Predicting Muscle Forces in the Human Lower Limb during ......Pauwels (1965) showed that muscles...

Universitat Ulm

Institut fur Unfallchirurgische Forschung und Biomechanik

Direktor: Professor Dr. Lutz Claes

Predicting Muscle Forces

in the Human Lower Limb

during Locomotion

Dissertation zur Erlangung des Doktorgrades

der Humanbiologie (Dr. biol. hum.)

der Medizinischen Fakultat der Universitat Ulm

Vorgelegt von Dipl.-Ing. Erik Forster

geboren in Hannover

Ulm 2003

Amtierender Dekan: Prof. Dr. R. Marre

1. Berichterstatter: Prof. Dr. L. Claes

2. Berichterstatter: Prof. PhD. J. Rasmussen

Tag der Promotion: 19. Dezember 2003

III

Acknowledgements

The work for this thesis was carried out at the Institute of Orthopaedic Research and

Biomechanis, University of Ulm. I wish to thank the head of the Institute, Professor Lutz

Claes for being my supervisor. As a member of the bone research group, I am also thankful

to the head of this group Privatdozent Peter Augat for supporting me.

Special thanks go to Professor John Rasmussen from the AnyBody group, University of

Aalborg, Denmark. It is a great honor to me that he agreed to examine this thesis. I would

also like to thank his colleague Professor Michael Damsgaard for the numerous interesting

and helpful discussions.

Thanks to the staff of the IT-Center of the University of Ulm for providing me with the

latest versions of various software for multiple computer platforms.

Thanks to all the colleagues that contributed to the friendly atmosphere at the Institute

of Orthopaedic Research and Biomechanics.

I am deeply grateful to Dr. Sandra Shefelbine for all her efforts and patience while proof-

reading this thesis.

Most of all I wish to thank my colleague and friend Dr. Ulrich Simon for his advice

concerning programming, mathematics, mechanics and musculoskeletal modelling. It was

a real pleasure working together with Dr. Ulrich Simon for the last three years. I appreciate

his contributions to the “dark and foggy factor” and always having time to discuss.

Finally, I wish to thank my parents for always being there for me.

Ulm, im Herbst 2003 Erik Forster

V

Contents

Nomenclature VII

1 Introduction 1

1.1 Redundancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Different Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Multiple Muscle Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Predicting Muscle Forces in the Human Lower Limb during Gait . . . . . . 9

1.5 Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Materials and Methods 11

2.1 Inverse Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Modelling of Muscles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Checking and Transforming the Calculated Quantities . . . . . . . . . . . . 34

2.5 Software and Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.6 Model of the Human Lower Limb . . . . . . . . . . . . . . . . . . . . . . . 37

3 Results 43

3.1 Mathematical and Mechanical Validation . . . . . . . . . . . . . . . . . . . 43

3.2 The Influence of the Optimization Criterion Employed . . . . . . . . . . . 47

3.3 Comparison of Activities Performed . . . . . . . . . . . . . . . . . . . . . . 50

3.4 The Influence of the Shift Parameter xs . . . . . . . . . . . . . . . . . . . . 53

3.5 The Sensitivity to Variations in Ground Reaction Forces . . . . . . . . . . 61

3.6 The Sensitivity to Variations in Muscle Attachment Points . . . . . . . . . 62

4 Discussion 67

4.1 Using Optimization Techniques to Predict Muscle Forces . . . . . . . . . . 68

4.2 Multibody-Dynamics Approach . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3 Musculoskeletal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.4 The Influence of the Optimization Criterion . . . . . . . . . . . . . . . . . 71

4.5 Comparison of Activities Performed . . . . . . . . . . . . . . . . . . . . . . 75

4.6 The Influence of Co-Contraction . . . . . . . . . . . . . . . . . . . . . . . . 77

VI CONTENTS

4.7 Sensitivity to Input and Model Parameters . . . . . . . . . . . . . . . . . . 78

4.8 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Summary 84

A Data and Parameter 86

A.1 Subject Specific Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

A.2 List of PCSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

References 92

VII

Nomenclature

Operators and functions

( )T Matrix or vector transpose.

( )−1 Matrix inverse.

( ) Build skew-symmetric matrix from a vector.

∇ The Nabla operator is used with multivariate functions. The result is

the gradient vector. The components of this gradient vector are the

first partial derivatives of the multivariate function with respect to all

variables.

G Objective function, a multivariate function.

g Separate, univariate function of objective function.

H Hessian matrix, consists of the second partial derivatives of G.

sign The sign function (sign(a) = a/abs (a)).

Variables

In general a small normal letter indicates a scalar, a small bold letter

indicates a vector and a capital bold letter indicates a matrix.

Scalars

t Time.

nB Number of bodies.

nM Number of muscles.

nJ Number of joints.

nDOF Number of degrees of freedom.

ci Weight factor for muscle i.

fMus,i Magnitude of the vector of muscle force of muscle i.

fMus,j Magnitude of the vector of muscle force of muscle j.

fmax,i Maximal magnitude of the vector of muscle force of muscle i.

σMus,i Magnitude of the vector of muscle tensile stress of muscle i.

σmax,i Maximal magnitude of the vector of muscle tensile stress of muscle i.

xMus,i Muscular activity of muscle i.

p Exponent of objective function or optimization criterion.

κi Eigenvalue.

mi Mass of body i.

λ Lagrangian Multiplier or vector of Lagrangian multiplier

β Additional bound for the min/max criterion.

ε Linear penalty.

εf Error tolerance for joint reaction forces.

VIII NOMENCLATURE

εm Error tolerance for joint reaction moments.

Vectors

z Vector that substitutes the vector fMus.

u Vector used for translating fMus.

p Vector of Euler parameters for a system.

pi Vector of four Euler parameters for body i.

qi Vector of coordinates for body i.

q Vector of coordinates for a system.

r Vector of positions for a system.

ri Vector from the origin of the inertial frame to the center of mass of

body i.

rj Vector from the origin of the inertial frame to the center of mass of

body j.

ri Velocity of center of mass of body i.

ri Acceleration of center of mass of body i.

ω′i Angular velocity of body i in components of the local reference frame

of body i.

ω′i Angular acceleration of body i in components of the local reference

frame of body i.

fa,i Vector of applied external forces on body i.

fa Vector of applied external forces on the system.

ma,i Vector of applied external moments on body i.

ma Vector of applied external moments on the system.

fMus Vector containing the magnitudes of all muscle forces.

fM,i Vector is the sum of applied muscle forces on body i.

fM This vector is the sum of applied muscle forces on the system.

mM,i Vector of the resultant moment of the applied muscle forces on body

i with respect to the center of mass of body i.

mM Vector of the resultant moment of the applied muscle forces on the

system.

fk Joint contact force transmitted by joint k.

mk Moment transmitted by joint k.

δr Vector of virtual translations.

δπ′ Vector of virtual rotations.

δWi Virtual work of body i.

bl Vector of lower bounds of muscle forces.

bu Vector of upper bounds of muscle forces.

NOMENCLATURE IX

d Vector that represents the right hand side of dynamic equilibrium.

Matrices

CR Coefficient matrix for reactions.

CM Coefficient matrix for muscles.

Cred Reduced coefficient matrix.

N Diagonal matrix for scaling the vector fMus.

D Diagonal matrix.

ΦK Kinematic constraints.

ΦKri

Jacobian matrix of kinematic constraints with respect to ri.

ΦKri

Matrix consists of time derivatives of the elements of matrix ΦKri

.

ΦKπ′

iTransformed Jacobian matrix of kinematic constraints with respect to

the Euler parameters pi.

ΦKπ′

iMatrix consists of time derivatives of the elements of matrix ΦK

π′i.

ΦKt First partial derivative of kinematic constraints with respect to time.

ΦD Driver constraints.

ΦDri

Jacobian matrix of driver constraints with respect to ri.

ΦDri

Matrix consists of time derivatives of the elements of matrix ΦDri.

ΦDπ′

iTransformed Jacobian matrix of driver constraints with respect to the

Euler parameters pi.

ΦKπ′

iMatrix consists of time derivatives of the elements of matrix ΦD

π′i.

ΦDt First partial derivative of driver constraints with respect to time.

ΦDtt Second partial derivative of driver constraints with respect to time.

ΦriJacobian matrix of kinematic and driver constraints with respect to

the systems positions r.

Φπ′i

Transformed Jacobian matrix of kinematic and driver constraints with

respect to the systems Euler parameters p.

ΦP Euler parameter normalization constraints.

Ai Transformation matrix of body i.

Aj Transformation matrix of body j.

I The Unity matrix.

L Lower triangular matrix.

U Upper triangular matrix.

Q Orthogonal Matrix.

R Upper triangular matrix.

0 Zero matrix.

Θi Inertia tensor of body i.

Θ Inertia matrix of a system.

X NOMENCLATURE

Mi Mass matrix of body i.

M Mass matrix of a system.

Abbreviations

BW Body weight.

CNS Central nervous system.

CT Computer Tomography.

MR Magnetic Resonance.

DOF Degrees of freedom.

EMG Electromyography.

PCSA Physiological Cross-sectional Area.

PCSAi Physiological Cross-sectional Area of muscle i.

HSR Abbreviation of a subject (Bergmann et al., 2001).

KWR Abbreviation of a subject (Bergmann et al., 2001).

IBL Abbreviation of a subject (Bergmann et al., 2001).

PFL Abbreviation of a subject (Bergmann et al., 2001).

WS Walking with slow speed.

WN Walking with normal speed.

WF Walking with fast speed.

SU Stair climbing upwards.

SD Stair climbing downwards.

KB Knee bend.

UFBSIM Software developed by the author at the “Institut fur

Unfallchirurgische Forschung und Biomechanik” used to simulate

musculoskeletal systems.

Gait Analysis

stance phase The period from heelstrike to toe-off.

swing phase The period from toe-off to heelstrike.

gait-cycle The period from one heelstrike to the next ipsolateral heelstrike.

1

Chapter 1

Introduction

The musculoskeletal systems of humans and animals have always been in the focus of humaninterest. Famous ancient scientists such as Aristotle and Galen spent much effort in tryingto understand and describe musculoskeletal systems (Singer and Underwood, 1962; Nigg,1994; Martin, 1999).

In the Renaissance the idea of iatro-physics devel-

Fig. 1.1: Figure from the second bookof Borelli: “De Motu Animalium”(Borelli, 1685).

oped. Iatro-physicists (Singer and Underwood,

1962) or iatro-mechanasists (Prendergast, 2000)

were scientists who tried to explain the bodily

functions on purely mechanical grounds. The

most prominent representatives of this idea were

da Vinci, Galileo and Borelli (Fig. 1.1). These

men contributed significantly to the understand-

ing of musculoskeletal systems (Martin, 1999;

Prendergast, 2000). Still today, when trying to

predict muscle forces we use some of the assump-

tions that were proposed by the iatro-physicists

(Crowninshield and Brand, 1981b).

Nigg titled the 19th century the “gait century”

(Nigg, 1994) because measuring methods were

developed to quantify kinematics and kinetics of

movement and were applied extensively to human

gait analysis. During this period the Weber brothers were the first to hypothesize that a

man walks in a way that muscular effort is minimized (Weber and Weber, 1836). The

hypothesis of the Weber brothers is particular relevant when using optimization techniques

2 CHAPTER 1. INTRODUCTION

to predict muscle forces.

Wolff (1892a,b) recognized the interdependence between form and function of bones. He

postulated that mechanical loading determines bone growth, what is known as Wolff’s law.

Pauwels (1965) showed that muscles influence the loading of the long bones. Muscles are

the active components in the musculoskeletal system. Furthermore, muscles may work as

traction braces and in this way help to reduce the bending moment transmitted directly

to the bone shaft.

From the work of Wolff and Pauwels it is obvious that the knowledge of musculoskeletal

loading is not only of interest for a general understanding of such systems but is also essen-

tial for the design of orthopaedic implants or fixation devices. However, due to practical

and ethical reasons muscle forces are hard to measure in vivo. Consequently, mathematical

models have been employed to calculate muscle forces.

1.1 Redundancy

Mathematical models of musculoskeletal systems typically consist of a linkage of rigid

bodies and actuators to describe the muscles (Fig. 1.2). A musculoskeletal system is usually

redundant (Crowninshield and Brand, 1981b) meaning that the number nM of the muscles

is greater than the number nDOF of degrees of freedom of the system. As a consequence a

desired motion can be achieved by an infinite number of activation patterns of the muscles.

In nature the central nervous system (CNS) controls the activation of the muscles (Vaughan,

1999).

1.2 Different Approaches

1.2.1 Forward versus Inverse Dynamics

Both, forward dynamics (Anderson and Pandy, 1999; Neptune, 1999) and inverse dynamics

have been used to calculate muscle forces. In this context it is important to note that for

many applications the time history of the system’s position is known in advance (e.g. from

gait analysis).

Using forward dynamics the motion of the system is computed by integrating the equations

of motion of the system. Computing a motion that is known in advance is a “tracking

1.2. DIFFERENT APPROACHES 3

mTg

fGRF

-mT Tr

-Q wT T

mSg

-mF Fr

-Q wF F

mFg

-mS Sr

-Q wS S

B B

A A

(a)

fM,3

fKnee

fM,4

fGRF

-fM,3-fM,4

-fKnee

fM,1

fHip

fM,2

mSg

-mF Fr

-Q wF F

mFg

-mS Sr

-Q wS S

mTg

-mT Tr

-Q wT T

(b)

mRes

-fRes

fRes

-mRes

mRes f

Res

fGRF

-mF Fr

-Q wF F

mFg

-mS Sr

-Q wS S

mTg

-mT Tr

-Q wT T

mSg

(c)

Fig. 1.2: a) A mechanical model of the musculoskeletal system usually consistsof bones, joints and muscles. The resultant volume forces of a body act atthe center of mass. The ground reaction force fGRF acts at the foot. b) Thesectional planes AA and BB lead to the free body diagram with muscle forces(fM,i i = 1, . . . , 4) and joint contact forces (fKnee, fHip). c) The muscle andjoint contact forces at the sectional planes AA and BB can be combined to aresultant joint force (fRes) and a resultant joint moment (mres).

problem”. Due to the redundancy of musculoskeletal systems the tracking problem has

generally no unique solution.

Using an inverse dynamics approach the muscle forces that have generated the motion are

calculated. The redundancy of musculoskeletal systems leads to a mathematical indeter-

minate problem1 (Herzog, 1994). There are only nDOF equations but nM unknown muscle

1Since the problem is to distribute the resultant joint forces and moments among the muscle forces, the

term “General Distribution Problem” has also been used (Crowninshield and Brand, 1981b).


forces.

Inverse dynamics has a number of attractive features including efficiency and numerical

stability (Rasmussen et al., 2001).

1.2.2 Eliminating the Problem of Mathematical Indeterminacy

Several approaches have been proposed to turn the mathematical indeterminate problem

into a determinate one when using inverse dynamics. In the reduction method , the number

of unknowns is reduced by grouping muscles together in functional units until the number

nM of unknown muscle forces matches the number nDOF of degrees of freedom. Paul (1966)

used this approach to calculate hip contact forces.

Conversely, the addition method increases the number of equations by introducing ad-

ditional constraint equations (e.g. Pierrynowski and Morrison, 1985). For example an

additional constraint may enforce that the force fMus,i of muscle i is always twice that

of the force fMus,j of muscle j: fMus,i = 2 fMus,j . While the reduction method suffers

from simplification of the musculoskeletal system the addition method implies non-trivial

assumptions about the muscle activation pattern.

Collins (1995) and Lu et al. (1998) used the Dynamically Determinate One-Sided Con-

strained (DDOSC) method. The indeterminate problem is resolved into a series of dynam-

ically determinate problems by considering only as many unknowns as number of equations

at a time. Solutions of these dynamically determinate problems are rejected when they

violate the restriction that muscle forces have to be tensile forces and joint contact forces

have to be compressive. From the remaining solutions a solution is chosen that is consistent

with EMG signals. However, the effort of solving all dynamically determinate problems

becomes huge with an increasing number of muscles and an increasing number of degrees

of freedom of a model.

Thirty years ago, Seireg and Arvikar (1973) were the first to use optimization techniques to

solve the mathematical indeterminate problem (Crowninshield and Brand, 1981b). After

that, optimization techniques became very popular in the analysis of musculoskeletal sys-

tems (Herzog, 1996; Tsirakos et al., 1997). A possible solution that minimizes a function is

considered to be the correct solution. The function to be minimized is called the objective

function (Gill et al., 1981, section 1.1), the cost-function (Herzog, 1994) or the optimization

criterion (Brand et al., 1986). The optimization criterion therefore reflects the strategy of

the central nervous system (CNS) in motor control. So far, all studies that used optimiza-

1.3. MULTIPLE MUSCLE SYSTEMS 5

tion techniques assumed that the CNS resolves the redundancy by some strategy that is

closely related to the hypothesis of the Weber brothers.

1.2.3 Analytical versus Numerical Optimization

Analytical methods (Dul et al., 1984b; Herzog, 1987; Herzog and Binding, 1993) as well

as numerical methods (Brand et al., 1994; Heller et al., 2001; Rasmussen et al., 2001;

Damsgaard et al., 2001) have been used to determine the optimal solutions. Analytical

approaches are fast and provide insight into the effects of the various parameters. However,

for complex systems with multiple degrees of freedom and many muscles, the analytical

solution becomes intricate (Herzog, 1994; Raikova and Prilutsky, 2001) and numerical

methods are required.

1.3 Multiple Muscle Systems

For the definition of the terms agonist, antagonist, synergist, active muscle and silent

muscle in a planar case we refer to Ait-Haddou et al. (2000):

“The term agonist (resp. antagonist) will be used for muscles, whose moment in

a two-dimensional system about a joint is in the same (resp. opposite) direction

as the resultant joint moment. [...] Muscles that help the agonist to perform a

desired action are called synergists. We defined an active muscle as one that

exerts force and a silent muscle as one that produces zero force.”

Co-contraction is defined as the presence of antagonistic muscle activity. There is no similar

definition for these terms in the three-dimensional case, because it is difficult to distinguish

between agonists and antagonists in three dimensions. We will suggest a new definition in

subsection 2.3.2.

1.3.1 Considerations about Design Variables

The unknown variables that the objective function depends on are called the design vari-

ables (Herzog, 1994). Most studies used the magnitudes of the muscle forces:

fMus,i (i = 1, . . . , nM) (1.1)


as design variables (e.g. Seireg and Arvikar, 1973; Pedotti et al., 1978; Heller et al., 2001;

Heller, 2002).

Another popular choice for the design variables are the magnitudes of the muscle stresses

σMus,i (e.g. Crowninshield and Brand, 1981a; An et al., 1984; Herzog and Binding, 1993).

In these studies the muscle stress σMus,i of muscle i is defined as the muscle force fMus,i

divided by the physiological cross sectional area PCSAi of muscle i,

σMus,i =fMus,i

PCSAi

(i = 1, . . . , nM) . (1.2)

Siemienski (1992); Rasmussen et al. (2001); Damsgaard et al. (2001) proposed using mus-

cular activities as design variables. The muscular activity xi of muscle i is defined as the

muscle force fMus,i divided by the maximal force fmax,i of muscle i,

xi =fMus,i

fmax,i

(i = 1, . . . , nM) . (1.3)

The maximal force fmax,i of a muscle i is normally considered to be proportional to its

physiological cross sectional area PCSAi. Some studies take also the momentary muscle

fibre length and the shortening velocity into account for the determination of the maximal

force fmax,i.

A muscle force is generated by the shortening of muscle fibres. The fatigue of a muscle

is related to the amount of shortening of these fibres. From this point of view it is a

reasonable choice to take muscle stresses or muscular activities as design variables because

muscle stresses and muscular activities are related to the shortening of the muscle fibres.

1.3.2 Enforcing Synergism

When using optimization techniques an optimization criterion must be defined. Most

studies have used the polynomial criterion

G (fMus) =

nM∑i=1

(ci fMus,i)p . (1.4)

The ci in the objective function (Eqn. 1.4) are the weight factors (Raikova, 1996). With an

appropriate choice of these weight factors, design variables other than the muscle forces can

be employed. Using ci = 1 is equivalent to use muscle forces (Eqn. 1.1) as design variables.

Using ci = 1/PCSAi and ci = 1/fmax,i is equivalent to use muscle stresses (Eqn. 1.2) and

muscular activities (Eqn. 1.3) as design variables.

1.3. MULTIPLE MUSCLE SYSTEMS 7

Seireg and Arvikar (1973) minimized the sum of muscle forces using a linear objective func-

tion (p = 1 in Eqn. 1.4) without any upper bounds on muscle forces. A major drawback of

a linear objective function without any upper bounds on muscle forces is that it generally

predicts only one active muscle per degree of freedom (Dul et al., 1984a; Siemienski, 1992).

Subsequently models were developed to predict a more physiologically reasonable syner-

gistic behavior of the muscles when using linear objective functions. Upper bounds on the

maximal muscle forces or the maximal muscle stresses were imposed (Crowninshield, 1978).

Bean et al. (1988) and Stansfield et al. (2003) used a double linear programming approach

to determine the upper bounds. First, they minimized maximal muscle stress without any

upper bounds. Second, they minimized the sum of muscle forces with the maximal mus-

cle stress times the PCSA calculated in the first step as upper bound. However, a linear

objective function with upper bounds generally still predicts only one active muscle per

degree of freedom. An additional muscle is only activated when another muscles reaches its

upper bound. Patriarco et al. (1981) enforced muscular synergism by additional equality

constraints.

Other approaches minimized the maximal muscle stresses (An et al., 1984) or activity

(Damsgaard et al., 2001; Rasmussen et al., 2001) which we will subsequently call the

min/max criterion

G (fMus) = max (ci fMus,i) . (1.5)

The min/max criterion is a highly non-linear objective function, however the optimal so-

lution can be found using a linear objective function (subsection 2.3.9).

Simultaneously non-linear polynomial objective functions were employed (p ≥ 2 in Eqn. 1.4,

see Crowninshield and Brand, 1981a; Herzog, 1987; Pedotti et al., 1978). Synergism is

predicted when using non-linear polynomial objective functions, however the load sharing

between synergistic muscles is still constant with discontinuities when a muscle becomes

saturated. To avoid these discontinuities Siemienski (1992) introduced a soft saturation

criterion. The relation between synergistic muscle using this criterion is non-linear with

muscles becoming saturated smoothly. Rasmussen et al. (2001) proposed a general form of

the soft saturation criterion

G (fMus) = −nM∑i=1

p

√1− (ci fMus,i)

p (1.6)

and showed that the need for upper bounds vanishes when using the polynomial criterion

with large exponents p. Note however, that the criterion in Eqn. 1.6 can only be used with

muscular activities. For large exponents p muscles are activated more equally. Finally,


both the polynomial and the soft saturation criterion converge to the min/max criterion

when the exponent p moves towards infinity (Rasmussen et al., 2001).

1.3.3 Enforcing Co-Contraction

It is commonly agreed that not only multiple synergistic muscles but also antagonistic

muscles are active (Collins, 1995; Herzog and Binding, 1993). Conventional criteria pre-

dict co-contraction only if the model includes multiple degrees of freedom joints (Jinha

et al., 2002) or bi-articular muscles (Ait-Haddou et al., 2000; Herzog and Binding, 1993).

However, this kind of co-contraction is not ”pure” co-contraction (Cholewicki et al., 1995).

Moreover, many groups reported that optimization methods failed to predict co-contraction

adequately (Hughes and Chaffin, 1988). Crowninshield (1978) reported that optimization

methods did not predict muscle forces in biceps and brachialis during forced elbow flexion

although muscle activity was indicated by Electromyography (EMG) signals. Brand et al.

(1994) and Collins (1995) also showed that antagonistic muscle activity indicated by EMG

signals was not predicted by optimization methods.

Only a few studies have tried to enforce co-contraction. Hughes et al. (1995) enforced co-

contraction by putting lower bounds greater than zero on muscle stresses. Raikova (1996,

1999) enforced the activation of antagonists by using negative weight factors for muscles

that counterwork the resultant joint moment. However, both approaches have disadvan-

tages. Lower bounds greater than zero prevent all muscles from exerting zero force. No

silent states of muscles as seen from EMG signals can be predicted. Assigning different

signs for the weight factors is not straightforward when applied to muscles spanning mul-

tiple joints or joints with more than one degree of freedom because a muscle may be a

contributor and a counterworker simultaneously.

Herzog and Binding (1993) showed analytically that co-contraction of a pair of one joint

antagonistic muscles is not predicted when using convex objective functions. However, Her-

zog and Binding (1993) assumed that the minimum of the unconstrained objective function

is the point where all design variables are zero (i.e. all muscles exert zero force). Forster

et al. (2004) developed an approach to predict co-contraction by shifting the minimum of

the objective function to small muscular activities. Forster et al. (2004) showed that with

this extension it is possible to predict muscular activity for a pair of one joint antagonistic

muscles.

1.4. PREDICTING MUSCLE FORCES IN THE HUMAN LOWER LIMB DURING GAIT 9

1.4 Predicting Muscle Forces in the Human Lower

Limb during Gait

Several studies were concerned with the prediction of muscle forces and joint contact forces

in the human lower limb during gait. In recent studies some groups compared the calculated

hip contact forces to in vivo measured hip contact forces (Brand et al., 1994; Pedersen

et al., 1997; Heller et al., 2001; Heller, 2002; Stansfield et al., 2003). All of these groups

used instrumented hip joint prostheses to measure hip contact forces in vivo.

Brand et al. (1994) calculated hip joint contact forces using a previously developed three-

dimensional model of the human lower leg (Brand et al., 1982). The kinematic data and

ground reaction forces were obtained by gait analysis. Pedersen et al. (1997) used the very

same model and input data and reported hip contact forces in a acetabulum based system.

Both authors reported good agreement between measured and calculated hip contact forces

using the minimum of muscle stresses cubed. However, the gait analysis in this two studies

was performed several weeks after the hip contact forces were recorded. Thus, no direct

comparison of calculated and measured results was possible.

Heller was the first to make a direct comparison between calculated hip contact forces and

measured hip contact forces (Heller et al., 2001; Heller, 2002). They used a general three-

dimensional model that was scaled to four individual subjects according to anthropometric

data. The kinematic data and ground reaction forces were also recorded by gait analysis.

Using the sum of muscle forces as optimization criterion they reported reasonable agreement

between measured and calculated hip contact forces.

Stansfield et al. (2003) reported good agreement between measured and calculated results

for two of the subjects used in the study by Heller et al.. They recorded new kinematic

data and ground reaction forces. Using a three-dimensional model and applying the double

linear programming approach of Bean et al. (1988) they reported good agreement between

calculated and measured results.

1.5 Aims

The aim of this study was to predict hip contact forces in the human lower limb during

various activities. To this end we developed a new software program that could be used

for general musculoskeletal systems (Forster et al., 2002). This program was then used


to analyze the model of Heller (2002) together with the input data from Bergmann et al.

(2001).

We investigated the coding, mathematics and mechanics of our program by comparing the

results of our program to the results of a similar program (Rasmussen et al., 2003) when

using the model of Heller (2002) and input data from Bergmann et al. (2001).

It is difficult to compare studies because they often use different optimization criteria and

different design variables. Therefore, we investigated the effect of the various optimization

criteria on the solution, which influences the level of muscle synergism. Additionally, we

enforced antagonistic muscle activity by the extension to existing optimization criteria

proposed by Forster et al. (2004).

Finally, we determined the influence of the input parameters on the calculated hip contact

and muscle forces.

11

Chapter 2

Materials and Methods

The classical approach for the calculation of muscle forces consists of two steps (Tsirakos

et al., 1997). First, in an inverse dynamics analysis the resultant joint forces and resul-

tant joint moments (Fig. 1.2) are calculated. Second, an optimal set of muscle forces is

determined. The set of muscle forces must equilibrate the resultant joints forces and mo-

ments that are not constrained by the joints. We will present a slightly different approach

recently proposed by Damsgaard et al. (2001), in which a multi-body dynamics approach

is used (Haug, 1989) to set up the equilibrium equations in matrix form. Deleting the

columns belonging to reactions that are caused by the muscles and performing a matrix

factorization they retrieved the nDOF linear constraint equations for the nM muscle forces.

This approach is very suitable for computer implementation.

2.1 Inverse Dynamics

For the inverse dynamics analysis we assume the bones to be rigid bodies that are connected

by kinematic joints. Forces acting on a bone are muscle forces, joint contact forces, forces

due to gravity and inertia as well as external loads such as the ground reaction force

(Fig. 1.2). When using inverse dynamics the position of the system over a period of time

must be prescribed completely (e.g. using data from gait analysis). The position and

orientation of each body can therefore be determined. Differentiating twice with respect

to time yields translational and angular accelerations of the bodies.

12 CHAPTER 2. MATERIALS AND METHODS

2.1.1 Position Analysis

For every body i = 1, . . . , nB we introduce a set of cartesian coordinates. The position of

the i ’th body in the global reference frame as well as its orientation relative to the global

reference frame is given by qi. In the general three-dimensional case qi can be written as:

qi =

[ri

pi

]. (2.1)

The vector ri in Eqn. 2.1 is the position of the center of mass of body i and pi is a vector of

the four Euler parameters that describe the orientation of the body (Haug, 1989; Nikravesh,

1988). The vectors qi (i = 1, . . . , nB) can be assembled to q:

q =

q1

q2

...

qnB

.

Two bodies can be connected by a kinematic joint that constrains the relative motion of

the bodies. Mathematically, these kinematic constraints can be expressed by scleronomous

constraint equations in the form:

ΦK (q) = 0.

Usually, the relative motion of two bodies is not completely constrained by the joints.

To prescribe the motion of the mechanism that was measured, we introduce additional

rhenomorous constraint equations in the form:

ΦD (q, t) = 0.

These constraints are called driver constraints . There are only three rotational degrees of

freedom but four Euler parameters to describe the orientation of a body. Thus, another

constraint is that the vector pi of Euler parameters of body i has to be a unit vector. These

constraints are called the Euler parameter normalization constraints:

ΦP (q) = 0.

Since the motion is completely prescribed, the sum of the kinematic constraints ΦK , the

driver constraints ΦD and the Euler parameter normalization constraints equals 7 nB. Thus

the number of equations matches the number of elements in the vector q. However,

Φ (q, t) =

ΦK

ΦD

ΦP

= 0 (2.2)

2.1. INVERSE DYNAMICS 13

is a system of non-linear equations.

We will use the relations in Eqn. 2.2 to derive the equations for velocity and acceleration

analysis (subsections 2.1.2 and 2.1.3). For the position analysis however there is a far more

efficient way if we restrict the system to open kinematic chain problems. Generally, a joint

links body i and body j. For open kinematic chain problems the transformation matrix Aij

that transforms vectors with respect to the local reference frame of body i to vectors with

respect to the local reference frame of body j can be computed from the driver constraints.

Additionally, the vector rij between the center of mass of body i and the center of mass of

body j can be computed from the driver constraints:

Aj = Aij Ai , with i 6= j ; i, j = 1, . . . , nB .

In a similar way the center of mass of body j is

rj = rij + ri , with i 6= j ; i, j = 1, . . . , nB .

2.1.2 Velocity Analysis

The next step is to determine the translational velocities ri and angular velocities ω′i for

every body. Differentiating Eqn. 2.2 with respect to time yields:

nB∑i=1

{[ΦK

ri

ΦDri

]ri +

[ΦK

π′i

ΦDπ′

i

]ω′i

}+

[ΦK

t

ΦDt

]= 0 (2.3)

The coefficient matrices ΦKri

and ΦDri

of the vector ri are the Jacobian matrices of the

kinematic constraints and driver constraints with respect to the vector ri, respectively.

The coefficient matrices ΦKπ′

iand ΦD

π′i

of the vector ω′i are transformed matrices of the

Jacobian matrices ΦKpi

and ΦDpi

of the kinematic constraints and driver constraints with

respect to the vector of Euler parameters pi, respectively (Haug, 1989). The column vector

ΦKt is the partial derivative of the kinematic constraints with respect to time and and the

column vector ΦDt is the partial derivative of the driver constraints with respect to time.

Since by definition time does not appear explicitly in the kinematic constraint equations

ΦKt is a zero vector. Using this fact and reordering Eqn. 2.3 so that the known quantities

are on the right hand side of the equation yields:

nB∑i=1

{[ΦK

ri

ΦDri

]ri +

[ΦK

π′i

ΦDπ′

i

]ω′i

}=

[0

−ΦDt

]. (2.4)


Eqn. 2.4 is a system of linear equations that can be solved for velocities. After assembling

the left hand side an LU factorization (see NAG, 2002; Golub and VanLoan, 1990, section

3.2) is performed such that:

LU = [ΦrΦπ′ ] , (2.5)

where L is a lower triangular matrix with unit diagonal elements and U an upper triangular

matrix. The coefficient matrix Φr is the Jacobian matrix of the kinematic and driver

constraints with respect to the system positions r = [r1, r2, . . . , rnB]T . Accordingly the

coefficient matrix Φπ′ is the transformed Jacobian matrix of the kinematic and driver

constraints with respect to the system Euler parameters p = [p1, p2, . . . ,pnB]T . Then the

right hand side of Eqn. 2.3 is assembled and the the velocities are retrieved by forward-

backward substitution (see NAG, 2002; Golub and VanLoan, 1990, section 3.1).

2.1.3 Acceleration Analysis

Differentiating of both sides of Eqn. 2.4 with respect to time yields:

nB∑i=1

{[ΦK

ri

ΦDri

]ri +

[ΦK

π′i

ΦDπ′

i

]ω′i

}=

[γK

γD

], (2.6)

with:

[γK

γD

]=

[0

−ΦDtt

]−

nB∑i=1

{[ΦK

ri

ΦDri

]ri +

[ΦK

π′i

ΦDπ′

i

]ω′i

}.

The right hand side consists of the second partial derivative ΦDtt of the driver constraints

with respect to time. The coefficient matrices ΦKri

, ΦDri, ΦK

π′iand ΦD

π′iare the time derivatives

of each component of the coefficient matrices ΦKri

, ΦDri, ΦK

π′iand ΦD

π′i.

Eqn. 2.6 is also a system of linear equations which can be solved for the translational

accelerations ri and angular accelerations ω′i. Moreover, the left hand side of Eqn. 2.6 is

identical to that of the velocity analysis in Eqn. 2.4. Since the left hand side of (Eqn. 2.5)

is already factored, we only need to assemble the right hand side and perform a forward-

backward substitution to retrieve the accelerations ri and ω′i.

2.1.4 Equations of Motion

From the principle of linear momentum and the principle of angular momentum (see Green-

wood, 1988, pp. 93) we can derive the Newton Euler equations of motion for every body:

2.1. INVERSE DYNAMICS 15

[Mi 0

0 Θi

] [ri

ω′i

]=

[fa,i − fM,i

m′a,i −m′

M,i − ω′iΘi ω′i

]. (2.7)

Mi in Eqn. 2.7 is a three by three diagonal matrix with the mass of the i ’th body on the

main diagonal, and Θi is the inertia tensor of the i ’th body:

Mi =

mi 0 0

0 mi 0

0 0 mi

, Θi =

Θxx,i Θxy,i Θxz,i

Θyx,i Θyy,i Θyz,i

Θzx,i Θzy,i Θzz,i

.

fa,i is the sum of all applied external forces, m′a,i is the sum of all applied external moments

on the i ’th body. The vector fM,i is the sum of all applied muscle forces on body i. The

vector m′M,i is the resultant moment of the applied muscle forces with respect to the center

of mass of body i. The matrix ω′i is a skew-symmetric matrix consisting of the elements of

the vector ω′i:

ω′i =

0 −ω′z,i ω′y,i

ω′z,i 0 −ω′x,i

−ω′y,i ω′x,i 0

. (2.8)

Using the principle of virtual displacements, the equations of motion for a constrained

system of bodies may be written as:

δrT [Mr− fa + fM ] + δπ′T [Θ ω′ + ω′Θω′ −m′a + m′

M ] = 0, (2.9)

with:

M =

M1 0 0 0

0 M2 0 0...

.... . .

...

0 0 . . . MnB

, Θ =

Θ1 0 0 0

0 Θ2 0 0...

.... . .

...

0 0 . . . ΘnB

,

fa =

fa,1

fa,2

...

fa,nB

, ma =

ma,1

ma,2

...

ma,nB

, fa =

fa,1

fa,2

...

fa,nB

and ma =

ma,1

ma,2

...

ma,nB

.

The virtual displacements δrT and the virtual rotations δπ′T in Eqn. 2.9 must be admissible

to the constraints. They are admissible if

Φrδr + Φπ′δπ′ = 0 (Haug, 1989) (2.10)


holds.

Introducing a vector of Lagrange multipliers λ, Eqns. 2.9 and 2.10 can be combined to:

δrT[Mr− fa + fM + ΦT

r λ]+ δπ′T

[Θ ω′ + ω′Θω′ −m′

a + m′M + ΦT

π′λ]

= 0 (2.11)

which must now hold for arbitrary virtual displacements δrT and arbitrary virtual rotations

δπ′T . Thus we can write the equations of motion for the whole mechanism as:[ΦT

r

ΦTπ′

]λ +

[fM

m′M

]=

[fa −Mr

m′a −Θ ω′ − ω′Θω′

](2.12)

Eqn. 2.12 is separated so that all unknown forces are on the left hand side and all known

forces are on the right hand side. Eqn. 2.12 represents a system of algebraic equations,

called dynamic equilibrium, because accelerations and velocities have already been com-

puted. The column vector λ introduced in Eqn. 2.11 represents the reaction of the kinematic

as well as the driver constraints. The matrix [Φr Φπ′ ]T is a quadratic matrix because the

bodies are fully supported (Damsgaard et al., 2001). Moreover, for meaningful physical

systems the matrix is a full rank matrix (Haug, 1989). As a consequence there is no work

for the muscles to do. Therefore we delete columns belonging to kinematic constraints or

drivers that are not active. The reduced coefficient matrix for the active constraints is

then:

CR =

[ΦT

r

ΦTπ′

]columns for active constraints

. (2.13)

The matrix CR has full column rank, because it is formed by deleting columns from a full

rank quadratic matrix.

2.2 Modelling of Muscles

The objective functions in subsection 1.3.2 depend on the magnitudes of the muscle forces

fMus,i , i = 1, . . . , nM . The modelling of the muscles must therefore provide us with a

coefficient matrix CMus similar to the coefficient matrix CR for the joint reactions λ

CR λ + CMus fMus = d .

The column vector fMus is a vector containing all magnitudes of the muscle forces fMus =

[fMus,1, fMus,2, . . . , fMus,nM]T . The vector d is the right hand side of Eqn. 2.12:

d =

[fa −Mr


]. (2.14)

2.2. MODELLING OF MUSCLES 17

2.2.1 Modelling the Lines of Action of Muscles

The muscles are modelled as line segments spanning between the point of origin and the

point of insertion. An arbitrary number of deviating points may be defined in between to

model a muscle’s path correctly. Origin, insertion as well as the deviating points are rigidly

attached to one of the bodies in the system.

ri

rj

sP

P

sQ Q

(a)

sP

P

sQ Q

uPQ

-uPQ

(b)

Fig. 2.1: The muscles are modelled as line segments. A path of a muscleconsists of the point of origin and the point of insertion (e.g. musculusgastrocnemius in the figure above). Additionally, an arbitrary numberof deviating points may be inserted to model the path correctly (e.g.musculus vastus in the figure above).

Let us consider a line segment of the k ’th muscle spanning between point P and Q

(Fig. 2.1). Let point P be attached to the body i and point Q be attached to body j

(i 6= j ; i, j = 1, . . . , nB). The vectors p and q point to the point P and point Q, respec-

tively:

p = ri + Ais′p ,

q = rj + Ajs′q .

The length lPQ of the segment is:

lPQ =√

(q− p)T (q− p) .


The action of this muscle segment on the i ’th body can be computed using the partial

derivatives of the length lPQ with respect to the bodies coordinates. This can be shown by

the principle of virtual work. The virtual work δWi of the muscle segment on the body i

is:

δWi = −(δlPQ, ri

+ δlPQ, π′i

)fMus,k ,

where fMus,k is the muscle force acting in the line segment. Additionally, uPQ is a unit

vector in the direction of q− p:

uPQ =q− p

lPQ

.

The virtual work δWi is then:

δWi =(δrT

i uPQ + δπ′Ti s′pATi uPQ

)fMus,k. (2.15)

The result of Eqn. 2.15 can be easily interpreted geometrically. The coefficients for the

virtual displacement δri are by definition a unit vector from point P to point Q. The unit

vector multiplied by the muscle force acting in the muscle segment yields the action on

body i (Fig. 2.1),

fM,ki = fMus,k uPQ .

The coefficients for the virtual rotations δπ′i are the cross-product of the vector sp and the

vector uPQ. The cross-product is carried out with respect to the local reference frame of

body bi. Multiplying the cross-product with the muscle force yields the moment:

m′M,ki = s′p × fMus,k u′PQ ,

that the muscle segment is exerting on the center of mass of body i. The length of a

muscle is the sum of the length of its line segments. Because the differential of a sum is

the sum of the differentials, each line segment of a muscle can be treated separately, and

their coefficients can be summed.

We retrieve the desired coefficient matrix CMus by considering the action of each muscle

on the bodies of the system such that:[fM

m′M

]= CMusfMus. (2.16)

2.3. OPTIMIZATION 19

2.2.2 Upper and Lower Bounds on Muscle Forces

Muscle forces fMus,i (i = 1, . . . , nM) are restricted to be positive (i.e. the muscles may only

pull not push) and not larger than a maximal force fmax,i. The maximal force of a muscle

i is considered to be the product of the physiological cross sectional area PCSAi and the

maximal tensile stress σmax that muscle fibres can exert. We therefore define a vector of

lower bounds bl and upper bounds bu

bl ≡

0

0

0...

0

≤

fMus,1

fMus,2

fMus,3

...

fMus,nM

≤

σmax · PCSA1

σmax · PCSA2

σmax · PCSA3

...

σmax · PCSAnM

≡

fmax,1

fmax,2

fmax,3

...

fmax,nM

≡ bu (2.17)

2.3 Optimization

Mathematically optimization means to find the minimum of an objective function G subject

to equality and inequality constraint equations. The variables that the objective functions

depends on are called the design variables (subsection 1.3.1). The muscle forces and muscu-

lar activities are the design variables. Because the physiological cross sectional area PCSAi

of muscle i is proportional to its maximal force (fmax,i in Eqn. 2.17), minimizing muscular

activities is equivalent to minimizing muscle stresses.

The optimization criteria have been introduced in subsection 1.3.2. We will use the polyno-

mial criterion (Eqn. 1.4), the soft saturation criterion (Eqn. 1.6) and the min/max criterion

(Eqn. 1.5).

The dynamic equilibrium of Eqn. 2.12 enter the optimization as linear constraint equations

(i.e. equality constraints). Additionally, the muscle forces must be within their bounds,

i.e. satisfy the inequality constraints of Eqn. 2.17. Taking Eqns. 2.13, 2.14 and 2.16 into

account we can write:

Minimize G (fMus) , (2.18)

subject to [CR CMus]

[λ

fMus

]= d (2.19)

and bl ≤ fMus ≤ bu . (2.20)

A set of muscle forces fMus that satisfies the linear constraint equations in Eqn. 2.19 and

the bounds in Eqn. 2.20 on the muscle forces is a feasible point. A set of muscle forces that


violates any of the linear constraint equations or any of the bounds is called an infeasible

point. A muscle fi , (i = 1, . . . , nM) is said to be active on its lower bound (resp. upper

bound) if fi,Mus = bli (resp. fi,Mus = bui). If a muscle is active on its lower or upper bound

respectively the particular inequality constraint is said to be part of the working set. The

linear constraint equations are always part of the working set. When calling an optimization

routine we must provide an initial guess of the muscle forces and optionally a working

set. The initial guess does not necessarily have to be a feasible point1. Accordingly, the

optimization is a two step procedure: first, identifying an initial feasible point (the feasibility

phase) and second, finding the minimum (the optimality phase). Finding an initial feasible

point is an optimization problem with the sum of infeasibilities as the objective function.

When a feasible point has been found, all subsequent points will also be feasible. During

the optimality phase the algorithms terminate when an optimal solution has been found.

Otherwise a feasible direction and a feasible step length along this feasible direction is

calculated such that the value of the objective function strictly decreases. Changes in the

working set are also considered in finding a minimum.

2.3.1 Separable Programming Problems

The polynomial criterion (Eqn. 1.4) and the soft saturation criterion (Eqn. 1.6) belong

to the class of separable programming problems (see Gill et al., 1981, section 6.8.2.2).

For separable programming problems, the multivariate objective function G(fMus) can be

written as a sum of separate, univariate functions:

G (fMus) =

nM∑i=1

gi (fMus,i) .

Consequently, the matrix of the second partial derivatives, the Hessian matrix

H (fMus) =

h1,1 . . . h1,nM

.... . .

...

hnM ,1 . . . hnM ,nM

, with hi,j =∂2G

∂fMus,i∂fMus,j

is a diagonal matrix. Moreover, for the polynomial criterion and the soft saturation criterion

all these diagonal elements will be positive for all p ≥ 2 and fMus ∈ [bl , bu] (Eqn. 2.30).

This means that the Hessian matrix is positive definite (see Golub and VanLoan, 1990,

1For the first instant in time we provide a zero vector as initial guess for the muscle forces. In the

following instances we provide the solution of the previous instant in time.


section 4.2) and the associated objective function is convex. For convex objective functions,

any local minimum will be a global and unique minimum (Gill et al., 1981; Herzog, 1994).

For muscular activity xi = fMus,i/fmax,i the objective function can be written as

G (x) =

nM∑i=1

gi (xi) .

The separable functions gi (xi) are shown in Fig. 2.2 for various exponents p. When using

the exponent p = 1 both criteria are linear functions and are identical except for a constant

shift in vertical direction. However, a shift in vertical direction does not influence the

location of the minimum.

With an increasing exponent p the separate functions of the polynomial criterion and the

soft saturation are more convex, i.e. the slope of the functions for small activities xi is

smaller than for large activities xi. This characteristic is more pronounced for the soft

saturation criterion. For exponents p ≥ 2 the slope of the separate functions of the soft

saturation criterion moves towards infinity for xi → 1. The slope of the separate function

is a measure of the increase in a muscle’s contribution to the objective function with an

increase in its activity xi. Consequently, increasing a muscle with a low activity xi yields

a smaller increase of the objective function than increasing a muscle with a large activity

xi.

0 1

Muscular activity

0.0

0.2

0.4

0.6

0.8

1.0

Co

ntr

ibu

tio

n o

f m

usc

le

to t

he

ob

ject

ive

fun

ctio

n

p = 1

p = 2

p = 3

p = 100

a)

0 1

Muscular activity

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

Co

ntr

ibu

tio

n o

f m

usc

le

to t

he

ob

ject

ive

fun

ctio

n

p = 1

p = 2

p = 3

p = 100

b)

Fig. 2.2: Contribution of a single muscle i to the objective function. a)Polynomial criterion gi (xi) = xp

i (p = 1, 2, 3, 100) b) Soft saturationcriterion gi (xi) = − p

√1− xp

i (p = 1, 2, 3, 100).

The min/max criterion does not belong to the class of separable programming problems.


However, both the polynomial and the soft saturation criterion converge to the min/max

criterion for p →∞ (Rasmussen et al., 2001).

2.3.2 A New Definition for Antagonistic Muscles

For multi-articular muscles and/or muscles that span joints with multiple degrees of free-

dom it is difficult to decide whether the muscles are synergists or antagonists. In sub-

section 2.3.4 we demonstrate that the min/max criterion activates every muscle that may

help to equilibrate the external load. Therefore we suggest that an antagonistic muscle is

a muscle with a zero muscle force predicted by the min/max criterion for one particular

instant in time.

2.3.3 Enforcing Co-Contraction

We have already demonstrated in subsection 2.3.1 that the polynomial criterion and the

soft saturation criterion are convex objective functions. Moreover, for all fMus ∈ [bl , bu]

the separate functions are strictly increasing functions. This prevents antagonistic muscles

from being active, because activity of an antagonistic muscle leads to higher activity of

the agonistic muscles in order to preserve the equilibrium equations. As all muscle activity

increases the objective function increases. Therefore, in order to minimize the objective

function antagonistic muscles are predicted to have no activity.

Introducing a new shift parameter xs we extend the objective function of the polynomial

criterion (Eqn. 1.4) to

G (fMus) =

nM∑i=1

(ci fMus,i − xs ci fmax,i)p , p ∈ 2n , n ∈ IN, (2.21)

and the the objective function of the soft saturation criterion (Eqn. 1.6) to

G (fMus) = −nM∑i=1

p

√1− (fMus,i − xs ci fmax,i)

p , p ∈ 2n , n ∈ IN, (2.22)

in order to enforce co-contraction. For xs = 0 Eqns. 2.21 and 2.22 represent the standard

polynomial and soft saturation criterion, respectively. For xs > 0 the univariate functions

g(xi) are shifted to the right (Fig. 2.3). Then small activity of an antagonistic muscle leads

to a decrease of its contribution to the objective function (Fig. 2.3a). If this decrease

together with the effect caused by the additional activation of the agonist muscles results

in a decrease of the objective function, antagonistic activity is predicted. The extension is


0 1

Muscular activity

0.0

0.2

0.4

0.6

0.8

1.0

Co

ntr

ibuti

on

of

musc

le

to t

he

ob

ject

ive

funct

ion

gstd,i( i) = i

2

gext,i( i) = ( i- s)2

a)

xs

0 1

Muscular activity

0

1

Musc

ula

r ac

tiv

ity

b)

xs

xs

Fig. 2.3: Effect of the shift parameter xs. a) Effect on the separate,univariate function b) Contour plot of the shifted objective functionin dependency of xi and xj.

applicable for the polynomial criterion and the soft saturation criterion for even exponents

p.

2.3.4 Force-Sharing among one Joint Synergistic Muscles

To demonstrate the effects of the various criteria let us consider a simple, planar example

where two synergistic muscles M1, M2 are originating at the upper arm and inserting at

the lower arm (Fig. 2.4). Both muscles are spanning the elbow joint modelled as a hinge

joint (center C). We apply an external load fext at the distal end of the lower arm.

Thus, the only linear constraint equation is the equilibrium of moments around joint center

C:

a fMus,1 + b fMus,2 = c fext , with sign(a c fext) = sign(b c fext) = 1 .

Using muscular activities x1 = fMus,1/fmax,1 and x2 = fMus,2/fmax,2 as design variables

and substituting, the optimization problem becomes

Minimize G (x1, x2) , (2.23)

subject to (a fmax,1) x1 + (b fmax,2) x2 = c fext (2.24)

and 0 ≤ x1 ≤ 1 , (2.25)

0 ≤ x2 ≤ 1 . (2.26)


ab

fM,1

fM,2

fext

C

fext

c

Upper Arm

Lower Arm

M1

M2

a) b)

Fig. 2.4: Planar example: Two synergistic muscles M1, M2 are spanningthe elbow joint (center C). b) Free body diagram of the lower arm.The moment arm of muscle M1, muscle M2 and the external load fext

is a, b and c, respectively.

The values of the objective function (Eqn. 2.23) are shown in contour plots (Figs. 2.5

and 2.6). In order to be feasible a point must be part of the solid lines that represent the

linear constraint equation (Eqn. 2.24). Additionally, the bound constraints (Eqns. 2.25

and 2.26) have the effect that only points in the first quadrant are feasible. The moment

generating capacity of the first muscle is a fmax,1, the moment generating capacity of the

second muscle is b fmax,2. We subsequently assume that

a fmax,1 > b fmax,2 .

The gradient vector ∇G(x1, x2) points towards the steepest increase of the objective func-

tion and is always orthogonal to the isolines where G(x1, x2) = const. By Taylor-Series

expansion of the objective function about a feasible point it can be shown that a necessary

condition for a local minimum is that the gradient vector is a linear combination of the

linear constraint equation and the active bound constraints (Gill et al., 1981, section 3.3.2).

When using a linear criterion (p = 1) only the first muscle is active at the optimal solution

for a moderate external loading fext (Fig. 2.5a). For an increasing external load the activity

of the first muscle increases while the second muscle stays silent until the activity of first

muscle reaches its upper bound (x1 = 1). Every additional external load must then be

equilibrated by the second muscle. The load sharing between the two muscles is determined

by the magnitude of the moment generating capacity of the two muscles. Note, that there

is no unique solution when the linear constraint and the isolines of the objective function

are parallel, i.e. magnitudes of the moment generating capacity of both muscles are equal.

When choosing the polynomial criterion with an exponent p = 2 or p = 3 both muscles are


0.160.30

0.450.60

0.80

1.101.30

1.50

0 1

Muscular activity

0

1

Mu

scu

lar

acti

vit

y

a)

0.05

0.30

0.60

1.10

1.50

0 1

Muscular activity

0

1

Mu

scu

lar

acti

vit

y

b)

0.16

0.45

0.80

1.30

0 1

Muscular activity

0

1

Mu

scu

lar

acti

vit

y

c)

0.1

6

0.4

5

0.8

0

0 1

Muscular activity

0

1

Mu

scu

lar

acti

vit

y

d)

Fig. 2.5: Contour plots of the polynomial criterion for a) p = 1 b) p = 2c) p = 3 and d) the min/max criterion. The solid lines representthe linear constraint equation. The dashed lines connect the optimalsolutions for varying external load fext. The arrows in b and c indicatethe gradient vector of the objective function.

activated at the optimal solution for the whole range of external loadings. When no muscle

is active at its upper bound the gradient vector of the objective function is orthogonal

to the linear constraint (the arrow in Figs. 2.5b and 2.5c) at the optimal solution. The

load sharing between the muscles for varying fext is constant with a discontinuity when

the first muscle becomes saturated. As in the linear case the load sharing is determined

by the moment generating capacity of the two muscles. The second muscle with the lower

moment generating capacity is activated more when using the exponent p = 3.

For p →∞ the polynomial criterion converges towards the min/max criterion (Rasmussen

et al., 2001). The objective function of the min/max criterion is a highly non-linear and

non-differentiable function. For x1 = x2 the partial derivatives do not exist (Fig. 2.5d).


-1.80-1.70

-1.55-1.30

-1.00

0 1

Muscular activity

0

1

Mu

scu

lar

acti

vit

y

a)

-1.92

-1.80-1.70

-1.55-1.30

-1.00

-1.0

0

0 1

Muscular activity

0

1

Mu

scu

lar

acti

vit

y

b)

Fig. 2.6: Contour plots of the soft saturation criterion. a) p = 1 b) p = 2.The solid lines represent the linear constraint equation. The dashedlines connect the optimal solutions for varying external load fext. Thearrows indicate the gradient vector of the objective function.

These corners however are the optimal solution, and consequently both muscles are equally

activated at an optimal solution. The load sharing between the two muscles is still constant

but does not depend on the moment generating capacity of the two muscles. More generally,

a muscle that can contribute to equilibrate the external loading is activated to the same

extent as the muscles with a large moment generating capacity.

For p = 1 and p →∞ the solution of the soft saturation criterion is identical to the solution

of the polynomial criterion. For p = 2 and p = 3 the situation is shown in (Figs. 2.6a) and

(2.6b), respectively. Both muscles are activated for the whole range of external forces fext.

For small external loads fext the optimal solutions are close to the optimal solutions for

the polynomial criterion with similar exponents. However, the load sharing between the

two muscles is not constant for varying external forces fext. The load sharing between the

muscles depend on the moment generating capacity of the two muscles and additionally on

the external force fext. Both muscles become saturated smoothly simultaneously. Thus,

the need for upper bounds vanishes because the upper bounds are defined implicitly in the

objective function.

2.3.5 Force-Sharing among one Joint Antagonistic Muscles

We showed in subsection 2.3.3 that conventional criteria generally do not predict co-

contraction. However, applying the extension (subsection 2.3.3) to the standard quadratic


polynomial criterion, we can predict co-contraction of antagonistic muscles. The stan-

dard quadratic polynomial criterion minimizes the distance from the origin (Fig. 2.5b).

The extended quadratic criterion, however, minimizes the distance from the point [xs, xs]

(Fig. 2.3b).

a

fext

C

fext

c

Upper Arm

Lower Arm

M1

V1

a) b)

fM,1

fM,2

V2 V2

M2

Fig. 2.7: Planar example: Two antagonistic muscles M1, M2 are spanningthe elbow joint (center C). a) The path of the muscle M2 is definedby four points: origin, insertion and additionally two deviating pointsV1, V2. The points V1 and V2 are rigidly attached to the upper armand lower arm, respectively. b) Free body diagram of the lower arm.The moment arm of muscle M1, muscle M2 and the external load fext

is a, b and c, respectively.

The example is similar to the example of the previous subsection, however the two muscles

are spanning opposite sides of the joint center C (Fig. 2.7).

Thus the optimization problem is identical to the previous section (Eqns. 2.23-2.26), except

that

sign(a c fext) 6= sign(b c fext).

For the standard quadratic criterion exactly one muscle is active depending on the sign

of the external force fext (Fig. 2.8a). For xs > 0 both muscles are active for a moderate

external loading (Fig. 2.8b). For large external loads fext the optimal solution is identical

to the optimal solution of the standard criterion.


0 1

Muscular activity

0

1

Mu

scu

lar

acti

vit

y

a)

0 1

Muscular activity

0

1

Mu

scu

lar

acti

vit

y

b)

Fig. 2.8: Contour plots of the polynomial criterion with exponent p = 2for a) xs = 0 and b) xs > 0. The solid lines represent the linearconstraint equation. The dashed lines connect the optimal solutionsfor varying external load fext. The arrow represents the normal of theobjective function.

2.3.6 Reducing the Number of Linear Equality Constraint Equa-

tions

The objective function in Eqn. 2.18 depends only on the muscle forces fMus. The linear

constraint equations however contain the unknown muscle forces and additionally the un-

known joint reactions λ (Fig. 2.9a). For efficiency and numerical stability it is advantageous

to reduce the number of constraints and eliminate the reaction forces λ (Damsgaard et al.,

2001).

The matrix C = [CR CMus] is a 6 nB × (6 nB − nDOF + nM) matrix (Fig. 2.9a). Since the

matrix CR has full column rank (subsection 2.1.4) we may apply a QR factorization (see

Golub and VanLoan, 1990, section 5.2)

CR = Q

[R

0

]

The matrix Q is a 6 nB × 6 nB orthogonal matrix such that QT Q = I and the matrix R is

a (6 nB − nDOF )× (6 nB − nDOF ) upper triangular matrix. The equation is multiplied by

QT from the left. Since the matrix Q is orthogonal and the matrix R is upper triangular,

this results in the system displayed in Fig. 2.9b. The last nDOF rows of the factorized

matrix C only depend on the unknown muscle forces fMus. The matrix Clead is the leading

(6 nB − nDOF ) × nM part of the resulting matrix QTCMus, and the matrix Cred is the


CR CMus

6 -n nB DOF

6n

B

nM

fMus

l

d

R

6n

B

fMus

l

0 Q dT

QTCMus

{

nD

OF

Cred

0

{

{{

Clead

dred

dlead

a)

b)

Fig. 2.9: Reducing thenumber of linearequality constraintequations. a) As-sembling the linearconstraints. b) Elim-inating the reactionforces by apply-ing a standard QRfactorization scheme.

remaining nDOF rows of QTCMus. Similarly, the leading 6 nB−nDOF elements of the vector

QTd are the vector dlead and the remaining elements are dred. Then it is sufficient to use

the matrix Cred together with the vector dred as linear equality constraint equations for the

optimization routine. After the muscles forces have been computed by the optimization

routine, their contribution to the remaining part of the vector dlead is added. The reactions

λ are then found by backward substitution:

Rλ = dlead −Clead fMus . (2.27)

The QR factorization of CR is computed by 6 nB − nDOF subsequent Householder trans-

formations (NAG, 2002; Golub and VanLoan, 1990, section 5.2.1). Neither the matrix Q

nor its transpose is calculated explicitly but the elements of the Householder vector are

stored in the lower triangular part of the matrix R. Computing QTCMus and QTd from

the Householder vectors is more efficient than building the matrix Q explicitly.

To eliminate the reaction forces it is also possible to use an LU factorization. Although the

QR factorization needs approximately twice the number of floating-point operations (NAG,


2002), the QR factorization has some distinct advantages (Golub and VanLoan, 1990, sec-

tion 5.7.1). In addition to these advantages the absence of row pivoting and orthogonality

makes the QR factorization convenient to implement for this particular problem.

2.3.7 Scaling the Design Variables

Real numbers are represented on a computer in floating-point format with a fixed number of

bits. The number of bits are divided into the sign bit, the mantissa and the exponent (Gill

et al., 1981, section 2.1.2). The sign bit usually indicates whether the represented number

is positive or negative. Large differences in the design variables can lead to round-off errors

and convergence problems in the optimization routine. To avoid numerical problems during

optimization we may substitute the vector of muscle forces fMus by a vector z:

z = N−1 fMus + u . (2.28)

The matrix N is a diagonal matrix, and u is a vector with the same length as the vector of

muscle forces fMus. According to Tab. 2.1 there are several possibilities for the matrix N

and the vector u. The choice of the actual substitution depends on the design variables that

are used and on the desired range of the elements of the vector z. Using a constant value

for the diagonal entries of N is equivalent to using muscle forces as design variables. Using

values that depend on the maximal force a muscle can exert for the diagonal entries of N

is equivalent to using muscular activities as design variables. Using the reduced system of

subsection 2.3.6 we may re-write the linear equality and inequality constraint equations

Cred Nz = dred + Cred Nu (2.29)

u ≤ z ≤ N−1 bu + u . (2.30)

Tab. 2.1: Possible choices for the matrix N and the vector u of Eqn. 2.28. Thematrix N is a diagonal matrix with the Ni,i, (i = 1, . . . , nM) on its diagonal.

Range For Muscle Forces For Muscular Activities

zi ∈ [0, fmax,i] Ni,i = 1 ui = 0 Ni,i =fmax,i

max(fmax)ui = 0

zi ∈ [0, 1] Ni,i = max (fmax) ui = 0 Ni,i = fmax,i ui = 0

zi ∈ [−1, 1] Ni,i = 12max (fmax) ui = −1 Ni,i = 1

2fmax,i ui = −1


To avoid round-off errors due to the floating-point format of numbers it is advantageous if

all zi (i = 1, . . . , nM) are of similar magnitude. This may be achieved by substituting in a

way that the zi ∈ [0, 1] , (i = 1, . . . , nM). It is even more effective to substitute such that

the zi ∈ [−1, 1] , (i = 1, . . . , nM). This approach makes use of the sign bit in the floating-

point format. However, some optimization algorithms are intended for use with positive

numbers only (e.g. the simplex algorithm used by Damsgaard et al., 2001).

After having solved the optimization problem for the substituted vector z, we find the

vector of muscle forces by the inverse transformation of Eqn. 2.28

fMus = N (z− u) . (2.31)

2.3.8 Implementation of the Polynomial Criterion and the Soft

Saturation Criterion

Implementing the polynomial and soft saturation criterion is straightforward, and available

routines can be used. We use linear programming routines for a linear objective function

(exponent p = 1) and quadratic programming routines for the standard quadratic poly-

nomial criterion. For p = 2 when using the soft saturation criterion and using p > 2 for

both criteria requires the use of general non-linear programming routines. However, from

a computational point of view it is advantageous to use linear programming or quadratic

programming routines when possible. Linear and quadratic programming routines are

guaranteed to find the global minimum and are far more efficient than general non-linear

programming routines.

Reducing the system of linear constraint equations and applying a substitution according

to the two previous subsections the optimization problem is:

Minimize G (z) =

nM∑i=1

(zi − ui)p or G (z) = −

nM∑i=1

p

√1− (zi − ui)

p , (2.32)

subject to Cred Nz = dred + Cred Nu (2.33)

and u ≤ z ≤ N−1 bu + u . (2.34)

2.3.9 Implementation of the min/max Criterion

Although highly non-linear, the min/max criterion is implemented using a linear program-

ming routine, by introducing a new variable β. Minimizing the maximal zi is equivalent to

minimizing β with the additional constraints that all zi must be smaller than β.


Minimize G (z, β) = β + ε

nM∑i=1

zi , (2.35)

subject to Cred Nz = dred + Cred Nu (2.36)

and u ≤ z ≤ β . (2.37)

Subsequently, we will refer to the ε in Eqn. 2.35 as “linear penalty”. A formulation of

the min/max criterion according to (Eqns. 2.35-2.37) causes some problems because it

only affects the maximally loaded muscles. This creates some indeterminacy because only

a subset of muscles is actually represented in the objective function. Damsgaard et al.

(2001) proposed solving the problem iteratively, by identifying and removing maximally

loaded muscles from the problem step by step. The iteration continues until a unique

solution is found (Fig. 2.10). See Gill et al. (1981, section 3.3.2) for optimality conditions

for linearly constrained optimization problems.

A linear inequality constraint is a hyperplane that divides the nM dimensional space into

two subspaces. Similar to the definition of active and inactive bound constraints in sec-

tion 2.3, an active inequality constraint occurs when the solution vector z is a point of the

hyperplane defined by the inequality constraint. An inactive constraint denotes a constraint

when the solution z satisfies the constraint but is not part of the hyperplane.

The maximally loaded muscles and the zero muscles can be identified by checking if

the corresponding constraint is active. However, the solution with a zero activated or

maximally activated muscle must be unique. Along with every active bound constraint and

inequality constraint a Lagrangian multiplier λ is defined. If a λ equal to zero is associated

with an active constraint there exist points in the neighborhood where the corresponding

constraint is not active and the value of the objective function remains unchanged (Gill

et al., 1981, section 3.3.2). Therefore, the solution is not unique, and we only remove

zero muscles and maximally activated muscles with a positive Lagrangian multiplier. A

positive Lagrangian multiplier indicates that the objective function is strictly increasing

when making a non-binding perturbation. Removal of these muscles removes the variables

along with their constraints from the problem. Additionally, the action of the muscle on

the system must be subtracted from the right hand side dred. The matrix Cjlin and the

vector djlin are the left hand side and right hand side of the linear constraint equations for

the j’th step, respectively. According to Eqn. 2.36 we see that for the first step C1lin and

d1lin are:

C1lin = Cred N and d1

lin = dred + Cred Nu .


Supposing that muscle i is removed in the j ’th step this means:

dj∗lin = dj

lin −Cjlin,iNizi ,

where Cjlin,i and Ni is the i ’th column of Cj

lin and N, respectively. As the iterative min/max

Return

Rank = 0 ?

Remove linear dependentconstraints

Activitysmall?

No moreMuscles?

RemovedMuscles?

Remove “Max.” Muscles

Remove “Zero” Muscles

Solutionunique ?

Solve min/max

Set up constraint equations

no

yes

yes

yes

no

no

yes

no

no

yes

Fig. 2.10: Flow chart of the iterativemin/max algorithm. The algorithm stopsif the solution is unique. It also terminatesif there are no muscles to remove, if allmuscles have been removed or the activ-ity of the remaining muscles is very small.Another termination criterion is that therank of the equality constraint equationsis zero.

algorithm proceeds not only the number of muscles nM decreases but also the number of

degrees of freedom nDOF . Mathematically, the rank of the nDOF×nM matrix Cjlin decreases.

To ensure the reliability and efficiency of the algorithm we determine the rank of Cjlin after

each iteration by a singular value decomposition (see Golub and VanLoan, 1990, section

8.3). If the rank of Cjlin is actually lower than the current number of degrees of freedom

njDOF we remove the linear dependent constraint equations using the results of the singular


value decomposition

Cjlin = QDPT ,

where Q is a nDOF × nDOF orthogonal matrix and PT is a nM × nM orthogonal matrix.

The leading nDOF × nDOF part of the matrix D is a diagonal matrix with the singular

values as diagonal elements κ1, κ2, . . . , κnDOF, ordered that

κ1 ≥ κ2 ≥ . . . ≥ κnDOF≥ 0.

The consecutive nDOF × nM part of the matrix D contains only zeros. The rank r of Cjlin

is the number of singular values greater than zero (κr > 0, κr+1 = 0). If r = 0, i.e. the

rank of the matrix Cjlin has vanished the algorithm terminates. Otherwise using the fact

that the matrix Q is orthogonal we can retrieve the linear constraint equations for the next

step j + 1

Cj+1lin = DPT and dj+1

lin = QTdj∗lin .

Multiplying matrix PT from the left with matrix D means to multiply the i ’th row of the

matrix PT with the i ’th singular value κi. If r < nDOF the r ’th and following rows are

multiplied by zero. These rows may be skipped and the number of degrees of freedom

nDOF for the consecutive step is set equal to the rank r of the matrix Clin .

2.4 Checking and Transforming the Calculated Quan-

tities

Before further processing we must transform the joint reactions λ computed in the previous

sections to the global reference frame.

2.4.1 Transforming Reaction Forces to the Global Reference

Frame

Considering a joint k (k = 1, . . . , nJ) that links body i to body j with the active constraints

Φk = 0 and associated Lagrange multipliers λk it can be shown by the principle of virtual

work that the joint contact force fk and the joint moment mk are

fk = ΦTk,rj

λk ,

mk = AjΦTk,π′

jλk .

(2.38)

2.4. CHECKING AND TRANSFORMING THE CALCULATED QUANTITIES 35

2.4.2 Checking the Accuracy of the Solution

The dynamic equilibrium equations of (Eqn. 2.12) enter the optimization routines as lin-

ear constraint equations. Because optimization is performed numerically the constraint

equations cannot be fulfilled exactly but to a certain tolerance. We check the tolerance of

the calculated set of muscle forces by assembling the system of Eqn. 2.12 again with the

calculated muscle forces and muscle moments on the right hand side:[ΦT

r

ΦTπ′

]λ =

[fa −Mr


]−CM fMus . (2.39)

In subsection 2.1.4 we pointed out that the coefficient matrix on the left hand side in

Eqn. 2.39 is a full rank quadratic matrix. Consequently Eqn. 2.39 can be solved for the

joint reactions λ. Applying Eqn. 2.38 we retrieve a new set of joint contact forces fk,new

and joint moments mk,new. We define two vector of discrepancies ∆f and ∆m:

∆f =

f1 − f1,new

f2 − f2,new

...

fJ − fJ,new

, ∆m =

m1 −m1,new

m2 −m2,new

...

mJ −mJ,new

.

Finally we define an error tolerance εf for joint contact forces and an error tolerance εm

for joint moments. The error tolerances are computed by taking the euclidian norm of the

vectors ∆f and ∆m, respectively:

εf =√

∆fT ∆f , and εm =√

∆mT ∆m . (2.40)

The higher the numerical accuracy, the smaller are the error tolerances εf and εm.

2.4.3 Calculating the Resultant Joint Reactions

Resultant joint reaction forces and resultant joint reaction moments (see Fig. 1.2) are

computed setting all constraints and drivers in Eqn. 2.13 to be active. Therefore, no

columns of the matrix CR are deleted and CR is a quadratic matrix. Moreover, CR is a

full rank matrix. Consequently, all muscles are predicted to be zero.

2.4.4 Calculating Internal Loads

In order to determine the internal loads that are acting on the bones, we define a plane by

means of a point P of the plane and a normal vector n (Fig. 2.11a). To determine whether


fint

mint

fjointfm

fm

a) b) c)

Fig. 2.11: Determination of internal loads.

a point P1 is on the same side that the normal vector n points or on the opposite side of

the plane we use the scalar product. The vector rPP1 is from the point P to the point P1.

If the scalar product s of the vector rPP1 and the normal vector n

s = rTPP1

n

is positive then the point is on the same side of the plane, otherwise it is on the opposite

side.

This allows us to determine if joint centers, muscle attachment points or external forces

must be considered in the calculation of the internal loads. The internal loads at the

sectional plane are the sum of all these forces.

2.5 Software and Hardware

The approach described in the sections 2.1- 2.4 was implemented into a software program

called UFBSIM using the object-orientated programming language C++. Numerical sub-

routines that were used are listed in Tab. 2.2.

In order to prove the results of UFBSIM we compared the results to the results retrieved by

using the software AnyBody r©(Rasmussen et al., 2003) that was developed at the University

of Aalborg and is currently in its beta release.

UFBSIM was compiled on a PC (1.3 GHz, Intel r©Pentium r©IV processor, 640 MB RAM,

Windows r©2000) and on a workstation (SunFire 6800, 900 MHz UltraSPARC-III processor,

192 GB RAM, Solaris 9). On the PC we used the Microsoft r©Visual C++ 6.0 compiler

and used the Intel r©Math Kernel Library for Pentium r©IV processors. On the worksta-

2.6. MODEL OF THE HUMAN LOWER LIMB 37

Tab. 2.2: Numerical subroutines from the NAG C library (NAG, 2002) used inUFBSIM .

Description Routine

Matrix FactorizationLU-factorization nag real luQR-factorization nag real qr & nag real apply qSVD-decomposition nag real svdOptimizationlinear programming nag opt lpquadratic programming nag opt qpnon-linear programming (sequentialquadratic programming)

nag opt nlp

tion we used the Sun r©ONE Studio 8 C++ compiler and the Sun r©performance library.

AnyBody r©was used on the PC only.

2.6 Model of the Human Lower Limb

We adapted a general model of the human lower limb (Heller, 2002; Heller et al., 2001) with

subject specific data to make an individual musculoskeletal model of a subject. The model

consisted of four segments: the pelvis, the thigh, the shank and the foot. The segments

were connected by three joints: the hip joint, the knee joint and the ankle joint. In total

51 muscles were included into the model (Tab. A.6). Using subject specific kinematic and

kinetic data from gait analysis (Bergmann et al., 2001) we computed muscle forces and

joint contact forces for an individual subject (Fig. 2.12).

2.6.1 Geometrical Model

A geometrical model of the bony surfaces must provide the geometric shape of the bone,

prominent bony landmarks and joint centers (Fig. 2.13). Heller reconstructed bony surfaces

from Computer Tomography (CT) using the Visible Human Dataset (Spitzer et al., 1996).


GeometricalModel

Scaling

Inverse Dynamics& Optimization

MusculoskeletalModel

GroundReaction Force

Driver

OptimizationCriterion

MuscleForces

MuscleParameters

GaitAnalysis

Gen

eral

Model

Indiv

idual

Dat

a

AnthropometricData

Fig. 2.12: The process of calculating muscle forces for an individual sub-ject when starting from a general model.

Fig. 2.13: Reconstructed surface of the thePelvis along with the muscle attachmentsites (small points) and hip joint center(large point, see Heller, 2002).

2.6.2 Muscle Parameters

The muscle parameters include the attachment sites of the muscles on the bony surfaces

(Fig. 2.13) and the cross-sectional area PCSA of the muscles. The attachment sites were

digitized by Heller (2002) from the Visible Human Dataset . Muscles with large attachment

sites (e.g. the gluteal muscles) were modelled as multiple separate strings. The data of

the PCSA (Tab. A.6) were taken from Brand et al. (1986). In order to better represent the

muscle path for the musculus rectus femoris, the musculus semitendinosus and the musculus

tensor fasciae latae during the whole movement cycle we added additional deviating points

for these muscles.


2.6.3 Anthropometric Data

Bergmann et al. (2001) determined anthropometric data including bony landmarks and

joint centers prior to gait analysis. The anthropometric data was determined making x-

rays, CT-scans and by palpating.

2.6.4 Gait Analysis

The gait analysis study was performed by Bergmann et al. (2001) and the data of this study

were published on a separate CD (Bergmann, 2001). In this study four subjects performed

several trials of daily activities. While performing the activities, kinematic data, ground

reaction forces and hip contact forces of the subjects were recorded simultaneously.

All four subjects had artificial instrumented hips. Via telemetry units Bergmann et al.(2001) measured hip contact forces. Two Kistler plates measured the ground reactionforces. A Vicon system with six cameras was used to measure the positions of bodymarkers.

The marker positions determined byP

1

P2

P3

P5

P7

P11

P17P

19

P21

Fig. 2.14: Body linkage used in Gait Analysis. TheLandmarks P1 - P21 have been calculated fromMarker Positions (Bergmann et al., 2001). Fig-ure adapted from Bergmann (2001).

the Vicon system were fitted to rigid-

body motion of the body linkage in

Fig. 2.14.

During gait and stair climbing there

are two different phases. The stance

phase starts when the heel touches

the ground (heel strike) and ends

when the toes leave the ground (toe

off). During the stance phase the foot

has permanent contact to the ground.

During the swing phase the foot has

no contact to the ground.

2.6.5 Anthropometrical

Scaling

The bony landmarks of the general geometrical model were transformed so that they match

the bony landmarks determined prior to gait analysis (Heller, 2002; Heller et al., 2001). The


transformation was then applied to the reconstructed bony surfaces, to the joint centers and

the muscle attachment sites. We thereby adapted the general model to the anthropometric

data to obtain a musculoskeletal model of an individual subject.

2.6.6 Musculoskeletal Model

As described in Fig. 2.12 the musculoskeletal model in this study is specific to the subject

and contains information about model topology, bony surfaces, muscle path and muscle

parameters.

2.6.7 Driver

The relative motion versus time between bodies that was not constrained by any joints was

prescribed using drivers (Eqn. 2.2). The driver data was calculated from the kinematic data

from gait analysis. The derivatives of the driver with respect to time for velocity analysis

(ΦDt in Eqn. 2.3) and acceleration analysis (ΦD

tt in Eqn. 2.6) were calculated numerically

with a central difference quotient scheme.

We deleted all columns in Eqn. 2.13 belonging to rotational driver constraints in the hip,

meaning that the hip joint cannot transfer any moments. We also deleted two columns

of the rotational driver constraint corresponding to the adduction/abduction and inter-

nal/external rotation moments for the knee joint and the ankle joint. Thus, the muscles

must equilibrate all three components of the resultant reaction moment in the hip joint,

whereas the muscles must only equilibrate the flexion/extension moment in the knee joint

and in the ankle joint.

2.6.8 Ground Reaction Forces

The force plates used by Bergmann et al. (2001) measured all three components of the

ground reaction force.

2.6.9 Calculating the Muscle Forces

Applying the methods of sections 2.1-2.3 to the musculoskeletal model driven by the drivers

and with the external loading of the ground reaction forces we calculated muscle forces with

different optimization criteria.


2.6.10 Subjects and Activities

The study of Bergmann et al. (2001) consisted of four subjects HSR, KWR, IBL and

PFL. The first two letters of the subjects were the initials of a subject, while the third

letter indicated the operated side (R=right, L=left). Details to the subjects are given in

section A.1.

Bergmann et al. (2001) recorded kinematic and kinetic data for several activities. These

activities were walking with normal speed (WN), walking with slow speed (WS), walking

with fast speed (WF), stair climbing upwards (SU), stair climbing downwards (SD), knee

bending (KB), sitting down (CD) and raising up from a chair (CU). We will investigate

all activities except for sitting down and raising up from a chair. Up to seven trials per

activity were performed. The kinematic and kinetic data was published for 201 equidistant

instances in time (Bergmann, 2001).

Heller (2002) modelled a lower left limb. Therefore, Bergmann (2001) mirrored the data

of the subjects HSR and KWR to the left side.

2.6.11 Coordinate Systems

Bergmann et al. (2001) defined a laboratory coordinate system and additionally segmental

coordinate systems. The laboratory coordinate system was defined as follows: x pointed in

walking direction, y from medial to lateral and z pointed vertically upwards. If not stated

explicitly otherwise, we refer to the laboratory coordinate system.

The x direction in the pelvis coordinate system pointed from the left hip joint center to the

right hip joint center. The y direction pointed in the ventral direction and the z direction

pointed upwards.

The z direction in the femur coordinate system was the idealized straight midline of the

femur, the x axis pointed medially and the y axis pointed in ventral direction.

2.6.12 Presentation of the Results

The vector of the measured hip contact force for instant i (i = 1, 2, . . . , 201) in time was

fMeas,i , while the vector of simulated hip contact force for instant i in time was fSim,i.

The absolute difference ∆xabs,i for instant i in time of the measured and calculated hip

contact force was determined by calculating the magnitude of the difference of the two


vectors fMeas,i and fSim,i:

∆xabs,i =

√(fMeas,i − fSim,i)

T (fMeas,i − fSim,i) . (2.41)

Analogously, the relative difference ∆xrel,i for instant i in time of the measured and cal-

culated hip contact force was determined by calculating the magnitude of the difference of

the two vectors fMeas,i and fSim,i divided by the magnitude of the vector fMeas,i :

∆xrel,i =

√(fMeas,i − fSim,i)

T (fMeas,i − fSim,i)√fTMeas,i fMeas,i

. (2.42)

In cases where we had to compare two vectors for multiple instances in time for multiple

trials of one activity, we assembled a vector ∆xabs consisting of all absolute differences

∆xabs,i of Eqn. 2.41. We then presented the vector ∆xabs in a boxplot comprising the min-

imum difference ∆xabs,i, the 25% quartile of the differences ∆xabs,i, the median difference

∆xabs,i, the 75% quartile of the differences ∆xabs,i and the maximum difference ∆xabs,i.

Additionally, we assembled a vector ∆xrel consisting of all relative differences ∆xrel,i for

all instances i (i = 1, 2, . . . , 201) in time for one trial. We defined the relative root mean

square (RMS) between the calculated and measured hip contact forces for this particular

trial as:

RMS =

√∆xT

rel ∆xrel√201

. (2.43)

43

Chapter 3

Results

Using anthropometric scaling described in subsection 2.6.5 we adapted the general geo-

metrical model of Heller (2002) to the four individual subjects HSR, KWR, IBL and PFL.

Computing the driver data according to subsection 2.6.7 we were able to investigate the

several trials of the activities walking with slow speed (WS), walking with normal speed

(WN), walking with fast speed (WF), stair climbing upwards (SU), stair climbing down-

wards (SD), and knee bend (KB) of the four subjects. For subject KWR Fig. 3.1 shows an

example musculoskeletal model for each activity investigated.

3.1 Mathematical and Mechanical Validation

All results that we will present, fulfill the dynamic equilibrium to a requested accuracy.

The error tolerances εf for joint contact forces and the error tolerances εm for joint moments

(Eqn. 2.40) were always smaller than the predefined limit (i.e. εf < 1 · 10−8N and εm <

1 · 10−8Nm).

In order to show that the methods in chapter 2 were implemented correctly, we compared

the output of UFBSIM to a similar software program called AnyBody r©(Rasmussen et al.,

2003). The comparison was carried out for all 201 instances in time for all trials of WN of all

four subjects. We performed the comparison stepwise. First, we compared the kinematic

results. Second, we compared the forces and moments predicted by the programs.

The results of the kinematic analysis computed with AnyBody r©and UFBSIM were identical

within numerical tolerances (Tab. 3.1). The comparison of the kinematic analysis included

the position analysis (subsection 2.1.1), the velocity analysis (subsection 2.1.2) and the

44 CHAPTER 3. RESULTS

(a) (b)

(c) (d)

Fig. 3.1: The musculoskeletal model for the activities that were performed bythe four subjects. a) Walking with normal (WN), slow (WS) and fast (WF)speed b) Knee bend (KB) c) Stair climbing upwards (SU) d) Stair climbingdownwards (SD).

acceleration analysis (subsection 2.1.3).

Before actually comparing calculated muscular activities of both programs, we compared

the resultant hip reaction moments (subsection 2.4.3). The hip reaction moments were

identical within numerical tolerances in AnyBody r©and UFBSIM (Tab. 3.1).

The muscular activities computed with AnyBody r©were compared to those computed with

UFBSIM for the min/max criterion with linear penalty ε = 0 and linear penalty ε = 1 · 103

(Eqn. 2.35). The latter case with ε = 1 · 103 is equivalent to the polynomial criterion

with exponent p = 1 without any upper bounds. We took muscular activities as design

variables. The course of the hip contact forces versus time for the two cases are shown as

an example for subject KWR, trial WN3 in Fig. 3.2.

3.1. MATHEMATICAL AND MECHANICAL VALIDATION 45

Tab. 3.1: Comparison of the results of AnyBody r©and UFBSIM. The discrepan-cies in kinematic data (∆r, ∆v, ∆a, ∆ω, ∆ω) are the maximal magnitudesof the differences of the particular vectors for all segments, for all instancesin time and for all trials of walking with normal speed. The discrepancyin hip reaction moments (∆MHip) is the maximum of the magnitude of thedifference in resultant hip reaction moments for all instances in time for alltrials of normal walking.

Differences HSR KWR IBL PFL

∆r in m 1.13 · 10−8 1.34 · 10−8 4.26 · 10−9 4.64 · 10−9

∆v in ms−1 1.69 · 10−5 1.57 · 10−5 1.63 · 10−5 1.57 · 10−5

∆a in ms−2 7.40 · 10−5 5.71 · 10−5 5.72 · 10−5 6.49 · 10−5

∆ω in s−1 2.10 · 10−5 1.98 · 10−5 2.09 · 10−5 2.22 · 10−5

∆ω in s−2 1.04 · 10−4 7.68 · 10−5 7.52 · 10−5 8.57 · 10−5

∆MHip in BWm 7.57 · 10−6 5.71 · 10−5 8.82 · 10−6 2.00 · 10−4

We summarized the discrepancies in calculated muscular activities and hip contact forces

for the two cases ε = 0 and ε = 1 · 103 in four boxplots (Fig. 3.3a-d).

The comparison of the muscular activity comprises the discrepancies in muscular activities

of all 51 muscles for all 201 instances in time for all trials of WN in each subject (Fig. 3.3a,b).

Analogously, the magnitudes of the discrepancies in hip contact forces (subsection 2.6.12)

stance phase swing phase

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xany

yany

zany

xufb

yufb

zufb

F

(a) min/max, ε = 0


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xany

yany

zany

xufb

yufb

zufb

F

(b) min/max, ε = 1 · 103

Fig. 3.2: Calculated hip contact forces for subject KWR, trial WN3. The differ-ences between the results of AnyBody r©and UFBSIM were so small that theyare not visible. The optimization criterion used was the min/max criterionwith muscular activities as design variables with different linear penalty a)ε = 0 and b) ε = 1 · 103.


between UFBSIM and AnyBody r©are shown for all 201 instances in time for all trials of

WN for each subject (Fig. 3.3c,d). The median of the discrepancies of muscular activities

was almost zero in both cases. Additionally, the interquartile range of the discrepancies

was small (maximally −5.13·10−5). However, there were some outliers for ε = 0 (Fig. 3.3a).

The maximal discrepancy was 0.11 for subject HSR.

The results for the hip contact forces were similar. The median of the discrepancies of

hip contact forces was almost zero in both cases. Additionally, the interquartile range was

small (maximally 0.015 BW ). Again, there were some outliers for ε = 0 (Fig. 3.3c). The

maximal discrepancy was 0.23 BW for subject HSR.

HSR

KW

RIB

LPFL

-0.15

-0.1

-0.05

0.0

0.05

0.1

0.15

Dif

fere

nce

of

Musc

ula

rA

ctiv

ity

x

(a) min/max, ε = 0

HSR

KW

RIB

LPFL

-0.15

-0.1

-0.05

0.0

0.05

0.1

0.15

Dif

fere

nce

of

Musc

ula

rA

ctiv

ity

x

(b) min/max, ε = 1 · 103

HSR

KW

RIB

LPFL

0.0

0.05

0.1

0.15

0.2

0.25

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(c) min/max, ε = 0

HSR

KW

RIB

LPFL

0.0

0.05

0.1

0.15

0.2

0.25

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(d) min/max, ε = 1 · 103

Fig. 3.3: Boxplots of the differences between calculated muscular activities (a,b)and calculated hip contact forces (c,d) with AnyBody r©and UFBSIM. The op-timization criterion used was the min/max criterion using muscular activitiesas design variables with different linear penalty ε. The boxplots comprise mini-mum, 25% percentile, median, 75% percentile and maximum. The interquartilerange for the boxplots a,b,d is almost zero

3.2. THE INFLUENCE OF THE OPTIMIZATION CRITERION EMPLOYED 47

3.2 The Influence of the Optimization Criterion Em-

ployed

We observed a reasonable agreement between measured hip contact forces and calculated

hip contact forces for walking with normal speed (WN) for all optimization criteria (Figs.

3.4, 3.5 and Tab. 3.2).

However, the optimization criterion employed had an effect on the calculated hip contact

forces (Figs. 3.4 and 3.5). In general a larger exponent p in the optimization criterion

produced larger hip contact forces (Fig. 3.4). Calculated hip contact forces using the soft

saturation criterion with exponents p = 2 and p = 3 showed only minor differences from

calculated hip contact forces using the polynomial criterion with analogous exponents p.

The median of the discrepancies was generally smaller then 0.55 BW . The relative root

mean square (Eqn. 2.43) of the discrepancies between calculated and measured hip contact

forces was generally smaller than 0.7 (Tab. 3.2). The min/max criterion produced the

largest discrepancies between measured hip contact forces and calculated hip contact forces

(maximally 2.28 BW for subject IBL, Fig. 3.5).

The patterns of the measured hip contact forces and the calculated hip contact forces over

time were comparable. Due to the absence of the ground reaction forces during the swing

phase, the calculated and measured hip contact forces were smaller during the swing phase

than during the stance phase (Fig. 3.6). Using an exponent p = 2 and minimizing muscular

activities the predicted number of peaks in hip contact forces during the stance phase was

identical to the number of measured peaks. Subjects HSR and KWR had two peaks during

the stance phase (Fig. 3.6a,b). The first peak was shortly after heel strike and the second

peak was shortly before toe off. Subject IBL had only one peak shortly after heel strike

(Fig. 3.6c) and subject PFL had no peak (Fig. 3.6d). Our models tended to over predict

measured hip contact forces during the stance phase while they under predicted hip contact

forces during the swing phase.

The choice of the design variables had also an effect on the calculated hip contact forces.

For the polynomial criterion with exponents p = 1 and p = 2 we calculated hip contact

forces using muscle forces and alternatively muscular activities as design variables. When

minimizing muscle forces, the level of activation of a muscle depends on its moment arm,

while the level of activation depends on the moment generating capacity when minimizing

muscular activities. Therefore, we expected that minimizing muscular activities would tend

to predict larger hip contact forces because muscles with a smaller moment arm might be


activated to a larger extent. However, for subjects HSR and KWR (Fig. 3.4) the predicted

hip contact forces for the polynomial criterion with p = 1 and p = 2 were larger when

minimizing muscle forces, especially the second peak during the stance phase.

Tab. 3.2: Arithmetic mean of the relative root mean square (Eqn. 2.43) ofthe discrepancies between measured hip contact forces and calculated hipcontact forces for all trials of WN.

Optimization Criterion Design Variables HSR KWR IBL PFL

Polynomial (p = 1) M. Forces 0.59 0.65 0.73 0.60Polynomial (p = 2) M. Forces 0.54 0.50 0.72 0.53Polynomial (p = 1) M. Activities 0.59 0.64 0.60 0.60Polynomial (p = 2) M. Activities 0.44 0.47 0.56 0.48Polynomial (p = 3) M. Activities 0.44 0.43 0.61 0.50min/max M. Activities 0.46 0.45 0.80 0.52

The optimization criterion employed also influenced the number of active muscles (Fig. 3.7).

Using a linear criterion (exponent p = 1) only five muscles were generally active at one

instant in time. The number of five active muscles corresponded to the number of degrees of

freedom. Only for instances in time when one muscle became saturated was an additional

muscle activated. This occurred only when using muscle forces as design variables. When

minimizing muscle forces, the linear criterion predicted maximally 7 active muscles. The

predicted number of active muscles increased significantly when using a non-linear criteria.

The differences between the non-linear criteria were small. We observed maximally between

27 and 35 active muscles for the non-linear criteria depending on the subject.

Using a linear criterion and muscle forces as design variables there was a period of 33 in-

stances in time during the stance phase of subject KWR, trial WN5 where the musculus

gastrocnemius lateralis was saturated. For these instances in time the musculus gastroc-

nemius medialis was recruited additionally. Both muscles are plantar flexors of the ankle

joint and knee flexors. Therefore, the reason for the recruitment of Musculus Gastrocne-

mius Medialis was either an additional need of a flexion moment in the knee joint or an

additional need of a plantar-flexion moment in the ankle joint.

This result supported the hypothesis that the linear criterion is physiologically not realistic

because the muscles are known to share load.

The method used was extremely efficient in terms of CPU time needed to perform the

calculations. The most efficient criterion was the linear criterion (Tab. 3.3) with about

0.5s of CPU time needed on the workstation and on the PC. The polynomial criterion

3.2. THE INFLUENCE OF THE OPTIMIZATION CRITERION EMPLOYED 49


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(a) Polynomial p = 1, M. Forces


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(b) Polynomial p = 2, M. Forces


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(c) Polynomial p = 1, M. Activities


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(d) Polynomial p = 2, M. Activities


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(e) Polynomial p = 3, M. Activities


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(f) min/max, M. Activities

Fig. 3.4: Calculated and measured hip contact forces versus time for subject KWRtrial WN5. The dashed lines represent the measured forces and the solid linesrepresent the calculated forces. Different criteria with different design variableswere taken to compute the hip contact forces.

with exponent p = 2 took about double the time. The min/max criterion took about five

times the time on both platforms. The increase for the other criteria was even higher and


Polynom

ial p

=1,

M. F

orces

Polynom

ial p

=2,

M. F

orces

Polynom

ial p

=1,

M. A

ctiv

ities

Polynom

ial p

=2,

M. A

ctiv

ities

Polynom

ial p

=3,

M. A

ctiv

ities

min

/max

,

M. A

ctiv

ities

0.0

0.5

1.0

1.5

2.0

2.5D

iffe

rence

of

Hip

Conta

ct F

orc

ein

BW

F

(a) HSR

Polynom

ial p

=1,

M. F

orces

Polynom

ial p

=2,

M. F

orces

Polynom

ial p

=1,

M. A

ctiv

ities

Polynom

ial p

=2,

M. A

ctiv

ities

Polynom

ial p

=3,

M. A

ctiv

ities

min

/max

,

M. A

ctiv

ities

0.0

0.5

1.0

1.5

2.0

2.5

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(b) KWR

Polynom

ial p

=1,

M. F

orces

Polynom

ial p

=2,

M. F

orces

Polynom

ial p

=1,

M. A

ctiv

ities

Polynom

ial p

=2,

M. A

ctiv

ities

Polynom

ial p

=3,

M. A

ctiv

ities

min

/max

,

M. A

ctiv

ities

0.0

0.5

1.0

1.5

2.0

2.5

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(c) IBL

Polynom

ial p

=1,

M. F

orces

Polynom

ial p

=2,

M. F

orces

Polynom

ial p

=1,

M. A

ctiv

ities

Polynom

ial p

=2,

M. A

ctiv

ities

Polynom

ial p

=3,

M. A

ctiv

ities

min

/max

,

M. A

ctiv

ities

0.0

0.5

1.0

1.5

2.0

2.5

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(d) PFL

Fig. 3.5: Differences between calculated and measured hip contact forces for allfour subjects and all trials of walking with normal speed. Different criteriawith different design variables were taken to compute the hip contact forces.

more pronounced on the PC. The choice of the design variables did not influence the CPU

times.

The polynomial criterion with exponent p = 2 produced good agreement between calculated

and measured hip contact forces, predicted a realistic number of active muscles and was

numerically efficient. We therefore used this criterion with muscular activities as design

variables throughout the rest of this study.

3.3 Comparison of Activities Performed

All activities investigated, except for knee bending (KB), showed a good agreement between

measured and calculated hip contact forces (Figs. 3.8 and 3.9). Bergmann et al. (2001)

3.3. COMPARISON OF ACTIVITIES PERFORMED 51


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5H

ip C

onta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(a) Subject HSR, trial WN3


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(b) Subject KWR, trial WN5


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(c) Subject IBL, trial WN3


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(d) Subject PFL, trial WN4

Fig. 3.6: Calculated and measured hip contact forces versus time for walking withnormal speed for all subjects. The dashed lines represent the measured forcesand the solid lines represent the calculated forces. The optimization criterionused was the polynomial criterion with exponent p = 2 and muscular activitiesas design variables.


0

5

10

15

20

25

30

35

40

Num

ber

of

acti

ve

musc

les

Polynomial p=1,

M. ForcesPolynomial p=1,

M. Activities

Polynomial p=2,

M. ActivitiesPolynomial p=3,

M. Activitiesmin/max,

M. Activities

Fig. 3.7: Number of activemuscles versus time forsubject KWR trial WN5,The number of degreesof freedom of the muscu-loskeletal model was 5.

reported measurements for walking with slow speed (WS), walking with fast speed (WF),

stair climbing upwards (SU), stair climbing downwards (SD) and knee bending (KB) for


Tab. 3.3: Arithmetic mean and standard deviation (std) of CPU Time forsolving the indeterminate problem for 201 instances in time. The model had5 independent equations of equilibrium and 51 independent unknown muscleforces. Computations were carried out on a SUN r© workstation and on a PCdescribed in section 2.5. The maximum usage of RAM was approximately140 MB.

Optimization Criterion Design VariablesCPU time SUN in s CPU time PC in s

mean std mean std

Polynomial (p = 1) M. Forces 0.53 0.03 0.55 0.04Polynomial (p = 1) M. Activities 0.53 0.03 0.56 0.04Polynomial (p = 2) M. Activities 1.06 0.06 1.09 0.08Polynomial (p = 3) M. Activities 5.07 0.88 13.47 4.54Soft Saturation (p = 2) M. Activities 2.96 0.14 6.04 2.09Soft Saturation (p = 3) M. Activities 5.55 0.47 17.31 3.64min/max M. Activities 2.72 0.06 2.66 0.05

all subjects, except for subject IBL who performed WN and SU only. The median of the

discrepancies in hip contact forces was about the same as for walking with normal speed

(Figs. 3.5 and 3.9). However, the interquartile range of the discrepancies and the maximal

discrepancies were larger for WS, WF, SU and SD than for WN.

In general the magnitudes of the measured hip contact forces were larger than the magni-

tudes of the calculated hip contact forces. Only for SU of subject IBL was the opposite

true.

Measured and calculated hip contact forces during stair climbing were larger during the

stance phase than during the swing phase (Fig. 3.8c,d). For walking with various speeds

(WS and WF) and stair climbing (SU and SD) the measured hip contact forces exceeded

the calculated hip contact forces during the whole swing phase. The effect was more

pronounced for stair climbing and was greater for SD than for SU.

As for WN there were two peaks in the measured and calculated hip contact forces during

WS and WF of subject KWR (Fig. 3.8a,b). With increasing walking speed these two peaks

became more pronounced.

Although the maximal discrepancies in hip contact forces were smaller for KB than for SD,

there was no reasonable agreement between measured and calculated hip contact forces

for KB. The median of the discrepancies in hip contact forces during KB ranged between

0.69 BW for subject PFL and 0.99 for subject KWR. The magnitudes of the measured hip

3.4. THE INFLUENCE OF THE SHIFT PARAMETER XS 53

contact forces exceeded the magnitudes of the predicted forces for all instances in time.

For subject KWR trial KB3 this is shown in Fig. 3.8e. Over the whole movement cycle the

measured forces were much larger than the predicted ones.

3.4 The Influence of the Shift Parameter xs

With an increasing shift parameter xs we observed an increase in predicted hip contact

forces for all subjects for WN. Fig. 3.10 illustrates this trend for subject KWR, trial WN5.

This increase in predicted hip contact forces was more pronounced in the swing phase of

gait.

The value of the shift parameter xs influenced the discrepancies between calculated hip

contact forces and measured hip contact forces (Fig. 3.11). The influence differed among

the four subjects, however a shift parameter between 0.015 and 0.025 led to smaller median

and maximal discrepancies for all subjects.

For those subjects who also performed SD and KB we investigated the influence of the

shift parameter on these activities (Figs. 3.12 and 3.13). We observed that larger values of

the shift parameter xs were needed to produce smaller maximal discrepancies and median

discrepancies between calculated and measured hip contact forces.

A larger value of the shift parameter xs, predicted more active muscles (Fig. 3.14). The

difference in the number of active muscles when using xs > 0 and the number of active

muscles when using xs = 0 were the additionally recruited antagonistic muscles.

Antagonistic muscle activity that was indicated by EMG signals could only be predicted

when using a shift parameter xs > 0 (Fig. 3.15b). Without any shift parameter (xs = 0)

“antagonistic”muscle activity was predicted only for secondary muscle functions (Fig. 3.14).

This is illustrated for subject KWR, trial WN4. For this trial we compared the measured

EMG signals with predicted activities for the gluteal muscles and the musculus tibialis

anterior. It is not possible to derive quantitative values from EMG signals, however a

comparison of high and low activity in a temporal way is possible. For instances in time

with high EMG activity the model also predicted large activities for both muscle groups

and vice versa. We observed activity of the gluteal muscles during late swing phase with

and without shift parameter. The gluteal muscles are mainly hip extensor muscles and it

might appear that the activity of these muscles was co-contraction. However, the activity

of the gluteal muscles can be explained by the secondary function as hip abductors. The

musculus tibialis anterior is a dorsi flexor of the ankle joint and is an antagonistic muscle



-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(a) subject KWR, trial WS2


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(b) subject KWR, trial WF5


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(c) subject KWR, trial SU5

swing phase stance phase

-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

F

(d) subject KWR, trial SD5

stance phase

-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

F

(e) subject KWR, trial KB3

Fig. 3.8: Calculated and measured hip contact forces versus time for subject KWRand one trial per activity performed. The dashed lines represent the measuredforces and the solid lines represent the calculated forces. The optimizationcriterion used was the polynomial criterion with exponent p = 2 and muscularactivities as design variables.

during late swing phase. Consequently, it was predicted to be silent without any shift


WS

WF

SU SD KB

0.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(a) HSR

WS

WF

SU SD KB

0.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(b) KWR

WS

WF

SU SD KB

0.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(c) IBL

WS

WF

SU SD KB

0.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(d) PFL

Fig. 3.9: Differences between calculated and measured hip contact forces for allsubjects and all trials. The optimization criterion used was the polynomialcriterion with exponent p = 2 and muscular activities as design variables. Theactivities walking with slow speed (WS), walking with fast speed (WF), stairclimbing downwards (SD), and knee bend (KB) were not performed by subjectIBL (Bergmann et al., 2001).

parameter xs. However, the EMG signals clearly indicated an activity as predicted with

the extended criteria (xs > 0).



-1

0

1

2

3H

ip C

onta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(a) xS = 0.010


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(b) xS = 0.015


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(c) xS = 0.020


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(d) xS = 0.025


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(e) xS = 0.030


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(f) xS = 0.035

Fig. 3.10: Influence of the shift parameter xs on the predicted hip contact forcesfor subject KWR trial WN5. The dashed lines represent the measured forcesand the solid lines represent the calculated forces. The optimization criterionused was the polynomial criterion with exponent p = 2 and muscular activitiesas design variables.


=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(a) HSR

=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(b) KWR

=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(c) IBL

=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(d) PFL

Fig. 3.11: Influence of the shift parameter xs on the calculated hip contact forcesduring walking with normal speed. The optimization criterion used was thepolynomial criterion with exponent p = 2 and muscular activities as designvariables.


=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(a) subject HSR, activity SD

=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(b) subject KWR, activity SD

=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(c) subject PFL, activity SD

=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0D

iffe

rence

of

Hip

Conta

ct F

orc

ein

BW

F

(d) subject HSR, activity KB

=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(e) subject KWR, activity KB

=0.0

00

=0.0

10

=0.0

15

=0.0

20

=0.0

25

=0.0

30

=0.0

350.0

0.5

1.0

1.5

2.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(f) subject PFL, activity KB

Fig. 3.12: Influence of the shift parameter xs on the calculated hip contact forcesduring SD and KB. The optimization criterion used was the polynomial crite-rion with exponent p = 2 and muscular activities as design variables.



-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(a) subject KWR, trial SU4 (xs = 0.000)


-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(b) subject KWR, trial SU4 (xs = 0.035)


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(c) subject HSR, trial SD4 (xs = 0.000)


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

F

xMeas

yMeas

zMeas

xSim

ySim

zSim

(d) subject HSR, trial SD4 (xs = 0.035)


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(e) subject PFL, trial KB3 (xs = 0.000)


-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Hip

Conta

ct F

orc

ein

BW

xMeas

yMeas

zMeas

xSim

ySim

zSim

F

(f) subject PFL, trial KB3 (xs = 0.035)

Fig. 3.13: Comparison of calculated hip contact forces with a shift parameter xs =0 (a,c,e) and with a shift parameter xs = 0.035 (b,d,f) during SU, SD and KB.The dashed lines represent the measured forces and the solid lines represent thecalculated forces. The optimization criterion used was the polynomial criterionwith exponent p = 2 and muscular activities as design variables.



0

5

10

15

20

25

30

35

40

45

50

55

Num

ber

of

acti

ve

musc

les

xS = 0.005

xS = 0.015

xS = 0.025

xS = 0.035

Fig. 3.14: Number of activemuscles versus time forsubject KWR trial WN5.


0.0

0.05

0.1

0.15

Musc

ula

rA

ctiv

ity

0

50

100

150

200

250

300

EM

G s

ignal

in r

elat

ive

unit

s

xS = 0.00

xS = 0.02

xS = 0.03

xS = 0.04

EMG

x

(a) gluteal muscles


0.0

0.05

0.1

0.15

Musc

ula

rA

ctiv

ity

0

500

1,000

1,500

EM

G s

ignal

in r

elat

ive

unit

s

xS = 0.00

xS = 0.02

xS = 0.03

xS = 0.04

EMG

x

(b) tibialis anterior

Fig. 3.15: Comparison of calculated muscular activity with measured EMG sig-nals.

3.5. THE SENSITIVITY TO VARIATIONS IN GROUND REACTION FORCES 61

3.5 The Sensitivity to Variations in Ground Reaction

Forces

In order to estimate the impact of a measuring error in the ground reaction forces (GRF)

we carried out two different variations. First, we changed the magnitudes of the vectors

of ground reaction forces without changing their direction. Second, we changed single

components of the vectors of GRF, thus changing the magnitudes and the directions.

The muscular activities and hip contact forces were almost linearly related to variations in

the magnitudes of the GRF during stance phase. The linear relationship held for variations

of the magnitudes of the GRF by ±40% for the polynomial and the soft saturation criterion

with arbitrary exponents p. The linear relationship also held for the min/max criterion

and the polynomial criterion with exponent p = 2 and a shift parameter xs > 0. During

swing phase there were no changes in muscular activities and hip contact forces because

the foot had no contact to the ground. Consequently, no GRF were transmitted during

the swing phase.

The discrepancies between calculated hip contact forces with measured GRF and calculated

hip contact forces with variations in the x and y components by ±10% of GRF were small

during walking with normal speed (Figs. 3.16 and 3.17). The median of these discrepancies

was less than 0.05 BW for 10% variations in the x and y components of the GRF. The

maximal discrepancies were 0.56 BW for subject PFL. The maximal discrepancies for the

other subjects were much smaller.

The discrepancies between calculated hip contact forces with measured GRF and calculated

hip contact forces with variations in the z components by ±10% were larger (Figs. 3.16 and

3.17). The median of the discrepancies was up to 0.15 BW and were maximally 0.78 BW

for subject PFL.

The differences between the magnitudes of the measured GRF and the varied GRF were

smaller than 10% because we changed single components of the vectors by ±10%. We

observed that the discrepancies in the magnitudes of the hip contact forces were sometimes

larger than 10%. This indicated that the directions rather than the magnitudes of the GRF

influenced the calculated hip contact forces.

We observed a similar behavior for the linear criterion and the min/max criterion.


3.6 The Sensitivity to Variations in Muscle Attach-

ment Points

In order to estimate the impact of a modelling error of a muscle path, we investigated the

effect of translating a deviating point for the musculus rectus femoris.

Changes of the deviating point were performed in the pelvis coordinate system (subsec-

tion 2.6.11). Changes in the y direction influenced the moment arm with respect to flex-

ion/extension. Therefore, we expected to find the largest discrepancies for the y direction.

However, the variations of the muscle attachment point in x direction resulted in the largest

discrepancies (Figs. 3.18 and 3.19). A variation in x direction influenced the moment arm

with respect to abduction/adduction.

The model for subject IBL was most sensitive to changes of ±2 cm (Fig. 3.19c). The

maximal discrepancies between calculated hip contact forces with original muscle path and

calculated hip contact forces with variations in the muscle path was 0.94 BW for subject

IBL. The corresponding difference in the muscular activity of the musculus rectus femoris

was 0.13.

For subject KWR trial WN5 the variations of ±2 cm showed only minor effects (Fig. 3.18)

on calculated hip contact forces. The effect was also largest for variations in x direction.

The maximal difference in calculated muscular activity for the musculus rectus femoris was

−0.05.

Variations in deviating points for musculus semitendinosus and the musculus tensor fasciae

latae showed similar results.

3.6. THE SENSITIVITY TO VARIATIONS IN MUSCLE ATTACHMENT POINTS 63


-1

0

1

2

3H

ip C

onta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(a) Fx+


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(b) Fx−


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(c) Fy+


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(d) Fy−


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(e) Fz+


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(f) Fz−

Fig. 3.16: The effect of varying single components of the ground reaction forceson calculated hip contact forces for subject KWR trial WN5 by ±10%. Thedashed lines are the hip contact forces calculated using the measured groundreaction forces. The solid lines represent the calculated hip contact forces whenvarying components of the vectors of ground reaction forces. The optimizationcriterion used was the polynomial criterion with exponent p = 2 and muscularactivities as design variables.


+ - + - + -0.0

0.2

0.4

0.6

0.8

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(a) HSR

+ - + - + -0.0

0.2

0.4

0.6

0.8

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(b) KWR

+ - + - + -0.0

0.2

0.4

0.6

0.8

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(c) IBL

+ - + - + -0.0

0.2

0.4

0.6

0.8

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(d) PFL

Fig. 3.17: The effect of varying single components of the ground reaction force by±10% on calculated hip contact forces for normal walking for all subjects. Theoptimization criterion used was the polynomial criterion with exponent p = 2and muscular activities as design variables.

3.6. THE SENSITIVITY TO VARIATIONS IN MUSCLE ATTACHMENT POINTS 65


-1

0

1

2

3H

ip C

onta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(a) rx+


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(b) rx−


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(c) ry+


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(d) ry−


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(e) rz+


-1

0

1

2

3

Hip

Conta

ct F

orc

ein

BW

xSim

ySim

zSim

xVar

yVar

zVar

F

(f) rz−

Fig. 3.18: The effect of translating a deviating point for the musculus rectusfemoris for subject KWR trial WN5 by ±2 cm. The dashed lines representthe calculated hip contact forces for the original model. The solid lines repre-sent the calculated hip contact forces when changing components of the muscleattachment point. The optimization criterion used was the polynomial crite-rion with exponent p = 2 and muscular activities as design variables.


+ - + - + -0.0

0.25

0.5

0.75

1.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(a) HSR

+ - + - + -0.0

0.25

0.5

0.75

1.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(b) KWR

+ - + - + -0.0

0.25

0.5

0.75

1.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(c) IBL

+ - + - + -0.0

0.25

0.5

0.75

1.0

Dif

fere

nce

of

Hip

Conta

ct F

orc

ein

BW

F

(d) PFL

Fig. 3.19: The effect of translating a muscle attachment point for the musculusrectus femoris ±2 cm on calculated hip contact forces for normal walking forall subjects. The optimization criterion used was the polynomial criterion withexponent p = 2 and muscular activities as design variables.

67

Chapter 4

Discussion

The aim of this study was to predict muscle forces and hip contact forces in the human lower

limb during various activities. We used inverse dynamics in combination with optimization

techniques and an adapted model previously developed by Heller et al. (2001) and Heller

(2002). The model was scaled to four individual subjects. For these subjects we compared

calculated hip contact forces to in vivo measured hip contact forces (Bergmann et al.,

2001).

The challenge when analyzing musculoskeletal systems is that musculoskeletal systems are

generally redundant and a given movement can be achieved by an infinite number of muscle

activation patterns. In nature the redundancy is resolved by the central nervous system

(CNS). Weber and Weber (1836) hypothesized that muscle recruitment is performed in a

way such that muscular effort is minimized during routine activities:

“Man binds his movements to certain rules even if he cannot express these

rules with words. These rules are based totally on the structure of his body and

on external conditions. The principle according to which only these rules are

deduced from these circumstances is apparently that of the smallest muscular

effort by which the goal of walking can be attained for a given structure of

the human body and in given external conditions. Based on this principle it

must be possible to determine not only laws of walking and running as observed

in trained walkers and runners but also the laws of countless other activities

and movements which are frequently carried out by man.”(translated by Paul

Maquet, Weber and Weber, 1991)

In mathematical terms a musculoskeletal system is indeterminate. Thirty years ago, Seireg

68 CHAPTER 4. DISCUSSION

and Arvikar (1973) were the first to define a multivariate function depending on the muscle

forces of a musculoskeletal system. They minimized the multivariate function subject to the

constraints that the muscle forces are non-negative and that the muscle forces must fulfill

the dynamic equilibrium. The multivariate function is called the objective function or the

optimization criterion. Thus, the objective function was the mathematical representation

of the rules in the hypothesis of Weber and Weber (1836). Determination of the minimum

of a multivariate function subject to constraints is called optimization.

4.1 Using Optimization Techniques to Predict Muscle

Forces

When using optimization techniques to predict muscle forces, we must be aware that the

hypothesis of Weber and Weber (1836) cannot be proven and the exact goal of the CNS

remains unknown. Crowninshield and Brand (1981b) pointed out that the goal of the CNS

may depend on the activity performed and the physical capabilities of an individual.

Bergmann et al. (2001) measured hip contact forces in vivo for four subjects via prostheses

equipped with telemetry units. Their results showed that the patterns of measured ground

reaction forces and measured hip contact forces differed between the subjects. For a sin-

gle subject performing an activity however, the measured ground reaction forces and the

measured hip contact forces were highly reproducible among trials. Therefore, it seems to

be appropriate to assume that muscle recruitment follows a strategy in these subjects.

So far, all optimization studies assumed that the goal of the CNS when recruiting muscles is

to minimize muscular effort. However antagonistic muscle activity is in clear contradiction

to the goal of minimizing muscular effort (Crowninshield, 1978). We postulated that in

addition to minimizing muscular effort another important goal is to ensure joint stability

because a muscle becomes stiffer when exerting a larger force (Collins, 1995). Thus, activ-

ity of agonistic and antagonistic muscles causes pre-loading of joints and thereby increases

joint stability. The need for stability in individuals may increase with unfavorable external

conditions or difficult tasks to perform. Forster et al. (2004) extended a standard opti-

mization criterion to predict and control the amount of co-contraction in a simple planar

model. Thus the extended criterion increased joint stability although the primary goal was

still to minimize muscular effort. In this work we have applied the extended criterion to

enforce antagonistic muscle activity for a complex model of the human lower limb.

4.2. MULTIBODY-DYNAMICS APPROACH 69

4.2 Multibody-Dynamics Approach

We developed a software program called UFBSIM to determine the muscular forces using

an approach recently proposed by Damsgaard et al. (2001).

Our program UFBSIM is capable of simulating general musculoskeletal systems (Forster

et al., 2002), nevertheless it is extremely efficient (Tab. 3.3) and robust. For this study

we executed the program approximately 500 times, each call with 201 instances in time.

We performed calculations for multiple trials of four subjects during various activities

using different optimization criteria on different computer platforms. However, we did not

encounter any crashes of our program.

Although, inverse dynamics in combination with optimization techniques has been widely

used for the last thirty years, there is no standard program to perform the computations.

Every group has written their own software programs (e.g. Brand et al., 1994; Heller et al.,

2001; Stansfield et al., 2003). There has been no attempt to validate the programs by

comparing the results to the results of similar programs developed by others. Moreover,

Herzog and Binding (1993) showed that Davy and Audu (1987) calculated muscle forces

that were in contradiction to the assumptions they had made. Therefore, we transferred

the musculoskeletal model that we used to another software (AnyBody r©, Rasmussen et al.,

2003) and performed a stepwise comparison of the results.

The results of the kinematic analysis in AnyBody r©and UFBSIM were identical within

numerical tolerances for all trials of walking with normal speed (WN) of all four subjects

(Tab. 3.1).

We employed the min/max criterion with a linear penalty ε = 0 (Eqn. 2.35) and a linear

penalty ε = 1 · 103. For large linear penalties the solution of the min/max criterion is

identical to the solution of a linear criterion (exponent p = 1) without any upper bounds.

In general the predicted muscular activities and hip contact forces with AnyBody r©and

UFBSIM for all trials of WN of all four subjects were identical within numerical tolerances

for both linear penalties (Fig. 3.3). Consequently, the coefficient matrix for the reactions

(Eqn. 2.13), the coefficient matrix for the muscle forces (Eqn. 2.16), the reducing of the

linear constraint equations (subsection 2.3.6), the scaling of the design variables (subsec-

tion 2.3.7) and finally the optimization criteria were implemented correctly into UFBSIM.

However, there were some outliers in the discrepancies between AnyBody r©and UFBSIM

when using no linear penalty (ε = 0). The maximal discrepancies occurred for subject HSR.

The maximal discrepancy in muscular activity was 0.11, while the maximal discrepancy in


hip contact forces was 0.23 BW . There are multiple possible reasons for these outliers when

using no linear penalty. First, the min/max criterion may amplify the small discrepancies

of the kinematic analysis. However, the min/max criterion tends to respond linearly to

changes in the input data (section 3.5) and the maximal discrepancies obtained were too

large to be caused by linear amplification of very small errors. Second, there might have

been some situations where the problem is ill conditioned (e.g. the linear constrained equa-

tions were almost linear dependent). Neither AnyBody r©nor UFBSIM have any measure

for the ill conditioning of a problem. However, it is likely that at least a single muscle

would be very active for such a solution. We did not observe single peaks in any of the

muscular activities. Third, there were differences in the implementation of the min/max

criterion in AnyBody r©and UFBSIM which was most likely the cause for the discrepancies.

The differences in the implementation of the min/max algorithm in AnyBody r©and UFB-

SIM concerned different numerical algorithms employed and different termination criteria

defined (Fig. 2.10). AnyBody r©used a Simplex algorithm to find the solution of the lin-

ear program while UFBSIM employed an interior point method (Tab. 2.2). Additionally,

AnyBody r©used an LU-factorization to remove the linear dependent constraint equations

while UFBSIM performed this by means of a singular value decomposition. A singular

value decomposition is numerically more costly but more stable, specifically in cases when

factoring rank deficient matrices. Because different numerical algorithms were employed

in AnyBody r©and UFBSIM the termination criteria cannot be compared exactly. The de-

cision if a solution was unique, if a muscle was maximally activated or not activated might

be different for some situations in AnyBody r©and UFBSIM. Although the discrepancies

were very small there is a need of further investigating this issue when using the min/max

criterion.

4.3 Musculoskeletal Model

We used a slightly modified musculoskeletal model compared to the originally developed

model by Heller et al. (2001) and Heller (2002). The model consisted of four segments: the

pelvis, the thigh, the shank and the foot. The segments were connected by three joints:

the hip joint, the knee joint and the ankle joint. The model was reconstructed from CT

images and scaled to the four individual subjects. In order to better represent the muscle

path of the musculus rectus femoris, the musculus semitendinosus and the musculus tensor

fasciae latae during the whole movement we added deviating points for these muscles.

4.4. THE INFLUENCE OF THE OPTIMIZATION CRITERION 71

The muscles had to equilibrate 5 degrees of freedom, namely all three resultant reaction

moments in the hip joint and additionally the flexion/extension moment in the knee joint

and the ankle joint. Although we prescribed the motion for the remaining rotational

degrees of freedom, the muscles did not need to equilibrate the resultant moments of these

movements. These movements were assumed to be enforced by the shape of the articular

surfaces and passive elements such as ligaments and joint capsules.

In the original model the muscles equilibrated the resultant abduction/adduction moment

in the knee joint as well (Heller et al., 2001; Heller, 2002). However, without the additional

deviating points of the muscles and with this additional degree of freedom, we calculated

hip contact forces that exceeded the measured hip contact forces considerably during heel

strike and toe off.

4.4 The Influence of the Optimization Criterion

It is well accepted that the optimization criterion employed strongly influences the load

sharing of muscles (Dul et al., 1984b; Pedersen et al., 1987; Tsirakos et al., 1997; Ras-

mussen et al., 2001). So far, all studies that compared calculated hip contact forces to

in vivo measured hip contact forces used different optimization criteria and different mus-

culoskeletal models. Nevertheless, all these studies reported a good agreement between

calculated and measured hip contact forces. Therefore, Stansfield et al. (2003) concluded

that the influence of the optimization criterion employed might be more severe on calcu-

lated muscle forces than on calculated hip contact forces. Our results showed that the

optimization criterion does affect the calculated hip contact forces. For all trials of WN of

all subjects we found an increase in hip contact forces with an increasing exponent p in the

optimization criterion (Fig. 3.4). These results are in good agreement to the findings of

Crowninshield and Brand (1981a), who compared linear and non-linear polynomial criteria

with exponents p = 1, 2, 3, 5 for a model of the human lower limb during walking. They

found relative differences of approximately 25% between the magnitude of calculated hip

contact forces with the linear criterion and the magnitude of hip contact forces calculated

with an exponent p = 5. The hip contact forces increase with an increasing exponent p

because of increasing muscle synergism. With increasing synergism muscles with smaller

moment arms are recruited. A muscle with a smaller moment arm must exert a larger force

in order to preserve the dynamic equilibrium. As the sum of muscle forces increases, the

joint contact forces increase.


We used muscle forces and muscular activities as design variables. When using muscle

forces as design variables, the recruitment of a muscle depends generally on its moment

arm. For muscular activities as design variables, the load sharing depends generally on the

moment generating capacity of a muscle. The moment generating capacity of muscle i is

the product of its maximal force fmax,i and its moment arm. Because minimizing muscle

forces tends to activate muscles with large moment arms, we expected that minimizing

muscle forces would result in equal or lower joint contact forces than minimizing muscular

activities for the same exponent p as shown by Dul et al. (1984b) in a simple one joint

model. However, for some cases (subjects HSR and KWR) we observed the opposite for

the hip contact forces. Therefore, we assume that the increase of hip contact forces when

minimizing muscle forces was caused by the complex interaction of multi-articular muscles.

For all trials of WN of all subjects the muscular activities and the hip contact forces cal-

culated with the soft saturation criterion and the polynomial criterion for same exponents

showed only minor differences because the maximal muscular activity during WN was ap-

proximately 0.3. For small muscular activities the separate, univariate functions (Fig. 2.2)

of the objective function are quite similar apart from a constant shift in vertical direction,

which can be shown by Taylor-Series expansion. A vertical shift does not influence the

location of the minimum. The differences for exponents p > 3 would be even smaller than

for p = 3 because muscles would be activated more equally for both criteria, and the maxi-

mal muscular activity would further decrease. With smaller muscular activity the separate

functions of the polynomial and soft saturation criterion become more similar. However,

we experienced convergence problems for both criteria when using exponents p > 3.

Using a linear optimization criterion (exponent p = 1) the number of muscles recruited

at one instant in time matched the number of degrees of freedom of the model (Fig. 3.7).

An additional muscle was only recruited when another muscle reached its upper bound.

Although Basmajian (1974) reported such a behavior during elbow flexion, it is commonly

agreed that muscles tend to share load (Dul et al., 1984b; Siemienski, 1992). Therefore, the

muscle recruitment predicted by the linear criterion seems to be unrealistic. For all non-

linear criteria the number of active muscles was much higher than the number of degrees of

freedom (Fig. 3.7). Furthermore, the differences in the number of muscles recruited at one

instant in time between the non-linear criteria was small. However, we expected that the

number of muscles recruited would be identical for all non-linear criteria and that these

muscles would be the synergists. Accordingly, we proposed calling antagonistic muscles

the muscles that are predicted to be silent from the min/max criterion in subsection 2.3.2.

The discrepancies between the number of active muscles was probably caused by numerical

4.4. THE INFLUENCE OF THE OPTIMIZATION CRITERION 73

inaccuracies.

The soft saturation and the polynomial criterion converge to the min/max criterion for ex-

ponents p →∞ (Fig. 3.4). The linear criterion and the min/max criterion are two extreme

cases. The linear criterion predicts no or only minimal synergism, while the min/max

criterion predicts maximal synergism. We assume that neither the linear criterion nor the

min/max criterion represents the strategy of the CNS correctly. Our results indicated that

an exponent of p = 2 or p = 3 seems to most accurately model the strategy of the CNS for

certain controlled movements.

Hip contact forces predicted with the min/max criterion were generally larger than mea-

sured hip contact forces. However, the min/max criterion is numerically very efficient

(Tab. 3.3) and stable and might serve as an estimation for maximal synergism. Addi-

tionally, the solution of the min/max criterion could be used as an initial guess for the

polynomial and the soft saturation criterion for large exponents p.

Another aspect to consider when choosing an optimization criterion is the aspect of control.

Rasmussen et al. (2001) pointed out that the linear relationship between external load and

muscle activation for the min/max criterion is attractive because a change in the external

load results in a linear increase or decrease in all muscle activities. This also applies to the

polynomial criteria without any upper bounds.

4.4.1 Comparison to Other Studies

Comparing calculated muscle forces to measured EMG signals for two subjects, Pedotti

et al. (1978) observed that the polynomial criterion with an exponent p = 2 and with muscle

tensile stresses as design variables resulted in the best agreement between measured EMG

signals and calculated muscle forces. We assumed that the maximal force of a muscle is

proportional to the maximal muscle stress (Eqn. 2.17). Therefore minimizing muscular

activities is equivalent to minimizing muscle stresses in our study. Similar to Pedotti

et al. (1978) we observed the best agreement between measured hip contact forces and

calculated hip contact forces without co-contraction for the polynomial criterion with an

exponent p = 2 and with muscular activities as design variables.

Many studies predicted hip contact forces in the human lower limb, however only a few

studies compared the calculated hip contact forces to in vivo measured hip contact forces.

Brand et al. (1994) compared calculated hip contact forces to measured hip contact forces

for one subject. They used the cubic polynomial criterion (p = 3) with muscle stresses as


design variables. However, a direct comparison between the measured hip contact forces

and calculated hip contact forces was not possible because the gait analysis was performed

six weeks after the hip contact forces were measured. Brand et al. (1994) found a reasonable

agreement between measured and calculated hip contact forces. Generally, the calculated

hip contact forces tended to be larger than the measured hip contact forces (Tab. 4.1).

A direct comparison of our study to the study of Brand et al. (1994) was not possible

because the subjects and the musculoskeletal model used were different. The maximal

discrepancies between measured and calculated hip contact forces was less than 1.5 BW

in our study during walking with normal speed among all subjects when using the same

criterion as Brand et al. (1994). This was about the same as reported by Brand et al.

(1994). Because Brand et al. (1994) were not able to make a direct comparison between hip

contact measurements and gait analysis, they only compared magnitudes of the measured

hip contact forces to magnitudes of the calculated hip contact forces. In our study we

compared the magnitudes of the differences between the vectors of measured hip contact

forces and the vectors of calculated hip contact forces (subsection 2.6.12). Comparing

the magnitude of the difference between two vectors is more rigorous than comparing the

difference between the magnitudes of two vectors. In the latter case a discrepancy between

the directions of the two vectors does not have an impact on the differences while it does

influence the differences in the first case.

Heller was the first to make a direct comparison between measured hip contact forces and

calculated hip contact forces for the exact same cycle for four subjects during walking and

stair climbing upwards (Heller et al., 2001; Heller, 2002). They used a linear criterion (ex-

ponent p = 1) with muscle forces as design variables and a quadratic criterion with muscle

stresses as design variables (Heller, 2002). They used the data collected by Bergmann et al.

(2001) as input data. Although using a linear criterion, the calculated hip contact forces

of Heller et al. (2001); Heller (2002) tended also to over predict the measured hip contact

forces (Tab. 4.1). Even though we used the same input data and adapted their model we

did not obtain the same results as Heller et al. (2001) and Heller (2002). We were able to

exclude programming errors in our program UFBSIM because we compared our program to

AnyBody r©(section 3.1). Heller (2002) calculated average discrepancies between calculated

and measured hip contact forces of 16% for the linear and 32% for the quadratic crite-

rion during walking with normal speed (WN). During stair climbing upwards (SU) these

average differences were 17% for the linear criterion and 29% for the quadratic criterion.

Although Heller (2002) did not state explicitly, it seems that they calculated their average

differences as arithmetic means of the differences between the magnitudes of the measured

4.5. COMPARISON OF ACTIVITIES PERFORMED 75

hip contact forces and the magnitudes of the calculated hip contact forces. We consider

our relative root mean square (RMS, Eqn. 2.43) to be more appropriate because it also

takes the differences in the directions of the measured and calculated hip contact forces

into account. Additionally, the arithmetic mean of differences that might have opposite

signs pretend a better agreement than actually exists. Because the relative RMS is more

rigorous, the values were much larger for comparable criteria (Tab. 3.2).

Stansfield et al. (2003) recorded kinematic data, ground reaction forces and measured hip

contact forces for two of the four subjects used in the study of Bergmann et al. (2001).

Thus, they were also able to make a direct comparison between measured and calculated

hip contact forces. They used the double linear approach originally proposed by Bean et al.

(1988). Their calculated hip contact forces were in reasonable agreement with measured hip

contact forces (Tab. 4.1). Stansfield et al. (2003) used also a different musculoskeletal model

and employed an optimization criterion that was not used in this study. We assumed that

the double linear programming approach they used behaves more or less like the min/max

criterion with a moderate linear penalty ε. However, a direct comparison to our study

was not possible. Stansfield et al. (2003) calculated average differences between calculated

hip contact forces and measured hip contact forces. These average differences were in the

same order of magnitude as the average differences calculated by Heller (2002). They also

calculated the relative discrepancies between measured hip contact forces and calculated

hip contact forces for the two measured peaks during stance phase. The discrepancies were

small (6.9% − 32.9%), however it appears that the calculated peaks did not occur at the

same time as the measured peaks and that the discrepancies for other instances in time

were much larger.

Generally, the discrepancies between calculated hip contact forces and measured hip contact

forces in our study were in the same range as for comparable studies (Tab. 4.1). We have

presented a comprehensive analysis of the most common criteria (Fig. 3.5), using the same

type of data and the same musculoskeletal model. This allows to directly compare the

different optimization criteria with respect to calculated muscular activities and hip contact

forces.

4.5 Comparison of Activities Performed

We calculated hip contact forces for walking with slow speed (WS), walking with normal

speed (WN), walking with fast speed (WF), stair climbing upwards (SU), stair climbing


Tab. 4.1: Overview of in vivo measured and calculated peak hip contact forcesin literature (adapted from Brand et al., 1994; Heller, 2002).

Activity Measurement Hip contact Calculation Hip contactforce in BW force in BW

Walking Rydell (1966) 1.6 - 2.5Walking Davy et al. (1988) 1.8 - 3.1Walking Kotzar et al. (1991) 2.7Walking Bergmann et al. (1993) 3.2 - 4.1Walking Paul (1966) 4.0 - 10.0Walking Seireg and Arvikar (1975) 4.0 - 6.0Walking Crowninshield and Brand (1981a) 3.0 - 6.0Walking Crowninshield et al. (1978) 3.0 - 6.0Walking Brand et al. (1994) 1.9 - 3.2 Brand et al. (1994) 2.7 - 4.0Walking Bergmann et al. (2001) 2.1 - 3.1 Heller et al. (2001); Heller (2002) 2.1 - 3.9Walking Stansfield et al. (2003) 2.2 - 2.8 Stansfield et al. (2003) 3.0 - 3.3Stair climbing Rydell (1966) 3.8Stair climbing Davy et al. (1988) 2.6Stair climbing Crowninshield et al. (1978) 6.0 - 8.0Stair climbing Bergmann et al. (2001) 2.3 - 3.7 Heller et al. (2001); Heller (2002) 2.6 - 5.0

downwards (SD) and knee bending (KB). We found a reasonable agreement between mea-

sured hip contact forces and calculated hip contact forces among all subjects and among

all activities except for KB (Figs. 3.5 and 3.9). Measured hip contact forces during knee

bending (KB) were much smaller than measured hip contact forces during walking or stair

climbing and smaller than the predicted hip contact forces. They were comparable to the

measured hip contact forces during the swing phase of the other activities. During walking

and stair climbing only a single leg supported the body for most instances in time, while

during KB both legs permanently supported the body. Thus the body weight was shared

by both legs. Accordingly, the peak ground reaction forces during KB were about half

the peak ground reaction forces during single leg support of walking and stair climbing.

Because GRF are input for our software, the effect of single and double leg support was

considered in the calculations.

We observed that calculated hip contact forces during SU, SD and especially during KB

were smaller than measured hip contact forces for most instances in time (Fig. 3.8). Stair

climbing and knee bending might require a larger extent of antagonistic muscle activity

(section 4.6).

The median of the discrepancies between measured hip contact forces and calculated hip

contact forces was about 0.5 BW for all activities except for KB (0.9 BW ). In addition

to co-contraction another possible explanation for the discrepancies during KB is that the

model of Heller (2002) was originally developed to calculate hip contact forces during gait

4.6. THE INFLUENCE OF CO-CONTRACTION 77

and stair climbing. The muscle paths of the model therefore accurately represented the

range of motion during walking and stair climbing, but perhaps not the extreme flexions

occurring during KB.

4.6 The Influence of Co-Contraction

All previous optimization studies have exclusively minimized muscular effort. Conse-

quently, a muscle contributed minimally to the objective function when it exerted no force.

Mathematically the separate, univariate functions had their minimums at the origin. We

enforced co-contraction by shifting this minimum to the right to small muscular activities

(Forster et al., 2004). Thus, a muscle contributed less to the objective function when it

exerted a small force than when it exerted no force. The amount of shifting was controlled

by a shift parameter xs. A large shift parameter produced more extensive co-contraction.

Accordingly, the calculated hip-contact forces increased (Fig. 3.10). Additionally, the num-

ber of active muscles increased (Fig. 3.14) because both the synergistic and antagonistic

muscles were recruited.

We compared calculated muscular activities to measured EMG signals (Fig. 3.15). Gen-

erally, measured EMG signals and calculated muscular activities showed the same course.

Predicted muscular activities with the extended criterion were more realistic. We were able

to demonstrate this for the example of the musculus tibialis anterior: Conventional criteria

predicted the musculus tibialis anterior to be silent during most of the stance phase, while

EMG signals clearly indicated an activity during this phase. The activity of the musculus

tibialis anterior during this phase was in contradiction to EMG signals of normal healthy

subjects (Basmajian, 1974; Collins, 1995; Pedotti et al., 1978). However, the activity of the

musculus tibialis anterior might have been a compensatory mechanism used by the four

subjects with hip implants. The activity of the musculus tibialis increased the pre-loading

and thereby the stability of the ankle joint.

The amount of co-contraction that produced the best results between measured and cal-

culated hip contact forces varied slightly among the subjects and highly among activities

(Figs. 3.11 and 3.12). The optimal amount of co-contraction was largest for stair climb-

ing and knee bending. These activities required increased coordination and joint stability

especially because no handrail was used (Bergmann et al., 2001).

We assumed the shift parameter to be constant for all muscles and constant over time.

However, it is likely that the amount of co-contraction varies between the muscles and


versus time. It is known that antagonistic muscular activity during gait mainly occurs at

the end of the swing phase (Brand et al., 1994). A reason for this antagonistic muscle

activity might be that the CNS stabilizes the joints to prepare the joints for the impact

at toe off. Therefore our assumption of a constant shift parameter xs was probably a

simplification of the real process of co-contraction. However, our extension was formulated

in a way that possible improvements can be easily applied.

4.7 Sensitivity to Input and Model Parameters

In section 3.5 we investigated the sensitivity of the calculated muscular activities and hip

contact forces to variations in the ground reaction forces (GRF). Muscles spanning the hip

joint were mainly activated in order to preserve the moment equilibrium around the hip

joint. From the differences between the hip contact forces during the stance phase and

the swing phase (e.g. Fig. 3.4) we can see that the GRF were the main contributors to

the resultant hip reaction moments during the stance phase. The resultant hip reaction

moments caused by the GRF depended on the magnitudes and the directions of the GRF.

The directions of the GRF influenced the length of the moment arms with respect to the

hip joint center.

We found an almost linear relationship between variations in the magnitudes of the GRF

and the calculated hip contact forces and muscular activities during the stance phase for all

criteria with and without a shift parameter. For the polynomial criterion and the min/max

criterion this behavior is expected because the relation between the muscular activities is

constant for varying resultant joint reaction moments as long as no muscle reaches its

upper bound (Fig. 2.5). For the soft saturation criterion the relation between the muscular

activities is generally not constant for varying resultant joint reaction moments (Fig. 2.6).

However, for small muscular activities this relation is almost linear. We showed that the

calculated muscular activities were small for walking with normal speed. Therefore, the soft

saturation criterion also produced a linear relationship between the magnitudes of the GRF

and calculated hip contact forces. If the muscular activities when using the soft saturation

criterion were higher than the calculated ones, an increase in external load would mainly be

equilibrated by muscle with small moment generating capacity in order to prevent muscles

with large moment generating capacity from becoming saturated.

The effect when varying single components of the vectors of GRF by ±10% was not linear.

The effect was larger when varying the z components than when varying the x and y

4.7. SENSITIVITY TO INPUT AND MODEL PARAMETERS 79

components of the vectors of GRF (Fig. 3.17). The GRF were given with respect to the

laboratory coordinate system (subsection 2.6.11), thus the z components were the vertical

components and were much larger than the x and y components. Accordingly a variation

by ±10% resulted in a larger absolute change in the z components.

The subject PFL was most sensitive to changes in GRF (Fig. 3.17) with maximal discrepan-

cies between calculated hip contact forces with measured GRF and calculated hip contact

forces with varied GRF of up to 0.8 BW . These discrepancies were much larger than the

linear relationship observed for varying only the magnitudes. We observed that during the

whole stance phase the vector of ground reaction forces was approximately directed from

the center of pressure of the foot to the hip joint center. Therefore, the moment arms and

the resultant hip reaction moments were small. By varying single components we varied

not only the magnitudes but also the directions and thereby changed the moment arms.

This analysis demonstrates that an accurate, three component force plate is mandatory

when performing gait analysis in order to analyze musculoskeletal loading.

In order to equilibrate a given hip reaction moment the activity of the muscles depends

mainly on their moment arms around the hip joint. Therefore we also investigated the

impact of translating a muscle attachment point of the musculus rectus femoris by ±2 cm

in the x,y and z direction of the pelvis coordinate system (subsection 2.6.11). The maximal

discrepancies between the calculated hip contact forces with the original muscle path and

the hip contact forces with the varied muscle path were comparable to the discrepancies

when varying the GRF by ±10% (Fig. 3.19). Most sensitive to variations in the muscle

path was the model for subject IBL.

Generally the influence of translating the attachment point of the musculus rectus femoris

in the x direction (medial-lateral) had the largest effect on the calculated hip contact forces.

This was contrary to our expectations. We expected that a variation in the y direction

(dorsal-ventral) and therefore a variation of the moment arm with respect to the flex-

ion/extension moment would result in the largest discrepancies. The results indicated that

an accurate modelling of the muscle paths is also very important when predicting muscle

forces for an individual subject. Because of the anthropometrical scaling and the modelling

of a muscle path (discussed in section 4.8), errors of ±2 cm in the muscle attachment points

were easily possible.


4.8 Limitations

The prediction of muscle forces in a living subject requires many assumptions (Crown-

inshield and Brand, 1981b). We assumed ligaments and joint capsules to be minor load

carrying structures and neglected them in our musculoskeletal model. It would be possible

to consider ligaments as passive structures using data about force-length relationships for

the ligaments of the knee (Durselen, 1990). However, generally ligaments are short com-

pared to muscles and tendons. Consequently the error in the strain of a ligament caused

by small inaccuracies during gait analysis can be huge. Because strain is related to force

in ligaments, small kinematical inaccuracies could result in large inaccuracies for the lig-

ament forces. The accuracy of the individual musculoskeletal model (Fig. 2.12) is limited

to the accuracy of the reconstruction of muscle attachment sites from CT, the muscle path

representation, the muscle parameters and the accuracy of the anthropometrical scaling.

Heller et al. (2001) and Heller (2002) reconstructed the bony surfaces and muscle attach-

ment sites from the “Visible Human”data set (Spitzer et al., 1996). For this data the slices

of the CT in z direction were taken in distances of 1 mm and the resolution in the xy plane

was 0.9 mm. The reconstructed muscle attachment sites were compared to anatomical

textbooks (Heller, 2002). Compared to the sensitivity of hip contact forces to variations

in the muscle attachment sites these possible errors were small. More problematic is the

representation of the path of a muscle. The geometric model of a muscle must provide the

correct path over the whole range of physiological movements. A muscle can change its

original function when using a model that only connects the origin and the insertion of a

muscle by a straight line. Heller (2002) showed that the musculus sartorius turned from a

knee flexor to a knee extensor when the knee was in extension. Therefore, they introduced

deviating points to model a muscle’s path correctly over the whole range of possible move-

ments (Heller et al., 2001; Heller, 2002). Delp et al. proposed defining deviating points

that became only active for certain positions of a joint (Delp et al., 1990; Delp and Loan,

1995). Some groups developed an approach in which a muscle can wrap around regular

surfaces such as cylinders and spheres (Garner and Pandy, 2000; Charlton and Johnson,

2001). Recently, Gao et al. (2002) proposed a model in which a muscle could wrap around

irregular shaped surfaces defined by a set of cross-sections. Particulary the last approach

could improve the description of a muscle’s path. However, even more important than the

wrapping of muscles around bony surfaces is the wrapping of muscles around each other.

This affects specifically the large muscles on the surface that have a large moment gener-

ating capacity. There are no models currently that allow wrapping of muscles around each

4.8. LIMITATIONS 81

other.

The muscle attachment sites were also strongly influenced by the anthropometrical scaling.

Analogous to Heller (2002) we applied a linear transformation based on three base points

determined in the “Visible Human” and determined in vivo by Bergmann et al. (2001) for

the pelvis, the femur, the tibia and the foot.

The maximal force a muscle can exert was assumed to be proportional to its PCSA. We

took the values of the PCSA from Brand et al. (1986) who determined the PCSA for all

muscles in the human lower leg in vitro for two subjects. Brand et al. (1986) calculated

muscle forces and hip contact forces during a gait cycle using the cubic polynomial criterion

and minimizing muscle stresses taking the data of the PCSA of the two subjects. While

the calculated muscle forces were sensitive regarding the differences in the PCSA among

the two subjects, the calculated hip contact forces were comparatively unsensitive. As

Heller (2002) we took the values from the subject who had a mass close to the mass of the

“Visible Human”. We scaled all values of the PCSA linearly to the weight of the subjects.

However, the relation between the PCSA for different muscles will vary within subjects

depending on the profession, fitness, age and many other factors. Although we assumed

that minimizing muscular activities is physiologically more reasonable than minimizing

muscle forces (subsection 1.3.1) an error in the PCSA is more severe in the first case

because the PCSA affects the moment generating capacity of a muscle. Varying the PCSA

has therefore the same effect as varying the moment arm of a muscle when minimizing

muscular activities.

We did not take the force-length and force-velocity relationships of a muscle into account

as described by Zajac (1989). This could certainly influence the solution because no rapid

changes in a force that a muscle exerts are possible. Additionally, a muscle can exert its

maximal force only if it acts at its optimal length and the optimal shortening velocity.

Otherwise the maximal force is smaller (Zajac, 1989). Generally this would lead to an

increase in predicted hip contact forces when using a linear optimization criterion because

the additional constraints on muscle properties decrease the space of possible solutions.

When using a non-linear optimization criterion and minimizing muscular activities, the

effect of the extended muscle properties depends on the muscles, which act at their optimal

working conditions. The maximal force and thereby the moment generating capacity is

only slightly influenced for muscles that operate close to their optimal working conditions.

Working conditions however can significantly decrease the moment generating capacity.

In situations when muscles with the large moment arms act at their optimal working

conditions and muscles with the small moment arms act at unfortunate working conditions,


the muscles with the larger moment arms are more favored and vice versa. Using similar

reasoning as for decreasing and increasing the exponent p, the hip contact forces decrease

or increase.

Another possible source of errors was the gait analysis performed by Bergmann et al.

(2001). The positions of markers fixed on the skin were recorded by an optical tracking

system. The segmental positions recorded were not the positions of the rigid bodies (i.e.

the bones) because of the soft tissue displacements. Additionally the data was matched to

a linkage connected by kinematic joints that do not represent the anatomical joints.

All efforts to improve musculoskeletal models would be wasted if the assumption of an

optimization criterion for the recruitment of muscles by the CNS was inadequate. However,

in section 4.1 we have thoroughly discussed why we assume an optimization criterion to be

adequate. The results of this study should be applied to normal, healthy subjects (without

implants) with caution. An optimization criterion and co-contraction that produced the

best agreement between measured hip contact forces and calculated hip contact forces for

the four subjects in this study might not be optimal for normal, healthy subjects. Although

the measurement was at least 11 months postoperatively (Tab. A.1), the subjects may have

been more cautious in movements than a normal subject. However, the subjects did not

report any pain during the movement. Consequently we assumed that avoiding pain was

not the primary goal when recruiting muscles.

4.9 Conclusion

While the degree of muscular synergism could be controlled by conventional optimiza-

tion criteria, it was usually considered to be a methodological drawback of inverse dy-

namics in combination with optimization techniques that antagonistic muscular activity

could not be predicted. For a complex model of the human lower limb we demonstrated

that co-contraction could be enforced when extending standard optimization criteria. Co-

contraction leads to a pre-loading of joints and thus decreases the risk of falling. The extent

of co-contraction that produced the best agreement between measured and calculated data

varied between activities and was larger for activities that we assumed to be more difficult.

We concluded that co-contraction must be included when analyzing activities that require

a high amount of coordination.

The results of this study indicate that the prediction of muscle forces during human loco-

motion is principally possible. However, there is a need of further investigating this issue.

4.9. CONCLUSION 83

To this end the software employed should be further developed. This includes improv-

ing the algorithms employed in order to ensure reliability and increase efficiency as well

as developing new facilities e.g. for muscle wrapping around bony surfaces. Future soft-

ware developments should also include improving the user interface, which should hide the

complexity of the program from the user and make it easier to develop new models. The

efficiency of the algorithms allows for even more detailed models which may include passive

structures such as ligaments. Finally, new input data should be recorded, considering the

results of this study concerning the sensitivity to measured input data. A dynamic mag-

netic resonance (MR) seems to have considerable advantages compared to optical tracking

systems. When using a dynamic MR the soft tissue displacement could be separated from

the rigid body motion by image processing. Additionally, the muscle paths could be com-

pared over the whole recorded range of motion. Programming, modelling and recording

input data are each challenging tasks and the efforts of the various research groups should

be coordinated in order to proceed as efficiently as possible.

The software UFBSIM developed in this study is not limited to musculoskeletal models of

the human lower limb. It can also easily be applied to models of the upper extremity or

full body models and can also be used for models of other species (e.g. sheep, see Forster

et al., 2002). Because UFBSIM is also able to determine the internal loading and the

interfragmentary movement, building models of animals can help to reduce the number of

animal experiments when testing new orthopaedic devices.

84

Summary

Knowledge of the musculoskeletal loading is essential for the design of orthopaedic implants

and surgical procedures. Because muscle forces are hard to measure in vivo, computer

models are employed. Inverse dynamics in combination with optimization techniques are

commonly used to predict muscle forces. This study presents an extremely efficient and ro-

bust approach to compute musculoskeletal loading using inverse dynamics and optimization

techniques and thereby predicting synergistic as well as antagonistic muscle activity.

Using this approach and a previously developed model of the human lower limb we pre-

dicted muscular activities and hip contact forces for four individual subjects during various

daily activities. A direct comparison of calculated hip contact forces to measured hip con-

tact forces showed good agreement for walking with various speeds, stair-climbing upwards

and downwards. The median of the discrepancies between measured and calculated hip

contact forces was always smaller than 0.5 BW (body weight). Using various state-of-

the-art optimization criteria we controlled the amount of muscular synergism and found

the best agreement of measured and calculated hip contact forces for moderate muscular

synergism. Antagonistic muscular activity is also known to occur. So far, antagonistic

muscular activity has only been enforced for simple, planar models. For the first time

we enforced antagonistic muscular activity for a complex, three-dimensional musculoskele-

tal model. Moderate muscular synergism accompanied by moderate antagonistic activity

improved the agreement between measured and calculated hip contact forces, specifically

during stair climbing and knee bending. Knee bending required the largest extent of an-

tagonistic muscle activity. Without antagonistic muscle activity no agreement between

measured and calculated hip contact forces could be found during knee bending.

Previous studies assumed that the central nervous system tries to minimize muscular effort.

Antagonistic muscle activity that was observed was in contradiction to this assumption.

We postulate that apart from minimizing muscular effort ensuring joint stability is another

simultaneous goal of the central nervous system. A greater need for stability and coordi-

nation could be the reason for the larger antagonistic muscle activity observed during stair

climbing and knee bending.

SUMMARY 85

Calculating muscle forces for living subjects requires many assumptions. We showed that

mainly three factors influence the calculated muscle and joint contact forces: first, in-

correct input data because of soft tissue displacement during gait analysis; second, the

anthropometrical scaling that is used to adapt a general anatomical model to the individ-

ual subjects; third, the limitations of the muscle path representation. However, the results

indicated that determining muscle forces is principally possible.

The software developed in this study is not limited to the musculoskeletal model of the

human lower limb, but can also be applied to full body models or models of other species.

The aim of building models of animals is to reduce the number of animal experiments.

86

Appendix A

Data and Parameter

A.1 Subject Specific Data

mTg

mSg

mFg

-mT Tr

-Q wT T

-mS Sr

-Q wS S

-mF Fr

-Q wF F

Fig. A.1: All volume forces act at the center of mass of a segment.The vector xT points from the hip joint center to the centerof mass of the Thigh. The vector xS points from the kneejoint center to the center of mass of the Shank. The vectorxF points from the center of the ankle joint to the center ofmass of the Foot.

A.1. SUBJECT SPECIFIC DATA 87

Tab. A.1: General Data of the four Subjects (Bergmann et al., 2001).

HSR PFL KWR IBL

Gender Male Male Male FemaleAge at implantation in years 55 51 61 76Operated joint Right Left Right LeftTime of measurement in months postopera-tively

14 11 12 31

Weights at measurement in kgTotal body (mTotal) 87.7 99.99 71.56 81.55Thigh (mT ) 7.72 8.18 6.34 10.03Shank (mS) 3.99 5.49 3.72 4.38Foot (mF ) 0.94 1.25 1.01 0.73

Lengths in cmBody height 174.0 175.0 165.0 170.0Thigh 43.4 41.0 39.3 47.5Shank 38.1 41.0 40.0 40.9Foot 30.0 27.5 29.0 26.0

Angles in degreeAV = Anteversion 4 23 -2 14S = Femur shaft-implant shaft 10 7 9 9

Tab. A.2: Inertia Tensors and Vectors to Center of Mass of Subject HSR(Heller, 2002).

Inertia Tensor in kg m2 Vector to center of mass in mm

ΘT =

0.11080 0.0 0.0

0.0 0.11080 0.0

0.0 0.0 0.02364

xT =

0.0

0.0

−182.01

ΘS =

0.04852 0.0 0.0

0.0 0.04852 0.0

0.0 0.0 0.00601

xS =

0.0

0.0

−182.94

ΘF =

0.00226 0.0 0.0

0.0 0.00098 0.00156

0.0 0.00156 0.00232

xF =

0.0

30.84

−63.26

88 APPENDIX A. DATA AND PARAMETER

Tab. A.3: Inertia Tensors and Vectors to Center of Mass of Subject PFL (Heller,2002).


ΘT =

0.10663 0.0 0.0

0.0 0.10663 0.0

0.0 0.0 0.02785

xT =

0.0

0.0

−174.99

ΘS =

0.07743 0.0 0.0

0.0 0.07743 0.0

0.0 0.0 0.01078

xS =

0.0

0.0

−179.08

ΘF =

0.00386 0.0 0.0

0.0 0.00153 0.00266

0.0 0.00266 0.00383

xF =

0.0

34.93

−71.66

Tab. A.4: Inertia Tensors and Vectors to Center of Mass of Subject KWR(Heller, 2002).


ΘT =

0.07198 0.0 0.0

0.0 0.07198 0.0

0.0 0.0 0.01794

xT =

0.0

0.0

−158.28

ΘS =

0.05045 0.0 0.0

0.0 0.05045 0.0

0.0 0.0 0.00504

xS =

0.0

0.0

−179.82

ΘF =

0.00278 0.0 0.0

0.0 0.00103 0.00192

0.0 0.00192 0.00269

xF =

0.0

33.14

−67.98

A.1. SUBJECT SPECIFIC DATA 89

Tab. A.5: Inertia Tensors and Vectors to Center of Mass of Subject IBL (Heller,2002).


ΘT =

0.16838 0.0 0.0

0.0 0.16838 0.0

0.0 0.0 0.03596

xT =

0.0

0.0

−197.87

ΘS =

0.06088 0.0 0.0

0.0 0.06088 0.0

0.0 0.0 0.00720

xS =

0.0

0.0

−172.20

ΘF =

0.00152 0.0 0.0

0.0 0.00066 0.00105

0.0 0.00105 0.00156

xF =

0.0

28.50

−58.45

90 APPENDIX A. DATA AND PARAMETER

A.2 List of PCSA

Tab. A.6: List of PCSA (adapted from Brand et al., 1986).

Muscle PCSA in mm2 Muscle PCSA in mm2

m. adductor brevis (s) 1152 m. superior gemellus 213m. adductor brevis (i) 534 m. biceps femoris caput longum 2734m. adductor longus 2273 m. gracilis 374m. adductor magnus 1 2252 m. rectus femoris 4296m. adductor magnus 2 1835 m. sartorius 290m. adductor magnus 3 1695 m. semimembranosus 4633m. gluteus maximus 1 2020 m. semitendinosus 1305m. gluteus maximus 2 1959 m. tensor fasciae latae 800m. gluteus maximus 3 2000 m. gastrocnemius medialis 5060m. gluteus medius 1 2500 m. gastrocnemius lateralis 1430m. gluteus medius 2 1621 m. biceps femoris caput brevis 814m. gluteus medius 3 2121 m. vastus intermedius 8200m. gluteus minimus 1 676 m. vastus lateralis 6441m. gluteus minimus 2 820 m. vastus medialis 6687m. gluteus minimus 3 1198 m. tibialis anterior 1688m. iliacus 2333 m. ext. digitorum longus 746m. psoas major 2570 m. ext. hallucis longus 649m. inferior gemellus 433 m. flex. digitorum longus 640m. obturator externus 271 m. flex. hallucis longus 1852m. obturator internus 907 m. peroneus brevis 1961m. pectineus 903 m. peroneus longus 2465m. piriformis 2054 m. peroneus tert. 414m. quadriceps femoris 2100 m. tibialis posterior 2627

m. soleus 18669

91

Curriculum Vitae

Personal Data

Date and Place of Birth 30st July 1974, Hannover

Citizenship German

Education

1981-1985 Elementary School

1985-1994 Gymnasium, finished with Abitur, leading to University

entry level

1995-2000 Graz, University of Technology, Study of Mechanical En-

gineering, finished with the degree of Diplomingenieur

Professional Development

since 10/2000 PhD Student at the Institute for Orthopaedic Research

and Biomechanics, University of Ulm

Practical Experience

Summer 1995 Eberspacher Highpressure Hydraulics

Summer 1996 Thyssen Elevators, Germany

Summer 1997 Thyssen Elevators, Germany

Summer 1998 Thyssen N.A. Inc., Detroit, USA

Summer 1999 Siemens Medical Systems Inc., Germany

Military Service

1994-1995 German Federal Armed Forces

92

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