Development of a Hill-Type Muscle Model With Fatigue for the ...

Development of a Hill-Type Muscle Model With Fatigue for
the Calculation of the Redundant Muscle Forces using
Multibody Dynamics
ANDRÉ FERRO PEREIRA
Dissertação para obtenção do Grau de Mestre em
ENGENHARIA BIOMÉDICA
Júri
Presidente: Prof. Helder Carriço Rodrigues
Orientador: Prof. Miguel Pedro Tavares da Silva
Prof. Mamede de Carvalho
Vogais: Prof. Jorge Manuel Mateus Martins
Prof. João Nuno Marques Parracho Guerra da Costa
Outubro 2009

Resumo
O objectivo deste trabalho é o desenvolvimento de um modelo muscular
versátil e a sua implementação de forma robusta e eficiente num código de
dinâmica de sistemas multicorpo com coordenadas naturais. São considera-
dos dois tipos de modelos: o primeiro é um modelo muscular do tipo Hill que
simula o comportamento das estruturas contrácteis, tanto para análises em
dinâmica directa como inversa. O segundo é um modelo dinâmico de fadiga
muscular que toma em consideração o historial de cada músculo, em termos
de força produzida, estimando o seu nível de aptidão física usando um mod-
elo multi-compartimentar triplo e a hierarquia de recrutamento muscular.
A formulação para as equação do movimento é adaptada, de forma a incluir
os modelos descritos, através do método de Newton. Isto permitirá, numa
perspectiva de dinâmica directa, que se proceda ao cálculo da cinemática do
sistema mecânico resultante para determinadas activações musculares previ-
amente conhecidas, ou, numa perspectiva de dinâmica inversa, a computação
das activações musculares necessárias para provocarem um movimento artic-
ular prescrito. Para ambas estas formulações, os sistemas são redundantes,
uma propriedade que é ultrapassada no segundo caso usando um algoritmo
de optimização.
As metodologias e modelos são aplicados para vários casos de estudo, de
forma a avaliar a sua robustez e precisão. Um modelo da extremidade su-
perior com sete músculos é considerado para evidenciar a aplicabilidade de
um modelo de fadiga muscular num sistema multicorpo. Um segundo mod-
elo, que inclui um aparato músculo-esquelético da extremidade inferior com
doze músculos, é utilizado com único propósito de calcular as activações
musculares. Os resultados associados a estes modelos são apresentados bem
como as respectivas conclusões. O trabalho conclui perspectivando eventuais
desenvolvimentos futuros.

Palavras-chave: Dinâmica multicorpo, dinâmica de contracção muscular,
dinâmica de fadiga muscular, forças musculares, activações musculares, op-
timização.

Abstract
The aim of this work is to develop a versatile muscle model and robustly
implement it in an existent multibody system dynamics code with natural
coordinates. Two different models are included: the first is a Hill-type muscle
model that simulates the functioning of the contractile structures both in
forward and in inverse dynamic analysis; the second is a dynamic muscular
fatigue model that considers the force production history of each muscle and
estimates its fitness level using a three-compartment theory approach and a
physiological muscle recruitment hierarchy.
The existent equations of motion formulation is rearranged to include the
referred models using the Newton’s method approach. This allows, in a
forward dynamics perspective, for the calculation of the system’s motion
that results from a pattern of given muscle activations, or, in an inverse
dynamics perspective, for the computation of the muscle activations that are
required to produce a prescribed articular movement. In both perspectives
the system presents a redundant nature that is overcome in the latter case
using an SQP optimization algorithm.
The methodologies and models are applied to several case studies to eval-
uate their robustness and accuracy. An upper extremity model with seven
muscles is designed to evidence the effectiveness of the implementation of
a muscle fatigue model in a multibody system. A second model, encom-
passing the lower extremity musculoskeletal apparatus with twelve muscles,
is proposed for the exclusive calculation of muscle activations. The results
are presented and conclusions are discussed. The work concludes with a
perspective of possible future developments.

Keywords: Multibody dynamics, muscle contraction dynamics, muscle fa-
tigue dynamics, muscle forces, muscle activations, optimization.

Acknowledgements
I would like to start by expressing my deepest gratitude to my supervisor Dr.
Miguel Silva, to whom I owe for his inspiration, knowledge, encouragement
and patience. This thesis would not have been possible without his wisdom
and total dedication. The uncountable meetings and brainstorming sessions
were the core of this project and are definitely the highlights of my academic
experience, so far.
To Prof. Dr. Mamede de Carvalho and Dr. João Costa, for providing their
highly motivational medical feedback and points-of-view.
A big thanks to Dr. Jorge Martins and to my colleagues Rita Malcata,
Pedro Moreira and Daniel Lopes for their help in the whole project process.
To Dr. Marko Ackermann, Dr. Maury Hull, Dr. Maxime Raison and Dr.
Ting Xia for their kindness in providing their papers and work.
To the University of Washington for providing the Musculoskeletal Images
from the "Musculoskeletal Atlas: A Musculoskeletal Atlas of the Human
Body" by Carol Teitz, M.D. and Dan Graney, Ph.D.
To all of my friends that supported me throughout these University years.
Nevertheless, some of them should be mentioned, due to their true friendship
and comfort: Nadir Abu-Samra, Diogo Almeida, Ana Barradinhas, André
Bento, João Cabaça, Luís Cabecinha, Akshay Chaudry, Fábio Coelho, João
Fayad, Artur Ferreira, Daniel Fitas, Diogo Geraldes, Maria João Lascas,
Gonçalo Marcelo, Daniel Martins, André Medeiros, Diana Nunes, João Maia
de Oliveira, Susana Palma, Manuel Rosa, Rafael Rosário, João Lála dos
Santos, Ricardo Serrano, César Silveira, João Venes.
To all my family, namely to my parents, my sister, my grandmother, cousins
and Maria.

The deepest acknowledgment goes to my mother. I thank her for being the
person who always was there for me, who motivated me all the way through,
and granted that I was guided by the same values of ambition and drive that
I recognise in herself. Indeed, this work is dedicated to her.

Para a minha mãe, Maria Isabel.
To my mother, Maria Isabel.

Contents
List of Figures vii
List of Tables xi
List of Symbols xiii
Glossary xv
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Musculoskeletal System Modelling 11
2.1 Skeletal Muscle Anatomy and Physiology . . . . . . . . . . . . . . . . . 12
2.1.1 Skeletal Muscle Anatomy . . . . . . . . . . . . . . . . . . . . . . 12
2.1.2 Skeletal Muscle Physiology . . . . . . . . . . . . . . . . . . . . . 18
2.2 Dynamics of the Muscle Tissue . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.1 Activation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Contraction Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 Muscle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
v

CONTENTS
3 Multibody Dynamics 39
3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Generic muscle forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Inverse Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6 Forward Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Biomechanical Models 63
4.1 Muscle model verification . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2 Muscle Fatigue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Human Gait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5 Conclusions and Future Developments 93
5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2 Future Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
A Apollo – Hill-type muscles manual 99
A.1 MDL File . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A.2 Simulation file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
B MHILL Data Visualizer manual 103
References 109
vi

List of Figures
2.1 Arrangement of the skeletal muscle structure, from an external level to
a molecular level [53]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Representation of the penation angle α between muscle fibers and ten-
dons [42]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3 Detail of a myosin molecule (A) and an actin filament (B) [53]. . . . . . 15
2.4 Illustration of the cross-bridges formed by the connections of actin fila-
ments with a myosin filament. Adapted from Reference [53]. . . . . . . . 15
2.5 Detail of a muscle fiber [62]. . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 Neural muscular junction with three different detail levels [62]. . . . . . 17
2.7 Contraction regulation mechanism by the troponin-tropomyosin complex,
dependent on the concentration level of Ca 2+ [62]. . . . . . . . . . . . . 18
2.8 Crossbridge theory steps [62]. . . . . . . . . . . . . . . . . . . . . . . . . 19
2.9 Relation between twitch frequency and effective muscle contraction [1]. . 21
2.10 Scheme of muscle tissue dynamics, with a series model of Activation
Dynamics and Contraction Dynamics. . . . . . . . . . . . . . . . . . . . 21
2.11 Activation dynamics model consistency [20]. . . . . . . . . . . . . . . . . 23
2.12 Hill-type muscle models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.13 Discrete length-tension diagram of the contractile contribution of a single
fully activated sarcomere [53] (a) and the same relationship scaled to the
whole muscle [1] (b). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.14 Force-length relationship of the passive element [1]. . . . . . . . . . . . . 27
2.15 Force-velocity relationship [18]. . . . . . . . . . . . . . . . . . . . . . . . 28
2.16 Activation scaling evidence: Force-length and force-velocity relationships
for different levels of muscle activation a(t) [18]. . . . . . . . . . . . . . . 28
vii

LIST OF FIGURES
2.17 Block diagram of the muscle contraction dynamics model implemented
in the forward dynamics multibody system routines. . . . . . . . . . . . 30
2.18 Block diagram of the muscle contraction dynamics model implemented
in the inverse dynamics multibody system routines. . . . . . . . . . . . . 30
2.19 Curve based on Rohmert’s relationship between %MVC (percentage of
maximum voluntary contraction) and endurance time (minutes) [56]. . . 32
2.20 Three-compartment theory flowchart. . . . . . . . . . . . . . . . . . . . . 33
2.21 Muscle recruitment hierarchy pile chart [59]. . . . . . . . . . . . . . . . . 36
3.1 Muscle representation for a biomechanical system. . . . . . . . . . . . . 45
3.2 Application of a force F m to pointo p, belonging to the rigid body defined
by points i and j and vectors u and v [1]. . . . . . . . . . . . . . . . . . 46
3.3 The different representations of a generic muscle force F m p . . . . . . . . . 48
3.4 Muscle force representation for the 3 via-point model in Figure 3.1. . . . 50
3.5 Used optimization methodology. . . . . . . . . . . . . . . . . . . . . . . . 56
3.6 Direct integration algorithm flowchart for a forward dynamics problem [1]. 60
4.1 Mechanical system with muscles m X and m Y for Hill-type muscle model
verification purposes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2 Activations pattern for muscles m X and m Y . . . . . . . . . . . . . . . . 65
4.3 The state of the mechanical model for different time instants when mus-
cles m X and m Y are activated as in Figure 4.2. . . . . . . . . . . . . . . 66
4.4 Muscle m X contractile element force output (black line) and possible
forces for the model (coloured surface). . . . . . . . . . . . . . . . . . . . 67
4.5 Results obtained for three cycles of isometric contractions of the described
model. The quantities MA + MR, MF and a(t) are scaled to F0. . . . . . 69
4.6 Simple upper extremity model with 7 elbow muscles and constant force
P applied at the hand. This image was conceived using the OpenSim
software [71]. Local reference frames only indicated in the lateral view. . 70
4.7 Resultant muscle forces of the contractile element F m CE
of the biomechan-
ical model of the upper extremity, when susceptible to fatigue dynamics. 71
4.8 Simple leg model with 12 muscles spanning the knee and ankle joints.
This image conceived using the OpenSim software [71]. Local reference
frames only indicated in the lateral view. . . . . . . . . . . . . . . . . . . 75
viii

LIST OF FIGURES
4.9 Scheme with relative progress of the gait cycle, indicating the stance and
swing phases [74]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.10 Part of the gait cycle with four points of reference [75]. . . . . . . . . . . 81
4.11 Point numbering used in the model: Point 1 - Lower torso, point 2 - hip
joint, point 3 - knee joint, point 4 - ankle joint, point 5 - metatarsopha-
langeal joint, 6 - heel position reference [75]. . . . . . . . . . . . . . . . . 81
4.12 Driver 1: Trajectory driver with the space coordinates in time of point 1. 82
4.13 Angular directions of Drivers 3, 4 and 5. . . . . . . . . . . . . . . . . . . 82
4.14 The motion obtained using the prescribed kinematic drivers for the model
of the lower extremity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.15 Reaction force vector components, acquired from a force platform. . . . 84
4.16 Ground reaction force application point coordinates, or centre of pressure
curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.17 Muscle activation patterns obtained from the solution of the EOM. . . . 86
4.18 Resultant total muscle forces of each considered muscle group in the lower
leg extremity model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.19 Comparison between the calculated total force of the tibialis anterior
muscle considering (Physiological case) and not considering (Non-physiological
case) a contraction dynamics model. . . . . . . . . . . . . . . . . . . . . 88
4.20 Comparison between the calculated activation of the tibialis anterior
muscle considering (Physiological case) and not considering (Non-physiological
case) a contraction dynamics model. . . . . . . . . . . . . . . . . . . . . 89
B.1 General aspect of the MHILL Data Visualizer. . . . . . . . . . . . . . . . 103
B.2 Operation boxes of the MHILL Data Visualizer. . . . . . . . . . . . . . . 104
B.3 Files addition for the folder system. . . . . . . . . . . . . . . . . . . . . . 105
B.4 Example of data plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
B.5 Selection of muscle data to be plotted. . . . . . . . . . . . . . . . . . . . 106
B.6 Plotting of muscle activations. . . . . . . . . . . . . . . . . . . . . . . . . 106
B.7 Removing some of the included files. . . . . . . . . . . . . . . . . . . . . 107
B.8 Time instant selection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
ix

LIST OF FIGURES
x

List of Tables
4.1 Fatigue parameters used for the muscle fatigue model employed, and the
same used in the work by Xia and Law [59]. The values for the F , R,
LD and LR are given in units of 1/s. . . . . . . . . . . . . . . . . . . . . 68
4.2 Body index number and local coordinates of the points that define the
rigid bodies for the upper extremity model. . . . . . . . . . . . . . . . . 70
4.3 Inertial data for rigid bodies considered in the upper extremity model. . 71
4.4 Geometrical and physiological parameters for the considered muscle in
the upper extremity model. Considered values were taken from the work
by Holzbaur et. al. [72]. Pictures reproduced with permission: "Copy-
right 2003-2004 University of Washington. All rights reserved including
all photographs and images. No re-use, re-distribution or commercial
use without prior written permission of the authors and the University
of Washington." [73]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 Fatigue parameters used for all the muscles in the upper extremity with
elbow muscles model. These have no correlation with experimental re-
sults [59]. Values given in units of 1/s. . . . . . . . . . . . . . . . . . . . 73
4.6 Body index number bn and local coordinates of the points that defined
each rigid body considered on the lower extremity model. . . . . . . . . 74
4.7 Geometrical and physiological properties inherent to the muscles existent
in the lower extremity model. Source: Reference [1]. . . . . . . . . . . . 76
4.8 Anthropometrical properties associated with the rigid bodies of the lower
extremity model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
xi

LIST OF TABLES
xii

List of Symbols
u(t) Neural signal a(t) Muscle activation
τrise Time constant of muscle activation
rise of a(t)
F m Muscle force F m CE
τfall Time constant of muscle activation
fall of a(t)
Muscle contractile element force
F m P E Muscle passive element force ˆ F m CE Muscle available contractile element
force
F m 0 Muscle maximum isometric force F m L
F m ˙ L
Muscle contractile element forcevelocity
relationship
Muscle contractile element forcelength
relationship
L m Muscle length
L m 0 Muscle length change rate ˙ L m Muscle rate of length change
˙v m Muscle speed ˙ L m 0 Muscle maximum contractile
velocity
F Muscle fatigue factor R Muscle recovery factor
BE Brain effort MA Fatigue model activated MU
compartment
MF Fatigue model fatigued MU compartment
RC Muscle residual capacity ˆ F f
CE
MR Fatigue model resting MU
compartment
Fatigued muscle available contractile
force
C(t) Fatigue model muscle activa- LD Muscle force development factor
tion–deactivationtroller
driving con-
LR muscle force relaxation factor T L Target load
xiii

LIST OF SYMBOLS
q Vector of generalised coordinates ˙q Vector of generalised velocities
¨q Vector of generalised accelerations Φ Kinematic constraints expressions
Φq Jacobian matrix of Φ in order to q ν Right-hand-side of the velocity
equation
γ Right-hand-side of the acceleration
equation
P ∗
Virtual power produced by the external
forces
˙q ∗ Virtual velocity vector M Global mass matrix
g Generalised force vector gΦ Internal constraint forces vector
λ Vector of Lagrange multipliers vp Number of via-points
ud Muscle direction d F m p Muscle force cartesian vector
representation
oξηζ Rigid body local reference frame 0xyz Global reference frame
rp Global coordinates of point p Cp Cartesian-generalised coordinates
transformation matrix for point p
g F e
gF m
CE
ˆg F m
CE
Rigid body generalised representation
of force F
Whole system generalised representation
of the contractile element
force
Whole system generalised representation
of the available contractile
element force
χ Set of generalised available contractile
element forces the considered
muscles
x Optimization problem control variables
F0
λ R
Optimization problem cost function
Lagrange multiplier without constraints
in the optimization problem
xiv
g F Whole system generalised representation
of force F
gF m
P E
Whole system generalised representation
of the passive element
force
g ext Generalised forces excluding muscle
forces
a Muscle activations vector
feq Optimization problem equality
constraints
λ ∗
Lagrange multipliers associated
with the kinematic drivers
g Gravity force

Glossary
ACh Acetylcholine
AD Activation dynamics
ADP Adenosine diphosphate
ATP Adenosine triphosphate
BIClong biceps brachii long head
BICshort biceps brachii short head
BRA brachialis
BRD brachioradialis
CE Muscle contraction model contractile element
CNS Central nervous system
DE Muscle contraction model damping element
EMG Electromyograph
EOM Equations of motion
FD Forward dynamics
HS Heel Strike
ID Inverse dynamics
MSO Modified Static optimization
MU Motor units
OHS Opposite Heel Strike
OTO Opposite Toe Off
PE Muscle contraction model passive element
SE Muscle contraction model series elastic element
TO Toe Off
TRIlat triceps brachii lateral head
TRIlong triceps brachii long head
TRImed triceps brachii medial head
xv

GLOSSARY
xvi

Chapter 1
Introduction
1.1 Motivation
The discipline of computational mechanics uses mathematical models to process the
state of a system regulated by the mechanical laws of physics. These models are math-
ematical formulations that emulate the behaviour of a determined system, and can
therefore be rendered to simple or complex networks of computational programs.
This way, a computational model simulation can predict the functioning of complex
systems, only by formulating its nature in a model. These simulations required an
immense set of algebraic calculations, impracticably solved by human beings. The
main advantage of an informatic application comes from the rapid development that
computer machines have exhibited in the last few decades. Nowadays, a simple personal
computer is capable of calculating a series of processes regarding the phenomena of any
computational science. Modern day computers not only have substantial capacity for
data processing, but also show a day-to-day improvement with respect to this potential.
The field of biomechanics, i.e., the scientific domain that concerns to the mechanics
of living systems, has additionally suffered a dramatic innovation in terms of its com-
putational applications. Multibody dynamics systems is one of the mechanical formula-
tions that emerged in this domain with the recent progress of new computational tools.
These systems model the dynamics of linked rigid (or flexible) bodies that experience
large displacements of position and rotation, and are easily adjusted to biomechanical
systems, with significance for the musculoskeletal complex of the human body.
Multibody dynamics is governed by its equations of motion (EOM) that will dictate
1

1. INTRODUCTION
the kinematics and kinetic properties of a certain mechanical system. From a classic
perspective, it is possible to solve these equations to obtain either [1]:
• The dynamic response of the multibody system experiencing known external force
solicitations;
• Or the internal and external forces that explain a given motion pattern.
The multibody formulation that is used to solve the first paradigm is called Forward
Dynamics (FD), the latter is known as Inverse Dynamics (ID).
In the human body, these formulation can be adapted to analyse the kinetics of
body articulations with the existence of bone stress, joint reactions or even muscle
forces. To implement such a template, some existent mathematical models that simu-
late the constitutive laws that rule the behaviour of muscle activation and contraction
are practicably employable. These have the potential of calculating the force output
of the contractile structures of a determined muscle, knowing some physiological and
geometrical parameters. In the same way, other models predict the fatigue state of a
muscle, for a given developed force history and can be included in a multibody system.
Implementing such models, the researcher is able to use them in all kinds of scientific
areas that deal with the mechanical interactions of the human body. The applications
of such a tool are undeniably appealing, regarding its competence for calculating, in a
non-invasive manner, muscle efforts, neural motor activations or even to predict how
physically exhausted is a person after an established activity. A wide range of areas
benefit from these kind of models, such as medicine, sports or ergonomics.
1.2 Objectives
The main goal of this thesis is to implement a muscle model that accounts for the con-
traction and fatigue dynamics inherent to the muscle tissue and to adapt this model in
the multibody system dynamics program APOLLO [1], in the interest of allowing the
inclusion of muscle actuators and incorporate its characteristics to the mechanical sys-
tem. The final scheme must be capable of calculating either the muscle activations for a
given kinematics, or the dynamics response of the system when the muscle activations
are prescribed, with the option of considering muscle fatigue. Therefore, it is desired to
use a versatile formulation, that will allow the operation in both forward and inverse
2

1.3 Literature review
dynamics paradigms. An important point of any inverse dynamics system that deals
with redundant muscle forces is the optimization process, that will select one of the in-
finite available combinations of muscle forces that justify a certain kinematic condition.
In addition, a special regard is given to the implementation of muscle fatigue dynamics,
as the implementation of a muscle fatigue model in multibody dynamics constitutes one
of the main novelties of this work.
1.3 Literature review
Muscle forces determination is a quintessential problem in the biomechanics field. Mus-
cle synchronisation and the estimation of the internal forces of a musculoskeletal sys-
tem (internal loads on both bones and joints) are the main targets in this kind of
calculation [2]. Direct measurement methods have been widely applied, as in Fujie et.
al. [3] and Nigg and Herzog [4], however these have the evident downside of its inva-
sive nature. Despite of their accuracy, these techniques take no advantage of numerical
methods whatsoever, lacking versatility and applicability. Therefore, multibody system
modelling became the widespread choice for representation of musculoskeletal systems.
Nikravesh and Attia [5] and Schiehlen [6] studied the application of multibody dy-
namics in general rigid body mechanics, describing its formulation and equations of
motion. The implementation of these methodologies in natural coordinates is distinctly
detailed in the work of Jalón and Bayo [7] and Silva [1]. This type of formulation has
a major importance for medical applications in rehabilitation prosthetics [8, 9], gait
disorders [2], sports [10], study of muscle diseases [11] and equipment ergonomics [12]
where its application can be specifically adapted to a certain subject by adjustment of
several anthropometrical and physiological parameters.
Multibody dynamics can be categorised in two important methodologies for solving
the equations of motion: inverse and forward dynamics. The inverse dynamics approach
calculates both external and internal forces based on the anthropometrical properties
and the motion of a given biomechanical system. Several inverse dynamics studies were
made uniquely concerning the calculation of joint torques for prescribed kinematics [13–
15]. In a musculoskeletal system, these external forces may include muscle efforts. In the
human body, there is an average number of 2.6 muscles per degree-of-freedom [16] which
act in synergy, in co-contraction or in overactuation around the spanned joints [17], i.e.,
3

1. INTRODUCTION
different muscle activations can generate the same mechanical model motion or posture,
leading to what is know as muscular redundancy [1]. From a multibody dynamics
perspective, this is numerically explained by the fact that the number of unknowns in
the equations of motions is larger than the number of available equations, resulting in
an indeterminate system with an infinite set of possible solutions. Numerically this
problem is tackled by optimization techniques [1, 18]. This way, the optimal solution
should minimise a specific cost function while fulfilling the equations of motion for
a set of prescribed kinematics and kinetics [19]. This approach is known as static
optimization and several studies have produced results regarding muscular efforts for
different physiological situations [1, 11, 17, 18, 20–27].
Static optimization has associated an important drawback: it only processes instant
frames and therefore only takes into account instantaneous performance criteria, that
do not simulate in the best way the behaviour of the central nervous system (CNS) when
choosing a muscle activation set [2]. In response to this aspect some authors [2, 28–
30] used what is know as dynamic optimization (a solution developed using forward
dynamics). This approach operates with a nodal point parametrisation of the neural
excitation, followed by several numerical integrations of the equations of motion [31].
This option allows the correct implementation of activation dynamics, since it relies on
multiple integrations, however it induces high computational times, which is its major
drawback. Despite of modern day personal computers being able to solve simple me-
chanical systems in less than 24 hours, these times can reach several days or weeks for
complex systems [2, 18]. Several biomechanical studies have been made for different
types of motion, such as normal gait [29], jumping [32] or cycling [33]. Anderson and
Pandy [30] concluded that, for normal gait, the use of dynamic optimization in not jus-
tified since the obtained results, when compared with the ones from static optimization,
are practically the same. In the present work, only static optimization procedures were
developed.
Cost functions used in static optimization, as the ones proposed by Crowninshield
and Brand [19], were widely used in classic works such as the ones by Collins [34],
Glitsch and Baumann [35], Patriarco [36] and Yamaguchi [18, 23]. An excelent review
regarding the various presented cost functions was made by Tsirakos et. al. [37]. Accord-
ing to Ackermann [2], these instantaneous functions are unable to accurately describe
the key performance criterion: the total energy expenditure. Therefore, two different
4

1.3 Literature review
approaches are introduced in reaction to the high computational times of dynamic op-
timization: Extended inverse dynamics and Modified static optimization. For further
reading on these methods please use reference [2].
Bearing in mind the second paradigm of multibody dynamics, forward dynamics, the
equations of motion are usually solved in order to obtain both the kinematic output and
internal reaction forces of a constrained biomechanical system when compelled to certain
external forces. Several authors developed studies regarding analysis of the forward
dynamics of musculoskeletal models for different motions, such as human gait [38, 39],
arm control [40], cycling [20] and high-jumping [41].
These types of multibody analysis require a physiological validation for the computed
muscle efforts. This is obtained by the use of mathematical models that mimic the
physical properties of excitable muscle tissue. Muscle models in this work are divided
in three different types of dynamics: activation dynamics, contraction dynamics and
fatigue dynamics. Activation dynamics model the time lag between the neural signal
that arrives at a motor neuron and the correspondent muscle activation level, and is
associated with the calcium dynamics of the muscle [1, 18, 20, 42]. Using such model, it
is possible to predict the neural signal that triggered a certain muscle activation pattern.
Several authors [18, 43] used bang-bang control models developed by He et al. [44],
where the neural signal is regarded as a control signal that switches steeply between
two states. These models fail when the control takes intermediate values between its
boundaries [20], a fact that lead to new model developments to account for neural
signal values amid 0 and 1 [31, 32]. In a numerical processing basis, some situations
need to have a stable model for negative values, therefore Raash et. al. [10] presented a
non-linear model, used by Neptune [45] and Kaplan [20], a model that is well-behaved
when handling neural signals that violate the physiological limits. Activation dynamics
models deploy two physiological parameters: the time constants of rise and fall of the
muscle activation [45].
Muscle contraction dynamics reports to the relation between muscle activation and
respective force applied, i.e., to the contractile behaviour of muscle tissue. Regular
tissue models are based on classical material models, such as the Maxwell model and
the Kelvin-Voigt model, and only simulate the response of soft tissues under compres-
sive and tensile loads, not being able to mimic the active behaviour of muscle tissue
at a macroscopic scale [1, 18]. Archibald Hill, in his experimental work, introduced an
5

1. INTRODUCTION
adaptation of the referred models, including a contractile component that explains the
functional nature of muscle tissue at a microscopic scale [46, 47]. Conversely, other com-
putational muscle models are aimed to simulate the microscopic behaviour of muscle
physiology, based on the cross-bridge theory by Huxley [48]. These are very accurate,
but highly complex and conditional on a considerable number of parameters that turn
it computationally impracticable for elaborated musculoskeletal models [42]. This lead
to the usage of the macroscopic Hill-based muscle models in the majority of studies that
deal with muscle efforts [1, 2, 10, 11, 17, 18, 24, 29, 38, 42, 44]. The contraction dynam-
ics model used in this work is composed by Hill-type model that comprises a passive
element and a contractile element, each with the respective mathematical formulation
equivalent to the one used in the work by Kaplan [20] and Silva [1]. This mathemat-
ical representation was conceived by relating the effective muscle force obtained for
variations of length, velocity and activation [18, 42].
Muscles are defined in the computational code by its geometry and force generating
properties [1]. They are geometrically defined by the locations where they blend with
tendons, the origin and insertion, and by eventual via-points [1, 42]. Some studies
consider additionally that muscles may be defined by wrapping around objects, in order
to define in a much more accurate manner the way that muscle geometry is arranged [49].
These can wrap around single objects [50] or multiple ones [51]. There is associated a
type of path optimization when processing the more lifelike form of joint wrapping [52].
The third component of the considered muscle model takes into consideration the dy-
namics of muscle fatigue. Physiologically and apart from other causes, peripheral fatigue
increases proportionally to the consumption level of intramuscular glycogen and Adeno-
sine triphosphate (ATP) [53]. Initial muscle fatigue studies by Scherrer and Monod [54]
and Rohmert [55] stated the first interpretations of the relationship between effort levels
and endurance times [56]. First models were based on empirical relationships, that lack
applicability to different situations due to the absence of parameters. The first theo-
retical models introduced a collection of physiological parameters and fatigue dynamic
laws [57, 58], and were only practicable for single muscle analysis regarding its com-
plexity and numerous parameters [59]. New models approach fatigue dynamics using
simple biophysical principles, such as compartment dynamics and muscle recruitment
hierarchy [59].
6

1.4 Contributions
Liu et al. [60] developed a muscle unit (MU) based fatigue model, where dynamic
fatigue laws dictate the evolution of the muscle fitness using compartment theory, and
solely three physiological parameters [60]. This model however does not account for
variable muscle efforts. Xia and Law [59] used Liu’s model and developed a model that
considers peripheral muscle fatigue for both plain and complex exercises at fluctuating
muscle force intensities [59]. In this model, the fitness state of a specific muscle varies
along a continuos scale, with each one of enclosed MUs having the possibility of being
ideally activated, ideally resting or ideally fatigued [59]. In addition, it optionally
takes into consideration the muscle recruitment hierarchy, described by Henneman [61].
Guyton and Hall [53] identifies two main types of muscle skeletal fibers: slow and fast
fibers. In his recruitment scheme, Henneman categorises muscle fibers in three different
sub-types that are recruited in a specific order: slow, fast fatigue-resistant, and fast
fatigable [59, 61]. In this work, a fatigue model based in the work by Xia and Law [59]
was included in the implemented muscle model of the multibody dinamics routines,
coupled with the muscle contraction dynamics model. It was tested for a single muscle
case and for a more complex musculoskeletal model of the upper limb.
1.4 Contributions
The contributions of this thesis are fourfold:
• To develop a flexible muscle contraction model, adapted from the work by Ka-
plan [20] and Silva [1], using known force-length-velocity and activation scaling
properties.
• To develop a muscle fatigue model, adapted from the work by Xia and Law [59],
using a three-compartment model and a muscle recruitment hierarchy model.
• To implement the coupling of the developed contraction and fatigue dynamics
muscle models in an existing FORTRAN-based multibody system dynamics pro-
gram, in way that allows their application in both forward and inverse dynamics
perspectives.
• To adapt the equations of motion of the multibody system, using the Newton
method [1, 7], and solve these equations by means of optimization procedures.
7

1. INTRODUCTION
1.5 Thesis Organization
This thesis is divided in five chapters:
Chapter 1 This first chapter explains the motivation and objectives of this work, and
introduces the reader to the outline of this document.
Chapter 2 The second chapter regards the underlying mechanism of muscle contrac-
tion and its mathematical model formulation, being divided in two main sections:
Skeletal Muscle Anatomy and Physiology, where the reader in introduced to the
anatomical and physiological properties of skeletal muscle tissue in order to un-
derstand the muscle contraction mechanism, and Dynamics of the Muscle Tissue,
where the muscle models are presented, with special consideration for the con-
traction, activation and fatigue models that were used in the developed routines.
Chapter 3 The third chapter starts by introducing the foundations of a multibody
system formulated with natural coordinates. Such preamble is followed by the
formulation of concentrated forces in our multibody approach. Subsequently, this
formulation is employed in the rearranging of the equations of motion. The next
two sections explain how the equations of motion will be used in the standards
of inverse and forward dynamic analysis. Then, the optimization problem that is
solved to deal with redundant muscle forces is described. The chapter closes with
a discussion regarding the used approach and methodology.
Chapter 4 The fourth chapter encloses the results obtained in this work. It begins by
explaining the method applied for the confirmation of the validity of the muscle
contraction dynamics model that was put into practice. It follows with two ex-
amples of success of the introduction of a fatigue model in multibody routines,
showing the obtained results. Furthermore, a gait cycle is analysed from an in-
verse dynamics perspective, where the muscle activations where determined for a
lower extremity model considering some of the muscles that span the knee and
ankle joints. The chapter terminates with a discussion contemplating the obtained
results.
8

1.5 Thesis Organization
Chapter 5 In the last chapter the author exposes some of the most important con-
clusions, suggesting some future developments related to the employed models,
methodologies and optimization techniques.
9

1. INTRODUCTION
10

Chapter 2
Musculoskeletal System Modelling
The complex formed by muscles and tendons, stimulated by the central nervous
system, will work as the biological actuator in order to perform a specific motor task.
As mentioned in [42]
"[...] muscles and tendons are the interface between the CNS and the
articulated body segments."
When excited, muscles apply a moment of force about specific joints. When ex-
erted in a synchronised way these moments will result in body balance or motion. A
muscle can be considered uniarticular or biarticular, when able to span one or two
joints [2] (some authors consider as biarticular the set of muscles that span more than
two joints [27]).
Muscle fibers can only produce contractile force, i.e., they can only pull [2]. The
tensile forces produced by muscle are transmitted through the tendons to the location
where tendons are attached to bone – muscle’s origin and insertion. Due to this limita-
tion, we find many muscles in sets of antagonist pairs, which produce opposite segment
movements. For instance, in the upper limb, we have the triceps brachii which is the
main muscle for elbow joint extension movements. Its antagonist is the biceps brachii,
responsible for the flexion of this joint (along with the supination of the forearm). In
biomechanical systems, degrees of freedom are usually overdriven, i.e., for a certain
joint movement there is more than one muscle that can deliver enough torque to pre-
scribe the movement in question. In addition, even the simplest antagonist pair has a
11

2. MUSCULOSKELETAL SYSTEM MODELLING
countless number of muscle force combinations requested by the CNS to perform joint
rotation, since there is the possibility of co-contraction. We call muscle redundancy to
this situation, where the minimum number of muscles required is outnumbered. This
will prompt one of the major aspects in the computation of muscle activations, since a
specific motion to be studied will have an infinite set of muscle forces combinations to
generate it.
There are three types of muscle tissue: cardiac, smooth and skeletal. The first is
found in most of the cardiac structure. Smooth muscle is accountable for the movements
of hollow inner structures. These are classified as involuntary muscles – that is, their
contraction happens without a conscious control. Skeletal muscle produces voluntary
anatomical segment displacements and promotes body balance, thus this is the relevant
type of muscle for this work [62].
In this chapter, an introduction to the underlying aspects of skeletal muscle anatomy
and physiology will be made. We will begin with a brief reference to its main structures
and design, and briefly describe the basic physical and biochemical phenomena in muscle
contraction. In order to formulate a computationally feasible model, the macroscopic
properties of the behaviour of muscle tissue must be assumed. The dynamics of muscle
contraction and muscle activation are formulated. The chapter closes with a simple
model for muscle fatigue.
2.1 Skeletal Muscle Anatomy and Physiology
Prior to any review of the employed muscle modelling methodology, a brief description of
the fundamentals of muscle anatomy and physiology is offered. This section is presented
to provide the essential background for the comprehension of the models presented
in Section 2.2.
2.1.1 Skeletal Muscle Anatomy
As illustrated by Figure 2.1, muscles have a fasciculated structure. An outer layer
covers the whole muscle, called epimysium. Inside it, assemblages of parallel muscle
fibers that are assumed to extend to the entire length of the muscle, embedded in
endomysium, constitute muscle fascicles [53]. Each one of these fascicles are delimited
by the perimysium. These three type of layers of connective tissue have elastic properties
12

2.1 Skeletal Muscle Anatomy and Physiology
and continue beyond the limits of muscle tissue, becoming more collagenous, to form
tendons [62].
Figure 2.1: Arrangement of the skeletal muscle structure, from an external level to a
molecular level [53].
Muscle fibers are stacked in parallel and oriented at an acute angle to the ten-
don, that can be different of 0 (zero). This angle is called pennation angle [42]. This
arrangement is shown in Figure 2.2.
In the same way that bundles of muscle fibers form fascicles, muscle fibers are com-
13

2. MUSCULOSKELETAL SYSTEM MODELLING
Figure 2.2: Representation of the penation angle α between muscle fibers and tendons
[42].
posed by myofibrils. These have a series arrangement of basic units called sarcomeres,
which are the fundamental structure responsible for muscle contraction. Each sarcomere
is composed by a complex of proteins comprised between two dark regions of proteins
called Z discs and three bands are visible – A, H and I. The A bands are darker due to
the existence of a brew of thick and thin filaments. Each A band includes a central H
band which corresponds to the region where only thick filaments are present. Finally,
the I band consists of a pale region where there are only thin filaments [62].
Thick filaments are formed by myosin proteins whose shape is shown in Figure 2.3-A.
Anchored to Z discs, thin filaments extend to connect to the myosin complex. These thin
filaments are composed by a system of actin, tropomyosin and troponin – Figure 2.3-B.
Myosin and actin proteins overlap in the A band, with a projection of myosin heads into
actin. These projections are known as cross-bridges – Figure 2.4 – and their interactions
will devise the contraction machinery. Troponin and tropomyosin form a complex that
inhibit the bonding of actin with the myosin of the thick filaments when significant
concentrations calcium ions (Ca 2+ ) are not present. This will impede contraction from
happening when not desired. The whole overlapped structure is supported by a frame-
work of titin molecules [53]. During contraction, thick and thin filaments glide without
changing length. Only the sarcomere length reduces due to this gliding. Therefore,
throughout the whole process, A band length is constant, and H and I lengths shortens.
Additionally, it should be referred that, an important thin membrane covers each
muscle fiber, vital to sustain electrical membrane potential – the sarcolemma. This
14

2.1 Skeletal Muscle Anatomy and Physiology
Figure 2.3: Detail of a myosin molecule (A) and an actin filament (B) [53].
Figure 2.4: Illustration of the cross-bridges formed by the connections of actin filaments
with a myosin filament. Adapted from Reference [53].
15

2. MUSCULOSKELETAL SYSTEM MODELLING
layer has extensions that passes through the fiber inner structures called T tubules. A
muscle fiber cytoplasm is known as sarcoplasm. Apart from several myofibrils, it also
contains a nucleus, various mitochondria associated with high concentrations of ATP,
and a substantial network of smooth sarcoplasmic reticulum, that will store Ca 2+ .
These structures are illustrated in Figure 2.5.
Figure 2.5: Detail of a muscle fiber [62].
Skeletal muscle contraction is voluntary, i.e., it happens under the conscious control
of the brain. Muscle fibers must then be innervated by a neuron – the motor neuron.
The set of fibers innervated by a motor neuron plus the neuron itself is called a motor
unit (MU ). The number of fibers in a MU may vary considerably. Muscles that are
responsible for precise movements will have few muscle fibers per MU, but a high number
of motor units. This example contrasts with the one of muscles that control gross
movements. The number of MU will be inferior, but each one will have a higher number
of muscle fibers [62].
Motor neurons ramify into several terminal axons that get close to a region of the
sarcolemma called motor end plate, keeping a small distance – the synaptic cleft [62].
This region is known as neuromuscular junction – Figure 2.6.
All the described structures will play a major role in the mechanisms of contraction,
that will be described in the following subsection.
16

2.1 Skeletal Muscle Anatomy and Physiology
Figure 2.6: Neural muscular junction with three different detail levels [62].
17

2. MUSCULOSKELETAL SYSTEM MODELLING
2.1.2 Skeletal Muscle Physiology
The general mechanism for muscle contraction – known as excitation-contraction cou-
pling – undergoes several stages, commencing with the progress of the action potential
from the nervous system to the sarcolemma – Figure 2.6 (a). When the action poten-
tial travelling across the myelinated motor nerve reaches the synaptic bulbs (the motor
axon end projections), a small quantity of acetylcholine or ACh (neurotransmitter) is
secreted to the synaptic cleft. The ACh combines with its receptors in channels located
in the motor end plate, allowing Na + ions to diffuse into the sarcoplasm, firing up a
action potential in the sarcolemma [53].
Once the action potential occurs, the contractile machinery must be activated –
Figure 2.7. The discharged action potential will cross through the sarcolemma and
widespread across the core of the fiber by the T tubules. This will lead to a release of
sizeable quantities of Ca 2+ from the sarcoplasmic reticulum. Calcium ions will bond to
troponin, changing the shape of the troponin-tropomyosin complex and exposing actin
double helix bonding sites to myosin cross-bridges [53].
Figure 2.7: Contraction regulation mechanism by the troponin-tropomyosin complex,
dependent on the concentration level of Ca 2+ [62].
18

2.1 Skeletal Muscle Anatomy and Physiology
Figure 2.8: Crossbridge theory steps [62].
Muscle contraction will happen solely in the presence of ATP and Ca 2+ . The sar-
comere shortening process, that depends fundamentally on these two components, can
be explained by the crossbridge theory and has its steps illustrated in Figure 2.8. Step
1 – In the relaxed form, myosin heads catalyse the decomposition of ATP (whenever
present), increasing its energy and becoming active. Adenosine diphosphate (ADP) and
phosphate ion resultant from this cleavage remain attached to the myosin head. If Ca 2+
is present (Step 2), by the action potential mechanism already described, cross-briges
will then be capable of bond to actin filaments with a perpendicular configuration,
loosing the phosphate ion (Step 3). Step 4 –Reaching this step, the energy stored will
provoke what is known as power stroke, i.e., the head of myosin molecules will suffer
a conformational modification slopping its orientation towards the centre of the sar-
comere. This ultimately results in the sliding of filaments and at this point the ADP
molecule is released. Step 5 – A new ATP binds causing detaching the thick from the
thin filament. Step 6 – This molecule is cleaved such as in Step 1, making the head
recover its perpendicular position to the actin filament and prompting the fiber for a
new power stroke. For as long as calcium ions are available (not to mention ATP),
19

2. MUSCULOSKELETAL SYSTEM MODELLING
fiber contraction will proceed – Step 7 – performing the previously mentioned steps,
providing that the physiological conditions are satisfied (presence of ATP and Ca 2+ ,
physical limits of the sarcomere and bearable muscle load [53]) [62].
In the absence of an action potential, the sarcoplasm tends to have low levels of
calcium ions. This is due to the existence of active transport membrane pumps that
constantly store Ca 2+ ions back to the sarcoplasmic reticulum hindering the bonding
of actin with myosin and therefore inhibiting muscle contraction [53].
The force applied by a certain muscle depends on various parameters and physical
conditions. Its regulation is defined by the number of recruited muscle fibers and the
stimulation frequency of the neural signal. If a muscle fiber is stimulated by the CNS
above a certain magnitude, then the sarcolemma will become depolarised and develop
a brief contraction, designated as twitch. An important aspect becomes apparent in
terms of the frequency of stimulation, shown in Figure 2.9. When the time interval
between successive twitches becomes smaller than the duration of a single twitch, then
contraction level is amplified by the sum of the twitches superposition. For increasing
twitching frequency, the produced muscle force will also increase until it reaches a
frequency threshold, where it is observed that the muscle contraction does not intensify
any more and holds its value in a plateau – this level is known as tetanic contraction.
2.2 Dynamics of the Muscle Tissue
In order to implement a computational model describing muscle behaviour, the dynam-
ics of muscle tissue play a vital role in the acquisition of physiologically valid muscle force
values, and therefore in the construction of such model. There are some reductionist
models [48, 63] that describe the microscopic properties introduced in Section 2.1. These
models are usually based on the crossbridge theory and regardless of their accuracy in
describing the mechanics and chemistry involved in contraction of muscle fibers, their
high complexity and numerous parameters lead to computationally inefficient routines
that prove to be disadvantageous. These drawbacks are specially evident when complex
mechanical structures, with many muscle structures, are considered. For a robust and
computationally efficient modelling, several authors [1, 2, 20, 42] employed a macro-
scopic approach for the muscle model, where some of its most important properties are
considered in order to identify the inherent parameters.
20

2.2 Dynamics of the Muscle Tissue
Figure 2.9: Relation between twitch frequency and effective muscle contraction [1].
In what muscle tissue modelling is concerned, two major types of dynamics are usu-
ally identified: Activation Dynamics (Section 2.2.1) and Contraction dynamics (Sec-
tion 2.2.2) – Figure 2.10. The former describes the conversion of a CNS neural signal
u(t) into a muscle tissue activation state a(t). The latter correlates a(t) with muscle
force development. A novel model to multibody methodologies is introduced: a muscle
fatigue dynamics model (Section 2.2.3).
u(t) a m (t) F m (t)
Activation
Contraction
✲ ✲ ✲
Dynamics
Dynamics
Figure 2.10: Scheme of muscle tissue dynamics, with a series model of Activation Dynamics
and Contraction Dynamics.
2.2.1 Activation Dynamics
Also known as excitation-contraction dynamics, activation dynamics (AD) models the
relation between the neural signal that arrives at a motor neuron u(t) and the muscle
21

2. MUSCULOSKELETAL SYSTEM MODELLING
activation level a(t), i.e., the dimensionless proportion of muscle fibers that are instan-
taneously active at time t [18]. By including an AD model, it is possible to predict the
neural signal that triggered a certain muscle activation pattern and therefore the motion
of the biomechanical system at issue. Despite the fact the developed routines in work
do not count with this sort of model, it is important to comprehend its characteristics
in order to implement it in a future work.
Physiologically there is a time lag between the neural signal u(t) and the corre-
sponding muscle activation a(t). The physical meaning of this lag is associated with
the muscle’s calcium dynamics [20, 42], described in Section 2.1.2. In the same manner
that the contraction dynamics model was developed, it is desired to simplify our AD
model by representing the behaviour observed from a macroscopic point-of-view. At
this level, AD is represented by first-order ordinary differential equations [1].
Several authors like Yamaguchi [18] and Pandy et al. [43] used a model developed by
He et al. [44], where the neural signal u(t) was regarded as bang-bang control, i.e., as a
control signal that switches steeply between two states (in this case taking the boundary
values – 0 or 1). These models report with precision the AD of a signal u(t) with a
binary nature, however failing when the control takes intermediate values between its
boundaries [20]. Developments from this model, such as the one in the work of Anderson
and Pandy [32], or the one developed in Pandy and Hull [31] account for values of u(t)
amid 0 and 1, but are unstable for negative values of the neural signal. These negative
values may occur, despite the fact that u(t) is bounded between 0 and 1, since some
iterative methods require its calculation below the lower bound. In response to that,
Raash et. al. [10] presented a non-linear model, used by Neptune [45] and Kaplan [20],
that is stable when handling neural signals that violate the physiological limits. This
model states that the change rate of muscle activation ˙a(t) is related to a(t) and u(t)
in the following way:
where
˙a(t) =
(u(t) − a(t))(c1u(t) + c2) u(t) ≥ a(t)
(u(t) − a(t))c2
c1 =
c2 =
1
τrise
1
τfall
22
− 1
τfall
u(t) < a(t)
(2.1)
(2.2)
(2.3)

2.2 Dynamics of the Muscle Tissue
The parameters τrise and τfall describe respectively the time constants of rise and fall
of the a(t), i.e., the time of activation and relaxation [45]. The latter term is generally
bigger than the first [64]. The consistency of this model is illustrated in Figure 2.11.
(a) Response to bang-bang control signal.
(b) Response to an intermediate value of neu- (c) Model consistency for negative values of
ral signal control u(t).
u(t).
Figure 2.11: Activation dynamics model consistency [20].
2.2.2 Contraction Dynamics
In order to simulate the active behaviour of muscles in body motion, an explicit model
of the contractile structures must be implemented to estimate the muscle forces. The
dynamics of muscle contraction are described in this subsection. These will bridge
values of muscle activation a(t) with muscle force F m .
The primary mathematical models used in generic tissue modelling include passive
ones (such as the Maxwell and the Kelvin-Voigt models), that are able to precisely mimic
the behaviour of soft tissues under compressive and tensile loads [1, 18]. However, due
to the absence of an active element, these models are incapable of representing the
dynamics of muscle contractile structures. Archibald Hill introduced an adaptation of
23

2. MUSCULOSKELETAL SYSTEM MODELLING
the Kelvin model, including an additional contractile element [46] that simulates the
active macroscopic action of the cross-brige cycle explained in Subsection 2.1.2. Due
to this active characteristic Hill-type models are widely used in the biomechanics field
to reproduce both contractile and passive behaviour of muscle tissue, since these have
accessible parameters and they are computationally tractable for systems with several
muscles.
The model used in this work is illustrated in Figure 2.12(a). It is composed by a
passive element (PE) that takes the non-linear passive elastic properties of the muscle
tissue and a contractile element (CE) that accounts for both the contractile structures
and the viscous force produced by intracellular and intercellular fluids enclosed in the
muscle [1]. This work, similarly to the works by Silva [1] and Kaplan [20], does not
consider the standard Hill-model – a Kelvin model with the CE inserted parallel to the
damping element (DE) as representated in Figure 2.12(b). It neglects the series elastic
element (SE) related to cross-bridge stiffness and includes the behaviour of the DE in
the expressions of the CE. The expression of the exerted muscle force for the model in
consideration for this work, is straightforwardly described by:
F m = F m CE + F m P E
(2.4)
(a) The used mathematical Hill-type muscle model [1] (b) Generic Hill-type muscle model [1].
Figure 2.12: Hill-type muscle models.
According to Zajac [42] and Yamaguchi [18], there are three key properties in the
muscle tissue. The first one, already mentioned in this chapter, states that it can only
produce tensile forces. Even in the possibility of muscles being able to push, tendons
would buckle, cancelling that effect [18]. The following two properties describe the force
24

2.2 Dynamics of the Muscle Tissue
output for a certain state of muscle length and muscle velocity. Muscle activation is
evidently also taken into consideration for the mathematical expressions correspondent
to the calculation of the muscle forces, since the amount of force produced by a muscle
depends on the number of muscle unites recruited.
Force-length relationship
Several authors, such as Hill [47] or more recently Zajac [42], exposed the evolution
of the forces produced by a fully activated muscle fiber (a m (t) = 1) with its length.
There were some evolutions concerning the bounding values for the region where active
muscle force is generated. The more recent studies state that this region for a muscle
length L m is comprised between 0.5L m 0 < Lm < 1.5L m 0 , where Lm 0
corresponds to the
muscle fiber resting length(or relaxed length). The muscle resting length L m 0 usually
verifies to have a close value to the length at which the active muscle force reaches its
maximum value when isometrically contracted [42], the muscle’s optimal length [18, 42].
The force developed at total muscle activation and fiber length L m = L m 0
is designated
maximum isometric force or optimal force F m 0 [1, 18]. This value does not correspond
to the maximum force that the muscle can exert, but to the maximum active force in an
isometric contraction. As we will see, there is a passive force that can sum to the active
muscle contribution, overtaking the value of F m 0
. The value of the maximum isometric
force may be estimated by multiplying the muscle’s average cross sectional area by the
maximal muscle specific tension (roughly 31.39N/cm 2 ) [18].
Figure 2.13(a) relates the overlap level between the actin and myosin fibers with
the contractile force of a fully activated muscle fiber. This overlapping varies along the
sarcomere length and it will have influence in the muscle’s capacity of generating con-
tractile force. Points A and D represent states where the muscle is unable of performing
its maximum force, respectively due to an adjacent positioning of the myosin with the
Z discs and no actin-myosin overlap [53]. The contractile force peaks when the highest
number of cross-bridges are formed. When reporting these facts to the whole muscle,
a curve similar to the one in Figure 2.13(b) is obtained. Its shape arises due to the
evolution of the actin-myosin overlapping conditions already mentioned.
A passive force also emerges from the length evolution of muscle fibers. It is
thought [42] that this force is due to intrafiber elasticity. For muscle lengths larger
25

2. MUSCULOSKELETAL SYSTEM MODELLING
(a) (b)
Figure 2.13: Discrete length-tension diagram of the contractile contribution of a single
fully activated sarcomere [53] (a) and the same relationship scaled to the whole muscle [1]
(b).
than the resting length, these tensions become apparent, reaching values larger than
F m 0
as observed in Figure 2.14.
Force-velocity relationship
The rate at which the length of muscle changes affects the magnitude of the obtained
muscle force. As illustrated in Figure 2.15, for concentric contraction, that is the short-
ening of activated muscle, the contractile force is less that the one observed at isometric
contractions with the same level of muscle activation [1, 42]. The opposite happens in
eccentric contractions when the activated muscle is lengthening. Empirically the muscle
apparatus shows a performance similar to a fluid "damper" [18] and this behaviour was
described by Hill [46] as
(F m + a) (b + v m ) = (F m 0 + a) b (2.5)
where v m is the muscle speed, that corresponds to the opposite of the the rate of length
change ˙ L m (or the muscle contraction velocity). A special attention should be taken
to this relation ( ˙ L m = −v m ), in order to relate the presented model graphs with the
used equations. ˙ L m is positive for eccentric contractions and negative for concentric
26

2.2 Dynamics of the Muscle Tissue
Figure 2.14: Force-length relationship of the passive element [1].
ones. Coefficients a and b are the values that define the asymptotes that define the
hyperbole curve in Figure 2.15. The observed range for the muscle rate of length
change is [− ˙ L m 0 , 0.5 ˙ L m 0 ], where ˙ L m 0
˙L m 0 ≈ 10Lm 0 [1].
is the maximum contractile velocity, normalised as
Only the active contribution of muscle will be affected by its velocity. The passive
forces mentioned previously will be considered to be the same for all muscle velocities,
i.e., they will only depend on the length of the muscle.
Activation scaling
In the model considered for this work, the forces developed by a muscle are classified
as passive element forces and contractile element forces. Both depend on the length of
muscle fibers L m , but the latter is additionally conditioned by the muscle’s contractile
velocity ˙ L m and activation level a(t). The assumption that all fibers respond to neural
activation in the same manner is made (which is not completely true, since it is known
that there are several types of muscle fibers with different contraction patterns) and
therefore it is considered that the force-length and force-velocity curves are linearly
scaled by the muscle activation a(t) as it can be observed in Figure 2.16 [42].
The force exerted by the contractile element is then composed by a factor dependent
on its relationship with the muscle length F m L (Lm (t)) and velocity F m ˙ L (L m (t)). This
27

2. MUSCULOSKELETAL SYSTEM MODELLING
Figure 2.15: Force-velocity relationship [18].
Figure 2.16: Activation scaling evidence: Force-length and force-velocity relationships
for different levels of muscle activation a(t) [18].
28

factor is then multiplied by a(t) as indicated in Equation 2.6.
2.2 Dynamics of the Muscle Tissue
F m CE(a(t), L m (t), ˙ L m (t)) = F m L (Lm (t))F m L ˙ ( ˙ Lm (t))
F m a(t) (2.6)
0
The factor to be multiplied by the muscle activation a m (t) corresponds to the avail-
able muscle contractile force which is referred hereafter as ˆ F m CE
tion 2.7.
and indicated in Equa-
ˆF m CE = F m L (Lm (t))F m L ˙ ( ˙ Lm (t))
F m . (2.7)
0
This term plays an important role in the following description of the fatigue model
(Section 2.2.3) and in the integration of the muscle constitutive model with the multi-
body methodology (Chapter 3). Equation 2.4 can be rewritten considering the activa-
tion term as an independent one:
Analytical expressions
F m = F m P E + ˆ F m CEa m . (2.8)
In order to analytically express the major characteristics of the contraction dynamics
model described before in its computational counterpart, the mathematical expressions
developed in the works of Kaplan [20] and Silva [1] are used. These take into account
the properties previously described.
The force-length and force-velocity relationships of the contractile element, referred
in Equation 2.6, are described in Equation 2.9 and Equation 2.10.
F m ˙ L ( ˙ L m (t)) =
F m L (L m (t)) = F m −
0 e
⎪⎩ − πF m 0
" »
− 9
4
„ L m (t)
L m 0
− 19
«– 4
− 20
1
»
− 4
9
„
L
m
(t)
4 Lm −
0
19
«– #
2
20
(2.9)
⎧
0
L˙ m (t) < −L˙ m
0
⎪⎨
− F m 0
arctan (5) arctan
−5 ˙ Lm (t)
˙L m
+ F
0
m 0 − ˙ Lm 0 ≤ ˙ Lm (t) ≤ 0.2 ˙ Lm 0
4 arctan (5) + F m 0 0.2 ˙ L m 0 < ˙ L m (t)
(2.10)
Equation 2.10 is the used expression that corresponds to the relation of Equation 2.5.
The passive element force will only depend on the muscle’s length L m (t) and the math-
ematical approximation used is
29

2. MUSCULOSKELETAL SYSTEM MODELLING
⎧
0 L m (t) < L m 0
F m P E(L m ⎪⎨
(t)) = 8
⎪⎩
F m 0
Lm 0 3 (Lm (t) − L m 0 ) 3 Lm 0 ≤ Lm (t) ≤ 1.63Lm 0
2F m 0 1.63L m 0 < Lm (t)
. (2.11)
This contractile model is introduced in the multibody system dynamics routines of
the software APOLLO, as schemed in the block diagrams of Figure 2.17 (on a forward
dynamics perspective) and Figure 2.18 (on an inverse dynamics perspective).
Figure 2.17: Block diagram of the muscle contraction dynamics model implemented in
the forward dynamics multibody system routines.
Figure 2.18: Block diagram of the muscle contraction dynamics model implemented in
the inverse dynamics multibody system routines.
30

2.2.3 Muscle Fatigue
2.2 Dynamics of the Muscle Tissue
Muscle fatigue is a common condition and empirically understood by the majority of
people as the debilitation of the muscle force production performance. The metabolic
conditions required for proper muscle contraction (described in Subsection 2.1.2) are
altered during sustained contractions and this will lead to an incapacity of the muscle
to deliver the same fitness levels. From a physiological point-of-view, this weakness
increases proportionally with the consumption level of intramuscular glycogen (where
glucose is stored) and ATP [53]. In addition, the conditions of neural signal transmis-
sion are also altered after extended muscle action, narrowing its output [53]. There are
several other justifications for muscle fatigue not related to the effects of prolonged mus-
cle activity, such as the restriction of blood supply (and consequent nutrient privation),
however these are not related to the scope of this work.
Initial empirical models were developed by Rohmert [55] who gave name to what
became known as the Rohmert curves, for which an example is depicted in Figure 2.19.
However, due to versatility limitations, these first models were overpowered by new
theoretical ones. Hawkins and Hull [57] studied the effects of fatigue in long-lasting
efforts by considering a set of empirical fatigue parameters and a model that calculates
muscle forces as the sum of individual fiber forces. The dependence of force with
time is then based on empirical input that requires an experimental consistence, which
leads to a foreseeable inaccuracy. The first theoretical models introduced a collection
of physiological parameters and fatigue dynamic laws, such as the ones presented in
Ding el al. [58]. Despite the precision of these models, their complexity and numerous
parameters lead them to be computationally disadvantageous when analysing multiple
muscles. In order to tackle these drawbacks, new theoretical models derive the muscle
force function from simple biophysical principles. This work uses an example of these
new theoretical models, adapted from the work by Xia and Law [59].
The three-compartment theory
The principle of the model used in this work derives from the MU-based fatigue model
proposed by Liu et al. [60]. In their work, a fatigue model where dynamic fatigue
laws prescribe the evolution of fiber conditions in three ideal states (resting, activated
and fatigued), over a certain period of time, is considered. This concept is known as
31

2. MUSCULOSKELETAL SYSTEM MODELLING
Figure 2.19: Curve based on Rohmert’s relationship between %MVC (percentage of
maximum voluntary contraction) and endurance time (minutes) [56].
compartment theory, which as been applied in a diverse set of scientific areas, namely
substance transport and chemical reaction phenomena modelling [59, 65, 66].
Liu’s model is based in simple biophysical principles and solely considers three pa-
rameters: a fatigue factor (F ), a recovery factor (R) and the total number of motor
units in the muscle [60]; and an input factor describing the brain effort (BE). The
BE variable plays an important role in this model, since it is analogous to the muscle
activation a m mentioned in the previous subsections. Note that this model becomes
appropriate for fitting experimental data, due to the few parameters that are involved,
and for a far more general use than the other mentioned models. However, this model
fails to consider conditions of non-constant muscle efforts, which is a major disadvan-
tage in our work, as it involves a non-linear application of muscle forces to a multibody
system.
Considering the consistency and computational efficiency of Liu’s model, and in
response to the fact that this model is not ready to deal with variable muscle efforts, Xia
and Law [59] developed a model that considers peripheral muscle fatigue for both plain
and complex exercises for fluctuating muscle force intensities. The same compartment
32

2.2 Dynamics of the Muscle Tissue
theory model of Liu’s was taken into consideration. To apply it, and despite the fact that
the fitness state of a specific MU varies along a continuous scale, each one of these MUs
are minded as being ideally activated, ideally resting of ideally fatigued. Physiologically
it is never expectable to have a MU either fully fatigued or fully comprised in one of
the other states. However, this model gives us the whole muscle fitness state by the
mixture of the set of MUs that form it and therefore mathematically it will be the sum
of the compartment proportions. Muscle units exerting maximum force will fall in the
activated compartment MA and the ones with zero available contractile force will be
consigned to the fatigued compartment MF . The compartment of the resting MUs is
symbolised as MR as schematically indicated in Figure 2.20.
Figure 2.20: Three-compartment theory flowchart.
For model flexibility purposes [59], the quantity of muscle fibers set in a particular
compartment are given in percent of maximum voluntary contraction (%MVC). In the
beginning of the exercise, it is considered that every muscle fiber is resting, i.e., it is
assumed that MR = 100%. The available contractile force will then be limited to the
fitness state of the whole muscle. The MUs that are available for force production will be
the ones laying in the activated and resting compartments (the fatigued compartment
is not considered). For that effect, a new term is introduced to describe the muscle
strength that still can be developed considering fatigue, the residual capacity (RC).
33

2. MUSCULOSKELETAL SYSTEM MODELLING
RC(t) = MA + MR = 100% − MF . (2.12)
When mathematically adapted to this work, the fatigued muscle available contractile
force ˆ F f
CE will be the product of the reposed muscle available contractile force ˆ F m CE with
RC and the muscle activation a(t). Hence:
ˆF f
CE = RC(t) × ˆ F m CE × a(t) (2.13)
The flow of MUs between compartments used by Xia and Law, i.e., the dynamic
process of muscle fatigue, depends on the same paramenters from Liu’s model: the
fatigue coefficient F and the recovery coefficient R, as indicated in Equations 2.14
to 2.16. These differential equations dictate the percentage of muscle fibers that are
transferred between compartments and used when a force solicitation is made, updating
the fitness state of the muscle structure. Accordingly:
dMR
dt
dMA
dt
dMF
dt
= −C(t) + R × MF (t) (2.14)
= C(t) − F × MA(t) (2.15)
= F × MA(t) − R × MF (t) (2.16)
where the muscle activation–deactivation driving controller C(t) (described in the next
paragraph) is the term that gives the competence of this model to process the dynamics
of muscle fatigue with variable efforts. The rate of resting MUs, in Equation 2.14, is
reduced by C(t) and increased by a multiplication factor between the recovery coefficient
R and the amount of fatigued MUs MF , a term that expresses the amount of fatigued
fibers that recovered in the course of the muscle effort history. Equation 2.15 dictates
the evolution of active muscle fibers and has a similar behaviour to the previous one.
This term will increase with the driving controller C(t) and decrease with the number
of freshly fatigued fibers (the multiplication between the fatigue coefficient F and the
active fibers MA). The rate of fatigued units, as indicated in Equation 2.16, is prescribed
by the difference between the amount of recently fatigued fibers and the the amount of
fatigued fibers that just recovered.
To define the driving controller C(t), stated in Equation 2.17, Xia and Law added
two additional parameters LD and LR, respectively the muscle force development and
34

2.2 Dynamics of the Muscle Tissue
relaxation factors, whose values are not of major relevance since these were added
merely to ensure a good system behaviour [59] (despite of the name resemblance of
these parameters with the activation and relaxation times in the Activation Dynamics
model of Section 2.2.1, there is no evidence of a correlation between these terms).
In Xia and Law’s model, C(t) is a bounded controller that depends on the relation
between the compartments’ state and a target load T L, i.e., the force that the muscle is
required to exert. In this work the target load is referred as the instantaneous solicited
contractile muscle force F m CE , since Xia and Law assume that the neuromuscular system
can produce T L [59], i.e., this term, just like F m CE , is limited by the available contractile
force ˆ F m CE . Moreover, the model used in this work assumes that the passive element force
contribution does not play a part in the fatigue process, therefore it is not considered
for the calculation of C(t).
⎧
⎪⎨ LR × (F
C(t) =
⎪⎩
m CE − MA × ˆ F m CE ) F m ≤ MA ˆ F m CE
LD × (F m CE − MA × ˆ F m CE ) MA ˆ F m CE < F m ≤ (MA + MR) ˆ F m CE
LD × MR × ˆ F m CE (MA + MR) ˆ F m CE < F m
(2.17)
Since F , R, LD and LR are the only parameters under consideration, then it becomes
accessible to infer expressions that express these factors by experimental data fitting,
granting a relevant flexibility and applicability to this model.
Muscle recruitment hierarchy
All muscles are composed by a brew of different fiber types that span from slow-twitch
to fast-twitch fibers. Muscles with a high percentage of fast fibers are able to rapidly
develop strong contractions, while slow fiber muscles have larger reaction times but
substantial endurance. There are several reasons [53] for this contrasts: fast fibers
have a bigger diameter and a higher activity of enzymes that catalyse a rapid release of
energy, leading to faster contractions (but in shorter periods); slow fibers contain a large
amount of mitochondria and myoglobin in the sarcoplasm, inducing higher amounts of
available AT P and higher rate of oxygen diffusion. In their work, Xia and Law [59]
identified three principal types of muscle fibers: slow, fast fatigue-resistant and fast
fatigable.
The recruitment of muscle fibers was described by Henneman [61] that stated that
slow MUs are the first to be recruited, followed by fast-resistant and fast-fatigable at
35

2. MUSCULOSKELETAL SYSTEM MODELLING
last. This mechanism is explained by the stack structure in Figure 2.21 where every level
includes an independent three-compartment fatigue sub-system indicated in Figure 2.20,
since each type of muscle fiber will have a characteristic fatigue development process.
Note that fast fatigue-resistant fibers are only activated when the RC of all slow fibers
is fully solicited. The same happens with fast-fatigable fibers in relation to fast-fatigable
ones.
Figure 2.21: Muscle recruitment hierarchy pile chart [59].
The importance of implementing such recruitment hierarchy model is that it allows
the same model to define, in a more specific way, muscles with different characteristics
and fiber constitution. Considering this hierarchy, it is possible to indicate the per-
centage of each type of fiber, improving the way the model recreates real-life fatigue
behaviour and approaching it towards a smaller (microscopic) scale. The muscle fatigue
model becomes characterised by a non-linear behaviour, even for constant target loads.
36

2.3 Discussion
2.3 Discussion
In this chapter, the fundamentals of muscle anatomy and physiology were introduced.
Skeletal muscle has a well defined fasciculated structure and known mechanical and
biochemical organisation, which voluntary contraction delivers tension that arises from
action of the myosin-actin cross-bridge cycling. The contraction process, starting with
action potential from the CNS, was described in order to acknowledge its microscopic
functioning when introducing the mathematical models that will simulate muscle con-
traction behaviour.
Three types of skeletal muscle tissue contraction dynamics were described: activa-
tion, contraction and fatigue dynamics. Activation dynamics models, that describe the
time lag between the neural signal u(t) and corresponding muscle activation a(t), have
evolved from bang-bang control models to more versatile and stable models, such as
the one described. In this kind of model, it is required to solve the activation dynamics
differential equations in order to obtain the neural signal u(t). These type of models
are usually included when using dynamic optimization, since integration is required.
The majority of the developed contraction dynamics models are based in a Hill-type
tissue model. Archibald Hill [46] introduced, to the classic tissue models, a contractile
element capable of developing active tension. Three important properties of muscle
tissue were reported: the force-length relationship, the force-velocity relationship and
activation scaling. The described contraction model, included in this work, is the one
proposed by Kaplan [20] and Silva [1].
Fatigue dynamics are a novelty to multibody dynamics, therefore a simple model
was considered, adapted from the work by Xia and Law [59]. This model is based in
a three-compartment theory, considering that muscle units are in one of three ideal
states: activated (exerting force), resting (recovering the contractile capabilities) and
fatigued (no muscle force can be delivered). The dynamics of the "transfer" of muscle
units between compartments are made with basis in differential equations ruled by a
recovery coefficient R, a fatigue coefficient F , a driving controller C(t) and the instan-
taneous state of the muscle (MR, MA and MF ). The driving controller C(t) allows the
model to consider a non-constant muscle force history, an aspect of major importance
for musculoskeletal multibody routines, which are highly non-linear. The model is com-
plemented by a muscle recruitment hierarchy model that enables the same model to be
37

2. MUSCULOSKELETAL SYSTEM MODELLING
more descriptive in terms of muscle fiber type composition.
38

Chapter 3
Multibody Dynamics
According to Jalón and Bayo [7], a multibody system is defined as an assembly of
two or more rigid bodies imperfectly connected (by what is known as kinematic pair or
joint), being able to move relatively to each others. Multibody routines are developed to
numerically analyse three-dimensional mechanical systems that undergo large displace-
ments and rotations, and are acted upon by external forces. These routines assemble
and solve the system’s equations of motion in a methodical manner [7]. This technique,
along with other computational mechanics methodologies, multiplied its relevance in
the last few decades due to the increasing capacities of modern computers, able to lead
with elaborate systems.
The formulation presented in this work employs fully Cartesian (or natural) coor-
dinates as introduced by Jalon and Bayo [7] and applied in Silva [1]. The system is
defined as the set of Cartesian coordinates of the points and vectors that define the
body elements, without making use of angular variables. Natural coordinates allows
the use of shared points and vectors when modelling kinematic joints, which leads to
a reduction of the system’s equations. This loss of information however conditions the
calculation of reaction forces in joints. Silva [1] overcomes this drawback by creating an
expanded system from the original, with no shared-point joints, when the calculation
of these reactions is needed.
This work adapts an already existent FORTRAN multibody code, developed by
Silva [1], to the models described in Chapter 2, in order to analyse biomechanical systems
where skeletal muscles are considered. This chapter starts with a first introduction to
the kinematics and equations of motion of multibody systems. The used formulation
39

3. MULTIBODY DYNAMICS
for muscle forces is described and the process of introduction of muscle equations in
the equations of motion is explained. The implication of such action is analysed in
both Inverse and Forward Dynamics paradigms. The vital optimization process that
is utilised for solving the equations of motion, is described. The chapter closes with a
brief discussion and conclusions about this methodology and the chosen approach for
muscle implementation.
3.1 Kinematics
The kinematic analysis of a multibody system comprises solely the geometrical as-
pects of position and orientation of each body, disregarding the forces and torques that
produce the observed movement [67].
Since we are working with natural coordinates, the configuration of the whole sys-
tem is characterised by the Cartesian coordinates of every point and vector use in the
description of the model. These are arranged in the column vector of generalised coor-
dinates q. For a system with n existing points and m existing vectors, the generalised
coordinates are described as shown below:
q = { xP1 yP1 zP1 ... xPn yPn zPn xV1 yV1 zV1 ... xVn yVn zVn } T
(3.1)
where x, y and z refer to the coordinates in the three Cartesian directions, and P and
V symbolises a point or vector, respectively. Vector q has a total size of nc = 3(n + m)
coordinates.
Taking into consideration that these coordinates show some dependencies that arise
from the anatomical and dynamic properties, supplementary algebraic equations must
be considered, in order to uphold the topology of the mechanical system. These equa-
tions are known as kinematic constraint equations and, to ensure the validation of the
system status, they must be fulfilled for every time instant in the analysis. In a multi-
body system with ns scleronomic constraints (equations with no explicit dependence
on the time variable) and nr rheonomic constraints (where time dependency is explicit,
usually related to driver actuators), the kinematic constraint equations are held by the
equality of the column vector Φ to the null vector:
Φ(q, t) = { Φ1(q) ... Φns(q) Φns+1(q, t) ... Φns+nr(q, t) } T = 0 T
40
(3.2)

3.1 Kinematics
The total number of equations will then be nh = ns + nr, where nh stands for the
number of holonomic constraints. Holonomic constraints are defined as constraints
with an exact differential and therefore integrable from the velocities equations. This
is important for the evaluation of the respective Jacobian entries (described in the next
subsection). For a detailed understanding of the characteristics of holonomic and non-
holonomic constraints, please refer to Jalón and Bayo [7] or Silva [1].
There are several types of constraints to be considered in the construction of Φ.
To model rigid bodies, constant lengths between points that belong to the same body
must be held constant, using rigid body constraints. For more complex structures,
linear combination constraints are used. Joint constraints are employed to describe
relative positions between components and driver constraints for motion prescription
purposes. These four types of constraints are the most relevant for the developed work.
For a detailed description of kinematic constraint equations the reader should refer to
References [7] and [1].
However, a special note should be taken regarding driver constraints. This is the type
of constraints that will correlate certain equations, numerically imposing the prescribed
motion. Driver constraints will play a major role in Section 3.5, where the proposed
optimization methods is described, and the calculation of muscle forces and activations
will be performed. As it will be described in that section, its importance comes as a
consequence of our muscle forces implementation in the equations of motion: it will be
required to "transfer" the numerical contribution that justifies the prescribed motion
from the kinematic drivers to the muscle equations.
Kinematic analysis
In our work, it is necessary on occasions to corroborate the consistency of prescribed
kinematics making use of a kinematic analysis routine. The position, velocity and
acceleration of the system’s driven elements are given in order to obtain the same
physical quantities of the remaining constituents of the system. This analysis is made
by solving Equation 3.2 in order to q using the first two terms of its Taylor series
expansion:
Φ(q, t) ∼ = Φ(qi, t) + Φq(qi)(q − qi) = 0 (3.3)
41

3. MULTIBODY DYNAMICS
where q corresponds to an initial approximation of the generalised coordinates for iter-
ation i. Matrix Φq is the Jacobian of Φ in order to q, defined as
Φq(q) = ∂Φi
⎡
∂Φ1
⎢ ∂q1 ⎢
= ⎢
∂qj
⎢ .
⎣∂Φnh
∂q1
...
. ..
...
⎤
∂Φ1
∂qnc ⎥
. ⎥ .
∂Φnh
⎦
∂qnc
(3.4)
where nc and nh are the already referred number of coordinates and number of holo-
nomic constraints, respectively. The Taylor expansion in Equation 3.3 will be employed
to implement the Newton-Raphson method. Since this method is iterative, the following
adaptations are required, where subscripts are the number of the iteration:
q = qi+1 (3.5)
∆qi = qi+1 − qi (3.6)
Adapting Equation 3.3 to the arrangement of Newton-Raphson method results in
Φq(qi)∆qi = −Φ(qi, t) (3.7)
being this equations used iteratively until ∆qi reaches a value considered negligible. At
this point, the algorithm is ready to increase for the next time step.
The automatic generation of the vector of constraints Φ(q, t) may occasionally lead
to an overconstrained system, due to the presence of redundant constraint equations
(equations that describe identical topological properties). In this case, the Jacobian
matrix will have linearly dependent lines, i.e., it is rank-defficient. To overcome this
aspect [15] and [7] apply the least-squares method below:
Φq(qi) T Φq(qi)∆qi = −Φq(qi) T Φ(qi). (3.8)
To work out the velocities of the bodies that constitute the multibody system, Equa-
tion 3.2 is differentiated in time, yielding:
˙Φ(q,
dΦ(q, t)
˙q, t) = =
dt
∂Φ(q, t)
+
∂t
∂Φ(q, t) ∂q
= 0 (3.9)
∂q ∂t
where the vector ∂Φ(q, t)/∂t corresponds to the partial derivatives of the constraints
with respect to time. Knowing that the term ∂Φ(q, t)/∂q = Φq and ∂q/∂t = ˙q,
Equation 3.9 can be then rewritten as
Φq ˙q = ν (3.10)
42

where
∂Φ(q, t)
ν(t) = −
∂t
3.2 Equations of motion
(3.11)
A similar procedure is taken for the calculation of the accelerations ¨q. The derivation
of Equation 3.9 in time yields:
that can be rewritten as
where
¨Φ(q, ˙q, ¨q, t) = d ˙Φ(q, ˙q, t)
dt
= Φq¨q + (Φq ˙q)q ˙q − νt = 0, (3.12)
Φq¨q = γ (3.13)
γ(q, ˙q, t) = νt − (Φq ˙q)q ˙q (3.14)
Note that Equations 3.10 and 3.13 will be used for verification of consistency of the
initial values in a forward dynamics analysis as it will be described in Section 3.6.
3.2 Equations of motion
Since the calculation of the muscle forces is the major goal of this work, the kinetic
laws used in multobody dynamics with natural coordinates must be examined. When
performing either forward and inverse dynamic calculations, the equations of motion of
a generic system must be worked out and modified. This section makes an overview of
some of the algebraic principles implicated in the physical laws of multibody systems.
The principle of the virtual power is one of the possible approaches that can be
used to obtain the equations of motion in constrained multibody systems [1]. It states
that, for a determined mechanical system, the sum of the virtual power produced by
the external forces must be zero for all instants of time, i.e.,
P ∗ = ˙q ∗T (M¨q − g) = 0 (3.15)
where ˙q ∗ is a virtual velocity vector, M¨q are the inertial forces (M is the global mass
matrix and ¨q the vector of generalised accelerations), and g the generalised force vector
(containing external and velocity-dependent forces) [1].
43

3. MULTIBODY DYNAMICS
Vector ˙q ∗ must contain for all time steps a set of fictional velocities consistent with
Equation 3.10 [1]. It belongs to the null space of the Jacobian matrix and therefore this
means that it must be orthogonal to the constraint manifold.
Equation 3.15 does not take into consideration the internal reaction forces related
to the kinematic constraints since these produce null virtual power. These forces are
developed by the system in order to assure that the system fulfils the imposed kine-
matic constraints [1]. Internal forces may be added to the equations of motion using
the Lagrange Multipliers method. This method correlates each internal force to its as-
sociated kinematic constraint, yielding nh equations to the equations of motion of the
mechanical system [1]. The internal constraint forces vector g Φ is given by:
gΦ = Φ T q
λ (3.16)
where λ is known as the vector of Lagrange multipliers that, from a physical perspec-
tive, provides the magnitude of the constraint forces, whereas the rows of the Jacobian
matrix Φq correspond to the directions of these forces. Combining Equation 3.15 with
Equation 3.16, the virtual power comes
P ∗ = ˙q ∗T (M¨q − g + Φ T q
λ) = 0. (3.17)
According to Silva [1], it is always possible to find nc − nh arbitrarily vitual velocities
and nh Lagrange multipliers (for the mentioned system with nc generalised coordinates
and nh holonomic constraints) that, using 3.17, will ensure that
M¨q − g + Φ T q
λ = 0. (3.18)
This equation embodies the equation of motion (EOM) to be solved in order to compute
the system unknowns.
3.3 Generic muscle forces
In order to insert a muscle model into the existing multibody routines, its data man-
agement must be carefully defined and implemented. As stated in Section 2.1.1, muscle
structures are spatially defined by the coordinates of their origin (subscript o), insertion
(subscript i) and eventual via-points (subscripts v1, v2, ... , vvp, with vp as the number
of via-ponts). This representation, for two muscle structures with 0 and 3 via-points,
44

3.3 Generic muscle forces
both applying a force F m , is shown in Figure 3.1 (a) and (b) respectivelly. Note that
some important geometrical assumptions are made, that do not correspond to the mus-
cle anatomy of humans: muscles have rectilinear orientations, constant cross-sectional
area and no wrap around structures in its via-points.
(a) Model with no via-points (b) Model with 3 via-points
Figure 3.1: Muscle representation for a biomechanical system.
Generically, a muscle exerts in the respective application points a number of forces
nf = 2 × vp + 2. The simplest case of a muscle with no via-points (vp = 0) will exert
a pair of forces F m at both the origin and insertion points, while the case with vp = 3
will have 4 pairs of opposite forces. A muscle with via-points will be defined by more
than one direction vectors that will be used to define the orientation of muscle forces
(u1, u2, ... , ud), where the number of direction vectors is d = vp + 1. The application
point for each applied force is associated to a rigid body, where the force is actually
being exerted. From a vectorial point-of-view, a muscle force with direction ud ′ and
applied in a point p will be expressed as
F m p = ud ′F m . (3.19)
This work has a formulation that uses forces for the whole muscle structures implemen-
tation, i.e., numerically muscles will be defined as forces. Differently from this, Silva [1]
45

3. MULTIBODY DYNAMICS
developed a methodology that uses Lagrange multipliers λmd associated with muscle
actuators. These multipliers were related to a force per unit of length (N/m, in SI
units). This approach considers a constant force per unit length in a muscle which is a
physiological assumption that requires additional proof. The advantage of working with
forces, rather that force per unit length, is inherent to the fact that the exerted muscle
force magnitude will be the same for different length locations in the same muscle, and
will correspond to the force exerted by the whole muscle F m .
Concentrated forces
Muscle forces must be adapted to the multibody system. Therefore, we must consider
an algebraic method to bridge vectorial with generalised forces. The way external forces
are processed in this work is the one described by Jalón and Bayo [7] and Silva [1]. Let
us consider the rigid body in Figure 3.2 with orthogonal local reference frame oξηζ
defined by two points i and j, and two non-coplanar vectors u and v, in an inertial
reference frame 0xyz. In this rigid body, a generic force F m is applied in a point p, as
illustrated.
Figure 3.2: Application of a force F m to pointo p, belonging to the rigid body defined
by points i and j and vectors u and v [1].
46

3.3 Generic muscle forces
Now, the global coordinates of point p, represented by vector rp, are related to the
Cartesian coordinates of ri, rj, u and v as follows
rp − ri = c1(rj − ri) + c2u + c3v (3.20)
where the coefficients c1, c2 and c3 correspond to the coordinates of vector rip in the
three-dimensional basis formed by vectors rij, u and v. Arranging Equation 3.20 to
obtain rp comes
or simply:
rp 3×1 = (1 − c1)I3 c1I3 c2I3 c3I3
3×12
⎧
⎪⎨
⎪⎩
ri
rj
u
v
⎫
⎪⎬
⎪⎭
12×1
(3.21)
rp = Cpqe. (3.22)
Matrix Cp plays an important role, since it is responsible to express the Cartesian
coordinates of any given point p, belonging to a rigid body e, as a linear combination
of the generalised coordinates used to describe that element 1 . Since Cp depends exclu-
sively of locally defined vectors, it will be independent of body motion, hence constant
during the whole analysis. It also should be noticed that, since this matrix transforms
generalised coordinates of a rigid body (arranged as 3 vectors in a column) in global
coordinates of a point, its dimensions are (3 × 12) and remain constant in time.
For every multibody dynamic analysis that involves any kind of external forces,
Cp must be assembled for every force application point. The c coefficients must be
calculated and it can be done by adapting Equation 3.21 to the local frame oξηζ:
(r ′ p − r ′ i) = r ′ ij u′ v
⎧
⎨
′
⎩
c1
c2
c3
⎫
⎬
⎭ = X′ c (3.23)
with X ′ = r ′ ij u′ v ′ . This matrix always has an inverse [1] and therefore Equa-
tion 3.23 can be solved in terms of c
c = X ′−1 (r ′ p − r ′ i) (3.24)
1 It should be noted that when rigid bodies with complex geometry are considered, a second trans-
formation matrix V must be used. However its analysis is out of the scope of the work. Please refer to
Silva [1] and Jalón and Bayo [7] for further study in this subject.
47

3. MULTIBODY DYNAMICS
Reaching this point, we are able to express the generic muscle force F m p in terms of
an equivalent term g F m p
e
expressed in terms of the generalised coordinates of the rigid
body in matter. It is considered that the virtual work carried out by concentrated force
F m p and its generalised counterpart g F m p
e
should be the same:
δW = δr T p F m p = δq T e g F m p
e
Using Equation 3.22 in the latter, yields that
δW = δq T e C T p F m p = δq T e g F m p
e
(3.25)
(3.26)
and relating with Equation 3.19, the expression of the concentrated muscle force ex-
pressed in the generalised coordinates that define the rigid body will be:
F m
ge = C T p F m p = C T p ud ′F m . (3.27)
This expression plays a vital role in this muscle modelling work, since it allows us to
transform a Cartesian representation of forces F m p in the generalised configuration g F m p
e .
Nevertheless the latter corresponds to a physical quantity that requires additional ma-
nipulation, since F m p needs to be expressed in terms of the generalised muscle force
vector of the whole system, the column vector g F m p of size nc. In his work, Silva [1]
developed natural-coordinates-specific computational routines whose function is to as-
semble g F m p
e into g F m p , expressing generic concentrated muscle forces in the scope of
multibody equations of motion. The three different representations of muscle force F m
are schemed in Figure 3.3.
Vectorial
F m p
Rigid body
generalised coordinates
g F m p
e
Whole system
generalised coordinates
Figure 3.3: The different representations of a generic muscle force F m p .
Since each generalised force vector g F m p
e
g F m p
is specific for a certain body (the body
containing the point where the force is applied), then a particular muscle framed in nb
bodies will have nb different g F m p
e
that need to be added to the equations of motion.
48

3.3 Generic muscle forces
Taking the case of the muscle in Figure 3.1 (b) as example, it is arranged by three dif-
ferent bodies I, II and III. Therefore three independent gF m
e force vectors (subscript
p will be dropped from this point) must be considered: gI e, gII e and gIII e . Each one
of these vectors will enclose information regarding the forces that are exerted in the
respectively attached via-points as depicted in Figure 3.4. For instance, gF m
I will be defined
by forces Fm 1 , Fm2 and Fm m
3 , while gF
III will only be determined by the contribution
of F m 8 . In mathematical terms and assuming that the muscle force F m has a constant
magnitude in all muscle extension, it comes that for body I
g I e = g F m 1
e
+ g F m 2
e
+ g F m 3
e
= C T o uov1F m + C T v1uv1oF m + C T m
v1uv1v2 F
(3.28)
(3.29)
and knowing that the direction vectors defined by any two points i and j will have
opposite directions
uij = −uji
then we can simplify Equation 3.29 and state that, for all ge:
g I e = (C T o − C T v1 )uov1 + CTv1 uv1v2
m
F
g II
e = −C T v2uv1v2 + (CTv2 − CTv3 )uv2v3 + CTv3 uv3i
m
F
g III
e = −C T m
i uv3i F
(3.30)
(3.31)
(3.32)
(3.33)
Having a numerical representation of all the forces of the muscle in Figure 3.4, it is
now possible to assemble gI e, gII e and gIII e
– the forces expressed in general coordinates
of the involved bodies – in the vectors of generalised muscle forces expressed in the
general coordinates of the whole system g I , g II and g III . Since these are not specific
to the bodies, but to the multibody system, they are summable, and therefore the
assembling of all muscle forces in terms of the system general coordinates gF m
as
F m
g = gI + gII + gIII
is given
(3.34)
Note that, as the term F m is present in all expressions as a multiplication factor, this
term can be divided in the Hill-type muscle model components. Using Equation 2.4 we
get
49

3. MULTIBODY DYNAMICS
Figure 3.4: Muscle force representation for the 3 via-point model in Figure 3.1.
F m
g
F m
= g
F m
P E + gCE (3.35)
where gF m
m
CE and gF
P E are respectively the generalised force vectors for the contractile and
passive elements contributions. In addition, and regarding the terminology of Equa-
tion 2.8, gF m
can be expressed in terms of the muscle activation a:
F m
g
F m
= g
F m
P E + ˆg
CE a (3.36)
where ˆg F m
CE corresponds to the generalised force vector of the maximum available contractile
force, i.e., the generalised representation of ˆ F m CE . These expressions will have
the utmost importance in the following sections, where the equations of motion will be
reshaped in order to include our muscle forces and calculate muscle activations.
3.4 Inverse Dynamics
Considering that the information about a mechanical system’s motion and anthro-
pometry (topology and kinematic restrictions) are available, then it will be possible to
calculate both internal and external forces by means of inverse dynamics based routines.
50

3.4 Inverse Dynamics
In this terms, it is possible to infer reaction forces and net-moments in articular joints,
by non-invasive procedures [1]. This is a major asset for biomechanics, specially when
a living mechanical system is being analysed. In this work, the methods considered
for solving our EOM are the Lagrange multipliers and the Newton method (for more
developments on other solutions refer to Jalón and Bayo [7]).
Taking the EOM (Equation 3.18) into the paradigm of inverse dynamics, the analysis
will be performed knowing the anthropometric data (specified by the mass matrix M and
the Jacobian Φq), the system motion (given by the vector of generalised accelerations
¨q), and both the external forces and velocity-dependent inertial forces (available in g).
The only unknown in this analysis, and therefore our output, is the Lagrange multipliers
column vector λ, that will provide the forces associated with each degree-of-freedom of
the system. Rewriting Equation 3.18 we get
Φ T q λ = g − M¨q. (3.37)
This equation corresponds to a system with nc equations with nh unknowns. If the
system is over-constrained (nh > nc), then Equation 3.37 will have an infinite set of
solutions. In his work, Silva [1] employed the minimum norm condition as a means to
acquire an unique solution. In this way, the best solution will be considered to be the one
orthogonal to the null-space of the matrix Φ T q [1], which mathematically corresponds
to
λ = Φ T q λ ∗
(3.38)
where λ ∗ will enclose the unique solution for λ, when replacing Equation 3.38 in Equa-
tion 3.37, since Φ T q Φq is always invertible [1].
There are several methods for calculation of muscle forces: some authors, such as
Raison [17] and Ackermann [2] use inverse dynamics to determine joint torques and
in a subsequent post-processing step, muscle forces are calculated using an optimizer
that relates these joint moments with muscles’ moment arm. When the goal is to
calculate muscle forces exclusively, this method assures a simple, efficient and flexible
implementation, along with excellent results. However, the values obtained for joint
reactions and inner tensions will not be accurate, since muscle forces are not taken into
account in the EOM.
51

3. MULTIBODY DYNAMICS
Silva [1] in his work, modelled muscle actuators as constraint equations describing
the muscle length development history. This method is quite similar to the one devel-
oped here in terms of muscle forces representation described in Section 3.2, but differs
in the way this information is added to the EOM. In Silva’s work, muscle actuators are
simply considered by adding the respective constraint equations to Φ. This method-
ology proved [1] to rectify the drawback of the previously described procedure, where
muscle forces are not considered for the calculation of joint reactions and inner forces,
since the system is solved taking into account the system mechanical constraints and
muscle contraction dynamics. This information is figured as a whole and the solution is
calculated in an integrated way. Nevertheless this representation lacks some versatility
when integrating the muscle model in both inverse and forward dynamic (Section 3.6)
approaches. In this work, muscles are represented by forces solely as it will be described.
Integrating muscle forces
The goal in this Section, is to include the referred muscle forces in the equations of mo-
tion and add muscle activations in the solutions of our problem. To do so, the Newton
method from Jalón and Bayo [7] is used. Employing this approach, muscle actuators
must be regarded as sets of concentrated external forces, avoiding the constraint repre-
sentation. Considering the existence of nm muscles in our model, g can be expressed
as
g = g ext + g M1 + g M2 + . . . + g Mnm (3.39)
where g ext is the remaining external forces (excluding muscle forces). Making use of
Equation 3.36, it is possible to explicitly express passive and active contributions of
muscle in the generalised force vectors, resulting in
g = g ext + ˆg M1
CEaM1 M1 + gP E + . . . + ˆgMnm
CE aMnm + g Mnm
P E
Replacing this equation in Equation 3.18, it comes that
. (3.40)
M¨q − (g ext + ˆg M1
CEaM1 M1 + gP E + . . . + ˆgMnm
CE aMnm + g Mnm
P E ) + ΦTq λ = 0. (3.41)
Now, the aim here is to compute the magnitude of the internal constraint forces λ
and muscle activations a M1 ...a Mnm . Reshaping our system to a suitable configuration,
52

Equation 3.41 suffers a rearrangement:
3.4 Inverse Dynamics
Φ T q λ − (ˆg M1
CEaM1 Mnm
+ . . . + ˆg CE aMnm ) = g ext + g M1
P E + . . . + gMnm
P E − M¨q. (3.42)
Since the inertial term M¨q, the passive contribution in muscle forces g m P E
and the
remaining external forces g ext are independent from both λ and the muscle activations,
they can be passed to the right hand side of the EOM. It is now possible to express the
equation in the matrix form as:
Φ T q −ˆg M1
CE
. . . −ˆgMnm
CE
or in a more compact fashion
⎧
⎪⎨
⎪⎩
λ
a M1
· · ·
a Mnm
Φ T q −χ T λ
a
⎫
⎪⎬
⎪⎭
= g ext + g M1
P E + . . . + gMnm
P E − M¨q (3.43)
= g ext + gP E − M¨q (3.44)
where χ is a matrix holding all the generalised maximum available contractile force
vectors of all the muscles included (Equation 3.45), and a is the column vector of all
muscle activations (Equation 3.46):
χ =
a =
⎡
⎣
⎡
⎣
ˆg M1
CE
· · ·
ˆg Mnm
CE
aM1 · · ·
aMnm ⎤
⎦ (3.45)
⎤
⎦ (3.46)
The final configuration of the EOM with muscle forces consists in a system of nc
equations and nh + nm unknowns. This will lead to a system with infinite solutions,
a fact that can be physiologically understood by the existence of muscular redundancy
(already mentioned in the previous chapters) understood by the infinite number of mus-
cle force combinations that will result in a specific motion. Optimization procedures are
used to solve this system, by minimising a function that will describe the muscle system
energy depletion. Further details about the optimization can be found in Section 3.5.
53

3. MULTIBODY DYNAMICS
3.5 Optimization
A musculoskeletal system is typically redundant. Even if a straightforward adductor-
abductor muscle pair is modelled, any movement of the joint that these muscles span
will have an infinite set of muscle force combinations that will produce it, i.e., the
CNS will pick one from a set of infinite muscle activation combinations to prescribe the
movement in question.
In our inverse dynamics implementation, due to the use of the Newton approach,
a redundant system will be created only by adding one muscle to span a certain joint.
This is due to the fact that the EOM of the system will hold information related to
the kinematic driver constraint that prescribes the joint motion, as well as information
related to the generalised force term ˆgCE, associated to the included muscle. This means
that there will be two unknowns related to each one of these terms that will constitute
our control variables, respectively: a Lagrange multiplier associated with the kinematic
driver λ ∗ and a muscle activation a m . Therefore, in any situation that considers muscles,
the number of equations will always be less than the number of unknowns in the EOM,
leading to an indeterminate system of equations with an infinite set of solutions. This
is a major issue in muscle modelling with multibody dynamics, i.e., to choose the best
possible solution.
The general optimization problem
The challenge in optimization is to make the mentioned solution choice. There are sev-
eral mathematical approaches to solve this problem. A typical one is trying to even out
the number of equations and unknowns either by decreasing the number of unknowns
(situation that will lead to loss of the system’s information) or by increasing the number
of equations, however it is difficult to obtain such equations). With optimization meth-
ods the objective is to find, from all the solutions, the one that, regarding a particular
set of constraints, minimises a stipulated objective or cost function.
The control variables of our optimization problem x are the terms that are meant
to be worked out, i.e., the unknowns of the inverse dynamics system, which are:
x =
λ
a
54
(3.47)

3.5 Optimization
The solution of x must respect the limitations imposed by the constraints of our system.
The first type of constraints are the equality constraints feq, that arise from the EOM
of the musculoskeletal system. Equation 3.37 defines this type of constraints.
⎧
⎪⎨
feq =
⎪⎩
f1
.
⎫
⎪⎬
⎪⎭ = ΦT q −χT
λ
+ M¨q − (g
a
ext + g M1
P E
fnc
+ . . . + gMnm
P E ) = 0 (3.48)
The gradient of feq will be required for the optimization routines, described in the
development of this section:
∇f n λ a
o T = Φq −χT (3.49)
An important point of our implementation, and a novelty in the usual procedure
of the calculation of muscle forces, is associated to the boundaries of control variables.
When muscles are not implemented and a motion is prescribed, the Lagrange multi-
pliers assigned to each kinematic driver hold the forces contribution that induces the
movement. Let us identify the Lagrange multipliers associated with the joint drivers
that prescribe the motion of the joints spanned by the muscles by λ ∗ . When muscles
are considered in the model, it is desired to eliminate that contribution from these La-
grange multipliers, shifting that value to the activations of the considered muscles. To
restrict their contribution, these must be kept within a bound of ε. This will be the
first inequality type constraints.
|λ ∗ | ≤ ε (3.50)
The optimizer, constrained by Equation 3.48 will mathematically justify the motion
prescription by assigning positive values for the muscle activations a, but limiting the
values between 0 and 1 as
0 ≤ a m ≤ 1 , for m = 1, . . . , nm (3.51)
The remaining existent Lagrange multipliers in the control variables, identified as λ R ,
will not have any special limitations and therefore can have any value:
− ∞ ≤ λ R ≤ +∞ (3.52)
This process of transferring the contributions of motion prescription from the La-
grange multipliers to the muscle activations is illustrated in Figure 3.5. Considering
55

3. MULTIBODY DYNAMICS
the system in Figure 3.5(a), where a musculoskeletal structure with two rigid bodies
and muscle m bears a load P [N] in the extremity of a vertical rigid body of length
l [m]. When this model is fed to an inverse dynamics solver that calculates the muscle
activation a m for a constant prescribed angle of 90 o between the two rigid bodies, the
correspondent kinematic driver will be assigned with the value of the angular moment
needed to sustain the desired position. The value of that correspondent torque in the
mentioned model will be P l ey[Nm], as indicated in Figure 3.5(b). Before the opti-
mization process, all muscle activations are 0 (zero). As mentioned, the optimizer is
directed to convey the numerical influence of λ ∗ to the respective muscle activations,
as depicted in Figure 3.5(c). Since all λ ∗ are kept in the interval in Equation 3.50, this
allows the optimizer to become more stable, since these terms will act moreover as a
scape for the optimizer in the accomplishment of the EOM.
(a) Simple mechanical system with mus-
cle m and exertion of load P in an ex-
tremity of a body with length l.
(b) Control variables x
before the optimization
procedure.
Figure 3.5: Used optimization methodology.
56
(c) Control variables x
after the optimization
procedure.

Ultimately, the general optimization problem can be then stated as
Given : x
Minimise : F0(x)
Subject to :
⎧
feq(x) = 0
⎪⎨ 0 ≤ am ≤ 1
−ε ≤ λ
⎪⎩
∗ ≤ +ε
−∞ ≤ λ R ≤ +∞
3.5 Optimization
(3.53)
The only aspect that is missing a description in the formulated optimization problem
is the cost function F0.
Cost functions
The cost function purpose is to reproduce the response of the CNS in terms of mus-
cle recruitment and muscle activation distribution, when required to provoke a certain
posture and articular motion. There is an interminable number of situations and param-
eters with significance depending in the type of movement and the physical conditions
inherent to the model. For instance, the gait pattern for an individual with a muscle
injury will differ from the gait pattern of an healthy one. The CNS of the former will
aim to minimise the force exerted in the damaged muscle, and the CNS of the latter
will probably try to minimise the effort deployed. In addition these functions should
assure a stable and efficient computational behaviour.
There are several works in this area regarding cost functions. Tsirakos [37] made an
extensive work regarding cost functions in linearly constrained optimization techniques.
Raison [17] in his work included muscle EMG function shape for the calculation of F0.
For further reading about the use of different objective functions, please refer to the
works by Crowninshield and Brand [19], Collins [34], Tsirakos [37] and Silva [1]. In the
present work, the considered cost functions are the following:
1. The sum of the square of the activations:
F0 =
nm
(a m ) 2
m=1
(3.54)
The usage of this functions aims to minimise the muscle activations, i.e., the
neural effort required for execution of a motor task.
57

3. MULTIBODY DYNAMICS
2. Sum of the square of the individual contractile element forces:
F0 =
nm
(F m CE) 2
m=1
This function usage aims to minimise energy consumption amounts [1, 37].
3. Sum of the cube of the individual average muscle stresses:
F0 =
nm
(σ m CE) 3
m=1
(3.55)
(3.56)
Crowninshield and Brand [19] proposed this function, by relating muscle force
with muscle endurance, as well as results from experimental procedures [1]. In
addition, physiologically accordant results were obtained regarding the prediction
muscle groups co-activation [1, 37].
Computational optimization routine
The optimizer used in our program is the DNCONG routine, the double precision ver-
sion of the NCONG routine. It is a FORTRAN routine available in the IMSL Library
developed by Visual Numerics, Inc. [68], based on a subroutine developed by Schit-
tkowski [69]. The DNCONG routine was developed in order to solve a general nonlin-
ear programming problem, by means of a successive quadratic programming algorithm
and a user-supplied gradient [68], in this case Equation 3.49. For detailed information
about this method, please refer to the IMSL Library Documentation [68] and the work
by Schittkowski [69].
3.6 Forward Dynamics
From a different perspective from the last sections, it is possible to compute the
dynamic reaction of a constrained biomechanical system when compelled by external
forces. This is the goal of standard forward dynamics: to calculate the system’s motion
and internal reaction forces, given the external forces that include, in our case, the
muscle forces. The way that forward dynamics (FD) methodology is implemented in this
work involves the prescription of the vector of the muscle activations a m . Nevertheless
58

3.6 Forward Dynamics
it is possible to use this kind of analysis prescribing the system’s motion in order to
withdraw both internal and external forces. The work by Anderson and Pandy [30]
explores this alternative approach.
The paradigm of forward dynamics implies different available information about
the system. As mentioned, the muscle activations a m are prescribed and the motion is
unknown, therefore ¨q is to be determined. This means that the generalised forces vector
g is fully known, i.e., all terms in Equation 3.39 are known, and Equation 3.18 becomes
a system of nc 2 nd order ordinary differential equations (ODE) with nc + nh unknowns
(that correspond to the accelerations vector ¨q and the Lagrange multipliers λ), and
therefore indeterminate. To make it determinate, the nh equations of Equation 3.13
are added, resulting in the following system
M Φ T q
Φq 0
¨q
λ
=
g
γ
. (3.57)
Once the accelerations are known, these can be integrated in time in order to obtain
the generalised coordinates of the system q. To do so, the initial conditions for position
q0 and velocity ˙q0 must be given and are required to be consistent with the kinematic
constraints of the system in analysis. This requirement is assured when Equations 3.58
and 3.59 are fulfilled for the initial value of the problem.
Φ(q0) = 0 (3.58)
Φq ˙q0 = ν(t0) (3.59)
Verifying the consistency of the initial conditions is the first step in a forward dynam-
ics analysis of a multibody system, illustrated in the flowchart of Figure 3.6. In his
work, Silva [1] uses an iterative method from the work of Jalón and Bayo [7] called
the Augmented Lagrange Formulation (ALF). This method consists in a penalty-type
formulation whose task is to stabilise the EOM when the Lagrange multipliers λ are
removed from Equation 3.57, so it becomes a 2 nd order ODE with nc equations, instead
of nc + nh. This way only the generalised accelerations vector ¨q is worked out, and the
Lagrange multipliers are only calculated when required.
Having the accelerations of the elements of the system, the process continues by the
integration in time of this information, so that the motion is computed. The generalised
velocities and accelerations vectors ( ˙q and ¨q) are assembled in a vector ˙yt
˙q
˙yt =
¨q
59
(3.60)

3. MULTIBODY DYNAMICS
Figure 3.6: Direct integration algorithm flowchart for a forward dynamics problem [1].
and integrated using the direct integration method [70], a numerical process that for a
time t will integrate ˙yt, obtaining yt+∆t, i.e., the vector that contains the generalised
coordinates q and velocities ˙q for the following time step:
q
yt+∆t =
˙q
(3.61)
With this information, the cycle ends by updating the positions and velocities for
the next iteration, as well as the time t = t + ∆t. As elucidated in Figure 3.6, the
algorithm feeds the new cycle with the calculated data whose initial conditions, due to
their nature, are not checked for consistency anymore.
3.7 Discussion
Some important points with respect to this work’s formulations must be mentioned.
The chosen approach incorporates the described muscle models in the kernel of multi-
body formulation. This way, the force network inherent to the kinetics of the mechanical
system in analysis will take in consideration muscle efforts together with the other ex-
ternal and internal forces.
As previously mentioned in this chapter, the majority of the studies concerning the
calculation of redundant muscle efforts do not carry out this inclusion, but separate
60

3.7 Discussion
the analysis in two parts. A first one where, by means of inverse dynamics routines,
the joint moments required to carry out an given motion are computed. A second part
uses optimization techniques to predict the shought-after muscle efforts. This approach
has the plus side of its simpler implementation and a broader range of optimization
methods adapted to this situation. Nevertheless, the calculated reaction forces and
structure stress values in the first part will not take into consideration eventual muscle
forces. So this approach is only applicable when the objective of the whole analysis is
to infer muscle forces exclusively.
The developed method on the other side, increases its complexity in terms of multi-
body implementation, but allows the usage of a formulation that carries the whole batch
of existent forces. Including muscle efforts in the system’s equations has the natural
asset of correctly calculating joint reactions and rigid body stresses, since it takes into
consideration the muscle forces that may alter their values. This is a very important
point, for instance, in cases where muscle pair co-contraction happens, a situation that
leads to an intensification of joint stiffening and bone stress.
Regarding the defined optimization problem, it will be proved in the next chapter
that this is a well defined problem and will lead in the desired way with the computation
of the the equation of motion’s solutions. Nevertheless, the used optimization algorithm,
DNCONG, is a very sensible one. For such an intricate and highly non-linar system,
it is expected to verify a hard convergence to the solution. In addition, this algorithm
only guarantees the acquisition of local minima solutions. This means that it is possible
to obtain solutions that do not correspond to the lowest energy combination of applied
muscle forces. However, this allows the prediction of muscle co-contraction of antagonist
pair muscle groups, a situation that is actually observed in human gait, for instance.
61

3. MULTIBODY DYNAMICS
62

Chapter 4
Biomechanical Models
The models and formulations described in the previous chapters were implemented in the
Apollo program. This inclusion involved the conception of new routines incorporating
a muscle Hill model, a muscle fatigue model, EOM manipulation and the usage of
optimization routines. In this chapter, case test examples are considered to evaluate
the robustness, accuracy and efficiency of the proposed methods. The first model is a
minimal upper limb musculoskeletal model with no physiological relevance, that is used
to test the validation of the muscle contractile model described in Section 2.2.2. The
second one is a more complex upper limb model with 7 muscles used to test the muscle
fatigue model described in Section 2.2.3, with an initial test for a single immobilised
muscle. The third model is a 12 muscles lower limb musculoskeletal model, used to
calculate the muscle activations and forces for a prescribed gait motion, testing the
validation of the muscle force calculation method introduced in Chapter 3.
4.1 Muscle model verification
After the complete implementation of the Hill-type muscle model described in Sec-
tion 2.2, the accuracy of the conceived routines is evaluated. To do so, a specific
mechanical model with muscles is conceived in order to evidence the relationship of
muscle force F m with muscle length L m and velocity ˙ L m , and confirm if the output
values fit the model’s equations. The conceived mechanical model has no physiologi-
cal relevance in both the used muscle parameters and anthropometric properties, apart
63

4. BIOMECHANICAL MODELS
from some slight physical resemblance to the musculoskeletal system of the elbow joint
– Figure 4.1.
It was required to have a situation where a certain muscle m X was fully activated
while the rigid bodies vary their relative positions in time. This way m X spans a cer-
tain range of fiber lengths and velocities while employing its full contractile capacity for
each state. This model is solved for a forward dynamics analysis, where muscle activa-
tions are prescribed. The relation between muscle m X ’s length and the correspondent
output contractile force, for a constant activation a X (t) = 1, is examined. In order
to control the system’s movement and therefore the muscle’s length, a second muscle
m Y was included and its variable activation a Y (t) pattern directs angle α while body
1 keeps its vertical orientation still. Only the active contribution of the muscle model
F X CE (Lm (t), ˙ L m (t), a(t) = 1) is tested, since it plays the most complex role in the model.
The activation patterns for both the modelled muscles is illustrated in Figure 4.2 and
the correspondent motion is depicted in Figure 4.3.
Figure 4.1: Mechanical system with muscles m X and m Y for Hill-type muscle model
verification purposes.
Analysing Figure 4.3, it should be noted that the muscle in question, muscle m X ,
starts performing an eccentric contraction increasing its length and velocity. This first
phase is followed by a deceleration that will lead to a concentric contraction with a
maximum of its velocity. The system halts its concentric contraction and returns to
the initial position where the analysis ends with an immobilised system. With this
64

4.1 Muscle model verification
Figure 4.2: Activations pattern for muscles m X and m Y .
description, the system should go through a set of muscle force values that travel in the
length-velocity (l ˙ l) space in the shape of a cardioid, and should lay in the surface that
define the muscle force model, as described in Section 2.2. By plotting this surface and
overlaying the muscle contractile force values calculated, this prediction proved to be
true, as illustrated in Figure 4.4.
The contractile force F X CE (Lm (t), ˙ L m (t), 1) in the l ˙ l space, appears as a cardioid
laying in the muscle model active force surface for a fully activated model with the con-
sidered parameters. This proved that our model is correctly implemented and responds
with accuracy to the parameters imposed.
65

4. BIOMECHANICAL MODELS
Figure 4.3: The state of the mechanical model for different time instants when muscles
m X and m Y are activated as in Figure 4.2.
66

4.2 Muscle Fatigue
Figure 4.4: Muscle m X contractile element force output (black line) and possible forces
for the model (coloured surface).
4.2 Muscle Fatigue
One of the novel aspects of this work is the couple of a fatigue model with the devel-
oped Hill-type muscle contraction model and its consequent integration in a multibody
dynamics system. The used fatigue model requires as input the parameters that will de-
fine the dynamics associated with the three-compartment theory described in Chapter 2,
and the response in terms of muscle force delivery by the musculoskeletal system.
At first, the muscle fatigue model is examined apart from the rest of the formulation,
without coupling it to the Hill-type muscle model. For this case, an immobilised muscle
with F0 = 300N and the fatigue parameters displayed in Table 4.1, is subjected to
cycles of isometric contractions to which is associated the fatigue model. It is assumed
that initially all muscle units are in the resting state. The values of the forces developed
by the isometric contractions of the muscle, are obtained by imposing a target muscle
force P to the muscle. This target force values are given, in Newtons, by the function
proposed in Equation 4.1
67

4. BIOMECHANICAL MODELS
P = F0
0.25 + 0.15g
where g(x) is a square wave function, given by:
1 sin(x) ≥ 0
g(x) =
−1 sin(x) < 0
2πt
[N] (4.1)
50
(4.2)
The values of the force P for an analysis time ta = 150 are plotted in Figure 4.5 (red
line).
Table 4.1: Fatigue parameters used for the muscle fatigue model employed, and the same
used in the work by Xia and Law [59]. The values for the F , R, LD and LR are given in
units of 1/s.
Composition F R LD LR
S 50 % 0.01 0.002 10 10
FR 25 % 0.05 0.01 10 10
FF 25 % 0.1 0.02 10 10
An analysis is performed and the obtained results are shown in Figure 4.5. Let
us first mind the available muscle units for force delivery, i.e., the residual capacity
RC = MA + MR (dark blue line). It is observable that, for the time intervals where
P = 0.4F0 = 120N that the available muscle force drops considerably and for the times
steps where P = 0.1F0 = 30N the recovery process is verifiable. The opposite happens
in terms of the evolution of fatigued fibers. In the periods where P is larger, it is
observable that MF has a steep increase, in opposition to the remaining time intervals,
where muscle recovery is observable. In response to the diminishing of RC, the muscle
activation a m must increase. This has a valid correspondence to what happens in the
human body: since muscle units can not deliver the same levels of contraction forces,
then the CNS must increase its level of fiber recruiting. Nevertheless, in the second
cycle of this analysis, slightly before the first 60s, the available muscle contractile force
is unable to keep the desired tonus, and the muscle force F m will be unfit to keep the
desired values of force. The activation a m becomes then saturated for the periods where
P = 120N and the fatigue state of the muscle will unfit it to provide the desired muscle
force: note in Figure 4.5 that F m (red line) displays occasionally values smaller than
the target force P (black line).
68

4.2 Muscle Fatigue
Figure 4.5: Results obtained for three cycles of isometric contractions of the described
model. The quantities MA + MR, MF and a(t) are scaled to F0.
The validated fatigue model block is incorporated into the multibody routines and
additional tests are carried out. To test its functionality, a simple right upper extremity
model was designed with an immobilised vertically standing humerus and considering
the most important muscles of the elbow joint. A graphic representation of the model is
shown in Figure 4.6. Notice that a concentrated force P = 150N was applied at 30cm
from the elbow joint.
The model is defined by three rigid bodies: torso, arm and a forearm-hand complex.
Table 4.2 holds the local coordinates of the points that define the referred rigid bodies
of the model, while Table 4.3 shows their mass and inertial properties. The torso rigid-
body is not considered in Table 4.3, since it was only considered to allow the insertion
of the origins of some of the muscles and is considered as the inertial (fixed) ground
body. The whole biomechanical system is also subjected to a constant gravitational
force g = −9.81ez [m/s].
The geometrical and physiological parameters of the considered muscle are displayed
in Table 4.4. The remaining muscles that cross the elbow joint, such as the extensor
carpi radialis longus or the pronator teres, are not considered in the model, since these
muscle’s functions are not related to the action in analysis.
This model is tested in a Inverse Dynamics analysis perspective where a kinematic
driver for the elbow joint is prescribed, imposing a constant angle of 90 o , i.e., the
69

4. BIOMECHANICAL MODELS
Figure 4.6: Simple upper extremity model with 7 elbow muscles and constant force P
applied at the hand. This image was conceived using the OpenSim software [71]. Local
reference frames only indicated in the lateral view.
Table 4.2: Body index number and local coordinates of the points that define the rigid
bodies for the upper extremity model.
Proximal point coordinates Distal point coordinates
Body bn ξ[m] η[m] ζ[m] ξ[m] η[m] ζ[m]
Torso 1 0.00 0.00 0.00 0.00 0.00 0.00
Arm 2 0.00 0.00 0.180496 0.00 0.00 -0.109904
Forearm
and hand
3 0.00 0.00 0.181479 0.00 0.00 -0.118521
70

4.2 Muscle Fatigue
Table 4.3: Inertial data for rigid bodies considered in the upper extremity model.
Inertial Moments [Kg.m 2 ]
bn Iξ Iη Iζ mass [Kg]
2 0.014810 0.004551 0.013193 1.864572
3 0.019281 0.001571 0.020062 1.534315
humerus is immobilised in a vertical position and the muscles in the biomechanical
model maintain the body that represents the complex radius-ulna-hand horizontal. The
fatigue parameters used in this case are the ones in Table 4.5. Similarly to the previous
example, these have no correlation with any experimental results or literature values.
They are only for example purposes. The analysis is performed till the time instant
when the fatigue state of the muscles caused the system to be incapable to hold the
load P and sustain a horizontal orientation of the forearm. The calculated muscle forces
of the contractile element F m CE
for the considered muscles are illustrated in Figure 4.7.
Figure 4.7: Resultant muscle forces of the contractile element F m CE of the biomechanical
model of the upper extremity, when susceptible to fatigue dynamics.
The optimizer computed that, in the first instants, the position is sustained by full
activation of the brachialis and brachioradialis muscles and a partial activation of the
long head of the biceps brachii. The first two loose the competence of maintaining
the tonus, since they are using the full capacity of the muscle, becoming importantly
71

4. BIOMECHANICAL MODELS
Table 4.4: Geometrical and physiological parameters for the considered muscle in the
upper extremity model. Considered values were taken from the work by Holzbaur et.
al. [72]. Pictures reproduced with permission: "Copyright 2003-2004 University of Washington.
All rights reserved including all photographs and images. No re-use, re-distribution
or commercial use without prior written permission of the authors and the University of
Washington." [73].
biceps brachii long head (BIClong)
F0[N] L0[N] LT [N] α[ o ] bn ξ[m] η[m] ζ[m]
624.3 0.1157 0.2723 0
1
1
2
2
2
2
2
2
3
-0.0392
-0.0289
0.0213
0.0238
0.0135
0.0107
0.0170
0.0228
0.0075
0.0221
0.0136
-0.0103
-0.0120
-0.0014
0.0017
-0.0002
0.0063
-0.0218
0.0035
0.0139
0.1984
0.1754
0.1522
0.1031
0.0593
0.0051
0.1331
biceps brachii short head (BICshort)
F0[N] L0[N] LT [N] α[ o ] bn ξ[m] η[m] ζ[m]
435.56 0.1321 0.1923 0
1
1
2
2
2
3
0.0047
-0.0070
0.0112
0.0170
0.0228
0.0075
0.0353
0.0249
0.0110
0.0108
0.0063
-0.0218
-0.0123
-0.0400
0.1047
0.0593
0.0051
0.1331
brachialis (BRA)
F0[N] L0[N] LT [N] α[ o ] bn ξ[m] η[m] ζ[m]
987.26 0.0858 0.0535 0
72
2
3
0.0068
-0.0032
0.0036
-0.0009
0.0066
0.1576

4.2 Muscle Fatigue
(Table 4.4 continuation)
brachioradialis (BRD)
F0[N] L0[N] LT [N] α[ o ] bn ξ[m] η[m] ζ[m]
261.33 0.1726 0.133 0
2
3
3
-0.0098
0.03577
0.0419
-0.00223
-0.02315
-0.0224
-0.019134
0.054059
-0.039521
triceps brachii lateral head (TRIlat)
F0[N] L0[N] LT [N] α[ o ] bn ξ[m] η[m] ζ[m]
261.33 0.1726 0.133 0
2
2
2
2
3
-0.00599
-0.02344
-0.03184
-0.01743
-0.0219
-0.00428
-0.00928
0.01217
0.01208
0.00078
0.054036
0.035216
-0.045874
-0.087074
0.191939
triceps brachii long head (TRIlong)
F0[N] L0[N] LT [N] α[ o ] bn ξ[m] η[m] ζ[m]
798.52 0.134 0.143 0.21
1
2
2
2
3
-0.05365
-0.02714
-0.03184
-0.01743
-0.0219
0.02277
0.00664
0.01217
0.01208
0.00078
-0.01373
0.066086
-0.045874
-0.087074
0.191939
triceps brachii medial head (TRImed)
F0[N] L0[N] LT [N] α[ o ] bn ξ[m] η[m] ζ[m]
624.3 0.1138 0.098 0.157
2
2
2
2
3
-0.00599
-0.02344
-0.03184
-0.01743
-0.0219
-0.00428
-0.00928
0.01217
0.01208
0.00078
0.054036
0.035216
-0.045874
-0.087074
0.191939
Table 4.5: Fatigue parameters used for all the muscles in the upper extremity with elbow
muscles model. These have no correlation with experimental results [59]. Values given in
units of 1/s.
F R LD LR
0.01 0.002 10 10
73

4. BIOMECHANICAL MODELS
fatigued. To compensate the loss in the potential for creating articular angular momen-
tum of these muscles, both long and short heads of the biceps brachii increase the force
output, and therefore their muscle activations. Note that, around the 14s of analysis,
the long head reaches its maximum level of activation, leading to an increase of the rate
of short head recruitment. At time = 22.4s the analysis terminates, since the horizontal
position of the forearm cannot be supported by the muscle framework. In addition, it
should be noted that the activation of the three heads of the triceps brachii muscle is 0
(zero) for the whole analysis period, since these work as antagonist muscles, not being
able to perform a practical action towards the bearing of load P .
4.3 Human Gait
A simple model of the right lower extremity with 12 muscles is conceived with the
purpose of testing both the practicability and accuracy of all the formulation described
in Chapter 3. The model consists in four rigid bodies: torso, thigh, shank and foot,
however the torso inertial properties are again not considered for this analysis, since it is
only included for the insertion of muscle origins. The model is illustrated in Figure 4.8.
Tables 4.6 to 4.7 display the anthropometrical data of the alluded rigid bodies, and the
physiological and geometrical parameters of the considered muscles.
Table 4.6: Body index number bn and local coordinates of the points that defined each
rigid body considered on the lower extremity model.
Proximal point coordinates Distal point coordinates
Body bn ξ[m] η[m] ζ[m] ξ[m] η[m] ζ[m]
Torso 1 - - - - - -
Thigh 2 0.00 0.00 0.215 0.00 0.00 -0.219
Shank 3 0.00 0.00 0.151 0.00 0.00 -0.288
Foot 4 -0.04980 0.00 0.06882 0.07529 0.00 -0.03042
74

(a) Anterior view. (b) Lateral view.
4.3 Human Gait
(c) Posterior view.
Figure 4.8: Simple leg model with 12 muscles spanning the knee and ankle joints. This
image conceived using the OpenSim software [71]. Local reference frames only indicated
in the lateral view.
75

4. BIOMECHANICAL MODELS
Table 4.7: Geometrical and physiological properties inherent to the muscles existent in
the lower extremity model. Source: Reference [1].
76

(continuation of Table 4.7)
77
4.3 Human Gait

4. BIOMECHANICAL MODELS
(continuation of Table 4.7)
78

(continuation of Table 4.7)
4.3 Human Gait
Table 4.8: Anthropometrical properties associated with the rigid bodies of the lower
extremity model.
bn mass [Kg]
Inertial Moments [Kg.m 2 ]
Iξ Iη Iζ
2 4.922 0.718 7.970 4.934
3 1.813 0.543 1.915 1.570
4 1.182 0.129 0.128 2.569
79

4. BIOMECHANICAL MODELS
The gait cycle is the movement output of the lower limbs of a walking subject, and
is mainly divided in two phases: the stance phase (period where the limb, in this case
the right leg in this case, is in contact with the ground) and the swing phase (the period
where the foot is off the ground surface), as depicted in Figure 4.9.
Figure 4.9: Scheme with relative progress of the gait cycle, indicating the stance and
swing phases [74].
The stance phase commences with the landing of the heel, the initial contact known
as Heel Strike (HS). Starting a period of double limb support that results in a loading
response stage. Around 20% of the gait cycle, the opposite limb toe leaves the ground,
an instant called Opposite Toe Off (OTO). The body advances while the right lower
extremity holds the weight, passing through a mid stance stage at 30% of the gait tread.
The terminal stance happens when the limb arranges itself to start its swing phase, just
before the Opposite Heel Strike (OHS). When OHS occurs (approximately at 50% of
the stride), a new double support period starts, and the pre swing phase takes place.
The right limb starts its swing period when Toe Off (TO) happens, and the right lower
extremity moves forward passing trough a mid swing instant (that corresponds to the
opposite mid stance, at 80% of the cycle) and ends when the succeeding HS comes
about. This completes the gait cycles.
Considering an Inverse Dynamics paradigm, the positions of the mechanical system
are provided. Therefore, it is mandatory to feed the routines with kinematic drivers
that will describe the positions of the anatomical segments in time. These positions
were taken from the work by Silva [1], and a 2 nd order Butterworth (low-pass) filter
was employed to eliminate the noise associated with the kinematic data. The move-
80

4.3 Human Gait
ment is considered to happen in the sagittal plane, similarly to the situation illustrated
in Figure 4.10. Figure 4.11 shows the point numbering employed.
Figure 4.10: Part of the gait cycle with four points of reference [75].
Figure 4.11: Point numbering used in the model: Point 1 - Lower torso, point 2 - hip
joint, point 3 - knee joint, point 4 - ankle joint, point 5 - metatarsophalangeal joint, 6 -
heel position reference [75].
Four kinematic different drivers are taken into account, that result in the motion
of Figure 4.14:
Driver 1 Trajectory driver describing the trajectory x and z of point 1 – Fig-
ure 4.12.
Driver 2 Rotational driver describing the angular trajectory of vector v21 with
the horizontal direction. This driver has a constant value of 90 o .
81

4. BIOMECHANICAL MODELS
Driver 3 Rotational driver describing the angular trajectory of vector v21 with
v23 – Figure 4.13(a).
Driver 4 Rotational driver describing the angular trajectory of vector v32 with
v34 – Figure 4.13(b).
Driver 5 Rotational driver describing the angular trajectory of vector v43 with
v45 – Figure 4.13(c).
(a) X-axis coordinates. (b) Z-axis coordinates.
Figure 4.12: Driver 1: Trajectory driver with the space coordinates in time of point 1.
(a) Driver 3. (b) Driver 4. (c) Driver 5.
Figure 4.13: Angular directions of Drivers 3, 4 and 5.
82

4.3 Human Gait
Figure 4.14: The motion obtained using the prescribed kinematic drivers for the model
of the lower extremity.
83

4. BIOMECHANICAL MODELS
Since this gait model involves an interaction with the ground, the reaction forces
resultant from foot-ground contact are included in our biomechanical system as external
forces, i.e., added to vector gext. The evolution of the reaction force vector in time is
illustrated in Figure 4.15 and its application point coordinates (or centre of pressure)
curve in Figure 4.16. These values are obtained using a force platform synchronised
with the camera acquisition settings [1] and provided to the Inverse Dynamic routines.
Figure 4.15: Reaction force vector components, acquired from a force platform.
(a) X-axis coordinate. (b) Y-axis coordinate. (c) Z-axis coordinate.
Figure 4.16: Ground reaction force application point coordinates, or centre of pressure
curve.
The developed routines processed the model information and the prescribed data,
resulting in the muscle activation patters of Figure 4.17. In this figure, muscles are set
84

4.3 Human Gait
together in functional groups. The total muscle force of each muscle in every functional
group is summed and plotted in Figure 4.18.
Now, examining Figure 4.18 and bearing in mind the gait phases of Figure 4.9, let us
identify the force contributions of muscles for the resulting dynamic patterns of the right
leg. The analysis start with the HS (all gait terms refer to the right leg), the beginning
of the stance phase, where three groups of muscles are active. The Hamstrings are
solicited to contribute for the weight transfer that happens on passage from the single
support of the left leg to the double support period, supplying stability for the body
foundations.
During the same period, the ankle dorsiflexors are contracted to control the position
of the heel for the initial contact, in order to allow the inferior surface of the calcaneus to
sustain the weight acceptance at the HS. In addition, a co-contraction of the antagonist
pair formed by the flexors group and the dorsiflexors group of the ankle joint is verified
for joint stabilisation. From an optimization point-of-view, this co-contraction is clearly
a local minima of the cost function, since there is an excess of energy expenditure in
terms of torque output. Nevertheless, this is something that resembles what happens in
fact on human gait [1]. In order to decrease the steep variation of ankle joint reaction
in the instant of HS, the CNS commands a co-contraction of these muscles, increasing
the stiffness of the ankle joint before and during ground contact.
This first weight acceptance and loading response phases are verified till somewhat
after time = 0.1s (8.3%) of the analysis, by the time that OTO occurs and the weight
of the whole body structure is transferred to the right limb. This means that the body
structure is in single support. Therefore, the Triceps surae group comes into picture,
to perform a slight extension of the knee joint, while the leg bears most of the body
weight, precipitating an initial period of forward body propulsion, by the time of the
mid stance.
After the mid stance, the ankle flexors start to develop a considerable muscle force
magnitude, swivelling the lower leg around the ankle joint and inclining the body.
Simultaneously, the area of the foot contacting with the floor diminishes and the body
weight is gradually shifted to the forefoot region. At this time, around t = 0.5s (41.5%),
the ankle flexors muscle group forces peak controlling the body impulse that is provided
in the stance phase. This stage corresponds to the terminal stance, when the OHS
happens, by t = 0.6s (50%).
85

4. BIOMECHANICAL MODELS
(a) Knee joint flexor muscles activations.
(b) Knee joint extensor muscles activations.
(c) Ankle joint flexor and extensor (Tibialis anterior) muscles activa-
tions.
Figure 4.17: Muscle activation patterns obtained from the solution of the EOM.
86

4.3 Human Gait
Figure 4.18: Resultant total muscle forces of each considered muscle group in the lower
leg extremity model.
The stance phase ends with a pre-swing stage, when double support comes about
again, corresponding to the drop in the angle flexor group forces that coincides with
the TO. Now, the right limb starts its swing phase by progressively recruiting the
fibers of the Triceps surae group, throughout the initial and mid swing stages. At the
same time, the Tibialis anterior is slightly contracted, providing a dorsiflexion of the
foot that will prepare it for the following HS. In addition, by the time of the terminal
swing and similarly to the beginning of the gait analysis, the ankle flexors are recruited
resulting in the mentioned co-contraction. The knee flexors also contract, preparing the
musculoskeletal framework of the right side for a new double support period.
87

4. BIOMECHANICAL MODELS
4.4 Discussion
Contraction dynamics model
An important point of our implementation is to determine wether the inclusion of the
contraction dynamics models is justified or not. A static optimization analysis for the
human gait musculoskeletal system where such model is not included, was preformed.
This means that the force-length-velocity properties were ignored, i.e., each muscle m ′
available contractile force ˆ F m CE was equal to the maximum isometrical force F m 0
passive element force was considered for the whole analysis.
ˆF m′
CE = F m′
0
and no
(4.3)
F m′
P E = 0 (4.4)
The obtained muscle force results for the non-physiological situation proved to be
fairly similar to the ones acquired in Section 4.3. This is evidenced in the example de-
picted in Figure 4.19, where the total muscle force of the the tibialis anterior muscle was
calculated and compared with the results obtained for the same muscle in Section 4.3.
Figure 4.19: Comparison between the calculated total force of the tibialis anterior muscle
considering (Physiological case) and not considering (Non-physiological case) a contraction
dynamics model.
88

4.4 Discussion
The similarity in muscle force results was discussed in the work by Anderson and
Pandy [30], where it is stated that using the time-independent performance criterions
described in Section 3.5 in a normal gait situation, the contraction model can be ne-
gleted, since muscles preform below the intrinsic physiological limits [30]. In terms of
muscle activations, the results differ in more relevant manner. The value of the available
contractile force ˆ FCE for the physiological case will vary throughout the analysis, hav-
ing values different from F0 depending in the length and velocity of the muscle. Apart
from the effect of the passive element force in the results, there is an additional scale
factor in the activations of the tibialis anterior for the two cases, due to the mentioned
differences in the values of ˆ FCE. When the contraction dynamics model is considered,
the system becomes more conservative, since it accounts for a muscular system which
competence to develop muscle forces is not the same for different muscle lengths and
velocities, leading to situations were larger activations may be needed to preform a given
muscle force.
Figure 4.20: Comparison between the calculated activation of the tibialis anterior muscle
considering (Physiological case) and not considering (Non-physiological case) a contraction
dynamics model.
The non-inclusion of muscle contraction dynamics may not lead to significant result
differences when predicting muscle forces [30], but the model’s importance becomes
apparent when it is intended to acquire muscle activations or when the prescribed motion
gives rise to situations where muscles work near the physiological limits (either when
89

4. BIOMECHANICAL MODELS
large muscle forces are produced, or muscles work at lengths/velocities considerably
distant from L0/ ˙ L0).
Muscle fatigue model
The muscle fatigue model was successfully implemented in the multibody dynamics
routines. The whole formulation of this model proved to be well suited and easily
integrated in the latter, and compatible with the considered contraction model. In
addition, such a fatigue model gains an uplift in its applicability when included in a
multibody dynamics system, since its usage becomes more accessible for situation where
complex musculoskeletal systems with intricate kinematic patterns are to be analysed.
Nevertheless, it should be noted that, in the employed fatigue model, there is an
inaccuracy in the used fatigue parameters F , R, LD and LR. These were random values
with no correlation with any experimental results, leading to the acquisition of mus-
cle fatigue-recuperation patterns that does not correspond necessarily to physiological
values. In the analysis of the upper extremity model, the subject with those muscle
physiological and fatigue parameters could only bear the 150N load for less than 25
seconds. The analysis can not be associated to any specific real-life situation, however
by carrying out experimental procedures, in order to estimate factual values the muscle
contraction and fatigue parameters, a good prediction system should be formulated.
Another very important point regarding the implementation of the fatigue model is
the way the fatigue components state is calculated. The most accurate procedure for
the computation of the compartments progress from a time instant t to t + ∆t would be
to integrate of the differential equations described by Equations 2.14 to 2.16 in order to
correctly update those values. However, this approach conditions the efficiency of the
programming routines, since it would imply a significant increase of the analysis time
and the programming code intricacy. The used solution was the difference quotient,
i.e., for a state M with derivative (dM(t)/dt) for time instant t, its updated value for
t + ∆t is given as
M(t + ∆t) = M(t) + dM
(t) × ∆t. (4.5)
dt
This means that there is an error factor associated with the loss of precision for the
calculation of M in the time intervals between t and t+∆t. This error is fairly noticeable
90

4.4 Discussion
in the situations when the term dM/dt has a large value, eventually leading to a bigger
variation of M than it was supposed.
When operating the implemented multibody routines associated with the fatigue
model, the user is entitled to provide a value for a time variation thresholds ∆ttresh.
If the time step frequency of the multibody analysis is larger than 1/∆tthresh, then a
state updating pre-processing is made, running Equation 4.5 with ∆t = ∆ttresh through
consequent time intervals, in order to reduce the error factor.
Human gait muscle activations
The used computational language, FORTRAN, is characterised as a high-level language,
nevertheless having a low level when compared with other more modern computing lan-
guages. This leads to a difficult implementation of the routines and code comprehension,
but substantially efficient in terms of algebraic calculations and matrix handling. All
the executed analysis performed, even in reasonably complicated musculoskeletal sys-
tems, were computed in intervals of few seconds. In addition, the possibility of working
with double precision variables supplies an extra precision on the numerical procedures.
However, some erratic performances from the optimizer DNCONG were verified.
This lead to a difficult interaction and result acquisition, since this optimization routine
is very sensible and only allows five iterations to search for a minimum. Furthermore,
the algorithm searches for local minima. This means that a particular configuration of
muscle forces, is not necessarily the one that minimises globally the cost function, i.e.,
it might not be the muscle force combination that reduces the most the physiological
cost. This situation is evident in the results of Figure 4.7 where there is a co-contraction
of the antagonist pair formed by the ankle joint flexors and dorsiflexors. As mentioned
in Section 4.3, this co-contraction situation is intelligible in a physiological point-of-
view. Nevertheless, the optimizer’s objective is to minimise the cost function and a
antagonist pair co-contraction is certainly not the lowest energetic configuration.
Another verified drawback from the optimizer is related to the method response to
the cost function applied. There are some basic cost functions, such as Equation 3.54,
that lead to situations where the optimizer could not find any solution for the system,
terminating the analysis before the final time step. This may suggest that some cost
functions exhibit a bolder physiological nature, when compared to others. The cost
91

4. BIOMECHANICAL MODELS
function in Equation 3.56 is indicated as the one that better relates to what is cogitated
by the CNS.
Considering the pattern of muscle activations obtained in Figure 4.8, the author
concludes that another type of values should be obtained. Empirically, for a simple
gait cycle, it is not expected to have some of the main muscles in our musculoskeletal
structure developing a full activation condition. Some studies [17] indicate that, even
for actions were maximum force is required, the muscle activations of the used subjects
in those studies never came close to the maximum value. This means that, in voluntary
contractions, muscle forces are never close to the maximum isometric force F0. Two pos-
sible reasons for the calculation of these high values include the usage of an out-of-date
F0 database and the misplace of the muscles’ points, leading to respective diminutive
moments arm (with a smaller moment arm, a muscle need to develop larger forces in
order to produce a certain joint torque). These points may invalidate the relationship
that exists between the contractile force F m CE and the muscle activation am terms of our
muscle model in Section 2.2.2.
Another relevant point, is the fact that not all muscles existent in the lower limb were
included. The absence of those muscles can make all the difference, even if those muscle
are not recruited. A muscle that has 0 (zero) activation throughout the whole analysis,
can still make a difference by exerting a forces related to the passive contribution of our
model. Over that, the hip muscles where not considered. These muscles play a major
role in the gait cycle, and its addition to the model is likely to change to some extent
the obtained activation patterns and muscle forces.
In addition to all the mentioned topics that may justify unfeasible values is the fact
that no tendon dynamics were considered. The muscle physiological parameters used in
Tables 4.7 were taken from a database available at Reference [76] that were developed
for multibody models that consider the dynamics of tendon structures, with variations
of length and an associated tendon tension. For instance, the upper arm model muscle
parameters, taken from the work of Holzbaur et. al. [72] and used in routines developed
by Delp et. al. [71], where the tendon dynamics consider length variation in the triceps
long head ranging from 147.59mm to 147.71mm that result in a tendon force variation
from 765N to 798N. The biceps brachii short head has larger tendon force variations
(around 295N) for length variations of 2mm. This means that an important accuracy
level is lost from the contribution of tendon dynamics.
92

Chapter 5
Conclusions and Future
Developments
5.1 Conclusions
The formulation of multibody system dynamics constitutes undoubtedly a powerful
method to numerically evaluate three-dimensional mechanical systems that undergo
large displacements and rotations, and are acted upon by external forces. Every single
tested biomechanical model involved muscle tissue modelling and was processed in a
personal computer laptop, with each execution never passing 10s of analysis time.
The developed mathematical tissue models regarding the muscle structure, suc-
cessfully simulated its physiological behaviour, accounting for the inherent voluntary
contractile properties and the dynamics of muscle fatigue. The first model, concerning
contraction dynamics, based on the macroscopic force-length-velocity relationships, has
proved by its considerable applicability that simulates with precision the process of force
production of muscles. Nevertheless, it was proven in Section 4.4 and in the work by
Anderson and Pandy [30] that the non-inclusion of muscle contraction dynamics, when
time-independent performance criterions are used, may not lead to significant result
differences when predicting muscle forces, when muscles are required to work below the
physiological limits. Such type of model turns to be fundamental when it is intended
to acquire muscle activations or when the physiological restrictions of muscles are to be
considered.
93

5. CONCLUSIONS AND FUTURE DEVELOPMENTS
The implementation of muscle fatigue models in multibody dynamics routines hap-
pen to be successful. The whole formulation of this model proved to be well suited and
easily integrated in the multibody system routines, and compatible with the considered
contraction model. The study of muscle fatigue using versatile mathematical models
in multibody system dynamics methodologies, gains a special motivation due to the
fact that these can be easily applied in complex musculoskeletal systems with intricate
kinematic patterns are to be analysed. In this work, the usage of the difference quotient
for the calculation of the MR, MA and MF compartments state induced the presence
of an error factor. This is not relevant when the rate of change of the compartments’
state dM/dt is relatively small compared with the analysis time step. In addition, the
small number of parameters of the model facilitates the fitting of experimental data to
the fatigue model. This same model may consider muscle fiber recruitment hierarchy,
allowing the same model to precisely define muscles with different characteristics and
fiber constitution.
Our inverse dynamics approach consisted in the adaptation of the equations of mo-
tion to the muscle formulation using the Newton’s method [1, 7]. With this methodol-
ogy, it is possible to calculate in the same analysis rigid body internal reactions, joint
reactions, and muscle and activations, with the nuisance of increasing the complexity
of its implementation and solving process. The developed formulation was successfully
included in the multibody routines, has observed in the results obtained in Chapter 4.
The optimization procedure, described in Section 3.5, proved to solve the developed
models in an accurate and efficient manner. The used optimizer DNCONG revealed
to be very sensible, searching solely for local minima. The detection of local minima
can lead to solutions that do not correspond to the configuration of muscle contractions
with minimum cost.
Static optimization itself only accounts for instantaneous performance criteria, that
may not simulate in the best way the behaviour of the CNS when choosing a muscle
activation set. According to Ackermann [2], these instantaneous functions are unable
to accurately describe the key performance criterion: the total energy expenditure. In
addition, static optimization seams to operate with a single muscle even when, for a
specific motion, has multiple muscles available able to exert the desired torque. These
points suggest that other approaches, such as dynamic optimization [32], EMG-driven
94

5.2 Future Developments
cost functions [17] or the methodologies proposed by Ackermann [2], should be consid-
ered for the calculation of individual muscle forces and activations. Static optimization
should be the favoured approach when it is intended to calculate the force exerted by
muscle groups.
5.2 Future Developments
Bearing in mind the discussed points in the previous chapters, there are some modules
in this work that can be refined and several others that are viable to be added. Muscle
modelling has a wide range of applications and formulations, but due to the limited
time frame available for this thesis, some of these were left aside.
Fatigue model and fatigue parameters
Firstly, the used fatigue model is relatively straightforward one, and such characteristic
was essential for its choice. No studies were found to date that included muscle fatigue
models in a multibody dynamics system, so the intention of keeping it simple is un-
derstandable. Other similar fatigue models coupled with experimental trials, such as
the one in Ma et. al. [77], should be considered and the parameters values analysed.
An engrossing procedure in the following of this fatigue model, would be to do some
experimental trials, in order to test the correlation of the fatigue parameters with actual
physiological situations, and infer the model’s validation.
Tendon dynamics model
One of the models lacking in this work, is the tendon dynamics model. Muscles and
tendons exist combined in nature, and tendons play a vital role in the foundations
of musculoskeletal biomechanics. Tendon models convey an additional computational
charge, and is far more complex that muscle modeling [42]. Nevertheless, as mentioned
in Section 4.4 tendons play a major role in musculotendon forces, since these have
associated considerable tensile force variations for small length differences. It would be
important to consider such a component, in order to accurately obtain proper muscle
fiber forces and activations.
95

5. CONCLUSIONS AND FUTURE DEVELOPMENTS
Activation dynamics model
An important additional functionality would be the determination of a neural signal u(t)
by means of an activation dynamics model. As introduced in Section 2.2.1, the time
lag between a neural signal u(t) and its corresponding muscle activation a(t) is ruled
by a first order ordinary diferential equation. The activation dynamics must then be
inverted [2], with the purpose of infering the value of the control signal, by calculating
˙a m (t), and solve Equation 2.1 in order to u(t). Additionally, the optimization problem
must consider the bounds of this signal, hence the following restriction
0 ≤ u m ≤ 1 , for m = 1, . . . , nm. (5.1)
Ackermann [2] refers to this formulation as Modified Static optimization (MSO). In
scope of this work, this method could be implemented by adding nonlinear constraints
to the individual activations a m , as additional upper and lower bounds for a specific
time instant, to ensure that the calculated muscle activations are compatible with the
activation and contraction dynamics. There are some drawbacks however. Similarly to
what happened in the fatigue model, the way the differential equation is worked out
plays a vital role to the accuracy of the model. A numerical integration procedure could
increase the order of the computational times considerably. On top of that, Equation 5.1
represent an additional constraint to our optimization problem, increasing the sensitivity
of the optimizer.
A solid global optimizer
The prime fragility of the developed FORTRAN routines is most probably the used op-
timizer. DNCONG is a very fast, however very sensible algorithm. When DNCONG is
close to the global solution, it reaches it rapidly and with sharp precision. Nevertheless,
it is quite demanding to find the region of the global minima. DNCONG calculates a
local-minima and, when far from any solution, the algorithm simply cannot converge.
A possible future development, regarding the precision of the optimizer, would be to
use a genetic algorithm for the computation of an initial guess and give this guess as
an input to DNCONG, letting DNCONG converge to the global minima. Another ap-
proach, and this one to a greater extent similar to physiological criteria, would be to
use electromyography signals to drive the obtained the muscle activation results, i.e.,
96

5.2 Future Developments
to use and EMG-driven models to estimate muscle forces. Several authors used this
approach [17, 78–80].
97

5. CONCLUSIONS AND FUTURE DEVELOPMENTS
98

Appendix A
Apollo – Hill-type muscles manual
In the Apollo software [1], Hill-type muscles are defined in both the .mdl and .sml
files. In the .mdl file, the user specifies the geometrical and physiological aspects of the
different muscles. In the .sml file, the user provides information regarding the type of
analysis that is intended to be done. In this appendix, templates of the relevant parts
of these files will be made inside frame boxes. The compulsory keywords are written
in monospace Courier New and the numeral parameters to be provided are shown in
emphasised text.
A.1 MDL File
The first step to include Hill-type muscles in an Apollo analysis, is to state the number
of muscles that will be modelled N total
muscle , in the 20th parameter of the generic *MAIN
PARAMETERS keyword.
A positive value of N total
muscle
*MAIN PARAMETERS
· · · N total
muscle
will inform the software that muscle information data
will be provided subsequently in the same file. The designated keyword that encloses
the information about the intrinsic properties of muscle will be *MUSCLE PARAMETERS.
99

A. APOLLO – HILL-TYPE MUSCLES MANUAL
*MUSCLE PARAMETERS
FATIGUE
OptF T
MUSCLE=Nmuscle
L0
˙L0
˙Ltendon
F0
αpenn
amin
amax
Nmp
P 1 number P 1 ξ P 1 η P 1 z P 1 body
P 2 number P 2 ξ P 2 η P 2 ζ P 2 body
· · ·
P Nmp
number
Nmp
Pξ P Nmp
η
P Nmp
ζ
[F R LD LR]
A brief description of each parameter is made:
P Nmp
body
OptF T The user starts by indicating whether muscle fatigue is to be considered or not.
OptF T = 0 Muscle fatigue is not considererd in the analysis.
OptF T = 1 Muscle fatigue is considererd in the analysis. In this case, the fatigue
parameters F , R, LD and LR must be provided in the last line of the *MUSCLE
PARAMETERS section.
L0 The resting length.
˙L0 The maximum contractile velocity.
˙Ltendon The tendon length, assumed to be constant.
F0 Maximum contractile force.
αpenn Pennation angle.
100

amin Muscle activation lower bound (equal to 0, in theory)
amax Muscle activation upper bound (equal to 1, in theory)
A.2 Simulation file
Nmp Number of points that define the muscle’s geometry. For every point n, the fol-
lowing five parameters must be provided:
P n number
P n body
Index number of point n.
Number of the body where point n is attached.
P n x Coordinate ξ in the local reference frame oξηζ.
P n η Coordinate η in the local reference frame oξηζ.
P n ζ
Coordinate ζ in the local reference frame oξηζ.
A.2 Simulation file
In the simulation file, different entries are expected for the type of analysis to be made.
In the keyword *MUSCLE ANALYSIS TYPE, the user defines what is meant to be calcu-
lated.
*MUSCLE ANALYSIS TYPE
type
The parameter type has two possible regular expressions:
MOTION When it is intended to calculate the motion resultant from the prescribed acti-
vations.
ACTIVATIONS When the user aims to calculate the muscle activations that account for
the prescribed motion.
If the indicated analysis type is MOTION, the software solves the equations of motion
in order to provide the kinematic response to a patter of muscle activation in time.
The user must then provide the file name with the information for the activations of
the different muscles in the model. This is made employing the *ACTIVATIONS FILE
keyword.
*ACTIVATIONS FILE
[filename].act
101

A. APOLLO – HILL-TYPE MUSCLES MANUAL
The indicated file must be a tab-separated text file, where the information is disposed
in the following manner:
nt
t0
t1
· · ·
tn
N total
muscle
actm1 t0
actm1 t1
act m1
tn
actm2 t0 · · · actm t0
actm2 t1 · · · actm t1
act m2
tn · · · act m tn
Where the value act m tn corresponds to the activation of muscle m for time instant tn. nt
is the number of provided time instants and N total
muscle is the number of muscles that are
displayed in the activation file.
If the user selects ACTIVATIONS as the analysis type, the muscle activations are
calculated, regarding the kinematic data available in the driver files [1]. The user is
entitled to indicate as follows the kinematic drivers to be labeled as Lagrange multipliers
λ ∗ , described in Section 3.5:
*BOUNDED DRIVERS
Nbd
εbd
dr1 dr2 · · · drn
where Nbd drivers, identified with the numbers indicated in the list of the third line, are
bounded by the optimizer, restricting their values to be inside the interval [−εbd; εbd].
102

Appendix B
MHILL Data Visualizer manual
An Apollo analysis where muscles are considered produces an output file with exten-
sion .mh for each modelled muscle in a folder MHILL_BIN. This output files store data
related to each muscle for different time instants of the analysis, data such as Length,
Velocity, Activation, Contractile Force, etc. To better analyse this data, a small appli-
cation developed in the GUI (Graphical User Interface) of MATLAB R○ . The general
aspect of this software is shown in Figure B.1.
Figure B.1: General aspect of the MHILL Data Visualizer.
103

B. MHILL DATA VISUALIZER MANUAL
The main window has two chart areas (identified in Figure B.1 as A and B) and
three operation boxes. Chart area A displays a 3D representation of the considered
muscles, and B is the chart area where the evolution of muscle data in time is shown.
The different operation sections are depicted in Figure B.2. The file box, where the
user is able to manage the muscle data files to be processed, is displayed in Figure B.2(a).
Two buttons are available, in order to include and remove the muscle data files to be
displayed, respectively the Add Files and the Delete Files buttons. The included files
are displayed in the list box bellow.
(a) File box. (b) Data box.
(c) Time box.
Figure B.2: Operation boxes of the MHILL Data Visualizer.
The data box, where the options regarding the manipulation of the data may be
found, is illustrated in Figure B.2(b). A pop-up menu is available to chose which data
is to be displayed from a set of options: Length, Velocity, Activation, Total Force, Force
CE/Activation (identified in this thesis as the available contractile force ˆ FCE), Force
CE, Force PE). In order to plot the data, the user uses the Plot! the button. To save
the plot or to export all the data to an Microsoft R○ Excel file, two options are available,
respectively the Save Plot and the Export to Excel buttons. The main window may be
cleared, to its initial form (Figure B.1), by pressing the Reset button.
In the time box, it is possible to select the time instant to be analysed, either by set-
ting its value in seconds in a text box or by means of a slider, as shown in Figure B.2(c).
104

Exemple of operation
The first step in the usage of this tool, is the inclusion of the muscle data files, by
selecting the Add Files button of Figure B.2(a). The user selects the muscle files to be
included in the analysis for the folder system, as in Figure B.3.
Figure B.3: Files addition for the folder system.
By clicking Push!, the results are plotted in both the chart areas. In Figure B.4,
the spatial representation and the chart of the Total Force curves for the muscles in the
lower limb example of Section 4.3 is shown.
Figure B.4: Example of data plots.
105

B. MHILL DATA VISUALIZER MANUAL
Using the pop-up menu of Figure B.2(b), it is possible to chose a different data to
be plotter. In the example of Figure B.5, the activations of the muscles are selected. By
pressing Plot! after this selection, the muscle activations are exhibited (Figure B.6).
Figure B.5: Selection of muscle data to be plotted.
Figure B.6: Plotting of muscle activations.
If the data chart are looks messy, the user is able to remove some of the represented
muscles, by deleting them from the file box and pressing Plot!. It is possible to make a
selection of the files to be excluded when using the Delete Files option (Figure B.7).
106

Figure B.7: Removing some of the included files.
To use the time box of Figure B.2(c), the user may opt to write the desired time
value in the text box or by using the slider. In Figure B.8(a) is noticeable that the new
time instant will lead to an update in the geometrical representation. This time instant
will be identified in the data chart by a coloured plot of the instants between t = 0 and
the selected time instant, as shown in Figure B.8(b).
(a) Using the time slider. (b) The plotting colors identify in the chart, the selected time in-
stant.
Figure B.8: Time instant selection.
107

B. MHILL DATA VISUALIZER MANUAL
108

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