Development of a Hill-Type Muscle Model with Fatigue for the Calculation of the Redundant Muscle Forces Using Multibody Dynamics

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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 ii 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 modelo 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 articular 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 superior com sete músculos é considerado para evidenciar a aplicabilidade de um modelo de fadiga muscular num sistema multicorpo. Um segundo modelo, 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 evaluate 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 fatigue 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. iv 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 tendons [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 filaments 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 Ca2+ [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 Fm to pointo p, belonging to the rigid body defined by points i and j and vectors u and v [1]...... 46 m 3.3 The different representations of a generic muscle force Fp ...... 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 mX and mY for Hill-type muscle model veriﬁcation purposes...... 64 4.2 Activations pattern for muscles mX and mY ...... 65 4.3 The state of the mechanical model for diﬀerent time instants when muscles mX and mY are activated as in Figure 4.2...... 66 4.4 Muscle mX 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 m 4.7 Resultant muscle forces of the contractile element FCE of the biomechanical 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 - metatarsophalangeal 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 results [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 τfall Time constant of muscle activation rise of a(t) fall of a(t)

m m F Muscle force FCE Muscle contractile element force m ˆm FPE Muscle passive element force FCE Muscle available contractile element force

m m F0 Muscle maximum isometric force FL Muscle contractile element force- length relationship F m Muscle contractile element force- Lm Muscle length L˙ velocity relationship m ˙ m L0 Muscle length change rate L Muscle rate of length change m ˙ m v˙ Muscle speed L0 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 com- MR Fatigue model resting MU partment compartment ˆf RC Muscle residual capacity FCE Fatigued muscle available contractile force

C(t) Fatigue model muscle activa- LD Muscle force development factor tion–deactivation driving controller LR muscle force relaxation factor TL 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 P∗ Virtual power produced by the ex- equation ternal 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

m ud Muscle direction d Fp 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

F F ge Rigid body generalised representa- g Whole system generalised repre- tion of force F sentation of force F

F m F m gCE Whole system generalised repre- gPE Whole system generalised representation of the contractile element sentation of the passive element force force

F m ext ˆgCE Whole system generalised repre- g Generalised forces excluding mus- sentation of the available contrac- cle forces tile element force χ Set of generalised available con- a Muscle activations vector tractile element forces the considered muscles

x Optimization problem control vari- feq Optimization problem equality ables constraints

∗ F0 Optimization problem cost func- λ Lagrange multipliers associated tion with the kinematic drivers

λR Lagrange multiplier without con- ~g Gravity force straints in the optimization problem

xiv 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 mathematical 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 computational applications. Multibody dynamics systems is one of the mechanical formulations 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 simulate 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 contraction 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 system. 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 system (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 invasive 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 dynamics 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 mechanical 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 optimization: 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 compressive 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 computational 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 dynamics 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 mathematical 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 dynamics 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 theoretical models introduced a collection of physiological parameters and fatigue dynamic laws [57, 58], and were only practicable for single muscle analysis regarding its complexity 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 program, 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 contraction 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 contraction, 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 examples of success of the introduction of a fatigue model in multibody routines, showing the obtained results. Furthermore, a gait cycle is analysed from an inverse 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 conclusions, 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 exerted 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 prescribe 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 tendon, 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 (Ca2+) are not present. This will impede contraction from happening when not desired. The whole overlapped structure is supported by a framework 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 Ca2+. 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 coupling – 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 potential 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 Ca2+ 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 Ca2+ [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 Ca2+. The sarcomere 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 Ca2+ 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 sarcomere. 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 Ca2+, 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 Ca2+ 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 dynamics 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 macroscopic 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 usually 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) am(t) F m(t) - Activation - Contraction - Dynamics Dynamics

Figure 2.10: Scheme of muscle tissue dynamics, with a series model of Activation Dy- namics 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 corresponding 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: ( (u(t) − a(t))(c u(t) + c ) u(t) ≥ a(t) a˙(t) = 1 2 (2.1) (u(t) − a(t))c2 u(t) < a(t) where 1 1 c1 = − (2.2) τrise τfall 1 c2 = (2.3) τfall

22 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:

m m m F = FCE + FPE (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 (am(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 m m m m m length L is comprised between 0.5L0 < L < 1.5L0 , where L0 corresponds to the m muscle fiber resting length(or relaxed length). The muscle resting length L0 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]. m m The force developed at total muscle activation and fiber length L = L0 is designated m maximum isometric force or optimal force F0 [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 m muscle contribution, overtaking the value of F0 . 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/cm2) [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 contractile 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 m F0 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 shortening 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

m m m (F + a)(b + v ) = (F0 + a) b (2.5) where vm 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 = −vm), 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 ˙ m ˙ m ˙ m change is [−L0 , 0.5L0 ], where L0 is the maximum contractile velocity, normalised as ˙ m m L0 ≈ 10L0 [1]. 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 Lm, 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(Lm(t)) and velocity F m(Lm(t)). This L L˙

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 2.2 Dynamics of the Muscle Tissue factor is then multiplied by a(t) as indicated in Equation 2.6.

F m(Lm(t))F m(L˙ m(t)) m m ˙ m L L˙ FCE(a(t),L (t), L (t)) = m a(t) (2.6) F0 The factor to be multiplied by the muscle activation am(t) corresponds to the avail- ˆm able muscle contractile force which is referred hereafter as FCE and indicated in Equa- tion 2.7. F m(Lm(t))F m(L˙ m(t)) ˆm L L˙ FCE = m . (2.7) F0 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 multibody methodology (Chapter 3). Equation 2.4 can be rewritten considering the activation term as an independent one:

m m ˆm m F = FPE + FCEa . (2.8)

Analytical expressions

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.

" # » „ m «–4 » „ m «–2 9 L (t) 19 1 9 L (t) 19 − − 4 m − 20 − 4 − 4 m − 20 m m m L0 L0 FL (L (t)) = F0 e (2.9)  0 L˙ m(t) < −L˙ m  0  m ˙ m !  F0 L (t) m ˙ m ˙ m ˙ m F m(L˙ m(t)) = − arctan −5 + F0 −L0 ≤ L (t) ≤ 0.2L0 L˙ arctan (5) L˙ m  0  πF m − 0 + F m 0.2L˙ m < L˙ m(t)  4 arctan (5) 0 0 (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 Lm(t) and the mathematical approximation used is

29 2. MUSCULOSKELETAL SYSTEM MODELLING

 m m 0 L (t) < L0  m m m  F0 m m 3 m m m FPE(L (t)) = 8 m3 (L (t) − L0 ) L0 ≤ L (t) ≤ 1.63L0 . (2.11)  L0  m m m 2F0 1.63L0 < L (t) 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 Dynamics of the Muscle Tissue

2.2.3 Muscle Fatigue

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 muscle 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 parameters: 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 am 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 ˆf ˆm force FCE will be the product of the reposed muscle available contractile force FCE with RC and the muscle activation a(t). Hence:

ˆf ˆm FCE = RC(t) × FCE × 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:

dM R = −C(t) + R × M (t) (2.14) dt F dM A = C(t) − F × M (t) (2.15) dt A dM F = F × M (t) − R × M (t) (2.16) dt A F 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 TL, i.e., the force that the muscle is required to exert. In this work the target load is referred as the instantaneous solicited m contractile muscle force FCE, since Xia and Law assume that the neuromuscular system m can produce TL [59], i.e., this term, just like FCE, is limited by the available contractile ˆm force FCE. 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).

 L × (F m − M × Fˆm ) F m ≤ M Fˆm  R CE A CE A CE m ˆm ˆm m ˆm C(t) = LD × (FCE − MA × FCE) MAFCE < F ≤ (MA + MR)FCE (2.17)  ˆm ˆm m LD × MR × FCE (MA + MR)FCE < F

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 percentage 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 contraction behaviour. Three types of skeletal muscle tissue contraction dynamics were described: activation, 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 instantaneous 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 displacements 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) coordinates 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 aspects 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 system 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 coordinates q. For a system with n existing points and m existing vectors, the generalised coordinates are described as shown below:

T q = { xP1 yP1 zP1 ... xPn yPn zPn xV1 yV1 zV1 ... xVn yVn zVn } (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 equations 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 multibody 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:

T T Φ(q, t) = { Φ1(q) ... Φns(q) Φns+1(q, t) ... Φns+nr(q, t) } = 0 (3.2)

40 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 iteration i. Matrix Φq is the Jacobian of Φ in order to q, deﬁned as  ∂Φ ∂Φ  1 ... 1  ∂q1 ∂qnc  ∂Φi  . . .  Φq(q) = =  . .. .  . (3.4) ∂q   j ∂Φ ∂Φ  nh ... nh ∂q1 ∂qnc where nc and nh are the already referred number of coordinates and number of holonomic 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:

T T Φq(qi) Φq(qi)∆qi = −Φq(qi) 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:

dΦ(q, t) ∂Φ(q, t) ∂Φ(q, t) ∂q Φ˙ (q, ˙q, t) = = + = 0 (3.9) dt ∂t ∂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 3.2 Equations of motion where ∂Φ(q, t) ν(t) = − (3.11) ∂t A similar procedure is taken for the calculation of the accelerations ¨q. The derivation of Equation 3.9 in time yields:

dΦ˙ (q, ˙q, t) Φ¨(q, ˙q, ¨q, t) = = Φ ¨q + (Φ ˙q) ˙q − ν = 0, (3.12) dt q q q t that can be rewritten as

Φq¨q = γ (3.13) where

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 kinematic 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 associated kinematic constraint, yielding nh equations to the equations of motion of the mechanical system [1]. The internal constraint forces vector gΦ is given by:

T gΦ = Φq λ (3.16) where λ is known as the vector of Lagrange multipliers that, from a physical perspective, 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

∗ ∗T T P = q˙ (M¨q − g + Φq 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

T M¨q − g + Φq 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 muscle 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 ud0 and applied in a point p will be expressed as

m m Fp = ud0 F . (3.19)

This work has a formulation that uses forces for the whole muscle structures implementation, 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 Fm 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

 r   i   r  r = (1 − c )I c I c I c I j (3.21) p 3×1 1 3 1 3 2 3 3 3 3×12 u    v  12×1 or simply:

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 1 of the generalised coordinates used to describe that element . Since Cp depends exclusively 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:

  c1 0 0 0 0 0   0 (rp − ri) = rij u v c2 = X c (3.23)  c3  0 0 0 0 with X = rij u v . This matrix always has an inverse [1] and therefore Equa- tion 3.23 can be solved in terms of c

0−1 0 0 c = X (rp − ri) (3.24) 1It should be noted that when rigid bodies with complex geometry are considered, a second transformation 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

m Reaching this point, we are able to express the generic muscle force Fp in terms of m Fp an equivalent term ge 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 m m Fp Fp and its generalised counterpart ge should be the same:

m T m T Fp δW = δrp Fp = δqe ge (3.25)

Using Equation 3.22 in the latter, yields that

m T T m T Fp δW = δqe Cp Fp = δqe ge (3.26) and relating with Equation 3.19, the expression of the concentrated muscle force expressed in the generalised coordinates that deﬁne the rigid body will be:

F m T m T m ge = Cp Fp = Cp ud0 F . (3.27)

This expression plays a vital role in this muscle modelling work, since it allows us to m m Fp transform a Cartesian representation of forces Fp in the generalised configuration ge . Nevertheless the latter corresponds to a physical quantity that requires additional ma- m nipulation, since Fp needs to be expressed in terms of the generalised muscle force m vector of the whole system, the column vector gFp of size nc. In his work, Silva [1] developed natural-coordinates-specific computational routines whose function is to as- m Fp F m semble ge into g 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.

Rigid body Whole system Vectorial generalised coordinates generalised coordinates

F m m m p Fp Fp ge g

m Figure 3.3: The diﬀerent representations of a generic muscle force Fp .

m Fp Since each generalised force vector ge is speciﬁc for a certain body (the body containing the point where the force is applied), then a particular muscle framed in nb m Fp bodies will have nb diﬀerent ge 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- F m ferent bodies I, II and III. Therefore three independent ge force vectors (subscript I II III p will be dropped from this point) must be considered: ge , ge and ge . Each one of these vectors will enclose information regarding the forces that are exerted in the F m respectively attached via-points as depicted in Figure 3.4. For instance, gI will be de- m m m F m ﬁned by forces F1 , F2 and F3 , while gIII will only be determined by the contribution m m of F8 . In mathematical terms and assuming that the muscle force F has a constant magnitude in all muscle extension, it comes that for body I

m m m I F1 F2 F3 ge = ge + ge + ge (3.28) T m T m T m = Co uov1 F + Cv1 uv1oF + Cv1 uv1v2 F (3.29) and knowing that the direction vectors deﬁned by any two points i and j will have opposite directions

uij = −uji (3.30) then we can simplify Equation 3.29 and state that, for all ge:

I T T T m ge = (Co − Cv1 )uov1 + Cv1 uv1v2 F (3.31) II T T T T m ge = −Cv2 uv1v2 + (Cv2 − Cv3 )uv2v3 + Cv3 uv3i F (3.32) III T m ge = −Ci uv3i F (3.33)

Having a numerical representation of all the forces of the muscle in Figure 3.4, it is I II III now possible to assemble ge , ge and ge – 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 gI , gII and gIII . 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 is given as F m g = gI + gII + gIII (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 F m F m g = gPE + gCE (3.35)

F m F m where gCE and gPE 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 F m F m g = gPE + ˆgCE a (3.36)

F m where ˆgCE corresponds to the generalised force vector of the maximum available con- ˆm tractile force, i.e., the generalised representation of FCE. 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

T Φq 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 T orthogonal to the null-space of the matrix Φq [1], which mathematically corresponds to T ∗ λ = Φq λ (3.38) where λ∗ will enclose the unique solution for λ, when replacing Equation 3.38 in Equa- T tion 3.37, since Φq 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 developed 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 methodology 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 motion 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 representation. Considering the existence of nm muscles in our model, g can be expressed as

g = gext + gM1 + gM2 + ... + gMnm (3.39) where gext 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

ext M1 M1 M1 Mnm Mnm Mnm g = g + ˆgCEa + gPE + ... + ˆgCE a + gPE . (3.40)

Replacing this equation in Equation 3.18, it comes that

ext M1 M1 M1 Mnm Mnm Mnm T M¨q − (g + ˆgCEa + gPE + ... + ˆgCE a + gPE ) + Φq λ = 0. (3.41)

Now, the aim here is to compute the magnitude of the internal constraint forces λ and muscle activations aM1 ...aMnm . Reshaping our system to a suitable conﬁguration,

52 3.4 Inverse Dynamics

Equation 3.41 suﬀers a rearrangement:

T M1 M1 Mnm Mnm ext M1 Mnm Φq λ − (ˆgCEa + ... + ˆgCE a ) = g + gPE + ... + gPE − M¨q. (3.42)

m Since the inertial term M¨q, the passive contribution in muscle forces gPE and the remaining external forces gext 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:

 λ    h i  aM1  ΦT −ˆgM1 ... −ˆgMnm = gext + gM1 + ... + gMnm − M¨q (3.43) q CE CE ··· PE PE    aMnm  or in a more compact fashion

λ ΦT −χT = gext + g − M¨q (3.44) q a PE 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):

 M1  ˆgCE χ =  ···  (3.45) Mnm ˆgCE  aM1  a =  ···  (3.46) aMnm

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 muscle 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 am. 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 several 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 methods 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:

λ x = (3.47) a

54 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.   f1   λ . T T ext M1 Mnm feq = . = Φq −χ + M¨q − (g + gPE + ... + gPE ) = 0 (3.48)   a  fnc 

The gradient of feq will be required for the optimization routines, described in the development of this section:

T T ∇fnλo = Φq −χ (3.49) a 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 multipliers 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 ≤ am ≤ 1 , for m = 1, . . . , nm (3.51)

The remaining existent Lagrange multipliers in the control variables, identiﬁed as λR, will not have any special limitations and therefore can have any value:

− ∞ ≤ λR ≤ +∞ (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 am for a constant prescribed angle of 90o 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 optimization 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- (b) Control variables x (c) Control variables x cle m and exertion of load P in an ex- before the optimization after the optimization tremity of a body with length l. procedure. procedure.

Figure 3.5: Used optimization methodology.

56 3.5 Optimization

Ultimately, the general optimization problem can be then stated as Given : x Minimise : F0(x)  f (x) = 0  eq  (3.53)  m 0 ≤ a ≤ 1 Subject to : ∗ −ε ≤ λ ≤ +ε   −∞ ≤ λR ≤ +∞ 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 muscle recruitment and muscle activation distribution, when required to provoke a certain posture and articular motion. There is an interminable number of situations and parameters 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: nm X m 2 F0 = (a ) (3.54) m=1 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:

nm X m 2 F0 = (FCE) (3.55) m=1

This function usage aims to minimise energy consumption amounts [1, 37].

3. Sum of the cube of the individual average muscle stresses:

nm X m 3 F0 = (σCE) (3.56) m=1

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 nonlinear 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 am. 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 am 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 2nd 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 MΦT ¨q g q = . (3.57) Φq 0 λ γ 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 dynamics 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 2nd 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 ˙y = (3.60) t ¨q

59 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 y = (3.61) t+∆t ˙q 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 multibody 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 external 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 multibody 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 Lm and velocity L˙ m, and confirm if the output values fit the model’s equations. The conceived mechanical model has no physiological 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 mX was fully activated while the rigid bodies vary their relative positions in time. This way mX spans a certain 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 activations are prescribed. The relation between muscle mX ’s length and the correspondent output contractile force, for a constant activation aX (t) = 1, is examined. In order to control the system’s movement and therefore the muscle’s length, a second muscle mY was included and its variable activation aY (t) pattern directs angle α while body 1 keeps its vertical orientation still. Only the active contribution of the muscle model X m ˙ m FCE(L (t), L (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 mX and mY for Hill-type muscle model veriﬁcation purposes.

Analysing Figure 4.3, it should be noted that the muscle in question, muscle mX , 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 mX and mY .

description, the system should go through a set of muscle force values that travel in the length-velocity (ll˙) 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. X m ˙ m ˙ The contractile force FCE(L (t), L (t), 1) in the ll space, appears as a cardioid laying in the muscle model active force surface for a fully activated model with the considered 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 mX and mY are activated as in Figure 4.2.

66 4.2 Muscle Fatigue

Figure 4.4: Muscle mX 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 developed 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

2πt P = F 0.25 + 0.15g [N] (4.1) 0 50 where g(x) is a square wave function, given by: ( 1 sin(x) ≥ 0 g(x) = (4.2) −1 sin(x) < 0

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 am 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 am 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 = −9.81~ez [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 90o, 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 3 0.00 0.00 0.181479 0.00 0.00 -0.118521 and hand

70 4.2 Muscle Fatigue

Table 4.3: Inertial data for rigid bodies considered in the upper extremity model.

Inertial Moments [Kg.m2]

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 m of the contractile element FCE for the considered muscles are illustrated in Figure 4.7.

m Figure 4.7: Resultant muscle forces of the contractile element FCE 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 Wash- ington. 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) o F0[N] L0[N] LT [N] α[ ] bn ξ[m] η[m] ζ[m] 1 -0.0392 0.0221 0.0035 1 -0.0289 0.0136 0.0139 2 0.0213 -0.0103 0.1984 2 0.0238 -0.0120 0.1754 624.3 0.1157 0.2723 0 2 0.0135 -0.0014 0.1522 2 0.0107 0.0017 0.1031 2 0.0170 -0.0002 0.0593 2 0.0228 0.0063 0.0051 3 0.0075 -0.0218 0.1331 biceps brachii short head (BICshort) o F0[N] L0[N] LT [N] α[ ] bn ξ[m] η[m] ζ[m]

1 0.0047 0.0353 -0.0123 1 -0.0070 0.0249 -0.0400 2 0.0112 0.0110 0.1047 435.56 0.1321 0.1923 0 2 0.0170 0.0108 0.0593 2 0.0228 0.0063 0.0051 3 0.0075 -0.0218 0.1331

brachialis (BRA) o F0[N] L0[N] LT [N] α[ ] bn ξ[m] η[m] ζ[m]

2 0.0068 0.0036 0.0066 987.26 0.0858 0.0535 0 3 -0.0032 -0.0009 0.1576

72 4.2 Muscle Fatigue

(Table 4.4 continuation) brachioradialis (BRD) o F0[N] L0[N] LT [N] α[ ] bn ξ[m] η[m] ζ[m]

2 -0.0098 -0.00223 -0.019134 261.33 0.1726 0.133 0 3 0.03577 -0.02315 0.054059 3 0.0419 -0.0224 -0.039521

triceps brachii lateral head (TRIlat) o F0[N] L0[N] LT [N] α[ ] bn ξ[m] η[m] ζ[m] 2 -0.00599 -0.00428 0.054036 2 -0.02344 -0.00928 0.035216 261.33 0.1726 0.133 0 2 -0.03184 0.01217 -0.045874 2 -0.01743 0.01208 -0.087074 3 -0.0219 0.00078 0.191939 triceps brachii long head (TRIlong) o F0[N] L0[N] LT [N] α[ ] bn ξ[m] η[m] ζ[m] 1 -0.05365 0.02277 -0.01373 2 -0.02714 0.00664 0.066086 798.52 0.134 0.143 0.21 2 -0.03184 0.01217 -0.045874 2 -0.01743 0.01208 -0.087074 3 -0.0219 0.00078 0.191939 triceps brachii medial head (TRImed) o F0[N] L0[N] LT [N] α[ ] bn ξ[m] η[m] ζ[m] 2 -0.00599 -0.00428 0.054036 2 -0.02344 -0.00928 0.035216 624.3 0.1138 0.098 0.157 2 -0.03184 0.01217 -0.045874 2 -0.01743 0.01208 -0.087074 3 -0.0219 0.00078 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 4.3 Human Gait

(a) Anterior view. (b) Lateral view. (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 4.3 Human Gait

(continuation of Table 4.7)

77 4. BIOMECHANICAL MODELS

(continuation of Table 4.7)

78 4.3 Human Gait

(continuation of Table 4.7)

Table 4.8: Anthropometrical properties associated with the rigid bodies of the lower extremity model.

Inertial Moments [Kg.m2] bn mass [Kg] 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 2nd 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 90o.

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.

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 m0 ˆm m available contractile force FCE was equal to the maximum isometrical force F0 and no passive element force was considered for the whole analysis.

ˆm0 m0 FCE = F0 (4.3) m0 FPE = 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 depicted 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 FˆCE for the physiological case will vary throughout the analysis, having 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 FˆCE. 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/L˙ 0).

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 muscle 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

dM M(t + ∆t) = M(t) + (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 systems, 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 possible 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 m m that exists between the contractile force FCE and the muscle activation a 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, successfully 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 happen 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 motion to the muscle formulation using the Newton’s method [1, 7]. With this methodology, 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 considered 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 ≤ um ≤ 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 am, 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 optimizer. 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 approach, 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 total th of muscles that will be modelled Nmuscle, in the 20 parameter of the generic *MAIN PARAMETERS keyword.

*MAIN PARAMETERS total ··· Nmuscle

total A positive value of Nmuscle 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

OptFT

MUSCLE=Nmuscle

L˙ 0

L˙ tendon

αpenn

amin

amax

Nmp 1 1 1 1 1 Pnumber Pξ Pη Pz Pbody 2 2 2 2 2 Pnumber Pξ Pη Pζ Pbody ···

Nmp Nmp Nmp Nmp Nmp Pnumber Pξ Pη Pζ Pbody

[FRLD LR]

A brief description of each parameter is made:

OptFT The user starts by indicating whether muscle fatigue is to be considered or not.

OptFT = 0 Muscle fatigue is not considererd in the analysis.

OptFT = 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.

L˙ 0 The maximum contractile velocity.

L˙ tendon The tendon length, assumed to be constant.

F0 Maximum contractile force.

αpenn Pennation angle.

100 A.2 Simulation ﬁle

amin Muscle activation lower bound (equal to 0, in theory) amax Muscle activation upper bound (equal to 1, in theory)

Nmp Number of points that deﬁne the muscle’s geometry. For every point n, the following ﬁve parameters must be provided:

n Pnumber Index number of point n. n Pbody Number of the body where point n is attached. n Px Coordinate ξ in the local reference frame oξηζ. n Pη Coordinate η in the local reference frame oξηζ. n Pζ 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 calculated. *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 activations.

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:

total nt Nmuscle m1 m2 m t0 actt0 actt0 ··· actt0 m1 m2 m t1 actt1 actt1 ··· actt1 ··· m1 m2 m tn acttn acttn ··· acttn

m Where the value acttn corresponds to the activation of muscle m for time instant tn. nt total is the number of provided time instants and Nmuscle 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 extension .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 application 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.

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 FˆCE), 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 instant.

Figure B.8: Time instant selection.

107 B. MHILL DATA VISUALIZER MANUAL

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