only recently been considered in literature [21], [70].<br />
Chetelat [21] considers the problem <strong>of</strong> convergence in a general way by considering reconstruction<br />
<strong>of</strong> coordinates <strong>of</strong> an implicit (matrix) function from which the image is known<br />
e.g. a Stewart platform pose from length measurements. He points out that if one considers<br />
functions, <strong>of</strong> possibly interconnected vector equations, <strong>of</strong> polynomial degree
2.3 Analysis <strong>of</strong> the Stewart platform kinematics 53<br />
and minimum gain <strong>of</strong> J and again the Lipschitz condition. These can be calculated for<br />
Stewart platforms. Exclusion <strong>of</strong> singular points <strong>of</strong> J is necessary to calculate a radius <strong>of</strong> the<br />
neighbourhood in which convergence is guaranteed.<br />
From this radius, the maximum gain <strong>of</strong> J and the maximum speed <strong>of</strong> the actuator, a<br />
sufficient update frequency <strong>of</strong> the iteration can be calculated above which (quadratic) convergence<br />
is guaranteed.<br />
The parameters <strong>of</strong> the new Simona research <strong>simulator</strong> are used as an example which<br />
shows that reasonable results can be obtained although the conditions derived are rather<br />
conservative.<br />
2.3.1 Convergence NR-iteration<br />
A weak version <strong>of</strong> the Newton-Kantovorich theorem [109] given by Stoer [141] will be<br />
used. From this theorem convergence <strong>of</strong> the NR-iteration can be inferred. It is stated as<br />
follows.<br />
Theorem 2.1 Given: a set D IR n , a convex set Do with exterior Do D and a function<br />
f : D ! IR n which is continuous on D and differentiable with derivative Df(x) on Do.<br />
If positive constants r, , , and h can be found for x o 2 Do such that Sr(xo) =<br />
fx jkx ; xok 0<br />
B) limk!1 xk = exists, 2 Sr(xo), and f( )=0<br />
C)<br />
8 k 0� kxk ; k<br />
h 2k;1<br />
1 ; h 2k<br />
With 0
54 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
The pro<strong>of</strong> <strong>of</strong> this theorem is given in [141].<br />
Roughly speaking, this theorem states that a solution can be found in the NR-iteration<br />
(B) if the differential Df (x) does not vary too much (a)), is far enough from singularities<br />
(b)) and eventually does not jump too close to the boundary <strong>of</strong> the defined neighbourhood,<br />
at the first iteration (c)).<br />
It also states that the iteration will not go out <strong>of</strong> a specified neighbourhood (A) and<br />
converges at a certain speed (C). Conditions for a NR-iteration towards the Stewart platform<br />
coordinates will be derived.<br />
Chetelat [21] uses the fact that if one ensures that the function, f (x), is not <strong>of</strong> polynomial<br />
degree higher than two, then the jacobian is Lipschitz and in this case the Lipschitz constant<br />
can be calculated using the infinity norm.<br />
2.3.2 NR-convergence Stewart platform<br />
Having stated the kinematical structure <strong>of</strong> the Stewart platform and given Theorem 2.1 on<br />
convergence <strong>of</strong> a NR-iteration, it is now possible to investigate under what conditions the<br />
specific NR-iteration <strong>of</strong> (2.58) will converge to the physical platform pose.<br />
First an appropriate definition <strong>of</strong> the coordinates has to be given. The defined jacobian,<br />
Jl�x, is the description <strong>of</strong> platform translational and angular speed/variation to actuator<br />
length variations. To go from orientational parameter variations to angular speed depends<br />
on the parametrization used.<br />
In this case the last three euler parameters, are used to parametrize orientation, sx T =<br />
[cT s T ] T , where s is a scaling factor which can be used to get less conservative results<br />
in specifying a variation <strong>of</strong> x since c and = sin( 1<br />
2 )n have different dimension (m and<br />
rad). The scaling factor s =2kakmax will shown to be appropriate in the sequel.<br />
It will be assumed that after each iteration will be reset to zero. In that case the<br />
relation between the angular velocity and the euler parameters is very simple, ! j =0= 2I _.<br />
Off course the rotation matrix in the iteration is now the multiplication <strong>of</strong> all the rotation<br />
matrices calculated, Tk+1 = T ( k+1)Tk. The continuous function, f, in Theorem 2.1, from<br />
which the platform coordinates have to be found can now be given by k l i k;klik = fi(sx)<br />
where kl i k is the measured length <strong>of</strong> the i th actuator (fixed value for each iteration) and klik<br />
is the length <strong>of</strong> the i th actuator given sx (li given by (2.38)). Since the measured length is<br />
fixed, the derivative function with scaled euler parameters, is only slightly different from<br />
the jacobian Jl�x given earlier in (2.42).<br />
Df (sx) T = Jl�sx(sx) T =<br />
Ln<br />
2(TA m Ln)=s<br />
(2.62)<br />
Since it is a function <strong>of</strong> unit direction lengths, it is only defined for k lk 6=0.<br />
To derive the conditions (a,b,c)-st(ewart)pl(atform) for convergence <strong>of</strong> the NR-iteration<br />
defined in Theorem 2.1 for the Stewart platform, the constants ( � � ) can be specified<br />
using the kinematics.<br />
a-stpl) In this case the 2-norm <strong>of</strong> the matrix is taken which is equal to the largest singular<br />
value, and in the formulas d means an finitely small difference.
2.3 Analysis <strong>of</strong> the Stewart platform kinematics 55<br />
kJl�sx(sx1) ; Jl�sx(sx2)k < stplksx1 ; sx2k (2.63)<br />
kJl�sx(sx1) ; Jl�sx(sx2)k =<br />
dLn<br />
2d(TA m Ln)=s<br />
= (dJ) (2.64)<br />
The Frobenius (semi)-norm is an upper bound on this norm and is advantageous in<br />
case <strong>of</strong> the Stewart platform since specific bounds can be derived on the matrix elements<br />
as will be shown later on.<br />
kdJkF =<br />
vu<br />
u<br />
t 6X<br />
i=1<br />
(dJ) kdJkF (2.65)<br />
(kdln�ik 2 + k2d(T ai ln�i)=sk 2 (2.66)<br />
To derive a constant, stpl, the separate elements <strong>of</strong> the last equation will be described<br />
as a function <strong>of</strong> sx. This will be done in the Section 2.3.3.<br />
b-stpl) To state the second condition from which a constant number stpl has to be calculated<br />
also the 2-norm is used which can be upper bounded by one over the minimal<br />
singular value, min, <strong>of</strong> the jacobian at some pose, xo. By using the maximal<br />
variation <strong>of</strong> the jacobian which has been calculated for the previous condition, the<br />
constant, , becomes an upper bound for the maximum gain <strong>of</strong> the inverse jacobian<br />
over a volume <strong>of</strong> poses, sx.<br />
kJ ;1<br />
l�sx (sx)k =( min(Jl�sx(sx))) ;1<br />
(2.67)<br />
(max( min(Jl�sx(sxo)) ;kdJkF � 0)) ;1 = stpl (2.68)<br />
c-stpl) To calculate stpl, also use can be made <strong>of</strong> the singular values. Since any point<br />
in the work space is a possible initial start <strong>of</strong> the iteration, the maximum condition<br />
number <strong>of</strong> the scaled jacobian over the work space can be taken as the constant, stpl.<br />
kJ ;1<br />
l�sx f (sx)k = min (Jl�sx)kdsxk = stpl (2.69)<br />
In the next part it will be shown that indeed the jacobian <strong>of</strong> a Stewart platform is Lipschitz,<br />
as a-stpl) requires, as long as the actuators <strong>of</strong> the platform have minimal stroke strictly<br />
larger than 0.
56 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
2.3.3 Lipschitz condition on jacobian<br />
By constructively analyzing the kinematics <strong>of</strong> the Stewart platform the following theorem<br />
can be derived.<br />
Theorem 2.2 The Stewart platform jacobian, Jl�sx, is lipschitz i.e. a can be found such<br />
that<br />
kJl�sx(sx1) ; Jl�sx(sx2)k ksx1 ; sx2k 8 sx1� sx2 2 Do (2.70)<br />
if kli(sx)k > >0 8 sx 2 D� i 2f1�::: �6g: (2.71)<br />
Note that the requirement <strong>of</strong> an actuator length larger than zero is an implicit constraint<br />
on the set <strong>of</strong> platform poses.<br />
To prove Theorem 2.2, it is observed that jacobian, J l�sx, in (2.62) consists <strong>of</strong> two<br />
elements: the unit actuator direction, ln, and the vector product, (ln T a), for each actuator.<br />
By bounding the difference between the unit actuator directions and vector products <strong>of</strong> two<br />
jacobians with a function linear in kdsxk, Theorem 2.2 can be proven and a can actually<br />
be constructed.<br />
In two steps dln will be bounded first.<br />
1. Difference <strong>of</strong> an upper gimbal connection coordinate, dx a, will be bounded given the<br />
finite difference between two platform coordinates dsx.<br />
2. Then the difference between two actuator unit vectors, d ln, givendxa, will be considered.<br />
See Fig. 2.11 in which the <strong>motion</strong> <strong>of</strong> actuator as a function <strong>of</strong> the moving<br />
gimbal point xa is schematically depicted.<br />
With<br />
the following bound directly follows<br />
xa�i =c + T a m i<br />
(2.72)<br />
kdxa�ik kdck +2kakkd k p 2kdsxk (2.73)<br />
as the <strong>motion</strong> <strong>of</strong> a point due to rotation can be bounded easily with the reseted euler parameter<br />
description as<br />
kd(T a)k =2kakkd k (2.74)<br />
As from Fig. 2.3, it can be observed that the absolute change <strong>of</strong> a vector due to rotation has<br />
length 2 j sin(1=2 ) j=j 2 j.<br />
The second bound takes into account that as the upper gimbal moves within a ball<br />
(Fig. 2.11), the maximum change <strong>of</strong> the actuator direction is achieved if the new l, just
2.3 Analysis <strong>of</strong> the Stewart platform kinematics 57<br />
l n<br />
l<br />
Φ<br />
dl n<br />
dx a<br />
Fig. 2.11: Change <strong>of</strong> the unit vector in the direction <strong>of</strong> the actuator, ln, as a function <strong>of</strong><br />
change <strong>of</strong> the upper gimbal vector, dx a.<br />
touches the ball. In that case (ln + dln) ? dxa. With some geometry, sin( )=kdxak=klk<br />
and<br />
cos( )=<br />
s<br />
b<br />
1 ; kdxak<br />
klk<br />
the following monotonous upper bounding function can be derived for d ln.<br />
r<br />
1 ; cos( ) p kdxak<br />
kdlnkmax =2<br />
2<br />
2<br />
klk<br />
2<br />
a<br />
(2.75)<br />
(2.76)<br />
Taking into account (2.73) gives kdlnk 2kdsxk=klk. So, the actuator (minimum) length<br />
directly influences the bound, which can be put on kd lnk. If the length <strong>of</strong> the actuator is not<br />
strictly larger than zero, one can show that arbitrary small variations <strong>of</strong> the pose can result<br />
in nondecreasing variations <strong>of</strong> dln i.e. not satisfying the Lipschitz condition.<br />
Now, bounding the vector product is also possible. In general ka bk kakkbk and<br />
(a +c) (b + d) = (a b) +(a d) +(c b) +(c d). Further, rotation does not<br />
change the 2-norm, kT ak = kak. Change <strong>of</strong> the moving gimbal due to rotation is bounded<br />
by kd(T a)k =2kakkd k = kds k. Now,<br />
kd(ln T a)=kakmaxk<br />
kdln T ank + kln d(T an)k + kdln d(T an)k
58 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
p 2<br />
klk kdxak + kds k +<br />
2kds kkdsxk<br />
: (2.77)<br />
klk<br />
So in this way every element <strong>of</strong> the jacobian matrix is explicitly bounded by the variation<br />
<strong>of</strong> the platform pose and the (explicitly platform pose dependent) actuator length. If a<br />
minimal actuator length is given, a specific stpl can be calculated.<br />
Adding the bounds, for each <strong>of</strong> the actuators in the F-norm <strong>of</strong> the jacobian, results in an<br />
explicit number for stpl. Assuming small platform pose variations i.e. second order effects<br />
in the last equation are relatively small e.g. kdsxk
2.3 Analysis <strong>of</strong> the Stewart platform kinematics 59<br />
Algorithm<br />
- Choose an expected minimal singular value <strong>of</strong> the jacobians, over the grid points.<br />
- Take a grid such that the boxes with radius rs�min around the grid points fill the whole<br />
work space.<br />
- Calculate the minimal singular value <strong>of</strong> the jacobian at every grid point.<br />
- If the minimal singular value is larger than the expected value, there are no singular points<br />
in the work space, the algorithm finishes. If not, start another iteration choosing a<br />
smaller expected minimal singular value.<br />
Lemma 2.3 If the work space <strong>of</strong> a robotic manipulator having a bounded Lipschitz constant,<br />
does not have any singular point this non singularity will be detected by Algorithm 7.1<br />
in a finite number <strong>of</strong> iterations.<br />
Of course the boundary <strong>of</strong> the work space (in six dimensions!) should be known which is a<br />
stand alone problem (treated by Haug et al. [86]). To calculate an upper bound for the gain<br />
<strong>of</strong> the inverse jacobian ( stpl) afiner grid should be taken. (This will increase calculation<br />
time tremendously, e.g. n 6 grid points extra.)<br />
2.3.5 Sufficient update frequency<br />
The bounds derived are rather conservative in most cases. However, by calculating a convergence<br />
radius for the NR-iteration in the practical example <strong>of</strong> the new Simona research<br />
<strong>simulator</strong> shows that it is possible to guarantee that this iteration can be used if a reasonable<br />
frequency is used to update the platform pose. Further, the singular points can really be<br />
excluded from the work space in this case.<br />
To satisfy NR-iteration convergence the following constants were obtained using the<br />
kinematics <strong>of</strong> the Simona <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> given in Table 2.1.<br />
a-simona Using (2.78), assuming a minimal length <strong>of</strong> the actuator <strong>of</strong> l min =2:08m, results<br />
in simona =5:2.<br />
b-simona With gridding a minimal radius <strong>of</strong> rs�min = :09 is found. (About 200000 grid<br />
points had to be calculated to exclude singularity from the work space). The smallest<br />
singular value found is 0:75 and with finer gridding a simona =2can be guaranteed.<br />
c-simona The first value <strong>of</strong> f(sxo) in any point can not be larger than f(sxo) (J)kdsxk<br />
Together with the smallest gain this gives simona = kdsxk 3:5kdsxk<br />
Now with a bound kdsxk 0:02, hsimona =( )=2 =(3:5 0:02 2 5:2)=2 =0:23 and<br />
rsimona = =(1 ; h) =(3:5 0:02)=(1 ; 0:23) = 0:11. Within the every ball with radius<br />
rsimona, [a-simona] and [b-simona] should be guaranteed which is the case in the work<br />
space given the operational stroke (not including actuator cushioning part). To guarantee<br />
convergence in the whole work space also non-singularity, etc. should be guaranteed further
60 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
outside the work space which needs lots <strong>of</strong> calculation (with an extra stroke <strong>of</strong> 0.15m,<br />
singularities can be obtained so rs�min becomes very small).<br />
With a bound on the maximal speed <strong>of</strong> the actuator it is possible to calculate a minimal<br />
update frequency which guarantees kdsxk < 0:02. Givenk _ lk < 1m=s, now<br />
simona t _ lmax < kdsxk (2.80)<br />
and t 100 Hz convergence <strong>of</strong> the<br />
NR-iteration is attained.<br />
Lemma 2.4 The Stewart platform with the Simona <strong>motion</strong> <strong>system</strong> parameters has no singular<br />
points in the work space and the NR-iteration with this platform will converge to the<br />
right platform pose if the update frequency is larger than 100 Hz.<br />
Summarising, to satisfy a general convergence theorem on Newton-Raphson iteration,<br />
one <strong>of</strong> the requirements is the Lipschitz condition on the derivative function, which was<br />
shown to be satisfied for Stewart platforms. Variations <strong>of</strong> the jacobian can be bounded by<br />
the variations <strong>of</strong> the platform coordinates.<br />
Next to this requirement, the jacobian should not be singular in any point <strong>of</strong> the work<br />
space. With the Lipschitz condition on the jacobian and gridding, volumes can be excluded<br />
from singularities.<br />
Although only sufficient (and thereby conservative) conditions could be derived for convergence,<br />
it was shown to be possible to obtain an important convergence result in the practical<br />
example <strong>of</strong> Simona <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>. It will be one <strong>of</strong> the requirements<br />
for safe application <strong>of</strong> model <strong>based</strong> feedback.<br />
2.4 <strong>Model</strong>ling the mechanical part <strong>of</strong> the <strong>system</strong> dynamics<br />
After the thorough analysis <strong>of</strong> the kinematics <strong>of</strong> the parallel driven Stewart platforms in the<br />
previous section, this section will focus on the dynamics <strong>of</strong> such <strong>system</strong>s. The aim <strong>of</strong> this<br />
section is to show how to derive a limited number <strong>of</strong> differential, and possibly also some<br />
algebraic, equations, which describe the <strong>motion</strong> <strong>of</strong> mechanical <strong>system</strong>s such as the <strong>motion</strong><br />
<strong>system</strong> <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> as a function <strong>of</strong> the forces acting upon this <strong>system</strong>. It will<br />
be assumed that these <strong>system</strong>s consist <strong>of</strong> rigid bodies. In the extensions these bodies are<br />
possibly interconnected by springs and dampers.<br />
2.4.1 General theory in modelling the dynamics <strong>of</strong> mechanical <strong>system</strong>s<br />
Most theory described in this section can be found in many textbooks on mechanics and<br />
robotics, rigid multi body <strong>system</strong>s like [64, 81, 102, 106, 120, 129, 134]. Based on this<br />
general theory, a rigid body model <strong>of</strong> the Stewart platform, possibly with actuator joint<br />
inertia or linear ground flexibility, will be derived.<br />
All mechanics treated here are <strong>based</strong> on the assumption <strong>of</strong> a semi-equilibrium given by<br />
Newton’s second law f ; mp =0in an inertial frame for any mass particle. To describe the
2.4 <strong>Model</strong>ling the mechanical part <strong>of</strong> the <strong>system</strong> dynamics 61<br />
acceleration, p, <strong>of</strong> all parts in a <strong>system</strong> as a function <strong>of</strong> a limited number <strong>of</strong> variables some<br />
kinematics has been introduced.<br />
Having defined the kinematics <strong>of</strong> a mechanical <strong>system</strong>, its dynamics can be specified. In<br />
the equations <strong>of</strong> <strong>motion</strong> the semi-equilibria are described in a compact form as a function <strong>of</strong><br />
generalized velocities or coordinates. The integrals over the mass-particles <strong>of</strong> a body result<br />
in inertia matrices and the active forces are projected along the velocities by a virtual work<br />
argument.<br />
The equations <strong>of</strong> <strong>motion</strong> can be stated in several ways. First the Lagrange equations will<br />
be given. Then it will be shown that from these equations the more simple Newton-euler<br />
equations follow if one rigid body is considered. Finally the method <strong>based</strong> on Kane [64]<br />
will be introduced. This method in which all forces, active and inertial, are projected along<br />
a limited set <strong>of</strong> generalized velocities, will be central in the derivation <strong>of</strong> the <strong>motion</strong> <strong>system</strong><br />
mechanical model.<br />
Considering generalized velocities or coordinates ( _z� z), with the method <strong>of</strong> Lagrange<br />
the difference between the kinetic energy, K(z� _z), and potential energy, P(z), <strong>of</strong> a <strong>system</strong><br />
is called the Lagrangian, L(z� _z): L = K ; P. For a <strong>system</strong>, which can be described by a<br />
minimal number <strong>of</strong> generalized coordinates z, the Lagrange equations are given by:<br />
d<br />
dt<br />
@L<br />
@ _z<br />
; @L<br />
@z<br />
= (2.81)<br />
The driving moments/forces, , are all the working forces (non inertial or conservative) projected<br />
along the variations <strong>of</strong> the generalized coordinates. These are called the generalized<br />
forces.<br />
The kinetic energy can in general be described by<br />
K = 1<br />
2 _z T M (z) _z� (2.82)<br />
where the mass matrix, M (z), is a symmetric positive semi definite matrix.<br />
It can be appropriate to apply a coordinate transformation by introducing the generalized<br />
momenta (assuming a positive definite and thus invertible mass matrix)<br />
m = @L<br />
@ _z = M (z) _z: (2.83)<br />
The implicit second order differential Lagrange equations (2.81) can now be transformed<br />
to a (nonlinear) state space description, i.e. the Hamiltonian equations <strong>of</strong> <strong>motion</strong> following<br />
Van der Schaft [147]. A key role is played by the hamiltonian or total energy function, H,<br />
consisting <strong>of</strong> the addition <strong>of</strong> kinetic and potential energy,<br />
H(m�z) = 1<br />
2 mT M ;1 (z)m + P(z) =m T _z ; L(z� _z): (2.84)<br />
Taking the partial derivative <strong>of</strong> (2.84) to m for the first set <strong>of</strong> state equations and filling in<br />
(2.84) into (2.81) eliminating L for the second, results in<br />
_z = @H<br />
(m� z)<br />
@m<br />
(2.85)<br />
_m = ; @H<br />
(m� z) +<br />
@z<br />
: (2.86)
62 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
In this non energy dissipating <strong>system</strong> the increase in energy can be shown to be equal to the<br />
amount <strong>of</strong> work delivered,<br />
d<br />
dt H = _z T<br />
(2.87)<br />
By observing that the work W, done by forces , does not change if a change <strong>of</strong> coordinates<br />
z, is applied, it is easy to show that projecting the forces along other coordinates is equal to<br />
multiplying by the transpose jacobian.<br />
W = T z1 = T Jz1�z2 z2 =(J T z1�z2 )T z2 (2.88)<br />
Considering such collocated pairs <strong>of</strong> forces and velocities as inputs ( = J T (z)u) and outputs<br />
y = J(z) _z, a hamiltonian <strong>system</strong> results. The Stewart platform rigid body mechanics<br />
will have this structure. The hamiltonian <strong>system</strong> structure does not require that the number<br />
<strong>of</strong> input/output pairs and the number <strong>of</strong> generalized momenta/velocity pairs is equal. This<br />
allows to take into account parasitic resonant effects without losing the structural properties<br />
<strong>of</strong> a hamiltonian <strong>system</strong>.<br />
Further, manipulation <strong>of</strong> the second order differential equation (2.81) can lead to<br />
M (z)z +<br />
Compactly written as<br />
d<br />
dt (M (z) _z ; _z T<br />
@<br />
@zi<br />
M (z) _z + @<br />
P(z) = : (2.89)<br />
@z<br />
M (z)z + C(z� _z) _z + G(z) = : (2.90)<br />
with a mass matrix, M, a non linear coriolis/centripetal matrix, C, and a gravity vector, G.<br />
If a rigid body (with mass, m, and inertia, Ic) is considered at its centre <strong>of</strong> gravity c, the<br />
Newton-euler equations result. The mass matrix is block diagonal in this case.<br />
Kbody = 1<br />
2<br />
_c !<br />
mI<br />
0<br />
0<br />
Ic<br />
_c<br />
!<br />
(2.91)<br />
From the two blocks, two independent equations follow from (2.81). The impulse law:<br />
f = d d<br />
(m (c)) = mIc (2.92)<br />
dt dt<br />
And the impulse moment law with the generalized forces f g c (moments in this case) is<br />
derived similarly from (2.81) taking the time derivative <strong>of</strong> the Lagrangians partial derivative<br />
to the second set <strong>of</strong> (angular) velocities. The Lagrangian, in this case, only consists <strong>of</strong> the<br />
kinetic energy, which does not depend on the position.<br />
f g c<br />
= d<br />
dt (g R m I m c ! m )= g R m I m c _! m + g R m ~ m I m c ! m<br />
= I g c _! g + ~ g I g c ! g = I g c _! g +! g<br />
I g c ! g<br />
(2.93)<br />
In this way, the Newton-euler equations are easily stated for each rigid body in a <strong>system</strong>.<br />
However, extra equations with unknown internal forces result, in using this method to state
2.4 <strong>Model</strong>ling the mechanical part <strong>of</strong> the <strong>system</strong> dynamics 63<br />
the equations <strong>of</strong> <strong>motion</strong> for a multi body <strong>system</strong>. Applying Lagrange, results in taking<br />
partial derivatives <strong>of</strong> complex energy functions if the whole <strong>system</strong> is considered. These are<br />
disadvantages which can be circumvented by using Kane’s method.<br />
With Kane also generalized variations or velocities have to be specified. The general<br />
formula<br />
f + f = 0 (2.94)<br />
states the (semi-)equilibrium <strong>of</strong> the active forces, f, and inertial forces, f , projected along<br />
the directions <strong>of</strong> the generalized velocities. Generalized velocities do not have to be derivatives<br />
<strong>of</strong> some positional coordinate e.g. in principle the angular velocity can be used as<br />
such.<br />
To calculate the over-all inertial forces, as in the Newton-euler approach, the specific<br />
inertial forces generated in the frame <strong>of</strong> each body can be stated. As in the Lagrangian<br />
approach, a minimal number <strong>of</strong> equations results by writing the <strong>motion</strong> <strong>of</strong> the bodies as a<br />
function <strong>of</strong> the generalized velocities and projecting each specific force from its local coordinates<br />
to the generalized ones. Also the active forces can first be stated in an appropriate<br />
frame after which projection follows. The projection in general consists <strong>of</strong> a change <strong>of</strong> coordinates<br />
i.e. a multiplication with a jacobian. This procedure can also be automated [65].<br />
Especially in fast modelling with the aim <strong>of</strong> simulation, this can be useful.<br />
With this method it is possible to start with a strongly simplified <strong>system</strong> by calculating<br />
part <strong>of</strong> the (inertial) forces and separately adding other forces if a more accurate model<br />
has to be taken into account. Many <strong>of</strong> the specific merits <strong>of</strong> the method will become clear<br />
considering the modelling <strong>of</strong> the mechanical part <strong>of</strong> the Stewart platform including actuator<br />
inertia.<br />
The amount <strong>of</strong> generalized coordinates can exceed the number <strong>of</strong> d.o.f. <strong>of</strong> a <strong>system</strong>. In<br />
that case, next to the differential equations given by the semi-equilibria <strong>of</strong> the forces (2.94)<br />
constraint equations <strong>of</strong> velocities and position, generally stated as<br />
A(x� t) _x + b(x� t) =0� (2.95)<br />
should be added. With mechanical <strong>system</strong>s having so called kinematic chains i.e. parallel<br />
manipulators, such additional equations easily appear if the <strong>motion</strong> <strong>of</strong> some parts <strong>of</strong> the<br />
<strong>system</strong> can not explicitly be written as functions <strong>of</strong> the minimal number <strong>of</strong> coordinates<br />
(amounting to the number <strong>of</strong> d.o.f.). This will not be the case with the Stewart platform apart<br />
from the additional coordinate necessary to attain a global description <strong>of</strong> the orientation.<br />
Systems where the constraint equations cannot be integrated to constraint position equations<br />
are called non-holonomic. A parallel manipulator, like the Stewart platform, will occur<br />
to be a holonomic <strong>system</strong>. Since there are kinematic chains, it is <strong>of</strong>ten easier to state the<br />
equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> a parallel manipulator by using constraint equations. In that case the<br />
manipulator is described as a serial <strong>system</strong> (with some <strong>of</strong> the joints disconnected). The parallel<br />
connections are incorporated by adding constraints. A combined differential/algebraic<br />
description results. This kind <strong>of</strong> description causes difficulties (index problems etc., [19])<br />
in simulation and model <strong>based</strong> <strong>control</strong>. With these goals in mind during modelling, it is<br />
more convenient to state the model in explicit differential equations if possible. Starting<br />
with Section 2.4.3 in the second part <strong>of</strong> this section it is shown that this can be done with
64 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
φ<br />
r<br />
x<br />
y<br />
m<br />
Fig. 2.12: Example <strong>system</strong> consisting <strong>of</strong> one mass rotating in the plane on the left. On the<br />
right also a variable length, r, exists to the centre <strong>of</strong> rotation.<br />
the Stewart platform. But first a simplified example will be given in Section 2.4.2 to clarify<br />
the modelling method used.<br />
2.4.2 Example in modelling using Kane’s method<br />
To clarify the method <strong>of</strong> Kane, the dynamics <strong>of</strong> a simple example <strong>system</strong> will be modelled.<br />
The <strong>system</strong> is given in Fig. 2.12. A simple mass, m, can be rotated in the plane around<br />
some point, which is taken as the origin in the inertial frame. It is assumed we do not know<br />
how to define the mass properties in rotation but only know the translational inertial force<br />
equations:<br />
f c =<br />
f x<br />
f y<br />
= ;<br />
m 0<br />
0 m<br />
φ<br />
x<br />
y<br />
r<br />
x<br />
y<br />
m<br />
= ;Mc (2.96)<br />
We, however, want to choose the angular velocity, _ ,defined in Fig. 2.12 as the generalised<br />
velocity for this 1-d.o.f.-<strong>system</strong>. The coordinates, x and y, and their derivatives can be<br />
formulated as a function <strong>of</strong> and _ with<br />
and by differentiation<br />
_c =<br />
c =<br />
_x<br />
_y<br />
x<br />
y<br />
= r cos( )<br />
r sin( )<br />
= ;r sin( )<br />
r cos( )<br />
_ = Jc�<br />
(2.97)<br />
_ : (2.98)<br />
The local inertial forces, f c can be projected along the generalised velocity by<br />
f = J T c� f c (2.99)<br />
Further differentiation <strong>of</strong> (2.98), provides us with the acceleration<br />
c = Jc� + _<br />
Jc�<br />
_ (2.100)
2.4 <strong>Model</strong>ling the mechanical part <strong>of</strong> the <strong>system</strong> dynamics 65<br />
with<br />
_<br />
Jc� =<br />
;r cos( )<br />
;r sin( )<br />
_ : (2.101)<br />
This equation can be filled into (2.96), which can be filled into (2.99) to arrive at the simple<br />
f = ;mr 2 � (2.102)<br />
since J T c� Jc�<br />
_ =0and J T c� Jc� = r2 in this example. The equilibrium equation <strong>of</strong> <strong>motion</strong>,<br />
considering an active force already projected along _ , i.e. a moment fa, will be<br />
f + f = fa ; mr 2 =0 (2.103)<br />
In principle, also _c could have been defined as the generalised velocity. The only constraint<br />
in this choice is the ability to describe the <strong>motion</strong> <strong>of</strong> any body with this set <strong>of</strong> velocities.<br />
In that case (2.96) would give the inertial forces and an additional constraint equation<br />
_c T c =0 (2.104)<br />
would have to be satisfied. I.e. a combined algebraic and differential set <strong>of</strong> equations would<br />
result. Simulating such a <strong>system</strong> is not trivial. E.g. in the example any finitely small step<br />
satisfying (2.104), a velocity orthogonal to c, would enlarge the radius r.<br />
In the case sketched, it is relatively easy to choose a minimal number <strong>of</strong> coordinates<br />
and end up with differential equations only. But already with planar <strong>system</strong>s such as the<br />
manipulator given in Fig. 2.6, it is not clear how to choose three coordinates (minimal local<br />
degrees <strong>of</strong> freedom) which globally describe this <strong>system</strong>.<br />
The method presented seems somewhat overdone if applied on such an example but it<br />
can easily be extended to the more complex cases. Consider for example the <strong>system</strong> <strong>of</strong><br />
Fig. 2.12 with an variable radius (e.g. due to a non rigid spring/damper) connection.<br />
If the force <strong>of</strong> the spring always acts along the radial direction it could be appropriate to<br />
choose the polar generalised velocities, _p, consisting <strong>of</strong> _r and _ .<br />
Now<br />
_c = Jxp _p =<br />
and the inertial forces follow from<br />
and using<br />
cos( ) ;r sin( )<br />
sin( ) r cos( )<br />
_r<br />
_<br />
(2.105)<br />
f p = ;J T xpm(Jxpp + _<br />
Jxp _p) (2.106)<br />
Jxp<br />
_ = ; sin( ) _ ;r cos( ) _ ; _r sin( )<br />
cos( ) _ ;r sin( ) _ +_r cos( )<br />
gives the inertial forces more explicitly as<br />
f xp = ;<br />
m 0<br />
0 mr 2<br />
r ; ;mr _2<br />
2mr _ _r<br />
(2.107)<br />
: (2.108)
66 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
In this case nonlinear velocity forces like the quadratic centripetal force, mr _2 , and the<br />
velocity product coriolis force, 2mr _ _r, naturally show up.<br />
Including an additional body is relatively easy. E.g. the inertia, i tube <strong>of</strong> the tube drawn<br />
in Fig. 2.12, can be taken into account by adding the inertial force, f tube = itube .<br />
In the calculations, the projections using the jacobians <strong>of</strong>ten enable geometric simplifications.<br />
This will also be the case in using this method in modelling the Stewart platform<br />
including the actuator inertial properties.<br />
With this example, the method <strong>of</strong> Kane has been made explicit and will be used in the<br />
next sections to derive a model <strong>of</strong> the mechanical part <strong>of</strong> the Stewart platform <strong>motion</strong> <strong>system</strong><br />
dynamics.<br />
2.4.3 Stewart platform dynamics<br />
The general procedure <strong>of</strong> stating the equations <strong>of</strong> <strong>motion</strong> has now been given. In this part,<br />
this will be applied in modelling the Stewart platform. Kane’s method <strong>of</strong> projecting local<br />
semi-equilibria (equations <strong>of</strong> <strong>motion</strong>) will be used to arrive at a compact description. To<br />
state the local equations, both Lagrange and Newton-Euler are applied wherever either one<br />
is most appropriate.<br />
With the choice <strong>of</strong> platform position/orientation as the generalized coordinates, all equilibria<br />
can be written as explicit functions <strong>of</strong> these coordinates (and its derivatives). The<br />
modelling approach, however, still enables a clear assignment <strong>of</strong> the contribution each part<br />
<strong>of</strong> the <strong>system</strong> has on the over all dynamics.<br />
First a simplified model, with the platform as the only (rigid) body, will be derived.<br />
Then, the influence <strong>of</strong> the actuator inertial forces is quantified.<br />
2.4.4 Basic Stewart platform mechanical model dynamics<br />
The basic structure <strong>of</strong> the Stewart platform model results if one considers the platform<br />
alone, not taking into account the inertial forces <strong>of</strong> the actuators. Since the <strong>system</strong> in this<br />
case consists <strong>of</strong> only one body, the equations <strong>of</strong> <strong>motion</strong> are easily derived with Newton-<br />
Euler taking the velocity <strong>of</strong> the platform coordinates as the generalized speed. With the<br />
impulse law, (2.92), the translational <strong>motion</strong> is described<br />
Lnfa + mcg = mcc� (2.109)<br />
where fa are the active forces generated by the actuators and g is the gravity vector.<br />
And with the impulse moment law, (2.93), the rotational <strong>motion</strong> is described by<br />
TA m Ln fa = Ic _! + ~ Ic!� (2.110)<br />
with Ic = TI m c T T . Combining these two results in the simplified model <strong>of</strong> the Stewart<br />
platform.<br />
Ln<br />
TA m Ln<br />
fa = mcI 0<br />
0 Ic<br />
c<br />
_!<br />
+ 0 0<br />
0 ~ Ic<br />
_c<br />
! ; mcg<br />
0<br />
(2.111)
2.4 <strong>Model</strong>ling the mechanical part <strong>of</strong> the <strong>system</strong> dynamics 67<br />
In short<br />
J T l�x (sx)fa = Mc(sx)x + Cc( _x� sx) _x + Gc� (2.112)<br />
from which the mass matrix, Mc, the coriolis/centripetal effects, Cc, and gravity, G, are<br />
clearly separated and the structure given in (2.90) is visible. Further, the jacobian, J l�x,<br />
already playing an important part in the sections on kinematics, also occurs in the equations<br />
<strong>of</strong> <strong>motion</strong> directing the actuator driving forces.<br />
In many cases, especially with large and heavy <strong>motion</strong> <strong>system</strong>s, these equations will sufficiently<br />
describe the rigid body dynamics <strong>of</strong> the mechanical part <strong>of</strong> the <strong>system</strong>. In practice,<br />
these dynamics will interact foremost with the hydraulic actuators, which will be modelled<br />
in the next chapter. This chapter concludes with some sections on extending the mechanical<br />
model.<br />
2.4.5 Influence <strong>of</strong> the actuator inertial forces<br />
In some cases, the inertial forces (i.e. mass properties) <strong>of</strong> the actuators can not be neglected.<br />
With many heavy <strong>flight</strong> simulation <strong>system</strong>s they can, but with the light weight <strong>motion</strong> <strong>system</strong>s<br />
requiring high performance <strong>motion</strong> this has to be reevaluated. Explicitly taking into<br />
account the actuator inertial forces will complicate the model. However, this model can be<br />
used to attain an appropriate approximate version as will be shown in the next chapter.<br />
Most <strong>of</strong> the assumptions considering the actuators have already been given considering<br />
their kinematical properties in Section 2.2.11. See e.g. Fig. 2.8. The gimbals are assumed to<br />
rotate frictionless. The inertia <strong>of</strong> both actuator parts rotating around the actuator direction<br />
is neglected and further assumed symmetric w.r.t. this axis. The mass properties <strong>of</strong> each<br />
actuator part can therefore be described by two parameters, the mass and the inertia around<br />
an axis orthogonal to the actuator direction. Mass m a and inertia ia are taken for the upper<br />
moving part <strong>of</strong> the actuator (mainly the piston), and (m b, ib) for the lower rotating part, the<br />
cylinder.<br />
The inertial forces <strong>of</strong> the actuators can be split up in three parts: the gravitational forces,<br />
the inertial mass forces and the influence <strong>of</strong> its inertia. These forces will first be projected<br />
on the upper gimbal points. Any force generated at this point is easily projected at the<br />
generalized platform velocities with (2.47) by<br />
f x a = J T a�xf a a : (2.113)<br />
The gravitational forces are easily projected along the platform velocities. The lower part<br />
<strong>of</strong> the actuator using (2.50),<br />
f a bg = J T bc�ambg = mbrb<br />
j l j Plng = Gmb : (2.114)<br />
Clearly, in a position where the gravity vector is directed along the actuator ( ln) this force<br />
will not contribute. The contribution <strong>of</strong> the moving part is in that case maximal at a as is<br />
shown by equivalently using (2.49),<br />
f a ag = J T ac�amag = ma(I ; ra<br />
Pln )g = Gma<br />
(2.115)<br />
j l j
68 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
The inertial force generated by the mass at the c.o.g. <strong>of</strong> the moving part <strong>of</strong> the actuator is<br />
easily described w.r.t. a frame at its c.o.g.<br />
fac = ma _vac<br />
(2.116)<br />
Projection <strong>of</strong> this force at the upper gimbal point using (2.49) and writing _vac as a<br />
function <strong>of</strong> _va using (2.55) and (2.57), results in<br />
f a ma = ma(I ; Pln<br />
+ ma<br />
ra<br />
2<br />
(j l j;ra)<br />
+<br />
j l j2 Pln ) _va ::: (2.117)<br />
j l j 2 j Plnva j 2 ln +2ma<br />
= Ma _va + Cmava<br />
(j l j;ra)ra<br />
j l j3 (l T n va)Plnva<br />
The first term consists <strong>of</strong> a part in the direction <strong>of</strong> the actuator (I ; P ln) where the mass<br />
directly acts on the gimbal point. Perpendicular to this direction the influence gets smaller<br />
with the squared ratio <strong>of</strong> distances to the lower gimbal point. The second and third term are<br />
the quadratic velocity dependent coriolis and centripetal force.<br />
The inertial forces generated by the inertias <strong>of</strong> the lower and upper part <strong>of</strong> the actuator<br />
can be taken together considering Lagrange and observing that their contribution to the<br />
kinetic energy is equivalent. With (2.48)<br />
Kia�ib<br />
= 1<br />
2 !T a !a(ia + ib) (2.118)<br />
= 1<br />
2 vT a<br />
(ia + ib)<br />
j l j2 Plnva = 1<br />
2 vT a M a ia�ibva The derivation <strong>of</strong> the following two equations can be found in Appendix B together with<br />
a proper choice for factoring the quadratic velocity terms. p a is the position <strong>of</strong> the upper<br />
gimbal point.<br />
d<br />
dt (Mia�ib )=; (ia + ib)<br />
j l j3 Kia�ib<br />
pa<br />
(2v T a lnPln + Plnval T n + lnv T a Pln )� (2.119)<br />
= ; (ia + ib)<br />
j l j3 ((l T n va)Plnva + lnv T a Plnva)� (2.120)<br />
With Lagrange (d=dt(M ) _v ; @K=@p g a ), (2.119) and (2.120) the inertial forces at p a result<br />
f a ia�ib = (ia + ib)<br />
j l j 2<br />
Pln _va ; 2(ia + ib)<br />
j l j3 (l T n va)Plnva = Mia�ib _va + Cia�ibva: (2.121)<br />
The contribution to the mass matrix only exists at <strong>motion</strong> perpendicular to the direction <strong>of</strong><br />
the actuator. Next to this, only coriolis and no centripetal terms appear. The coriolis force<br />
is generated as a result <strong>of</strong> the inertia points in the opposite direction as the one generated by
2.4 <strong>Model</strong>ling the mechanical part <strong>of</strong> the <strong>system</strong> dynamics 69<br />
the mass. This is due to the fact that the influence <strong>of</strong> the inertia decreases while that <strong>of</strong> the<br />
mass increases as the actuator gets longer.<br />
All the separate inertial force contributions <strong>of</strong> each actuator represented by f a bg , f a ag ,<br />
f a ma and f a ia�ib together with the active actuator force generated by the hydraulics, result in<br />
total forces at the upper gimbals. Projection to the platform coordinates via the jacobian,<br />
Ja�x, puts these forces into place for the equations <strong>of</strong> <strong>motion</strong>.<br />
2.4.6 The Stewart platform model including actuator inertia<br />
The equation <strong>of</strong> <strong>motion</strong> <strong>of</strong> the Stewart platform including the inertia <strong>of</strong> the actuators can<br />
still be put in form <strong>of</strong><br />
Where Mt, Ct and Gt are given by<br />
Ct _x = Cc _x +<br />
6X<br />
i=1<br />
J T l�x (sx)fa = Mt(sx)x + Ct( _x� sx) _x + Gt(sx) (2.122)<br />
Mt = Mc +<br />
6X<br />
i=1<br />
J T ai�x (Mma�i + Mia�i�ib�i )Jai�x<br />
(2.123)<br />
(J T ai�x (Cma�i + Cia�i�ib�i )Jai�x _x; j! j 2 (Mma�i<br />
Gt = Gc +<br />
6X<br />
i=1<br />
+ Mia�i�ib�i )P!ai)<br />
(2.124)<br />
J T ai�x (Gma�i + Gmb�i ) (2.125)<br />
In this model, the platform coordinates, velocities and acceleration are the only variables<br />
and the effect <strong>of</strong> each term (mass, coriolis, centripetal, gravity, driving forces) and parameter<br />
(mass, inertia, centres <strong>of</strong> gravity, gimbal point) can clearly be distinguished. The model is<br />
well defined in each state except for those in which any <strong>of</strong> the actuator lengths becomes<br />
zero. Note that this puts an implicit limitation to the validity <strong>of</strong> the model.<br />
Adding the inertial influence <strong>of</strong> the actuators to the simplified model did not change the<br />
compact form <strong>of</strong> six coupled second order differential equations. The equations, however,<br />
became much more complex which is not favourable in model <strong>based</strong> <strong>control</strong> in which the<br />
model has to be calculated at high speed.<br />
Ji claims in [61] that the actuator inertial effects can be seen as a change <strong>of</strong> the platform<br />
mass, inertia and c.o.g. This claim should be carefully interpreted as this change is not only<br />
dependent on the position, but also on the direction <strong>of</strong> the <strong>motion</strong> i.e. not even valid at one<br />
operating point. E.g. in case <strong>of</strong> the Simona <strong>flight</strong> <strong>simulator</strong>, the mass <strong>of</strong> the actuators add<br />
more to the mass matrix <strong>of</strong> <strong>simulator</strong> in heave than in the lateral directions <strong>of</strong> surge and<br />
sway.<br />
With the equations given, it is possible to give bounds on the forces not taken into account<br />
if the inertial forces <strong>of</strong> the actuators would be neglected. Although with conventional
70 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
<strong>motion</strong> <strong>system</strong>s this is <strong>of</strong>ten justified, the tendency towards light weight platforms makes<br />
the actuator inertial forces more evidently come into play. Total neglection would result<br />
in a too rough approximation in that case. Approximation by a constant additive term will<br />
shown to be more convenient.<br />
2.4.7 Parasitic mechanical aspects: modelled as linear flexibility<br />
The models presented only take into account the mechanical <strong>system</strong> as the rigid bodies.<br />
Together with the hydraulic actuators, this forms the most relevant part <strong>of</strong> the <strong>system</strong> dynamics.<br />
However, it appeared that two kinds <strong>of</strong> parasitic mechanical effects due to flexibility<br />
could not be neglected.<br />
First, the foundation on which the Simona <strong>motion</strong> <strong>system</strong> was tested in the Central<br />
Workshop was not entirely rigidly attached to the inertial world. Though relatively heavy<br />
(45 tons), the foundation formed by a block <strong>of</strong> concrete appeared to be in <strong>motion</strong> in the<br />
plane orthogonal to z if the <strong>simulator</strong> <strong>motion</strong> contained too much energy in the frequency<br />
area above > 5 Hz [41]. The foundation was not ideal for these tests, but in countries<br />
like the Netherlands, where buildings are <strong>of</strong>ten build on large piles in weak ground, similar<br />
effects can be expected if no special precautions are being taken.<br />
Flexible effects like this, can be modelled by considering additional mass/ spring/ damper<br />
<strong>system</strong>s in connection with the rigid body model. In Fig. 2.13, a representation <strong>of</strong> a foundation<br />
is given, which has two translational and one torsional spring/damper pot connecting<br />
the mass/inertia <strong>of</strong> the foundation to the inertial world. Further, the <strong>motion</strong> <strong>system</strong> introduces<br />
forces through the lower gimbal points.<br />
If the foundation is given, the three degrees <strong>of</strong> freedom: x f along the surge (x-)direction,<br />
yf along the sway (y-)direction and the angle f around the z-axis (yaw-<strong>motion</strong>), three<br />
additional equations <strong>of</strong> <strong>motion</strong> will result. At first instance, the interaction <strong>of</strong> the foundation<br />
with the inertial world will be modelled by decoupled spring/damper connections with<br />
spring stiffnesses cxf, cyf, c f, and damping coefficients bxf , byf, b f, which are used for<br />
the respective directions. Now, the equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> the foundation with mass m f and<br />
inertia If can be modelled by using the additional matrix equation:<br />
2<br />
4 mf 0 0<br />
0 mf 0<br />
0 0 If<br />
32<br />
5<br />
4 xf<br />
yf<br />
f<br />
3<br />
5 +<br />
2<br />
4 bxf 0 0<br />
0 byf 0<br />
0 0 b f<br />
2<br />
4 cxf 0 0<br />
0 cyf 0<br />
0 0 c f<br />
32<br />
5<br />
32<br />
5<br />
4 _xf<br />
_yf<br />
_ f<br />
4 xf<br />
yf<br />
f<br />
3<br />
5 + (2.126)<br />
3<br />
5 = J T l�xf fa<br />
The jacobian <strong>of</strong> the general (six) foundation body parameters, x fg to actuator lengths<br />
has a similar structure as Jl�x.<br />
Jl�xfg = ;LT n ;(B Ln) T (2.127)<br />
In case only the aforementioned three degrees <strong>of</strong> freedom in the plane are possible a second<br />
projection is required through the selection matrix P f with unit vectors selecting x, y and
2.4 <strong>Model</strong>ling the mechanical part <strong>of</strong> the <strong>system</strong> dynamics 71<br />
Fig. 2.13: Physical representation <strong>of</strong> the model used for the flexible attached foundation<br />
.<br />
P T f =<br />
2<br />
4<br />
1 0 0 0 0 0<br />
0 1 0 0 0 0<br />
0 0 0 0 0 1<br />
Jl�xf = Jl�xfgPf (2.129)<br />
3<br />
5 (2.128)<br />
Finally, the equation for the platform velocity has to be adapted since b will start to vary.<br />
_<br />
l = L T n (Va ; Vb) (2.130)<br />
With the lower gimbal point velocities, _vb stacked in Vb. If the model with the platform<br />
alone (or a constant mass matrix with respect to the moving <strong>simulator</strong>) is taken, the model<br />
equation (2.111) does not have to be adapted. All together these equations form a model <strong>of</strong><br />
the <strong>system</strong> with a platform attached to a non-inertial ground, which has the same validity as<br />
the simple model (for all <strong>simulator</strong> poses except those for which (klk i =0). This problem<br />
has not seen to be considered in literature apart from cases in space technology [93, 108] in<br />
which the moving platform has very limited manoeuvrability and the actuators are used to<br />
supress vibrations.<br />
At first sight, the foundation, though also moving if the actuators force the <strong>simulator</strong><br />
to move, does not have too much direct interaction with <strong>simulator</strong> <strong>motion</strong>, since the basic<br />
mechanical model equation (2.112) looks the same. Indirectly, the following dependencies<br />
can be observed.
72 2 Mechanics <strong>of</strong> parallel driven <strong>motion</strong> <strong>system</strong>s<br />
- The <strong>motion</strong> <strong>of</strong> the foundation has influence on the actuator velocities. The hydraulic<br />
actuators form a port connection with the mechanical <strong>system</strong> through the active forces<br />
and velocity. Generally, the active force will be influenced by the velocity due to a<br />
coupling in the dynamics <strong>of</strong> the (<strong>control</strong>led) hydraulic <strong>system</strong>. This influence will<br />
therefore affect the <strong>simulator</strong> <strong>motion</strong>. Note that this influence will be much larger if<br />
the actuators are position/velocity <strong>control</strong>led than with force <strong>control</strong> in the relevant<br />
frequency area.<br />
- The platform pose can not be calculated from the actuator lengths anymore without correcting<br />
for the moving foundation.<br />
- The jacobian, Jl�x, depends on the <strong>motion</strong> <strong>of</strong> the foundation since it depends on the unit<br />
direction vectors <strong>of</strong> the actuators. These directions will change as the foundation<br />
moves resulting in change <strong>of</strong> direction <strong>of</strong> the active forces acting on the <strong>simulator</strong>.<br />
This is a second order effect and is mostly compensated for by reconstructing the<br />
Jacobian from the actuator length measurements. The <strong>motion</strong> in the plane <strong>of</strong> the<br />
foundation will not influence the forces necessary to compensate for gravitational<br />
effects <strong>of</strong> the gross moving platform.<br />
- The <strong>motion</strong> <strong>of</strong> the actuator (inertias) is directly influenced by the foundation. The equations<br />
<strong>of</strong> <strong>motion</strong> for these parts have to be adapted and become somewhat more complex.<br />
If the actuator inertia can be neglected or taken as part <strong>of</strong> the constant mass<br />
matrix w.r.t. moving platform, this effect can also be neglected.<br />
Given influences, both in the choice <strong>of</strong> the <strong>control</strong> strategy as in identifying the dynamics<br />
<strong>of</strong> the <strong>system</strong>, the parasitic behaviour <strong>of</strong> the foundation should be taken into account.<br />
The second important parasitic mechanical effect to be analyzed are the structural modes<br />
<strong>of</strong> the <strong>simulator</strong> itself. Especially in optimising the ability to realistically simulate the high<br />
frequency vibrations in a vehicle, the unwanted resonances <strong>of</strong> the <strong>simulator</strong> should not be<br />
hit up to noticeable level. In the design <strong>of</strong> the shuttle <strong>of</strong> the Simona <strong>simulator</strong>, the ratio<br />
<strong>of</strong> stiffness and mass has been optimized [8]. However, finite element models pointed out<br />
that with this <strong>system</strong> one will not be able to extend the lowest structural vibrations above<br />
15-20 Hz<br />
As a rule <strong>of</strong> thumb the flexible modes should only appear in a frequency area well<br />
distinct from the area where the <strong>motion</strong> to be simulated has relevant energy and thus usually<br />
form an upper bound on the achievable bandwidths. Since there are only a limited number<br />
<strong>of</strong> actuators, the moving platform loses functional <strong>control</strong>lability if structural deformation<br />
takes place. In theory, one could well be able to apply a desired acceleration pr<strong>of</strong>ile to the<br />
pilots head but one would in these cases structurally have to settle for vibration in other parts<br />
<strong>of</strong> the platform e.g. the large visual projection screen. Additional active damping devices<br />
can probably be <strong>of</strong> help in making this restriction less tight.<br />
The unneglectable deformations in the moving platform drastically change the structure<br />
<strong>of</strong> the <strong>system</strong>. <strong>Model</strong>ling using the projection methods like Kane’s is still possible if one can<br />
approximate the modes by splitting up the previously rigid bodies in a limited number mass/<br />
spring/ damper <strong>system</strong>s i.e. smaller rigid parts connected by springs and dampers ([146]).<br />
Deriving these equations by hand, however, becomes very tedious and does in general not
2.5 Chapter Resume 73<br />
result in a model from which analysis can take place more easily than from automatically<br />
derived models. In a master’s project as part <strong>of</strong> this research Rijnten [119] modelled the<br />
<strong>system</strong>, including a structural vibration, using the symbolic equation modelling s<strong>of</strong>tware,<br />
Autolev [65]. Though simulation with such models can clarify <strong>system</strong>s behaviour, not much<br />
structural understanding <strong>of</strong> <strong>system</strong>s like a deformed Stewart platform can be gained since<br />
the number <strong>of</strong> variables in the model well goes above a thousand.<br />
Other, more advanced, methods in modelling flexible structures [62, 63, 66, 129] <strong>of</strong>ten<br />
used in space applications with large flexible deformations, usually result in models, which,<br />
due to their complexity, can not directly be used in model <strong>based</strong> <strong>control</strong>. It was decided to<br />
do the actual analysis <strong>of</strong> the structural deformations in the Simona <strong>motion</strong> <strong>system</strong> using the<br />
actual data measured at the <strong>system</strong>. This will be discussed in Chapter 4. Further analysis<br />
and also simulation <strong>of</strong> the models derived will take place after introducing and integrating<br />
the models <strong>of</strong> the hydraulic <strong>system</strong> to be discussed in the next chapter. This will be more<br />
realistic since the integration <strong>of</strong> the hydraulics and the mechanics forms the most relevant<br />
part <strong>of</strong> the dynamics considering the whole <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>.<br />
2.5 Chapter Resume<br />
Since the kinematic model <strong>of</strong> the Stewart platform construction is fully parallel, the platform<br />
(end effector) coordinates and velocities can globally be used to describe the state <strong>of</strong><br />
the mechanical <strong>system</strong>. It appeared most convenient to state orientation applying an euler<br />
parameter (quaternion) like parametrization. Apart from the numerical advantages, an algorithm<br />
could be defined to evaluate the appearance <strong>of</strong> singular points in the work space <strong>of</strong> a<br />
parallel manipulator like the Stewart platform.<br />
With this method, it finally became possible to pro<strong>of</strong> that the SRS kinematical construction<br />
has no singular points. This is not trivial, since it would have been possible to reach<br />
such a point with only 15 % more stroke <strong>of</strong> the actuators. Moreover, it could be proven that<br />
the iterative Newton-Raphson procedure to calculate the platform pose from the actuator<br />
lengths converges anywhere in the SRS work space during any kind <strong>of</strong> possible (velocity<br />
limited) <strong>motion</strong>. It converges sufficiently fast for safe use in a real time digital feedback<br />
loop as will be done in the model <strong>based</strong> <strong>control</strong>ler discussed in Chapter 5.<br />
The dynamics <strong>of</strong> the Stewart platform was stated as a set <strong>of</strong> differential equations without<br />
algebraic constraints resulting from the kinematic chains in the <strong>system</strong>. This is possible<br />
by writing the actuator <strong>motion</strong> explicitly as a function <strong>of</strong> platform <strong>motion</strong>. By using Kane’s<br />
method <strong>of</strong> projecting forces onto the generalized velocities, the contribution <strong>of</strong> each inertial<br />
or active force e.g. stemming from the actuators piston mass, etc., can be quantified separately<br />
using local (simple) Newton-Euler like equations and (by projection) ending up with<br />
a limited number <strong>of</strong> equations <strong>of</strong> <strong>motion</strong>.<br />
The models derived only describe the dynamics <strong>of</strong> the mechanical <strong>system</strong>. The observed<br />
dynamics <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> will shown to be much better approximated by<br />
models which incorporate both the hydraulic and mechanical characteristics and their (two<br />
sided) interaction. This chapter provided for the mechanical modules for such models. The<br />
next chapter will treat the modelling <strong>of</strong> the hydraulic <strong>system</strong> itself and the interconnected<br />
structure <strong>of</strong> hydraulically driven mechanical <strong>system</strong>s.
Chapter 3<br />
Hydraulically driven <strong>motion</strong><br />
<strong>system</strong>s<br />
In the previous chapter, the mechanical <strong>system</strong> was modelled with input actuator forces<br />
from which accelerations, velocities and positions <strong>of</strong> the <strong>motion</strong> <strong>system</strong> and possibly its<br />
parasitic dynamics could be calculated. The energy necessary to move is supplied by the<br />
actuators. In robotics both electrically as hydraulically driven <strong>system</strong>s are seen to be applied<br />
and the mechanical model can be merged easily to models <strong>of</strong> either kind <strong>of</strong> <strong>system</strong>s.<br />
With the larger <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s almost invariably use is made <strong>of</strong> hydraulic<br />
actuators as is also the case with the Simona <strong>motion</strong> <strong>system</strong>. Since both the energy<br />
and the cooling with fluid power technology can be supplied at an exterior hydraulic pump,<br />
a favourable high ratio <strong>of</strong> force delivered over the weight and size <strong>of</strong> the actuator can be<br />
sustained well into the high power area. But even more importantly, the use <strong>of</strong> hydraulics<br />
enables smoothly running (low friction, low wear) long linear actuators, which is necessary<br />
in simulation requiring an unnoticeable change <strong>of</strong> sign in the direction <strong>of</strong> <strong>motion</strong>.<br />
An important aspect <strong>of</strong> the <strong>motion</strong> <strong>system</strong>s under consideration is the fact that the dynamics<br />
<strong>of</strong> the actuators heavily interact with the mechanics. So modelling either one <strong>of</strong><br />
these phenomena does not provide much insight in the dynamics <strong>of</strong> such <strong>system</strong>s. In most<br />
textbooks treating the modelling and <strong>control</strong> <strong>of</strong> mechanical <strong>system</strong>s the dynamics <strong>of</strong> the<br />
actuators are at best considered as parasitic influences to be taken into account at high frequencies<br />
[11, 31, 113]. However, with (heavy) <strong>system</strong>s requiring a high amount <strong>of</strong> power,<br />
especially if driven hydraulically, the interaction between the actuators and mechanical <strong>system</strong><br />
determines the ’rigid’ resonant modes. By using an integrated model, which will be<br />
presented in Section 3.3, the most relevant dynamics is captured as will be shown in the<br />
subsequent Chapter 4 in which evaluation with the experimental data takes place.<br />
The basic integrated model can still have a relative simple structure although the response<br />
<strong>of</strong> such <strong>system</strong>s may already seem quite complex. The basic structure <strong>of</strong> the hydraulically<br />
driven mechanical <strong>system</strong> allows the derivation <strong>of</strong> some important properties<br />
like passivity, discussed in Section 3.3.1, which can be <strong>of</strong> use in <strong>control</strong> and experimental<br />
identification. In this chapter, the dynamics <strong>of</strong> the hydraulic actuators will be modelled with<br />
a bilateral coupling to the mechanics through the energetic pairs <strong>of</strong> force and velocity at the<br />
mechanical side and flow and pressure in the hydraulical part <strong>of</strong> the <strong>system</strong>.<br />
75
76 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 3.1: Schematic drawing <strong>of</strong> a symmetrical hydraulic actuator.<br />
The long stroke hydraulic actuators, which are typical for <strong>flight</strong> <strong>simulator</strong>s, have been<br />
modelled in detail for <strong>motion</strong> <strong>control</strong> design by Van Schothorst in [124]. In Section 3.1, an<br />
overview will be given <strong>of</strong> the parts <strong>of</strong> those models which proved to be essential as either<br />
being an explicit part in the model <strong>based</strong> <strong>control</strong>ler considered in this research or in analysis<br />
exploring the limitations <strong>of</strong> such <strong>control</strong>lers. This last part usually requires an additional<br />
degree <strong>of</strong> detail in modelling discussed in Section 3.2. In this case, the dynamics <strong>of</strong> the<br />
hydraulic servo valve together with the transmission lines had to be taken into account.<br />
For a more fundamental introduction into hydraulic <strong>system</strong>s, one is referred to textbooks<br />
like [99] and [153]. Most <strong>of</strong> the notation used here follows Van Schothorst [124]. The<br />
modelling for <strong>control</strong> approach in hydraulic <strong>system</strong>s can also be found in references like<br />
[48] and [146]. In these references, however, the dynamics <strong>of</strong> the transmission lines was<br />
assumed to be neglectable. In this research, this was not the case due to the long stroke<br />
and high performance requirements, and therefore a modelling for <strong>control</strong> approach with<br />
the interplay <strong>of</strong> transmission lines dynamics and servo valve as the limiting factor, had to be<br />
chosen.<br />
3.1 The basic structure <strong>of</strong> hydraulic actuators<br />
The basic notion <strong>of</strong> the functionality <strong>of</strong> a hydraulic actuator can be observed by looking<br />
at Fig. 3.1, although the actually used actuator looks far more complicated as shown in<br />
Fig. 3.3. In Fig. 3.1, the piston connected to the load can be made to move by letting oil<br />
flow into the compartment <strong>of</strong> the cylinder on the left, o1, and out <strong>of</strong> the compartment on<br />
the right, o2, or v.v. This flow will result in a pressure build up, P o1 ; Po2, which through<br />
the reflected areas, Ap, will force the load to move. In a symmetrical actuator, this area,<br />
Ap, is equal on each side. In <strong>motion</strong>, the oil flows will also have to compensate for the<br />
change in volume in the respective actuator chambers. This effect accounts for the twosided<br />
coupling <strong>of</strong> flow/ pressure and/ or velocity/ force. Usually one also has to account for<br />
some dissipation resulting from leakage flows.<br />
The dynamics due to the hydraulic part <strong>of</strong> the <strong>system</strong> is caused by the compressibility<br />
<strong>of</strong> the oil. The relative decrease in volume, ; _ V=V, causes a pressure rise, P _ , linear with
3.1 The basic structure <strong>of</strong> hydraulic actuators 77<br />
the effective bulk modulus <strong>of</strong> the oil, E.<br />
P _ = ;E _ V<br />
(3.1)<br />
V<br />
By taking into account the oil flow into and out <strong>of</strong> the compartments, the following equations<br />
can be obtained, which describe the pressure dynamics <strong>of</strong> each chamber.<br />
Po1<br />
_ = E<br />
( o1 ; l1 ; Ap _q)<br />
V1<br />
Po2<br />
_ = E<br />
( l2 ; 02 + Ap _q) (3.2)<br />
V2<br />
The sign <strong>of</strong> the oil flows due to leakage, l1, l2, are taken negative w.r.t. the main flows,<br />
o1, o2, usually <strong>control</strong>led by a valve. The position, q, <strong>of</strong> the piston is set to zero at the<br />
point where the volumes <strong>of</strong> the actuator chambers are equal, V m = V1 = V2. This will<br />
typically be at or near half stroke position as drawn and will be referred to as mid position.<br />
As a result<br />
V1 = Vm + Apq<br />
V2 = Vm ; Apq (3.3)<br />
At this point it is most appropriate to apply a change <strong>of</strong> coordinates by taking mean and<br />
difference pressures, flows and oil stiffnesses (Cm and dC), which are defined as follows<br />
Pm m lm Cm<br />
dP d d l dC =<br />
1<br />
2<br />
1<br />
2<br />
1 ;1<br />
The pressure dynamics <strong>of</strong> (3.2) can now be transformed to<br />
Po1 o1 l1 E=V1<br />
Po2 o2 l2 E=V2<br />
_<br />
dP = 2Cm( m ; lm ; Ap _q) +dC=2(d ; d l)<br />
(3.4)<br />
_<br />
Pm = Cm=2(d ; d l) +dC=2( m ; lm ; Ap _q) (3.5)<br />
The change <strong>of</strong> coordinates can be motivated by the fact that only the pressure difference,<br />
dP , directly affects the mechanics through the actuator force, f a = ApdP . Further analysis<br />
reveals that even more advantage can be taken <strong>of</strong> this transformation. First <strong>of</strong> all, dC =0<br />
in the mid position, which fully decouples the two equations. Also in non mid position,<br />
the term d ; d l is small or even zero which makes the second equation not only badly<br />
observable but also almost un<strong>control</strong>lable. So, in most cases a sufficiently accurate model<br />
can be obtained by only taking into account the structure given by<br />
_<br />
dP =2Cm(q)( m ; LlmdP ; Ap _q)� (3.6)<br />
assuming a laminar leakage flow proportional to the pressure difference, lm = LlmdP .<br />
Equation (3.6) provides one <strong>of</strong> the two main equations to describe the basic hydraulically<br />
driven mechanical <strong>system</strong> as given in the block scheme <strong>of</strong> Fig. 3.2.<br />
Removing the dynamics <strong>of</strong> Pm not only simplifies the model but also reduces problems<br />
(e.g. numerical) in model <strong>based</strong> <strong>control</strong> <strong>of</strong> a hydraulic actuator due to the minor effect on
78 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 3.2: Basic block scheme <strong>of</strong> a hydraulic actuator connected to a mechanical <strong>system</strong>.<br />
the input/ output relation <strong>of</strong> this part <strong>of</strong> the dynamics. This model reduction step is <strong>of</strong>ten<br />
performed without consistently showing what is to be neglected i.e. not considering the<br />
change <strong>of</strong> coordinates.<br />
The first order differential equation is nonlinear due to the position dependence <strong>of</strong> the<br />
oil stiffness, using (3.3) and (3.4),<br />
Cm(q) =<br />
EVm<br />
V 2 m ; A2 p<br />
q2 = E<br />
Vm<br />
which attains the minimum at the mid position (q =0).<br />
3.1.1 Leakage<br />
1<br />
1 ; A 2 p q2 =V 2 m<br />
� (3.7)<br />
The leakage flows are considered laminar, which results in flows lineary dependent on the<br />
pressure drops. Assuming two reference pressures: P s as the higher level supply pressure<br />
and Pt as the lower level tank pressure (which can chosen to be the 0-level), the leakage<br />
flows can depend on five different parameters, L i, given in the following equations.<br />
l1 = L1t(Po1 ; Pt) ; L1s(Ps ; Po1) +Ln(Po1 ; Po2) (3.8)<br />
l2 = ;L2t(Po2 ; Pt) +L2s(Ps ; Po2) +Ln(Po1 ; Po2) (3.9)<br />
Not all terms will appear in every construction. E.g. in the double concentric actuator used<br />
for the SRS, schematically given in Fig. 3.3, the terms Ln and L1s do not exist.<br />
Also in this case it is appropriate to consider a change in coordinates<br />
Ltm Lsm<br />
dLt dLs =<br />
1<br />
2<br />
1<br />
2<br />
1 ;1<br />
L1t L1s<br />
L2t L2s<br />
(3.10)
3.1 The basic structure <strong>of</strong> hydraulic actuators 79<br />
The leakage can now be described as function <strong>of</strong> the new coordinates<br />
lm<br />
d l<br />
=<br />
+<br />
Ln +(Lsm + Ltm)=2 dLt + dLs<br />
dLt + dLs 4(Ltm + Lsm)<br />
;dLs<br />
;dLt<br />
;4Lsm ;4Ltm<br />
Ps<br />
Pt<br />
dP<br />
Pm<br />
(3.11)<br />
It can be analyzed at what value the mean pressure, P m, will stabilize, using these equations<br />
together with (3.5) with _ dP = _ Pm =0.<br />
In dropping Pm, only the main leakage parameter, Llm, influencing the resulting basic<br />
hydraulic equation (3.6), is relevant. It can be constructed as<br />
Llm = Ln +(Lsm + Ltm)=2 (3.12)<br />
3.1.2 Basic hydraulically driven mechanical <strong>system</strong> model<br />
In Fig. 3.2, a 2-port block scheme <strong>of</strong> (3.6) is given in connection with a mechanical <strong>system</strong>.<br />
This mechanical <strong>system</strong> with mass, M, viscous/ damping force constant, B p, and possibly<br />
some additional disturbance forces, Fext, can be described by the following equation<br />
Mq = fa ; Bp _q ; Fext<br />
(3.13)<br />
As all the parameters given are strictly positive, the hydraulic <strong>system</strong> can viewed as a velocity<br />
to force feedback with a first order <strong>system</strong>. The loop gain over theqtwo integrators,<br />
which is usually dominating, determines the undamped eigenfrequency 2CmA2 pM ;1 .<br />
Both leakage, Llm, and viscous friction/ damping, Bp will dissipate energy.<br />
It is not advisable to model the physical hydraulic actuator as a fully separate module.<br />
E.g. some parts <strong>of</strong> the hydraulic <strong>system</strong> to be modelled like the mass <strong>of</strong> the piston/ cylinder<br />
and the viscous friction should be made part <strong>of</strong> the mechanical <strong>system</strong> model to be<br />
connected.<br />
In case <strong>of</strong> the multivariable <strong>flight</strong> <strong>simulator</strong>, basically Fig. 3.2 holds, considering vector<br />
paths and matrices. To include platform coordinates, a mapping using the jacobian has to<br />
be applied. The actuator forces fa = ApdP will effect the platform coordinates through<br />
the transpose jacobian J T l�x and the platform velocities will have to be projected along the<br />
actuators by using _q = _ l = Jl�x _x. The undamped eigenfrequencies <strong>of</strong> the hydraulically<br />
driven mechanical <strong>system</strong> can be predicted by calculating the square roots <strong>of</strong> the eigenvalues<br />
<strong>of</strong> the loop matrix gain, p (2CmA2 pJl�xM ;1J T l�x ), assuming equal Ap and Cm for all<br />
actuators.<br />
If this is not the case one should consider taking matrices with the specific actuator<br />
values for Ap and Cm on the diagonal. All these variables, except for A p, depend on the<br />
<strong>simulator</strong> pose. These mass and oil stiffness dependent eigenvalues will be shown to reflect<br />
the most prominent part <strong>of</strong> the dynamics <strong>of</strong> a hydraulically driven <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong><br />
<strong>system</strong>.<br />
The structure given in Fig. 3.1 is a very rough simplification <strong>of</strong> an actual hydraulic<br />
actuator although the model derived will describe the main dynamical effects which can be
80 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 3.3: Schematic drawing double concentric symmetric hydraulic actuator.<br />
observed in practice. However, the way the <strong>control</strong>led oil flow, m (d ), behaves, has not<br />
yet been given and should also be studied to arrive at an appropriate model. This requires<br />
the modelling <strong>of</strong> the servo valve and transmission lines as will be discussed in the next<br />
section.<br />
3.2 Extensions<br />
In Fig. 3.3, a schematic drawing is given <strong>of</strong> the so-called double concentric symmetric<br />
hydraulic actuator, which is used in case <strong>of</strong> the SRS. This construction enables a symmetric<br />
actuator without the disadvantage <strong>of</strong> a piston at both sides. The arrows are drawn in<br />
the direction where the flows and velocity take a positive sign. Note that e.g. the leakage<br />
flow, l2;, is always negative. Next to the cylinder, the picture shows two other parts <strong>of</strong> a<br />
hydraulic servo actuator, which haven’t been discussed yet, the servo valve and the transmission<br />
lines.<br />
3.2.1 Servo valve<br />
The input flows, i1 and i2, given in Fig. 3.3 are determined by the servo valve. <strong>Model</strong>ling<br />
the relevant dynamics between our <strong>control</strong> input signal (voltage), u, to the servo valve and<br />
these oil flows is the subject <strong>of</strong> this section. The valve consists <strong>of</strong> an pilot valve and main<br />
spool as given in Fig. 3.4. The main spool is <strong>control</strong>led electronically using a <strong>control</strong> scheme<br />
given in Fig. 3.5.<br />
With the external connection to two reference oil compartments to the servo valve, the<br />
oil flow can be <strong>control</strong>led by the valve opening. One external reference is the hydraulic<br />
pump supply pressure, Ps, which is 160 Bar higher than the second reference in case <strong>of</strong> the<br />
SRS. This second reference, the tank pressure, Pt, can be considered zero. Both references<br />
are assumed to be constant in the sequel.
3.2 Extensions 81<br />
Fig. 3.4: Schematic drawing <strong>of</strong> a three stage valve. The main spool position, X m, is measured<br />
electronically and <strong>control</strong>led by feedback to the flapper positioning current,<br />
ia. The flapper/ nozzle mechanism internally stabilizes the positioning <strong>of</strong> the pilot<br />
valve with position, Xs, with oil flows ni. This is used to enable large oil flows<br />
at the main spool, which is positioned with oil flows mi. For a valve model,<br />
describing the most relevant high frequent dynamics as given in (3.17), only a<br />
limited number <strong>of</strong> variables given in the figure have to be taken into account.
82 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 3.5: Valve main spool <strong>control</strong> scheme<br />
Considering a turbulent oil flow through the servo valve, the main behaviour <strong>of</strong> this<br />
flow can be described by a nonlinear algebraic relation depending on spool position and the<br />
square root <strong>of</strong> the pressure drop. In the higher frequency area (> 100 Hz in case <strong>of</strong> the SRS<br />
servo valves), the dynamics <strong>of</strong> the spool positioning mechanism have to be considered too.<br />
The spool is an important part <strong>of</strong> the servo valve. By positioning the spool, the valve<br />
opening can be set. The valve opening and pressure drop over the valve determine the<br />
oil flow through the valve to the actuator cylinder compartment itself. If large oil flows are<br />
required (typically larger than 100 l=min), the spool positioning itself requires an additional<br />
hydraulic circuit, in itself an extra servo actuator <strong>control</strong>led by a pilot spool. This multistage<br />
concept is applied in the SRS as flows up to 150 l=min are needed.<br />
A schematic drawing <strong>of</strong> the valve is given in Fig. 3.4. The pilot spool (with position<br />
xs), is positioned by an internal hydromechanical feedback mechanism called the flapper/<br />
nozzle. As already noted, the positioning <strong>of</strong> the main spool is performed by an electronic<br />
feedback given in Fig. 3.5. This feedback uses the main spool position x m, or positioning<br />
error m to the current ia which forces the flapper position, xf , aside.<br />
Over a broad frequency area, usually including the area <strong>of</strong> interest in positioning and/<br />
or forcing a mechanical <strong>system</strong> (servo-valve bandwidth in case <strong>of</strong> the SRS is 150 Hz), the<br />
transfer function from desired to actual spool position, x m, can be considered a static unity<br />
gain.<br />
So the main servo valve phenomena can be captured by considering the turbulent flow<br />
equations <strong>of</strong> the main four-way valve. With symmetric hydraulic actuators, one usually<br />
applies symmetric 4-way valves. The main spool position, x m, together with the pressure<br />
drops determine the flows, i1 and i2, to and from the actuator chambers from P s and to<br />
Pt, as is drawn in Fig. 3.4.<br />
The equation describing a turbulent flow restriction is given by Merrit [99]<br />
=CdAm(x)<br />
s 2(Pin ; Pout)<br />
(3.14)<br />
In this case the discharge coefficient, Cd, and the oil density, , can be considered constant.<br />
The geometric properties, Am(x) <strong>of</strong> the flow restriction depend (almost linearly) on the<br />
spool position. The flow gain evaluated w.r.t. the spool position varies with the square root<br />
<strong>of</strong> the pressure drop and, moreover, this results in a hard non-linear change in gain if the<br />
flow changes sign and Pi1 or Pi2 are not both equal to 1=2Ps.
3.2 Extensions 83<br />
x_s, P_n, f_p, f_v<br />
2<br />
1.5<br />
1<br />
0.5<br />
0<br />
P_n<br />
Step response pilot valve<br />
−0.5<br />
0<br />
f_v<br />
0.001 0.002 0.003 0.004 0.005<br />
time (s)<br />
0.006 0.007 0.008 0.009 0.01<br />
Fig. 3.6: A state step response <strong>of</strong> the model (3.17) <strong>of</strong> the pilot valve with the identified<br />
parameters given in Table 3.1.<br />
As observed earlier, a more compact description can be arrived at at the actuator side in<br />
applying a change <strong>of</strong> coordinates, using difference and mean flows and pressures. This is<br />
also the case with the valve. With the important additional assumptions that<br />
x_s<br />
Pi1 + Pi2 = Ps� (3.15)<br />
and unaltered geometric properties A(xm), and considering normalised pressures and flows,<br />
^P = P=Ps, ^xm = xm=xx�max, ^ = = max, the maximum flow can be found by calculating<br />
or measuring max := f (xm�dPi�nom) =f (xm�max� 0). Now, one equation describes<br />
the oil flow through the four-way valve<br />
q<br />
^<br />
im = f2(^xm) 1 ; sgn(^xm)d ^ Pi� (3.16)<br />
where the mean normalised valve flow is defined by ^ im =( ^ i1 + ^ i2)=2. Again note that,<br />
due to the sign convention taken, this mean valve flow, im, means the average amount <strong>of</strong><br />
oil going into transmission line 1 and out <strong>of</strong> transmission line 2.<br />
With this equation, as proposed in [124] by Van Schorhorst, static geometric nonlinearities<br />
can be captured by the function f 2(^xm), which have been identified experimentally for<br />
all the six SRS servo valves. The sign-function (sgn) accounts for the switch in ports at<br />
the four-way valve. Note that a reduction step by omitting one state in taking away mean<br />
pressure and difference flow at the actuator is also motivated at the input, the valve, since<br />
these variables are not involved in (3.16).<br />
The main limitation in only using (3.16) in describing a valve lies in the high frequency<br />
area, as the spool position, xm, can not be positioned infinitely fast. The positioning mechanism,<br />
as depicted in Fig. 3.4, can be modelled following Van Schothorst [124] by using the
84 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
variable value variable value<br />
c1 4:29e3 c6 1e4<br />
c2 2:96e7 c7 9:93e-1<br />
c3 2:3e10 c8 4e3<br />
c4 1:13e11 c9 2:68e10<br />
c5 1e2 c10 1:27e2<br />
Kp 8 Kd 4:9e-4<br />
Table 3.1: The Simona three stage valve parameters identified by Van Schothorst [124]<br />
state space equation with normalized states given for the pilot valve:<br />
2<br />
6<br />
4<br />
^xf<br />
_^xf<br />
_^xs<br />
d _ ^ P n<br />
3<br />
7<br />
5 =<br />
2<br />
6<br />
4<br />
;c1 ;c2 ;c3 ;c4<br />
1 0 0 0<br />
0 ;1 ;c5 c6<br />
0 c7 1 ;c8<br />
3 2<br />
7<br />
6<br />
5<br />
6<br />
4<br />
_^xf<br />
^xf<br />
^xs<br />
d ^ Pn<br />
3<br />
7<br />
5 +<br />
2<br />
6<br />
4<br />
c9<br />
0<br />
0<br />
0<br />
3<br />
7<br />
5 ia<br />
(3.17)<br />
All the coefficients, c1:::9 are positive and the numerical values, identified on the actual<br />
actuators <strong>of</strong> the SRS by Van Schothorst [124], can be found in Table 3.1. The model takes<br />
into account the force balance and acceleration <strong>of</strong> the flapper, x f , the limited stiffness <strong>of</strong><br />
the oil at Pn and a reduced force balance (velocity only) <strong>of</strong> the pilot spool, x s. The model<br />
incorporates a right half plane zero as the flapper forces the pilot spool to move in negative<br />
direction, which can be compensated only after pressure build up has taken place.<br />
Further, the dynamics <strong>of</strong> the SRS pilot valve has four poles, among which is one lightly<br />
damped resonance, at ca. 900 Hz and one pole at ca. 150 Hz. The step response <strong>of</strong> the pilot<br />
valve is given in Fig. 3.6. Next to each other the flapper velocity (f v), position, xf , and<br />
pressure difference peak, Pn, in resonating at 880 Hz. The pilot spool moves slightly non<br />
minimum phase and resonating along a first order response in approx. 6 ms to its final value.<br />
In this simulation the four states have been scaled to achieve a numerically more favourable<br />
description. xsc = diag(1e ; 7� 1e ; 3� 1� 10): The pilot spool lets oil flow ( m) tomove<br />
the main spool. Considering a dPm 0, this can be modelled by a additional equation for<br />
the third stage adding an integrator.<br />
_^xm = c10f1(^xs) (3.18)<br />
Also the quasi static functions f1(^xs) and constants c10, have all been experimentally identified<br />
[124].<br />
As already noted, the position <strong>of</strong> the main spool is <strong>control</strong>led with an electronic PD<strong>control</strong>ler<br />
(Fig. 3.5) with proportional (K p) plus derivative (Kd) feedback given by<br />
ia = Kp ^xm ; Kd _ ^xm = Kp(u ; ^xm) ; Kdc10f1(^xs) (3.19)<br />
where u is the input signal sent to the valve.<br />
The additional state <strong>of</strong> this <strong>system</strong> in feedback results in a slightly resonating behaviour<br />
( =0:3) at 150 Hz as opposed to the mainly first order like response <strong>of</strong> the pilot valve.
3.2 Extensions 85<br />
x_m, x_s, P_n, f_p, f_v<br />
1.4<br />
1.2<br />
1<br />
0.8<br />
0.6<br />
0.4<br />
0.2<br />
0<br />
x_s<br />
Step response three stage valve<br />
x_m<br />
−0.2<br />
f_v<br />
−0.4<br />
P_n<br />
−0.6<br />
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02<br />
time (s)<br />
Fig. 3.7: <strong>Model</strong> state step response <strong>of</strong> the three stage servo valve.<br />
As can be seen from the state step response <strong>of</strong> the valve in Fig. 3.7, the high frequency<br />
behaviour can hardly be noted in the input/ output relation. Up till 200 Hz, the valve can<br />
also well be approximated by a second order <strong>system</strong> (e.g. via balanced reduction), which<br />
will lack physical interpretation, however. Taking a bode plot <strong>of</strong> the linearised version <strong>of</strong><br />
the valve shows the close correspondence as shown in Fig. 3.8.<br />
In hydraulic servo actuators, the valve characteristics are <strong>of</strong>ten limiting the achievable<br />
performance <strong>of</strong> the <strong>system</strong>. In this case, the high bandwidth valves still introduce a phase<br />
lag <strong>of</strong> approx. 10 at 25 Hz and double this at 50 Hz. The combination with the long<br />
transmission lines, which will be described next, form a risk to stability.<br />
3.2.2 Transmission lines<br />
A disadvantage <strong>of</strong> requiring long stroke actuators is the unavoidable use <strong>of</strong> relatively long<br />
transmission lines. Transmission lines <strong>of</strong>ten show a lightly damped resonating behaviour<br />
and if the transmission lines are longer, the eigenfrequencies tend to be lower which is a<br />
disadvantage in striving towards a smooth high bandwidth <strong>control</strong> by the servo valve <strong>of</strong> the<br />
flow to and from the actuator. Moreover, if actuator pressure <strong>control</strong> (dP o) is intended to<br />
be applied measuring the pressure, dPi, at the valve, the transmission line dynamics are in<br />
between. It will be shown that the pressure at the other side <strong>of</strong> a long transmission line, the<br />
output pressure, Po1�2, <strong>of</strong> the actuator, can deviate considerably from the measured input<br />
pressure, Pi1�2, at frequencies ten times lower than the transmission line eigenfrequencies.<br />
The transmission line dynamics can <strong>of</strong>ten be neglected if properly designed and with<br />
moderate performance requirements. In [48, 146] practical examples can be found <strong>of</strong> hydraulically<br />
driven mechanical <strong>system</strong>s (Brick laying robot, RRR-robot) in which the transmission<br />
lines can be omitted in modelling and <strong>control</strong>. In that case, the bandwidth <strong>of</strong> the<br />
servo-valve can be made reasonably lower than the lowest resonance resulting from the
86 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Amplitude<br />
Phase (deg)<br />
10 1<br />
10 0<br />
10 −1<br />
10 1<br />
10 −2<br />
0<br />
−100<br />
−200<br />
−300<br />
10 1<br />
−400<br />
Bodeplot <strong>of</strong> full and reduced (2nd order) model valve<br />
10 2<br />
10 2<br />
Frequency (Hz)<br />
Fig. 3.8: Bode plot <strong>of</strong> 5 th -order linearised servo valve model and 2 nd -order reduced version.<br />
transmission lines. With [153], the lowest transmission line eigenfrequency, ! t can roughly<br />
be approximated by<br />
Lt<br />
!t =<br />
co 2<br />
10 3<br />
10 3<br />
(3.20)<br />
With line length, Lt =1:4m in case <strong>of</strong> the SRS and the wave propagation velocity, c o<br />
given by<br />
s r<br />
E 109 co = =<br />
1000� (3.21)<br />
850<br />
where E is the bulk modulus <strong>of</strong> the oil and the density, this results in an estimated eigenfrequency<br />
<strong>of</strong> 180Hz.<br />
In [124] it has been shown that these dynamics can not be neglected if one wants to apply<br />
high performance <strong>motion</strong> <strong>control</strong> on long stroke hydraulic actuators typically used in <strong>flight</strong><br />
simulation. The lightly damped resonances <strong>of</strong> the transmission lines together with the phase<br />
lag introduced by the limited bandwidth <strong>of</strong> the servo valve easily result in stability problems.<br />
By taking these phenomena into account in modelling, proper model <strong>based</strong> <strong>control</strong> can be<br />
applied.<br />
The main steps in modelling the transmission lines will now be described, following Van<br />
Schothorst [124] and Yang and Tobler [158]. Details can be found in these references. The<br />
main conclusion will be that transmission lines can be modelled as a (theoretically infinite)<br />
set <strong>of</strong> parallel connected linear second order <strong>system</strong>s. Each second order <strong>system</strong> appoints a
3.2 Extensions 87<br />
φ i<br />
P i<br />
Fig. 3.9: A causal two-port configuration for a transmission line.<br />
specific vibrational mode with increasing eigenfrequency and, in approximating, all modes<br />
<strong>of</strong> interest are easily picked.<br />
In choosing the input flow and output pressure as input and the input pressure and output<br />
flow as output, a proper causal solution will result as discussed by Van Schothorst in<br />
[124]. This two-port representation <strong>of</strong> the transmission line is given in Fig. 3.9. The exact<br />
causal solution to a one dimensional distributed parameter model <strong>of</strong> a uniform rigid fluid<br />
transmission line with laminar flow in the Laplace domain, is given by Yang and Tobler in<br />
[158] by<br />
Pi(s)<br />
o(s)<br />
=<br />
1<br />
cosh(;(s))<br />
1 ;Zc(s) sinh(;(s))<br />
; sinh(;(s))=Zc(s) 1<br />
φ o<br />
oP<br />
Po(s)<br />
i(s)<br />
(3.22)<br />
with the characteristic impedance, Zc(s), and wave propagation operator, ;(s), <strong>of</strong> a single<br />
transmission line, as in [158]. One can approximate this <strong>system</strong> by a finite dimensional state<br />
space model by using the first n terms <strong>of</strong> an exact infinite modal sequence, which can be<br />
derived from (3.22).<br />
This modal approximation technique leads to<br />
Pi(s)<br />
o(s)<br />
=<br />
=<br />
n<br />
k=1<br />
Pik(s)<br />
ok(s)<br />
n 1<br />
k=1<br />
s2 + cks + dk<br />
acks + bck ;azks + bzk<br />
ask ; bsk acks + bck<br />
Po(s)<br />
i(s)<br />
(3.23)<br />
Considering each second order <strong>system</strong>, an even more compact state space representation<br />
can be derived [158], requiring only four parameters for the first mode and two additional<br />
(due to parameter dependencies) for each higher harmonic to be taken into account.<br />
Each harmonic can then separately be described by<br />
_<br />
Pik<br />
_ ok<br />
=<br />
0 (;1) k+1 1k<br />
(;1) k 2k ; 3k<br />
Pbk<br />
ak<br />
+<br />
0 4k<br />
; 2k<br />
1k 4k 0<br />
Pok<br />
ik<br />
:<br />
(3.24)<br />
These parameters still have a clear physical interpretation, as given in [124]. A block scheme<br />
representing this model is given in Fig. 3.10 with for k =1, t j = j j =1�::: �4� t5 =<br />
; 4 2= 1. Only one mode is given in this block scheme, as the structure <strong>of</strong> the others is
88 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 3.10: Basic block scheme <strong>of</strong> the parallel modal transmission line model. Note that each<br />
additional mode, i, can be accounted for by an equivalent scheme connected in<br />
parallel through the sigma signs.<br />
equivalent. The sigma-terms show the connection <strong>of</strong> these additional harmonics, which can<br />
be taken into account by a parallel connection <strong>of</strong> similar block schemes.<br />
To maintain approximate steady state gain for n =1, which will be lost due to simple<br />
neglection <strong>of</strong> all higher order terms, 4 can be set to ; 1 if pressure and flow have been<br />
normalised. The term p 1 2 is equal to the undamped eigenfrequency <strong>of</strong> each mode and<br />
3 is the dissipating term, damping the resonance. In the hydraulic actuators <strong>of</strong> the SRS<br />
the lowest eigenfrequencies <strong>of</strong> the transmission lines could be found in the range <strong>of</strong> 160 ;<br />
220 Hz and the first higher harmonic at a factor three higher. Given the bandwidth <strong>of</strong> the<br />
valve <strong>of</strong> 150Hz, in this research each transmission line was modelled by taking only one<br />
mode into account<br />
Apart from the resonating behaviour, the transmission lines result in a difference between<br />
the pressure measured over the valve, dP i and the pressure dPo, over the actuator<br />
chambers, which actually drive the mechanical <strong>system</strong>. This difference can become a problem<br />
if one wants to apply a well defined force to accelerate with high precision. As will be<br />
shown, these differences occur at much lower frequencies than those where the transmission<br />
line resonances occur.<br />
In [124], it was argued that the state describing the mean pressure in the hydraulic actuator,<br />
Pm, should be retained if transmission lines are to be taken into account and the<br />
model is to be used over the whole stroke including the position dependent stiffness <strong>of</strong> the<br />
hydraulic actuator. But in applying model <strong>based</strong> <strong>control</strong>, it is unfavourable to include an<br />
almost un<strong>control</strong>lable and unobservable state. Therefore it is proposed to use an alternative<br />
model.<br />
There are two separate transmission lines as shown in Fig. 3.3. The model <strong>of</strong> transmission<br />
line 1 has inputs i1 and Po1 and outputs Pi1 and o1. The model <strong>of</strong> transmission line<br />
2 has inputs i2 and Po2 and outputs Pi2 and o2. The transmission line inputs Po1 and<br />
Po2 are constructed from the reduced hydraulic actuator model, (3.6), by<br />
Po1 = 1=2(Ps + dPo)<br />
Po2 = 1=2(Ps ; dPo)� (3.25)
3.3 Integrated <strong>system</strong> 89<br />
which is a similar assumption as given in (3.15). Further, the other connection <strong>of</strong> this two<br />
sided coupling between the transmission lines and the hydraulic actuator is given by<br />
m =( o1 + o2)=2 (3.26)<br />
At the other side <strong>of</strong> the transmission lines, the servo valve is connected. The models <strong>of</strong> the<br />
two can be merged by retaining (3.16) and using i1 = i2 = im and dPi = Pi1 ; Pi2.<br />
By evaluating the differences between the reduced and full order model after connecting<br />
the building blocks, this step can and will be validated. Integrating the set <strong>of</strong> submodels<br />
introduced for the mechanical and hydraulical part <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> will<br />
be the subject <strong>of</strong> the next section.<br />
In this section two relevant extensions, the valve characteristics and transmission line<br />
dynamics, to the basic hydraulically driven mechanical <strong>system</strong> model were introduced. Of<br />
course many more effects could be investigated. One, which should be mentioned here,<br />
but will not be discussed much further, are the hydraulic actuator cushioning areas at both<br />
ends <strong>of</strong> the stroke. By gradually disabling oil flow to and from the actuator, the cushioning<br />
prevents from high accelerations (max. 2g) occurring if an actuator for some reason runs out<br />
<strong>of</strong> stroke. This mechanism, however, also influences the <strong>system</strong> dynamics for at least the<br />
first and last 0:15m stroke <strong>of</strong> all the actuators. Design and evaluation <strong>of</strong> the SRS cushioning<br />
is discussed in [124].<br />
3.3 Integrated <strong>system</strong><br />
Having introduced models <strong>of</strong> the mechanical part <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> in<br />
Chapter 2 and <strong>of</strong> the hydraulic servo actuators which drive the <strong>system</strong> in Section 3.1 and<br />
Section 3.2, the two can be connected. This results in a nonlinear model describing the<br />
hydraulically driven mechanical <strong>system</strong>.<br />
An important <strong>system</strong> analytic tool, passivity analysis, will be used to derive some interesting<br />
properties <strong>of</strong> these non-linear models. E.g. it will be shown that the full multivariable<br />
nonlinear <strong>system</strong> model is passive from input oil flows to input valve pressure differences<br />
irrespective <strong>of</strong> parasitic effects like transmission line resonances and/ or flexible mechanical<br />
modes. As a result proportional (or either other passive) input pressure feedback to valve<br />
oil flow can be applied without destabilising the <strong>system</strong>, as is <strong>of</strong> course already long known<br />
heuristically.<br />
In the sequel, the <strong>system</strong> characteristics due to the hydraulic <strong>system</strong> are best clarified by<br />
first considering a one degree-<strong>of</strong>-freedom actuator driving a single mass. Also the effect <strong>of</strong><br />
a parasitic mass or a non rigidly connected foundation is best understood by using such a<br />
simplified model first. The complexity <strong>of</strong> the fully non-linear six degree-<strong>of</strong>-freedom Stewart<br />
platform model driven by its hydraulic actuators is considerable since it easily includes over<br />
fifty states. Most <strong>of</strong> the <strong>system</strong> properties seen in practise are, however, well understood<br />
by using these kinds <strong>of</strong> models because the physical structure <strong>of</strong> the whole <strong>system</strong> has been<br />
maintained.
90 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
3.3.1 Passive input/output pairs in <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s<br />
Passivity is a strong input/ output <strong>system</strong>s property, which can be derived for a large class<br />
<strong>of</strong> <strong>system</strong>s also inhibiting nonlinearities. It roughly tells that a <strong>system</strong> does not generate<br />
energy by itself. From this property, stability is <strong>of</strong>ten easily derived, also in combination<br />
with feedback.<br />
There exist a large amount <strong>of</strong> <strong>control</strong> strategies <strong>based</strong> on the passivity property e.g.<br />
found in [11, 66, 128, 147]. Mathematically it is defined here following Van der Schaft<br />
[147]. The inner product for the vector signals, f and g in the L n 2 -space, is taken as<br />
Z 1<br />
< f�g>=<br />
0<br />
n<br />
i=1 fi(t)gi(t)dt: (3.27)<br />
Definition 3.1 Let the causal input/ output <strong>system</strong> G map L n 2 ! L n 2 (thus stable). Then G<br />
is passive if there exists some constant such that<br />
� 8 u 2 L n 2 : (3.28)<br />
By introducing the extended L2e-space, the definition can be generalised to a larger signal<br />
set and noncausal/ nonstable <strong>system</strong>s. A causal or nonanticipating <strong>system</strong> means that<br />
G(u)T = G(uT ) 8 u 2 Ln 2e where the subscript T gives the truncation at time T . In the<br />
extended space, integration <strong>of</strong> (3.27) is required only up till time T.<br />
Starting with small sub<strong>system</strong>s from which passivity can be derived, complex <strong>system</strong><br />
passivity can be evaluated by using the fact that feedback <strong>of</strong> a passive <strong>system</strong> with a passive<br />
<strong>system</strong> results in a over all passive <strong>system</strong> and two parallel passive <strong>system</strong>s together form<br />
one passive <strong>system</strong>. This does not hold for passive <strong>system</strong>s in a series connection.<br />
Consider the feedback <strong>system</strong> <strong>of</strong> Fig. 3.11. With the assumption that (e 1� e2) 2 Ln 2 !<br />
(u1� u2) 2 Ln 2 and passive G1 and G2 i.e. < yi� ui > i for i = 1� 2, the feedback<br />
<strong>system</strong> with inputs (e1� e2) and outputs (y1� y2) is also passive since<br />
< y1� e1 > + < y2� e2 > = < y1� e1 +y2 > + < y2� e2 ; y1 >=<br />
< y1� u1 > + < y2� u2 > 1 + 2 8u1�2 2 L n 2 (3.29)<br />
Notice that since this holds for a multivariable <strong>system</strong>, any input/ output combination, for<br />
which a <strong>system</strong> is passive, can be fed back by a passive <strong>system</strong> without influencing the<br />
passivity <strong>of</strong> some passive input/ output combination somewhere else. Passivity <strong>of</strong> a parallel<br />
connection <strong>of</strong> some passive G1 and G2 with output y =y1 +y2 and u =u1 =u2 is proven<br />
by<br />
< y� u>= < y1� u1 > + < y2� u2 > 1 + 2 8 u 2 L n 2 (3.30)<br />
Passivity can be proven easily for some basic <strong>system</strong> structures. For the time varying<br />
positive gain ’<strong>system</strong>’ described by y(t) =k(t)u(t) with k(t) 0, it can be proven by<br />
Z 1 Z 1<br />
y(t)u(t)dt = u(t)k(t)u(t)dt 0 8 u 2 L2 (3.31)<br />
0<br />
0
3.3 Integrated <strong>system</strong> 91<br />
e1<br />
y2<br />
+<br />
- d<br />
- 6<br />
u1<br />
-<br />
G1<br />
G2<br />
u2<br />
Fig. 3.11: A general feedback structure<br />
For the stable linear first order <strong>system</strong> described by _y =(u;ky) with k 0,oru = _y +ky,<br />
it follows from<br />
Z 1 Z 1<br />
y(t)u(t)dt = y(t)ky(t) +<br />
0<br />
0<br />
1<br />
Z 1<br />
2 0<br />
d<br />
dt y2 (t)dt > ; 1<br />
2 y2 (0) 8 u 2 L2 (3.32)<br />
For linear <strong>system</strong>s passivity means that the real part <strong>of</strong> the transfer function in the frequency<br />
domain will be positive. For passive linear single input single output <strong>system</strong>s the phase will<br />
thus vary between =2.<br />
The collocated input torque - output velocity pairs <strong>of</strong> a mechanical <strong>system</strong>, which can be<br />
described in hamiltonian form, are passive for every initial condition. From the definition <strong>of</strong><br />
the Hamiltonian (2.84) and assuming that the potential energy function is chosen as P(z)<br />
0 follows H 0. With the mechanical <strong>system</strong> described by the generalised coordinates,<br />
z, consider an output mapping y = J(z) _z, and the input, u, defined by the dual map,<br />
= J T (z)u. Now the <strong>system</strong> from input u to output y will shown to be passive. Consider<br />
(2.87), then<br />
< y� u>T =< _z� >T =<br />
0<br />
d<br />
H(m(t)� z(t))dt =<br />
dt<br />
H(m(T )� z(T )) ; H(m(0)� z(0)) > ;H(m(0)� z(0)) 8 u 2 L2e (3.33)<br />
Take y = _q, u = fa, z = sx and J(z) = Jl�sx(sx) and this establishes passivity <strong>of</strong> the<br />
rigid body Stewart platform (2.122) for the <strong>system</strong> considered from input actuator force,<br />
fa to output actuator velocity, _q. Moreover, it tells that passivity will be preserved in case<br />
<strong>of</strong> connecting a large class <strong>of</strong> parasitic mass- spring- damper <strong>system</strong>s connected e.g. a non<br />
rigidly attached foundation.<br />
Having established passivity <strong>of</strong> the mechanical <strong>system</strong>, the <strong>system</strong> can be extended to<br />
the hydraulics retaining the passivity property.<br />
-Viscous friction/ damping Every Rayleigh function, R satisfying _z T (@R)=(@ _z)( _z) 0<br />
is a passive <strong>system</strong>. Rayleigh functions <strong>of</strong>ten occur as dissipative feedback over passive<br />
mechanical (flow) <strong>system</strong>s retaining passivity. A simple example <strong>of</strong> a Rayleigh<br />
function is the viscous friction, bi _q i in each <strong>of</strong> the actuators:<br />
Rvact( _x� sx) =J T l�x (sx)diag(bi _qi) =J T l�x (sx)diag(bi)Jl�x(sx) _x: (3.34)<br />
Z T<br />
+<br />
d?<br />
+<br />
y1<br />
e2<br />
-
92 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Amplitude<br />
Phase (deg)<br />
10 2<br />
10 1<br />
10 0<br />
10 −1<br />
10 0<br />
10 −2<br />
100<br />
0<br />
−100<br />
dPo−dPi<br />
−200<br />
10 0<br />
−300<br />
10 1<br />
10 1<br />
Bodeplot Phi_im−>dP_i,o<br />
dPo−dPi<br />
Frequency (Hz)<br />
Fig. 3.12: Bode plot <strong>of</strong> 7 th -order linearised hydraulically driven mechanical <strong>system</strong> model<br />
including transmission lines. Input valve flow, im, outputs, dPi (full), dPo<br />
(dashed), dPo ; dP i (dotted).<br />
Equivalently damping induced through resonances can also be included without destroying<br />
passivity.<br />
-The hydraulic actuator The model <strong>of</strong> the hydraulic part <strong>of</strong> an actuator, given in (3.6)<br />
can be considered as a stable first order <strong>system</strong> with a varying gain C m(q). Since<br />
Cm(q) > 0 each linearisation <strong>of</strong> the hydraulics described by (3.6) results in a passive<br />
<strong>system</strong> for every position q. Further, as shown as a part in the block scheme <strong>of</strong><br />
Fig. 3.2, it is connected to the mechanical <strong>system</strong> as a negative feedback over a passive<br />
<strong>system</strong>. So each linearised hydraulic actuator connected to the nonlinear mechanical<br />
<strong>system</strong> preserves passivity. The passive input/ output <strong>system</strong> <strong>of</strong> the actuator can be<br />
chosen from m to dPo respectively. With the varying gain, the situation is somewhat<br />
different:<br />
< m�dPo > =<br />
+<br />
Z 1<br />
0 Z 1<br />
0<br />
d<br />
dt dPo(t)<br />
10 2<br />
10 2<br />
dPi<br />
dPo<br />
dPi<br />
dPo<br />
dPo(t) Llm ; d<br />
dt<br />
1<br />
4Cm(q(t)<br />
1<br />
4Cm(q(t))<br />
10 3<br />
10 3<br />
dPo(t) dt<br />
dPo(t)dt (3.35)<br />
Due to the time derivative <strong>of</strong> the inverse actuator stiffness, this <strong>system</strong> is not always<br />
passive. With an upper bound on the actuator velocity <strong>of</strong> e.g. 1 m=s and with a<br />
polynomial fit Cm(q) = b2q 2 + b1q + b0 =99:2q 2 ; 9:8q + 104 [124], a leakage<br />
term <strong>of</strong> 0.002 (0.01 is normal) will passify this <strong>system</strong>.<br />
-The transmission lines The structure <strong>of</strong> the transmission lines models in the block scheme<br />
given in Fig. 3.10 is clearly a parallel connection <strong>of</strong> passive <strong>system</strong>s which can be considered<br />
as two stable linear first order <strong>system</strong>s in feedback. The actuator side <strong>of</strong> the
3.3 Integrated <strong>system</strong> 93<br />
10 2<br />
10 0<br />
10 −2<br />
10 2<br />
10 0<br />
10 −2<br />
10 2<br />
10 0<br />
10 0<br />
10 −2<br />
10 1<br />
10 2<br />
Frequency (Hz)<br />
10 3<br />
100<br />
0<br />
−100<br />
100<br />
0<br />
−100<br />
100<br />
0<br />
10 0<br />
−100<br />
10 1<br />
10 2<br />
Frequency (Hz)<br />
Fig. 3.13: Bode plots from input valve flow ( im) to pressure difference (dPi) <strong>of</strong> linearised<br />
hydraulically driven mechanical <strong>system</strong> models including transmission lines in<br />
upper/ mid/ lower position (q = :47� 0� ;:47m) for full (full), reduced (n =6,<br />
dashed) order models and difference (dotted).<br />
port feeds back over the passive input/ output pair dP o and m respectively and at<br />
the valve side the input/ output pair is given by im and dPi, considering the parallel<br />
transmission lines over the pairs Pi1 and i1 and Pi2 and i2.<br />
-The valve port dynamics The valve port dynamics modelled as turbulent flow port can<br />
be considered as a positive time varying gain. If j dP i j 1,<br />
p<br />
1 ; sgn(xm(t))dPi(t) =k(t)xm� (3.36)<br />
im(t) =xm(t)<br />
implies k(t) 0. This is a passive <strong>system</strong> which preserves passivity if used in a feedback<br />
connection with e.g. constant gain from dP i to xm. As shown in Section 3.2.1,<br />
the valve spool positioning <strong>system</strong> cannot be considered a constant gain over an infinitely<br />
large frequency area but inhibits dynamics which is clearly not passive from<br />
input voltage (u) to output spool position (x m). At this point passivity <strong>based</strong> <strong>control</strong><br />
strategies may cause possible (stability) problems.<br />
Summarising, it has been established that the hydraulically driven nonlinear mechanical<br />
<strong>system</strong> is passive with respect to a large number <strong>of</strong> input/ output pairs. Under moderate<br />
conditions, which are typically satisfied in practice, constant feedback <strong>of</strong> the pair (valve<br />
spool position, pressure difference measured at the valve) leaves the passivity properties <strong>of</strong><br />
the nonlinear <strong>system</strong> unaltered. In fact, this kind <strong>of</strong> feedback will dissipate energy, which<br />
drives the <strong>system</strong> with badly damped resonances faster to its stable equilibrium.<br />
10 3
94 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Variable Description Value Unit<br />
n Max. valve flow 25e ;4 [m3 =s]<br />
Ps Supply Pressure 160e5 [N=m2 ]<br />
Fmax = Ps Ap Max. actuator force 40e3 [N ]<br />
_qmax = n=Ap Max. velocity 1 [m=s]<br />
Normalised Operators Description Value Unit<br />
An = Ap _qmax= n Piston area 1 [;]<br />
C1 = E n=(V1Ps) Stiffness chamber 1 101 [1=s]<br />
C2 = E n=(V2Ps)<br />
L1 = L1tPs= n<br />
Stiffness chamber 2<br />
Leakage 1<br />
104<br />
4:32e<br />
[1=s]<br />
;2<br />
L2 =(L2s + L2t)Ps= n Leakage 2 5:06e<br />
[;]<br />
;2<br />
Wpn = wp=Fmax Visc. friction 3:5e<br />
[;]<br />
;4<br />
M<br />
[;]<br />
;1<br />
n = Fmax=(Mp _qmax) Inv. Mass 10 [1/s]<br />
Table 3.2: Parameters taken for a 1-d.o.f. hydraulic mechanical <strong>system</strong>. First the four normalising<br />
quantities are given. Then all the normalised values for <strong>of</strong> the operators<br />
are specified.<br />
Variable Value Variable Value<br />
t11 = 1737 t12 = 1254<br />
t21 = 650 t22 = 1184<br />
t31 = 30 t32 = 54<br />
Table 3.3: Parameters taken for the two transmission lines.<br />
3.3.2 The 1-d.o.f. hydraulically driven mechanical <strong>system</strong><br />
Further insight into the characteristics <strong>of</strong> the hydraulic mechanical models, derived in the<br />
previous sections, can be obtained by filling in realistic values into the parameters, e.g. SRS<br />
design numbers, and performing an evaluation in the frequency (linearised models) and<br />
time domain. First a one-degree-<strong>of</strong>-freedom hydraulically driven mechanical <strong>system</strong> with<br />
transmission line dynamics will be considered. For these models, the difference in the<br />
dynamics <strong>of</strong> the actual driving pressure in the actuator and the measured pressure over the<br />
valve will be evaluated. Further, the implications <strong>of</strong> the reduction step proposed by taking<br />
(3.6) instead <strong>of</strong> (3.2), assuming (3.25) and (3.26), can be quantified.<br />
It is important to normalise the variables such as pressures (typically 10 7 in SI-units)<br />
and flows (typical values <strong>of</strong> 10 ;3 in SI) when building numerical models which contain<br />
hydraulics. In Table 3.2, first the normalising values <strong>of</strong> these variables are given. Then the<br />
normalised values <strong>of</strong> the operators taken for the models consisting <strong>of</strong> (3.13) together with<br />
(3.6) or (3.2) are specified at neutral position (at full stroke q =0:625). With the mass taken<br />
<strong>of</strong> 4000kg and the typical values <strong>of</strong> the hydraulic actuators <strong>of</strong> the SRS, an undamped rigid
3.3 Integrated <strong>system</strong> 95<br />
body mode eigenfrequency, f rm <strong>of</strong> approx. 7Hz results.<br />
frm = 1<br />
2<br />
r<br />
2Cm<br />
= 1<br />
2<br />
r<br />
2 4:1 106 4000<br />
= 1 p<br />
(101 + 104)10 = 7 Hz<br />
2<br />
(3.37)<br />
Mn<br />
This corresponds with the lowest eigenfrequencies found for the SRS. Due to the normalising<br />
velocity <strong>of</strong> one, the normalised inverse mass can also be interpreted as the maximum<br />
acceleration to be attained, which in this case approx. amounts to gravity. The leakage term<br />
related to the damping in the actuator driving direction amounts to L lm =2:35%, which is<br />
equal to the value experimentally found by Van Schothorst [124]. This is also the case for<br />
the viscous friction term.<br />
The model is completed by connecting the transmission lines using (3.24) and assuming<br />
an ideal valve input, i1 = i2 = im. The parameters taken for the transmission lines<br />
correspond to the values derived in [125] adapted with 10 % additional line length to arrive<br />
at the eigenfrequencies found in practice. The three resulting parameters, t 1k:::3k, for the<br />
two lines (k = 1� 2) are given in Table 3.3. Eigenfrequencies <strong>of</strong> the unconnected transmission<br />
lines therefore amount to 169 Hz and 194 Hz with damping <strong>of</strong> t1 = 0:01 and<br />
t2 =0:02 respectively.<br />
Influence <strong>of</strong> the transmission lines on input and output pressure dynamics<br />
Bode plots <strong>of</strong> this <strong>system</strong> from input m to the outputs dPi and dPo are given in Fig. 3.12.<br />
Connection <strong>of</strong> the impedance <strong>of</strong> the hydraulic actuator to the transmission lines shifted<br />
the transmission line eigenfrequencies approx. 10 Hz higher. This shift <strong>of</strong> approx. 5%<br />
is also observed for the lower (rigid body) mode and in that case to lower values for the<br />
eigenfrequency resulting from the load together with the oil spring stiffness. This effect<br />
<strong>of</strong> interaction between the transmission lines and the actuator should not be neglected. It<br />
means e.g. that the effective oil stiffness with the SRS-actuators will be 12% lower than<br />
the theoretically found value in case one identifies this hydraulically driven <strong>system</strong> with a<br />
model structure not accounting for the transmission lines.<br />
The phase <strong>of</strong> the passive input/ output pair ( im�dPi) clearly remains within 90 .It<br />
is important to notice that although the pressures have a one-to-one correspondence in the<br />
lower frequency area, they start to deviate much earlier than the transmission line eigenfrequencies.<br />
This is due to the anti-resonance in the transfer function from im to dPi at<br />
approx. 60 Hz.At20 Hz this amounts to 16% and at 30 Hz already accounts for a difference<br />
<strong>of</strong> 25%. Further, the resonating behaviour <strong>of</strong> the transmission lines is expected to be<br />
much (10 times) heavier observed in the measurable output, dP i, than it will be felt by the<br />
mechanical <strong>system</strong> (through dPo).<br />
Influence <strong>of</strong> the varying actuator stifness and model reduction step<br />
Since the stiffness <strong>of</strong> the oil column depends on the actuator position, the eigenvalues, also<br />
those related to the transmission lines, will shift if the actuator moves. As argued in [124],<br />
this shift will be slightly different if the actuator model is reduced to (3.6). However, the<br />
reduced model describes the <strong>system</strong> reasonably well as can be observed in Fig. 3.13. Bode<br />
plots <strong>of</strong> the <strong>system</strong> from input, im, to the measurable output dPi in an upper/ mid and<br />
lower position are taken for the reduced and nonreduced models. Also the differences are
96 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Amplitude<br />
Phase (deg)<br />
10 2<br />
10 1<br />
10 0<br />
10 −1<br />
10 0<br />
10 −2<br />
0<br />
−200<br />
−400<br />
10 0<br />
10 1<br />
10 1<br />
Bodeplot u−>dPi,dPo<br />
Frequency (Hz)<br />
Fig. 3.14: Bode plot <strong>of</strong> 8 th -order linearised hydraulically driven mechanical <strong>system</strong> model<br />
including transmission line and valve dynamics. Input valve input voltage, u,<br />
outputs dP i (full), dPo (dashed).<br />
shown. The upper and lower positions are just within the operational area where the cushioning<br />
does not influence the actual <strong>system</strong>. Since the eigenvalues are relatively lightly<br />
damped, very small differences in the eigenvalue shift locally result in larger deviations but<br />
the general behaviour is described fairly well by the reduced order model. Both for the<br />
upper as the lower position the anti-resonance between the hydromechanical mode at approx.<br />
7 Hz and the first transmission line related mode moves up from 60 Hz to maximally<br />
90 Hz.<br />
Influence <strong>of</strong> the valve dynamics<br />
The model can be extended by adding the reduced linear second order model <strong>of</strong> the three<br />
stage valve dynamics derived from (3.19) and (3.17). In Fig. 3.14 the Bode plot <strong>of</strong> this<br />
model is given. The main effect <strong>of</strong> the valve is an additional lag in phase, which becomes<br />
prominent for frequencies > 100 Hz. Together with the high gains introduced with the<br />
transmission lines resonances, the valve phase lag forms an important stability risc in feedback<br />
design, which has to be taken into account.<br />
In <strong>system</strong>s with moderate requirements, the valve bandwidth can be decreased in order<br />
to filter most <strong>of</strong> the transmission line resonances. But also in that case, the phase lag <strong>of</strong><br />
the valve will be dangerous in some frequency area e.g. in combination with resonances<br />
due to structural deformation. In the lower frequency region, the nonlinearities <strong>of</strong> the valve,<br />
e.g. due to (3.16), will form the most important factor.<br />
In the time domain, the step response <strong>of</strong> this model is given in Fig. 3.15 on a 200 ms<br />
range. The steady state response to the input and output pressures is neglectable. This<br />
means it is not very feasible to induce sustained accelerations with hydraulically driven<br />
10 2<br />
10 2<br />
10 3<br />
10 3
3.3 Integrated <strong>system</strong> 97<br />
5<br />
4<br />
3<br />
2<br />
1<br />
0<br />
−1<br />
−2<br />
−3<br />
Step response u−>dPi,dPo,dPo−dPi<br />
−4<br />
0 0.02 0.04 0.06 0.08 0.1<br />
time (s)<br />
0.12 0.14 0.16 0.18 0.2<br />
Fig. 3.15: Step response <strong>of</strong> 8 th -order linearised hydraulically driven mechanical <strong>system</strong><br />
model including transmission line and valve dynamics. Input valve input voltage,<br />
u, outputs dPi (full), dPo (dashed). dPo ; dPi (dash-dot). The high frequency<br />
resonances result from the transmission lines, the slower vibration with<br />
approx. 0:16s period time is the rigid body mode.<br />
mechanical <strong>system</strong>s. A quasi static gain <strong>of</strong> a moving actuator is attained from valve input<br />
voltage to actuator velocity. Of course in practice, the limited stroke is the main limiting<br />
factor for the low frequency accelerations. In <strong>control</strong> design for acceleration (simulation),<br />
however, the virtually zero gain from valve input to force/ acceleration will be an important<br />
factor to be taken into account.<br />
In both pressures the 160 ms time period (7Hz) <strong>of</strong> the hydromechanical mode is most<br />
prominent. The difference in response is mainly due to the transmission line resonances to<br />
which the pressure measured over the valve is much more sensitive. Due to the interference<br />
<strong>of</strong> the two transmission line modes, the high vibrational amplitude comes up and falls within<br />
a 40 ms time range.<br />
3.3.3 Additional mechanical modes<br />
The models presented in the previous paragraphs include the dynamics <strong>of</strong> the hydraulic actuator(s)<br />
into the higher frequency areas <strong>of</strong> approx. 200 Hz. In deriving accurate models <strong>of</strong> a<br />
hydraulically driven mechanical <strong>system</strong>s over this broad frequency area, one can usually not<br />
assume the mechanics to be entirely rigid. In this section, two <strong>of</strong> such cases are examined<br />
for a 1-d.o.f. <strong>system</strong>. First, the possible nonrigidness <strong>of</strong> a foundation is taken into account.<br />
In the second case, the driven <strong>system</strong> consists <strong>of</strong> a main mass connected to a parasitic mass/<br />
spring/ damper.<br />
A schematic drawing <strong>of</strong> hydraulically driven mass, m s, connected to a nonrigid foundation<br />
is given in Fig. 3.16. The foundation with mass, m f is fastened to the ground with a
98 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 3.16: Schematic drawing <strong>of</strong> a hydraulic actuator moving a consisting <strong>of</strong> a single mass<br />
ms in 1-d.o.f. pushing from a non rigidly attached foundation, acting as a mass/<br />
spring/ damper <strong>system</strong> mf �bf �cf .<br />
spring <strong>of</strong> stiffness cf and a damper, bf . The equations <strong>of</strong> <strong>motion</strong> become<br />
msxs = AdPo (3.38)<br />
mf xf = ;AdPo ; bf _xf ; cf xf<br />
Although the actual force fa = ApdPo necessary to accelerate the mass, ms does not<br />
change with the nonrigid foundation, the feedback path relating _q = _x s ; _xf changes. In<br />
case <strong>of</strong> the structure with simple mass given in Fig. 3.2, the feedback path becomes _q =<br />
ApdPo=(mss), assuming no viscous friction. Every pole <strong>of</strong> the feedback path becomes a<br />
transmission zero in the transfer function from input (valve flow) to output pressure, dP o.<br />
With the foundation the feedback path changes to<br />
considering the additional equation <strong>of</strong> <strong>motion</strong>,<br />
_q = (ms + mf )s 2 + bf s + cf<br />
mss(mf s 2 + bf s + cf ) ApdPo� (3.39)<br />
mf xf + bf _xf + cf xf + fa =0: (3.40)<br />
p<br />
cf =mf ,<br />
A complex pair <strong>of</strong> transmission zeros with the undamped eigenfrequency, ! f =<br />
<strong>of</strong> the foundation alone results. The structure <strong>of</strong> the poles <strong>of</strong> this interconnected <strong>system</strong> is<br />
somewhat more complex. The foundation will add two complex poles and the transmission<br />
zeros will always lie in between, since the passivity properties will be preserved. Usually the<br />
mass <strong>of</strong> the foundation will be much higher than the mass to be moved. In this case, although<br />
the respective eigenfrequencies can be close, there will not be a large shift in connecting the<br />
models <strong>of</strong> the hydraulically driven mass, ms and the foundation. In Fig. 3.17 the Bode plots<br />
<strong>of</strong> this model structure with three different parameter values are given. In all cases (1f,2f,3f)<br />
the model <strong>of</strong> the hydraulically driven mass is as specified in the previous paragraphs. The<br />
parameters concerning the foundation are given in Table 3.4. f is the relative damping<br />
given by 2 f cf = !fbf .<br />
In both cases 1f and 2f, in which the mass <strong>of</strong> the foundation is relatively large (the<br />
realistic cases), the nonrigid foundation results in a relatively small distortion <strong>of</strong> the open
3.3 Integrated <strong>system</strong> 99<br />
Amplitude<br />
Phase (deg)<br />
10 2<br />
10 1<br />
10 0<br />
10 −1<br />
10 0<br />
10 −2<br />
100<br />
0<br />
−100<br />
−200<br />
10 0<br />
−300<br />
Case mf =ms !f f<br />
1f 10 2 10 0:05<br />
2f 10 2 4 0:05<br />
3f 1 2 10 0:05<br />
Table 3.4: Parameters taken for the foundation.<br />
10 1<br />
10 1<br />
Bodeplot Phi_im−>dP_i<br />
Frequency (Hz)<br />
Fig. 3.17: Bode plots <strong>of</strong> input valve flow, im, to pressure difference over the valve, dP i,<br />
with 1-d.o.f. hydraulically driven mass connected to a nonrigid foundation. Case<br />
1f (full), 2f (dashed), 3f (dash-dotted).<br />
loop characteristics <strong>of</strong> the original <strong>system</strong>. The additional resonance and anti-resonance are<br />
clearly visible and also the, mostly oil spring stiffness-moveable mass related resonance is<br />
seen to be shifted slightly. In the third case, where the mass <strong>of</strong> the foundation and moveable<br />
mass are equal, the interaction between the modes is shown to be much higher. Although<br />
this example is highly hypothetical in case <strong>of</strong> a foundation, these typical effects can occur<br />
if one would place two platforms on top <strong>of</strong> each other like McInroy et al. [93]. In all<br />
cases the transfer functions from im to dPi clearly remain strict positive real i.e. passive.<br />
For the high and low frequencies, the response is not sensitive to the characteristics <strong>of</strong> the<br />
foundation.<br />
The second example <strong>of</strong> a possible parasitic resonating behaviour is schematically drawn<br />
in Fig. 3.18. The construction to be moved itself, e.g. the <strong>simulator</strong>, is nonrigid. In practice<br />
such a situation can occur if one connects a large projection screen in front <strong>of</strong> and heavy<br />
projectors on top <strong>of</strong> the cockpit <strong>of</strong> a <strong>simulator</strong>. In this 1-d.o.f. case the parasitic behaviour<br />
is modelled as two masses connected through a spring/ damper. Although this schematic<br />
example will not catch the nonlinear behaviour <strong>of</strong> a flexible distributed mass with gross<br />
spatial movement, some typical, linear parts <strong>of</strong> the characteristics can be investigated.<br />
10 2<br />
10 2<br />
10 3<br />
10 3
100 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Fig. 3.18: Schematic drawing <strong>of</strong> a hydraulic actuator moving a mass m s in 1-d.o.f., which<br />
is connected to a parasitic mass/ spring/ damper <strong>system</strong> mp�bp�cp.<br />
Case mp=ms !p p<br />
1p 1 2 10= p 2 0:05= p 2<br />
2p 1 2 10 p 2 0:05= p 2<br />
3p 3 2 10 p 2 0:05= p 2<br />
Table 3.5: Parameters taken for the parasitic mass.<br />
The equations <strong>of</strong> <strong>motion</strong> for this <strong>system</strong> become<br />
msxs = cp(xp ; xs) +bp(_xp ; _xs) +ApdPo (3.41)<br />
mpxf = cp(xs ; xp) +bp(_xs ; _xp)<br />
In this case the velocity measured at the actuator is equal to the inertial velocity <strong>of</strong> the<br />
mass, ms. The transfer function from applied actuator force, f a = ApdPo, to the velocity,<br />
_q = _xs changes to<br />
_q =<br />
mps 2 + bps + cp<br />
mss(mps 2 + rmbps + rmcp) ApdPo� (3.42)<br />
with the mass ratio, rm = (ms + mp)=ms, since the actuator force is not the only force<br />
on ms as also the spring and damper force are taken into account. Again, the zero at zero<br />
frequency is retained and an additional complex pair is introduced. This time predicted at<br />
p cprm=ms. In principle, the structure <strong>of</strong> (3.39) and (3.42) is equal. As a result, one can not<br />
discriminate between the case <strong>of</strong> a nonrigid foundation or a nonrigid structure to be moved,<br />
given the measurement from valve input to a pressure difference (dP i or dPo) or actuator<br />
velocity, _q. To determine what causes the parasitic resonance, additional measurements<br />
have to be taken e.g. from accelerometers attached to the foundation or <strong>simulator</strong>. This<br />
identification problem will be considered in the next chapter.<br />
In Table 3.5 the parameters <strong>of</strong> three cases are given for the flexible structure <strong>of</strong> Fig. 3.18.<br />
All the parameters <strong>of</strong> the hydraulic actuator are also taken equal to the ones presented in<br />
the previous section (Table 3.2). The total moveable mass (m s + mp) is taken equal to
3.4 Hydraulically driven Stewart platform 101<br />
Amplitude<br />
Phase (deg)<br />
10 2<br />
10 1<br />
10 0<br />
10 −1<br />
10 0<br />
10 −2<br />
100<br />
0<br />
−100<br />
−200<br />
10 0<br />
−300<br />
10 1<br />
10 1<br />
Bodeplot Phi_im−>dP_i<br />
Frequency (Hz)<br />
Fig. 3.19: Bode plots <strong>of</strong> input valve flow, i, to pressure difference over the valve, dP i, with<br />
1-d.o.f. hydraulically driven structure consisting <strong>of</strong> two masses interconnected by<br />
spring and damper. Case 1p (full), 2p (dashed), 3p (dash-dotted).<br />
4000 kg in all cases. The first case (1p) results in exactly the same model parameters<br />
as case 3f, considering (3.39) and (3.42). The transfer function from i to dPi for these<br />
parameters, plotted in Fig. 3.19, is almost the same (the viscous friction <strong>of</strong> the actuator<br />
results in very small differences). The other cases shown have somewhat more realistic<br />
behaviour considering a resonant frequency twice as high (2f) and further taking 25% <strong>of</strong><br />
the part <strong>of</strong> the moving object nonrigidly connected (3f). In reducing this part, the zeros<br />
and poles mostly related to the parasitic resonance shift back to lower values towards the<br />
original eigenfrequency <strong>of</strong> the hydraulically driven load since there is less interaction.<br />
The effect <strong>of</strong> interaction will be similar to the parasitic resonating spring/ damper/<br />
masses treated in this section if one measures the flow to pressure/ velocity transfer function<br />
<strong>of</strong> one actuator in case <strong>of</strong> a hydraulically driven Stewart platform where more than one<br />
actuator is connected to the <strong>system</strong>. This structure will be treated in the next section.<br />
The parasitic resonance shows a moderate peak in the frequency domain in the more<br />
or less realistic setting <strong>of</strong> case 3p. In open loop the characteristics <strong>of</strong> the standard hydromechanical<br />
mode will dominate. With feedback this will change and will change more<br />
drastically if performance specifications require a bandwidth close to these parasitic effects.<br />
This problem will be more closely looked at in Chapter 5 where the closed loop <strong>system</strong> is<br />
validated.<br />
3.4 Hydraulically driven Stewart platform<br />
Having considered the hydraulic actuator and its most relevant properties to be taken into<br />
account in modelling in a 1-d.o.f. setting and considering the description <strong>of</strong> the 6-d.o.f. me-<br />
10 2<br />
10 2<br />
10 3<br />
10 3
102 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
dP o<br />
R<br />
b J T fa - A - - - b<br />
l�x<br />
Mt<br />
?<br />
-L 6<br />
C<br />
-B<br />
6<br />
R?<br />
?<br />
6<br />
-A<br />
_q _x<br />
- b - b<br />
Jl�x -<br />
om<br />
Fig. 3.20: Basic structure hydraulically driven <strong>motion</strong> <strong>system</strong>.<br />
chanical <strong>system</strong> model <strong>of</strong> the Stewart platform introduced in Chapter 2, the two can be<br />
combined. As interaction will be the most prominent additional effect to be taken into account,<br />
the structure <strong>of</strong> this phenomenon will be analyzed first, using a simplified model. The<br />
inertial effect <strong>of</strong> the actuators will be treated in the hydraulically driven setting. Finally, the<br />
full model will be considered performing analysis in the frequency domain (through local<br />
linearisation) for several platform poses using numerical design values <strong>of</strong> the SRS for the<br />
model parameters.<br />
3.4.1 Basic model structure hydraulically driven <strong>system</strong>s<br />
As a resume, the basic structure <strong>of</strong> a hydraulically driven parallel <strong>motion</strong> <strong>system</strong> is given<br />
in Fig. 3.20. Through the servo valves and transmission lines, oil flows to the actuator<br />
compartments, om, can be regulated. Strong coupling with the mechanics results through<br />
actuator velocities, _q, requiring a in/decreasing amount <strong>of</strong> oil in the compartments. Together<br />
with a small leakage term, ;L, which provides for some damping, the nett flow will result<br />
in in/decreasing pressure rises, dP<br />
_<br />
o, through the oil stiffness, C. Actuator pressure times<br />
piston area, A, will force the <strong>system</strong> to accelerate. With parallel <strong>system</strong>s the actuator forces,<br />
fa, have to be transformed to platform forces though the transposed jacobian, J T l�x . Some<br />
additional damping force results from the viscous friction along the actuators, B. The acceleration<br />
<strong>of</strong> the actuators can be calculated as the nett sum <strong>of</strong> forces times the inverse mass<br />
matrix, M ;1<br />
t , considered along the platform coordinates. Coulomb, coriolis and centripetal<br />
forces are not shown in this <strong>system</strong> and neither are the gravity terms.<br />
The mass matrix as seen from the actuators consists <strong>of</strong> the <strong>simulator</strong> pose, sx, dependent<br />
jacobian, Jl�x(sx) as defined by (2.42), and an almost constant <strong>simulator</strong> mass matrix, M t,<br />
6<br />
M ;1<br />
;1<br />
act = Jl�xM t J T l�x : (3.43)<br />
This matrix determines the coupling between the actuators. Given similar hydraulic actuators<br />
in equal position, all other matrix operators can be considered as one operator times<br />
the identity matrix. So, the eigenvectors <strong>of</strong> the linearised version <strong>of</strong> this <strong>system</strong> can be calculated<br />
from the mass matrix. This still holds if decentralized feedback (diagonal feedback<br />
structure) is applied. As the mass matrix is symmetric the eigenvector matrix can be made<br />
?<br />
-1
3.4 Hydraulically driven Stewart platform 103<br />
unitary. This results from the singular value decomposition<br />
M ;1 T<br />
act = U U (3.44)<br />
With the basic model for the hydraulically driven Stewart platform, it will be shown that the<br />
matrix, U, can be used in a state transformation matrix, which decouples the 12 th order <strong>system</strong><br />
into 6 second order <strong>system</strong>s each having the same structure as the 1-d.o.f. hydraulically<br />
driven mechanical <strong>system</strong>.<br />
Considering for a moment dP o and _q as state1 , the state space equations for Fig. 3.20<br />
become<br />
dP<br />
_<br />
o<br />
q<br />
=<br />
;CL<br />
M<br />
;CA<br />
;1<br />
actA ;M ;1<br />
actB<br />
The state transition matrix can be decomposed as<br />
dP o<br />
_q<br />
+ C<br />
0<br />
om<br />
(3.45)<br />
;CL<br />
M<br />
;CA<br />
;1<br />
actA ;1<br />
;M actB =<br />
U<br />
0<br />
0<br />
U<br />
;CL<br />
A<br />
;CA<br />
; B<br />
U T<br />
0<br />
0<br />
U T : (3.46)<br />
This can be used to decouple the <strong>system</strong> using the assumption that all matrices involved,<br />
apart from the mass matrix, have a scalar times identity structure, I. From this observation,<br />
a decoupled <strong>system</strong> results if the <strong>system</strong> is described w.r.t. a new set <strong>of</strong> inputs ^ om<br />
U<br />
=<br />
T ^<br />
om and outputs ^ dP o = U T dP o, _ ^q = U T _q.<br />
If lower captions are used for the diagonal elements <strong>of</strong> the respective operators, e.g. b<br />
from B = bI, the six decoupled second order <strong>system</strong>s can be described by the state space<br />
equations<br />
"<br />
_^<br />
dP o�i<br />
^q i<br />
#<br />
= ;cl ;ca<br />
ia ; ib<br />
dP ^<br />
o�i<br />
_^q i<br />
+<br />
c<br />
0<br />
^<br />
om�i� (3.47)<br />
with i =1�::: �6. So the <strong>system</strong> can be described with six second order <strong>system</strong>s in parallel.<br />
Moreover, exciting the <strong>system</strong> with an oil flow, om, directed along one <strong>of</strong> the columns, ui,<br />
<strong>of</strong> U will only result in response <strong>of</strong> the pressure, dP o, and velocity, _q along this very same<br />
direction, ui. Therefore the following definition makes sense.<br />
Definition 3.2 Each column <strong>of</strong> the matrix, U, is defined as the rigid body modal direction<br />
<strong>of</strong> the hydraulically driven Stewart platform. The mode, eigenfrequency/ damping, <strong>of</strong><br />
each sub<strong>system</strong>, (3.47), is defined as a rigid body mode <strong>of</strong> the hydraulically driven Stewart<br />
platform.<br />
The eigenvalues <strong>of</strong> the sub<strong>system</strong>s can be calculated from the respective determinants,<br />
i = ; p ic( l p<br />
c= i +<br />
2<br />
bp<br />
i=c<br />
2<br />
ja)� (3.48)<br />
1 The most appropriate choice for the state equations in modelling e.g. in numerical simulation, explicit model<br />
<strong>based</strong> <strong>control</strong>, are <strong>of</strong> course the platform pose, sx and velocity, _x as state instead <strong>of</strong> q and its derivative and require<br />
variational actuator stiffness. The model which results from that choice will be given with equation (3.66) in<br />
Section 3.4.4
104 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
where the product <strong>of</strong> l and b is assumed to be neglectable. The eigenfrequencies can be<br />
considered the rigid modes and are mainly determined by the hydraulic stiffness, c, and<br />
the singular values <strong>of</strong> the inverse mass matrix, i, taken w.r.t the actuators. These singular<br />
values can be interpreted as the inverse <strong>of</strong> the generalised masses, which the actuators will<br />
have to accelerate moving along the six orthogonal directions described by the columns <strong>of</strong><br />
the unitary decoupling matrix, U.<br />
The characterisation with U can already be found by H<strong>of</strong>fman [51], who uses this decoupling<br />
to analyse a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>, a Stewart platform, in combination<br />
with a decentralised feedback. It has not seen to be applied after that time. However, as<br />
will be shown in the sequel, both in model analysis, identification and decoupling <strong>control</strong> it<br />
can be very appropriate, since the decoupling is seen to hold in practice even in the face <strong>of</strong><br />
moderate additional variations/ dynamics.<br />
The parallel structure makes the calculation <strong>of</strong> the directions in U simpler. It can be<br />
calculated from a ’scaled’ jacobian, Jm without inversion by assuming M ;1<br />
t does not vary<br />
and can be calculated in advance<br />
Jl�x(sx)M ;1<br />
t J T l�x (sx) = Jm(sx)J T m (sx)<br />
= U (sx) 1=2 (sx)V T (sx)V (sx) 1=2 (sx)U T (sx) (3.49)<br />
As shown in the previous sections, the stiffness <strong>of</strong> the actuators, C, also varies with<br />
the position <strong>of</strong> the actuator. With different stiffness values, the decoupling with U will<br />
not be exact anymore. However, in many practical cases, as will be analyzed by taking<br />
numerical design values in the next sections and experimentally found parameters in the next<br />
chapter, a strongly decoupled <strong>system</strong> results calculating U and only from the platform<br />
pose dependent Jacobian and a constant inverted platform mass matrix M t.<br />
In the next section, first, the specific properties <strong>of</strong> the SRS <strong>motion</strong> <strong>system</strong> will be discussed.<br />
Then, possible neglection <strong>of</strong> the variation <strong>of</strong> M t and appropriate choice <strong>of</strong> its value<br />
will be treated.<br />
3.4.2 Dynamical and kinematical properties <strong>of</strong> the SRS-<strong>motion</strong> <strong>system</strong>;<br />
Design aspects<br />
Having introduced the generic formulas to describe the mechanics and hydraulics <strong>of</strong> a (<strong>flight</strong><br />
<strong>simulator</strong>) <strong>motion</strong> <strong>system</strong>, additional insight can be gained by considering a specific design<br />
example <strong>of</strong> such a <strong>system</strong> and analyzing the Simona Research Simulator characteristics.<br />
The analysis presented in this section can be done before actually building a <strong>motion</strong> <strong>system</strong><br />
and can/ should therefore be taken into account in the process <strong>of</strong> design. The parameters for<br />
the kinematics <strong>of</strong> the SRS were already given in Table 2.1.<br />
The inertial properties <strong>of</strong> the SRS-<strong>motion</strong> <strong>system</strong> discussed here will be the values calculated<br />
in the design <strong>of</strong> a fully operational <strong>flight</strong> <strong>simulator</strong>. In that case the centre <strong>of</strong> gravity<br />
(cog) <strong>of</strong> the moving body excluding the actuators is predicted 1 m above the upper centre<br />
gimbal point (cgb). The cog will be taken as the origin <strong>of</strong> the moving reference frame. The<br />
vector from cog to cgb will thus be given by<br />
x m T<br />
cgb = 0 0 1<br />
(3.50)
3.4 Hydraulically driven Stewart platform 105<br />
It is not required to take the c.o.g. as the origin but usually specific design values, such as<br />
the mass matrix are more easily interpreted. Further, an important point in simulation, the<br />
design eye point (DEP) <strong>of</strong> the pilot is predicted very close to this point, only :2 m higher.<br />
x m T<br />
DEP = 0 0 ;0:207<br />
(3.51)<br />
As discussed, <strong>flight</strong> simulation usually requires a neutral position <strong>of</strong> the <strong>motion</strong> <strong>system</strong>. The<br />
high pass wash-out filter characteristics should keep the <strong>motion</strong> <strong>system</strong> not to far away from<br />
this point to prevent the actuators from running out <strong>of</strong> stroke. The most straight forward<br />
choice <strong>of</strong> the neutral position is at half stroke <strong>of</strong> all the actuators. The moving reference<br />
frame <strong>of</strong> the SRS is in that case 3:38 m above the lower centre gimbal point, which is taken<br />
as the origin <strong>of</strong> the inertial grounded frame (or its nominal position in case <strong>of</strong> a flexibly<br />
attached foundation).<br />
x g<br />
T<br />
neutral = 0 0 ;3:38 (3.52)<br />
Since the <strong>motion</strong> <strong>system</strong> will normally be close to this point 2 the (dynamic) characteristics<br />
are <strong>of</strong>ten specified at this point. The jacobian in neutral position, sx n, is given by:<br />
Jl�x(sxn) =<br />
2<br />
6<br />
4<br />
0:1943<br />
;0:4668<br />
0:2725<br />
0:2725<br />
;0:4668<br />
;0:4269<br />
;0:0452<br />
0:3817<br />
;0:3817<br />
0:0452<br />
;0:8832<br />
;0:8832<br />
;0:8832<br />
;0:8832<br />
;0:8832<br />
0:3386<br />
;1:2204<br />
;1:5589<br />
1:5589<br />
1:2204<br />
1:6047<br />
;1:0955<br />
;0:5091<br />
;0:5091<br />
;1:0955<br />
;0:7011<br />
0:7011<br />
;0:7011<br />
0:7011<br />
;0:7011<br />
7<br />
5<br />
0:1943 0:4269 ;0:8832 ;0:3386 1:6047 0:7011<br />
(3.53)<br />
Each column <strong>of</strong> this matrix specifies at what velocity the actuator should run in case only<br />
one <strong>of</strong> the platform states is to be moved with unit velocity ( _ l = Jl�x _x). The columns,<br />
in this specific pose, are orthogonal except for the pitch/ surge and roll/ sway direction.<br />
This strongly influences the hydromechanical modes <strong>of</strong> the <strong>system</strong> evaluated along the platform<br />
directions. The rows <strong>of</strong> the matrix specify the platform forces felt in case <strong>of</strong> a unit<br />
force <strong>of</strong> one <strong>of</strong> the actuators. These rows are not orthogonal showing strong coupling <strong>of</strong><br />
neighbouring actuators and strongly influence the mass matrix evaluated along the actuator<br />
coordinates.<br />
Recalling the discussion on the jacobian in Section 2.2.9, it is possible to calculate what<br />
can be the maximum velocity in a certain platform direction given a maximum actuator<br />
velocity, given the jacobian. If the infinity norm (max) <strong>of</strong> the actuator velocity vector is<br />
given, the 1-norm (added absolute values) <strong>of</strong> the rows <strong>of</strong> the inverse jacobian specify the<br />
maximum attainable platform velocity.<br />
k _xik1 = kJ ;1<br />
i� k1k _ l1 (3.54)<br />
Given a maximal applicable actuator force, a similar formula can be used to calculate the<br />
maximal attainable platform force (take 1-norm <strong>of</strong> each row <strong>of</strong> the jacobian). In both cases<br />
2Directing the gravitational vector properly <strong>of</strong>ten results in a certain <strong>of</strong>f set from neutral as an airplane is usually<br />
operated slightly pitched.<br />
3
106 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Acceleration limits [m/s2,100 deg/s2]<br />
10 2<br />
10 1<br />
10 0<br />
10 −1<br />
10 −1<br />
10 −2<br />
x,y z m_z m_x,m_y<br />
10 0<br />
Frequency [Hz]<br />
Fig. 3.21: Approximate accelerational limits for sinusoidal platform <strong>motion</strong> considered at<br />
the neutral position, along the platfrom directions, surge (x), sway (y), heave (z),<br />
roll (mx), pitch (my) and yaw (mz).<br />
it should be noted that the other platform variables are allowed to vary. A more appropriate<br />
bound is therefore given by the maximal attainable velocity/ force in a specific direction not<br />
moving/ forcing the others.<br />
For the velocity this bound can be found by<br />
And for the force<br />
_xi�max = _ lmax=kJ �ik1<br />
10 1<br />
(3.55)<br />
fxi�max = fa�max=kJ ;1<br />
i� k1 (3.56)<br />
The maximal actuator force and velocity are parameters, which are determined by the specific<br />
design <strong>of</strong> the hydraulic <strong>system</strong>. The maximum valve flow, max, together with the<br />
hydraulic actuator operational area, Ap, determine the maximum velocity<br />
_qmax = max=Ap =SRS 1 m=s (3.57)<br />
The maximal applicable actuator force can be found by multiplication <strong>of</strong> the supply pressure,<br />
Ps, with this area, Ap,<br />
fa�max = PsAp =SRS 40 kN (3.58)<br />
The operational area is clearly a trade-<strong>of</strong>f parameter in design for velocity vs. force given a<br />
specific available amount <strong>of</strong> energy. There are, however, also other constraints in choosing<br />
this area e.g. radial stiffness, and <strong>of</strong> course also the stroke <strong>of</strong> the actuator limits <strong>motion</strong>.<br />
A first estimate <strong>of</strong> this last limit can be found by using (3.55) from which the jacobian to
3.4 Hydraulically driven Stewart platform 107<br />
Pos. displacement x y z x y z<br />
[m� deg] 1.34 1.10 0.74 22.3 20.9 45.2<br />
Neg. displacement x y z x y z<br />
[m� deg] -1.05 -1.10 -0.69 -22.3 -30.0 -45.2<br />
Est. displacement (1 it.) x y z x y z<br />
[m� deg] 1.34 1.46 0.71 23.1 22.5 52.9<br />
Max. velocities _x _y _z !x !y !z<br />
[m=s� deg=s] 2.14 2.34 1.13 36.8 35.7 81.7<br />
Max. force fx fy fz mx my mz<br />
[kN,kNm] 47.7 46.3 212.0 159.1 174.1 168.1<br />
Max. acceleration x y z _!x _!y _!z<br />
[m=s 2 �kdeg=s 2 ] 13.5 12.9 55.4 1.39 1.24 1.01<br />
Freq. <strong>of</strong> pos. to vel. constr. x y z x y z<br />
[Hz] 0.32 0.39 0.29 0.31 0.32 0.35<br />
Freq. <strong>of</strong> vel. to acc. constr. x y z x y z<br />
[Hz] 0.99 0.87 7.71 5.85 5.56 2.01<br />
Table 3.6: SRS-properties considered at neutral position and c.o.g.<br />
euler parameter velocities for the rotational variation can be calculated. Iteratively using this<br />
formula (Newton-Raphson) and lmin�max, the least available stroke, a precise limit can<br />
be found very fast as opposed to a bilinear search. This Newton-Raphson iteration usually<br />
requires three or four steps, ending up far below measurable error in a well designed <strong>system</strong>.<br />
With a given mass matrix, it is also possible to calculate the maximal attainable acceleration.<br />
The specific mass matrix (design values) <strong>of</strong> the SRS is discussed in the next section. All<br />
these values calculated for the SRS in the neutral position are specified in Table 3.6.<br />
Given sinusoidal inputs, the velocity and positional constraints limit the maximal attainable<br />
acceleration for lower frequencies. For the 1-d.o.f. hydraulic actuator, Viersma [153]<br />
specifies exact performance limit diagrams in the velocity domain. In Fig. 3.21 approximate<br />
performance diagrams for the 6-d.o.f. <strong>motion</strong> <strong>system</strong> are given in the platform coordinates<br />
using the limits specified in Table 3.6. The precision <strong>of</strong> the approximate velocity and acceleration<br />
constraints for the sinusoidal <strong>motion</strong> is favourably influenced since the maximum<br />
velocity <strong>of</strong> the sinusoid is attained at neutral position where the velocity constraint is exact<br />
and the maximal acceleration is attained (with limited positional <strong>of</strong>fset at these high frequencies)<br />
at zero velocity with no centripetal and coriolis accelerations. Gravity, however,<br />
results in a more serious <strong>of</strong>fset to be taken into account. Given the force which needs to<br />
be applied to compensate gravity, the least available maximal force required to move in a<br />
specific direction determines the maximal attainable acceleration.<br />
The frequencies where the positional constraint, p max, and velocity constraint, vmax,<br />
meet, fpv, and velocity and accelerational constraint meet, f av, are given by fpv = vmax<br />
2 pmax<br />
and fpv = amax<br />
2 vmax .Afirst estimate <strong>of</strong> fpv is equal for all platform directions and does only<br />
depend on the actuator specifications: ~ fpv = vact�max<br />
2 pact�max =SRS<br />
1<br />
2 0:625<br />
=0:25Hz. From
108 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
−z [m]<br />
3.5<br />
3<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
0<br />
Upper Gimbal<br />
Lower Gimbal 1 Lower Gimbal 2 (6)<br />
−1.5 −1 −0.5 0<br />
−y [m]<br />
0.5 1 1.5<br />
Fig. 3.22: Envelope and largest circle within this envelope <strong>of</strong> a 2-d.o.f. parallel planar manipulator.<br />
Parameters taken as for actuator 1 and 6 <strong>of</strong> the SRS except for merging<br />
<strong>of</strong> the upper gimbals.<br />
Fig. 3.21 it can be observed that the actual constraint will be attained at a frequency which<br />
is somewhat higher. It is interesting to observe that already from :3 Hz and higher it is not<br />
so much the finite stroke <strong>of</strong> the actuators which limits the (<strong>flight</strong>) simulation but the velocity<br />
constraint. For these frequencies it is <strong>of</strong> no use to build a larger <strong>motion</strong> <strong>system</strong>. One should<br />
first choose a larger valve.<br />
Only looking at heave, roll and pitch, the operational area <strong>of</strong> the actuators could have<br />
been chosen somewhat smaller as velocity is the main constraint for these directions. For<br />
surge and sway, however, the force constraint comes in at much lower frequencies. In this<br />
respect the jacobian (kinematical structure) <strong>of</strong> the Stewart platform is also a trade-<strong>of</strong>f design<br />
parameter where e.g. with the SRS neutral position high surge velocity can be attained<br />
(twice as high as the maximal actuator velocity since the actuators are mainly rotating)<br />
compromised with limited applicable force in this direction.<br />
The specific values given in Table 3.6 should be treated with extreme care since they<br />
can vary considerably over the <strong>motion</strong> envelope. Especially the <strong>motion</strong> envelope itself has<br />
a complex non convex structure. Consider a simplified (planar) <strong>system</strong> with two actuators<br />
having limited stroke manipulating a point. In Fig. 3.22 this structure is drawn with the<br />
lower gimbals the same distance (2:5 m) apart as actuator 1 (2,4) and 6 (3,5) in the SRS<br />
and the same constraints on the minimal and maximal length (2:08 m,3:33 m). The upper<br />
gimbals have been merged (are :2 m apart in the SRS). Choosing the neutral point somewhat<br />
lower in this construction, would lead to a considerable growth <strong>of</strong> the maximal attainable<br />
sway moving straight away from this point. But the constraint towards the attainable heave<br />
would become even more asymmetric in doing so. Thus changing the neutral point in this<br />
way will probably not be a appropriate decision.<br />
Appropriately describing the <strong>motion</strong> envelope <strong>of</strong> the general 6-d.o.f. Stewart platform
3.4 Hydraulically driven Stewart platform 109<br />
in task space is still an open problem [27]. The inverse problem <strong>of</strong> checking whether a<br />
specific task space envelope fits in a specific kinematical structure can be interesting for<br />
(<strong>flight</strong>) simulation and can sometimes be solved. Consider e.g. a normed task space for the<br />
2-d.o.f manipulator <strong>of</strong> Fig. 3.22. With the 2-norm, ( p x T x), this space describes a ball and<br />
the maximal radius (norm) <strong>of</strong> the circle can be found in the half stroke neutral position and<br />
is exactly equal to this half stroke (0:625 m).<br />
With the general Stewart platform, exactly the same bound can be found if the platform<br />
is not allowed to rotate since the upper gimbals in that case are also only allowed to move in<br />
this same ball. So the maximal ball <strong>of</strong> translational <strong>motion</strong> <strong>of</strong> the SRS which is applicable<br />
can be found in the all actuator half stroke neutral position and has radius r =0:625 m. For<br />
any ball larger than this, translations can be found which run (some <strong>of</strong>) the actuators out <strong>of</strong><br />
stroke.<br />
In case <strong>of</strong> orientational variations together with translation, a radius can be found which<br />
satisfies the constraints but is not necessarily maximal. Recall the <strong>motion</strong> <strong>of</strong> any upper<br />
gimbal<br />
which can be bounded by<br />
xa�i(sx) =c + T ( 13)ai<br />
kxak2<br />
(3.59)<br />
p 2ksxk (3.60)<br />
In case <strong>of</strong> the SRS kak = 1:6m when using the centre upper gimbal point as origin (and<br />
gets larger/worse in using DEP or cog). So in case <strong>of</strong> the SRS r = :44m and this means a<br />
maximal rotation <strong>of</strong> =5:7 . It would be interesting to know whether it is possible to use<br />
more structure concerned with the interaction between rotation and translation in order to<br />
arrive at possibly less conservative results.<br />
In some cases it is favourable to define a (hyper) ellipsoid instead <strong>of</strong> a ball in which<br />
the task space states are allowed to move. E.g. in case <strong>of</strong> Fig. 3.22 it would be appropriate<br />
to choose a larger allowable variation in y-direction. Also this problem is in some cases<br />
tractable in optimising a norm given quadratic constraints, which can be transformed to<br />
linear matrix inequalities. The problem has to made convex, however. This can be attained<br />
by replacing the constraints w.r.t. the minimal actuator lengths by convex approximations<br />
in the task space e.g. a planes in the tangent space <strong>of</strong> the l min balls. The ellipsoidal form is<br />
attained by multiplying xa by a (positive definite) scaling matrix S.<br />
E.g. maximize =x T x such that<br />
(Sx ; b) T (Sx ; b) 0 8kxk (3.62)<br />
By use <strong>of</strong> the S-procedure [18], the ’uncertainty’ ball <strong>of</strong> x’s can be removed from the equations,<br />
leaving a tractable problem.
110 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Parameter Description Value Unit<br />
ma Mass upper part actuator (piston) 120 [kg]<br />
mb Mass lower part actuator (cylinder) 135 [kg]<br />
ra Distance upper gimbal to cog piston 0:7 [m]<br />
rb Distance lower gimbal to cog cylinder 0:5 [m]<br />
ia Inertia upper part actuator wrt cog 20 [kgm 2 ]<br />
ib Inertia lower part actuator wrt lower gimbal 36 + mbr 2<br />
b<br />
Table 3.7: Inertial parameters SRS-actuators.<br />
[kgm 2 ]<br />
Limitations <strong>of</strong> a hydraulically driven <strong>motion</strong> <strong>system</strong> with specific kinematical and dynamical<br />
properties have been quantified in this section. This knowledge can be used in<br />
design <strong>of</strong> the <strong>system</strong> itself but also in the design <strong>of</strong> experiments in which it is <strong>of</strong>ten important<br />
not to approach such limits to closely.<br />
3.4.3 Actuator inertial effects<br />
In model <strong>based</strong> <strong>control</strong> choice <strong>of</strong> an appropriate model is important. The model has to<br />
describe the <strong>control</strong> relevant aspects <strong>of</strong> the <strong>system</strong>. On the other hand it should not be too<br />
complex since it has to be calculated on a real-time <strong>control</strong> computer. W.r.t. modelling, the<br />
question arises whether the mass properties <strong>of</strong> the actuators have to be taken into account.<br />
And if so, how this should be done in a convenient way.<br />
With the currently used <strong>simulator</strong> <strong>motion</strong> platforms the weight <strong>of</strong> the actuators is relatively<br />
low (ca. 250 kg) w.r.t. the platform (up to 12:5 tons). In these cases the weights do<br />
not contribute significantly to the <strong>motion</strong> <strong>of</strong> the total <strong>system</strong>. As the designed weight <strong>of</strong> the<br />
<strong>simulator</strong> was reduced drastically (to ca. 3:2 tons) with the Simona research <strong>simulator</strong>, one<br />
has to reevaluate this assumption.<br />
Let’s look into the mass contribution <strong>of</strong> the <strong>system</strong>. The one body <strong>simulator</strong> has according<br />
to the design (finite element modelling) roughly the following mass matrix, M c,<br />
in platform coordinates w.r.t the c.o.g. which is calculated 1 meter above the centre upper<br />
gimbal point.<br />
Mc =<br />
2<br />
6<br />
4<br />
3200 0 0 0 0 0<br />
0 3200 0 0 0 0<br />
0 0 3200 0 0 0<br />
0 0 0 7000 0 500<br />
0 0 0 0 7000 0<br />
0 0 0 500 0 8000<br />
3<br />
7<br />
5<br />
(3.63)<br />
The inertia <strong>of</strong> the <strong>simulator</strong> is somewhat higher than the weight but this does not necessarily<br />
mean that the impact is higher since the actuators are connected on a circle with radius<br />
larger than 1m. The main axes <strong>of</strong> inertia do not exactly correspond to the platform body<br />
axes. The y-axis does, as the <strong>system</strong> can be mirrored w.r.t the xz-plane. The projectors on<br />
the upper back <strong>of</strong> the <strong>simulator</strong> are the main reason for the xz cross term.
3.4 Hydraulically driven Stewart platform 111<br />
Case/Actuator 1 2 3 4 5 6<br />
Neutral position lmid lmid lmid lmid lmid lmid<br />
Extreme pitch/surge lmin lmin lmax lmax lmin lmin<br />
Extreme twist (yaw) lmax lmin lmax lmin lmax lmin<br />
Extreme roll/sway lmid lmin lmin lmax lmax lmid<br />
Table 3.8: Actuator lengths for platform poses considered.<br />
Case/Platform pose x y z 1 2 3<br />
Neutral position 0 0 -3.38 0 0 0<br />
Extreme pitch/surge 2.00 0 -2.43 0 -0.32 0<br />
Extreme twist (yaw) 0 0 -3.02 0 0 -0.52<br />
Extreme roll/sway 0.14 0.41 -3.25 0.20 -0.01 0<br />
Table 3.9: Platform poses considered.<br />
Recalling the additive mass matrix term, Ma�n, in (2.123), given by<br />
Ma�n =<br />
6X<br />
i=1<br />
J T ai�x (Mma�i + Mia�i�ib�i )Jai�x� (3.64)<br />
which consists <strong>of</strong> the contribution <strong>of</strong> all six actuators. In neutral position it becomes the following,<br />
using the relations determining (2.122), and the inertial parameters for the actuator<br />
given in Table 3.7.<br />
Ma�n =<br />
2<br />
6<br />
4<br />
500 0 0 0 ;525 0<br />
0 500 0 525 0 0<br />
0 0 675 0 0 0<br />
0 525 0 1400 0 0<br />
;525 0 0 0 1400 0<br />
0 0 0 0 0 1325<br />
3<br />
7<br />
5<br />
(3.65)<br />
The actuators contribute considerably to the total mass matrix (about 15 %). The terms in<br />
the upper right and lower left block contribute to a lower cog <strong>of</strong> approx. .1 m. Remarkable<br />
is the fact that the weight experienced in moving in z-direction will be higher than in x<br />
or y-direction, due to the actuators. The actuator mass matrix will vary as a function <strong>of</strong><br />
platform position since the actuators move w.r.t. the platform. These variations, however,<br />
are rather small. By taking a constant mass matrix in platform coordinates into the model<br />
<strong>based</strong> <strong>control</strong>ler, a relatively simple but accurate choice is made.<br />
If a Stewart Platform with even less relative weight is used, the full model should be<br />
taken into account. Apart from extra calculational effort this does not result in additional<br />
problems if the modelling steps described, are taken.<br />
With respect to the nonlinear quadratic velocity terms in the model describing the coriolis<br />
and centripetal terms, the constraints on the velocities are important. The most restrictive
112 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
10 2<br />
10 0<br />
−500<br />
−1000<br />
10 2<br />
10 0<br />
0<br />
100<br />
0<br />
−100<br />
10 0<br />
10 2<br />
10 0<br />
10 2<br />
Fig. 3.23: Bode plots from input valve voltage (u) to pressure difference (dP i) <strong>of</strong> linearised<br />
multivariable hydraulically driven mechanical <strong>system</strong> models. Masses: Simona<br />
design values. Rows 1,2 gives response <strong>of</strong> input 1 to outputs 1 (full), 2-6 (dotted).<br />
Rows 3,4 give responses <strong>of</strong> the diagonal elements along the decoupled rigid body<br />
modal directions (1,1:full), (2,2-6,6:dotted). Column 1: mid position, Column<br />
2: extreme pitch/ surge, Column 3: extreme twist, Column 4: extreme roll/<br />
sway. X-axis: frequency (Hz).<br />
constraint, which holds for both pitch and roll <strong>of</strong> 35 =s =0:61rad=s in the neutral position,<br />
was taken as the rotational velocity bound in calculating the influence <strong>of</strong> these nonlinear<br />
terms in [119] and it was found that these only contribute up to ca. 500 N for the SRS.<br />
3.4.4 Analyzing the SRS hydraulically driven <strong>motion</strong> <strong>system</strong> model<br />
Having introduced the specific parameters <strong>of</strong> the SRS, the SRS-model can be analyzed considering<br />
the interaction between the hydraulics and mechanics in this 6-d.o.f. <strong>system</strong>. It will<br />
be evaluated whether the hydromechanical rigid body modes describe the main characteristics<br />
<strong>of</strong> the <strong>system</strong>. Along the rigid body modal directions, it is expected that the <strong>system</strong><br />
locally behaves as six independent 1-d.o.f. hydraulic actuators moving a mass. This has to<br />
be checked also as is the predictability <strong>of</strong> these directions. The influence <strong>of</strong> the main mechanical<br />
nonlinearity, the platform pose dependent variation <strong>of</strong> the jacobian, J lx, has to be<br />
quantified.<br />
The state equations, constructed from Fig. 3.20, describing the nonlinear dynamical<br />
10 0<br />
10 2<br />
10 0<br />
10 2
3.4 Hydraulically driven Stewart platform 113<br />
frb 1 2 3 4 5 6<br />
1: (y� x) (x� - y) ( z) ( y� -x) ( x�y) (z)<br />
[Hz] 4.5 4.5 7.6 13.3 13.5 14.8<br />
2: (x� z� - y) (y� x� - z) ( z� - x) ( y� -x� z) ( x�y) (z)<br />
[Hz] 4.1 5.8 9.1 15.5 16.1 17.4<br />
3: ( z� -z ) (y� x� y) (x� - y� x) ( y�x) ( x� z� -y) (z� z� y)<br />
[Hz] 3.6 4.5 4.5 16.1 16.3 17.2<br />
4: (x� -y� - y) (y� x� x� - y) ( z) ( x�z�-y) ( y�x) (z� - x�x)<br />
[Hz] 4.9 5.4 7.7 12.0 14.3 18.0<br />
Table 3.10: <strong>Model</strong> eigenfrequencies, frb, in Hz <strong>of</strong> the rigid body modes and roughly the<br />
rigid body modal directions. 1: Neutral position, 2: Extreme pitch/ surge, 3:<br />
Extreme twist (yaw), 4: Extreme roll/ sway. i is the rotation around the i th<br />
axis.<br />
model from the output oil flows, om, are given by<br />
x<br />
_<br />
dP o<br />
;bI aI<br />
;aI ;lI<br />
=<br />
;1<br />
M<br />
t J T lx (sx) 0<br />
Jlx(sx) 0<br />
0 I<br />
:::+ M ;1<br />
t<br />
0 Ch(sx)<br />
_x<br />
dP o<br />
+ 0<br />
I<br />
G(sx) +C(sx� _x)<br />
0<br />
::: (3.66)<br />
om + :::<br />
In this case it is assumed that there are similar actuators (operational area, a, viscous friction,<br />
b and leakage, l) easily replaced by the respective matrices, A� B� L otherwise. By taking<br />
the platform pose, sx� as state variable, the model is valid for singular poses as well. Only<br />
if actuators would reach zero length, the jacobian is not defined.<br />
The two-sided coupling with all ( om,dPo) pairs to the transmission lines and valves can<br />
readily be made. For a reduced but fairly good model, each transmission line has four states<br />
and each valve has two. Further, the six platform pose coordinates, sx, have to be calculated<br />
through integration <strong>of</strong> the velocities. The model thus consists <strong>of</strong> 12 + 6(2 + 4) + 6 = 54<br />
states.<br />
This model with the SRS-parameters has been analyzed for several poses. Together<br />
with the neutral position, three extreme poses were taken into account. Actuator lengths<br />
are given in Table 3.8. The extreme pitch/ surge and twist only consist <strong>of</strong> extreme actuator<br />
positions. As the oil stiffness varies parabolically with the position, the extreme roll/ sway<br />
pose is important since it contains actuators with maximal (extreme position) and minimal<br />
(mid position) stiffness.<br />
The platform poses, which are given in Table 3.9, do result from these actuator lengths.<br />
The euler parameters can be translated to angles by taking 12 per :1 up to :6 with 1
114 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Case/mode 1 2 3 4 5 6<br />
Neutral position 8.8 8.8 3.0 1.0 1.0 0.8<br />
Extreme pitch/surge 14.8 7.4 3.0 1.0 0.9 0.7<br />
Extreme twist (yaw) 19.0 12.2 12.2 0.9 0.9 0.8<br />
Extreme roll/sway 9.0 7.8 3.6 1.8 1.0 0.6<br />
Table 3.11: Generalised masses (tons). The large differences in (generalised) mass easily<br />
result in interaction. E.g. consider a force along the 8.8 ton direction in neutral<br />
position, which is only 1 misaligned into the 0.8 ton direction. This will result<br />
in 20% interaction. In more extreme positions e.g. pitch/ surge this will even<br />
amount to more than 40%!<br />
accuracy. In the first two rows <strong>of</strong> Fig. 3.23 the Bode plots <strong>of</strong> the linearised models in these<br />
positions are given from the first valve input voltage to the various pressures, dP i, taken<br />
over the valves. The hydromechanical modes are visible and are highly interacting in the<br />
frequency area between 1 Hz and 30 Hz. The phase remains within 90 for the (1� 1)<br />
transfer function up till the point where the valve itself starts to be relevant (> 70 Hz).<br />
Above 30 Hz the measurable interaction strongly fades as the nondiagonal terms lack zeros<br />
in the transfer functions, which can be interpreted as springs in between input/ cause and<br />
output/ point <strong>of</strong> measurement.<br />
In the last two rows the transfer functions have been considered from the other inputs and<br />
outputs (only diagonal terms) resulting from the approximate decoupling matrix U defined<br />
in (3.44). This matrix was calculated from the approximate mass matrix (not taking into<br />
account the variation due to the specific inertial properties <strong>of</strong> the actuators) and does not<br />
take into account the varying actuator stiffness. Still the modes in the <strong>system</strong> can clearly<br />
be distinguished. In the hydromechanical frequency interaction area, only one mode per<br />
transfer function results in a single shift in phase <strong>of</strong> 180 . All the nondiagonal transfer<br />
functions in these coordinates are small over the full frequency area considered.<br />
Considering stability from a <strong>control</strong> point <strong>of</strong> view, the connection <strong>of</strong> a hydraulic circuit<br />
still enables a simple passivity preserving dissipating feedback from pressure/ force supplied<br />
to the mechanical <strong>system</strong> to the oil flow through the valve instead <strong>of</strong> velocity feedback to<br />
force in a mechanical <strong>system</strong>.<br />
The eigenfrequencies related to the hydromechanical rigid body modes <strong>of</strong> the <strong>system</strong> are<br />
given in Table 3.10. There is considerable variation for the different rigid body modes both<br />
for a specific pose as for the different poses considered. As each extreme position considered<br />
was only given a specific variation <strong>of</strong> the platform pose w.r.t the neutral position, the rigid<br />
body modal directions in platform coordinates, given by J ;1<br />
lx U, can still be interpreted.<br />
E.g. in extreme pitch/ surge, symmetric involving x� z� y and asymmetric modes involving<br />
y� x� z. If all platform coordinates are varied this structure will be lost.<br />
Looking at the Bode plots in Fig. 3.23, it can be observed that building up a unit pressure<br />
in a specific direction will take very different input values (more than a factor 10 variation).<br />
Very slight misalignment (wrong input) will easily result in large coupling. This is roughly<br />
linear with the generalised masses, which affect the eigenfrequencies through a square root.
3.4 Hydraulically driven Stewart platform 115<br />
Case/actuator 1 2 3 4 5 6<br />
Neutral position 18 18 18 18 18 18<br />
Retraction gain -1.08 -1.08 -1.08 -1.08 -1.08 -1.08<br />
Extraction gain +0.90 +0.90 +0.90 +0.90 +0.90 +0.90<br />
Extreme pitch/surge 26 67 -42 -42 67 26<br />
Retraction gain -1.12 -0.57 -1.19 -1.19 -0.57 -1.12<br />
Extraction gain +0.86 +1.29 +0.76 +0.76 +1.29 +0.86<br />
Extreme twist (yaw) -24 48 -24 -24 48 -24<br />
Retraction gain -0.87 -1.21 -0.87 -1.21 -0.87 -1.21<br />
Extraction gain +1.11 +0.72 +1.11 +0.72 +1.11 +0.72<br />
Extreme roll/sway 21 32 25 6 11 18<br />
Retraction gain -1.10 -1.15 -1.12 -1.03 -1.05 -1.09<br />
Extraction gain +0.89 +0.83 +0.86 +0.97 +0.95 +0.90<br />
Table 3.12: Static load as percentage <strong>of</strong> maximal load F max = PsA and the resulting normalized<br />
valve gain discontinuity.<br />
The generalised masses, 1= i from (3.44), are given in Table 3.11.<br />
Already in the neutral position, the SRS is expected to have very different characteristics<br />
along the rigid body modal directions. In the sway/ roll and surge/ neg. pitch directions the<br />
<strong>system</strong> will experience approx. 9 tons where the sway/ neg. roll and surge/ pitch directions<br />
only actuator forces to accelerate 1 ton <strong>of</strong> load have to be applied. In the extreme positions<br />
the largest mass ratio (mass matrix conditioning) along the different modes grows from nine<br />
to more than twentyfour in extreme twist. Although the jacobian, J lx, has a main influence<br />
on this variation, its conditioning is not a very well predictor for this factor. It grows from<br />
4:5 to 5:2 going from the neutral position to extreme twist.<br />
With the SRS design parameters, gravity requires approx. 18 % <strong>of</strong> supply pressure in<br />
the neutral position. The platform contributes 15 % and the moving parts <strong>of</strong> the actuators<br />
3 %. The contribution <strong>of</strong> the lower parts is almost neglectable. This percentages can rise<br />
extensively in other platform poses. In the extreme surge/ pitch position considered, actuator<br />
2 and 5 will have to carry two third <strong>of</strong> the theoretically maximal weight. In practice, two<br />
third is relatively high to take care <strong>of</strong> the static forces only. This means that the SRS design<br />
weight and height <strong>of</strong> the c.o.g. are in fact maximal for this <strong>motion</strong> <strong>system</strong> to operate safely.<br />
Another important aspect <strong>of</strong> the static loads on the actuators is the nonlinear characteristic<br />
<strong>of</strong> the valve as given in (3.16). A preload results in a discontinuous valve voltage to flow<br />
gain as im = u p (1 ; sgn(u)dPi) for normalized variables. In Table 3.12 the required<br />
preload for the four platform poses considered is given with the resulting valve gains for<br />
actuator extension and retraction. Negative percentages for the load mean that retraction<br />
preload has to be applied to statically balance the platform. The all actuator minimal length<br />
position is not the global minimum for the potential gravitational energy. As a result, the<br />
<strong>system</strong> will not always go to this pose if for some reason the preload disappears e.g. pump<br />
supply pressure drop to tank pressure in shut down.<br />
All together, the hydraulically driven Stewart platform model can be transformed to six
116 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
Direction Mfii [tons� tons m 2 ] !fi [Hz] fi<br />
xf 45 2 7 0:05<br />
yf 45 2 7:5 0:05<br />
!f 80 2 6 0:10<br />
Table 3.13: Parameters taken for the foundation.<br />
independent 1-d.o.f. hydraulically driven dynamic and static masses for each pose using<br />
the variable unitary decoupling matrix, U (sx), describing the rigid body modal directions.<br />
Analysis <strong>of</strong> the SRS <strong>motion</strong> <strong>system</strong> design showed considerable differences in the rigid<br />
body eigenfrequencies among these 1 d.o.f.-<strong>system</strong>s for the SRS.<br />
3.4.5 Connection <strong>of</strong> a flexible foundation to the SRS <strong>system</strong> model<br />
The hydraulically driven Stewart platform model <strong>of</strong> (3.66) can be extended with a model<br />
including parasitic dynamics such as a flexible foundation. This model is used to predict<br />
in what way the dynamics <strong>of</strong> the <strong>system</strong> is influenced by such distortion. A model <strong>of</strong> a<br />
flexible foundation was introduced in Section 2.4.7 described by (2.126). Combination <strong>of</strong><br />
these models leads to the following state equation<br />
22<br />
6<br />
6<br />
4<br />
6<br />
4<br />
2<br />
6<br />
4<br />
_xf<br />
xf<br />
x<br />
_<br />
dP o<br />
3<br />
7<br />
5 =<br />
2<br />
6<br />
4<br />
I 0 0 0<br />
0 M ;1<br />
f 0 0<br />
0 0 M ;1<br />
t 0<br />
0 0 0 Ch<br />
0 I 0 0<br />
Cf ;Bf ; J T f BJf ;J T f BJlx J T f A<br />
0 ;J T lx BJf ;J T lx BJlx J T lx A<br />
0 ;AJf ;AJlx ;L<br />
32<br />
7<br />
6<br />
5<br />
6<br />
4<br />
3<br />
7<br />
5<br />
xf<br />
_xf<br />
_x<br />
dP o<br />
::: (3.67)<br />
3<br />
7<br />
5 +<br />
2<br />
6<br />
4<br />
0<br />
0<br />
0<br />
I<br />
3<br />
7<br />
5 om<br />
The foundation is allowed to move in the horizontal plane and introduces three additional<br />
modes, foundation surge, xf ,sway,yf , and yaw, !f , described by six states (xf ,<br />
_xf ). As an example, the parameters given in Table 3.13, are filled into the equations. These<br />
describe roughly the characteristics <strong>of</strong> the foundation on which the <strong>system</strong> was tested in the<br />
central workshop. The mass matrix <strong>of</strong> the foundation has numbers on the diagonal, which<br />
are ten times higher than those <strong>of</strong> the platform. The stiffness matrix and damping matrix<br />
are chosen such that the undamped eigenfrequencies and damping factors specified for the<br />
foundation alone result. The high mass <strong>of</strong> the foundation in relation to the platform lets<br />
these frequencies only slightly altered in connecting the <strong>motion</strong> <strong>system</strong>.<br />
In Fig. 3.24 <strong>of</strong> Appendix A, the <strong>system</strong> is evaluated in the frequency domain from 1 Hz<br />
to 50 Hz. The upper left bode amplitude plots shows the model actuator pressure difference<br />
responses to the oil flow into the first actuator (not all 36 responses as the other 30 are<br />
3<br />
7<br />
5
3.4 Hydraulically driven Stewart platform 117<br />
10 1<br />
10 0<br />
10 −1<br />
10 1<br />
10 0<br />
10 −1<br />
10 0<br />
10 0<br />
10 1<br />
10 1<br />
Frequency (Hz)<br />
10 1<br />
10 0<br />
10 −1<br />
10 1<br />
10 0<br />
10 −1<br />
10 0<br />
10 0<br />
10 1<br />
10 1<br />
Frequency (Hz)<br />
Fig. 3.24: Bode amplitude plots <strong>of</strong> linearised multivariable hydraulically driven mechanical<br />
<strong>system</strong> models without transmission lines but with flexible foundation. Upper<br />
left: neutral position from first input actuator flow om1 to all output pressures<br />
(dP o). Upper right: neutral position along the (original) rigid body modal direc-<br />
tions from all ^ om to ^<br />
dP o, nondiagonal terms dashdotted. Lower left: neutral<br />
position along the (original) rigid body modal directions from all ^ om to the<br />
accelerations (m=s 2 )(xf � yf� _!f ) <strong>of</strong> the foundation, nondiagonal terms dashed.<br />
Lower right: extreme pitch/ surge along the (original) rigid body modal directions<br />
from all ^ om to ^ dP o, nondiagonal terms dashdotted.<br />
similar) in the neutral position. The complex dynamics <strong>of</strong> the hydromechanical modes<br />
prevents one from recognising the parasitic dynamics. Using the decoupling transformation<br />
U calculated from the platform mass matrix the bode amplitude plots in the upper right part<br />
<strong>of</strong> the figure results. In the diagonal responses the fact that there are parasitic modes is<br />
directly visible. Three additional modes show up separately along the eigenvectors <strong>of</strong> the<br />
lowest three original platform modes. Further, the nondiagonal terms are not zero anymore<br />
though still quite small.<br />
As discussed in Section 3.3.3, both a flexible foundation as a mass/ spring/ damper<strong>system</strong><br />
connected to the platform result in similar characteristics evaluated at (combinations<br />
<strong>of</strong>) actuator coordinates. To evaluate whether the foundation is causing parasitic dynamics,<br />
one can measure accelerations at the foundation itself. The model predicts the acceleration<br />
frequency response at the neutral position given in the lower left part <strong>of</strong> Fig. 3.24 <strong>of</strong> Appendix<br />
A. This was calculated by multiplying the velocity frequency response <strong>of</strong> the model<br />
by j!. Remarkable is the fact that the peaks in the response, which are most prominent, are
118 3 Hydraulically driven <strong>motion</strong> <strong>system</strong>s<br />
not in all cases mainly the original modes <strong>of</strong> the foundation. Input along the higher platform<br />
eigenfrequencies are expected to result in relatively high (approx. 15 Hz) frequency vibrations<br />
and <strong>of</strong> the three main transfer functions the yaw platform direction to yaw foundation<br />
modal response peaks are maximal at approx. the original platform yaw mode.<br />
The fact that the parasitic modes influence only three platform modes is not generally<br />
true. In the lower right part <strong>of</strong> Fig. 3.24, the <strong>system</strong> is evaluated again along the original<br />
platform eigendirections. But now the platform has been put into the extreme surge/ pitch<br />
mode. The fact that the platforms c.o.g. is now approx. 2 m in front <strong>of</strong> the c.o.g. <strong>of</strong> the<br />
foundation makes the sway mode <strong>of</strong> the platform interact with both the foundation yaw<br />
and sway mode and through the foundation yaw mode with the platform yaw mode. The<br />
response along this direction shows four resonances. Also the decoupling is almost completely<br />
lost between 4 Hz and 15 Hz.<br />
So, parasitic dynamics such as a flexible foundation can have an important influence<br />
on the <strong>motion</strong> <strong>system</strong> characteristics. This is not always directly visible in the measurable<br />
responses (inputs and outputs <strong>of</strong> the actuators).<br />
3.5 Chapter Resume<br />
The <strong>motion</strong> <strong>system</strong>s rigid bodies interacting with the hydraulic actuator stiffnesses, the<br />
hydromechanical rigid body modes, describe the main characteristics <strong>of</strong> the hydraulically<br />
driven <strong>motion</strong> <strong>system</strong> dynamics. The eigendirections <strong>of</strong> these modes can be quite accurately<br />
predicted by taking the eigenvectors <strong>of</strong> the approximate actuator mass matrix consisting <strong>of</strong><br />
the variable jacobian, Jlx(sx), and the platform mass matrix. Along the rigid body modal<br />
directions the <strong>system</strong> locally behaves as six independent 1-d.o.f. hydraulic actuators moving<br />
a mass in every platform pose.<br />
Under mild conditions, these <strong>system</strong>s are passive evaluating pairs <strong>of</strong> the oil flow and<br />
pressure difference at the valve or at the actuator <strong>of</strong> the hydraulic servos or (collocated)<br />
combinations <strong>of</strong> these pairs, no matter what parasitic flexible mechanical <strong>system</strong>s are to be<br />
included. Any passive feedback over these pairs will result in stability <strong>of</strong> the full nonlinear<br />
<strong>system</strong>.<br />
The valve dynamics causes loss <strong>of</strong> passivity mostly relevant in combination with the<br />
high frequency (e.g. transmission line) modes. The transmission lines, though mainly resulting<br />
in high frequency dynamics, cause a considerable shift <strong>of</strong> the hydromechanical rigid<br />
body modes and introduce large differences at relative low frequencies (10 times smaller<br />
than the transmission line modes) between the measurable pressure at the valve and the<br />
equivalent force (pressure difference times operational area) the actuator induces into the<br />
<strong>motion</strong> <strong>system</strong>.<br />
Parasatic dynamics such as a flexible foundation or an additional mass/ spring/ damper<br />
<strong>system</strong> connected to the moving platform is easily included in the models introduced. This<br />
dynamics can have considerable influence on the <strong>system</strong>s characteristics and is most easily<br />
recognized evaluating along the hydromechanical rigid body modes. Only measuring at the<br />
actuators, the cause <strong>of</strong> the parasitic dynamics is not identifiable.
Chapter 4<br />
Parameter identification and<br />
model validation<br />
The models presented in the two previous sections were derived from the laws <strong>of</strong> physics.<br />
Where numerical examples were given, the parameter values were taken from design values<br />
or rough approximations <strong>of</strong> what could be expected in practice. These models can be<br />
valuable for the structural analysis <strong>of</strong> (parallel, hydraulically driven) <strong>motion</strong> <strong>system</strong>s. The<br />
strengths <strong>of</strong> these models should, however, be proven by experimental validation. Proper<br />
choice <strong>of</strong> model parameter values is required in using the models to predict the characteristics<br />
<strong>of</strong> the actual <strong>system</strong> and enable direct comparison <strong>of</strong> model and experimental response.<br />
Usually, the parameter choice is also to be <strong>based</strong> on experiments. Experiments from<br />
which the parameters can be identified. Small differences between model and practice,<br />
i.e. a well validated model structure with proper parameter values is also important in the<br />
use <strong>of</strong> a model <strong>based</strong> <strong>motion</strong> <strong>control</strong>ler.<br />
Identification can also be performed with more general model structures which are especially<br />
suited for parameter identification. In nonlinear <strong>system</strong>s it is very hard to find<br />
convenient model structures. The physically motivated structure <strong>of</strong> the nonlinear models<br />
presented in this thesis, however, enables one to investigate what influence changes in the<br />
design <strong>of</strong> the <strong>system</strong> will have on its dynamics and helps in pointing at the relevant parts<br />
<strong>of</strong> the statics and dynamics in a complex <strong>system</strong> as the <strong>flight</strong> simulation <strong>motion</strong> <strong>system</strong>s at<br />
hand. This also helps in the design <strong>of</strong> proper experiments.<br />
As already discussed, the <strong>motion</strong> <strong>system</strong> (model) consists <strong>of</strong> kinematics and dynamics<br />
related to the mechanics i.e. the Stewart platform, and the hydraulic actuators which drive<br />
the <strong>system</strong>. Further some additional parasitic dynamics can be expected from a nonrigid<br />
foundation and a flexible platform/<strong>simulator</strong>. The identification <strong>of</strong> the dynamics specifically<br />
related to the hydraulic actuators has already been discussed in detail by Van Schothorst<br />
[124] and will not form an explicit part <strong>of</strong> this thesis though the hydraulic model structure<br />
presented does. Of course the dynamics resulting from the interaction between the<br />
hydraulics and mechanics will be, since the models presented in the previous chapter appointed<br />
this effect as the most relevant.<br />
This chapter will have to provide a good starting point for model <strong>based</strong> <strong>control</strong> on the<br />
actual <strong>system</strong>. The gap between the theoretically derived model structures in the previous<br />
119
120 4 Parameter identification and model validation<br />
chapter and the actual <strong>motion</strong> <strong>system</strong> to be <strong>control</strong>led will be bridged in three steps.<br />
Parameter estimation. The calibration method for the Stewart platform kinematical<br />
model will be presented, implemented and evaluated in Section 4.1. Next, in Section<br />
4.2, the experiments to infer the dynamic model parameters for the mechanics<br />
are set up, implemented and evaluated. Dynamic experiments to infer the position<br />
and orientation <strong>of</strong> measurement equipment consisting <strong>of</strong> accelerometers and rate gyros<br />
will be treated in Section 4.3.2.<br />
<strong>Model</strong> validation. The model characteristics with the parameters estimated will be<br />
validated in the frequency domain in Section 4.3. Especially the conclusion in the<br />
previous chapter that the hydromechanic rigid body modes are most relevant and that<br />
the <strong>system</strong> can be considered locally as six independent 1-d.o.f.-hydromechanic <strong>system</strong>s<br />
along the rigid body modal directions, will have to be evaluated experimentally.<br />
If this conclusion is justified in practise, the models, which describe this dynamics,<br />
are expected to provide a good starting point for model <strong>based</strong> <strong>control</strong>.<br />
Quantification <strong>of</strong> parasitic dynamics. The cause <strong>of</strong> the main parasitic dynamics<br />
observed in Section 4.3 will be clarified in the final part <strong>of</strong> this chapter, Section 4.3.2,<br />
by additional measurements showing vibration <strong>of</strong> the foundation and non rigid shuttle.<br />
Together with the hydraulically generated parasitic dynamics stemming from the<br />
valve and transmission lines, these parasitic dynamic effects are to be considered the<br />
limitations to the <strong>control</strong> scheme <strong>based</strong> on the basic model.<br />
All experiments discussed in this chapter are open loop <strong>based</strong> i.e. no feedback is involved<br />
apart from a very low bandwidth <strong>control</strong>ler, which is to stabilise the <strong>system</strong> without<br />
relevant influence on the information inferred from measurements for calibration/identification<br />
(higher frequencies or static relations).<br />
4.1 Calibration<br />
In this section the parameters <strong>of</strong> the kinematical model <strong>of</strong> the <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> will<br />
be identified (calibrated) and the kinematical model itself will be validated. In robot calibration<br />
[15], parameter identification <strong>of</strong> both the kinematics and dynamics <strong>of</strong> the mechanical<br />
<strong>system</strong> is referred to if calibration is considered. In this thesis, calibration will assume to be<br />
concerned with the identification <strong>of</strong> the kinematical model parameters or <strong>of</strong> specific measurement<br />
equipment. Parameter identification will generally refer to the model parameters.<br />
It is most convenient to perform calibration first as this step can be made independent <strong>of</strong> the<br />
<strong>system</strong> dynamics.<br />
Calibration is a very important topic in robotics. Especially if positional accuracy is<br />
relevant since this can <strong>of</strong>ten be improved by an order <strong>of</strong> a magnitude [15]. Further, one<br />
can expect that most experimental effort will have to be put into calibration as related to<br />
the time spend in testing the <strong>motion</strong> <strong>control</strong> strategies in practice [52]. In <strong>flight</strong> <strong>simulator</strong><br />
<strong>motion</strong> <strong>system</strong>s positional accuracy is clearly not the main issue but indirectly it affects the<br />
accelerational accuracy and is therefore considered in this thesis.
4.1 Calibration 121<br />
The basics <strong>of</strong> calibration can be found in textbooks such as [101] and books containing<br />
selections <strong>of</strong> articles published such as [15]. The framework adapted in this thesis is <strong>based</strong><br />
on an article <strong>of</strong> Hollerbach and Wampler [52]. In this article, a large set <strong>of</strong> robot calibration<br />
methods is unified considering joints and selecting closed kinematic loops. Further, a number<br />
<strong>of</strong> numerical issues is discussed, which provide a theoretical basis on the identifiability<br />
problems <strong>of</strong> kinematical parameters in practice. This method easily integrates the parallel<br />
driven structures but does not explicitly do so by presenting equations which are explicit<br />
in the end effector coordinates. The approach <strong>of</strong> Zhuang in [161] generalizes the idea <strong>of</strong><br />
making use <strong>of</strong> residuals in redundant equations for parallel structures in identifying the unknown<br />
parameters i.e. calibrating such <strong>system</strong>s and these ideas will also be incorporated in<br />
the calibration procedure.<br />
4.1.1 Calibrating the Stewart platform<br />
In this research no specific new theory will presented considering the calibration <strong>of</strong> a Stewart<br />
platform. The contribution will be a fully worked out applicable method consisting <strong>of</strong> a<br />
combination <strong>of</strong> known only slightly modified steps, which, not <strong>of</strong>ten seen in literature, is<br />
brought into practice and is put into a general framework. A large part <strong>of</strong> the procedure<br />
followed has been worked out as a part <strong>of</strong> two master projects [16] [151] in calibrating the<br />
SRS <strong>motion</strong> <strong>system</strong> with the dummy platform <strong>of</strong> Fig. 1.5. Here the results for the practical<br />
part <strong>of</strong> the calibration will involve the calibration <strong>of</strong> the <strong>motion</strong> <strong>system</strong> with the SRS-shuttle<br />
<strong>of</strong> Fig. 1.4.<br />
The calibration procedure proposed consists <strong>of</strong> only two basic steps.<br />
A redundant measurement procedure for platform pose reconstruction.<br />
Identification <strong>of</strong> the kinematical parameters <strong>of</strong> each platform leg separately using the<br />
redundancy <strong>of</strong> the actuator length measurement.<br />
Only a small part <strong>of</strong> the literature on calibration is devoted to the parallel manipulators.<br />
These <strong>system</strong>s differ from most serial <strong>system</strong>s considering the number <strong>of</strong> kinematical<br />
parameters involved, which is high. For the Stewart platform, a kinematical model,<br />
schematically given by Fig. 2.7, was presented by (2.38) from which<br />
klmi + loik 2 =(T ai +c ; bi) T (T ai +c ; bi) i =1�::: �6 (4.1)<br />
follows, which involves 42 unknown kinematical parameters. Seven for each equation consisting<br />
<strong>of</strong> an unknown <strong>of</strong>fset length, loi, which together with the measured length, lmi represents<br />
the total length <strong>of</strong> the i’th leg. The unknown upper and lower gimbal vectors, a i<br />
and bi, each consist <strong>of</strong> three parameters. These parameters will be identified in this section<br />
after redundantly measuring the platform pose consisting <strong>of</strong> the rotation matrix, T and the<br />
translation, c.<br />
This kinematical model assumes that all the gimbals are perfect ball and u-joints as<br />
schematically given by Fig. 2.8 and the prismatic joint axis <strong>of</strong> the sliding actuator exactly<br />
intersects the two gimbal points. In [154, 155] a more general kinematical structure is<br />
considered, allowing each axis <strong>of</strong> rotation in one gimbal to cross the other at a specific
122 4 Parameter identification and model validation<br />
distance not perfectly orthogonal. This model involves 132 unknown kinematic parameters,<br />
which would heavily complicate the calibration problem. Validation <strong>of</strong> the simple model<br />
after calibration can point out whether this is necessary.<br />
Reducing the number <strong>of</strong> unknown parameters to be identified in one step usually helps<br />
the conditioning <strong>of</strong> the problem to be solved and therefore improves the accuracy <strong>of</strong> the<br />
results. In [162] by Zhuang and Roth, the separate calibration <strong>of</strong> each leg <strong>of</strong> the Stewart<br />
platform is made possible by assuming a redundant measurement <strong>of</strong> the platform pose. This<br />
will leave one redundant actuator length variation measurement per actuator from which<br />
the seven unknown kinematical parameters in a i, bi and loi will have to be identified. In<br />
[162] also the calibration <strong>of</strong> the <strong>of</strong>fset length, l oi, is decoupled from the others, but as this<br />
strongly reduces the number <strong>of</strong> measurements, which can be taken into account, this part <strong>of</strong><br />
the procedure has not been adopted.<br />
The solution method to identify the kinematical parameters is iterative with only local<br />
convergence, like the forward kinematical problem discussed in Section 2.3. In [59, 60] a<br />
method is presented to calculate every solution to the calibration <strong>of</strong> a i and bi nonredundantly<br />
taking only six measurements into account and using a seventh measurement in deciding<br />
which solution is the right one. As it involves rootfinding <strong>of</strong> a twentieths order polynomial<br />
equation, this method is usually numerically not suitable for full (noisy) calibration in<br />
practice but it can be used for validation <strong>of</strong> the calibration procedure used here.<br />
The mobility <strong>of</strong> a <strong>system</strong>, roughly its degrees <strong>of</strong> freedom, can be specified with the<br />
mobility index, M as done by Hollerbach and Wampler [52]. It can be calculated given the<br />
number <strong>of</strong> links, L, including the base link and the number <strong>of</strong> constraints, D i, for each joint.<br />
Putting the method <strong>of</strong> Zhuang and Roth [162] for the Stewart platform actuator <strong>of</strong> Fig. 2.8,<br />
in the framework given in [52], the mobility index, M, <strong>of</strong> a separate leg <strong>of</strong> the platform is<br />
M =6(L ; 1) ; J i=1 Di = 3(4 ; 1) ; (4+5+6)=6 (4.2)<br />
The four links consist <strong>of</strong> base, lower and upper actuator part and the platform. There is<br />
a u-, prismatic and a ball joint. The calibration index, C, a redundancy measure given by<br />
Hollerbach and Wampler [52], can also be calculated as the total number <strong>of</strong> sensors, S, is<br />
seven, assuming an independent pose measurement, with<br />
C = S ; M =1: (4.3)<br />
This means there is only one redundant measurement involved from which all seven kinematical<br />
parameters have to be constructed. As in <strong>system</strong> identification <strong>of</strong> a dynamic miso<br />
<strong>system</strong> with several parameters, this is only possible if more than one, here at least seven,<br />
measurements are taken and if the inputs, the platform poses chosen, can be made sufficiently<br />
rich to make the parameters identifiable.<br />
As opposed to the discussion <strong>of</strong> serial manipulator examples given in [52] for <strong>system</strong>s<br />
with C =1, the number <strong>of</strong> measurements in the joints in a parallel <strong>system</strong> is one instead<br />
<strong>of</strong> the number <strong>of</strong> measurements for the end-effector. An advantage <strong>of</strong> C = 1 is that all<br />
equations involved and the residuals resulting, are <strong>of</strong> the same type. This is favourable in<br />
conditioning.<br />
In applying a calibration method in practice, the desired precision <strong>of</strong> the result should<br />
be known since as a rule <strong>of</strong> thumb the measurement equipment should be an order <strong>of</strong> a
4.1 Calibration 123<br />
−z<br />
−z<br />
2<br />
1<br />
0<br />
−1<br />
−2<br />
3<br />
2<br />
1<br />
0<br />
View in perspective<br />
0<br />
x<br />
Side view<br />
2 −2<br />
0<br />
−y<br />
−1<br />
−2 −1 0<br />
x<br />
1 2<br />
2<br />
−z<br />
−y<br />
3<br />
2<br />
1<br />
0<br />
Front view<br />
−1<br />
−2 −1 0<br />
−y<br />
1 2<br />
2<br />
1<br />
0<br />
−1<br />
Top view<br />
−2<br />
−2 −1 0<br />
x<br />
1 2<br />
Fig. 4.1: Two calibration measurement frames for the case <strong>of</strong> the Stewart platform. The<br />
fixed frame consists <strong>of</strong> three points. With the SRS they were chosen somewhere<br />
between the lower gimbals. The moving frame consists <strong>of</strong> three points chosen<br />
somewhere on the upper gimbal blocks. Knowledge on the exact positioning <strong>of</strong><br />
the measurement frame points is not necessary for this method.<br />
magnitude more precise [15]. For the measurement <strong>of</strong> the relative position <strong>of</strong> the prismatic<br />
joint a linear position transducer <strong>of</strong> the Temposonic type is used. The accuracy <strong>of</strong> these<br />
sensors is 0:1 mm (resolution with [55] 0:009 mm). A calibrated <strong>motion</strong> <strong>system</strong> with<br />
an positional accuracy <strong>of</strong> 1 mm will be strived for. The independent measurement <strong>of</strong> the<br />
platform pose discussed next will have to be done with equipment <strong>of</strong> :0:1 mm accuracy<br />
also.<br />
4.1.2 Redundant measurement <strong>of</strong> the platform pose<br />
The redundant measurement <strong>of</strong> the platform pose is <strong>based</strong> on the method presented by Geng<br />
and Haynes [43]. One defines two measurement frames e.g. as in Fig. 4.1. One fixed on<br />
the ground and one on the moving platform/<strong>simulator</strong> using three points for each frame.<br />
The <strong>motion</strong> <strong>of</strong> a point with respect to the fixed frame can be determined by measuring the<br />
distance <strong>of</strong> this point to the three points fixed to the ground if the distance between the<br />
fixed points is known and if the moving point remains in a known area w.r.t. plane spanned<br />
by the three points. With three additional length measurements (two to the second moving<br />
point and one to the third) the orientation can also be determined if the distance between the<br />
moving points is known.<br />
In case <strong>of</strong> the calibration <strong>of</strong> the shuttle <strong>of</strong> the SRS the distance between the moving<br />
points fixed to the shuttle are not precisely known and can not easily be measured directly<br />
since the shuttle shape does not allow so. This problem is solved by taking three distance<br />
measurements for each moving point (nine in total) from which the relative position <strong>of</strong> the<br />
moving points can be calculated. The relative distance <strong>of</strong> these points is redundant if more
124 4 Parameter identification and model validation<br />
Fig. 4.2: Determination <strong>of</strong> the position <strong>of</strong> a moving point in space using three length measurements.<br />
In case <strong>of</strong> the SRS, the three fixed points, Gi, can be thought in between<br />
the lower gimbal blocks. Each moving point, P i, was taken as part <strong>of</strong> the<br />
upper gimbal blocks.<br />
than one pose measurement is taken. This redundancy can and will be used to decide on<br />
a confidence level for the measurements in the weighted least squares calibration <strong>of</strong> the<br />
kinematical parameters. As only one distance to a specific point can be measured at the<br />
time this redundancy does not take significant additional effort to be measured since the<br />
platform has to be moved to a specific pose three times at least (using three pairs <strong>of</strong> three<br />
measurements instead <strong>of</strong> three, two and one measurement respectively).<br />
In Fig. 4.2 a top view <strong>of</strong> a three point distance measurement with three lengths, l i,<br />
i = 1�::: �3 is given to determine the position <strong>of</strong> a moving point P i with respect to the<br />
frame defined by the three points Gi, i =1�::: �3. Point, G1, is taken as the origin and the<br />
line through G1 and G2 as the y-axis. Using the cosine formula:<br />
l 2<br />
2 = l2<br />
1 + b2<br />
2 ; 2l1b2cos( )=l 2<br />
1 + b2<br />
2 ; 2b2y1� (4.4)<br />
y1 can be calculated. By letting the third fixed point, G 3, define the fixed measurement<br />
xy-plane, similar application <strong>of</strong> the cosine rule will lead to a parameter, s. This parameter,<br />
s, is the distance and direction <strong>of</strong> the origin to the point S(s x�sy� 0) in Fig. 4.2. So also the<br />
following holds<br />
s<br />
p 2 a3 + b2<br />
3<br />
= sx<br />
=<br />
a3<br />
sy<br />
b3<br />
(4.5)<br />
Further, the line in the xy-plane through S, orthogonal to the line through G 1 and G3, runs<br />
through (x1�y1� 0), i.e. direction @y=@x = ;a3=b3 can also be given by<br />
y1 ; sy<br />
= ;<br />
x1 ; sx<br />
a3<br />
b3<br />
Combining these equations to get rid <strong>of</strong> sx and sy leads to<br />
q<br />
2<br />
b3y1 + a3x1 = s a<br />
3<br />
: (4.6)<br />
+ b2<br />
3<br />
(4.7)
4.1 Calibration 125<br />
So in applying the cosine rule, s can be replaced by x 1 and y1, from which x1 can be solved<br />
l 2<br />
3 = l2<br />
1 + a2<br />
3 + b2<br />
3 ; 2l1<br />
=<br />
q<br />
2<br />
cos( ) a3 + b2<br />
3<br />
l 2<br />
q<br />
2<br />
1 + a2<br />
3 + b2<br />
3 ; 2s a3 + b2 3<br />
Finally j z1 j can be determined from<br />
= l 2<br />
1 + a2<br />
3 + b2<br />
3 ; 2(b3y1 + a3x1) (4.8)<br />
l 2<br />
1 = x21<br />
+ y2 1 + z2 1<br />
(4.9)<br />
The sign <strong>of</strong> z1, above or below the plane <strong>of</strong> measurement, should be known in advance. In<br />
this way the three vectors from the fixed frame to the moving points, p i, can be calculated.<br />
The frame defined through Gi given in Fig. 4.2 is convenient in determining the vectors to<br />
the moving reference points pi. In identifying the kinematical parameters it will be more<br />
suitable to define a slightly different fixed and moving frame from G i and Pi respectively,<br />
which are almost equal to the ground frame and moving frame defined for the SRS (Fig. 2.7).<br />
The new fixed reference measurement frame origin, G m, is given by a translation<br />
oGm =(g1 +g2 +g3)=3 (4.10)<br />
with respect to the frame given in Fig. 4.2. The orientation remains unchanged. The moving<br />
reference measurement frame origin, Pm, is specified w.r.t Gm accordingly with an origin<br />
(translation cpg)<br />
oPm =(p1 +p2 +p3)=3 ; oGm =cpg<br />
(4.11)<br />
and the orientation, with the line through P 1 and P2 chosen parallel to the y-axis, consists<br />
<strong>of</strong> three unit vectors along the new axes. With pij = pj ; pi, npy = p12= p p T 12 p12,<br />
npz = (npy p13)= p p T 13 p13 and npx = npy npz. These together form the rotation<br />
matrix from Pm to Gm<br />
TGm�Pm =[npx npy npz ]: (4.12)<br />
And this is the pose measurement we were looking for. In short it is given by (c pg, Tpg),<br />
which basically can be interpreted along the same lines as the pose defined for the Stewart<br />
platform.<br />
All together the used platform pose measurement requires nine absolute length measurements<br />
and the knowledge <strong>of</strong> the absolute distance between the fixed reference points.<br />
The quality <strong>of</strong> the calibration procedure heavily depends on these measurements. The positioning<br />
<strong>of</strong> the reference points themself is not that important. A rough idea will help in<br />
selecting a proper initial estimate in the calibration and the conditioning <strong>of</strong> the optimization<br />
problem to be solved is improved if all the distances to be measured (e.g. in Fig. 4.2) are<br />
about the same. Further the platform reference points have to reside at a predefined side <strong>of</strong><br />
the plane defined by the fixed reference points.
126 4 Parameter identification and model validation<br />
Fig. 4.3: In practise, the measurement device could not be used to take direct measurement<br />
<strong>of</strong> the length between two points at an angle ( 6= 0). A funnel (a cone shaped<br />
utensil) had to be placed in between. The picture shows that the measurement<br />
cable will not run directly from Gi to Pi but will follow the rounded edges <strong>of</strong> the<br />
funnel from Gi to D. This requires funnel length measurement compensation.<br />
4.1.3 Stewart platform pose measurement in practice<br />
The length measurements to reconstruct an independent, redundant platform pose in case <strong>of</strong><br />
the SRS-<strong>motion</strong> <strong>system</strong> have been performed with relatively low cost string encoders made<br />
by AMS Gmbh. A measurement cable winds onto an accurately machined cable drum. A<br />
coil spring delivers a constant pull in force to maintain cable tension. An incremental rotary<br />
encoder on the drum gives a digital output. The resolution <strong>of</strong> these encoders is 0.03 mm and<br />
the calibrated relative precision 0.1 mm. The standard version <strong>of</strong> this equipment does not<br />
allow the cable to be used at different angles w.r.t. the output point. This would damage the<br />
device.<br />
To allow calibration for an absolute length measurement, also for measurements taken<br />
at an angle, a funnel was designed. As shown in Fig. 4.3, this slightly affects the length<br />
measurement between Gi, which is chosen at the bottom <strong>of</strong> the funnel and P j. In practice<br />
this difference can take values up till 4 mm (R=15 mm) and should be corrected.<br />
Considering Fig. 4.3, lpg, the length between Pj and Gi is equal to<br />
p<br />
lpg = (X0 ) 2 + Z2 (4.13)<br />
the measured length will be equal to<br />
d1 + R<br />
p<br />
= (X0 ) 2 + Z2 ; 2RX0 + R (4.14)<br />
By using the measured lengths to reconstruct a first estimate <strong>of</strong> p i the required correction
4.1 Calibration 127<br />
can be approximated by taking<br />
0 = tan X 0<br />
Z<br />
(4.15)<br />
or more precise (in this case not necessary since R=lpg was small, only used for evaluation)<br />
reconstructing from<br />
Z tan( )=X 0 + R<br />
1 ; cos( )<br />
cos( )<br />
(4.16)<br />
It is assumed that the direction (Z) <strong>of</strong> the funnels (in the frame G m) is known.<br />
The choice for the approximate positions <strong>of</strong> the reference measurement points is given<br />
in Fig. 4.1. Considering Fig. 2.10, the fixed reference points (the encoders) have been<br />
placed in between the lower gimbal pairs (4,5), (2,3), (6,1) respectively. This allows the<br />
funnels to point along the z-axis <strong>of</strong> the measurement (and ground) frame. The upper, moving<br />
reference points have been taken in between (slightly lower: 80 mm) the same pairs <strong>of</strong><br />
upper gimbal points. In most platform poses, the lengths l pg will therefor not be the same<br />
(slightly less favourable conditioning) but in practice prevention from actuators running into<br />
the measurement cables is more important. As a result, the fixed ground frame and platform<br />
frame almost coincide with the measurement reference frames. This helps in interpretation<br />
<strong>of</strong> numerical results but does not improve the calibration itself.<br />
To allow measurement in between a favourable maximum and minimum length the limited<br />
(2 m) range string encoders were enlarged with steel cables (as used for fishing!) which<br />
were calibrated with the string encoders themselves. Two sets <strong>of</strong> these additional <strong>of</strong>fset cables<br />
were used (for Gi to Pj with i = j approx. 1.5 m and 2.0 m in case <strong>of</strong> i 6= j). The <strong>of</strong>fset<br />
<strong>of</strong> the string encoders themselves is also necessary to allow absolute measurement and was<br />
measured at a small, approx. 60 mm, retracted up right position with a sliding calipers to<br />
the funnels. The distance ( 1:8 m) between the string encoders could be measured with a<br />
large (2 m range) sliding calipers.<br />
Careful attention in performing these (<strong>of</strong>fset) measurements largely improved the calibration.<br />
By calibration before and after a measurement session also some drift compensation<br />
(1.4 mm for one string encoder) could be applied. Another important point <strong>of</strong> consideration<br />
was the dry cable friction in the funnels. Before each measurement, the cables had to be<br />
vibrated a little bit to get rid <strong>of</strong> play.<br />
To allow proper choice <strong>of</strong> the sensor range to be used, the set <strong>of</strong> platform poses, approximate<br />
since the actual kinematics is not known in advance, used for calibration can be<br />
simulated. In Fig. 4.4 all the positions <strong>of</strong> upper gimbal 1 are given for all 64 combinations<br />
<strong>of</strong> setting the six actuators in a minimum length position <strong>of</strong> 20 cm above minimal and below<br />
maximal stroke. These positions are sufficiently far away from the cushioning area <strong>of</strong> the<br />
actuators to be able to hold the <strong>system</strong> stable at these positions in taking a measurement.<br />
It is assumed that this set <strong>of</strong> positions will give relatively high variation in order to<br />
enhance the identifiability <strong>of</strong> the kinematical parameters. From the front view it can be observed<br />
that the positions are roughly the extremes <strong>of</strong> the 2-d.o.f. structure given in Fig. 3.22<br />
which are rotated if considered from the side view. The fact that some points are relatively
128 4 Parameter identification and model validation<br />
−z<br />
3<br />
2<br />
1<br />
0<br />
−2<br />
3<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
View in perspective<br />
0<br />
Side view<br />
2 −2<br />
0<br />
−2 0 2 4<br />
x<br />
0<br />
2<br />
−z<br />
−y<br />
3<br />
2.5<br />
2<br />
1.5<br />
1<br />
0.5<br />
Front view<br />
0<br />
−2 −1 0<br />
−y<br />
1 2<br />
2<br />
1<br />
0<br />
−1<br />
Top view<br />
−2<br />
−2 0 2 4<br />
x<br />
Fig. 4.4: Positions <strong>of</strong> upper gimbal 1 for calibration.<br />
close to each other in this picture does not imply that they will contain the same information<br />
since the orientation <strong>of</strong> the platform, which will be different, is not shown.<br />
If compared to the unconstraint actuator, it can be observed that especially the range <strong>of</strong><br />
angles to be reached by the platform actuators is relatively limited. E.g. the angles <strong>of</strong> the line<br />
between the lower gimbal point 1 and the dots in Fig. 4.4. This will affect the conditioning<br />
<strong>of</strong> the calibration as will be discussed in the next sections.<br />
4.1.4 Identification <strong>of</strong> the kinematical parameters<br />
After the determination <strong>of</strong> the platform poses, this information plus the actuator displacement<br />
measurements can be used to identify the kinematical parameters. Taking an initial<br />
estimate <strong>of</strong> these parameters in (4.1), each measurement regarding the j’th actuator, will<br />
have a residual. One can try to change the parameters in order to minimize these residuals.<br />
Using the quadratic form <strong>of</strong> (4.1), the jacobian <strong>of</strong> kinematical parameter variations, k, to<br />
residual variation, r, has a favourable linear structure<br />
Jrk = @r<br />
@k = Xo + 7<br />
i=1 Xiki<br />
with the kinematical parameters, k, <strong>of</strong> one actuator defined by<br />
and the i’th row (measurement) <strong>of</strong> the jacobian<br />
(4.17)<br />
k T =[a T b T loj] (4.18)<br />
Jrk(i� ) = 2[(lm(i) +lo(i)) (c T i Ti + b T Ti +a T ) (c T i +aT T T<br />
i + bT )]: (4.19)
4.1 Calibration 129<br />
It is immediately clear from the last equation that every row can be split in a non kinematical<br />
parameter dependent row<br />
Xo(i� )=2[lm(i) c T i Ti c T i ] (4.20)<br />
from which the matrix, Xo, can be constructed and parts which are linearly dependent on<br />
only one kinematical parameter e.g.<br />
X1(i� )k(1) = 2[0 1 0 0 T T<br />
i ( � 1)]a(1): (4.21)<br />
A weighted least squared solution to the linearised problem <strong>of</strong> minimising the residuals<br />
is given by<br />
kwls =(J T rk WJrk) ;1 JrkW r (4.22)<br />
In robot calibration, this is usually effective since the equations are only mildly nonlinear<br />
[52]. The weighting matrix, W , can be used to discriminate between the measurements.<br />
This can be appropriate since the confidence level for the different platform pose measurements<br />
can differ. If one <strong>of</strong> the string encoders showed e.g. a little hysteresis due to friction<br />
once in a while, the redundancy in the platform pose measurement can be used to detect<br />
this. With<br />
Wii =<br />
1<br />
0:5 + 1000 j lPP ; mlpp j<br />
(4.23)<br />
the confidence is upper bounded by 2 and goes to zero if high difference is detected between<br />
the measured distance, lpp, and the mean distance, mlpp, between two points, Pi.<br />
Iteration on (4.22) is similar to a Newton Raphson iteration. For convergence, it helps if<br />
the jacobian is Lipschitz, far from singular, well conditioned and starts with a well chosen<br />
initial estimate. Due to the linearity in the parameters, the jacobian is clearly Lipschitz.<br />
Further the structure <strong>of</strong> each parameter (Xi) is almost equal. The choice <strong>of</strong> platform poses<br />
can be used for conditioning. E.g. not varying the orientation, T , will result in a singular<br />
jacobian, since all solutions with only a translational vector added to the actual solutions <strong>of</strong><br />
both a and b will have equal residuals.<br />
The singular values <strong>of</strong> the jacobian, Jrk, are also called the observability indices <strong>of</strong> the<br />
calibration problem. In a comparative study cited by Hollerbach and Wampler in [52] it was<br />
concluded that optimal conditioning <strong>of</strong> these indices or maximising the minimal index gave<br />
good quantative results.<br />
The calibration problem <strong>of</strong> the actuator kinematics was idealised by an unconstrained<br />
simulation i.e. no constrains <strong>of</strong> platform poses to be chosen and all actual parameters zero.<br />
Choosing all (729) combinations <strong>of</strong> the tri state (-1,0,1) for translation and (-.5,0,.5) for the<br />
euler parameters results in a jacobian with condition number two at the solution, which is<br />
almost optimal. By Hollerback and Wampler [52] it is referred to a research concluding the<br />
condition number should be better (lower) than 100 to have reasonable results. With the<br />
SRS Stewart Platform to be calibrated with constrains on the actuator lengths the condition<br />
number increases to sixty using the 64-state set given in Fig. 4.4.
130 4 Parameter identification and model validation<br />
−z<br />
0.03<br />
0.02<br />
0.01<br />
0<br />
−0.01<br />
0.03<br />
0.02<br />
0.01<br />
0<br />
view in perspective<br />
0<br />
0.01<br />
side view<br />
0.01<br />
0<br />
−0.01<br />
−0.01<br />
−0.02 −0.01 0<br />
x<br />
0.01 0.02<br />
−z<br />
−y<br />
0.03<br />
0.02<br />
0.01<br />
0<br />
front view<br />
−0.01<br />
−0.02 −0.01 0<br />
−y<br />
0.01 0.02<br />
0.02<br />
0.01<br />
0<br />
−0.01<br />
top view<br />
−0.02<br />
−0.02 −0.01 0<br />
x<br />
0.01 0.02<br />
Fig. 4.5: SRS calibration result with the design structure on a 1:100 scale and the differences<br />
1:1.<br />
e_length [mm]<br />
e_trans [mm]<br />
e_rot [meps]<br />
5<br />
0<br />
−5<br />
0 20 40 60<br />
5<br />
0<br />
−5<br />
0 20 40 60<br />
5<br />
0<br />
−5<br />
0 20 40 60<br />
non calibrated <strong>system</strong><br />
5<br />
0<br />
−5<br />
0 20 40 60<br />
5<br />
0<br />
−5<br />
0 20 40 60<br />
5<br />
0<br />
−5<br />
0 20 40 60<br />
calibrated <strong>system</strong><br />
Fig. 4.6: Positional errors for all 64 platform poses used for calibration made by the noncalibrated<br />
and the calibrated <strong>system</strong>. The first row shows length measurement<br />
prediction errors <strong>of</strong> the actuators length measurement given the redundantly measured<br />
pose. The next rows give the transformation <strong>of</strong> these errors to translational<br />
(2 nd row) and rotational (3 rd row) prediction errors <strong>of</strong> the platform pose.
4.1 Calibration 131<br />
4.1.5 Results in calibrating the SRS <strong>motion</strong> <strong>system</strong><br />
The calibration procedure leads to kinematical parameter (vectors) defined in the reference<br />
measurement frames. By construction <strong>of</strong> the forward kinematics <strong>of</strong> the newly found parameters<br />
for the actuators in half stroke, the transformation, sx gp, between the two measurement<br />
frames can be calculated. All upper gimbal positions can now also be referred to the ground<br />
measurement frame, Gm.<br />
a Gm =cgp + Tgpa Pm (4.24)<br />
As the parameters were not constrained to a plane, the <strong>simulator</strong> ground and moving<br />
frame are not trivially defined anymore. As for the reference measurement frames, the<br />
ground frame origin is chosen as the mean <strong>of</strong> the lower gimbal points. The ground frame xdirection<br />
is chosen as the addition <strong>of</strong> the vectors between the origin and gimbal point 1 and<br />
6 and the xy-plane is formed by making the vector from the origin to the mean <strong>of</strong> gimbals 2<br />
and 3 part <strong>of</strong> it. This defines the transformation to a ’design’ ground frame.<br />
Both the upper as the lower gimbals can now be referred to the design frames. In Fig. 4.5<br />
the gimbal points found through calibration are given in a design frame which is a 100<br />
times compressed making the differences, which are less than 0.1 %, visible. As opposed<br />
to the dummy platform [16] [151], no large deviations are found in the construction <strong>of</strong> the<br />
shuttle. It is now interesting to see what the differences are in predicting the actuator length<br />
measurements i.e. using the design values and the calibrated values. These deviations for<br />
all six actuators can also be transformed to the deviations in platform pose using<br />
sx = J ;1<br />
l�sx (sx) l (4.25)<br />
In Fig. 4.6 it is shown that the positional accuracy <strong>of</strong> the calibrated platform has almost<br />
improved an order <strong>of</strong> a magnitude.<br />
The accuracy <strong>of</strong> the noncalibrated <strong>system</strong> is not better than 5 mm in actuator lengths,<br />
2 mm in translation and 10 mrad in orientation. With the calibrated <strong>system</strong> this improves<br />
as shown in detail in Fig. 4.7 to .75 mm in actuator length predictability, apart from one<br />
outlier also predicted with upper reference point distance measurement. In the more important<br />
platform pose translational and orientational accuracy the errors are reduced to less<br />
than 0.5 mm and 2 mrad respectively.<br />
The initial estimate in the calibration procedure was more than another magnitude worse<br />
(> 100 times) since the pose <strong>of</strong> the measurement reference frame was only roughly (within<br />
100 mm) taken into account. Still the iteration on the kinematical parameters converged<br />
quickly within 4 steps for each actuator leg. Convergence is not guaranteed with such<br />
initial errors, but since the procedure can be done <strong>of</strong>f line, as opposed to the platform pose<br />
reconstruction treated in Section 2.2.11, this is not so important.<br />
One important aspect, the direction <strong>of</strong> the gravitational field has not been identified yet.<br />
This can be done using an additional measurement device (e.g. an inclinometer) as has been<br />
done in [16] [151]. But the pressure (force) measurements in the actuators can also be used<br />
to detect this as shown in the next section.
132 4 Parameter identification and model validation<br />
residual [mm]<br />
error [mm]<br />
Variation upper triangle measurement<br />
3<br />
2<br />
1<br />
0<br />
−1<br />
−2<br />
0 20 40 60<br />
0.5<br />
−0.5<br />
Translational pose prediction errors<br />
1<br />
0<br />
−1<br />
0 20 40<br />
pose no.<br />
60<br />
residuals in [mm]<br />
residual [meps]<br />
0.5<br />
0<br />
−0.5<br />
−1<br />
−1.5<br />
Errors in actuator length prediction<br />
1<br />
−2<br />
0 20 40 60<br />
Orientational pose prediction errors<br />
1.5<br />
1<br />
0.5<br />
0<br />
−0.5<br />
−1<br />
−1.5<br />
0 20 40<br />
pose no.<br />
60<br />
Fig. 4.7: Positional errors for all 64 platform poses used for calibration made by the calibrated<br />
<strong>system</strong> in detail. Also showing the variation in upper reference point distance<br />
measurement, which is used to calculate the confidence levels.<br />
Resume<br />
The following conclusions can be drawn w.r.t. the calibration procedure described.<br />
It has been shown that the kinematic mechanical parameters <strong>of</strong> a Stewart platform<br />
used as a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> can be calibrated resulting in a <strong>system</strong> with a<br />
positional accuracy better than 1 mm (for DEP). This is an order <strong>of</strong> a magnitude more<br />
precise than before calibration.<br />
The fact that the kinematical model itself is accurate within these error bounds for the<br />
SRS is another important conclusion.<br />
Further, it means that with calibration such <strong>motion</strong> <strong>system</strong>s can be made more accurate<br />
without putting more extreme tolerances on the fabrication and constructional<br />
process.<br />
The two step calibration procedure has been shown to work in practice.<br />
For the first step, measuring relative platform pose variations w.r.t. unknown, but<br />
platform and inertial world fixed, reference frames is sufficient redundant information<br />
necessary for calibration.<br />
The parallelism <strong>of</strong> the platform can be used in the second step to identify the 42<br />
kinematical parameters in six separate actuator groups. This enhances accuracy.<br />
It is important to realise that the result strongly depends on the accuracy <strong>of</strong> some<br />
<strong>of</strong> the measurements taken. These are the absolute length measurement <strong>of</strong>, and the<br />
distances in between, the redundant sensors.
4.2 Stewart platform model parameter identification 133<br />
As the calibration is concerned with sub mm precision on a sup m <strong>motion</strong> <strong>system</strong>, it is<br />
recommended that the procedure is rerun after each reconstruction <strong>of</strong> a platform e.g. in<br />
moving the SRS <strong>motion</strong> <strong>system</strong> to another site.<br />
4.2 Stewart platform model parameter identification<br />
Now the kinematic model has been validated and its parameters have been calibrated, the<br />
parameters concerning the dynamics <strong>of</strong> the <strong>motion</strong> <strong>system</strong> can be identified. The main<br />
unknowns in the Stewart platform model are its mass properties.<br />
First, the experiments to reconstruct the parameters related to the gravity forces, the<br />
location <strong>of</strong> the centre <strong>of</strong> mass and the weight itself will be discussed. Then, the identification<br />
<strong>of</strong> the inertial parameters is treated. These can be determined experimentally without much<br />
concern about the precise dynamics <strong>of</strong> the hydraulics. In this way, a reasonably accurate<br />
approximate model, i.e. (2.112) with a general mass matrix, <strong>of</strong> the mechanics presented in<br />
Chapter 2 taking into account the actuator intertia as in (2.122) can be identified.<br />
In the model <strong>based</strong> <strong>control</strong> structure to be presented in the next chapter, this part <strong>of</strong> the<br />
model will be important since the forces required for the accelerations to be applied for<br />
simulation will for most be generated by feed forward which requires an accurate model.<br />
Identification <strong>of</strong> the hydraulics, considering the separate actuators, has been discussed in<br />
[124]. The model parameters concerning the hydraulics are therefore assumed to be known.<br />
Validation <strong>of</strong> the dynamics resulting from the interaction in the hydraulically driven Stewart<br />
platform will be discussed in the next section.<br />
The set up <strong>of</strong> the hydraulically driven <strong>motion</strong> <strong>system</strong> supplied with the six valve inputs<br />
and measurement equipment to determine both pressure and positional information <strong>of</strong> the<br />
actuators, the legs, provides a complex but sufficiently rich environment by itself to perform<br />
almost all experiments required. The only additional measurement equipment which is<br />
used in this research are accelerometers and rate gyros which do more directly reflect the<br />
simulation performance on the platform and with which one can more accurately appoint<br />
the cause <strong>of</strong> parasitic dynamics resulting from flexible modes in the shuttle or foundation.<br />
4.2.1 Gravitational force determination<br />
The parameters to be identified, the static mass experienced and the position <strong>of</strong> the centre <strong>of</strong><br />
gravity, can be used to incorporate a gravity compensation in a <strong>control</strong>ler. This is important<br />
since gravity forces form a considerable part <strong>of</strong> the total force (up to 70 % <strong>of</strong> full pressure<br />
required as specified in Table 3.12). Further, a reasonable mass matrix results from taking<br />
the centre <strong>of</strong> gravity as the origin <strong>of</strong> the moving body axes in the dynamic model.<br />
Considering the model (2.122) in a static pose, leaves<br />
Gt =<br />
fg<br />
mg =(fg co) = J T l�x (sx)fa (4.26)<br />
with the gravity force, fg = Tgg[0 0 mg] T , transformed to the fixed lower gimbal frame by<br />
the rotation matrix, Tgg. Usually the matrix Tgg is approximately the identity matrix.
134 4 Parameter identification and model validation<br />
z<br />
f 1<br />
co<br />
f 2<br />
fg l1 l2<br />
c1<br />
f 3<br />
l<br />
3<br />
X<br />
Fig. 4.8: Simplified planar example <strong>of</strong> identifying mass and centre <strong>of</strong> gravity. Through the<br />
three actuator forces <strong>of</strong> a parallel 3-d.o.f. <strong>system</strong> the gravity vector (length and<br />
direction) plus line <strong>of</strong> action can be identified with two force equilibria and one<br />
moment equation. By rotating the <strong>system</strong> and identifying another (rotated) line<br />
<strong>of</strong> action, the intersection <strong>of</strong> both lines provides for the position <strong>of</strong> the centre <strong>of</strong><br />
gravity. In a spatial 6-d.o.f.-<strong>system</strong>, also two measurements are sufficient but the<br />
two lines <strong>of</strong> action can cross in practice. By identifying the point, which is closest<br />
to a set <strong>of</strong> lines <strong>of</strong> action, a more robust estimation can be attained.<br />
Further, the moments, mg, are induced by the vector product <strong>of</strong> this force with the vector<br />
<strong>of</strong> the upper centre gimbal point to the centre <strong>of</strong> gravity, c o. The vectors, fg, the gravity force<br />
i.e. platform mass times gravity and the centre <strong>of</strong> gravity, c o, have to be identified.<br />
From Fig. 4.8, it can be observed that one can only identify the part <strong>of</strong> c o, which is<br />
orthogonal to fg considering one pose as the moments result from a vector product i.e. a<br />
line <strong>of</strong> action can be identified on which the centre <strong>of</strong> gravity point resides. By measuring<br />
the forces required in other static poses in which the platform has been rotated (around a<br />
vector which is at least partly orthogonal to the gravitational direction), a set <strong>of</strong> lines can<br />
be determined from which co can be obtained. In theory all these lines will intersect at this<br />
point but in practice they will cross. The point which is closest to all lines in some sense<br />
can be taken as best guess for the centre <strong>of</strong> gravity. Here the least squares optimal solution<br />
<strong>of</strong> all (Euler) distances will be taken.<br />
Consider a specific static pose measurement <strong>of</strong> the pressure differences at the valves,<br />
[f T g mT T<br />
g ]=AdP i J(sx), assuming the input and output pressures are equal statically. A<br />
valid point <strong>of</strong> the line containing the centre <strong>of</strong> gravity, o, can be found by<br />
with ng = fg=(mg) and<br />
c1<br />
z<br />
l<br />
1<br />
f 1<br />
o = ; 1<br />
f T g fg<br />
fg Pngmg� (4.27)<br />
q<br />
f T<br />
g fg = mg. Now, the line <strong>of</strong> action can be described by<br />
o + ng. Every line, ( o + ng)i, found for a pose (ci�Ti), can be transformed back to the<br />
platform coordinates in one specific pose e.g. the neutral pose.<br />
( on + nngn)i = T T<br />
i ( o + ng) ; ci (4.28)<br />
f g<br />
co<br />
l<br />
2<br />
c2<br />
f 2<br />
f 3<br />
l<br />
3<br />
X
4.2 Stewart platform model parameter identification 135<br />
Configuration A B C<br />
m [tons] 2.60 2.10 4.25<br />
co(x) [m] 0.00 -0.12 0.00<br />
co(y) [m] 0.00 0.00 0.00<br />
co(z) [m] -0.23 -0.26 -0.45<br />
Table 4.1: The Simona Research Simulator gravitational parameters<br />
The vector, dlp, with smallest distance <strong>of</strong> some point, xp, to such a line is given by<br />
dlp = Pnng (xp ; on) (4.29)<br />
The vector, co, to the point, which minimizes the sum <strong>of</strong> squared distances to n <strong>of</strong> such<br />
lines, is given by<br />
co = 1<br />
n<br />
n<br />
i=0 Pnng�i<br />
;1 1<br />
n<br />
n<br />
i=0 Pnng�i oni (4.30)<br />
The conditioning <strong>of</strong> this optimization problem greatly depends on the ability to perform a<br />
measurement in which the summation <strong>of</strong> the projection matrices, P nng�i, equally span the<br />
3D-space i.e. whether the platform has been rotated sufficiently.<br />
Due to the limited stroke <strong>of</strong> the actuators <strong>of</strong> the Stewart platform, rotation can not be<br />
performed freely. In fact, performing valid pressure measurements can only be done if<br />
the actuators are out <strong>of</strong> the cushioning area (approx. 15 cm from max or min). Further,<br />
incorporating the mass <strong>of</strong> the actuators in the total mass is less tolerable with large rotations.<br />
With the SRS, the fast gravitational force identification procedure consisted <strong>of</strong> measuring<br />
pressures in five poses: neutral, +roll, -roll, +pitch and -pitch from neutral considering the<br />
limits: (lmin + :15,lmax ; :15).<br />
The more elaborate procedure takes into account all 64 poses used in the kinematic<br />
calibration. Over time, the mass and position <strong>of</strong> the centre <strong>of</strong> gravity <strong>of</strong> a <strong>simulator</strong> under<br />
construction, as the SRS, is likely to change. Therefore this procedure is to be repeated<br />
more <strong>of</strong>ten than full kinematic calibration.<br />
The conditioning <strong>of</strong> the matrix to be inverted in (4.30) is 12 and as a result the x- and<br />
y-position <strong>of</strong> the centre <strong>of</strong> gravity can be found approx. 10 times as accurate as the zposition.<br />
Fortunately, as this point is mainly used for gravitational compensation, the x- and<br />
y-position are also much more important.<br />
The gravitational forces <strong>of</strong> the three SRS <strong>system</strong> configurations evaluated in this research,<br />
A) The dummy platform alone,<br />
B) The shuttle and<br />
C) The dummy platform with additional load
136 4 Parameter identification and model validation<br />
have been identified. The results are given in Table 4.1.<br />
In the poses considered, the observed gravity forces are varying related to a mass <strong>of</strong><br />
approx. 20 kg due to the relative <strong>motion</strong> <strong>of</strong> actuators w.r.t. the platform. In all cases the<br />
c.o.g. is found above the centre gimbal plane. As the dummy platform has symmetricity, it<br />
is natural to find the c.o.g. in the origin <strong>of</strong> the xy-plane. The c.o.g. <strong>of</strong> the shuttle, which is<br />
somewhat to the back, is expected to move to the front if the visual <strong>system</strong> will be added.<br />
Measurements are best taken with a very slowly moving platform (e.g. sinusoidal <strong>motion</strong><br />
in z-direction < :5Hz, amplitude < 2mm over a whole number <strong>of</strong> periods) to eliminate<br />
most <strong>of</strong> the friction forces. Coulombic friction forces are small for the prismatic joint, the<br />
actuator with hydrostatically bearings, [124] but the rotation <strong>of</strong> the gimbal joint also has<br />
friction. These friction forces together are seen to be as large as 250 N.<br />
Both the calibration procedure as the gravitational force determination require a simple<br />
<strong>control</strong>ler to stabilize the <strong>system</strong> and moving the <strong>system</strong> to a specific platform pose and hold<br />
it there. The standard local (actuator per actuator) pressure feedback [153] in combination<br />
with positional PI-feedback with low bandwidth (1/10 <strong>of</strong> lowest eigenfrequency in neutral<br />
position i.e. approx. 0.5 Hz) will usually be sufficient.<br />
To determine the inertial properties <strong>of</strong> the <strong>system</strong>, discussed in the next section, experiments<br />
preferably require vibrations <strong>of</strong> higher frequency, e.g. approx. 4 Hz. This will still<br />
be attainable with a local <strong>control</strong> structure as no harsh requirements will be stated towards<br />
cross talk in the identifying procedure.<br />
4.2.2 Identification <strong>of</strong> the inertial properties<br />
Basic concept<br />
The main unknown parameters left to be determined are those contained in the mass matrix,<br />
the inertial properties <strong>of</strong> the <strong>system</strong>. In short, these parameters will be estimated by<br />
considering force and acceleration (through position) measurements and evaluating these<br />
signals in a domain where the basic equation force is equal to mass times acceleration in<br />
matrix form, f = Mx, holds. The mass matrix, M, can be deduced by performing six<br />
approximately sinusoidal vibrations which span the 6-d.o.f. coordinate space with sufficient<br />
acceleration (position) and force amplitude to be measured accurately and sufficiently small<br />
positional amplitude to have a <strong>system</strong> with only moderate nonlinear effects. It is assumed<br />
that no relevant parasitic flexible modes are hit at this harmonic.<br />
Basically from calculation <strong>of</strong> the platform forces fp = J T l�x (sx)AdP i and position<br />
(sx = f ;1 (l)), the amplitudes <strong>of</strong> the ground harmonic signal parts can be determined. The<br />
amplitudes <strong>of</strong> the accelerations, xa, resulting from the ground harmonic with frequency ! s<br />
are easily obtained from the positional as j xa j=j ! 2<br />
s xa j. Now, taking all experiments<br />
together in a matrix equation,<br />
Mt = FapX ;1<br />
a � (4.31)<br />
Mt can be identified, with for the i’th experiment M txai fapi and the force basic harmonic<br />
amplitudes, fap, stacked in the columns <strong>of</strong> Fap and stacking the accelerational amplitudes,<br />
xa, into Xa.
4.2 Stewart platform model parameter identification 137<br />
mpp (40kNm,kN)<br />
pitch (3eps,m)<br />
6<br />
4<br />
2<br />
0<br />
−2<br />
−4<br />
−6<br />
0.2<br />
0.1<br />
0<br />
−0.1<br />
x 10 −3<br />
0 0.5 1<br />
−0.2<br />
0 0.5<br />
time (s)<br />
1<br />
1<br />
0.5<br />
0<br />
−0.5<br />
−1<br />
0.03<br />
0.02<br />
0.01<br />
0<br />
−0.01<br />
−0.02<br />
x 10 −4<br />
0 0.5 1<br />
−0.03<br />
0 0.5<br />
time (s)<br />
1<br />
Fig. 4.9: Decomposition <strong>of</strong> platform forces and position into ground harmonic and residual<br />
signals in a vibrational (pitch) test <strong>of</strong> the dummy platform without additional load<br />
(configuration A.) for mass matrix determination. Upper left plot gives positional<br />
ground harmonics for the second euler parameter (pitch in three times the euler<br />
parameter, eps) and the ground harmonics for the other d.o.f.’s. Upper right are<br />
the residual positional measurements during the test. Lower left are the platform<br />
forces/moments ground harmonics, normalised to the maximum force <strong>of</strong> one actuator<br />
(times 1m for the moments). The lower right plot provides the measured<br />
residual forces and moments.
138 4 Parameter identification and model validation<br />
Estimation in the presence <strong>of</strong> nonlinear terms<br />
To evaluate under what conditions a reasonable estimate <strong>of</strong> the SRS <strong>motion</strong> <strong>system</strong> mass<br />
matrix, Mt, results, the nonlinear model will be considered. As proposed in the previous<br />
chapter, the mass matrix is assumed constant in the platform frame i.e. the relative <strong>motion</strong> <strong>of</strong><br />
the actuator parts is not taken into account. Referring to (2.122) and including a dissipating<br />
velocity term, Bt(sx) _x,<br />
Mt(sx)x + Ct(sx� _x) _x + Bt(sx) _x + Gt(sx) =J T l�x (sx)fa<br />
it is immediately clear that even in order to identify the mass matrix, M t, locally at some<br />
sx, dynamic experiments (x 6= 0) have to be performed in which nonlinear terms are<br />
unavoidable. Fortunately, as the term, Ct, is quadratic in the velocity it does for most<br />
have second and higher harmonic signal components since 2 sin( ) cos( ) = sin(2 ) and<br />
2cos 2 ( ) = (1 + cos(2 ))). So in convolution with the ground harmonic, C t almost disappears.<br />
Further, the ground harmonic function can be split up into two parts. One in phase<br />
with the position and one shifted over =2 rad, or the part which is in phase with velocity.<br />
By only considering the part <strong>of</strong> the signal (force, position) which is in phase with position,<br />
also the dissipating forces drop out.<br />
In Fig. 4.9 the necessary measurements are provided for one typical experiment out <strong>of</strong><br />
six in estimating the mass matrix. Only taking the ground harmonic into account allows<br />
accurate determination <strong>of</strong> the ground harmonic accelerational amplitude. As the ground<br />
harmonics along all degrees <strong>of</strong> freedom are taken into account, no severe requirements hold<br />
for the <strong>control</strong>ler during the experiment.<br />
Another cause left <strong>of</strong> calculating a biased estimate <strong>of</strong> the mass matrix will be the dry<br />
friction terms. Also these are velocity (sign) related and will also mainly influence the<br />
ground harmonic which is =2 out <strong>of</strong> phase.<br />
In practice, the <strong>system</strong> will not move in one platform coordinate only. Often the parasitic<br />
<strong>motion</strong> will also be out <strong>of</strong> phase. The <strong>system</strong> will therefore move along an ellipsoidal<br />
trajectory instead <strong>of</strong> a line as can be observed from Fig. 4.9, which shows phase shifted<br />
positional ground harmonics. This will be a possible cause <strong>of</strong> still having friction like terms<br />
in phase with the ground harmonic used in the estimation procedure.<br />
Another nonlinear term to be considered is the position dependent jacobian, J l�x(sx).<br />
Linearising in a static pose, sxo, with static gravitational actuator forces, fao results in<br />
Mt x = J T (sxo) fa + 6 @Jl�sx(sxo)<br />
i=0<br />
(sxo) sxifao<br />
(4.32)<br />
@sxi<br />
With sxi sufficiently small, these terms can be neglected. To check what is sufficiently<br />
small, the partial derivative <strong>of</strong> the jacobian, Jl�sx to the states sx is to be evaluated. As<br />
given in (2.42), this jacobian consists <strong>of</strong> two kind <strong>of</strong> vectors. All the unit direction vectors,<br />
lni <strong>of</strong> the actuators and the vector products, T a i lni. Using the time derivative <strong>of</strong> ln given<br />
in (2.52), and decomposition with (2.47) the partial derivative to sx is given by<br />
@ln 1<br />
=<br />
@sx klk PlnJai�x<br />
I 0<br />
0 J!<br />
(4.33)
4.2 Stewart platform model parameter identification 139<br />
with the transform from euler parameter variations to angular velocity given by J ! =<br />
2G( ) (2.23). Now<br />
@(T a ln)<br />
@sx<br />
= T a<br />
@ln @T<br />
+<br />
@sx @sx a ln (4.34)<br />
where @T=@sx follows from _<br />
T = ~!T. Validation is easily done by calculation <strong>of</strong> the<br />
difference between two jacobians for which the pose is slightly altered.<br />
Experiment design<br />
Calculation <strong>of</strong> each @Jl�sx=@sxi-matrix in the neutral position gives matrices with roughly<br />
the same gains as Jl�sx itself. With the gravity forces at the same magnitude as the dynamics<br />
forces, the relative effect <strong>of</strong> the variation (to compensate for gravity) in the linearisation<br />
is nearly the same as the ratio <strong>of</strong> positional over accelerational amplitudes. As the test<br />
frequency becomes higher, this ratio decreases quadratically.<br />
In practice the test frequency <strong>of</strong> the ground harmonic, ! t is chosen somewhat below<br />
the lowest open loop ’rigid’ eigenfrequency. In this manner, it possible to successfully<br />
perform the test with a simple non model <strong>based</strong> <strong>control</strong>ler. In case <strong>of</strong> configuration A.<br />
and B., !t = 2 4 and for the dummy with additional load it was taken somewhat lower<br />
!t =2 =0:3. At these frequencies it is possible to perform the measurements with reasonable<br />
pressure (up till 25 % <strong>of</strong> full load) and velocity (up till 20 % <strong>of</strong> maximum velocity)<br />
amplitudes by requiring harmonic reference signals for each platform coordinate in a separate<br />
test <strong>of</strong> positional amplitude <strong>of</strong> 10mm in the translational and :003 for k 13k in<br />
rotational directions. Also an equivalent ratio <strong>of</strong> measurement errors in position (0.1 mm)<br />
and pressure ( 25 mV noise on 10V scale) in relation to the amplitudes is attained in this<br />
way (both 1%). Typical measurement signals, for the forces, S f , and positions, Sp, for such<br />
a test are given in Fig. 4.9. The actual test length was tl =5s and the measurements were<br />
sampled at 500 Hz.<br />
Determination <strong>of</strong> the ground harmonics<br />
The harmonic coefficients <strong>of</strong> the positional and force (pressure) measurements contain the<br />
information <strong>of</strong> the mass properties. They are calculated by using a matrix, H, with a basis<br />
function in each column over the time span 0�::: �tl sampled at the same frequency<br />
H =[1t ; :5tl sin(!tt) cos(!tt) sin(2!tt) cos(2!tt) :::] (4.35)<br />
Now the coefficient matrices for the platform forces C f and positions Cp are the least square<br />
estimates found by<br />
[Cf Cp] =(H T H) ;1 H T [Sf Sp] (4.36)<br />
For the ground harmonic (third, c 3, and fourth, c4 rows <strong>of</strong> the coefficient matrices), the<br />
phase, p, <strong>of</strong> the platform coordinate, which is tested, is calculated. The columns fap and<br />
xap <strong>of</strong> the matrices Fap and X <strong>of</strong> (4.31) are now found by<br />
[f T ap x T<br />
ap ]=real(e;j p ([c3f + jc4f c3p + jc4p])) (4.37)
140 4 Parameter identification and model validation<br />
Configuration Place in 6x6 A B C<br />
Mass matrix Mt<br />
mx [tons] (1,1) 2.6 2.0 4.3<br />
my [tons] (2,2) 2.6 2.0 4.3<br />
mz [tons] (3,3) 2.7 2.3 4.3<br />
Ixx [tons m 2 ] (4,4) 2.3 2.6 4.1<br />
Iyy [tons m 2 ] (5,5) 2.3 2.8 4.0<br />
Izz [tons m 2 ] (6,6) 3.8 3.5 6.7<br />
Ixz�zx [tons m 2 ] (4,5),(5,4) 0.0 0.3 0.0<br />
M15�51 [tons m] (1,5),(5,1) 0.0 0.0 0.1<br />
Table 4.2: The Simona Research Simulator identified nonzero inertial parameters. These<br />
are mainly appearing on the diagonal <strong>of</strong> the mass matrix, M t, apart from<br />
the Ixz�zx term in case <strong>of</strong> the shuttle (configuration B.). At least 1 % error<br />
(25 ; 50 kg) can be expected since this is the maximum singular value <strong>of</strong> the<br />
nonsymmetric part <strong>of</strong> the estimated mass matrix.<br />
In this way, a maximal part <strong>of</strong> the ground harmonic for each test direction is taken into<br />
account and only the part <strong>of</strong> the force and positional signal which are in phase with this<br />
<strong>motion</strong>.<br />
In Fig. 4.9 it is shown that a significant part <strong>of</strong> the forcing signal does not belong to the<br />
ground harmonic and will not be taken into account in the mass matrix estimation.<br />
Experimental results <strong>of</strong> the mass matrix estimation<br />
The estimated mass matrices for the three configurations are given in Table 4.2. An advantage<br />
<strong>of</strong> taking the identified centre <strong>of</strong> gravity as origin in the moving body frame is that the<br />
mass matrix becomes almost diagonal. The dummy platform has the advantage <strong>of</strong> having<br />
its main axes <strong>of</strong> inertia along the platform frame axes. For the shuttle, configuration B, the<br />
xz-plane is a plane <strong>of</strong> symmetry (making the y-axis a main axis <strong>of</strong> inertia, running I yz�zy<br />
and Ixy�yx zero) but also has a cross term Ixz�zx <strong>of</strong> approx. 250 kg in case <strong>of</strong> an empty<br />
shuttle.<br />
In principle, the mass matrix should be symmetric. The nonsymmetricity <strong>of</strong> the identified<br />
mass matrix can give some impression <strong>of</strong> the errors made due to measurement inaccuracy,<br />
etc. In case <strong>of</strong> configuration A., the maximum singular value <strong>of</strong> the non symmetric<br />
part is 26 kg(m 2 ) and in B. and C. it amounts to 50 kg i.e. 1 % <strong>of</strong> the estimate itself for the<br />
dummy platform and somewhat higher for the shuttle. This means no more accurate results<br />
were obtained than specified in Table 4.2.<br />
Resume<br />
A very compact test sequence was presented, which allows estimation <strong>of</strong> the relevant<br />
dynamic mechanical parameters <strong>of</strong> the Stewart platform in the presence <strong>of</strong> hydraulic
4.3 Frequency response model validation 141<br />
actuators.<br />
In practice five static poses (at least two in theory) measuring pressure and position,<br />
allowed identification <strong>of</strong> the total mass experienced and the position <strong>of</strong> the centre <strong>of</strong><br />
gravity.<br />
With six persistently exciting periodic <strong>motion</strong> tests, the inertial properties <strong>of</strong> the <strong>motion</strong><br />
<strong>system</strong> could be identified through a (6 by 6) mass matrix.<br />
By only taking the ground harmonic in phase with positional <strong>motion</strong>, velocity related<br />
(friction) and higher order terms (nonlinearities) have minimal influence on the<br />
procedure.<br />
Together with the earlier identified hydraulic actuator parameters, these mechanical<br />
parameters allow a reasonably tight fit to practice <strong>of</strong> the complex multivariable nonlinear<br />
hydraulically driven mechanical <strong>system</strong> model.<br />
In the next section, this will be validated in comparing the responses <strong>of</strong> the models derived<br />
with the parameters determined with ’open loop’ <strong>motion</strong> <strong>system</strong> frequency response<br />
measurements.<br />
4.3 Frequency response model validation<br />
At this point, the model structures have been formulated and the model parameters have<br />
been estimated. With this information, the calibrated models can be evaluated by comparison<br />
<strong>of</strong> model and actual responses. In this section the model open loop <strong>system</strong> characteristics<br />
will be validated in the frequency domain. Especially with mechanical <strong>system</strong>s, also<br />
hydraulically driven, many important properties are revealed considering responses to sinusoidal<br />
inputs.<br />
Characterising nonlinear <strong>system</strong>s by describing functions<br />
With frequency response measurements not only the input-output relation <strong>of</strong> a linear <strong>system</strong><br />
can uniquely be described but also many nonlinear <strong>system</strong> structures can be characterized<br />
by performing harmonic excitation. E.g. a sinusoidal input describing function identification<br />
procedure approximately characterises a nonlinear <strong>system</strong> by considering amplitude<br />
and phase <strong>of</strong> the response as a function <strong>of</strong> frequency and input sinusoidal amplitude, as<br />
extensively discussed in Van Schothorst [124]. This only requires a filter hypothesis, which<br />
says that the non-linear <strong>system</strong> has low-pass characteristics.<br />
Further, the linear dynamics <strong>of</strong> a non-linear <strong>system</strong> (for some operating point) can be<br />
defined as the describing function <strong>of</strong> the <strong>system</strong> as the amplitude approaches zero i.e. is<br />
sufficiently small. The describing function can be found measuring a frequency response<br />
such that the amplitude <strong>of</strong> the response is flat. This requires the design <strong>of</strong> an input amplitude<br />
filter found e.g. by iteratively measuring the frequency response.<br />
Appropriate experiments characterising hydraulically driven mechanical <strong>system</strong>s<br />
Systems experiencing structural vibrations such as mechanical <strong>system</strong>s are <strong>of</strong>ten charac-
142 4 Parameter identification and model validation<br />
terised experimentally using modal analysis techniques. A large part <strong>of</strong> the modal analysis<br />
also heavily relies on measuring frequency responses [39, 90]. With modal analysis, the velocity<br />
response <strong>of</strong> a mechanical <strong>system</strong> to force excitation is called mobility. With a single<br />
degree <strong>of</strong> freedom (one mode, vibration) e.g. a mass-spring-damper <strong>system</strong>, the mobility<br />
frequency response has a zero at zero and at infinite frequency and a complex pole pair<br />
assuming reasonably low damping with an undamped eigenfrequency at the square root <strong>of</strong><br />
the mass vs. spring stiffness ratio.<br />
Similar responses can be expected measuring the response <strong>of</strong> our <strong>motion</strong> <strong>system</strong> from<br />
input valve voltage to the pressure difference over the valve. As discussed in the previous<br />
chapter, the hydraulically driven mechanical <strong>system</strong> has roughly a mobility like characteristic<br />
if the response <strong>of</strong> the pressure difference over the actuator compartments is considered<br />
from input actuator valve flow. Measuring from input valve voltage sets a low pass filter<br />
in series with this <strong>system</strong> thereby fulfilling the filter hypothesis. Viscous damping not only<br />
damps the resonance but also shifts the zero at zero frequency to the left in the complex<br />
plane. As shown in Section 3.3.1 considering the models, the response <strong>of</strong> the multi degree<br />
<strong>of</strong> freedom multi input valve/output pressure difference <strong>of</strong> the <strong>motion</strong> <strong>system</strong> at hand is expected<br />
to have a mobility like characteristic with multiple modes. Also with parasitic effects<br />
from transmission lines or mechanical flexibility, this does not change, apart from inducing<br />
additional anti-resonance and resonance pairs.<br />
4.3.1 Frequency response dummy platform<br />
The models <strong>of</strong> the previous chapter appointed the rigid body modes as the main characteristics<br />
<strong>of</strong> the hydraulically driven mechanical <strong>system</strong>. In this section, this will be evaluated by<br />
the actual frequency responses <strong>of</strong> the dummy platform (configuration A. with no additional<br />
load). In this configuration relatively little influence <strong>of</strong> parasitic effects, e.g. stemming from<br />
flexible deformation, is expected to enter the frequency area <strong>of</strong> interest i.e. the frequency<br />
area up till 50 Hz where the rigid body modes reside.<br />
With the inertial parameters found in the previous section and the parameters for the hydraulic<br />
actuators given in Table 3.2, the theoretic basic model structure <strong>of</strong> the hydraulically<br />
driven <strong>motion</strong> <strong>system</strong> given in Section 3.4.1 is specified. Calculation <strong>of</strong> the eigenfrequencies<br />
<strong>of</strong> the rigid body modes in the neutral position for the dummy platform, configuration A.,<br />
results in 6:8 Hz for a surge/pitch and sway/roll mode, 13:6 Hz for a yaw mode, 17:7 Hz<br />
for heave and 23:4 Hz for the different sign surge/pitch and sway/roll modes.<br />
With a HP frequency analyzer ([91]) the relation at the neutral pose from each valve<br />
steering voltage to each actuator pressure difference was measured between 5 Hz and<br />
50 Hz by a swept sine at 1:5% <strong>of</strong> full input voltage. With the analyzer 201 logarithmically<br />
spaced datapoints per input output relation were obtained. As this analyzer only resolves<br />
one input output relation at the time, 36 separate experiments were performed. Most <strong>of</strong> the<br />
frequency domain measurements are given in 6x6 form in Appendix A.<br />
The <strong>system</strong>s pose was stabilized by a moderate position feedback <strong>of</strong> gain k p = 0:2<br />
resulting in a bandwidth <strong>of</strong> 0:05 Hz. At 5 Hz the influence <strong>of</strong> this feedback can be<br />
1% and grows at the first resonance to maximally 2%. At the frequencies larger than<br />
10 Hz it will be as little as 0:1% and decrease quadratically. This influence is estimated
4.3 Frequency response model validation 143<br />
Amplitude<br />
Phase (deg)<br />
10 1<br />
10 0<br />
0<br />
−200<br />
−400<br />
−600<br />
10 1<br />
10 1<br />
Frequency (Hz)<br />
10 1<br />
10 0<br />
0<br />
−200<br />
−400<br />
−600<br />
10 1<br />
10 1<br />
Frequency (Hz)<br />
Fig. 4.10: Bode plots <strong>of</strong> the highly interacting responses from the first valve input voltage<br />
to the six pressure input differences (dPi), measured at the dummy platform<br />
(configuration A.) at the left. On the right side the model response (18 th order)<br />
is given incorporating the rigid body modes and three additional modes for the<br />
foundation. Six by six separated Bode plots can be found in Fig. A.1 and Fig. A.2<br />
<strong>of</strong> Appendix A<br />
by taking the measured ’closed loop’ gain, k m, and recalling the open loop response gain,<br />
kg, to be measured is found through kg = km(1 ; kckm) -1 with kc, the position loop gain<br />
including the transfer function from (normalized) pressure to position. For small (complex)<br />
values, kckm is the relative gain variation.<br />
j kckm j = kpAdpnormkm(m! 2 ) ;1<br />
= 0:2 25 10 -4 200 10 5 7=(8:8 10 3 (2 5) 2 )<br />
= 0:008<br />
(4.38)<br />
The measured Bode plot <strong>of</strong> the measurement from the first valve input to the six pressure<br />
differences is given in the left column <strong>of</strong> Fig. 4.10. The predicted rigid body modes are<br />
present and additionally some parasitic resonances are present between 10 ; 13 Hz and at
144 4 Parameter identification and model validation<br />
Direction Mfii [tons� tons m 2 ] !fi [Hz] fi<br />
xf 45 2 10:5 0:05<br />
yf 45 2 10:0 0:035<br />
f 140 2 12:5 0:05<br />
Table 4.3: Parameters taken for the foundation.<br />
approx. 30 Hz.<br />
Additional measurements at the floor, which will be discussed in the next section,<br />
pointed out that the vibrations at 10 ; 13 Hz could be assigned to the foundation. By modelling<br />
the foundation as a 3-d.o.f mass/spring/damper <strong>system</strong> as modelled in Section 3.4.5<br />
with the parameters <strong>of</strong> Table 4.3, an 18 th -order model results.<br />
In general the damping structure <strong>of</strong> mechanical <strong>system</strong>s is hard to predict theoretically.<br />
The damping <strong>of</strong> the first three rigid body modes is well described by the experimentally<br />
found values for leakage and viscous friction <strong>of</strong> the hydraulic actuators. For the three highest<br />
frequency modes some additional damping had to be introduced.<br />
This can be explained by the fact that the gimbal rotation is probably also dissipating<br />
energy and not explicitly taken into account. Further, it is observed in modal analysis that<br />
damping does not always act as the linear viscous effect, which is part <strong>of</strong> the model used<br />
here. Alternatively, hysteretic damping structures are proposed in [39, 90], in which the<br />
frequency dependence <strong>of</strong> the dissipation is changed. A drawback <strong>of</strong> that structure is that it<br />
can not be incorporated in a state space (time-domain) model. Therefore hysteretic damping<br />
has not been included in the models proposed in this thesis for <strong>motion</strong> <strong>system</strong>s.<br />
The Bode plot from the first valve input to the six pressure differences <strong>of</strong> the (6 6)<br />
18 th -order model is given in the right column <strong>of</strong> Fig. 4.10. The measured transfer functions<br />
are fully shown in Fig. A.1 and Fig. A.2 <strong>of</strong> Appendix A. The model gives a reasonably<br />
close description <strong>of</strong> the characteristics observed from the measurements. Both the phase as<br />
amplitude frequency responses are very much alike. Note that only the phase <strong>of</strong> the first<br />
input to the first output remains within 90 necessary for passivity.<br />
The largest discrepancy can be observed at 30 Hz where the measurements point out<br />
an additional resonance, which is not incorporated in the model. Limited stiffness <strong>of</strong> the<br />
actuators in the radial direction are a possible cause <strong>of</strong> this vibration but this could not be<br />
confirmed by some additional measurements <strong>of</strong> an accelerometer waxed at the actuators.<br />
Further, the additional phase lag coming in at frequencies 40 Hz can be assigned to<br />
the valve characteristic. The model including this effect will be evaluated with the shuttle<br />
(configuration B.). The frequency responses <strong>of</strong> that <strong>system</strong> were measured up till 500 Hz.<br />
Decoupled response by coordinate transformation<br />
In the frequency responses <strong>of</strong> Fig. 4.10 is it not immediately clear how many modes are to be<br />
used to the describe the 6 6 <strong>system</strong>. Now we can use the basic model structure we obtained<br />
for hydraulically driven mechanical <strong>system</strong>s in Section 3.4.1. An important <strong>system</strong> property<br />
<strong>of</strong> the model is the coordinate transformation along the rigid body modal directions, which
4.3 Frequency response model validation 145<br />
Amplitude<br />
Phase (deg)<br />
10 1<br />
10 0<br />
100<br />
50<br />
0<br />
−50<br />
−100<br />
10 1<br />
10 1<br />
Frequency (Hz)<br />
10 1<br />
10 0<br />
100<br />
50<br />
0<br />
−50<br />
−100<br />
10 1<br />
10 1<br />
Frequency (Hz)<br />
Fig. 4.11: Bode plots along the six estimated decoupling rigid body modal directions (without<br />
parasitic vibrations) from redirecting the input valve voltages and valve pressure<br />
differences measured (left) and modelled (right). In the upper left, the<br />
largest singular value over the frequency <strong>of</strong> the error <strong>system</strong> regarding all the<br />
nondiagonal elements is given in the dashed plot. Six by six separated Bode<br />
plots can be found in Fig. A.3 and Fig. A.4 <strong>of</strong> Appendix A<br />
decouples the <strong>system</strong> into six independent <strong>system</strong>s. Each <strong>of</strong> which incorporates one <strong>of</strong> the<br />
rigid body modes.<br />
With additional parasitic modes this decoupling is not exact anymore but in the previous<br />
chapter it was found that at least in the neutral position, this transformation still puts the<br />
main modes on the diagonal. This transformation with an unitary matrix, U, can be derived<br />
from the mass matrix evaluated from the actuators as defined in (3.44). A quick impression<br />
for these modes can attained by evaluating the singular values for each separate frequency<br />
measured.<br />
In Fig. 4.11, the frequency responses <strong>of</strong> the diagonal elements <strong>of</strong> the 6 6 transformed<br />
transfer function are plotted. (Full transfer functions are shown in Fig. A.3 and Fig. A.4<br />
<strong>of</strong> Appendix A. In the left column, the measured response is given and on the right, the<br />
model response is drawn. Still, the model and the measured response have very much the<br />
same characteristics. Having the rigid body modes decoupled on the diagonal <strong>of</strong> the transfer<br />
function enables a much easier interpretation. E.g. the phases <strong>of</strong> the <strong>system</strong> are now clearly<br />
90 from which the conclusion can be drawn that the linearised <strong>system</strong> is passive at this<br />
point <strong>of</strong> operation as was predicted from the non-linear <strong>system</strong>. As predicted, the dynamics
146 4 Parameter identification and model validation<br />
<strong>of</strong> the foundation and other additional mechanical resonances do not influence this property.<br />
Of course, the valve characteristics will make the <strong>system</strong> lose its property <strong>of</strong> being strict<br />
positive real and cause the additional phase shift at the higher frequencies <strong>of</strong> the measured<br />
response.<br />
In this decoupled form, the two frequency responses, the measured response and the<br />
one derived from the model, can be compared in more detail. Further, it is easier to adjust<br />
the model in this form in order to let the model more tightly fit reality. For example the<br />
damping structure is most conveniently changed in these coordinates since each rigid mode<br />
can be varied independently. Also parameters like massmodes (eigenvalues massmatrix)<br />
can be adjusted. In the model, the mass viewed along the z-direction can be changed to<br />
a 13% higher value to have higher correspondence <strong>of</strong> the modelled and measured z-mode.<br />
Probably the actuator masses are somewhat higher than expected.<br />
Important is the fact that the non diagonal part <strong>of</strong> the responses is almost neglectable<br />
as the largest singular value <strong>of</strong> this part <strong>of</strong> the transfer is much smaller than the rigid body<br />
modes, which appear separated on the diagonal transfer functions. In the measured responses<br />
it is clear that there is some distortion at the resonance frequencies coming into the<br />
low gain parts <strong>of</strong> the other modes.<br />
Resuming, the model which describes<br />
the rigid body mechanical <strong>system</strong><br />
together with the basic hydraulic <strong>system</strong>,<br />
and in this environment some additional dynamics from the non rigid foundation,<br />
gives a reasonable description <strong>of</strong> the <strong>system</strong> over the frequency range, which will be important<br />
for <strong>control</strong> (5 ; 50 Hz). Both the rigid modal eigenfrequencies and directions can<br />
be predicted from the model structure together with the kinematic and inertial model parameters<br />
identified. The damping structure had to be adapted somewhat using the measured<br />
frequency responses. Something which is not unusual since the damping structure <strong>of</strong> a<br />
mechanical <strong>system</strong> is hard to predict.<br />
Frequency domain identification<br />
A more tightly fitted local description <strong>of</strong> the <strong>system</strong> at the neutral pose can be obtained by<br />
frequency domain identification. A flexible tool was used to identify a multivariable model<br />
from frequency domain data with the method <strong>of</strong> De Callafon et al [28] as already reported<br />
in [75]. No reasonable results could be obtained, however, from directly fitting the original<br />
6 by 6 by 201 datapoints. Results improved drastically by separately identifying the rows <strong>of</strong><br />
the data transformed by the coordinate transformation, U. Rows instead <strong>of</strong> columns were<br />
taken as the singular directions could be predicted more closely at the directly measured<br />
output than at the input through the valve.<br />
As these models will usually contain redundant modes from non exact decoupling, especially<br />
at the dominant lowest rigid modes at 6:8 Hz, the full multivariable model can be<br />
reduced. By using the block balancing approach <strong>of</strong> Wortelboer [157], the reduction problem<br />
can be transformed to finding a good multivariable representation <strong>of</strong> each observed mode.
4.3 Frequency response model validation 147<br />
Amplitude<br />
10 0<br />
10 −1<br />
10 1<br />
Input filter shuttle frequency response measurements<br />
Frequency (Hz)<br />
Fig. 4.12: Bode amplitude plot <strong>of</strong> the 6 th -order experiment notch filter.<br />
The resulting 20 th -order model describes both the six rigid modes as four additional<br />
parasitic modes, including the modes at 30 Hz which were not incorporated in the more<br />
global model with physical interpretation. The largest singular value <strong>of</strong> the difference between<br />
model and data at each frequency at which the data was measured is equivalent with<br />
the dashed line in the upper left <strong>of</strong> Fig. 4.11. Thus the error remains well below the relevant<br />
dynamics.<br />
With the inverse coordinate transformation, U T , this model can be transformed to the<br />
actuator coordinates from which the original measurements were taken. The error is approximately<br />
10%.<br />
More research into identification <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> has to point out<br />
in what way models or model parameters can be identified which provide a tight description<br />
<strong>of</strong> the <strong>system</strong> on a global scale, e.g. over all platform positions.<br />
4.3.2 Additional dynamics into the higher frequency area<br />
With the shuttle on top <strong>of</strong> the <strong>motion</strong> <strong>system</strong>, Configuration B., it was decided to measure<br />
the frequency responses over the wide frequency area between 5 Hz and 500 Hz again<br />
with a sine (up) sweep <strong>of</strong> 201 logarithmically spaced frequencies. In this way the possible<br />
flexible modes <strong>of</strong> the shuttle and the influence <strong>of</strong> the transmission lines on the pressure<br />
dynamics, could be observed and evaluated with the <strong>system</strong> characteristics predicted by the<br />
models obtained in the previous chapter.<br />
In performing the experiments, more care w.r.t. the dummy platform measurements had<br />
to be taken not to hit resonances to badly. Especially because the pilot seats, experiencing<br />
some play in the mounting, appeared to have their resonant behaviour close to the main<br />
rigid modes <strong>of</strong> the shuttle. Moreover, the high frequency area, containing the transmission<br />
line dynamics, did not allow, already for the audible noise level alone, input voltage levels<br />
necessary to measure the low frequecy rigid mode area with sufficient signal/noise ratio.<br />
A 6 th -order experiment filter was designed to obtain a sufficiently flat response level,<br />
which is also favourable in spreading the relative effect <strong>of</strong> possible nonlinearities. The measured<br />
bode amplitude plot <strong>of</strong> this filter is drawn in Fig. 4.12. Having the poles <strong>of</strong> the filter<br />
10 2
148 4 Parameter identification and model validation<br />
!p1 !p2 !p3 !n1 !n2<br />
2 5 2 35 2 100 2 7:8 2 42<br />
p1 p2 p3 n1 n2<br />
0:7 0:7 0:7 0:25 0:25<br />
Table 4.4: The parameters <strong>of</strong> the input frequency response measurement filter.<br />
appear just before the lightly damped (notching) zeros on 7:8Hz and 42Hz, gradually reduces<br />
the amplitude level and above 100Hz filters more rigorously. The filter was designed<br />
in the continuous time domain and converted into a digital version with a sample frequency<br />
<strong>of</strong> 5kHz by a bilinear transformation (in Matlab) for implementation on the <strong>motion</strong> <strong>control</strong><br />
computer. By measuring the response <strong>of</strong> the filter implemented on the computer, the <strong>system</strong><br />
frequency responses can be adequately compensated for. The input level <strong>of</strong> the experiments<br />
could be set on 2 % but the filter already starts to reduce this level at 6Hz.<br />
The continuous version <strong>of</strong> the experiment filter, U f (s) is given by<br />
Uf (s) = (s2 =! 2 n1 +2 n1s=!n1 +1)<br />
(s 2 =! 2<br />
p1 +2 p1s=!p1 +1)<br />
(s2 =! 2<br />
n2 +2 n2s=!n2 +1)<br />
(s2 =! 2<br />
p2 +2 p2s=!p2 + 1)(s2 =! 2<br />
p3 +2 p3s=!p3 +1)<br />
(4.39)<br />
with the parameters specified in Table 4.4.<br />
The direct measurements <strong>of</strong> the <strong>system</strong> with the shuttle from the actuator inputs to the<br />
input pressure differences show an even higher number <strong>of</strong> resonances than those with the<br />
dummy platform. However, the structure <strong>of</strong> the <strong>system</strong> characteristics is revealed by decomposition<br />
along the eigen directions <strong>of</strong> the rigid modes predicted through the (rigid body)<br />
model <strong>of</strong> the shuttle using the mass properties identified by the experiments discussed in the<br />
previous section. This requires pre and post multiplication <strong>of</strong> the measured 6 by 6 transfer<br />
function by a constant unitary matrix Ush�n and its transpose.<br />
Also with the shuttle and over the larger frequency area, this transformation puts the<br />
dominant responses on the diagonal, which show only one rigid mode per direction, while<br />
all 36 directly measured responses contain every mode in principal. This result and the<br />
model predicted frequency responses are given in Fig. 4.13 and in Fig. A.7 and Fig. A.8<br />
<strong>of</strong> appendix A all plots are given seperately, while Fig. A.5 and Fig. A.6 <strong>of</strong> Appendix A<br />
provide the Bode plots for the untransformed frequency data.<br />
Due to the nondiagonal inertia matrix, there are more platform directions involved in<br />
each mode considering the neutral pose than in case <strong>of</strong> the dummy platform. Still, in this<br />
pose, one can discriminate between symmetric (x� z� -related) and non-symmetric (y� � -<br />
related) modes. Premultiplying the Ush�n-matrix, whose columns describe actuator displacements,<br />
by the inverse jacobian results in a column wise description <strong>of</strong> the modal direction<br />
in the platform coordinates. Rescaling the columns to a (2-)norm <strong>of</strong> one, by postmulti-
4.3 Frequency response model validation 149<br />
Amplitude<br />
Phase (deg)<br />
10 1<br />
10 0<br />
10 −1<br />
100<br />
0<br />
−100<br />
−200<br />
−300<br />
10 1<br />
10 1<br />
10 2<br />
10 2<br />
Frequency (Hz)<br />
10 1<br />
10 0<br />
10 −1<br />
100<br />
0<br />
−100<br />
−200<br />
−300<br />
10 1<br />
10 1<br />
10 2<br />
10 2<br />
Frequency (Hz)<br />
Fig. 4.13: Bode plots <strong>of</strong> the shuttle response redirected along the six estimated decoupling<br />
rigid body modal directions considering the valve input voltages and the valve<br />
pressure difference responses measured (left) and modelled (right). Six by six<br />
separated Bode plots can be found in Fig. A.7 and Fig. A.8 <strong>of</strong> Appendix A<br />
plying with Ncsc gives:<br />
J ;1<br />
lx Ush�nNcsc =<br />
2<br />
6<br />
4<br />
0:987 0:000 0:000 0:083 ;0:078 0:000<br />
0:000 0:988 ;0:032 0:000 0:000 ;0:064<br />
;0:026 0:000 0:000 0:828 0:676 0:000<br />
0:000 0:153 0:089 0:000 0:000 0:988<br />
;0:159 0:000 0:000 0:554 ;0:733 0:000<br />
0:000 ;0:022 0:996 0:000 0:000 ;0:139<br />
3<br />
7<br />
5<br />
(4.40)<br />
The eigenfrequencies, which belong to these modal directions, are 7:3, 7:3, 14:7, 20:3, 22:0,<br />
24:2 Hz. The (60 th -order) model predicts these frequencies (and its directions) appropriately<br />
as are the transmission lines anti-resonances at approximately 70Hz and the characteristic<br />
due to the (anti)-resonances at 200Hz. This does not include the exact behaviour at<br />
the resonant transmission line frequencies since only the actuator cylinder pressure difference<br />
and not the absolute pressures were taken into account as was discussed in the previous<br />
chapter.
150 4 Parameter identification and model validation<br />
The transfer function from valves flow to input pressure differences is passive despite<br />
the transmission lines, flexible shuttle modes and moving foundation. In Fig. 4.13 this<br />
clearly shows by the phase moving between =2 rad. The valve voltage to valve flow<br />
characteristic with two poles modelled at 150Hz results in the additional phase shift, which<br />
(together with the transmission line resonances and flexible shuttle modes) can result in<br />
stability problems when using pressure feedback.<br />
Another property, which clearly shows, is that the hydraulic actuators are velocity generators<br />
at low frequencies below the rigid body resonance. The Bode plots show that this<br />
requires different pressure levels since the mass experienced along the modal directions<br />
varies. Above the resonant frequencies, the actuators become pressure difference derivative<br />
generators and in this respect the fact that all actuators are similar causes indifference for<br />
direction considering the amplitudes. A dual effect will show in considering the velocity<br />
(accelerating) behaviour over the frequency using constant valve input amplitude.<br />
The characteristics <strong>of</strong> the foundation having resonant frequencies at 10:2, 10:7 and<br />
12:7Hz do show the same characteristics in the model and measurements. The model parameters<br />
and the evaluation <strong>of</strong> the foundation causing these resonances were obtained by<br />
performing measurements at the foundation itself, which will be discussed in the next part<br />
<strong>of</strong> this section.<br />
The main discrepancy between the model and the actual <strong>system</strong> is the omission <strong>of</strong> the<br />
flexible modes which show at 33Hz, again, as with the dummy platform probably due to<br />
the radial stiffness <strong>of</strong> the actuators. Further, new resonances show up most prominently at<br />
43� 57 and 65Hz and are still visible around 100Hz. These resonances were most probably<br />
caused by the flexibility <strong>of</strong> the shuttle. To confirm this, accelerometers measurements were<br />
performed at several points <strong>of</strong> the shuttle.<br />
Although the characteristics <strong>of</strong> the pressure measurements in Fig. 4.13 are for most<br />
well contained in the model, the model does not predict the measurements very accurately<br />
considering the numerical values. This probably requires a more thorough identification<br />
procedure and/or more research into the nonlinear behaviour <strong>of</strong> the <strong>system</strong>. The Bode response<br />
is <strong>based</strong> on a linearised version <strong>of</strong> the model. E.g. the nonlinear valve characteristics<br />
could be a cause <strong>of</strong> some <strong>of</strong> the discrepancies.<br />
The vibrating foundation<br />
As already shown in several Bode plots, the vibrating foundation is part <strong>of</strong> the model according<br />
to the equations presented in Section 2.4.7. In principle, the parasitic resonance,<br />
which shows up in e.g. Fig. 4.13, can be caused by any flexibly attached mass at either<br />
side <strong>of</strong> the platform. In experiments and demonstrations it was readily verifiable that also<br />
the floor was moving if the <strong>motion</strong> <strong>system</strong> was made to vibrate at frequencies approaching<br />
10Hz.<br />
To confirm that it was actually the concrete foundation, which is approx. 20 times as<br />
heavy as the shuttle <strong>of</strong> Configuration B., which causes the specific behaviour, and to help<br />
identifying the model parameters and structure, experiments were set up measuring the<br />
acceleration at the floor itself. Creating the experimental environment to perform the measurements<br />
for closer analysis <strong>of</strong> parasitic mechanical behaviour <strong>of</strong> the <strong>system</strong> (so also those
4.3 Frequency response model validation 151<br />
10 0<br />
10 −1<br />
10 −2<br />
10 0<br />
10 −1<br />
10 −2<br />
10 1<br />
10 1<br />
Fig. 4.14: Bode plot amplitude responses <strong>of</strong> measured (dashed) and modelled accelerations<br />
along the xf (upper left), yf (upper right), x2f at (0,-2) (lower right) and constructed<br />
angular acceleration around the z-axis, ! (lower left) due to input valve<br />
vibration along the first three ’rigid modes’.<br />
100<br />
0<br />
−100<br />
−200<br />
−300<br />
100<br />
0<br />
−100<br />
−200<br />
−300<br />
10 1<br />
10 1<br />
Fig. 4.15: Bode plot phase responses <strong>of</strong> measured (dashed) and modelled accelerations<br />
along the x (upper left), y (upper right), x 2 at (0,2) (lower right) and constructed<br />
angular acceleration around the z-axis, ! (lower left) due to input valve vibration<br />
along the first three ’rigid modes’.<br />
10 0<br />
10 −1<br />
10 −2<br />
10 0<br />
10 −1<br />
10 −2<br />
100<br />
0<br />
−100<br />
−200<br />
−300<br />
100<br />
0<br />
−100<br />
−200<br />
−300<br />
10 1<br />
10 1<br />
10 1<br />
10 1
152 4 Parameter identification and model validation<br />
in the shuttle itself) have been part <strong>of</strong> a masters’ project described in [41].<br />
Use was made <strong>of</strong> the (at that time) new compact accelerometers <strong>of</strong> Analog Devices, the<br />
ADXL05 1 , which evolved from airbag <strong>system</strong> initiators and now measure accelerations at a<br />
range <strong>of</strong> 5g at a resolution <strong>of</strong> 5mg from DC into reasonable high frequencies (10 kHz).<br />
Three <strong>of</strong> these sensors were mounted orthogonally in a small package, such that acceleration<br />
measurement in all three directions, xf , yf and zf was possible.<br />
This package was mounted on the floor in between the <strong>motion</strong> <strong>system</strong> on three places<br />
performing 6(x6) experiments in total assuming that the floor itself was rigid but flexibly<br />
attached to the inertial frame. Right in between the lower gimbal points as the frame given<br />
in Fig. 2.13, all three accelerations were measured, again while performing a sine sweep<br />
at the valve input voltage for each actuator at the time (between 5Hz and 500Hz usually<br />
stopping the experiment if no <strong>motion</strong> could be observed anymore above 50Hz). Again, the<br />
<strong>motion</strong> <strong>system</strong> provides a nice 6-d.o.f. force inductor to perform the required experiment 2 .<br />
Recording both accelerations and the forces induced is required in identifying the mass<br />
and spring stiffness, which can hardly be identified if only the resonant behaviour in the<br />
e.g. pressure dynamics is taken into account.<br />
At the coordinates (0� -2)[m] and (2� 0)[m] taken in the frame <strong>of</strong> Fig. 2.13, measurements<br />
were performed in x and z and z direction respectively. All measurements in z-direction<br />
did not show any acceleration (apart from gravitation), allowing to take only three directions<br />
(xf ,yf, f ) into account.<br />
In Fig. 4.14 and Fig. 4.15, the bode amplitude and phase plots <strong>of</strong> the model and measurements<br />
in the xf and yf direction are given as well as for the x2f and reconstructed f<br />
direction due to the inputs along the rigid modal directions most concerned with the <strong>motion</strong><br />
to be observed. Clearly, in Fig. 4.14 and Fig. 4.15 both the rigid shuttle modes can be observed<br />
along with the mode due to the fact that the foundation is not rigidly attached to the<br />
floor. In the neutral state, there is no interaction between the rigid shuttle modes due to the<br />
floor but in other platform poses it can.<br />
Although the accelerometers were relatively noisy, the characteristics observed in the<br />
models can be recognized in the measurements. The model fails to identify a secondary<br />
mode in x-direction (at approx 12 Hz) which is hardly affecting the yaw mode (as it does<br />
with the model). The most relevant behaviour is, however, well described by the model. No<br />
additional effort has been done to more accurately describe the dynamics <strong>of</strong> the foundation<br />
as the <strong>system</strong> had to be moved to another building in the end.<br />
The parameters used in the model <strong>of</strong> the foundation are given in Table 4.5. The total <strong>system</strong><br />
eigenvalues, which are due to this part <strong>of</strong> the mechanics, are shifted somewhat w.r.t. the<br />
values given here for the foundation alone. The direction <strong>of</strong> this shift naturally depends on<br />
the eigenfrequencies <strong>of</strong> the rigid body modes <strong>of</strong> the platform (up in case <strong>of</strong> x and y, down<br />
in case <strong>of</strong> ).<br />
The experiments point at the importance <strong>of</strong> taking into account the dynamic behaviour<br />
<strong>of</strong> the construction <strong>of</strong> the building, foundation, best already in design, in order to prevent<br />
1This sensor is not available anymore and has been replaced by the ADXL105 having higher resolution and<br />
temperature compensation<br />
2As already put forward, the hydraulic actuators are no force generators by themselves. Probably, the experiments<br />
could be enhanced by using some sort <strong>of</strong> force <strong>control</strong> loop.
4.3 Frequency response model validation 153<br />
Phase (deg)<br />
Amplitude (m/s2/10V)<br />
Direction Mfii [tons� tons m 2 ] !fi [Hz] fi<br />
xf 45 2 10:5 0:035<br />
10 2<br />
10 1<br />
10 0<br />
0<br />
−200<br />
−400<br />
−600<br />
yf 45 2 10:0 0:03<br />
f 182 2 12:5 0:07<br />
Table 4.5: Parameters taken for the foundation.<br />
10 1<br />
10 1<br />
Frequency (Hz)<br />
Fig. 4.16: A typical frequency response with a Bode plot <strong>of</strong> measured and reconstructed<br />
(from the other measurements) acceleration approx. along the y-axis in the back<br />
left floor position due to a sinusoidal sweep <strong>of</strong> the input voltage at the 4 th actuator.<br />
this part <strong>of</strong> the <strong>system</strong> influence the characteristics <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> in a<br />
relevant frequency area as it did with the experiments performed at the Central Workshop<br />
<strong>of</strong> mechanical engineering.<br />
Accelerating a flexible shuttle<br />
At this point only measurements were presented, which indirectly provide insight in the behaviour<br />
<strong>of</strong> acceleration in the shuttle. By mounting a (minimal) number (<strong>of</strong> six) accelerometers<br />
onto the shuttle, the rigid body <strong>motion</strong>, felt by a person sitting in the <strong>simulator</strong>, can be<br />
observed. Additional sensors can provide insight into the parasitic flexibility resulting from<br />
the shuttle itself.<br />
10 2<br />
10 2
154 4 Parameter identification and model validation<br />
10 2<br />
10 1<br />
10 0<br />
10 2<br />
10 1<br />
10 0<br />
10 2<br />
10 1<br />
10 0<br />
10 1<br />
10 1<br />
10 1<br />
10 2<br />
10 2<br />
10 2<br />
Frequency (Hz)<br />
10 2<br />
10 1<br />
10 0<br />
10 2<br />
10 1<br />
10 0<br />
10 2<br />
10 1<br />
10 0<br />
10 1<br />
10 1<br />
10 1<br />
10 2<br />
10 2<br />
10 2<br />
Frequency (Hz)<br />
Fig. 4.17: Bode amplitude plots <strong>of</strong> the platform acceleration response due to the valve inputs<br />
along the rigid body modal input directions. Left column x� y� z, and on the<br />
right _!x� _!y� _!z. For each plot one <strong>of</strong> the modal direction is not dashed. It is<br />
clear that strong interaction and hard to analyze responses result from evaluation<br />
along the platform directions. Much stronger decoupling will result evaluating<br />
along the rigid body modal directions at the output also.<br />
Reconstruction <strong>of</strong> platform acceleration<br />
To reconstruct the platform acceleration, first the gain, direction and positional parameters<br />
<strong>of</strong> the accelerometers used, have to be identified. By using a method in which the procedure<br />
in identifying the platform mass matrix and centre <strong>of</strong> gravity are combined, the sensors can<br />
be characterised. In general, an accelerometer with gain k, mounted at position, r, w.r.t. a<br />
reference frame and directed along the unit vector, n, will output a signal, s, determined by<br />
s = kn T (c ; g + _! r +! (! r)) (4.41)<br />
where c and _! provide the reference frame translational and rotational acceleration.<br />
By performing sinusoidal <strong>motion</strong>, the accelerational amplitudes, ( c� _!)i, in six directions,<br />
which span the <strong>motion</strong> space, can robustly be derived from the position measurements.<br />
By correlating with the basic sinusoidal frequency, ! t, noise and the quadratic
4.3 Frequency response model validation 155<br />
velocity term will be very small as in (4.36) and will be neglected. (4.41) also holds for<br />
the ground harmonic amplitudes. With some manipulation using (4.41), the accelerometer<br />
ground harmonic output amplitudes <strong>of</strong> all the experiments, s, can be written as<br />
s = Am� (4.42)<br />
the multiplication <strong>of</strong> a known matrix, A, <strong>of</strong> platform accelerational ground harmonic amplitudes<br />
<strong>of</strong> which each i’th row is given by<br />
h<br />
Ai� = aT c�i aT i<br />
_!�i � (4.43)<br />
and the measurement device vector, m, with the yet unknown accelerometer parameters,<br />
direction, n, gain, k, and line <strong>of</strong> action, r, given by<br />
h<br />
m = knT (kn ( g<br />
! 2 t ; r))T i T<br />
(4.44)<br />
Gravity gives an <strong>of</strong>fset to the line <strong>of</strong> action <strong>of</strong> the sensor which is inversely proportional with<br />
the squared frequency, ! 2 t , at which the test has been performed. This <strong>of</strong>fset results from<br />
the fact that gravity is constant in the inertial frame and the direction <strong>of</strong> the accelerometers,<br />
n, moves (and thus rotates) with the platform. If e.g. pitch rotation occurs, gravity will<br />
show at the accelerometer measuring in x-direction proportionally to the sine <strong>of</strong> the angle<br />
rotated. For small angles the sine operation can be dropped and the angle decreases with<br />
the squared frequency with constant rotational acceleration. By inversion <strong>of</strong> A, the gain, k,<br />
and direction, n, <strong>of</strong> the accelerometers can be determined.<br />
m = A ;1 s (4.45)<br />
The position, r, can not, just as in the determination <strong>of</strong> the centre <strong>of</strong> gravity. Using only<br />
the linear terms (not the quadratic velocity term), the sensor can be anywhere on a line<br />
parametrized by the free variable , described by<br />
with<br />
r( )= o + n (4.46)<br />
1<br />
o = ;<br />
mT 11m11 (m11 Pnm21) (4.47)<br />
where [m T 11 mT 21 ]T = m T .<br />
With at least two accelerometers, not measuring in parallel directions, in a measurement<br />
device, the position <strong>of</strong> this device can be defined by the point closest to both lines identified<br />
as r( ). As in the identification <strong>of</strong> the centre <strong>of</strong> gravity (4.30), incorporation <strong>of</strong> more<br />
measurements (accelerometers) can easily be done. In case <strong>of</strong> the shuttle measurements, a<br />
device with three accelerometers was attached in several positions, p o, as given in Fig. 4.19.<br />
For each position, the accelerometer lines <strong>of</strong> measurement were identified and from these<br />
lines po.
156 4 Parameter identification and model validation<br />
Amplitude (160Bar/10V)<br />
Phase (deg)<br />
10 1<br />
10 0<br />
0<br />
−100<br />
−200<br />
−300<br />
−400<br />
10 1<br />
10 1<br />
Frequency (Hz)<br />
Fig. 4.18: Bode plot <strong>of</strong> the diagonal elements <strong>of</strong> the input valve voltage to output pressure<br />
transfer function along the rigid body modal directions reconstructed from the<br />
accelerometer measurements. As in Fig. 4.13 the <strong>system</strong> is strongly decoupled<br />
and each plot shows only one rigid body mode. The transfer function to output<br />
(actuator) pressure difference lacks the transmission zeros at approx. 70Hz and<br />
shows phase shift much earlier in comparison with input pressure difference as<br />
output plotted in Fig. 4.13.<br />
Three positions were chosen at the shuttle floor, the part <strong>of</strong> the shuttle which was expected<br />
to be the most rigid (highest flexible eigenfrequencies). Just behind the pilot’s chairs<br />
in the middle, in the back at port and starboard. At the front side, the ’chin’ <strong>of</strong> the shuttle,<br />
just below the place where the window is to be placed, in the middle and at the starboard<br />
side, the redundant measurements were taken. The identified positions, using 4Hz sinusoidal<br />
<strong>motion</strong>, are given in Table 4.6. These positions give some idea <strong>of</strong> the placing <strong>of</strong> the<br />
accelerometers. In further calculation also the relative positioning <strong>of</strong> the measurement lines,<br />
r( ), and their direction should be taken into account.<br />
Assuming a rigid body shuttle floor, the rigid body frequency response can be identified<br />
by measuring at least six accelerometer responses at the floor. To include some redundancy<br />
for evaluation, seven sensors were used in measuring the response to the sine sweep <strong>of</strong> all<br />
the actuators independently. x, y and z for the mid and y and z for the (port and star) back<br />
10 2<br />
10 2
4.3 Frequency response model validation 157<br />
Position 1:mid 2:port back 3:star back 4:front mid 5:front star<br />
x [m] -0.44 -1.33 -1.32 1.21 1.11<br />
y [m] -0.18 -0.40 0.43 0.07 0.89<br />
z [m] -0.11 -0.11 -0.12 -0.39 -0.39<br />
Table 4.6: Positions identified in the shuttle for the accelerometer device relative to the<br />
centre <strong>of</strong> gravity in platform coordinates.<br />
4<br />
5<br />
x<br />
y<br />
Fig. 4.19: Approximate positions <strong>of</strong> the accelerometer device in the shuttle xy plane as also<br />
given by Table 4.6.<br />
positions. Port y is used for evaluation. The other six are used to construct the platform<br />
response by<br />
1<br />
3<br />
2<br />
x = M ;1<br />
p s (4.48)<br />
where each row <strong>of</strong> Mp is given by m T <strong>of</strong> (4.45). With the platform response, x, it is possible<br />
to reconstruct all the (linear) parts <strong>of</strong> the acceleration resulting from the rigid body <strong>motion</strong><br />
at all positions <strong>of</strong> the shuttle.<br />
Reconstructing rigid body acceleration, pressure and velocity dynamics using accelerometers.<br />
Sine sweeps from 5Hz up to 500Hz with 201 logarithmically spaced frequency points were<br />
made with 150mV amplitude send through the experiment filter <strong>of</strong> Fig. 4.12. All measurements<br />
presented have been compensated for this filter.<br />
Reconstruction <strong>of</strong> rigid body redundant accelerometer measurement.<br />
In Fig. 4.16, a typical accelerometer frequency response to a sinusoidal valve voltage input<br />
is given. It is the response <strong>of</strong> the accelerometer measuring in y-direction at the port back<br />
position. In dashed form the reconstructed response is plotted.<br />
Clearly visible are the rigid body resonance at 8Hz (sway/roll), 15Hz (yaw) and 24Hz<br />
(roll/sway). At 10Hz the resonance due to the foundation can be observed. Modes, which<br />
are not identified yet, are visible at 35� 57 and 80Hz. Irrespective <strong>of</strong> the high number <strong>of</strong><br />
resonances, the two responses correspond quite well. Apart from the valve phase drop, the
158 4 Parameter identification and model validation<br />
Amplitude ("m"/s/10V)<br />
10 1<br />
10 0<br />
10 −1<br />
10 1<br />
Frequency (Hz)<br />
Fig. 4.20: Bode amplitude plot along the rigid body modal directions <strong>of</strong> valve input to the<br />
velocities along these directions reconstructed by the accelerometer measurements.<br />
Again these plots are decoupled showing only one rigid mode per plot<br />
and further the convergence to the typical hydraulic actuator static velocity gain<br />
going to frequency zero shows.<br />
transfer function phase start to drop below =2 around 57Hz and further at 80Hz possibly<br />
due to flexibility between the input (actuator) and the sensor.<br />
Reconstructed dynamics <strong>of</strong> input valve voltage to platform accelerations.<br />
Step by step the responses can be manipulated such that a structured response results which<br />
can be analyzed more easily. In a first step, Fig. 4.17 provides the responses <strong>of</strong> the platform<br />
pose accelerations to the inputs along the rigid body directions. Almost all platform<br />
directions (apart from the yaw-direction) are influenced by more than one rigid body mode.<br />
In the neutral pose, where the measurements where taken, it is possible to discriminate between<br />
symmetric and non-symmetric modes (x� z� ) and (y� � ).<br />
Output pressure dynamics.<br />
More fully decoupled responses (up till the flexible resonances) can be achieved by considering<br />
the reconstructed output pressure differences or velocities along the rigid body<br />
directions. The output pressures are reconstructed using a constant premultiplication given<br />
by<br />
dP<br />
^<br />
o = U T A ;1 J ;T<br />
l�x Mx (4.49)<br />
containing the unitary modal direction matrix, U, the actuator area matrix, A, the jacobian,<br />
Jl�x, and the identified mass matrix, M.<br />
10 2
4.3 Frequency response model validation 159<br />
actuator 1 2 3 4 5 6<br />
yaw 1 -1 1 -1 1 -1<br />
chin up 1 0 0 0 0 1<br />
torsional x-axis 1 -1 -1 1 1 -1<br />
Table 4.7: Input directions flexible mode measurements.<br />
In Fig. 4.18 it is shown that all the rigid modes have been decoupled in this way. It<br />
looks like the six accelerometer measurements allow a reconstruction <strong>of</strong> the (linear part<br />
<strong>of</strong> the) output pressure difference characteristics. The rigid modes appear on the diagonal<br />
elements <strong>of</strong> the 6x6-transfer function. For the lower frequencies, there is a considerable<br />
difference in gain between the different directions due to the variation in mass (and thus the<br />
pressure necessary to move with the same speed). Above the rigid mode eigenfrequencies<br />
the gains are expected to be equal apart from the effects due to the parasitic modes.<br />
The three assumably flexible modes at 43, 57 and 65Hz mainly appear along the direction<br />
<strong>of</strong> the most stiff two rigid modes. Along the pitch/surge mode direction the 43<br />
and 65Hz resonances are experienced. The 57Hz vibration comes in along the roll/sway<br />
direction.<br />
The pressure transfer function constructed from the accelerometers assuming a rigid<br />
construction should be passive apart from the influence <strong>of</strong> the transmission lines (and <strong>of</strong><br />
course the valve characteristics). Passive behaviour is certainly lost at approx. 60Hz where<br />
the phase along the direction <strong>of</strong> the pitch/surge mode (6) drops another rad.<br />
Velocity dynamics.<br />
Another alternative to reconstruct a decoupled transfer function is to use the velocities along<br />
the rigid body modal directions. These frequency responses are plotted in Fig. 4.20. These<br />
plots show the fact that for low frequencies, the hydraulic actuator is a velocity generator.<br />
This enables easy evaluation <strong>of</strong> the differences in the rigid mode resonance gains. It also<br />
shows that for the most stiff modes, the acceleration at the frequencies above the rigid body<br />
modes is the highest. Probably the (high frequency) flexible modes appear mainly along<br />
these directions due to the fact that they are best measured along these directions.<br />
Flexible phenomena.<br />
A closer inspection <strong>of</strong> these flexible modes was done by evaluation <strong>of</strong> the additional accelerometer<br />
measurements taken at the front <strong>of</strong> the shuttle. From visual inspection <strong>of</strong> the<br />
shuttle, it can be expected that the weakest part <strong>of</strong> the shuttle appears at this point where<br />
a large hole has been provided for the front window. Possibly torsional or chin up like<br />
deformation could be induced. By comparison <strong>of</strong> the actual accelerations measured at the<br />
front and those reconstructed from the (back) floor assuming rigid <strong>motion</strong>, this has been<br />
investigated.<br />
In Fig. 4.21 three examples <strong>of</strong> actually measured and reconstructed frequency responses<br />
from inputs along some specific directions are given, some <strong>of</strong> which are expected to induce<br />
energy for the flexible modes. These directions are given in Table 4.7.
160 4 Parameter identification and model validation<br />
10 1<br />
10 0<br />
10 −1<br />
10 1<br />
10 0<br />
10 −1<br />
10 1<br />
10 0<br />
10 −1<br />
10 1<br />
Frequency (Hz)<br />
10 2<br />
0<br />
−200<br />
−400<br />
0<br />
−200<br />
−400<br />
−600<br />
−800<br />
0<br />
−200<br />
−400<br />
−600<br />
−800<br />
10 1<br />
Frequency (Hz)<br />
Fig. 4.21: Bode plots <strong>of</strong> measured (dotted) and reconstructed rigid part <strong>of</strong> the acceleration at<br />
front middle in approx. y (upper row), x (middle row) and y (lower row) direction<br />
due to input vibration along the yaw, ’chin up’ and ’torsional x-ax’ directions<br />
respectively. Left column provides amplitude and right column provides phase<br />
responses. Differences in measured and reconstructed plots can point at flexible<br />
modes.<br />
10 2
4.4 Chapter Resume 161<br />
In the yaw direction it can be observed that the accelerometer output at front can be predicted<br />
fairly well by the accelerometers at the back <strong>of</strong> shuttle on the floor. However, none<br />
<strong>of</strong> the flexible modes to be investigated (43,57,65 Hz) is very pronounced in this direction.<br />
For the chin up input, the reconstructed and measured output for the accelerometer mounted<br />
for the x-direction result in totally different frequency responses. In the reconstructed frequency<br />
response, it is not expected that the modes at 43 and 65Hz will be observed while<br />
they appeared to be very pronounced in the measurements at this point. Finally, with the<br />
torsional x-axis input it is shown that around 57Hz the phase <strong>of</strong> the bode plots starts to<br />
differ 180 . The measurements have a complex zero pair like characteristic at 50Hz while<br />
the reconstructed response has not.<br />
In many ways, the reconstructed and measured responses differ. Too much to closely<br />
predict the characteristics <strong>of</strong> the flexible modes very accurately at this point. In some directions,<br />
it seemed the accelerometers were not sufficiently excited by the experiments performed.<br />
If exact knowledge <strong>of</strong> the flexibilities is required, more thorough modelling together<br />
with further experiments is necessary. In this research the parasitic resonances were<br />
mainly investigated in order to give a proper view on the extent to which the models not<br />
including these effects are valid.<br />
Resume<br />
Through frequency response model validation it was shown that the model structure<br />
proposed fairly well describes the characteristics <strong>of</strong> the actual <strong>system</strong>.<br />
Especially the rigid body modes and directions resulting from the interaction <strong>of</strong> hydraulics<br />
and mechanics, which were predicted rightly to dominate the response, are<br />
well caught by the model.<br />
Low frequency ( 10Hz) parasitic behaviour could be pinpointed to the dynamics <strong>of</strong><br />
the floor, also well described by the model including the floor.<br />
Accelerometers could be used to reconstruct rigid body platform accelerations, velocity<br />
and pressure dynamics and point at possible flexibilities, after careful identification<br />
<strong>of</strong> accelerometer line <strong>of</strong> action and gain.<br />
The shuttle flexible modes could be assigned to part <strong>of</strong> the high frequent parasitic<br />
resonances but could not yet be included in the model structure. This will also require<br />
more rigorous measurement to allow accurate mode shape identification.<br />
4.4 Chapter Resume<br />
In this chapter it has been shown that the parameters <strong>of</strong> the models derived on physical laws<br />
could be identified by performing a very limited number <strong>of</strong> experiments. The responses<br />
<strong>of</strong> the kinematical model <strong>of</strong> the <strong>motion</strong> <strong>system</strong> did have a very close match with the measurements<br />
taken. The characteristics <strong>of</strong> the dynamics modelled for the hydraulically driven
162 4 Parameter identification and model validation<br />
mechanical <strong>system</strong> does correspond fairly well to the actually observed dynamics up till<br />
reasonable high frequencies ( 30Hz) if the dynamics <strong>of</strong> the foundation is taken into account.<br />
The properties due to the valve and transmission lines in the hydraulic <strong>system</strong> are<br />
still important for the over all <strong>motion</strong> <strong>system</strong>. But at the frequencies where these effect have<br />
their main influence ( 30Hz) additional parasitic resonances due to the mechanics have<br />
to be taken into account, especially in case <strong>of</strong> the light weight shuttle.
Chapter 5<br />
<strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong><br />
<strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
5.1 Introduction<br />
In previous chapters it was shown through theoretical modelling and experimental verification<br />
that in <strong>system</strong>s like the Simona <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>, the dynamics due to the<br />
hydraulic actuators and the parallel driven mechanical <strong>system</strong> with six degrees <strong>of</strong> freedom<br />
form an integrated structure with typical characteristics, which can be modelled with reasonable<br />
complex equations. This is in practice the most elaborate part <strong>of</strong> design in using a<br />
model <strong>based</strong> <strong>control</strong> strategy as laid down in this chapter.<br />
Referring to the discussion in Chapter 1, the eventual task <strong>of</strong> high performance <strong>simulator</strong><br />
<strong>motion</strong> <strong>control</strong> is improved <strong>motion</strong> realism. With <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s one wants<br />
to provide the pilot in the <strong>simulator</strong> with appropriate generalized specific forces i.e. both<br />
rotational and translational accelerations plus gravity [92]. Particularly in case <strong>of</strong> tight pilot<br />
<strong>control</strong>, e.g. landing, phase lag <strong>of</strong> realized versus desired accelerations should be minimal.<br />
In this way smaller differences between <strong>simulator</strong> and in-the-air <strong>flight</strong> conditions will exist.<br />
In <strong>flight</strong> simulation, the pilot relies on the perception <strong>of</strong> self-<strong>motion</strong> through several<br />
stimuli, and uses this perception to exercise <strong>control</strong> over the aircraft. Stimuli cueing <strong>system</strong>s<br />
include the visual <strong>system</strong>, the <strong>motion</strong> <strong>system</strong>, the audio <strong>system</strong>, <strong>control</strong> loading and the<br />
aircraft instruments stimulation. In the <strong>control</strong> task, the pilot uses his visual perception to<br />
make a good estimation <strong>of</strong> the aircraft’s attitude and velocity. The approximate frequency<br />
response <strong>of</strong> visual perception can be modelled as a first order low-pass filter with a break<br />
frequency <strong>of</strong> 0.1 Hz [92], which is rather slow.<br />
Motion in the higher frequency area is, however, mostly perceived by the pilot’s vestibular<br />
and tactile sensors, which are sensitive to specific forces and angular accelerations.<br />
These signals are processed rapidly by the central nervous <strong>system</strong> and, therefore, give the pilot<br />
lead information. With this lead information, the pilot can react more quickly to changes<br />
in the vehicle <strong>motion</strong> state. This information is most important at the higher frequencies<br />
(above the bandwidth <strong>of</strong> visual perception). These high-frequency <strong>motion</strong>s are, in simulation,<br />
<strong>of</strong>ten called onset-cues.<br />
163
164 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Because <strong>of</strong> the high-frequency nature <strong>of</strong> these onsets, it is important that the <strong>motion</strong><br />
<strong>system</strong> has sufficient bandwidth. Moreover, it is important that the time-delay in the <strong>motion</strong><br />
simulation is kept as small as possible, since an onset is also time-critical. If the onset is<br />
simulated noticeably late, <strong>simulator</strong> training quality will be decreased considerably. Time<br />
delays, furthermore, lower the pilot-vehicle crossover frequency and may require the pilot<br />
to adapt by applying lead compensation. The reduction <strong>of</strong> the adaptation required is<br />
precisely one <strong>of</strong> the goals <strong>of</strong> future <strong>flight</strong> training [7]. Therefore, a primary objective <strong>of</strong><br />
a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> is to provide the pilot in the <strong>simulator</strong> with appropriate<br />
representations <strong>of</strong>, what one could call, the ”generalized forces” i.e. both translational and<br />
rotational accelerations and gravity. This has to be attained by driving the <strong>simulator</strong> with<br />
six parallel hydraulic servo actuators. With the models <strong>of</strong> the SRS, an important basis for<br />
high performance <strong>motion</strong> has been attained further helped by construction design for <strong>control</strong><br />
including low-mass, low centre-<strong>of</strong>-gravity, and a high-stiffness structure. Its potential<br />
can only be fully exploited by advanced <strong>control</strong>ler design techniques such as used in the<br />
area <strong>of</strong> robotics.<br />
Another necessary requirement is the application <strong>of</strong> appropriate hardware components,<br />
like fast multi-processor Digital Signal Processor (DSP) boards, and s<strong>of</strong>tware which performs<br />
automatic DSP-code generation from higher level programs e.g. Matlab/Simulink.<br />
Thereby, it is possible to use highly structured complex <strong>control</strong> strategies which take model<br />
knowledge into account. This includes fast iteration on <strong>control</strong>ler design methods consisting<br />
<strong>of</strong> setting the specifications, analyzing the <strong>system</strong>, synthesising the <strong>control</strong>ler and<br />
implementing on the actual experimental set up.<br />
5.2 Another look at the <strong>control</strong> problem<br />
Since the <strong>motion</strong> <strong>control</strong>ler to be designed has to be implemented on a complex <strong>system</strong>, the<br />
previous chapters dealt with extensive modelling, which took place with the <strong>control</strong> objective<br />
in mind, even before construction [74], [124]. In the earlier stages measurements could<br />
be done on an experimental set up in which each hydraulic actuator was tested separately.<br />
Further, tests were performed with a dummy platform and the empty shuttle replacing the<br />
eventual <strong>simulator</strong> on top <strong>of</strong> the <strong>motion</strong> <strong>system</strong> [75].<br />
<strong>Model</strong> <strong>based</strong> <strong>control</strong> strategies such as the computed torque like methods are directly<br />
applicable for mechanical <strong>system</strong>s modelled in the standard structure <strong>of</strong> explicit differential<br />
equations whose states can be measured and whose <strong>control</strong> inputs are the torques which are<br />
aligned to the model coordinates. So, in applying a computed torque like technique, at least<br />
the following issues have to be regarded.<br />
- <strong>Model</strong> equations: A model-<strong>based</strong> method requires modelling. <strong>Model</strong>ling parallel <strong>system</strong>s<br />
typically can result in combined constraining algebraic and differential equations.<br />
For instance, the <strong>motion</strong> <strong>of</strong> an actuator mass in the Stewart Platform, will be<br />
fully dependent on the <strong>motion</strong> <strong>of</strong> the platform itself. Preferably the model equations<br />
will have to be parametrized in such a way that an explicit differential structure results.
5.2 Another look at the <strong>control</strong> problem 165<br />
- <strong>Model</strong> coordinates: Parametrization <strong>of</strong> the model also includes choice <strong>of</strong> the model coordinates.<br />
In the Stewart Platform one can choose end-effector/ platform coordinates,<br />
input coordinate/actuator length or any other coordinate <strong>system</strong> to describe the 6 degrees<br />
<strong>of</strong> freedom. Parallel <strong>system</strong>s are usually and most conveniently modelled in<br />
end-effector coordinates. In general these are not the coordinates which can be directly<br />
measured or steered.<br />
- Extent <strong>of</strong> modelling: Every <strong>system</strong> can be modelled in several ways and in various degrees<br />
<strong>of</strong> accuracy and complexity. In case <strong>of</strong> the mechanics <strong>of</strong> the Stewart Platform<br />
the simplified model <strong>of</strong> (2.112) could be taken. But next to the <strong>simulator</strong> body also<br />
the actuators themselves have mass properties which can not always be neglected.<br />
- Measurements: A full state measurement is assumed. In mechanical <strong>system</strong>s this means<br />
that a number <strong>of</strong> independent positions and velocities equal to the number <strong>of</strong> degrees<strong>of</strong>-freedom<br />
<strong>of</strong> the <strong>system</strong> has to be measured. On top <strong>of</strong> this, not only position and<br />
velocity measurement are assumed to be taken, but it is also assumed that these quantities<br />
directly represent the model coordinates.<br />
With the Simona <strong>motion</strong> <strong>system</strong> only actuator lengths (next to actuator pressure) are<br />
measured. So not only the velocity but also the appropriate model position or platform<br />
pose have to be constructed somehow.<br />
- Control inputs: The same goes for the <strong>control</strong> inputs. These are assumed to be the generalised<br />
forces along the model coordinates. These would be the platform forces in<br />
case <strong>of</strong> the Stewart Platform. Since the actuator forces have several (position dependent)<br />
components in platform coordinates, these can not be considered to be the<br />
required <strong>control</strong> inputs using a standard feedback linearisation scheme without any<br />
further manipulation.<br />
Also the forces will have to be generated by an actuator which in practice cannot<br />
apply required forces instantaneously. Actuator dynamics has to be looked into. In<br />
case <strong>of</strong> the Simona <strong>motion</strong> <strong>system</strong>, the actuators are hydraulic servo <strong>system</strong>s. This<br />
kind <strong>of</strong> actuators can not be considered force generators as such.<br />
Direct application <strong>of</strong> the feedback linearising <strong>control</strong> on the <strong>motion</strong> <strong>system</strong> requires<br />
consideration <strong>of</strong> the various problems observed. Other aspects also need to be looked into.<br />
In the sequel several issues will be discussed, and where necessary the <strong>control</strong> strategy will<br />
be modified.<br />
By analysis <strong>of</strong> both theoretic models derived from basic physical laws as experimental<br />
models <strong>based</strong> on measurements taken, an inventory <strong>of</strong> the relevant <strong>control</strong> problems can be<br />
put together.<br />
Control objectives<br />
Flight simulation or fooling a pilots <strong>motion</strong> awareness basically forms a <strong>control</strong> problem<br />
with mixed objectives. The <strong>system</strong> should provide for the accelerations being<br />
simulated without running out <strong>of</strong> stroke. This problem is mainly left to a host which<br />
has to come up with feasible trajectories but the <strong>motion</strong> <strong>control</strong>ler still has to both<br />
track reference accelerations as to stabilize platform pose.
166 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Hydraulics.<br />
Control <strong>of</strong> the long-stroke hydraulic actuators used in <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s<br />
is not easy since the phase lag introduced by the servo valve together with the nonnegligible<br />
high frequent transmission line resonances form a stability problem [125].<br />
Hydraulics/Mechanics<br />
Further, the bilateral coupling <strong>of</strong> the hydraulic mechanical <strong>system</strong> with strong energy<br />
exchange via pressure/flow and force/velocity introduces mechanical pose dependent<br />
resonances with interaction over the actuators. As was shown, the dynamics resulting<br />
from this interaction forms the most relevant part <strong>of</strong> the <strong>system</strong>.<br />
Mechanics<br />
With the actuators mounted in parallel to the <strong>simulator</strong>, the construction forms a Stewart<br />
platform. Due to the resulting kinematic loops, care should be taken to model this<br />
<strong>system</strong> with only explicit differential equations [74]. This required modelling in appropriate<br />
coordinates i.e. the platform pose. As only the actuators lengths are being<br />
measured, a transformation to platform pose was required to be able to apply model<br />
<strong>based</strong> <strong>control</strong>. This transformation is, dual to the actuator trajectory generation <strong>of</strong> serial<br />
robots, explicitly known from platform pose to actuator length but not injective.<br />
Trajectory generation<br />
The reference acceleration and pose will be calculated at a low sample rate host computer<br />
which incorporates a complex airplane model. References have to be introduced<br />
smoothly to the <strong>motion</strong> <strong>control</strong>ler. But smoothing the signal should not result in responses<br />
with too much phase lag since the timing <strong>of</strong> on-set <strong>motion</strong> is an important<br />
part <strong>of</strong> simulation quality.<br />
In the next section, a <strong>control</strong> strategy will be described, which takes into account the<br />
afore mentioned problems and resulted in a <strong>control</strong>ler which could be implemented on a<br />
real-time <strong>control</strong> computer connected to the <strong>motion</strong> <strong>system</strong>.<br />
5.3 Control Strategy<br />
The <strong>control</strong> strategy will be targeted at generating the appropriate accelerations for the simulation<br />
and at the same time stabilising the platform pose. Taking into account the reference<br />
generating <strong>system</strong>, e.g. the airplane, enables sufficiently smooth accelerating trajectories.<br />
Now, acceleration <strong>of</strong> the <strong>simulator</strong> mechanical <strong>system</strong> results from the forces applied.<br />
Through a model <strong>of</strong> the mechanical <strong>system</strong> the necessary forces can be calculated. Minimal<br />
invasive but stabilising position feedback is attained through use <strong>of</strong> a decoupled coordinate<br />
<strong>system</strong>. The hydraulic actuators can generate the required forces but in the previous<br />
chapters we observed that they face strong interaction from the mechanics. By supply <strong>of</strong><br />
appropriate oil flow the hydro-mechanical coupling can be minimized and local pressure<br />
feedback enables the desired force generation.<br />
By design <strong>of</strong> a <strong>control</strong> structure which has four levels, each <strong>of</strong> which have their own<br />
specifications in close relation with the other levels, one can circumvent having to solve
5.3 Control Strategy 167<br />
Fig. 5.1: Multi level <strong>control</strong> <strong>of</strong> the Simona Motion System.<br />
1. Inner loop feedback: Each hydraulic actuator is feedback transformed into a<br />
force generator by the inner loop feedback using the input pressure, dp i, and estimated<br />
velocity, ^ _q i, <strong>of</strong> an actuator. Input to these sub<strong>system</strong>s are the desired input<br />
pressures, dpi�d.<br />
2. Feedback linearisation: The desired pressures are calculated using a model (2a)<br />
<strong>of</strong> the multivariable mechanics, which calculates gravity, centripetal and coriolis<br />
force compensation and uses the mass matrix to calculate the required forces for<br />
acceleration. This block uses the measured platform state (pose, sx, and estimated<br />
velocity, ^ _x) to do this feedback linearisation step. This state is calculated on line<br />
from the measured actuator position and velocity in an iterative manner (2b).<br />
3. Outer loop feedback: This is outer loop state feedback <strong>of</strong> the platform coordinate<br />
and velocity errors, the difference between the desired xs dr and _xdr and<br />
reconstructed coordinates. Corrective accelerations, xc, result from this block and<br />
are fed to to feedback linearisation block (2.).<br />
4. Reference model-<strong>based</strong> feedforward: Smooth and possibly predictive acceleration,<br />
velocity and position, xdr� _xdr� xs, are reconstructed from the desired accelerations<br />
supplied by the host, xd. They are output to the feedback linearisation<br />
block to be used for feed forward.<br />
a too complex set <strong>of</strong> problems at once. Further, it is shown that this structure leads to an<br />
implementable <strong>control</strong>ler.<br />
General idea<br />
Looking at Fig. 5.1, in which the <strong>control</strong> structure is schematically depicted and described,<br />
we consider the following <strong>control</strong> levels with reference to the applied <strong>control</strong> theory [76].<br />
Level 1. Local hydraulic pressure <strong>control</strong> loops [49], [124], [127].<br />
Level 2. Multivariable feedback linearisation [71], [133].<br />
Level 3. Outer loop position stabilisation [113].
168 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
-Llm -B<br />
R R<br />
- V - ? d- - d?<br />
- M - -<br />
6<br />
1<br />
C<br />
- Ap J -<br />
u dp<br />
f<br />
_x<br />
T -<br />
-Ap<br />
Fig. 5.2: Basic structure hydraulically driven <strong>motion</strong> <strong>system</strong>.<br />
Level 4. Reference model <strong>based</strong> feed-forward [72], [111], [144].<br />
In short, the actuators are turned into pressure generators by local <strong>control</strong>lers. These<br />
<strong>control</strong>lers receive their reference pressure from a feedback linearisation loop in which pressures<br />
can be calculated necessary to track desired accelerations. Desired accelerations are<br />
partly corrections which are required to stabilise the pose <strong>of</strong> the <strong>simulator</strong> and for most<br />
the cues generated to provide the pilot with reasonable <strong>motion</strong> awareness. As these cues<br />
have to be smoothed but not delayed, a reference model <strong>based</strong> <strong>control</strong>ler has to calculate<br />
appropriate cues for the feedback linearisation <strong>control</strong>ler.<br />
The next sections will describe the different <strong>control</strong> levels more closely.<br />
5.4 Inner loop pressure <strong>control</strong><br />
To turn the hydraulic actuators into nice force generators two <strong>of</strong> the afore mentioned <strong>control</strong><br />
problems have to be solved at this level. Feedback <strong>of</strong> the pressure can result in stability problems<br />
since the relatively long transmission lines cause badly damped resonances together<br />
with phase lag <strong>of</strong> the valve. Further, the coupling between the mechanics and hydraulics<br />
results in the pose and load dependent rigid modes <strong>of</strong> the <strong>system</strong>.<br />
As shown in Fig. 4.13, for the shuttle the rigid modes can be observed in the frequency<br />
area between 7Hzand 25 Hz. At200 Hz the transmission lines cause peaking and at 75 Hz<br />
also a notch results. The valve has a bandwidth <strong>of</strong> 150 Hz which can clearly be observed by<br />
looking at the phase. Flexibility in the mechanics caused some additional parasitic modes<br />
between 40 Hz and 80 Hz.<br />
The coupling from which the rigid modes result, can be dealt with using the <strong>control</strong><br />
method introduced by Sepehri [127] and successfully applied by Heintze [49]. A hydraulically<br />
driven <strong>motion</strong> <strong>system</strong> basically has the structure given in Fig. 5.2. Through the valves,<br />
V , the oil flows, , can be steered by the inputs, u. The required oil flows are mainly determined<br />
by the speed at which the volumes in the actuators have to be filled. These are<br />
equal to the velocities, _q, <strong>of</strong> the actuators times the area <strong>of</strong> the piston, Ap. Together with<br />
the oil loss due to the leakage Llm, the net oil flow difference cause the pressures dp to rise<br />
through the hydraulic oil stiffness, C.<br />
fa<br />
_q<br />
J<br />
_<br />
dp = C(V u ; Llmdp ; Ap _q) (5.1)
5.4 Inner loop pressure <strong>control</strong> 169<br />
dp d<br />
?<br />
-1<br />
V ;1<br />
d<br />
d?-<br />
- -<br />
6<br />
- Kdp<br />
Inner loop<br />
<strong>control</strong>ler<br />
A ;1<br />
p<br />
6<br />
-<br />
u<br />
-Llm<br />
V - ? d-<br />
C -<br />
6<br />
Fig. 5.3: Basic structure inner loop <strong>control</strong>lers. On the right, the hydraulic part <strong>of</strong> the <strong>system</strong><br />
structure <strong>of</strong> Fig. 5.2 is given. On the left, the inner loop <strong>control</strong>ler, the cascade<br />
dp structure, compensates for the influence <strong>of</strong> the mechanics and <strong>control</strong>s the<br />
pressure, dp, which results in a theoretically arbitrarily fast first order response dependent<br />
on the <strong>control</strong>ler feedback gains in K dp. In case <strong>of</strong> the SRS, _q can not be<br />
measured and has to be reconstructed.<br />
The acceleration <strong>of</strong> the actuators is determined by the inverse mass matrix, M ;1 , which<br />
causes the interaction, times the forces supplied by the actuators minus the viscous friction<br />
e.g. b due to each hydraulic bearing, B = J T l�x bIJl�x, with Jl�x as the pose dependent<br />
jacobian.<br />
R<br />
dp<br />
-<br />
Ap<br />
-Ap<br />
Mx = J T l�x Apdp ; B _x (5.2)<br />
The actuator and platform velocities are related through _q = Jl�x _x. Gravity forces and the<br />
less relevant coriolis and centripetal forces are assumed to be dealt with at the higher levels.<br />
In Fig. 5.3, the basic structure <strong>of</strong> an inner loop pressure <strong>control</strong>ler, which decouples the<br />
mechanics from the hydraulics, is given together with the hydraulic part <strong>of</strong> the <strong>system</strong> <strong>of</strong><br />
Fig. 5.2. By compensation <strong>of</strong> the oil flow due to actuator velocity, the hydraulics can be<br />
decoupled from the mechanics.<br />
A smooth 50 Hz bandwidth pressure generator was obtained by filtering the pressure<br />
feedback signal properly for a one degree <strong>of</strong> freedom set up in which the hydraulic actuator<br />
were all tested separately. As the inner loop <strong>control</strong>ler should not interact with the mechanics<br />
its characteristics could be designed and tested in this setting at first instance [124]. In<br />
this way the hydraulic servo actuators, which usually are considered velocity engines, are<br />
approximately turned into force generators. However, as already put forward, the measured<br />
pressure difference (at the valve) does not reflect the applied force over the full relevant<br />
frequency area due to the transmission lines. Already at 30 Hz, this difference amounts to<br />
25%.<br />
The structure <strong>of</strong> the implemented inner loop <strong>control</strong>ler as given in Fig. 5.4 is somewhat<br />
different from the basic structure given in Fig. 5.3. The structure implemented on the single<br />
hydraulic actuator as was used by Van Schothorst [124] is taken. As the velocity <strong>of</strong> the<br />
hydraulic actuator is not measured directly it has to be reconstructed and three filters, C 1,<br />
C2 and C3 have been added to deal with the transmission lines and the valve dynamics.<br />
-<br />
fa<br />
_q
170 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Roughly the filters remove signal content from the frequency area above 100Hz where<br />
the valve and transmission line dynamics are most prominent and only mildly change the<br />
characteristics up to 30Hz where the rigid body dynamics resides.<br />
The linear dynamic filters, C1�C2�C3, used for the platform are given in the frequency<br />
domain by<br />
and for the shuttle<br />
C1(s) =<br />
C2(s) =<br />
C3(s) =<br />
1<br />
1<br />
(2 95) 2 s 2 +<br />
1<br />
1<br />
2 120<br />
C2s(s) =<br />
s +1<br />
1<br />
(2 200) 2 s 2 +<br />
1<br />
(2 400) 2 s 2 +<br />
1<br />
2 0:3 s +1<br />
2 95<br />
2 0:2<br />
2 200<br />
1 s +1<br />
2 120<br />
s +1<br />
2 0:2<br />
s +1<br />
2 400<br />
1<br />
(2 65) 2 s 2 +<br />
1<br />
(2 65) 2 s 2 +<br />
1<br />
1<br />
(2 150) 2 s 2 +<br />
2 0:015<br />
2 65<br />
2 0:13<br />
2 65<br />
2 0:6<br />
s +1<br />
2 150<br />
s +1<br />
s +1<br />
(5.3)<br />
(5.4)<br />
(5.5)<br />
(5.6)<br />
a notch had to be added due to the parasitic resonance <strong>of</strong> the mechanical <strong>system</strong> at this<br />
frequency, which should not be hit. Implementation was done through the standard bilinear<br />
transformation to digital filters running at 5kHz.<br />
The flexibility <strong>of</strong> the shuttle, which enters well above 30 Hz has still to be taken into<br />
account. An illustrative picture is given in Fig. 5.5. Nyquist plots <strong>of</strong> the characteristic<br />
loci are given reconstructed from the 6x6 frequency domain measurements taken for the<br />
shuttle by taking the eigenvalues <strong>of</strong> the measured transfer function, T , (T (j!)), for each<br />
frequency, !. Recall that since the multivariable <strong>system</strong> can approximately be decoupled<br />
by a unitary matrix at each operating point, Nyquist plots can be constructed, which can be<br />
analyzed as in the SISO case as given in (3.47).<br />
In the upper left part <strong>of</strong> Fig. 5.5, no feedback filter is applied and a pressure feedback<br />
gain <strong>of</strong> kdp = 0:35 is chosen. All rigid body modes and flexible resonances remain<br />
’circling’ in the right half plane as our passivity analysis <strong>of</strong> Section 3.3.1 predicted. The<br />
additional phase lag <strong>of</strong> the valve rotates the transmission line dynamics into the left half<br />
plane encircling the stability point, ;1.<br />
The gain <strong>of</strong> kdp =0:35, is well below the achievable gain <strong>of</strong> kdp =0:75 reported on<br />
in the SISO case [124]. For the SISO case this gain could safely be used in combination<br />
with the dynamic filters. The parasitic dynamics due to the flexibility <strong>of</strong> the shuttle, however,<br />
complicates the separation <strong>of</strong> low (rigid body modes, foundation) and high frequent<br />
phenomena (valve and transmission lines). The upper right plot <strong>of</strong> Fig. 5.5 still shows a<br />
resonance (at 65Hz) encircling the stability point. An additional notch solves the stability<br />
problem as shown in the lower left and right plots <strong>of</strong> Fig. 5.5 although the resonances at 43<br />
and 57Hz become more prominent.<br />
In theory, measured velocity compensation should remove all influences, also the flexibilities,<br />
from the pressure dynamics. As velocity could only be estimated with limited<br />
accuracy, a considerable amount <strong>of</strong> this compensation was done with the required velocity<br />
instead. This leaves the pressure feedback dynamics for a large part unchanged.
5.4 Inner loop pressure <strong>control</strong> 171<br />
dpi�d<br />
C1<br />
- -+ ? - -+ -<br />
+<br />
6<br />
C3<br />
e Kdp<br />
e<br />
C2<br />
-<br />
Fig. 5.4: Implemented structure inner loop <strong>control</strong>lers with velocity observer and additional<br />
filters, Ci. The output, ^ i, <strong>of</strong> this <strong>control</strong>ler part is still an intermediate <strong>control</strong><br />
variable, which is input to the valve flow compensation module <strong>of</strong> (5.7).<br />
IMAG<br />
IMAG<br />
5<br />
0<br />
−5<br />
−10<br />
−10 0 10 20<br />
REAL<br />
5<br />
0<br />
−5<br />
−10<br />
−5 0 5 10<br />
REAL<br />
IMAG<br />
5<br />
0<br />
−5<br />
K _q<br />
6<br />
_qobs<br />
−10<br />
−5 0 5 10<br />
REAL<br />
−1.5<br />
−1 −0.5 0 0.5<br />
REAL<br />
Fig. 5.5: Largest characteristic locus for pressure feedback with just feedback gain (k dp =<br />
:35) (upper left), feedback filter applied (upper right), feedback filter with notch<br />
for the flexible platform (lower left and right).<br />
IMAG<br />
0.5<br />
0<br />
−0.5<br />
−1<br />
dpi<br />
- ^ i<br />
_qref<br />
dp i<br />
q
172 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
10 0<br />
10 −1<br />
0<br />
−100<br />
−200<br />
−300<br />
10 1<br />
10 1<br />
Fig. 5.6: Bode plot <strong>of</strong> pressure feedback filter (C 2KdpC1) with flexible platform.<br />
Given the normalized stiffness, C =2Cm 220, similar as in Table 3.2, this results in<br />
a bandwidth <strong>of</strong> 2Cmkdp<br />
=12:2Hz in case only the first order pressure <strong>system</strong> is considered.<br />
2<br />
A Bode plot <strong>of</strong> the resulting feedback <strong>control</strong>ler is shown in Fig. 5.6.<br />
As discussed in Section 3.2.1, the valve has a nonlinear characteristic given by (3.16),<br />
especially in the face <strong>of</strong> load variations. Van Schothorst [124] proposes a <strong>control</strong> structure<br />
u =<br />
10 2<br />
10 2<br />
f ;1<br />
2 (^ i)<br />
q 1 ; sgn( ^ i)dpi<br />
to compensate for these effects. As this was shown to be successful experimentally, it was<br />
made part <strong>of</strong> the inner loop <strong>control</strong>ler for the SMS.<br />
(5.7)<br />
Velocity estimation<br />
The positive velocity compensation loop requires knowledge <strong>of</strong> the actuator velocity. Position<br />
and (indirectly through pressure difference) applied force are measured. Further, it<br />
is known at what velocity the actuators should run. Again, in the thesis <strong>of</strong> Van Schothorst<br />
[124] a range <strong>of</strong> velocity observers is presented. The most simple version consists <strong>of</strong> differentiating<br />
the position signal (with some low pass filtering). With the application <strong>of</strong> the<br />
increased resolution position measurement [55], this version seemed feasible. However, the<br />
position measurement appeared to have a (complex deterministic) error <strong>of</strong> 100 m in amplitude<br />
(and 4:3mm period length) which could result in severe vibrations running at higher<br />
velocities.<br />
Therefore it was decided to use the other sources <strong>of</strong> information as well, i.e. the measured<br />
pressures and the required actuator velocities, which can be calculated by v des =<br />
Jl�x _xdr. A simple second order filter was used to attain cut <strong>of</strong>f at both low (no bias) and
5.5 Multivariable feedback linearisation 173<br />
high (less noise) frequencies for all three inputs <strong>of</strong> the velocity reconstructor.<br />
1<br />
2 0:7<br />
2 6 svdes + sq<br />
(2 6) ^_qobs =(1; )<br />
2 s^q dpi +<br />
1<br />
(2 6) 2 s2 + vdes (5.8)<br />
2 0:7<br />
+ s +1<br />
2 6<br />
The acceleration <strong>of</strong> the actuator, q, had to be reconstructed at the higher level using the<br />
information on the mass properties <strong>of</strong> the <strong>system</strong> and the measured pressure differences<br />
at the valves. The velocity compensation gain was set at k _q = 0:9, just safely below the<br />
theoretic value <strong>of</strong> 1 as the velocity valve gain is still slightly nonlinear. Further, in practice<br />
the feedforward part, vdes, had to be set to 50% , i.e. =0:5.<br />
The parametrization <strong>of</strong> the inner loop <strong>control</strong>, which was implemented, has more or less<br />
been attained through loop shaping. A more structural approach has been taken in [150] by<br />
doing inner loop <strong>control</strong> design using robust <strong>control</strong> techniques. This unfortunately did not<br />
lead to <strong>control</strong>lers which could be implemented safely yet. Additional effort in this direction<br />
is required.<br />
5.5 Multivariable feedback linearisation<br />
Considering the actuators as smooth force generators, a task <strong>of</strong> the upper level is to come<br />
up with proper reference forces. These can be calculated using the model <strong>of</strong> the mechanical<br />
part <strong>of</strong> the <strong>system</strong>. A feedback linearisation structure results.<br />
5.5.1 Feedback linearising robotic manipulators<br />
General feedback linearisation as described in [105] can for rigid body robotic manipulator<br />
models be given in a computed torque <strong>control</strong> structure [133] or an inverse dynamics <strong>control</strong><br />
structure [31]. In [77] it is shown that many <strong>of</strong> such differently named <strong>control</strong> strategies in<br />
fact lead to the same <strong>control</strong> structure.<br />
For a rigid body serial robotic manipulator, the <strong>control</strong>ler is constructed as follows. See<br />
Fig. 5.7. Given the general equations <strong>of</strong> a mechanical model <strong>of</strong> such a <strong>system</strong><br />
M (q)q + C(q� _q) +G(q) = (5.9)<br />
in which similar terms appear as in (2.112). The torques/forces, are assumed to drive the<br />
<strong>system</strong>. In the computed torque <strong>control</strong>ler all terms in (5.9) are compensated for according<br />
to<br />
= M (q)(q d + c) +C(q� _q) +G(q) (5.10)<br />
which leaves decoupled double integrators.<br />
This linear <strong>system</strong> is achieved by compensation with terms, which depend on the measured<br />
state and is therefore called feedback linearisation.<br />
The desired accelerations, q d, can be steered directly, and the double integrators can be<br />
stabilised independently by the outer loop, e.g. a PD position feedback <strong>control</strong>ler for each<br />
input-output channel
174 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Fig. 5.7: Feedback linearising <strong>control</strong> <strong>of</strong> a robotic manipulator<br />
c = K(s)(qd ; q): (5.11)<br />
As the feedback linearising <strong>control</strong> strategy provides a suitable feedforward path for<br />
desired accelerations and turns the non-linear multivariable structure into an easy to <strong>control</strong><br />
decoupled double integrator <strong>system</strong>, the method should be able to deal with most <strong>of</strong> the<br />
specifications set for the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> [83].<br />
However, in the theory presented quite some assumptions are implicitly made. These<br />
have to be checked in applying the method. In the next section it will be put forward that<br />
with parallel robotic <strong>system</strong>s the assumed model structure does not fit in general. The<br />
<strong>control</strong> method has to be modified, which requires additional analysis.<br />
5.5.2 Feedback linearisation <strong>of</strong> a Stewart platform<br />
Also with parallel <strong>motion</strong> <strong>system</strong>s, the force generators should be used to <strong>control</strong> the nonlinear<br />
and multivariable mechanics. With model <strong>based</strong> calculation <strong>of</strong> the required forces to<br />
accelerate along the desired path, xd, given a measured pose and velocity, the <strong>system</strong> is to<br />
be provided with both feed forward and decoupled feedback linearised correction paths to<br />
be used by the higher level <strong>control</strong>lers.<br />
The proposed <strong>control</strong> structure is given in Fig. 5.8. This structure differs from the standard<br />
computed torque <strong>control</strong>ler <strong>of</strong> a mechanical <strong>system</strong>. In modelling for <strong>control</strong> the parallel<br />
Stewart platform configuration, one has to take care <strong>of</strong> generating an explicit set <strong>of</strong><br />
differential equations [74]. Since this is only possible taking the platform pose as the generalised<br />
coordinates, the <strong>control</strong>ler has to incorporate an algorithm which calculates these
5.5 Multivariable feedback linearisation 175<br />
Fig. 5.8: Modified feedback linearising <strong>control</strong> structure<br />
coordinates from the measured actuator lengths, l, and translates desired platform forces<br />
into required actuator pressures.<br />
The actuator lengths, l can be calculated from the platform pose sx.<br />
l = f(sx) (5.12)<br />
As discussed in Chapter 2, in measuring the actuator lengths, the platform pose has to be<br />
reconstructed iteratively e.g. by (2.58).<br />
With the jacobian, Jl�sx, which was defined by<br />
sxk+1 = sxk + J ;1<br />
l�sx (sxk)(lmeasured ; lk) (5.13)<br />
Jl�sx(x) = @l<br />
@sx<br />
(5.14)<br />
In Chapter 2 it was shown that this iteration converges sufficiently fast if some requirements<br />
are fulfilled.<br />
The desired actuator pressures can be constructed by calculation <strong>of</strong> a platform mass<br />
matrix M, coriolis and centripetal forces C _x and gravity G and filling in the desired platform<br />
accelerations, xd for the simulation plus the corrective accelerations, xc, from the outer loop<br />
position <strong>control</strong>. These are functions <strong>of</strong> the reconstructed platform pose and velocity. The<br />
simplified model <strong>of</strong> the Stewart platform taking into account the actuator inertia in addition<br />
to the platform mass matrix given in (2.112) will be considered and further the actuator<br />
forces are approximated by fa = ApdP o ApdP i, where Ap is the normalised operational<br />
area <strong>of</strong> the hydraulic actuator. Multiplying (2.112) by J ;T<br />
l�x , the desired pressure is given by<br />
dP i�d =(ApJl�x) ;T (Mt(xd + xc) +Cc( _x� sx) _x + Gc): (5.15)
176 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
In the first approach, filling in design values for these parameters like masses, inertias and<br />
centres-<strong>of</strong>-gravity already results in reasonable performance by a considerable reduction in<br />
interaction. This was further improved by taking identified model parameters. With the<br />
parameters <strong>of</strong> the kinematical model <strong>of</strong> (4.1) calibrated in Section 4.1.4 the jacobian, J l�x,<br />
can be calculated, as follows from (2.42). The identified platform mass matrix M t <strong>of</strong> (4.31),<br />
including actuator intertia constant w.r.t. the platform, is taken instead <strong>of</strong> the more limited<br />
one body mass matrix Mc <strong>of</strong> (2.112). By using the identified center <strong>of</strong> gravity in (4.30) as<br />
origin <strong>of</strong> the platform coordinates, the model parameters become relatively simple. E.g. as<br />
seen in (4.26), the gravity vector force, G c only consists <strong>of</strong> the identified mass, mg, times<br />
gravitational acceleration g in z-direction.<br />
The inverse jacobian calculation was performed at 1kHz twice as required for the implicit<br />
state measurement. Inversion was implemented through a LU-factorisation with full<br />
pivoting as given in [112].<br />
Gravity compensation is easily made explicit for the actuators by use <strong>of</strong> (the third column<br />
<strong>of</strong>) the inverse jacobian information as<br />
G = ;J ;T<br />
l�x ( � 3)mg� (5.16)<br />
Easily derived from (2.111).<br />
For coriolis and centripetal forces, only the nonlinear term given in (2.111) at platform<br />
force level by<br />
fc =! (Ic!) = ~ Ic! (5.17)<br />
was taken into account. Thereby neglecting the (small) actuator related nonlinear inertial<br />
terms.<br />
5.5.3 Implicit state measurement requirements<br />
As the iteration (5.13) reconstructing the state is part <strong>of</strong> the feedback loop, it has to converge<br />
at all times and moreover sufficiently fast in order to prevent the <strong>system</strong> from going<br />
unstable. This problem has been considered in detail in Section 2.3. Summarising, the convergence<br />
properties <strong>of</strong> the implicit state measurement by the NR-scheme can be looked into<br />
by considering the weak Newton-Kantovorich theorem given by Stoer [141], stating that a<br />
NR-scheme results in a well-defined sequence in a limited area with a solution (limit point)<br />
to which it converges with a guaranteed speed if the following three properties hold.<br />
1. The Lipschitz condition on the jacobian This condition implies that a limited difference<br />
between two platform poses should result in a limited difference between the<br />
two jacobians at those points.<br />
kJl�sx(sx1) ; Jl�sx(sx2)k ksx1 ; sx2k (5.18)<br />
2. ’Away’ from singular positions This means that the smallest gain <strong>of</strong> the jacobian should<br />
be sufficiently far away from zero:
5.5 Multivariable feedback linearisation 177<br />
kJ ;1<br />
l�sx (sx)k (5.19)<br />
With Stewart Platforms this is not trivially true. Singular points exist in the construction<br />
if the actuator lengths are not constrained by limited stroke [97]. For example, if<br />
a platform is moving down until all legs are in the XY-plane, no forces in Z direction<br />
or moments around X and Y axis are possible.<br />
Singularity can be excluded by considering a nonsingular point and the Lipschitz<br />
condition. This excludes a (possibly small) volume from singularity, and the whole<br />
workspace can be proved to be free <strong>of</strong> singularities by gridding. In this way the<br />
workspace <strong>of</strong> the Simona <strong>motion</strong> <strong>system</strong> was proved to be free <strong>of</strong>, and even far enough<br />
from singular points.<br />
3. ’Near’ to solution The third and last condition requires an initial guess which is not too<br />
far away from the real solution:<br />
kJ ;1<br />
l�sx (lmeasured ; lk=0)k : (5.20)<br />
With the described <strong>control</strong> structure implemented on a digital computer with given<br />
sample rate (> 100 Hz) and limited speed <strong>of</strong> the <strong>system</strong>, the previous solution can<br />
be used as an initial guess which can be proved to be close enough. The hydraulic<br />
actuators are physically limited in speed.<br />
Since the three conditions can be proved to be true for the Simona <strong>motion</strong> <strong>system</strong>, as was<br />
shown in detail in Chapter 2, the proposed <strong>control</strong> structure can be used without stability<br />
problem concerning the iterative part <strong>of</strong> the <strong>control</strong>ler. Given the speed <strong>of</strong> convergence, the<br />
solution is close enough (within sensor accuracy) after two steps. The update frequency is<br />
easily attained by calculating two iterations at 1 kHz.<br />
5.5.4 Outer loop <strong>control</strong><br />
Considering Fig. 5.8, the transfer function from our input, the desired accelerations, x d, to<br />
the platform pose without the feedback path K(s), does contain unstabilized double integrator<br />
paths. The outer loop <strong>control</strong>ler, K(s), will have to stabilise the <strong>simulator</strong> pose to prevent<br />
the actuators from running out-<strong>of</strong>-stroke. As the feedback linearising <strong>control</strong>ler decouples<br />
the mechanics into separate double integrators, the outer loop can generate correction accelerations<br />
resulting from a PD-structure (kpos + kvels) to stabilise these integrators.<br />
The output <strong>of</strong> the outer loop <strong>control</strong>ler, K(s), can be considered as an extra desired<br />
acceleration, additional to xd, to correct the errors in platform pose. These correction accelerations,<br />
xc, should not exceed human sensory thresholds [54] i.e. generate no noticeable<br />
false cues. Therefore the correction should ideally be sufficiently smooth (filtered) and<br />
only requires limited bandwidth (well below 1Hz). In practice, a bandwidth <strong>of</strong> 2Hz was<br />
chosen necessary to achieve sufficient suppression <strong>of</strong> the disturbances and for most the unmodelled<br />
dynamics. With the normalised signals, the feedback gains are easily chosen at
178 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
kpos =158=(2 2) 2 and kvel =17:5 for a damping <strong>of</strong> =0:7 for each platform direction.<br />
The measured platform velocity is constructed from the actuator velocity estimation<br />
multiplied by the inverse jacobian. The angular velocity, which follows from this multiplication<br />
can then be used to calculate euler parameter velocities.<br />
The outer loop <strong>control</strong>ler are split up into translational and rotational platform coordinate<br />
<strong>control</strong>lers. The outer loop for the translational directions is given by<br />
cc = kpos(cd ; c) +kvel(_cd ; _c) (5.21)<br />
For the rotational directions, the nonlinear structure for the parameters used for the description<br />
<strong>of</strong> the rotation should be used. In this case the euler parameters are taken and as<br />
= G T s _!=2, equivalent to (2.22), a double integrator structure <strong>of</strong> the euler parameters can<br />
also be considered in the platform structure <strong>of</strong> Fig. 5.8. The appropriate corrective angular<br />
acceleration to generate a corrective euler parameter acceleration can be found through<br />
where the corrective euler parameter acceleration, T<br />
<strong>control</strong> structure on the reduced euler parameters<br />
and the reconstruction <strong>of</strong> 0�c is found by<br />
_!c =2G( e) e�c� (5.22)<br />
e�c = [ 0�c<br />
T<br />
c ], is taken as the P(I)D<br />
c = kpos( d ; )+kvel(_ ; _) (5.23)<br />
0�c = ;( T c + _ T<br />
e _ e)= 0� (5.24)<br />
twice differentiating (2.19). Of course, using this strategy, rotations, , should satisfy<br />
< . A disadvantage <strong>of</strong> this structure is the nonlinear structure <strong>of</strong> the euler parameters.<br />
This results in different response on similar rotational errors in different poses. In [45]<br />
an alternative method is proposed, which <strong>control</strong>s difference rotation matrices but it is not<br />
clear yet how to incorporate integral <strong>control</strong> in this structure.<br />
As the pressure generators could not be made an order <strong>of</strong> a magnitude faster than the<br />
outer loop, the references for the outer loop (the desired velocity and position) were made<br />
to lag by using a second order filter with the estimated bandwidth <strong>of</strong> the acceleration generation<br />
(12Hz). Otherwise the outer loop would start to compensate errors, which were<br />
already taken into account.<br />
Although a P(I)D-structure <strong>of</strong> the outer loop <strong>control</strong>ler robustifies the <strong>system</strong> [113],<br />
explicit robust <strong>control</strong> will have to be used to more accurately deal with the varying <strong>system</strong><br />
conditions one encounters working within a real-time environment. A first step towards<br />
such a <strong>control</strong>ler was done as part <strong>of</strong> a master thesis research [122], but this did not lead to<br />
<strong>control</strong>lers, which could be implemented safely. So also at this point many research aspects<br />
are still open.<br />
5.6 Reference model <strong>based</strong> feed forward<br />
The <strong>motion</strong> <strong>control</strong> computer is provided with desired <strong>simulator</strong> accelerations ( xd) and<br />
poses (sxd) by a host, which <strong>control</strong>s the over all simulation (including visual, instrumental
5.6 Reference model <strong>based</strong> feed forward 179<br />
and acoustic stimuli). These signals are generated by a model <strong>of</strong> the vehicle to be simulated<br />
and the sub<strong>system</strong> with the wash-out filters which translate vehicle <strong>motion</strong> into feasible<br />
<strong>simulator</strong> <strong>motion</strong>. Although the host comes up with new set points at a relative low update<br />
frequency (ca. 60 Hz), the fact that the <strong>system</strong>s being simulated are known, will enable a<br />
reasonable prediction <strong>of</strong> the next set point.<br />
The reference model <strong>based</strong> <strong>control</strong> has the task to deal with the set points and future<br />
predictions in a proper way. Using knowledge <strong>of</strong> the set points supplied (mainly the fact<br />
that the signals do not contain information at frequencies higher than 30 Hz) a smooth<br />
interpolation filter provides a suitable reference acceleration to the feedback linearisation<br />
level together with a smooth jerk (derivative acceleration) signal which can be used as lead<br />
signal in the same feed forward channel.<br />
With the reference model <strong>based</strong> <strong>control</strong> considerable improvement can be achieved<br />
w.r.t. phase lag or delay in simulating on-set <strong>of</strong> abrupt (e.g. landing bump) and fast varying<br />
<strong>motion</strong> (e.g. turbulence) [111].<br />
The objective <strong>of</strong> predictive reference model <strong>based</strong> feed-forward <strong>control</strong> is to reduce<br />
latencies to effectively zero. This is achieved by using knowledge <strong>of</strong> the vehicle to be simulated,<br />
known as the reference model, to guide the <strong>simulator</strong> by feed-forward and feedback<br />
<strong>control</strong>, which also accounts for the <strong>motion</strong> <strong>system</strong> dynamics.<br />
Providing the <strong>motion</strong> <strong>control</strong>ler solely with the desired <strong>system</strong>’s output results in latencies,<br />
since pure feedback has limited bandwidth. This observation led to the key idea<br />
that predictive knowledge from the fully available simulation model should be obtained and<br />
used in an appropriate way.<br />
To obtain a solution that can be implemented on and properly integrated with the available<br />
sub<strong>system</strong>s, namely the host and <strong>motion</strong> computer, the problem was split into the<br />
following:<br />
reference model prediction<br />
reference model <strong>based</strong> feed forward <strong>control</strong> design<br />
<strong>system</strong>s interconnection<br />
These problems will be treated subsequently now.<br />
5.6.1 Reference model <strong>based</strong> predictors<br />
The simulation model can be used as a reference signal generator. The only uncertain factor<br />
is the pilot. However, due to the relatively slow response by the human operator, it is quite<br />
easy to predict with reasonable accuracy the future accelerations over a short period <strong>of</strong> time<br />
(30-50 ms). The major problem is due to the complexity <strong>of</strong> the vehicle model, making it<br />
difficult to calculate future values within the real-time environment. Approximate models<br />
can however be used, and several methods have been studied and evaluated [29].
180 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Approaches<br />
The model approximation methods ranged from the very general method <strong>of</strong> series expansion<br />
(with little knowledge <strong>of</strong> the simulation model), to the fast-time modelling approach, in<br />
which the structure <strong>of</strong> the simulation model is taken explicitly into account.<br />
Taylor series expansion.<br />
With the most recent data points <strong>of</strong> the simulated signals, a polynomial is fitted. Extrapolation<br />
<strong>of</strong> this polynomial results in a prediction <strong>of</strong> the future values <strong>of</strong> the signal.<br />
Identified linear model predictor.<br />
A linear state-space model is identified from the non-linear time-variant vehicle model.<br />
With the identified model and its state, a future prediction can be made.<br />
Self tuning predictor.<br />
The previous model parameters can be updated by an adaptation using a stochastic<br />
model and correlation methods.<br />
Fast time modelling (FTM).<br />
The simplified model <strong>of</strong> the equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> the aircraft can be updated by the<br />
coefficients calculated by the original (and more complex) simulation model. In the<br />
construction <strong>of</strong> the predicted <strong>motion</strong>, these coefficients are assumed to be constant.<br />
The advantage <strong>of</strong> the first three methods is that they can be used more independent <strong>of</strong><br />
the specific vehicle to be simulated. However, the expansion method and self tuning predictor<br />
appeared to suffer heavily from high frequent distortion like a turbulence model. One<br />
constant linear approximation model was not able to come up with high quality predictions<br />
at all operation points. The FTM method proved to be best suited for prediction over limited<br />
time horizons. It revealed constant performance under different test conditions.<br />
The parameters transferred to a linear model <strong>of</strong> the equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> the aircraft<br />
are:<br />
Aerodynamic coefficients,<br />
Thrust,<br />
Mass parameters,<br />
Aircraft state vector.<br />
With these parameters together with the wash out filter parameters (which do not have much<br />
effect on the short horizon as they merely act as high pass filters) the required acceleration<br />
<strong>of</strong> the <strong>simulator</strong> can be obtained<br />
The total <strong>of</strong> forces and moments considered acting on the aircraft is a summation <strong>of</strong><br />
those generated by the engine model, aerodynamic model, landing gear model and turbulence<br />
model. As forces immediately result in accelerations considering the equations <strong>of</strong><br />
<strong>motion</strong>, the prediction horizon should be obtained by the ability to predict the future forces
5.6 Reference model <strong>based</strong> feed forward 181<br />
smoothly over some time frame. As thrust, aileron, elevation and rudder steering and landing<br />
gear model have reasonable time constants, this can be done with respect to the engine,<br />
aerodynamic and landing gear forces. The turbulence related forces vary considerably faster<br />
but as the dependence on the pilots actions can be assumed constant over short horizons, the<br />
stochastic nature can be simulated in advance to deliver future predictions.<br />
5.6.2 Construction <strong>of</strong> the reference model <strong>based</strong> <strong>control</strong>ler<br />
An important aspect with respect to the reference model <strong>based</strong> <strong>control</strong> is that there is no<br />
direct response <strong>of</strong> the acceleration (or force generation) resulting from the <strong>control</strong> inputs,<br />
u. This is due to the presence <strong>of</strong> the valve dynamics and the finite oil stiffness in the<br />
column. With the application <strong>of</strong> model-<strong>based</strong> <strong>control</strong>, the desired <strong>motion</strong> outputs (namely,<br />
the specific forces and angular accelerations) can be driven more directly. Therefore the<br />
design <strong>of</strong> a feed forward signal becomes more straight forward.<br />
First, the requirements for the reference model <strong>based</strong> <strong>control</strong> task will be given. Then,<br />
the different approaches will be outlined.<br />
Requirements<br />
In the ideal case, the multiple-level <strong>control</strong>ler would result in a transfer function from the desired<br />
platform accelerations, xd to actual accelerations x equal to a fast first-order response,<br />
determined by the time constant <strong>of</strong> the pressure feedback.<br />
In practice, however, several other aspects lead to deficiencies in the <strong>system</strong>. First <strong>of</strong> all,<br />
the velocity compensation is not perfect and, as a result, the pressure feedback gains cannot<br />
be increased to arbitrarily high values. Nonetheless, the hydraulic actuators can be turned<br />
into ’force generators’ with a bandwidth <strong>of</strong> about 2-3 times the lowest natural frequency <strong>of</strong><br />
the rigid-mass <strong>system</strong> with finite oil spring stiffness. In the case <strong>of</strong> the SRS, this natural<br />
frequency, with a design load <strong>of</strong> 4000 kg, is 4-7 Hz. Finally, the limited bandwidth <strong>of</strong> the<br />
servo valves accounts for an additional latency <strong>of</strong> about 5 to 10 ms as was shown in Fig. 3.7.<br />
Furthermore, the reference accelerations should not have a frequency content higher<br />
than 20-30 Hz: Undesirable deformations <strong>of</strong> the <strong>simulator</strong> result in parasitic resonances<br />
slightly above this frequency range. The pilot’s visual <strong>system</strong> is highly sensitive to vibrations<br />
<strong>of</strong> the visual display <strong>system</strong> optics.<br />
Approach<br />
A study <strong>of</strong> literature resulted in three options for a trajectory tracking <strong>control</strong> approach.<br />
Servo Control. If little use can be made <strong>of</strong> knowledge <strong>of</strong> a reference model or <strong>of</strong> the<br />
<strong>system</strong> to be <strong>control</strong>led, the servo <strong>control</strong> <strong>of</strong> Desoer [33] still results in the robust<br />
asymptotic tracking <strong>of</strong> a reference signal. Transient response quality, which is considerably<br />
relevant in <strong>motion</strong> simulation where onsets play an important role, cannot<br />
be guaranteed however. The hydraulically driven <strong>motion</strong> <strong>system</strong> has tracking quality<br />
w.r.t. step signals <strong>of</strong> the position if position feedback is applied due to the physical<br />
structure given in Fig. 5.2.
182 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
<strong>Model</strong> Matching. Vehicle simulation can also be seen as trying to match the dynamics<br />
<strong>of</strong> the ’washed out’ vehicle model by the dynamics <strong>of</strong> the <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>.<br />
<strong>Model</strong> matching [156] can be applied with feed-forward, feed back, or both.<br />
Preview Control. With preview <strong>control</strong> [144], predicted references can be used in a<br />
<strong>control</strong>ler in order to yield limited phase lags for <strong>system</strong>s not having stable inverses.<br />
The scheme, given in Fig. 5.1, results in tracking for the long term <strong>of</strong> the platform pose<br />
which is stabilized by the outer loop position feedback. Further, as already pointed out, the<br />
inner loop feedback and feedback linearising <strong>control</strong> result in first order responses <strong>of</strong> the<br />
<strong>system</strong>, from desired to actual accelerations nominally. This <strong>system</strong> can be described by<br />
Gnom(s) =<br />
1<br />
I� (5.25)<br />
s +1<br />
where the time constant is determined by one over the product <strong>of</strong> pressure feedback and<br />
the oil stiffness times valve gain, CV . The oil stiffness, C, varies slightly with the position<br />
<strong>of</strong> the actuator[124]. The valve gain also shows some non-linearity.<br />
The immediate unit (step) response <strong>of</strong> a <strong>system</strong> can be obtained by precompensation<br />
with an inverse <strong>system</strong>. Because the <strong>system</strong> Gnom(s) is strictly proper, the inverse is not a<br />
proper stable <strong>system</strong>. Two approaches can be taken to overcome this problem.<br />
First, knowledge <strong>of</strong> the reference <strong>system</strong> can be used to determine the derivative acceleration,<br />
the ’jerk’ signal d=dt(xd). If<br />
xr = d<br />
dt (xd) +xd<br />
(5.26)<br />
is used as feed forward, both the phase lag and high frequent amplitude decrease are<br />
compensated for. As high frequency components tend to be amplified in this way, one<br />
needs to be sure that the reference signal does not contain high frequent components<br />
which could excite the parasitic resonances <strong>of</strong> the <strong>motion</strong> <strong>system</strong>. This could be<br />
reduced somewhat by filtering the reference acceleration with first order filter with<br />
a shorter time constant than , which results in a faster but still lagging response.<br />
However, as is not even exactly known and is not constant, the compensation can<br />
be wrong in the mid-frequency area i.e. the bandwidth area <strong>of</strong> the <strong>motion</strong> <strong>system</strong> <strong>of</strong><br />
10-20 Hz.<br />
Alternatively, using a prediction <strong>of</strong> the future acceleration over a horizon equal to<br />
the time constant <strong>of</strong> the <strong>control</strong>led <strong>motion</strong> <strong>system</strong> can be used to compensate for the<br />
phase lag over the relevant frequency area without unnecessarily amplifying the high<br />
frequency area. The reference to the feedback linearising <strong>control</strong>ler would then be<br />
xr(t) =xd(t + ): (5.27)<br />
Also with this approach high frequency reference signal will not be compensated for<br />
properly, but at least will not be amplified.
5.6 Reference model <strong>based</strong> feed forward 183<br />
In either case, the connection <strong>of</strong> the host to the <strong>motion</strong> computer has to be such that<br />
the reference signal does not have a significant high frequency content. This is especially<br />
relevant since the host is operating at a relative low sample rate w.r.t. the <strong>motion</strong> computer.<br />
Caution has to be taken to prevent aliasing effects from mapping into this area. This will be<br />
discussed next.<br />
Host/Motion computer connection<br />
The host computer generates the <strong>simulator</strong> feasible trajectories on a sample frequency which<br />
is usually set at 60 Hz. In earlier days even 30 Hz was seen to be used. In near future, one<br />
is likely to be able to set this frequency at 120 Hz.<br />
According to Shannon’s theorem, reference signal frequencies higher than half the sample<br />
frequency, > 0:5fn, can not be discriminated from the area 0 :::0:5fn. In practice,<br />
aircraft model time constants should be well below this frequency. This alone motivates a<br />
higher sampling frequency as also flexible effects <strong>of</strong> large aircraft or rotorcraft need to be<br />
taken into account in simulation models in the future.<br />
Extensive literature [26] towards signal processing and filtering in the area <strong>of</strong> multirate<br />
signal processing exists. Since the <strong>motion</strong> <strong>system</strong> <strong>control</strong> computer requires (smooth)<br />
signals at 1-5 kHz, the lower-frequency sampled signals <strong>of</strong> the host computer will have to<br />
be interpolated. Ideally, if the signal x(k) to be interpolated would be known from time, t,<br />
in the interval ;1 to 1, filtering with the well known anti-causal filter h(t),<br />
xc(t) =<br />
=<br />
1X<br />
h(t ; k=fn)x(k)<br />
k=;1<br />
1X<br />
k=;1<br />
sin( (fnt ; k))<br />
x(k)� (5.28)<br />
(fnt ; k)<br />
would exactly reconstruct a continuous signal x c(t) with only frequency content in relevant<br />
area without any deformation in this area. In practice none (or at best only part) <strong>of</strong> the future<br />
is known; different techniques exist to approximate h(t). Every filtering technique then<br />
typically not only reduces high frequency signal contents, but also introduces phase lags.<br />
There are always shortcomings, which the designer must be aware <strong>of</strong>.<br />
Choice <strong>of</strong> polynomial interpolation technique<br />
In this research, several filtering techniques like Finite Impulse Response (FIR) filters, Infinite<br />
Impulse Response (IIR) filters and Cubic Polynomial Reconstruction (CPR), have been<br />
considered. Unlike FIR and IIR filters which are designed in the frequency domain, polynomial<br />
interpolation techniques can be designed in time domain. This has some advantages:<br />
time domain criteria can be used,<br />
non-linear design is possible,<br />
interpolation functions are continuous time functions, so they are independent <strong>of</strong> the<br />
interpolation factor.
184 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
The last point is considered important because in case <strong>of</strong> inter-connecting <strong>system</strong>s with<br />
large (> 50) ratios between their sampling rate (as in the case <strong>of</strong> host-<strong>motion</strong> computer<br />
integration), CPR interpolation techniques can still be implemented as low order filters.<br />
This is unlike the FIR-filters. Further, no redesign has to done as the interpolation factor<br />
changes, unlike the IIR-filter design.<br />
In polynomial interpolation on the interval t k to tk+1 <strong>of</strong> order p, a function<br />
f ( )=<br />
pX<br />
i=0<br />
ai i<br />
(5.29)<br />
is defined on the local time interval from 0 (tk) to1(tk+1). The coefficients ai <strong>of</strong> the<br />
polynomial are chosen such that the value <strong>of</strong> the signal at the sample time points, and<br />
possibly the time derivatives there<strong>of</strong>, satisfy certain constraints. With a third-order CPR,<br />
which was used to filter the acceleration signal, the four coefficients can for example be<br />
specified by defining the value f (0) = x(k ; 1) and f (1) = x(k) and their first time<br />
derivative _<br />
f(1) = x(k) ; x(k ; 1) and _<br />
f(0) = x(k ; 1) ; x(k ; 2). If a prediction <strong>of</strong> the<br />
signal x(k +1)is known. the interpolation can be shifted.<br />
With the position signal interpolation <strong>of</strong> a fifth-order function, (C)PR can be used to<br />
allow the accelerations (second order time derivatives) to correspond to the desired values<br />
also. In case signals from the host would arrive late, or not at all, precaution should be taken<br />
not to extrapolate the polynomials. This could run the signal <strong>of</strong>f-line very quickly. Here,<br />
by taking the next sample equal to the previous one for the position the <strong>system</strong> gradually<br />
comes to a stop.<br />
With CPR, the host and <strong>motion</strong> computer communicate the reference signals with a simple<br />
low-order method, which has relatively little high-frequency contents. The method can<br />
easily be combined with the use <strong>of</strong> the time-derivative <strong>of</strong> the acceleration for feed forward.<br />
This ’jerk’-signal can be extracted explicitly at every time point. Predicted time points can<br />
be incorporated directly to reduce the phase lag.<br />
Resume concerning the reference model <strong>based</strong> <strong>control</strong><br />
In this section a number <strong>of</strong> approaches to reduce time delays or lags in the response <strong>of</strong><br />
simulation <strong>motion</strong> <strong>system</strong>s were presented. A synergistic procedure was proposed in which<br />
<strong>system</strong> knowledge, <strong>of</strong> both the vehicle model to be simulated and the <strong>motion</strong> <strong>system</strong> to<br />
be accelerated, is used to have virtually immediate response to the desired <strong>motion</strong>. By a<br />
division <strong>of</strong> sub-tasks, the modular simulation structure can be maintained.<br />
Further research with a full operational <strong>simulator</strong> will be needed in order to point out<br />
exactly what delays are allowed while having no influence on the perceived realism <strong>of</strong> simulation.<br />
Having a <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> as the SRS, which is able to operate nominally<br />
with low latencies, will help enabling this research.<br />
Resume concerning the multi level <strong>control</strong> structure<br />
Motion <strong>control</strong> <strong>of</strong> a complex non-linear multivariable <strong>system</strong> like the Simona Research<br />
Simulator can be performed by using a <strong>control</strong> strategy in which the <strong>control</strong>ler is structured<br />
in multiple levels. Each level has its own specifications in close relation with the level it
5.7 Implementational issues 185<br />
communicates with. In this way a set <strong>of</strong> various <strong>control</strong> problems can be solved more or<br />
less separately. Further, the <strong>control</strong> structure enables implementation on a multi-processor<br />
dsp-<strong>system</strong>.<br />
5.7 Implementational issues<br />
The multiple level <strong>control</strong>ler has been implemented on a real-time multi-processor DSP<br />
<strong>motion</strong> computer [37] connected to the <strong>motion</strong> <strong>system</strong> with configuration C., a temporary<br />
dummy platform (ca. 4tons). In this set-up, as shown in Fig. 5.9, one C40-processor has to<br />
perform all communication with the outside world (bottle-neck w.r.t. sampling frequency)<br />
and could be run at 5 kHz. Also the coprocessor, which calculates the inner loop <strong>control</strong>,<br />
runs at 5 kHz, necessary to deal with the relevant fast actuator dynamics.<br />
In this respect the multiple level structure pays-<strong>of</strong>f since the other levels, especially the<br />
feedback linearising <strong>control</strong> ran on yet another coprocessor at 1 kHz which is just sufficient<br />
to go through all the algorithms involved.<br />
Design <strong>of</strong> the <strong>control</strong> structure was performed in the user friendly environment <strong>of</strong> Matlab/Simulink<br />
1 from which c-code can be generated automatically and connected with userwritten<br />
code in so called S-functions. The four processor c-code for the full model <strong>based</strong><br />
<strong>control</strong>ler consisted <strong>of</strong> about 9500 lines automatically generated code from the simulink<br />
model and 1850 lines <strong>of</strong> user written code (<strong>of</strong> which 600 lines came from the standard templates).<br />
Though probably not optimal in efficiency, a very user friendly environment has<br />
been enabled, which allows reasonable complex <strong>control</strong> structures to be implemented. In<br />
this way, rapid prototyping <strong>of</strong> complex <strong>control</strong>lers, as presented in this research, becomes<br />
feasible. Going from a Simulink model to a <strong>control</strong>ler running in a real time environment<br />
takes about 10 minutes.<br />
5.8 Performance quantification<br />
The evaluation procedure <strong>of</strong> the multiple level <strong>control</strong>ler implemented on the <strong>flight</strong> <strong>simulator</strong><br />
<strong>motion</strong> <strong>system</strong> SRS is presented in this section. The results, as partly also described<br />
in [73], were obtained through experimental measurements with the dummy platform with<br />
additional weights added, which amounted to the predicted final weight <strong>of</strong> 4 tons as in<br />
Fig. 1.5.<br />
The <strong>motion</strong>-base must be able to guarantee a high level <strong>of</strong> performance throughout its<br />
workspace. To do this, first a test procedure and a framework was defined whereby the<br />
dynamic characteristics could be quantified. A basis for the procedure is the long existing<br />
AGARD Advisory Report AR-144 [1], which quantifies several independent tests including<br />
the measurement <strong>of</strong> describing functions, dynamic thresholds, noise levels and the hysteresis.<br />
This set <strong>of</strong> tests gives insight into several linear and non-linear properties <strong>of</strong> a <strong>control</strong>led<br />
<strong>motion</strong> <strong>system</strong> in both the time and frequency domains. The AGARD tests had to be modified<br />
to include the high-frequency dynamics, and extended to enable characterisation <strong>of</strong><br />
1 Matlab and Simulink are registered trademarks <strong>of</strong> the MathWorks, Inc.
186 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Fig. 5.9: Motion <strong>control</strong>ler hardware setup: The model <strong>based</strong> <strong>control</strong>ler is implemented on<br />
a multi-C40-processor board. A master processor has to deal with all the I/O and<br />
forms the bottle-neck w.r.t. the necessary 5 kHz sampling time for the hydraulic<br />
loops. Apart from I/O the master only performs some safety checks on the outgoing<br />
signals. The two slave processors calculate through the multi level <strong>control</strong><br />
structure given in Fig. 5.1 On the first coprocessor (slave 1), the six inner loop<br />
<strong>control</strong>lers (module 1 <strong>of</strong> Fig. 5.1) are calculated at 5 kHz. They receive their reference<br />
(pressure) from the second coprocessor (slave 2), whose task is to process<br />
the mechanical model <strong>based</strong> <strong>control</strong> (modules 2 to 4 <strong>of</strong> Fig. 5.1). Finally, a third<br />
co-processor (slave 3), was used to generate the host signal (at 100 Hz).<br />
the <strong>system</strong> throughout its operating area. In order to arrive at a standardised approach, it is<br />
proposed that the latter be achieved by performing a set <strong>of</strong> path tracking tests. Furthermore,<br />
it was noticed that in order to enable an accurate characterisation <strong>of</strong> a high-performance<br />
<strong>motion</strong> <strong>system</strong>, both the experimental set up and test method have to be critically observed.<br />
Since the goal <strong>of</strong> the SRS simulation facility is to develop and validate new simulation<br />
technologies and to investigate human-machine interface design concepts, the performance<br />
demands on the <strong>motion</strong> cueing capability are higher than current devices. The high level <strong>of</strong><br />
performance has to be guaranteed throughout the workspace <strong>of</strong> the <strong>motion</strong> <strong>system</strong>.<br />
Qualifying or even quantifying the performance <strong>of</strong> a non-linear, multivariable, physical<br />
<strong>system</strong> is complex, especially if one wishes to compare the results with other configurations<br />
or <strong>simulator</strong>s. Ideally, first a model structure is chosen which describes the characteristics<br />
<strong>of</strong> such <strong>system</strong>s. Then, the parameters <strong>of</strong> this model structure are identified, which requires<br />
experiments or testing. Finally, the models are classified by assignment <strong>of</strong> a measure or cost<br />
function. The performance classification will at best produce results as good as the method<br />
and devices used to measure it. The identification <strong>of</strong> rigorous models, which describe all
5.8 Performance quantification 187<br />
<strong>system</strong>s characteristics, generally does not lead to accurate results [84]. Therefore, <strong>motion</strong><br />
<strong>system</strong>s are <strong>of</strong>ten classified by performing several tests to identify specific parameters,<br />
sometimes assigned within independent simplified model structures.<br />
In case <strong>of</strong> <strong>motion</strong> simulation, the objective cueing performance should preferably be<br />
measured by the ability to properly stimulate the human vestibular <strong>system</strong> [54]. This consists<br />
<strong>of</strong> two organs; the otoliths and the semicircular canals. Consider the perception models<br />
given in Section 1.2 and the parameters provided in Table 1.1. The otholiths are primarily<br />
and almost proportional sensitive to linear specific forces, the so-called ”regular” unit. The<br />
time derivatives <strong>of</strong> the specific forces, called the ”irregular” unit or ”jerk”, are also detected.<br />
The semicircular canals are proportionally sensitive to rotational acceleration in some frequency<br />
area (1:44 ; 30Hz) but are also proportional to rotational speed at lower frequency<br />
(0:026;1:44Hz). These sensitivities form the basis <strong>of</strong> the use <strong>of</strong> acceleration as the primary<br />
metric to measure the performance <strong>of</strong> a <strong>motion</strong> <strong>system</strong>.<br />
5.8.1 Motion <strong>system</strong> evaluation methods and requirements<br />
At present, only one standard method to characterise the performance <strong>of</strong> a <strong>motion</strong> <strong>system</strong><br />
is known to exist. This is described in the AGARD Advisory Report 144 [1]. Although this<br />
report dates back to the end <strong>of</strong> the seventies, the technique is still up to date. Additional<br />
aviation requirements, like JAR-STD-1A8 [118] and FAA 120-40C5 [3] generally, overlap<br />
AR-144 [1] and are basically less stringent. In robotics, even though rapid developments are<br />
underway in design and testing parallel robots, these have not led to a generally accepted<br />
characterisation standard. The AGARD report defines tests to measure<br />
- the describing function as a frequency domain evaluation,<br />
- dynamic threshold as the time domain response,<br />
- noise levels to characterise parasitic <strong>motion</strong>,<br />
- and hysteresis to identify hard non-linearities.<br />
It uses these parameters to describe the performance in one prescribed point within the<br />
workspace. As this is an international standard, which has been applied to several <strong>motion</strong><br />
<strong>system</strong>s [46], [123], it will be used as a basis in further expansion to evaluate the performance<br />
<strong>of</strong> our mechanism.<br />
The AGARD method does have some limitations.<br />
- First <strong>of</strong> all, it operates in only one point <strong>of</strong> the workspace, the neutral point. The dexterity<br />
in this point is <strong>of</strong>ten minimal [7], implying that the excursion forces are well within<br />
the operational limits <strong>of</strong> the <strong>motion</strong> <strong>system</strong>, and the limited non-linearity in this point<br />
implies that simple <strong>control</strong>lers can still perform reasonably well.<br />
- Secondly, the <strong>simulator</strong> upper-gimbal centroid is taken as the point <strong>of</strong> reference, but this<br />
point is almost never the pilot’s head reference location.<br />
- Furthermore, only a limited bandwidth <strong>of</strong> up to 10 Hz is tested in the AR-144 procedures.
188 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
- No evaluation <strong>of</strong> the limitations <strong>of</strong> the experimental set up is included. Especially in<br />
measuring the noise levels and hysteresis in high-performance <strong>motion</strong> <strong>system</strong>s, the<br />
discrimination between measurement noise and <strong>system</strong>-generated parasitic <strong>motion</strong><br />
should be possible.<br />
- Finally, the separate tests do not guarantee a level <strong>of</strong> performance <strong>of</strong> the <strong>system</strong> being<br />
operated during actual <strong>flight</strong> <strong>motion</strong> simulation. This evaluation is lacking in AR-<br />
144.<br />
To improve the performance characterisation this section will therefore propose some<br />
modifications to the original AGARD method in order to enable the calibration <strong>of</strong> highperformance<br />
<strong>motion</strong> <strong>system</strong>s. The newly proposed test should not be unnecessary extensive<br />
to remain economically feasible.<br />
The objective <strong>of</strong> this section on performance quantification is to characterise the performance<br />
<strong>of</strong> a high-fidelity <strong>flight</strong> simulation <strong>motion</strong> <strong>system</strong> throughout its workspace in an<br />
efficient manner. This can not be achieved with existing methods. Therefore, a first attempt<br />
is made to modify the test procedure in order to overcome the current limitations. For the<br />
sake <strong>of</strong> practicality and standardisation purposes, the experimental set up is considered to be<br />
part <strong>of</strong> the procedure. Application to the Simona <strong>motion</strong> <strong>system</strong> will lead to a preliminary<br />
characterisation and an evaluation <strong>of</strong> the procedure.<br />
Although tests exist which are independent <strong>of</strong> the <strong>system</strong> at hand, it is preferred to<br />
take the specific characteristics <strong>of</strong> a <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> into account in designing a<br />
procedure.<br />
- A <strong>motion</strong> <strong>system</strong> is a hydraulically or electrically-driven mechanism in which the un<strong>control</strong>led<br />
dynamics usually have resonance frequencies with a low level <strong>of</strong> damping.<br />
A test procedure should point out whether the degree <strong>of</strong> damping <strong>of</strong> the <strong>control</strong>led<br />
<strong>system</strong> is sufficient.<br />
- A purely mechanical <strong>system</strong> is considered to have immediate force generation and, as<br />
a result, direct acceleration. If the dynamics <strong>of</strong> the actuators is relevant, and this is<br />
certainly the case with hydraulics (no direct feed-through), then there will be a limited<br />
bandwidth to be quantified.<br />
- The use <strong>of</strong> a synergistic Stewart Platform implies that all actuators have to move to perform<br />
pure <strong>motion</strong> <strong>of</strong> one platform degree-<strong>of</strong>-freedom. Interaction in moving along<br />
the different degrees <strong>of</strong> freedom easily results if not compensated for. This should be<br />
taken into account in testing. Further, the interaction and reflected masses depend on<br />
the platform pose.<br />
- The hydraulic actuators are symmetric, enabling similar force generation in both directions.<br />
Asymmetricity can still occur if there is a pre-load, as is usually the case if<br />
gravitational forces exist. Hydrostatic bearings are applied to reduce friction, thereby<br />
reducing hysteresis, turn-around bump and parasitic <strong>motion</strong>. As a result, there are<br />
several sources <strong>of</strong> possible non-linear behaviour <strong>of</strong> the <strong>system</strong>, which should be considered.
5.8 Performance quantification 189<br />
In the design <strong>of</strong> a modernised test procedure an attempt will be made to take these<br />
characteristics into account.<br />
5.8.2 New test procedure<br />
In setting up the test procedure to describe the dynamic properties <strong>of</strong> a <strong>control</strong>led <strong>flight</strong><br />
<strong>simulator</strong> <strong>motion</strong> <strong>system</strong>, the AGARD Advisory Report 144 [1] is taken as a basis. In<br />
the following it is discussed, which part <strong>of</strong> AR-144 is considered important, what relevant<br />
modifications were made, and what extensions were taken into account.<br />
Describing function<br />
With the describing function method, the response <strong>of</strong> a <strong>system</strong> is given along two axes, in<br />
frequency and amplitude. As extensively discussed in Van Schothorst [124], the describing<br />
function method can be used to characterise a large class <strong>of</strong> non-linear <strong>system</strong>s in the<br />
frequency domain. It is stated there that hydraulically-driven mechanical <strong>system</strong>s belong to<br />
this class even if hard non-linearities such as Coulomb friction are relevant. If no effects<br />
like Coulomb friction are present, as should be the case with high performance <strong>motion</strong> <strong>system</strong>s<br />
with hydrostatic bearings, then the sinusoidal input describing function converges to<br />
the frequency response <strong>of</strong> the linear dynamics for small input signals. This assumption has<br />
to be checked, e.g. by the hysteresis and threshold tests described below.<br />
Considering the frequency response <strong>of</strong> a linear <strong>system</strong>, several properties can be deduced.<br />
Bandwidth can be calculated, stated as the -3 dB and/or the ;45 point. The ”peaking”<br />
or degree <strong>of</strong> damping <strong>of</strong> the <strong>control</strong>led <strong>system</strong> can be found, and cross talk can be<br />
analyzed. By taking a sinusoidal input amplitude <strong>of</strong> about 10 percent <strong>of</strong> the <strong>system</strong>’s positional,<br />
speed or acceleration limits, the non linear effects are observed to have a relatively<br />
minor influence.<br />
By using an analog frequency analyzer, a broad frequency spectrum can be taken into<br />
account and highly accurate frequency responses can be measured. This is considered to be<br />
preferable to the method described in AGARD in which an extensive number <strong>of</strong> discretefrequency<br />
measurements have to be taken. The limitations <strong>of</strong> this technique are however<br />
that one should measure within the (amplitude) range where the <strong>system</strong> can be considered<br />
linear, and that no higher harmonic response is taken into account.<br />
Threshold response<br />
With step response measurements, the response <strong>of</strong> the <strong>system</strong> can be analyzed in the time<br />
domain. It is considered an easily applicable test to infer the non-linearity <strong>of</strong> the <strong>system</strong><br />
with respect to different amplitudes. Originally, these kinds <strong>of</strong> tests were used to measure<br />
the lowest amplitude or threshold acceleration to which the <strong>system</strong> still responded. With<br />
modern <strong>motion</strong> <strong>system</strong>s having virtually no friction, the <strong>motion</strong>-base will respond to any<br />
amplitude and the problem at very low amplitude responses mainly results from the accuracy<br />
<strong>of</strong> the measurement apparatus.<br />
By considering a very simple first-order linear <strong>system</strong> response model with time delay,
190 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
acceleration (m/s2)<br />
position (m)<br />
1<br />
0.5<br />
0<br />
−0.5<br />
−1<br />
0.1<br />
0.05<br />
0<br />
0 0.5 1 1.5 2 2.5 3<br />
0 0.5 1 1.5<br />
time (s)<br />
2 2.5 3<br />
Fig. 5.10: Representative input signal, acceleration and position, for the dynamic threshold<br />
test.<br />
the following structure results:<br />
G(s) =<br />
K<br />
s +1 e- ds<br />
(5.30)<br />
This is <strong>of</strong>ten a convenient structure to approximate the <strong>motion</strong> <strong>system</strong> accelerational response.<br />
If e.g. a feedback linearisation and inner loop cascade-dp is applied, the <strong>motion</strong><br />
<strong>system</strong> acceleration roughly follows the first order presssure <strong>control</strong>led dynamics. The dynamics<br />
at higher frequencies and computational implementation mostly influences phase<br />
and can therefore <strong>of</strong>ten appropriately be described by one time delay.<br />
The parameters <strong>of</strong> this model are identified from the different step responses. The time<br />
delay, d, is the time that the <strong>system</strong> takes to respond, and the time constant, ,isgivenby<br />
the time taken from this point to reach 63% <strong>of</strong> the final value.<br />
With <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s, the acceleration step response is considered important<br />
for good cueing. To measure this response, a trajectory as given in Fig. 5.10 can<br />
be applied. One should notice that this cycled trajectory actually consists <strong>of</strong> eight different<br />
acceleration step responses. In this research, a response starting with zero velocity is<br />
considered.<br />
Path tracking<br />
To evaluate the <strong>system</strong> in its normal operating mode, and to analyze whether the properties<br />
measured at one operating point can be extended to a relevant part <strong>of</strong> the workspace, a new
5.9 Experimentally evaluating performance 191<br />
test is introduced. Advani [7] already introduced a set <strong>of</strong> benchmark manoeuvres to check<br />
the design <strong>of</strong> the kinematics <strong>of</strong> a <strong>motion</strong> <strong>system</strong>. These include standard manoeuvres during<br />
<strong>flight</strong> simulation training which are considered critical in the utilisation <strong>of</strong> the <strong>simulator</strong>.<br />
Some 30 manoeuvres, responses to heavy turbulence, rejected take-<strong>of</strong>f, landing, etc., are<br />
included. The ability <strong>of</strong> the <strong>control</strong>led <strong>system</strong> to track these manoeuvres is considered. At<br />
this moment, the mean and standard deviation <strong>of</strong> the acceleration error, after being compensated<br />
by the time delay, and the time delay itself are taken as the parameters to be measured.<br />
With this test, two aforementioned limitations <strong>of</strong> the AGARD procedure are accounted for.<br />
Further tests<br />
The AGARD procedure also includes noise levels and hysteresis tests. With high performance<br />
<strong>system</strong>s, the acceleration noise levels are very close to perceivable thresholds <strong>of</strong> 1%<br />
<strong>of</strong> gravity, and become hard to measure. This is because the measurement apparatus also<br />
introduces measurement noise, which should be discriminated from <strong>system</strong>-generated parasitic<br />
<strong>motion</strong>. With the gyro measurement taken with respect to rotation, the situation even<br />
becomes more severe as this signal has to be differentiated. With the hysteresis test, in<br />
which the <strong>system</strong> is made to track very low-frequency sinusoids, the situation is the same.<br />
It was concluded that at this moment these tests can (and should) only be used to check<br />
whether the <strong>system</strong>’s noise and hysteresis are below measurable thresholds e.g. to investigate<br />
if turn-around-bumps can be noticed.<br />
5.9 Experimentally evaluating performance<br />
In this section the results <strong>of</strong> the model <strong>based</strong> <strong>control</strong>ler will be evaluated using the proposed<br />
test method and with the reference <strong>of</strong> a conventional <strong>control</strong>ler with a decentralised<br />
pressure/position feedback for each actuator.<br />
5.9.1 Experimental set up<br />
As explained in the previous sections, the test procedure consists <strong>of</strong> several trials in which<br />
the <strong>control</strong>led <strong>motion</strong> <strong>system</strong> is made to perform a series <strong>of</strong> manoeuvres. During these<br />
tests, the response <strong>of</strong> the <strong>system</strong> has to be measured. First, the design and calibration <strong>of</strong> the<br />
experimental set up will be considered.<br />
Performance test set up<br />
The experimental configuration is given in Fig. 5.11. It consists <strong>of</strong> the Simona <strong>motion</strong><br />
<strong>system</strong>, which is <strong>control</strong>led by the <strong>motion</strong> computer. With the actuator extensions being<br />
measured, together with the kinematics <strong>of</strong> the <strong>system</strong>, the platform pose (position and orientation)<br />
can be calculated on-line. In fact the <strong>system</strong>’s response with respect to the accelerations<br />
can also be calculated from these measurements. However, as the signals then<br />
need to be differentiated twice, the high-frequency component <strong>of</strong> the reconstructed acceleration<br />
becomes corrupted by measurement noise. Therefore, an independent data acquisition<br />
<strong>system</strong> is used to measure the <strong>system</strong>s accelerating response.
192 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Fig. 5.11: Motion <strong>control</strong> evaluation test setup: Independent <strong>of</strong> the <strong>motion</strong> <strong>control</strong> <strong>system</strong><br />
given in Fig. 5.9 implemented on the <strong>motion</strong> computer, the accelerations and<br />
angular velocities <strong>of</strong> the <strong>system</strong> are measured at the <strong>motion</strong> platform by another<br />
data acquisition <strong>system</strong> (also a dSpace computer). The <strong>motion</strong> <strong>control</strong>ler and<br />
data acquisition <strong>system</strong> are synchronised by a frequency generator which is used<br />
to trigger the start <strong>of</strong> each run and detect possible discrepancies between the<br />
<strong>system</strong> clocks.<br />
With a reference measurement package consisting <strong>of</strong> three accelerometers and three<br />
rate gyros attached to the dummy platform, the local specific forces and angular velocities<br />
are measured. This still requires one differentiation step in order to calculate the angular<br />
acceleration.<br />
As measurements <strong>of</strong> the <strong>motion</strong> computer and the data acquisition computer will be<br />
used in the evaluation <strong>of</strong> the test procedures, the time scale <strong>of</strong> both <strong>system</strong>s has to be synchronised.<br />
To achieve this, an external reference signal from a frequency generator was fed<br />
to both <strong>system</strong>s. This sinusoid was used to trigger the start <strong>of</strong> each run. The test setup is<br />
depicted in Fig. 5.11.<br />
Calibration measurement <strong>system</strong><br />
Reconstruction <strong>of</strong> the actual acceleration <strong>of</strong> the platform in all six degrees-<strong>of</strong>-freedom cannot<br />
be done with the measurements taken by the reference package alone. Some additional<br />
information has to be known in order to process the data properly. The gain and orientation<br />
<strong>of</strong> the measurement devices, and also their relative position, are required. Some (but not all)<br />
<strong>of</strong> these parameters can be calibrated by <strong>of</strong>f-line measurements. As part <strong>of</strong> this research,
5.9 Experimentally evaluating performance 193<br />
another approach was taken.<br />
In the low-frequency region, the accelerations <strong>of</strong> the platform can be reconstructed with<br />
the actuator extensions and the kinematics <strong>of</strong> the platform as also done in Section 4.2.2.<br />
By performing a manoeuvre at a specific frequency (e.g. at f=4 Hz), a high signal-to-noise<br />
ratio can be obtained. Especially if only this component <strong>of</strong> the signal is taken into account,<br />
the non-linear components can be neglected. This frequency should be chosen such that<br />
both the accelerometers and the position transducers have an approximately equal signalto-noise<br />
ratio, taking into account the accuracy and resolution <strong>of</strong> the measurement devices.<br />
Manoeuvring in six independent platform directions assures spanning the whole <strong>motion</strong><br />
space. Here, we assume only rigid-body <strong>motion</strong> <strong>of</strong> the platform, and a rigidly-attached<br />
base. This assumption is not valid in the case <strong>of</strong> the current placement <strong>of</strong> the Simona<br />
<strong>motion</strong> <strong>system</strong> (on the floating concrete floor plate) at frequencies higher than 8 Hz.<br />
The position and orientation <strong>of</strong> the reference package in platform coordinates can be reconstructed<br />
taking the procedure presented in the previous chapter (4.46) for the accelerometers<br />
used in identifying the flexible dynamics. The identification was improved further by<br />
incorporating the dynamics <strong>of</strong> the anti-aliasing filters.<br />
This procedure was applied to the reference package attached to the dummy platform<br />
moved by the Simona <strong>motion</strong> <strong>system</strong>. The accelerometers and gyros appeared almost perfectly<br />
aligned with the axes <strong>of</strong> the moving reference frame. The gains also corresponded<br />
to the earlier <strong>of</strong>f-line calibrated values (approx. 3.6V/g). The position <strong>of</strong> the lines <strong>of</strong> measurement<br />
for the accelerometers were also very much the same with respect to their relative<br />
design positions. The absolute position <strong>of</strong> the reference package was identified at<br />
683mm 1mm below the centre <strong>of</strong> gravity (which is 233mm below centre upper gimbal<br />
point). The x-y coordinates (forward and to the side) depend on the accelerometer at hand<br />
which should be compensated for in the following test procedure. As this easy to apply<br />
procedure was successful, it could be used in the future to reconstruct the relative position<br />
in the interior <strong>of</strong> the <strong>simulator</strong>.<br />
5.9.2 Characteristics <strong>of</strong> the Simona <strong>motion</strong> <strong>system</strong><br />
Given the description <strong>of</strong> the test procedure and the experimental set up, the resulting performance<br />
characteristics <strong>of</strong> the Simona <strong>motion</strong> <strong>system</strong> will be outlined.<br />
Frequency response<br />
First, the application <strong>of</strong> the describing function test will be discussed. In Fig. 5.12, both<br />
the amplitude and phase frequency characteristics <strong>of</strong> the Simona <strong>motion</strong> <strong>system</strong> with the<br />
multiple level <strong>control</strong>ler from 0.5 to 50 Hz are shown. All plots are separately shown in<br />
Fig. A.9 and Fig. A.10 <strong>of</strong> Appendix A with direct comparison <strong>of</strong> the model <strong>based</strong> <strong>control</strong>ler<br />
to a conventional <strong>control</strong>ler and in Fig. A.11 <strong>of</strong> Appendix A also the error response <strong>of</strong> both<br />
<strong>control</strong>lers to reference signals is given.<br />
It can be seen that the -3 dB point can be found at 13 to 15 Hz for the ’non horizontal’<br />
directions <strong>of</strong> pitch (omy), roll (omx) and heave (z). In surge (x), sway (y) and yaw (om z),<br />
the bandwidth is lower due to the <strong>motion</strong> <strong>of</strong> the current floor foundation. Except for yaw,
194 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Direction Response time (ms) Direction Response time (ms)<br />
Surge 32 Roll 35<br />
Sway 32 Pitch 35<br />
Heave 33 Yaw 34<br />
Table 5.1: The Simona Research Simulator latency with a model <strong>based</strong> <strong>control</strong>ler.<br />
the ;45 bandwidth can be found at approx. 5 Hz. The peak amplitudes remain well below<br />
2 dB. The most relevant parasitic <strong>motion</strong> is measured between pitch due to surge and vice<br />
versa, and roll due to sway and vice versa.<br />
For reference purposes, a conventional approach was used to <strong>control</strong> the same <strong>motion</strong><br />
<strong>system</strong>. The frequency response is given in Fig. 5.13. The same pressure feedback gain<br />
is applied, however, since there is no compensation for the different natural frequencies,<br />
some responses, like surge and sway, demonstrate peaks much more than others (like pitch<br />
and roll, which are over damped). The bandwidth is twice as low, and the peaking and<br />
interaction are twice as high with this <strong>control</strong> approach. The describing function enables<br />
the characterisation <strong>of</strong> some <strong>of</strong> the parameters, which are considered most important in<br />
<strong>control</strong>, like bandwidth, damping and interaction in a multivariable <strong>system</strong>.<br />
Threshold response<br />
Threshold tests have been performed for all platform directions for four different amplitudes<br />
in acceleration (1, 0.4, 0.1, 0.05 m/s2) for both negative as positive directions. The response<br />
in the surge direction is shown in Fig. 5.14. Note that the signal amplitudes have been scaled<br />
to a desired final value <strong>of</strong> 1. As can be seen the response becomes less regular for the lower<br />
amplitudes, but the time delay (approx. 10 ms) and time constant (approx. 22 ms) do not<br />
vary too much and correspond with the ;45 bandwidth ( =1=(2 fbandwidth)) measured<br />
with the frequency response. Also, the smooth response without too much peaking was<br />
predicted by the frequency response.<br />
The characteristic parameters for the other platform directions are given in Table 5.1. In<br />
all cases, a time delay <strong>of</strong> about 10 ms can be observed. Given the sample frequency <strong>of</strong> the<br />
host processor <strong>of</strong> 100 Hz, such values can be expected. It should be noted that the responses<br />
in the rotational directions were much more corrupted by measurement noise, as the signal<br />
had to be differentiated. Especially for the low amplitude responses <strong>of</strong> 0.05 and 0.1 rad/s2,<br />
no exact time constant or delay could be extracted.<br />
Path tracking<br />
To check whether the characteristics as measured in one operating point would still be preserved<br />
while performing relevant <strong>flight</strong> manoeuvres, the path tracking test was introduced.<br />
Testing Characteristic Manoeuvres<br />
At the KLM <strong>flight</strong> crew training centre, 31 <strong>simulator</strong> training-critical manoeuvres were<br />
flown in a Boeing 747-400 <strong>simulator</strong> and the aircraft model responses registered. These
5.9 Experimentally evaluating performance 195<br />
Amplitude m/s2−>m/s2<br />
Phase (deg)<br />
10 0<br />
10 −1<br />
0<br />
−50<br />
−100<br />
−150<br />
10 0<br />
10 0<br />
om_z<br />
z<br />
x y<br />
Frequency (Hz)<br />
om_y<br />
om_x<br />
om_z<br />
Fig. 5.12: Actual model <strong>based</strong> <strong>control</strong> <strong>motion</strong> <strong>system</strong> frequency response. The Bode<br />
plot <strong>of</strong> the closed loop <strong>system</strong> from required to actual (translational and rotational)<br />
accelerations applying the model <strong>based</strong> <strong>control</strong>ler in the neutral position.<br />
The most important interactions are also given (dashed) and rotations are multiplied<br />
by the upper gimbal radius. Highest bandwidths are attained with pitch<br />
(omy), roll (omx) and somewhat lower with heave (z). For the directions surge<br />
(x), sway (y) and yaw (omz), where the foundation is flexible and not included<br />
in the model used to design the <strong>control</strong>ler, some energy does not accelerate the<br />
platform above 4 Hz. The phase <strong>of</strong> the yaw direction does lag the others. Maybe<br />
the gyro filter was not compensated for properly. All responses are reasonably<br />
flat without peaking more than a few percent above 10 0 . Interaction, especially<br />
between pitch to surge and roll to sway and vice versa is unexpectedly high up<br />
till 30 % above 4Hz. The six by six Bode response plots directly comparing the<br />
model <strong>based</strong> and conventional <strong>control</strong>ler can be found in Fig. A.9 and Fig. A.10<br />
<strong>of</strong> Appendix A<br />
10 1<br />
10 1
196 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Amplitude m/s2−>m/s2<br />
Phase (deg)<br />
10 0<br />
10 −1<br />
0<br />
−50<br />
−100<br />
−150<br />
−200<br />
10 0<br />
10 0<br />
y<br />
x<br />
x<br />
y om_z<br />
z<br />
om_z<br />
Frequency (Hz)<br />
10 1<br />
10 1<br />
om_y<br />
om_x<br />
Fig. 5.13: Actual conventionally <strong>control</strong>led <strong>motion</strong> <strong>system</strong> frequency response. The<br />
Bode plot <strong>of</strong> the closed loop <strong>system</strong> from required to actual (translational and rotational)<br />
accelerations applying the basically conventional <strong>control</strong> strategy in the<br />
neutral position. The most severe interactions, between pitch (om y) and surge (x)<br />
and roll (omx) and sway (y) are given (dashed) and the rotations are multiplied<br />
by the upper gimbal radius. As expected, with the conventional <strong>control</strong>ler it is not<br />
possible to tune the bandwidths <strong>of</strong> the platform accelerational loops w.r.t. each<br />
other. Further, a compromise has to be found between peaking (up till 6 dB)<br />
<strong>of</strong> the lowest bandwidth loops <strong>of</strong> surge (x) and sway (y) and the overdamped<br />
response <strong>of</strong> pitch (omy) and roll (omx). The 4 Hz bandwidth <strong>of</strong> surge (x) and<br />
sway (y) is also a maximum attainable bandwidth using this <strong>control</strong> structure, the<br />
amount <strong>of</strong> peaking given and the lowest eigenfrequency <strong>of</strong> the platform. The six<br />
by six Bode response plots directly comparing the model <strong>based</strong> and conventional<br />
<strong>control</strong>ler can be found in Fig. A.9 and Fig. A.10 <strong>of</strong> Appendix A
5.9 Experimentally evaluating performance 197<br />
1.5<br />
1<br />
0.5<br />
0<br />
−0.5<br />
−1<br />
−1.5<br />
+/−1.0 m/s2<br />
+/−0.4 m/s2<br />
+/−0.1 m/s2<br />
0.05 m/s2<br />
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14<br />
Fig. 5.14: Short response times <strong>of</strong> the model <strong>based</strong> <strong>control</strong>ler <strong>of</strong> a large range <strong>of</strong> accelerational<br />
amplitudes. The normalised threshold or acceleration step response<br />
measurements are given over a range <strong>of</strong> amplitudes for the most difficult surge<br />
direction.<br />
manoeuvres included a wide range <strong>of</strong> dynamic conditions, including take-<strong>of</strong>fs, normal and<br />
hard landings, engine seizures, and response to heavy turbulence. These manoeuvres were<br />
used for research into kinematical design <strong>of</strong> <strong>motion</strong> platforms by Advani [7] and reference<br />
model <strong>based</strong> feedforward <strong>control</strong> within a Masters project as part <strong>of</strong> this research by Piatkiewitz<br />
[111].<br />
It was decided to use a set <strong>of</strong> these manoeuvres, which were dynamically most challenging,<br />
as a benchmark for the SRS <strong>motion</strong> platform. Five <strong>of</strong> the thirty-one manoeuvres were<br />
selected, since these represented the most <strong>motion</strong>-demanding conditions.<br />
These were:<br />
1. Response to maximum clear air turbulence<br />
The <strong>motion</strong> <strong>system</strong> is made to move in a stochastic manner experiencing vibrations<br />
up to 10 Hz. Also an airpocket is included which requires almost full speed <strong>of</strong> the<br />
actuators (:9 m/s with vmax=1 m/s).<br />
2. Taxiing<br />
Fast vibrations introduced through the landing gear have to be experienced in <strong>simulator</strong><br />
during which it has to move gradually through surge, sway and yaw <strong>motion</strong>.<br />
3. Landing with cross wind<br />
In the air, the <strong>simulator</strong> has to experience the stochastic nature <strong>of</strong> cross wind. At
198 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
landing three instances <strong>of</strong> landing gear touch down bumps require fast acceleration<br />
peak generation after which the landing slide introduces even faster vibrations up to<br />
15 Hz.<br />
4. Rejected take <strong>of</strong>f<br />
Immediate negative surge together with tilt coordination (in which gravity is made<br />
to introduce long term breaking force) move the <strong>simulator</strong> through a wide area <strong>of</strong><br />
operating conditions w.r.t the platform pose.<br />
5. Rotation during take <strong>of</strong>f roll<br />
Limited amount <strong>of</strong> <strong>motion</strong> also has to be introduced smoothly in the asymmetric<br />
(roll) <strong>motion</strong>. Faster resonances (before take <strong>of</strong>f) during the first seconds have to be<br />
replaced by the more smooth vibrations in flying.<br />
To test the <strong>system</strong> with the load conditions <strong>of</strong> a full operational SRS the <strong>control</strong> strategy<br />
proposed was evaluated with a dummy platform <strong>of</strong> 4000 kg on top <strong>of</strong> the SRS <strong>motion</strong> <strong>system</strong><br />
depicted in Fig. 1.4. The host computer was replaced by an additional processor on the<br />
<strong>motion</strong> computer running at a lower sampling rate <strong>of</strong> 100 Hz. In the ’host’-processor the<br />
tracks for platform pose and acceleration are made available and send to the <strong>motion</strong> computer<br />
<strong>control</strong> processors at demand. The original tracks were aircraft model output. By a<br />
plain wash out these trajectories were made fit for the SRS. Actuator stroke had to be within<br />
1.25 m and velocity within 1 m/s. To test the <strong>motion</strong> <strong>system</strong> with the <strong>control</strong> strategy these<br />
signals were considered the feasible trajectories which had to be tracked as close as possible.<br />
Evaluation <strong>of</strong> the path tracking<br />
Each manoeuvre has a duration <strong>of</strong> 16 seconds, however, due to the limited hydraulic power<br />
supply (temporarily) available at the Central Workshop <strong>of</strong> Mechanical Engineering this duration<br />
is just manageable. In Fig. 5.15, the measured heave response together with the<br />
reference acceleration, are depicted.<br />
The measured heave signals correspond very closely to the desired acceleration. If it<br />
is considered on a short time frame an almost constant lag <strong>of</strong> approximately 30 m can be<br />
observed. The main contribution <strong>of</strong> the lag in response is due to the smoothing filter at 30 Hz<br />
(63 % rise time <strong>of</strong> 20 ms). This motivates the use <strong>of</strong> a prediction horizon <strong>of</strong> 20-30 ms, in<br />
which case the lag in response could very well be reduced to a situation <strong>of</strong> ’virtual zero time<br />
delay’. In this chapter a number <strong>of</strong> approaches to achieve this were presented.<br />
In Fig. 5.15 part <strong>of</strong> the response during a landing with cross wind is shown. Acceleration<br />
peaks can be observed as the three landing bumps occur. These are well represented by the<br />
<strong>control</strong>led <strong>motion</strong> <strong>system</strong>. The frequency content <strong>of</strong> the signal is clearly changing as the<br />
wheels start to roll over the landing track (more high frequency rumble required). The<br />
errors are well below 2% <strong>of</strong> gravity. As the high-frequency part <strong>of</strong> the error signal can be<br />
significant (due to measurement noise and also anti-alias filtering, starting at 30 Hz), it is<br />
hard to draw conclusions with respect to this frequency region.<br />
The figure shows that the <strong>motion</strong> <strong>system</strong> reacts without gain decrease on the stochastic<br />
nature <strong>of</strong> wheel landing track contact at least up to vibrations <strong>of</strong> ca. 8 Hz (which can be seen<br />
during the time span 8.05 s:::8.45 s). The ’noise’ in the measured signal is mainly caused
5.9 Experimentally evaluating performance 199<br />
acceleration (m/s2)<br />
2<br />
1<br />
0<br />
−1<br />
−2<br />
−3<br />
−4<br />
landing with cross wind, desired/actual heave<br />
−5<br />
0 2 4 6 8 10<br />
time (s)<br />
12 14 16 18 20<br />
Fig. 5.15: Path tracking in landing with cross wind manoeuvre with the model <strong>based</strong> <strong>control</strong>ler<br />
considering accelerations in the platform heave (z-direction). Upper plot<br />
gives the desired accelerations in heave direction and the lower one gives the<br />
actually measured accelerations (<strong>of</strong>f-set 2m=s 2 brought in on purpose for comparison)<br />
by the application <strong>of</strong> servo valve dither. As said, the top-top values remain however well<br />
below 0.02 g, which almost unnoticeable.<br />
The other manoeuvres, not depicted here, show comparable characteristics. This proves<br />
that this <strong>motion</strong> <strong>system</strong> <strong>control</strong> and design strategy resulted in a high-quality <strong>system</strong>. The<br />
use <strong>of</strong> a predictive reference model <strong>based</strong> <strong>control</strong> will even enable a synergistic connection<br />
between host and <strong>motion</strong> computer, which will considerably reduce latencies.<br />
The two aforementioned <strong>control</strong>lers, (1) the multiple level <strong>control</strong>ler and (2) the conventional<br />
<strong>control</strong>ler, were compared using the manoeuvres. The characteristic parameters,<br />
given in Table 5.2, show again that the multiple level <strong>control</strong>ler has higher performance with<br />
respect to the conventional one.<br />
Further tests<br />
Further, as already pointed out, noise level and hysteresis tests are also part <strong>of</strong> this procedure.<br />
As both were near or below measurable or noticeable levels, no exact values can be<br />
given here. The hysteresis tests showed values less than 0.3 mm. This value was measured<br />
going through a sinusoid with 0.4 m <strong>of</strong> amplitude and period time <strong>of</strong> 5 minutes, and was<br />
still decreasing as the period time was increased. It was observed that higher values <strong>of</strong> hys-
200 5 <strong>Model</strong> <strong>based</strong> <strong>control</strong> <strong>of</strong> the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong><br />
Manoeuvre Control Delay error Mean error Std error<br />
(ms) (m=s 2 ) (m=s 2 )<br />
Landing <strong>Model</strong> <strong>based</strong> 32 0.032 0.037<br />
Conventional 62 0.042 0.041<br />
Taxiing <strong>Model</strong> <strong>based</strong> 34 0.022 0.019<br />
Conventional 62 0.041 0.033<br />
Table 5.2: Comparison path tracking results in heave (delay, mean error, standard deviation<br />
(std)).<br />
teresis appear if the dither - a high-frequency vibration in the pilot valve <strong>of</strong> the actuator - is<br />
not properly tuned. Therefore, this test should be part <strong>of</strong> the procedure to guarantee some<br />
upper bound.<br />
The noise levels measured with respect to the translational acceleration are acceptable,<br />
though close to measurable levels <strong>of</strong> 0.01 to 0.02 g. It should be noted that with the experimental<br />
set-up used, no predictions with respect to high-frequencies ( 50 Hz) can be<br />
given. The noise levels in the rotational directions could not be measured as there was too<br />
much measurement noise caused by differentiating the angular velocity. Therefore, more<br />
accurate gyros should be used here. The frequency responses measured in Fig. 5.12 do not,<br />
however, predict higher sensitivity in the rotational directions compared to the translations.<br />
5.10 Chapter Resume<br />
Almost invariantly, in every test performed, the model <strong>based</strong> <strong>control</strong>ler did a far better job<br />
than the reference conventional <strong>control</strong>ler. Higher bandwidths (8 Hz for 90 phase lag)<br />
were measured in the frequency domain along with less peaking (max: 2 dB). Further,<br />
higher suppression <strong>of</strong> interaction was also attained, though somewhat less than expected. In<br />
the time domain, fast responses (response time between 30 and 35 ms) were observed over<br />
a large range <strong>of</strong> amplitudes including 5 mg.<br />
In the conventional <strong>control</strong>ler, a compromise had to be settled for in which some loops<br />
have high bandwidth (still not more than 4 to 5 Hz) and too severely damped response<br />
and other loops show already relatively high peaking up till 6 dB. Only in heave direction,<br />
in the neutral position, performance is almost equal to the model <strong>based</strong> <strong>control</strong> approach.<br />
Fortunately, for the conventional <strong>control</strong> approach, most accelerations in the simulated manoeuvres<br />
were required along the heave direction.<br />
In this chapter also a test procedure was proposed to characterise the performance <strong>of</strong> a<br />
high-fidelity <strong>flight</strong> simulation <strong>motion</strong> <strong>system</strong> throughout its workspace in an efficient manner.<br />
The procedure was evaluated with the Simona <strong>motion</strong> <strong>system</strong>, which, although in preliminary<br />
form, already shows favourable properties due to its design and the model <strong>based</strong><br />
<strong>control</strong> approach. The main parameters <strong>of</strong> such a <strong>system</strong> appear to be given by its linear<br />
dynamics. Measuring the frequency response is an efficient way to characterise this. Other<br />
tests such as threshold step responses, path tracking and hysteresis, can be used to quantify
5.10 Chapter Resume 201<br />
to what extent these properties are maintained throughout the full operating area. Further<br />
research has to be done to identify the noise levels with higher accuracy.<br />
Some modifications to the current standard outlined in AGARD-144 can yield a better<br />
quantification <strong>of</strong> high-performance <strong>motion</strong> <strong>system</strong>s.<br />
Performance <strong>of</strong> the <strong>motion</strong> <strong>system</strong> was defined by the degree <strong>of</strong> <strong>motion</strong> realism attained.<br />
There are no measures known which exactly quantify this. More research into human perception<br />
has to point out how this has to be done. The Simona Research Simulator could<br />
play a role in attaining this goal.
Chapter 6<br />
Review and discussion on the<br />
results<br />
The previous chapters went over the full <strong>system</strong>/<strong>control</strong> design process <strong>of</strong> <strong>flight</strong> <strong>simulator</strong><br />
<strong>motion</strong> <strong>system</strong>s. Before drawing the final conclusions towards this research, the main<br />
considerations in this process will be discussed.<br />
It has been investigated to what extent the use <strong>of</strong> relevant <strong>system</strong> knowledge in the <strong>motion</strong><br />
<strong>control</strong> strategy improves the <strong>control</strong>led dynamics <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>.<br />
This required structuring the research in several subproblems. Defining quality i.e. setting<br />
specifications, obtaining <strong>system</strong> knowledge through theoretical modelling, quantifying and<br />
verifying these models and the relevant dynamics by experiments. Definition <strong>of</strong> a model<strong>based</strong><br />
but implementable <strong>control</strong> strategy and a test to validate and compare the obtained<br />
closed loop dynamics with conventionally <strong>control</strong>led <strong>motion</strong> <strong>system</strong>s. In literature various<br />
solutions to many <strong>of</strong> these subproblems are proposed. Integratibility in the full design<br />
scheme and actual applicability have been the main arguments in this research in the choice<br />
for the most suitable available alternative or, in some cases, newly proposed variant.<br />
6.1 Flight simulation<br />
Flight simulation is about creating a complex environment, which can be used to train pilots,<br />
evaluate <strong>flight</strong> <strong>system</strong> characteristics, etc. in a <strong>system</strong> which is not actually flying.<br />
Using a <strong>flight</strong> <strong>simulator</strong> is cheaper, safer, environmentally less harmful than flying an aircraft<br />
and training or evaluation can more easily be defined, standardised, modularised and<br />
repeated. Since these advantages become more and more important, there is a continuing<br />
market driven tendency towards the use <strong>of</strong> <strong>flight</strong> <strong>simulator</strong>s. However, requiring higher standards<br />
in simulation quality. Major breakthrough in technology, mainly due to the increase<br />
in computer power, concern the visual <strong>system</strong> and the ability to take complex <strong>flight</strong> characteristics<br />
into account in the aircraft (environment) modelling. Motion is one the stimuli<br />
which can never be perfect given the limited dexterity <strong>of</strong> <strong>motion</strong> <strong>system</strong>s kinematics. This<br />
foremost constrains the ability to duplicate the low frequency part <strong>of</strong> manoeuvres. This can<br />
partly be compensated for by current visual <strong>system</strong>s. The attainable quality <strong>of</strong> simulating<br />
203
204 6 Review and discussion on the results<br />
high frequent vibration or onset <strong>motion</strong> is determined by the <strong>control</strong>led dynamics <strong>of</strong> the<br />
<strong>motion</strong> <strong>system</strong> together with the simulation model. By design <strong>of</strong> the Simona <strong>motion</strong> <strong>system</strong><br />
construction, expectedly favourable properties such as low mass and inertia, low centre <strong>of</strong><br />
gravity and high rigidity were strived for. Design <strong>of</strong> a suitable <strong>control</strong>ler to exploit this, is<br />
the next step. This has been investigated in this research.<br />
6.2 Motion <strong>system</strong> specifications<br />
Perceived <strong>motion</strong> or realism <strong>of</strong> <strong>motion</strong> simulation is a subjective measure which can not<br />
exactly be specified. Research into <strong>motion</strong> perception models is one <strong>of</strong> the arguments in<br />
trying to attain a relatively high performance <strong>motion</strong> <strong>system</strong>. Flight <strong>simulator</strong> users explicitly<br />
ask for quantification <strong>of</strong> required <strong>motion</strong> cues and accuracy necessary to attain a certain<br />
training quality. Literature provides for quite some indications in what range specifications<br />
should be tighter. First <strong>of</strong> all, predictable characteristics <strong>of</strong> the <strong>control</strong>led <strong>motion</strong> <strong>system</strong><br />
over its full working area should be strived for to be able to perform well defined <strong>simulator</strong><br />
training experiments. This alone motivates the use <strong>of</strong> as much relevant <strong>system</strong> knowledge<br />
as possible.<br />
Control specifications should result from the requirements set, the internal structure <strong>of</strong><br />
the <strong>system</strong> to be <strong>control</strong>led and the properties <strong>of</strong> the external signals which are expected<br />
to perturb this <strong>system</strong>. A fundamental difference in <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s w.r.t. usual<br />
robotics, is the fact that acceleration instead <strong>of</strong> positional accuracy is most important. In the<br />
low frequency range (preferably below :1 Hz) position stabilisation to prevent the actuators<br />
running out <strong>of</strong> stroke should prevail, but as from this frequency on, acceleration should<br />
be tracked. Since the <strong>simulator</strong> is running free, tracking <strong>of</strong> acceleration reference signals<br />
is the major task, which has to be compromised with unmodelled dynamics, uncertainties<br />
and measurement noise. Disturbance rejection is less evidently necessary. If the <strong>system</strong> is<br />
known, feed forward tracking can be used. As uncertainty grows, feedback will have to be<br />
applied.<br />
Although there is no difference in the ratio between desired and attained position or<br />
acceleration, errors or sensitivity to external signals are to be weighted with the squared<br />
frequency (! 2 ) if one evaluates accelerational w.r.t. positional accuracy in the frequency<br />
domain. This means sensitivity to noise is much more important in <strong>simulator</strong>s than it is in<br />
robotics. As a <strong>system</strong> is typically sensitive to noise around the bandwidth, it will be more<br />
difficult to obtain a high bandwidth <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>. Use <strong>of</strong> <strong>system</strong> knowledge<br />
through model-<strong>based</strong> <strong>control</strong>, can prevent from unnecessary amplification <strong>of</strong> noise.<br />
In the usual setting, the <strong>control</strong>led <strong>motion</strong> <strong>system</strong> is in serial closed loop connection<br />
with the aircraft model and pilot. Both the aircraft and pilot are not exactly specified, however,<br />
an aircraft typically has its highest relevant frequency modes at 2-5Hz and pilots will<br />
close the loop in order to obtain a bandwidth <strong>of</strong> .3 to 1Hz. This is attained by providing<br />
lead over a decade around this frequency (so up till 3 Hz). To have minimal influence on<br />
the pilot-aircraft(-model) loop, phase lag <strong>of</strong> the <strong>control</strong>led <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> should<br />
be virtually zero in this frequency area, which otherwise would influence relative stability<br />
<strong>of</strong> this loop. This requires a bandwidth <strong>of</strong> an order <strong>of</strong> a magnitude higher ( 10 Hz).<br />
The frequency response <strong>of</strong> an aircraft evaluated w.r.t. position can be misleading as this
6.3 Theoretical modelling 205<br />
response quickly drops above the highest mode while acceleration response remains constant.<br />
Apart from these loop characteristics, realism might also be improved by simulation<br />
<strong>of</strong> vibrations well above these frequencies such as turbulence, rotorcraft resonance, taxiing<br />
wheel to ground rumble and impact on landing bump. Amplitude characteristics should be<br />
unchanged in simulation <strong>of</strong> these kind <strong>of</strong> manoeuvres.<br />
Finally the multivariable <strong>system</strong> should not introduce parasitic coupling in the other<br />
degrees <strong>of</strong> freedom and as this depends on the relative position, the response in each degree<br />
<strong>of</strong> freedom should be similar.<br />
To summarise, a <strong>control</strong>led <strong>motion</strong> <strong>system</strong> should preferably be able to simulate references<br />
with frequency content up to 1 Hz with virtually zero time delay and an order <strong>of</strong><br />
a magnitude higher without amplitude attenuation. Use <strong>of</strong> <strong>system</strong> knowledge can help in<br />
attaining a predictable response without unnecessary noise amplification.<br />
6.3 Theoretical modelling<br />
<strong>Model</strong>ling is the first step towards model <strong>based</strong> <strong>control</strong> but also helps in analysing the characteristics<br />
<strong>of</strong> the open loop <strong>system</strong>, the response <strong>of</strong> the conventionally <strong>control</strong>led <strong>system</strong><br />
and picking the relevant dynamics. Theoretic physical modelling enables analysis before<br />
actually building a <strong>system</strong> and helps in attaining an experimentally fitted model <strong>of</strong> limited<br />
complexity describing the relevant dynamical features and pointing at the physics from<br />
which these features result. In describing the dynamics <strong>of</strong> a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>,<br />
the most relevant dynamics can be found in the mechanical <strong>system</strong>, hydraulic drives and the<br />
connection between the two.<br />
The mechanical <strong>system</strong> is, unlike the usual robotic manipulators, a fully parallel <strong>system</strong>.<br />
This leads to some dual properties if compared with the serial connected <strong>system</strong>s. In<br />
singular points, the manipulator becomes unsupported in one or more degrees <strong>of</strong> freedom,<br />
while in serial connected <strong>system</strong>s these become fixed. Kinematically, the parallelism poses<br />
problems in describing the position as an explicit function <strong>of</strong> the actuated coordinates, while<br />
in series connections these problems can occur using end effector coordinates. In describing<br />
the dynamics, the equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> a mechanical <strong>system</strong>, not choosing appropriate coordinates<br />
then leads to combined differential algebraic equations which results in additional<br />
problems in simulation or model <strong>based</strong> <strong>control</strong>. In the most general mechanical <strong>system</strong>s<br />
both problems occur together with possible nonholonomy, in which case it is not clear if<br />
or which coordinates can be chosen to have a global description in explicit form. In the<br />
Stewart Platforms, i.e. the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s, only parallelism has to be taken<br />
care <strong>of</strong>f and its dynamics can be described explicitly as a function <strong>of</strong> platform coordinates.<br />
Of course still choices have to be made in describing the spatial rotation. The use <strong>of</strong> quaternions<br />
or Euler-parameters leads to a computationally simple description also favourable in<br />
<strong>system</strong> analysis although an additional but easy to handle, constraint equation results.<br />
Symbolic multibody equation <strong>of</strong> <strong>motion</strong> solvers are suitable to generate a simulation<br />
model in short time and without to much difficulties. These solvers, however, as they handle<br />
the most general form <strong>of</strong> mechanical <strong>system</strong>s, do not easily recognise special properties, like<br />
the possible explicit form <strong>of</strong> certain <strong>system</strong>s like the Stewart Platforms. Large numbers <strong>of</strong><br />
scalar variables result with <strong>system</strong>s <strong>of</strong> limited complexity like the <strong>motion</strong> <strong>system</strong>. They are
206 6 Review and discussion on the results<br />
not suited to represent automatically appropriate vector and matrix equations or to take into<br />
account equivalence between the different actuator legs. This makes <strong>system</strong> analysis more<br />
difficult.<br />
Using a projection method, the equations <strong>of</strong> <strong>motion</strong> <strong>of</strong> hexapod <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong><br />
<strong>system</strong> can be written down preserving insight. Further, this shows that in this case both<br />
Euler-Newton as Lagrange, whatever is appropriate, methods can be used in stating the<br />
local equations <strong>of</strong> <strong>motion</strong>.<br />
System analysis <strong>of</strong> multibody mechanical <strong>system</strong> shows the following. The strong coupling<br />
in the <strong>system</strong> evaluated along the platform coordinates is mainly due to the coupled<br />
transposed jacobian w.r.t. the centre <strong>of</strong> gravity mapping the actuator forces onto the platform<br />
with specific mass and inertial properties. The jacobian varies with the platform position<br />
and forms the main non-linearity in the mechanical <strong>system</strong>. Choice <strong>of</strong> appropriate kinematical<br />
parameters, mass and inertial properties in the design <strong>of</strong> the construction should result in<br />
reasonable conditioning <strong>of</strong> the mass matrix evaluated at the actuators. Inertial properties <strong>of</strong><br />
actuators are not very relevant in conventional <strong>system</strong>s due to high mass payload. With the<br />
Simona <strong>motion</strong> <strong>system</strong> the one body mass matrix structure is not enough. Varying relative<br />
actuator inertia is, however, still neglectable. The centripetal and coriolis forces will remain<br />
relatively small due to the limited velocity <strong>of</strong> the actuators and reasonably low centre <strong>of</strong><br />
gravity.<br />
Connection <strong>of</strong> hydraulics and mechanics results in the most relevant modes, the platform<br />
pose dependent rigid modes. Direction <strong>of</strong> eigenvectors <strong>of</strong> these modes only depend<br />
on the mass matrix <strong>of</strong> mechanical <strong>system</strong> in which the jacobian plays an important part.<br />
Eigenfrequencies typically vary three to four times in magnitude as the eigenvalues <strong>of</strong> the<br />
mass matrix vary ten to twenty times.<br />
The collocated input/output pair <strong>of</strong> valve position and valve pressure difference <strong>of</strong> a<br />
basic set <strong>of</strong> hydraulic actuator models connected to a general non-linear mechanical <strong>system</strong><br />
is passive. This is not the case with taking resulting actuator output pressure or platform<br />
acceleration as an output. The property is also lost if valve dynamics can not be neglected<br />
as is the case with the range where transmission line dynamics plays a role.<br />
Passive feedback <strong>of</strong> pressure difference will not cause stability problems with any flexible<br />
mode (e.g. position dependent rigid modes, fundament or <strong>simulator</strong> parasitic dynamics,<br />
(transmission lines dynamics)). Direct feedback <strong>of</strong> pressure to flow e.g. leakage provides<br />
for an additional dissipative term like viscous friction as feedback from velocity to force<br />
does in purely mechanical <strong>system</strong>s. Total energy consists <strong>of</strong> both the mechanical kinetic<br />
part as the hydraulic pressure oil stiffness part.<br />
6.4 Experimental modelling and validation<br />
The kinematical model forms a basic part <strong>of</strong> the dynamics. Validation <strong>of</strong> the kinematics<br />
and identification <strong>of</strong> the parameters is the first step in identifying a full model. Redundant<br />
measurements are required to identify the parameters. It is important to distinguish measurements<br />
in calibration, which should be accurate vs. those which do not matter. Combining<br />
two existing methods with limited adaptation, first redundant measurement <strong>of</strong> platform pose
6.4 Experimental modelling and validation 207<br />
and secondly identification <strong>of</strong> the kinematic parameters, results in closely fit (within evaluation<br />
accuracy) kinematics. This improves positional accuracy by an order <strong>of</strong> a magnitude<br />
and can be attained by low cost length measuring devices. The location <strong>of</strong> the devices and<br />
their connection on moving and rigid base is basically not important though it can influence<br />
the sensitivity <strong>of</strong> the method. Important are the absolute lengths between the devices and<br />
the absolute measurement <strong>of</strong> length itself.<br />
In evaluating <strong>simulator</strong> acceleration response, the location <strong>of</strong> the accelerometers is highly<br />
relevant. Gain, direction and location can be identified using the kinematical model <strong>of</strong> the<br />
<strong>system</strong> with reasonable accuracy. Conditioning filters should also be known and taken into<br />
account in evaluating measurements.<br />
In this research, the <strong>motion</strong> <strong>system</strong> has been tested with three different load conditions.<br />
The unloaded dummy platform (2250 kg), the empty shuttle (1680 kg) and finally the<br />
dummy platform with full operational <strong>flight</strong> <strong>simulator</strong> load (4000 kg).<br />
Already with a basic hydraulic/mechanical <strong>system</strong> model, the most relevant open loop<br />
dynamics can quite closely be predicted. This was confirmed by measuring the result <strong>of</strong><br />
sine sweeps with a frequency analyser. With the Simona <strong>motion</strong> <strong>system</strong> this rigid body<br />
dynamics typically showed around five to thirty Hertz. Dynamics, earlier obtained from a<br />
single actuator and resulting from the valve and transmission lines is still valid, confirmed,<br />
in the multivariable <strong>system</strong> but appears for most in the high frequency area (larger than fifty<br />
Hertz) .<br />
With the dummy platform additional resonances could be observed in the mid frequency<br />
area, eight to fifteen Hertz, which resulted from a non rigid foundation at the test site as<br />
confirmed by additional measurements with accelerometers attached to the ground. By<br />
assuming a planar mass/spring/damper <strong>system</strong> in the fundament this additional dynamics<br />
could be modelled, which led to a reasonable fit to the measurements taken at both the<br />
pressure dynamics in the actuator and the acceleration measurements at the ground. With<br />
the loaded dummy platform the dynamics <strong>of</strong> the foundation changes since a relatively higher<br />
dissipation (possibly relatively more work done by friction) can be observed leading to<br />
overdamped characteristics. Design <strong>of</strong> the foundation at a <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> test site<br />
should be done in careful consideration and always be evaluated properly after the <strong>motion</strong><br />
<strong>system</strong> has been set up.<br />
Accelerometer measurements in the newly designed shuttle reveal deformation at the<br />
frequencies where resonances occur in the pressure dynamics. Though stiff (> 40 Hz), the<br />
modes <strong>of</strong> the <strong>system</strong> can be very lightly damped and occur in a frequency area, which was<br />
previously used to attenuate pressure feedback (introducing phase lag losing the passivity<br />
property). Further, one <strong>of</strong> the modes <strong>of</strong> the shuttle (at 65 Hz) appeared to be almost unobservable<br />
in the valve pressure dynamics as the transmission line dynamics block (pair <strong>of</strong><br />
complex zeros) at this point. The number <strong>of</strong> measurements and the ability <strong>of</strong> the data acquisition<br />
<strong>system</strong> should be enlarged to identify such mode shapes and other characteristics<br />
<strong>of</strong> a flexible <strong>system</strong> more closely. Flexible deformations can be expected to occur at much<br />
lower frequencies with a large projection screen attached. Further, a less exciting experiment<br />
should be designed in case <strong>of</strong> a full operational <strong>simulator</strong> to prevent such a <strong>system</strong><br />
from any harm done.<br />
If flexibility <strong>of</strong> the fundament can be modelled precisely, it can be compensated for,
208 6 Review and discussion on the results<br />
but this is doubtful. The flexibility <strong>of</strong> the <strong>simulator</strong> results in a principle compromise since<br />
only six coordinates can be functionally <strong>control</strong>led. Problems with these kinds <strong>of</strong> parasitic<br />
modes should for most be dealt with in design <strong>of</strong> the <strong>system</strong>. Not afterwards in <strong>control</strong>.<br />
6.5 Control strategy and evaluation<br />
With a conventional <strong>control</strong> strategy one uses pressure feedback to introduce extra damping<br />
to the rigid mode resonances. With additional position feedback, a bandwidth close to this<br />
resonance frequency can be obtained. As in this case, decentralised feedback is used, no<br />
explicit compensation for each direction, which typically has a range <strong>of</strong> the rigid mode<br />
frequencies, is taken into account and worst case compensation, usually overdamping the<br />
<strong>system</strong> with respect to the variation in platform pose, will have to be settled for. Feed<br />
forward can at best locally provide for adequate amplification <strong>of</strong> desired high frequency<br />
references. If the <strong>system</strong> is operated and testing is only performed in or close to any neutral<br />
position, this can be sufficient but still can require quite some tuning. Tuning can be reduced<br />
if a model is taken into account in design.<br />
With a model-<strong>based</strong> <strong>control</strong> strategy, one tries to compensate for the directionally different<br />
rigid modes varying as a function <strong>of</strong> the position by taking into account the varying<br />
jacobian and platform mass matrix. Taking into account the integration <strong>of</strong> hydraulics and<br />
mechanics will <strong>of</strong> course form the basic part <strong>of</strong> this strategy. To attain a <strong>control</strong>ler, which is<br />
structured and can be implemented in different modules, it was split up in the following sub<br />
tasks.<br />
- Inner loop decentralised actuator pressure <strong>control</strong><br />
- Partial feedback linearisation<br />
- Coordinate reconstruction<br />
- Reference model <strong>based</strong> feed forward<br />
- Outer loop position stabilisation<br />
As acceleration in the mid-frequency area is important, the hydraulic actuators should<br />
be able to deliver the required force to attain this acceleration. With the so-called cascadedp<br />
<strong>control</strong> one compensates for the oil necessary to move at actual velocity and feeds back<br />
the pressure necessary to attain required force. With this principle, a bandwidth <strong>of</strong> two to<br />
three times the rigid mode frequency can be obtained. Velocity compensation principally<br />
decouples the influence <strong>of</strong> the mechanics on the hydraulics.<br />
The lowest rigid mode frequency can be found at 5 Hz with the fully loaded Simona<br />
<strong>motion</strong> <strong>system</strong> in the neutral position. It will typically change to lower values moving to<br />
other positions.<br />
As the velocity is not measured directly, it has to be reconstructed by an observer. Both<br />
position and pressure measurements and the desired velocity can be used to calculate the<br />
required compensation. Differentiation <strong>of</strong> position is simple but very limited in frequency<br />
due to measurement errors. Pressure can be used with higher frequencies but requires the
6.5 Control strategy and evaluation 209<br />
full multivariable mechanical model which is not exact, adds multiple uncertainties and is<br />
further very sensitive to friction which is small but existent in the <strong>motion</strong> <strong>system</strong>. Even with<br />
exact velocity reconstruction, velocity compensation will not be exact due to uncertainty in<br />
the valve characteristics.<br />
Also platform state reconstruction has to be performed since the model is only explicit<br />
in platform coordinates. An iterative scheme is used to calculate the platform pose from<br />
actuator position measurement. Use in feedback requires convergence at all time for stability<br />
and guaranteed accuracy in a very limited amount <strong>of</strong> iterations if applied in real time.<br />
Sufficient conditions on the convergence (speed) <strong>of</strong> a general Newton-Raphson scheme can<br />
be translated to the general Stewart Platform. The jacobian used in the scheme is shown to<br />
be Lipschitz under mild conditions. With the Simona <strong>motion</strong> <strong>system</strong>, convergence in two<br />
iterations within measurement device accuracy can be guaranteed given the limited actuator<br />
speed and a sufficiently high update frequency (100 Hz), which was easily attainable.<br />
With partly feedback linearising <strong>control</strong> <strong>of</strong> the mechanical <strong>system</strong>, one strives for two<br />
objectives. First, from feed forward <strong>of</strong> desired accelerations, required forces are calculated<br />
taking the varying mass matrix, non-linear velocity terms and gravity into account. Further,<br />
feedback <strong>of</strong> positional and velocity error is performed in a theoretically decoupled set <strong>of</strong><br />
coordinates. Translation <strong>of</strong> desired platform forces to actuator forces requires inversion <strong>of</strong><br />
the jacobian, which, however, can also be used with the coordinate reconstruction.<br />
In the conventional structure <strong>of</strong> <strong>motion</strong> simulation, the series connection <strong>of</strong> <strong>control</strong>ler<br />
and <strong>motion</strong> <strong>system</strong> ideally requires an infinitely fast <strong>control</strong>led <strong>motion</strong> <strong>system</strong>. Since the<br />
aircraft model and wash-out filters are incorporated in the simulation program, reference<br />
model knowledge can be used to generate a predicted acceleration over a certain amount <strong>of</strong><br />
time. 30-50 ms would be sufficient to attain a situation in which the acceleration onset can<br />
be felt exactly at the required instant <strong>of</strong> time with a finitely fast <strong>motion</strong> <strong>system</strong>. This will be<br />
possible since the model <strong>based</strong> <strong>control</strong>led <strong>motion</strong> <strong>system</strong> behaves in a predictable manner.<br />
As positional accuracy is not the main objective <strong>of</strong> the <strong>control</strong>ler, the position feedback<br />
should only be used in preventing the <strong>system</strong> from running out <strong>of</strong> stroke and attaining the<br />
right accelerations in the low frequency area. Especially at the bandwidth <strong>of</strong> the positional<br />
feedback, care should be taken not to distort the appropriately generated accelerations.<br />
A model <strong>based</strong> <strong>control</strong>ler is able to enlarge the bandwidth <strong>of</strong> the accelerations to be simulated<br />
by a factor two w.r.t. the directions <strong>of</strong> the rigid mode with the lowest frequency and<br />
also compared to the conventional <strong>control</strong>led <strong>system</strong>. This was confirmed in this research.<br />
By limiting the bandwidth in the other directions to attain equal response in any direction,<br />
unnecessary excitation <strong>of</strong> unmodelled dynamics, amplification <strong>of</strong> noise and coupling between<br />
rotation and translation is avoided.<br />
The dynamics <strong>of</strong> the foundation at a test site, which is not explicitly taken into account,<br />
limits the performance <strong>of</strong> the <strong>system</strong>. Success <strong>of</strong> compensation is doubtful since the effect<br />
is seen to vary a unpredictable manner w.r.t. payload characteristics.<br />
System feedback <strong>control</strong> enforces reasonable linear response. With this assumption,<br />
the characteristics <strong>of</strong> the <strong>system</strong> can be tested by measuring the frequency response. The<br />
assumption can be verified by performing some additional tests. In case <strong>of</strong> a <strong>motion</strong> <strong>system</strong>,<br />
the linearity can be tested with respect to a set <strong>of</strong> different amplitude acceleration<br />
step responses and a standardised number <strong>of</strong> realistic manoeuvres representative for <strong>motion</strong>
210 6 Review and discussion on the results<br />
through the workspace <strong>of</strong> the <strong>system</strong>. Exact quantification <strong>of</strong> the noise would require more<br />
sophisticated methods and measurement devices. Moreover, the noise seems to vary over<br />
time.<br />
A modern standardised test method is required to evaluate the performance <strong>of</strong> a <strong>simulator</strong><br />
<strong>motion</strong> <strong>system</strong>. In this research some building blocks to set up such a test were provided.<br />
Main aspect in such a test should be the predictability <strong>of</strong> a <strong>system</strong> i.e. guaranteed simulation<br />
quality. The model <strong>based</strong> approach taken in this research helps to attain this.
Chapter 7<br />
Conclusion and recommendations<br />
7.1 Conclusion<br />
Central to this conclusion is the main problem statement to investigate what relevant <strong>system</strong><br />
knowledge should be used in a model <strong>based</strong> <strong>control</strong> strategy for a <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong><br />
<strong>system</strong> and to what extent this results in improved <strong>control</strong>led dynamics.<br />
In this thesis, the path <strong>of</strong> choice became physical modelling through balance equations<br />
<strong>of</strong> the hydraulically driven mechanical <strong>system</strong>, identification <strong>of</strong> the model parameters, experimental<br />
evaluation <strong>of</strong> the model structure and explicit use <strong>of</strong> the model in the <strong>control</strong>ler.<br />
An important first observation, considering the mechanical construction <strong>of</strong> Stewart platform<br />
like <strong>motion</strong> <strong>system</strong>, is their limited complexity if compared to general higher degree<br />
<strong>of</strong> freedom manipulators. The dynamics can explicitly be described as a function <strong>of</strong> a fixed<br />
set <strong>of</strong> platform coordinates unlike many combined serial/parallel constructions. This also<br />
limits the complexity <strong>of</strong> a model <strong>based</strong> <strong>control</strong> strategy.<br />
The mapping between the variations along the platform coordinates and actuator coordinates,<br />
the jacobian, is an important parameter in <strong>control</strong> <strong>of</strong> multi degree <strong>of</strong> freedom manipulators<br />
such as the Stewart platform. Most <strong>of</strong> the calculations in the model <strong>based</strong> <strong>control</strong>ler<br />
presented for the <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong> are directed towards this parameter.<br />
In parallel <strong>system</strong>s, <strong>control</strong>lability is lost if the jacobian loses rank in some position,<br />
unlike serial robotic <strong>system</strong>s. In this thesis, a method was presented to exclude these kind<br />
<strong>of</strong> singular positions from the reachable envelope <strong>of</strong> a parallel <strong>system</strong> considering limited<br />
stroke actuators. For the Simona Research Simulator (SRS) it was proven that such points,<br />
although not far away, can just not be reached.<br />
If the position <strong>of</strong> a parallel <strong>system</strong> is only measured along the actuator coordinates,<br />
an inverse problem reconstructing the platform coordinates has to be solved in real time<br />
using the model <strong>based</strong> <strong>control</strong> strategy. In this thesis a method was presented to prove<br />
convergence <strong>of</strong> the specific Newton-Rapshon iteration chosen anywhere in the reachable<br />
envelope <strong>of</strong> the platform. It was proven that it does and it does sufficiently fast for the<br />
SRS. The platform coordinate reconstructor could therefore be made a central module in<br />
the model <strong>based</strong> <strong>control</strong>ler, which was implemented.<br />
The platform pose dependent jacobian together with the platform coordinate related<br />
mass matrix <strong>of</strong> the <strong>system</strong> recover the specific masses as seen through the actuators. In<br />
211
212 7 Conclusion and recommendations<br />
most <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s, and also the SRS, large variations <strong>of</strong> the specific<br />
masses, more than an order <strong>of</strong> a magnitude, can be expected. Therefore, only misdirecting<br />
a force 1% easily results in a ten times higher accelerational error <strong>of</strong> 10%. Feedforward<br />
<strong>control</strong> therefore requires highly accurate models.<br />
The kinematical model structure <strong>of</strong> the <strong>system</strong> was shown to be accurate within tens<br />
<strong>of</strong> mm. Calibration <strong>of</strong> the model parameters improved the positional accuracy an order <strong>of</strong><br />
a magnitude. The structural model properties <strong>of</strong> the dynamics <strong>of</strong> the hydraulically driven<br />
mechanical <strong>motion</strong> <strong>system</strong> were also confirmed. However, estimation <strong>of</strong> the <strong>system</strong> mass<br />
properties can probably be improved despite the scheme developed to robustly derive the<br />
mass properties <strong>of</strong> such <strong>system</strong>s in the presence <strong>of</strong> dissipation or some nonlinear effects.<br />
The specific mass directions are almost equal to the rigid body modal directions connecting<br />
the hydraulic actuators. In analysis and identification <strong>of</strong> hydraulically driven <strong>motion</strong><br />
<strong>system</strong>s, knowledge <strong>of</strong> these directions proved to be very helpful to decouple the, at first<br />
sight, highly interactive dynamics. Un<strong>control</strong>led hydraulically driven <strong>system</strong>s are usually<br />
lightly damped and this increases the already large difference along the rigid body modal<br />
directions in the neighbourhood <strong>of</strong> the resonance frequencies. The rigid body resonance<br />
modes form the main dynamics in the most relevant frequency area for <strong>flight</strong> simulation up<br />
till 30Hz.<br />
The conventional <strong>control</strong> structure robustifies the decentralised positional feedback through<br />
dissipating energy by pressure feedback thereby damping the resonance peaks. In this thesis<br />
this strategy proved right by showing that feedback <strong>of</strong> the input valve pressure difference<br />
to valve flow is feedback <strong>of</strong> a passive transfer function, also in the presence <strong>of</strong> parasitic<br />
dynamics e.g. due to transmission line resonances or mechanical flexibilities. However,<br />
no directionality can be accounted for in this structure. It was shown that this leads to<br />
limited bandwidth <strong>system</strong>s, which still have to compromise overdamped directions with<br />
some peaking <strong>of</strong> others.<br />
With the model <strong>based</strong> strategy, the already known cascade dp structure was chosen,<br />
which maintained local decentralised dissipating pressure feedback by inner loops and decoupled<br />
hydraulics from mechanics. This leaves the directionality to be dealt with by the<br />
multivariable feedback <strong>of</strong> the mechanics. Through implementation <strong>of</strong> this <strong>control</strong>ler on<br />
the SRS it was shown that this indeed leads to a <strong>system</strong> with higher bandwidth and more<br />
equalised response in each degree <strong>of</strong> freedom.<br />
The attainable improvement through the model <strong>based</strong> <strong>control</strong>ler chosen is limited in several<br />
ways. Not all dynamics was or could be made part <strong>of</strong> the model used in <strong>control</strong>. In testing<br />
the SRS, <strong>of</strong> course, the dynamics <strong>of</strong> the foundation appeared in the relevant frequency<br />
area due to the non ideal experimental environment. But this should not be a problem in a<br />
specific <strong>simulator</strong> building. However, it can be concluded that the absence <strong>of</strong> such dynamics<br />
should be confirmed, tested, in any place where <strong>motion</strong> <strong>system</strong>s are to be used or evaluated.<br />
The flexible deformations <strong>of</strong> the <strong>system</strong> are a more severe problem. This causes loss <strong>of</strong><br />
functional <strong>control</strong>lability, strived for in simulation. Adding additional objects, e.g. projection<br />
screens, will further limit the attainable performance.<br />
The phase lag <strong>of</strong> the hydraulic valve usually limits the attainable feedback pressure<br />
bandwidth. With fast valves one can choose this bandwidth in between the rigid body<br />
modes and the transmission lines resonances but exactly in this frequency area the flexible
7.2 Recommendations 213<br />
resonances show up if velocity is not precisely compensated for, as is impossible in practice.<br />
The physical modelling in this research enabled a limited complexity model <strong>based</strong> <strong>control</strong><br />
structure for <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s by choosing only the most relevant dynamical<br />
effects to be incorporated. Also the parameter identification procedure could be made<br />
very compact in this way. The model <strong>based</strong> <strong>control</strong>ler proved better in almost all respects if<br />
compared to a conventional <strong>control</strong> structure implemented on the actual <strong>motion</strong> <strong>system</strong>.<br />
Summarising, the full design process proposed was shown to be applicable and leads to<br />
more predictable characteristics and higher performance <strong>of</strong> <strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s.<br />
7.2 Recommendations<br />
This thesis <strong>system</strong>atically went through the full design process <strong>of</strong> <strong>motion</strong> <strong>control</strong> design <strong>of</strong><br />
<strong>flight</strong> <strong>simulator</strong> <strong>motion</strong> <strong>system</strong>s but it still leaves many points <strong>of</strong> research open for more<br />
thorough investigation. Some at the high abstract level <strong>of</strong> the over all simulation, some at<br />
the lower level <strong>of</strong> detail though in practise <strong>of</strong>ten the most troublesome.<br />
First <strong>of</strong> all, specifications towards the design <strong>of</strong> <strong>motion</strong> <strong>control</strong>lers in simulation should<br />
become more precise in stating in what way the fundamentally non exact <strong>motion</strong> simulation<br />
characteristics should be compromised for. E.g. how much more accelerational noise can<br />
be settled for in attaining one additional Hertz <strong>of</strong> bandwidth? What class <strong>of</strong> signals will<br />
have to be tracked by the <strong>motion</strong> <strong>system</strong>? Usually there is much information in this area,<br />
structure in the trajectories, but not yet very well specified. Short time spectral properties<br />
<strong>of</strong> the reference signals is relevant in this sense since otherwise the important aspects, high<br />
frequency content, <strong>of</strong> e.g. a landing bump, would not be revealed.<br />
These specifications should form the basis for a standardised test to measure the performance<br />
and predictability i.e. robustness <strong>of</strong> a <strong>motion</strong> <strong>system</strong>. In this research a number<br />
<strong>of</strong> tests were introduced to measure the characteristics <strong>of</strong> a <strong>control</strong>led <strong>motion</strong> <strong>system</strong> in a<br />
more modernised manner as compared to the current <strong>flight</strong> simulation standards. But only<br />
providing more specific guidelines, weighting factors w.r.t. desired properties <strong>of</strong> a <strong>motion</strong><br />
<strong>system</strong>, will enable the design and analysis <strong>of</strong> the right test and can lead the way towards<br />
the most effective <strong>control</strong> strategy.<br />
Information from the <strong>flight</strong> simulation model can also be used more conveniently in the<br />
<strong>control</strong> scheme itself. Limited short response time <strong>of</strong> the required pressure results in lagged<br />
accelerational response <strong>of</strong> the <strong>motion</strong> <strong>system</strong> as compared to what was required in simulation.<br />
As the spectrum <strong>of</strong> the desired accelerations has almost always limited bandwidth,<br />
predictive information theoretically is sufficient to exactly attain the required response. In<br />
this thesis it was shown how this information can be used. Flight simulation models should<br />
be set up in such a way that this predictional information can be communicated to the <strong>motion</strong><br />
<strong>system</strong> <strong>control</strong>ler. It is the most viable way to attain virtual zero time delay response in<br />
simulation.<br />
Dynamics, not taken into account by the model <strong>based</strong> <strong>control</strong>ler, occurred in practise.<br />
The most important part was shown to be caused by the flexible deformations <strong>of</strong> the <strong>system</strong>,<br />
which in future will become even more relevant. Procedures to identify the specific structure<br />
<strong>of</strong> these parasitic resonances will have to be set up and applied. With more specifically
214 7 Conclusion and recommendations<br />
directed identification techniques in general, parameter estimation <strong>of</strong> the <strong>system</strong> model can<br />
probably be improved further.<br />
The <strong>control</strong> strategy should be extended in order to deal effectively with the uncertain<br />
or identified flexible dynamics. One could even consider applying additional actuators,<br />
to dissipate energy stemming from local resonant modes, to solve the current problem <strong>of</strong><br />
functional <strong>control</strong>lability in the presence <strong>of</strong> deformations. Also additional sensors, such<br />
as the accelerometers applied in identification, could be considered in an extended <strong>control</strong><br />
structure. After all, acceleration felt in the cockpit is what really matters in <strong>motion</strong> cues for<br />
<strong>flight</strong> simulation.<br />
Dealing with uncertainty in general in <strong>control</strong> requires robust feedback/feed forward<br />
schemes. <strong>Model</strong>s can never be exact. Unfortunately many modern robust <strong>control</strong> techniques<br />
are not mature enough at the moment to deal efficiently with nonlinear six d.o.f. hydraulically<br />
driven <strong>motion</strong> <strong>system</strong>s. In the side line <strong>of</strong> this research some first steps towards robust<br />
<strong>control</strong> <strong>of</strong> such <strong>system</strong>s were taken but this did not lead to satisfying results yet. Still, attaining<br />
robustly guaranteed properties in simulation is important in <strong>flight</strong> training and makes<br />
more effort in this direction worthwhile.<br />
W.r.t. the specific model <strong>based</strong> <strong>control</strong> structure chosen, the interaction between the inner<br />
and outer loops appeared to be more severe than expected. More exact velocity compensation,<br />
central in the decoupling <strong>of</strong> mechanics and hydraulics, would directly enhance the<br />
proposed <strong>control</strong> structure. This requires more accurate velocity estimation first. E.g. due<br />
to a structural deviation in the position measurement the increased resolution could not yet<br />
be used in more precise velocity reconstruction though it can probably be compensated for.<br />
Next, enhanced <strong>control</strong> over the valve, the oil flow to and from the actuators, would be <strong>of</strong><br />
help. Especially the nonlinear characteristics <strong>of</strong> the valves <strong>of</strong>ten appeared to be troublesome<br />
in practise. E.g. each valve has to be activated at all time by a high frequent dither signal to<br />
prevent stick <strong>of</strong> one <strong>of</strong> the spools. In setting the dither amplitude one had to be very careful<br />
not to hit transmission line dynamics (too high) or otherwise let sticking effects occur (too<br />
low). An adaptive setting can improve this.<br />
Over all this thesis provides on one hand many leads towards more scientific investigations<br />
and on the other hand laid down an improved design procedure, which can be applied<br />
directly by the practioneer. And that’s what’s research is all about.
Appendix A<br />
Frequency domain measurements<br />
215
hdp_i1<br />
hdp_i2<br />
hdp_i3<br />
hdp_i4<br />
hdp_i5<br />
hdp_i6<br />
216 A Frequency domain measurements<br />
hu_1<br />
hu_2 hu_3 hu_4 hu_5 hu_6<br />
Fig. A.1: Bode amplitude plots dummy platform from valve voltage to valve pressure<br />
measured and modelled (dashed). Frequency runs form 5Hz to 50Hz with<br />
ticks at 10Hz, 20Hz and 30Hz. Amplitude from 0:5 to 50 with ticks at 1 and 10.
hdp_i1<br />
hdp_i2<br />
hdp_i3<br />
hdp_i4<br />
hdp_i5<br />
hdp_i6<br />
hu_1<br />
hu_2 hu_3 hu_4 hu_5 hu_6<br />
217<br />
Fig. A.2: Bode phase plots dummy platform from valve voltage to valve pressure measured<br />
and modelled (dashed). Frequency runs form 5Hz to 50Hz with ticks at<br />
10Hz, 20Hz and 30Hz. Phase from ;700 to 100 with ticks from ;135 to<br />
90 every 45 .
hdp_i1<br />
hdp_i2<br />
hdp_i3<br />
hdp_i4<br />
hdp_i5<br />
hdp_i6<br />
218 A Frequency domain measurements<br />
hu_1<br />
hu_2 hu_3 hu_4 hu_5 hu_6<br />
Fig. A.3: Bode amplitude plots dummy platform from valve voltages to valve pressures<br />
along the rigid body modal directions measured and modelled (dashed). Frequency<br />
runs form 5Hz to 50Hz with ticks at 10Hz, 20Hz and 30Hz. Amplitude<br />
from 0:5 to 50 with ticks at 1 and 10.
hdp_i1<br />
hdp_i2<br />
hdp_i3<br />
hdp_i4<br />
hdp_i5<br />
hdp_i6<br />
hu_1<br />
hu_2 hu_3 hu_4 hu_5 hu_6<br />
219<br />
Fig. A.4: Bode phase plots dummy platform from valve voltages to valve pressures<br />
along the rigid body modal directions measured and modelled (dashed). Frequency<br />
runs form 5Hz to 50Hz with ticks at 10Hz, 20Hz and 30Hz. Phase<br />
from ;700 to 100 with ticks from ;135 to 90 every 45 .
hdp_i1<br />
hdp_i2<br />
hdp_i3<br />
hdp_i4<br />
hdp_i5<br />
hdp_i6<br />
220 A Frequency domain measurements<br />
hu_1<br />
hu_2 hu_3 hu_4 hu_5 hu_6<br />
Fig. A.5: Bode amplitude plots shuttle from valve voltage to valve pressure measured<br />
and modelled (dashed). Frequency runs form 5Hz to 500Hz with ticks at 10Hz<br />
and 100Hz. Amplitude from 0:03 to 30 with ticks at :1, 1 and 10.
hdp_i1<br />
hdp_i2<br />
hdp_i3<br />
hdp_i4<br />
hdp_i5<br />
hdp_i6<br />
hu_1<br />
hu_2 hu_3 hu_4 hu_5 hu_6<br />
221<br />
Fig. A.6: Bode amplitude plots shuttle from valve voltage to valve pressure measured<br />
and modelled (dashed). Frequency runs form 5Hz to 500Hz with ticks at 10Hz<br />
and 100Hz. Phase from ;700 to 100 with ticks from ;180 to 90 every 90 .
hdp_i1<br />
hdp_i2<br />
hdp_i3<br />
hdp_i4<br />
hdp_i5<br />
hdp_i6<br />
222 A Frequency domain measurements<br />
hu_1<br />
hu_2 hu_3 hu_4 hu_5 hu_6<br />
Fig. A.7: Bode phase plots shuttle from valve voltages to valve pressures along the<br />
rigid body modal directions measured and modelled (dashed). Frequency<br />
runs form 5Hz to 500Hz with ticks at 10Hz and 100Hz. Amplitude from 0:03<br />
to 30 with ticks at :1, 1 and 10.
hdp_i1<br />
hdp_i2<br />
hdp_i3<br />
hdp_i4<br />
hdp_i5<br />
hdp_i6<br />
hu_1<br />
hu_2 hu_3 hu_4 hu_5 hu_6<br />
223<br />
Fig. A.8: Bode phase plots shuttle from valve voltages to valve pressures along the<br />
rigid body modal directions measured and modelled (dashed). Frequency<br />
runs form 5Hz to 500Hz with ticks at 10Hz and 100Hz. Phase from ;700 to<br />
100 with ticks from ;180 to 90 every 90 .
ax<br />
ay<br />
az<br />
aex<br />
aey<br />
aez<br />
224 A Frequency domain measurements<br />
ax_d<br />
ay_d az_d aex_d aey_d aez_d<br />
Fig. A.9: Bode amplitude plots closed loop around heavy weight dummy platform<br />
from desired to actual ccelerations model <strong>based</strong> <strong>control</strong>ler (-) and conventional<br />
<strong>control</strong>ler (dashed). Frequency runs form 0:5Hz to 50Hz with ticks at<br />
1� 2� 3� 5� 10� 20� 30Hz. Amplitude from 0:025 (;32dB)to2:5 (8dB) with ticks<br />
at ;30� ;20� ;6� ;3� 0� 3� 6dB.
ax<br />
ay<br />
az<br />
aex<br />
aey<br />
aez<br />
ax_d<br />
ay_d az_d aex_d aey_d aez_d<br />
Fig. A.10: Bode phase plots closed loop from desired to actual accelerations model<br />
<strong>based</strong> <strong>control</strong>ler (-) and conventional <strong>control</strong>ler (dashed). Frequency runs<br />
form 0:5Hz to 50Hz with ticks at 1� 2� 3� 5� 10� 20� 30Hz. Phase from ;180<br />
to 180 with ticks from ;180 to 180 every 45 .<br />
225
e_ax<br />
e_ay<br />
e_az<br />
e_aex<br />
e_aey<br />
e_aez<br />
226 A Frequency domain measurements<br />
ax_d<br />
ay_d az_d aex_d aey_d aez_d<br />
Fig. A.11: Bode amplitude plots closed loop from desired to error accelerations model<br />
<strong>based</strong> <strong>control</strong>ler (-) and conventional <strong>control</strong>ler (dashed). Frequency runs<br />
form 0:5Hz to 50Hz with ticks at 1� 2� 3� 5� 10� 20� 30Hz. Amplitude from<br />
;32dB to 14dB with ticks at ;20� ;6� ;3� 0� 3� 6dB.
Appendix B<br />
Derivation <strong>of</strong> actuator inertial<br />
properties<br />
In this appendix essentially the derivation <strong>of</strong> (2.119) and (2.120) is performed. Further, it<br />
is shown how to choose a factor N in the quadratic velocity term to make _ M ; 2N skew<br />
symmetric. This last property is important in passivity <strong>based</strong> <strong>control</strong>.<br />
First the derivation <strong>of</strong> the derivative mass matrix equation (2.119).<br />
and<br />
With<br />
d<br />
dt (Mia�ib )=; (ia + ib)<br />
j l j3 (2v T a lnPln + Plnval T n + lnv T a Pln ) (B.1)<br />
d d p l<br />
j l j= lT l =<br />
dt dt<br />
T va<br />
j l j = lT n va<br />
_<br />
ln = d l<br />
dt j l j =<br />
l<br />
_ j l j;l d dt j l j<br />
j l j2 the required steps can be taken<br />
d<br />
(Mia�ib )<br />
dt<br />
=<br />
d<br />
dt ( (ia + ib)<br />
j l j2 Pln )=(ia + ib)<br />
(B.2)<br />
= (I ; lnlT n )<br />
va =<br />
j l j<br />
1<br />
j l j Plnva: (B.3)<br />
d<br />
dt<br />
1<br />
j l j2 I + d<br />
dt<br />
lnl T n<br />
j l j 2<br />
= (ia + ib) ;2lT n va j l j<br />
j l j4 1<br />
(<br />
jlj<br />
I ; PlnvalT 1<br />
n +<br />
jljlnvT a Pln ) j l j2 ;2lT n va j l j lnlT n<br />
j l j4 = ; (ia + ib)<br />
j l j3 (2v T a lnPln + Plnval T n + lnv T a Pln ): (B.4)<br />
N is defined as a factor in the factorisation <strong>of</strong> the nonlinear velocity term C(x� _x). But<br />
this factorisation is not uniquely defined by requiring that C(x� _x) =N (x� _x) _x.<br />
227
228 B Derivation <strong>of</strong> actuator inertial properties<br />
Taking<br />
f a ia�ib = Mia�ib _va ; 2(ia + ib)<br />
j l j3 (l T n va)Plnva: (B.5)<br />
The nonlinear velocity term, ; 2(ia+ib)<br />
jlj3 (lT n va)Plnva can be factored into N (x� va)va in an<br />
infinite number <strong>of</strong> ways. Two possible factors N are in this case:<br />
and<br />
N1 = ; 2(ia + ib)<br />
j l j3 (l T n va)Pln<br />
(B.6)<br />
N2 = ; 2(ia + ib)<br />
j l j 3 Plnval T n (B.7)<br />
And all factorisations N1 +(1; )N2, with 2 IR are valid. Only one factorisation<br />
results in a skew _ M ; 2N.<br />
Take<br />
Then<br />
N = 1<br />
2 (N1 + N2): (B.8)<br />
_M ; 2N = 2(ia + ib)<br />
j l j3 (Plnval T n ; lnv T a Pln ): (B.9)<br />
Check through _ M ; 2N = ;( _ M ; 2N ) T that the skew symmetry is there (P T<br />
ln<br />
= Pln).<br />
Now the derivation <strong>of</strong> the partial derivative <strong>of</strong> the kinetic energy actuator inertia given<br />
by<br />
@Kia�ib<br />
@pa<br />
= ; (ia + ib)<br />
j l j3 Define a partial derivative <strong>of</strong> a vector, x to a vector, y n 1 as:<br />
@x<br />
@y =<br />
h @x<br />
@y1<br />
@x<br />
@y2<br />
l T n (vT a Plnva) +v T a Pln (lT n va) : (B.10)<br />
:::<br />
@x<br />
@yi<br />
:::<br />
@x<br />
@yn<br />
Kinetic energy <strong>of</strong> the actuator intertia was derived as the scalar term:<br />
with<br />
Kia�ib<br />
i : (B.11)<br />
1<br />
=<br />
2 vT (ia + ib)<br />
a<br />
j l j2 Plnva (B.12)<br />
= 1<br />
2 vT a<br />
(ia + ib)<br />
j l j2 va ; 1<br />
2 (ia + ib)<br />
v T a l<br />
j l j 2<br />
v T a l<br />
j l j 2<br />
� (B.13)<br />
Pln =(I ; lnl T n )� ln = l= j l j : (B.14)
The upper actuator gimbal point, pa, depends on the variables c and in the same way as l:<br />
Therefore<br />
Now<br />
229<br />
pa =c + T ( )a� (B.15)<br />
l =c + T ( )a ; b: (B.16)<br />
@l<br />
= I: (B.17)<br />
@pa<br />
= 1<br />
2 (ia + ib)v T a va<br />
h<br />
1<br />
@<br />
jlj2 i<br />
; (ia + ib)<br />
@pa<br />
h<br />
vT a l<br />
jlj2 i<br />
@Kia�ib<br />
@pa<br />
vT a l<br />
j l j2 @<br />
@pa<br />
: (B.18)<br />
So two partial derivatives to scalars have be examined further. The first<br />
and the second<br />
@<br />
h v T a l<br />
jlj 2<br />
@pa<br />
i<br />
@<br />
h 1<br />
jlj 2<br />
@pa<br />
= vT a j l j 2 ;2v T a ll T<br />
j l j 4<br />
i<br />
= @ 1<br />
l T l<br />
@pa<br />
= vT a<br />
j l j2 ; 2vT a lnlT n<br />
j l j2 = ;2lT<br />
j l j4 = ;2lT n<br />
� (B.19)<br />
j l j3 Filling in these equations in (B.18) results in, using (x T y)=(y T x),<br />
@Kia�ib<br />
@pa<br />
=vT 1<br />
a Pln<br />
j l j2 ; vT a lnlT n<br />
: (B.20)<br />
j l j2 = ; 1<br />
2 (ia + ib)v T 2l<br />
a va<br />
T n<br />
j l j3 ; (ia + ib) v T 1<br />
a Pln<br />
j l j2 ; vT a lnlT n<br />
j l j2 = ; (ia + ib)<br />
j l j 3<br />
= ; (ia + ib)<br />
j l j 3<br />
And this was what was to be determined.<br />
v T a Ival T n ; v T a lnl T n val T n +vT a Pln (vT a ln)<br />
v T a ln<br />
j l j<br />
l T n (vT a Plnva) +v T a Pln (lT n va) : (B.21)
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Glossary <strong>of</strong> symbols<br />
241
242 Glossary <strong>of</strong> symbols<br />
Arabic symbols (large)<br />
Variable Unit Explanation<br />
A [m 2 ] Actuator operational area<br />
[m] 3x6 matrix with upper gimbal (to c.o.g.) vectors stacked<br />
Ap [m 2 ] Actuator operational area<br />
B [m] 3x6 matrix with lower gimbal vectors stacked<br />
B,Bp [Ns=m�Ns] Viscous friction (matrix)<br />
C [kg=s,kgm 2 =s] Nonlinear coriolis and centripetal terms<br />
Ci [N=m 5 � ;] Oil column stifness chamber i<br />
dC [N=m 5 ] Difference oil column stifness chambers 1 and 2<br />
Cm [N=m 5 ] Mean oil column stifness chambers 1 and 2<br />
E [N=m 2 ] Effective bulk modulus <strong>of</strong> the oil<br />
G [ ] Ground frame or intertial frame<br />
[;] 3x4 transformation as part <strong>of</strong> rotation R = GL T<br />
G T s [;] 3x3 ! to transformation<br />
I [ ] Identity matrix<br />
Iz [kgm 2 ] 3x3 platform inertia matrix evaluated at c.o.g.<br />
Jy�x [ ] Jacobian matrix between two sets <strong>of</strong> variables y and x<br />
Jl�x [;�m] Jacobian from platform velocities to actuator velocities<br />
K [ips=( c irc=s 2 �m=s 2 )] Semicircular and vestibular organ model gain<br />
[m=(sA)] Hydraulic actuator model gain<br />
Kd As=m Valve main spool differential feedback gain<br />
Kp A=m Valve main spool proportional feedback gain<br />
Kh [nd] Crossover model pilot gain<br />
L [ ] 4x3 eul. par. transformation as part <strong>of</strong> rotation R = GL T<br />
Ln [;] 3x6 matrix with all unit actuator directions vectors stacked<br />
Llm [m 5 =N] Oil leakage constant due to pressure difference actuator chambers<br />
Lt [m] Transmission line length<br />
M [kg,kgm 2 ] Mass matrix<br />
[ ] Moving frame<br />
[-] Mobility index<br />
Mact [kg� kgm 2 ] Simulator mass matrix actuator coordinates<br />
Mt [kg� kgm 2 ] Simulator mass matrix platform coordinates
Glossary <strong>of</strong> symbols 243<br />
Variable Unit Explanation<br />
dP [N=m 2 ] Net generated actuator pressure<br />
Pi1 [N=m 2 ] Oil pressure at valve side transmission line 1<br />
Pi2 [N=m 2 ] Oil pressure at valve side transmission line 2<br />
Pm [N=m 2 ] Mean actuator chamber pressure<br />
Pm1�2 [N=m 2 ] Main valve spool pressures<br />
Pn1�2 [N=m 2 ] Pilot valve spool pressures<br />
Po1 [N=m 2 ] Oil pressure at actuator chamber 1<br />
Po2 [N=m 2 ] Oil pressure at actuator chamber 2<br />
Ps [N=m 2 ] Supply oil pressure<br />
Pt [N=m 2 ] Tank oil pressure<br />
Pxn [;] 3x3 projection matrix onto plane with normal x n<br />
R [;] 3x3 rotation matrix<br />
B R A [;] 3x3 rotation matrix from frame A to B<br />
T [;] 3x3 rotation matrix platform<br />
Tx�y�z [;] 3x3 rotation matrix around one axis <strong>of</strong> the frame<br />
U [;] Unitary (rigid body modal direction) matrix<br />
V [m 3 ] General volume<br />
V1 [m 3 ] Oil volume actuator chamber 1<br />
V2 [m 3 ] Oil volume actuator chamber 2<br />
Vm [m 3 ] Mean oil volume actuator chambers 1 and 2<br />
Va [m=s] 3x6 matrix with upper gimbal velocity vectors stacked<br />
X general matrix (capital)<br />
~X vector product matrix<br />
Arabic symbols (small)<br />
Variable Unit Explanation<br />
ac [;] Upper actuator body c.o.g.<br />
ai [m] Vector from c.o.g. <strong>simulator</strong> to i th upper gimbal point<br />
bi [Ns=m] Viscous friction coefficient actuator i<br />
bxf�yf� f [Ns=m� Nms=rad] Viscous friction coefficients foundation<br />
bc [;] Lower actuator body c.o.g.<br />
bi [m] Vector from ground frame origin to i th lower gimbal point<br />
c [N=m] Hydraulic actuator stifness<br />
c1�::: �10 [ ] Valve model parameters<br />
co [m=s] Wave propagation velocity<br />
cxf�yf� f [N=m� Nm=rad] Stifness coefficients foundation<br />
c [m] Vector from ground frame origin to c.o.g. <strong>simulator</strong><br />
d [ ] Finitely small variation<br />
dl [m] Lower gimbal spacing<br />
du [m] Upper gimbal spacing<br />
e [ ] Error vector<br />
[ ] General external signal<br />
dp [N=m 2 ] Pressure difference over hydraulic actuator compartments
244 Glossary <strong>of</strong> symbols<br />
Variable Unit Explanation<br />
f N�Nm Inertial force<br />
fa N 6x1 vector active actuator forces<br />
f (:) [ ] Vector function<br />
g(:) [ ] Vector function<br />
g [N,Nm] Gravitational terms<br />
fd [N] Actuator driving force<br />
frm [1=s] Rigid body mode eigenfrequency<br />
i [A] Valve input current<br />
ia [kgm2 ] Inertia upper actuator body around<br />
ib [kgm2 j<br />
]<br />
[;]<br />
Inertia lower actuator body around around gimbal point<br />
Imaginairy unit, p ;1<br />
kdp [Am2 =N, ] (Normalised) pressure feedback gain<br />
li [m] Length ith joint (serial), actuator (parallel) manipulator<br />
l [m] Vector with all actuator platform lengths<br />
li [m] Vector form the i th lower to upper gimbal<br />
ln [;] Normalised unit direction vector actuator<br />
nG�M x�y�z [m] Unit direction vectors<br />
direction in ground, G, and moving frame, M<br />
m [kg] Mass (platform)<br />
ma [kg] Mass upper actuator body<br />
mb [kg] Mass lower actuator body<br />
mf [kg] Mass <strong>of</strong> the foundation<br />
m [kgm=s� kgrad=s] Vector <strong>of</strong> generalised momenta<br />
n [ ] Euler paramter unit vector along axis <strong>of</strong> rotation<br />
pg [m] Vector to point p in frame G<br />
pm [m] Vector to point p in frame M<br />
q [m] Actuator extension<br />
qi [m] Rotation angle i th joint serial manipulator<br />
q [m,rad] Robotic manipulator positional coordinates<br />
qd [m,rad] Desired positional coordinates<br />
ra [m] Length from c.o.g. upper actuator body to upper gimbal<br />
rb [m] Length from c.o.g. lower actuator body to lower gimbal<br />
s [rad=s] Sloppy Laplace operator as j!<br />
[m] Scaling factor, ka jmax<br />
sx [m] Vector with platform coordinates<br />
t [s] Time<br />
t1�::: �5 [ ] Transmission line model parameters<br />
tg [m] Translational vector in frame G<br />
u [V ] Actuator input voltage<br />
[ ] General <strong>system</strong> input<br />
va [m=s] Upper gimbal point velocity vector<br />
vm m m=s Velocity vector <strong>of</strong> a point in a moving frame
Glossary <strong>of</strong> symbols 245<br />
Variable Unit Explanation<br />
x [ ] General scalar<br />
[m] Surge direction<br />
xf [m] Valve flapper position<br />
[m] Foundation surge displacement<br />
xm [m] Valve main spool position<br />
xs [m] Valve pilot spool position<br />
xa [m] Vector from ground origin to upper gimbal point<br />
xn [ ] General normalized vector (index n) with length one<br />
x [ ] General vector (bar on top)<br />
_x [m=s� rad=s] Vector with translational and angular velocity platform<br />
x g [m] Vector in inertial (ground) frame<br />
x m [m] Vector in platform (moving) frame<br />
xin [(rad� m)=s 2 ] Semicircular and vestibular input (rotational) acceleration<br />
xout [(rad� m)=s 2 ] Noticed semicircular and vestibular (rotational) acceleration<br />
y [m] Sway direction<br />
[ ] General <strong>system</strong> output<br />
z [m] Heave direction<br />
z [m� rad] Vector with generalised (positional) coordinates<br />
Greek symbols (small)<br />
Variable Unit Explanation<br />
[ ] Bound on first Newton Raphson iteration<br />
[ ] Relative damping factor<br />
[ ] Bound on inverse lowest singular value NR-iteration<br />
[rad] Vector <strong>of</strong> euler angles<br />
[] Infinitely small variation<br />
0�1�2�3 [;] The specific four euler parameters<br />
[;] Vector with last three euler parameters 1, 2, 3<br />
e [;] Vector with all four euler parameters<br />
m [V ] Valve spool position error<br />
~ [;] Cross product matrix with euler parameters<br />
[ ] Lipschitz constant<br />
[rad] Angle in general<br />
[rad] Last euler angle around airplane roll axis<br />
[rad] Euler parameter angle <strong>of</strong> rotation<br />
[rad] Second euler angle around intermediate pitch-axis<br />
[kg=m 3 ] Oil density<br />
[ ] Singular value
246 Glossary <strong>of</strong> symbols<br />
Variable Unit Explanation<br />
[N,Nm] Generalized torques, forces<br />
1 [s] Crossover model pilot lead time constant<br />
[s] First semicircular and vestibular lag time constant<br />
2 [s] Crossover model pilot lag time constant<br />
[s] Second semicircular and vestibular lag time constant<br />
L [s] Semicircular and vestibular lead time constant<br />
d [s] Crossover model pilot pure time delay<br />
Nm�N Generalised torques<br />
! [rad=s] Frequency<br />
!o [rad=s] Hydraulic actuator eigenfrequency<br />
! [rad=s] Angular velocity vector platform<br />
!a [rad=s Angularo velocity vector actuator orth. to actuator length<br />
1�::: �5 [ ] Transmission line model parameters<br />
[rad] First euler angle around ground frame yaw axis<br />
Greek symbols (large)<br />
Variable Unit Explanation<br />
i1 [m 3 =s] Oil flow from valve into transmission line 1<br />
i2 [m 3 =s] Oil flow from valve out <strong>of</strong> transmission line 2<br />
l1 [m 3 =s] Leakage oil flow out<strong>of</strong> actuator chamber 1<br />
l2 [m 3 =s] Leakage oil flow into actuator chamber 2<br />
lm [m 3 =s] Leakage oil flow from actuator chamber 2 into 1<br />
m [m 3 =s] Mean oil flow into chamber 1 and out <strong>of</strong> chamber 2<br />
d [m 3 =s] Difference oil flow o1 ; o2<br />
~ [rad=s] Cross product matrix with angular velocity<br />
Caligraphic symbols (large)<br />
Variable Unit Explanation<br />
G [N�Nm] Generalised gravitational forces<br />
H [kgm 2 =s 2 ] Hamiltonian function (addition kinetic and potential energy)<br />
K [kgm 2 =s 2 ] Kinetic energy function<br />
L [kgm 2 =s 2 ] Langrangian function (difference kinetic and potential energy)<br />
P [kgm 2 =s 2 ] Potential energy function
Samenvatting en CV<br />
De hoge veiligheidseisen in de luchtvaart vereisen een goed begrip van het gedrag van de<br />
piloot in extreme weersomstandigheden. Vernieuwend onderzoek op dit gebied moet gebruik<br />
kunnen maken van vlucht<strong>simulator</strong>en die in staat zijn om met hoge nauwkeurigheid<br />
kritische condities tijdens een vlucht te simuleren. Verschijnselen zoals turbulentie en zijwind<br />
in de lagere delen van de atmosfeer vertonen een snelheidsspectrum dat grote krachten<br />
introduceert over een breed scala aan frequenties. Het reproduceren van de benodigde beweging,<br />
het voelbaar maken van de inertiële versnellingen en het juist uitrichten van de<br />
zwaartekracht in geavanceerde vlucht<strong>simulator</strong>en, is dus een uitdagende regeltaak die het<br />
uiterste uit het systeem moet halen. Voor vergelijkbare <strong>system</strong>en, zoals robots, bestaan<br />
moderne modelgebaseerde regelstrategieën, die hogere prestaties kunnen behalen dan minder<br />
gestructureerde methoden, meer inzicht geven in de begrenzingen van het systeem<br />
en wellicht op constructieve eigenschappen van het systeem kunnen wijzen waar gebruik<br />
van kan worden gemaakt in het ontwerp. Deze regelstrategieën worden nog vrijwel niet<br />
toegepast bij bewegings<strong>system</strong>en van vlucht<strong>simulator</strong>en. Eén van de redenen zou kunnen<br />
zijn dat deze <strong>system</strong>en zo complex zijn dat exacte <strong>of</strong> zeer gedetailleerde modellen geen<br />
onderdeel van de regelaar konden en kunnen worden gemaakt. Toepassing ervan in dit<br />
onderzoek vereiste een tussenstap waarin een minder gedetailleerd model moest worden<br />
geëxtraheerd dat toch de meest relevante dynamica moest beschrijven.<br />
De vraag welke relevante systeemkennis gebruikt zou moeten worden in een modelgebaseerde<br />
regelstrategie en in welke mate deze strategie bijdraagt aan het geregelde gedrag<br />
van een bewegingssysteem van een vlucht<strong>simulator</strong>, is het centrale onderzoeksthema van<br />
dit onderzoek geweest.<br />
Om hierop een antwoord te geven moest het volledige regelaarontwerpproces worden<br />
gedefinieerd, gestructureerd en geëvalueerd. Veel van de tussenstappen toegepast in dit proces<br />
kunnen al in de literatuur worden gevonden maar een integrale aanpak, waarin ook de<br />
specifieke eisen ten aanzien van een vlucht<strong>simulator</strong> werden meegenomen, ontbrak nog. Bij<br />
de keuze van bestaande dan wel nieuw voorgestelde bouwstenen in dit proces van systeemen<br />
regelaarontwerp, vormden integreerbaarheid en praktische toepasbaarheid de voornaamste<br />
argumenten.<br />
Door middel van modelvorming op basis van fysische balansvergelijkingen is eerst een<br />
relevante modelstructuur ten behoeve van systeemanalyse en regeling afgeleid. Daarna is<br />
deze structuur geëvalueerd met behulp van experimenten. Daarbij vond identificatie van de<br />
modelparameters en het in kaart brengen van het geldigheidsgebied van de gekozen structuur<br />
plaats. Vervolgens is de meest geschikte modelgebaseerde regelstrategie gekozen en<br />
aangepast voor toepassing op dit bewegingssysteem met zijn specifieke parallelle struc-<br />
247
248 Samenvatting en CV<br />
tuur. Ontwerp en implementatie van deze regeling op het bewegingssysteem van de Simona<br />
vlucht<strong>simulator</strong> was de volgende stap. Tenslotte is een testprocedure gedefinieerd om de<br />
prestatie van dit geregelde systeem te kunnen kwantificeren en te kunnen afzetten tegen de<br />
prestatie van een conventioneel geregeld systeem. Gegeven de specifieke toepassing van<br />
vluchtsimulatie moesten verschillende stappen in dit generieke ontwerpproces nader worden<br />
uitgewerkt.<br />
Opbouwen van vergaand inzicht in de kinematica en dynamica van het systeem was<br />
nodig om zowel de robuustheid van een conventionele regeling te behouden als ook de<br />
prestatie en de voorspelbaarheid te verhogen. Met fysische modelvorming kon kwalitatief<br />
inzicht worden opgebouwd en gelijkertijd gaf deze structuur de mogelijkheid om de parameters<br />
van het zich niet-lineair gedragende bewegingssysteem experimenteel te identificeren<br />
en de te implementeren regelaarstructuur te modulariseren. Met deze identificatie en experimentele<br />
evaluatie kon de vereiste nauwkeurigheid voor het model worden bereikt die<br />
nodig was om de prestatieverhoging met de modelgebaseerde regelaar ook in de praktijk te<br />
kunnen behalen.<br />
Fysische modelvorming van bewegings<strong>system</strong>en begint met de analyse van de systeemkinematica.<br />
In dit onderzoek wordt een nieuwe methode gepresenteerd waarmee het<br />
voor het eerst mogelijk werd om te garanderen dat singuliere punten buiten het werkgebied<br />
van het systeem vallen. Hierop verder bouwend kan ook nagegaan worden <strong>of</strong> een iteratief<br />
schema om de platformcoördinaten te berekenen, gegeven de gemeten lengte van de actuatoren,<br />
convergeert en snel genoeg convergeert voor implementatie in de werkelijke regeling.<br />
Hiermee kon voor het Simona bewegingsysteem de garantie worden geboden dat dit deel<br />
van de regelstructuur veilig toegepast kon worden. Inzicht in de kinematica gaf verder de<br />
mogelijkheid om door middel van calibratie de positioneringsnauwkeurigheid van het al<br />
nauwkeurig gespecificeerde systeem met een factor tien te verhogen waarmee direct ook<br />
het kinematisch model geëvalueerd was.<br />
Van het bewegingssysteem in de vorm van een Stewart platform is afgeleid dat deze tot<br />
de klasse van volledig parallelle <strong>system</strong>en behoort waardoor de dynamica als een stelsel expliciete<br />
differentiaalvergelijkingen beschreven kan worden vanuit de platformcoördinaten.<br />
Zowel in theorie als in de praktijk kon aangetoond worden dat de modes van de stijve<br />
lichamen volgend uit de interactie tussen hydraulica en mechanica de meest relevante systeemdynamica<br />
vormen bij <strong>simulator</strong>toepassingen. De relatie tussen platform- en actuatorcoördinaten<br />
beschreven door een jacobiaan vormt hierbij de centrale operator. Volgend<br />
uit deze analyse kan het systeem, met op het eerste gezicht sterke interactie, in elke stand<br />
lokaal vanuit de juiste coördinaten gezien worden als zes hydraulische actuatoren die elk<br />
vrijwel onafhankelijk een massa aandrijven.<br />
Met de voorgestelde modelgebaseerde regeling kunnen een aantal zaken bereikt worden<br />
die conventioneel niet mogelijk zijn. Ten eerste kunnen de krachten benodigd voor de<br />
gewenste versnelling aangepast worden voor de standsafhankelijke richtingen waarin ontkoppeld<br />
de sterk verschillende massa’s gevoeld worden. Vervolgens kunnen de toch nog<br />
optredende positioneringsfouten ook langs deze richtingen zonder interactie teruggekoppeld<br />
worden. Tenslotte wordt de interactie tussen de hydraulica en de mechanica geminimaliseerd<br />
door te compenseren voor de oliestroom die nodig is om de gewenste snelheid<br />
aan te houden. Implementatie van deze structuur heeft aangetoond dat hiermee een hogere
Samenvatting en CV 249<br />
bandbreedte, minder opslingering en een meer over de vrijheidsgraden geëgaliseerde respons<br />
kan worden bereikt dan met de minder gestructureerde conventionele regeling. Verder<br />
is aangegeven hoe nog meer bereikt kan worden als ook voorspellende informatie vanuit het<br />
voertuigsimulatiemodel aan de regeling wordt aangeleverd.<br />
De te behalen prestaties van de voorgestelde regelaarstructuur worden eerst begrensd<br />
door het optreden van parasitaire effecten ten gevolge van mechanische vervormingen, flexibiliteiten.<br />
Ook met de combinatie van de altijd iets vertraagde olietoevoer en zingende<br />
leidingen moet op nog wat hogere frequenties rekening gehouden worden. Door in het<br />
ontwerp van een <strong>simulator</strong> een laag gewicht en hoge stijfheid na te streven is echter aangetoond<br />
dat met de voorgestelde regelaarstuctuur over een breed spectrum kritische simulatieexperimenten<br />
zeer realistisch uitgevoerd kunnen worden.
250 Samenvatting en CV<br />
Curriculum Vitae<br />
Naam Sjirk Holger Koekebakker<br />
25 oktober 1967 Geboren te Vierhouten (Gemeente Ermelo).<br />
1980 - 1986 VWO aan het Agnes College te Leiden.<br />
1986 - 1993 Werktuigbouwkunde aan de Technische Universiteit Delft.<br />
Afgestudeerd vanuit de vakgroep Systeem- en Regeltechniek.<br />
Titel afstudeerwerk: ”Decoupling by means <strong>of</strong> H 1�F ;<br />
application to a wafer stepper”. Afstudeerwerk uitgevoerd<br />
op het Philips Natuurkundig Laboratorium te Eindhoven.<br />
1993 - 1994 Onderzoeker bij het TNO Productcentrum i.h.k.v.<br />
de vervangende dienst.<br />
1994 - 1998 Promovendus bij de vakgroep Systeem- en Regeltechniek<br />
van de faculteit Werktuigbouwkunde<br />
aan de Technische Universiteit Delft.<br />
1999 - Onderzoeker bij de Group Research Technology<br />
te R & D Venlo van Océ Technologies B.V.