NONLINEAR MODELING OF A QUANSER FLEXIBLE LINK Monica ...

NONLINEAR MODELING OF A QUANSER FLEXIBLE LINKMonica Roman, Eugen Bobasu, Dorin SendrescuUniversity of Craiova, Faculty of Automation, Computers and ElectronicsDepartment of Automation, e-mail: monica@automation.ucv.roAbstract: This paper presents an approach on the bond graph modeling applied tomechanical oscillating systems, Quanser flexible link, followed by the simulation of theresulted mathematical model. The system was decomposed into smaller parts –subsystems – that were modeled separately. The obtained subsystems generatedsubmodels and the overall model was then built up by combining these separatestructures. The linear model was obtained by modifying the flexible link displacementfactor. We compared the results of linear and nonlinear models simulation to identify thenonlinear factor influence upon the output parameters evolution.Keywords: Bond Graphs, Flexible Arm, Modeling, Simulation.1. INTRODUCTIONThe study of system dynamics resides in modeling itsbehavior. Systems models are simplified, abstractedstructures used to predict the behavior of the studiedsystems. Our interest is pointing towards themathematical model used to predict certain aspects ofthe system response to the inputs. In mathematicalnotations a system model is described by a set ofordinary differential equations in terms of statevariables and a set of algebraic equations that relatethe state variable to other system variables.In order to model a system it is usually necessary todecompose the system into smaller parts –subsystems - that can be modeled separately. Thesubsystem is a part of the system that can be modeledas a system itself obtaining submodels. The overallmodel can then be built up by combining the separatesubmodels.System models will be constructed using a uniformnotation for all types of physical systems which isbond graph method based on energy and informationflow (Karnopp D., 1975). The method uses the effortflowanalogy to describe physical processes. A bondgraph consists of subsystems linked together by linesrepresenting power bonds. Each process is describedby a pair of variables, effort (e) and flow (f), and theirproduct is the power. The direction of power isdepicted by a half arrow. In a dynamic system theeffort and the flow variables, and hence the powerfluctuate in time. Using the bond graph approach it ispossible to develop models of electrical, mechanical,magnetic, hydraulic, pneumatic, thermal, and othersystems using a small set of variables.It is remarkable how models of various systemsbelonging to different engineering domains can beexpress using a set of only nine elements, calledelementary components. These elements aresufficient to describe any physical system regardlessof the energy types processed by it.A classification of bond graph elements can be madeup by the number of ports. The ports are placeswhere interactions with other processes take place.There are one port elements represented by inertialelements (I), capacitive elements (C), resistiveelements (R), effort sources (SE) and flow sources(SF), two ports elements represented by transformerelements (TF) and gyrator elements (GY), and multiports elements - effort junctions (J0) and flowjunctions (J1).Two other types of variables are very important indescribing dynamic systems and these variables,sometimes called energy variables, are thegeneralized momentum (p) as time integral of effortand the generalized displacement (q) as time integralof flow.

134 56891012131416 1727111518Fig. 9. The complete Bond Graph model of the systemBy joining together these three models, we obtainedthe complete bond graph model of the Quanserflexible link system.Gawthrop and Bevan (2007) presented a model ofthis system resuming to a simplified bond graphrepresentation.Before writing the constitutive equations it isdesirable to name all the bonds in the graph, to assignto each bond a reference power direction and toassign causality to each bond. The study of causalityis an important feature of bond graph method and itis specified by means of the causal stroke. The causalstroke is a short, perpendicular line made at one ofthe bond line ends indicating the direction in whichthe effort signal is directed.It can be easily seen that we have four elements inintegral causality which means that we have fourstate variables in terms of generalized momentumsand generalized displacements ( p2, p10,q13,p15).The constitutive equations of I and C elements inintegral causality are given by the followingrelations:f21 11= p2, f10= p10,f15= p15(6)L JJme13hg1= q13(7)kstiffFor the inertial elements in derivative causality and Relements, the constitutive equations are as follows:e= R , e7 = Bmf7, e11 = Bhgf11(8)p6 = Jmf 6, p15 = mlf15(9)3 mf 3The effort junctions (J1) and the flow junctions (J0)are characterized by, the flows on all bonds equal tozero and the algebraic sum of the efforts equal tozero, respectively the efforts on all bonds equal tozero and the algebraic sum of the flows equal to zero.For the TF and GY elements we have the followingrelations:lf(TF)e891= f9n, (GY)1= e8nee45= kt= ktff54(10)There is one more transformer element, a modulatedor controlled transformer, denoted MTF. Thiscomponent satisfies the power conservationrequirement. This is satisfied not only by thetransformer modulus, but also by the ratiosdependent on a control variable. In our case themodulated transformer modulus depends on therelative angular velocity of the flexible link. Thus,the modulated transformer is described by thefollowing constitutive relations:ef1617= k e1= k117f16(11)Combining all these equations and using thefollowing notationsJ +e= n2 JhgJm, Ren Rhg+ RmJeb= k2ml+ Jl1Lcos(α)= 2 (12)1(13)k = (14)we arrive at the state space equations in terms ofenergy variables p and q.⎧⎪ p&⎪⎪⎪ p&⎪⎨⎪q&⎪⎪⎪⎪ p&⎩2101318R= −LnJhgk=J L1=Jehgpp10k1ml=J kebmmmtstiff2k−nJp2r−J1−k mq131thgeleppp101018+ e12k J−k Jstiffhgeq13(15)

Deriving the constitutive equations of I and Celements we will obtain the state space equations interms of power variables. Taking into account thephysical significance of the effort e and flow f:f2= i a, f10= ω , f18 = ωl(16)e = τ(17)13we arrive at the final form of state space equations:⎧diaRmkt1⎪ = − ia− ω + udt LmnLmLm⎪2⎪dωnktren⎪ = ia− ω − τdt JeJeJe⎨⎪dτ1 1= ω − ωl⎪ dt kstiffkstiff⎪⎪dωl1= τ⎪⎩ dt Jeba(18)Where: ia- the armature current, ua- the armaturevoltage, ω - angular velocity of gear shaft, τ - thetorque action on the rotor, ω l- angular velocity ofthe flexible link.4. SIMULATION RESULTSFor the simulation the following values were used:u a =2V motor input voltage, R m =2.6 Ω armatureresistance, L m= 0.18e− 3Harmature inductance,k t= 0 . 00767 N ⋅mmotor torque constant,2J m= 3.87e− 7 Kg ⋅ m motor inertia,2J hg= 2e− 3Kg⋅mequivalent high gear inertia,B hg= 4e− 3Nm/( rd / s)viscous dampingcoefficient, L = 0. 45mthe flexible link length,m l= 0. 08 Kg flexible link mass; f c= 3. 2 Hz naturalfrequency;Figures 10 – 14 represent the variations of mainmodel parameters.Angular velocity4.543.532.521.510.500 0.5 1 1.5 2 2.5 3TimeFig. 11. The gear shaft angular velocity as timefunctionIn the diagrams the continuous line was used topresent the results of linear system model. Withdashed lines were represented the results of nonlinearsystem model. The nonlinearity is introduced by theangular factor sin(α ) in the free end of flexible linkdisplacement (eq. 4).Torque0.250.20.150.10.050-0.05-0.10 0.5 1 1.5 2 2.5 3TimeFig. 12. The shaft torque variation54.543.50.80.70.60.5Angular velocity32.521.51Current0.40.30.20.10.500 0.5 1 1.5 2 2.5 3TimeFig. 13. The angular velocity of the flexible linkdeflection0-0.10 0.5 1 1.5 2 2.5 3TimeFig. 10. The current variation in the DC motorPrevious works on oscillating mechanical units suchas the flexible beam, ball and beam, gyro, rotaryinverted pendulum etc. carried out (Ionete C., 2003)showed similar evolutions of the system behavior.

Amplitude654321Impulse Responsethese three models, we obtained the complete bondgraph model of the Quanser flexible link system.The model was simulated in the MatLab environmentthe results showing differences between linear andnonlinear systems in terms of amplitude for theoutput parameters due to angular factor existent inthe mathematical expression of the last one. Despitethese differences the settling time is almost the samefor both systems.0-1REFERENCES-20 0.5 1 1.5 2 2.5 3 3.5 4 4.5Time (sec)Fig. 14. The impulse response of the flexible linkBased on parameters evolution in the diagrams abovewe notice a slight difference between linear andnonlinear response. The settling time is almost thesame for both systems even if the amplitude fornonlinear model is higher. The influence of systemnonlinearity is more visible in the variation ofangular velocity of the flexible link deflection due toangular factor existent in its mathematicalexpression.5. CONCLUDING REMARKSIn this work our interest was pointed towards themathematical model used to predict certain aspects ofthe system response to the inputs.The system model was constructed using a uniformnotation for all types of physical systems, the bondgraph method based on energy and information flow.Using this method, models of various systemsbelonging to different engineering domains can beexpress using a set of only nine elements.First we wrote a word bond graph containing wordsinstead of standard symbols for the main componentsand bonds for power and signal exchange. The nextstep was to replace words by standards elements thatcontain precise mathematical or functional relations.The system was decomposed into three subsystemsthat were modeled separately. By joining togetherDamic V., Montgomery J. (2002). Mechatronics byBond Graphs. Springer, Germany.Dauphin-Tanguy G. et al. (2000). Les Bond Graphs.Hermes Science, Paris.Fossard A. J. (1993). Sistemes Nonlineaires. Masson,Paris.Gawthrop P. J., Bevan G. P. (2007). A TutorialIntroduction for Control Engineers. IEEE ControlSystems, Vol. 27, No. 2, pp. 24-45.Ionete C. (2003). LQG/LTR Controller Design forFlexible Link Quanser Real-time Experiment, inProc. of Int. Symp. SINTES11, Craiova,Romania, vol. 1, pp. 49-54.Ivanescu M. (1994). Roboti industriali. Ed.Universitaria, Craiova.Karnopp D., Rosenberg R. (1975). System Dynamics:A Unified Approach. John Wiley & Sons, NewYork.Pastravanu O., Ibanescu, R. (2001). Bond-graphlanguage in modeling and simulation of physicaltechnicalsystems (in Romanian). Gh. Asachi,Iasi.Thoma J. (1990). Simulation by bond-graphs.Introduction to a Graphical Method. SpringerVerlag, New York.*** Quanser Consulting Inc., MultiQ, ProgrammingManual*** Quanser Consulting Inc., WinCon 3.2, Real-Time Digital Signal Processing and Control underWindows 95/98 and Windows NT using Simulinkand TCP/IP Technology*** Quanser Consulting Inc., Flexible LinkExperiment ROTFLEX, 1998