Css 2013 temperature controlled transport - risk mitigation - luc huybreghts - pauwels consulting
Predicting Muscle Forces in the Human Lower Limb during ......Pauwels (1965) showed that muscles...
Transcript of 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).
4 CHAPTER 1. INTRODUCTION
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)
6 CHAPTER 1. INTRODUCTION
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,
8 CHAPTER 1. INTRODUCTION
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
10 CHAPTER 1. INTRODUCTION
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)
14 CHAPTER 2. MATERIALS AND METHODS
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)
16 CHAPTER 2. MATERIALS AND METHODS
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
m′a −Θ ω′ − ω′Θω′
]. (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) .
18 CHAPTER 2. MATERIALS AND METHODS
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
20 CHAPTER 2. MATERIALS AND METHODS
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.
2.3. OPTIMIZATION 21
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.
22 CHAPTER 2. MATERIALS AND METHODS
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
2.3. OPTIMIZATION 23
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)
24 CHAPTER 2. MATERIALS AND METHODS
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
2.3. OPTIMIZATION 25
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).
26 CHAPTER 2. MATERIALS AND METHODS
-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
2.3. OPTIMIZATION 27
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.
28 CHAPTER 2. MATERIALS AND METHODS
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
2.3. OPTIMIZATION 29
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,
30 CHAPTER 2. MATERIALS AND METHODS
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
2.3. OPTIMIZATION 31
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.
32 CHAPTER 2. MATERIALS AND METHODS
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 .
2.3. OPTIMIZATION 33
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
34 CHAPTER 2. MATERIALS AND METHODS
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
m′a −Θ ω′ − ω′Θω′
]−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
36 CHAPTER 2. MATERIALS AND METHODS
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).
38 CHAPTER 2. MATERIALS AND METHODS
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. MODEL OF THE HUMAN LOWER LIMB 39
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
40 CHAPTER 2. MATERIALS AND METHODS
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. MODEL OF THE HUMAN LOWER LIMB 41
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
42 CHAPTER 2. MATERIALS AND METHODS
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
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
(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.
46 CHAPTER 3. RESULTS
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
48 CHAPTER 3. RESULTS
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
stance phase swing phase
-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
stance phase swing phase
-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
stance phase swing phase
-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
stance phase swing phase
-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
stance phase swing phase
-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
stance phase swing phase
-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
50 CHAPTER 3. RESULTS
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
stance phase swing phase
-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
stance phase swing phase
-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
stance phase swing phase
-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
stance phase swing phase
-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.
stance phase swing phase
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
52 CHAPTER 3. RESULTS
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
54 CHAPTER 3. RESULTS
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(a) subject KWR, trial WS2
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(b) subject KWR, trial WF5
stance phase swing phase
-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
3.4. THE INFLUENCE OF THE SHIFT PARAMETER XS 55
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).
56 CHAPTER 3. RESULTS
stance phase swing phase
-1
0
1
2
3H
ip C
onta
ct F
orc
ein
BW
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(a) xS = 0.010
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(b) xS = 0.015
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(c) xS = 0.020
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(d) xS = 0.025
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(e) xS = 0.030
stance phase swing phase
-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.
3.4. THE INFLUENCE OF THE SHIFT PARAMETER XS 57
=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.
58 CHAPTER 3. RESULTS
=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.
3.4. THE INFLUENCE OF THE SHIFT PARAMETER XS 59
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
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(a) subject KWR, trial SU4 (xs = 0.000)
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
xMeas
yMeas
zMeas
xSim
ySim
zSim
F
(b) subject KWR, trial SU4 (xs = 0.035)
stance phase swing phase
-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)
stance phase swing phase
-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)
stance phase swing phase
-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)
stance phase swing phase
-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.
60 CHAPTER 3. RESULTS
stance phase swing phase
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.
stance phase swing phase
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
stance phase swing phase
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.
62 CHAPTER 3. RESULTS
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
stance phase swing phase
-1
0
1
2
3H
ip C
onta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(a) Fx+
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(b) Fx−
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(c) Fy+
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(d) Fy−
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(e) Fz+
stance phase swing phase
-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.
64 CHAPTER 3. RESULTS
+ - + - + -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
stance phase swing phase
-1
0
1
2
3H
ip C
onta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(a) rx+
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(b) rx−
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(c) ry+
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(d) ry−
stance phase swing phase
-1
0
1
2
3
Hip
Conta
ct F
orc
ein
BW
xSim
ySim
zSim
xVar
yVar
zVar
F
(e) rz+
stance phase swing phase
-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.
66 CHAPTER 3. RESULTS
+ - + - + -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
70 CHAPTER 4. DISCUSSION
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.
72 CHAPTER 4. DISCUSSION
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
74 CHAPTER 4. DISCUSSION
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
76 CHAPTER 4. DISCUSSION
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
78 CHAPTER 4. DISCUSSION
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.
80 CHAPTER 4. DISCUSSION
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,
82 CHAPTER 4. DISCUSSION
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).
Inertia Tensor in kg m2 Vector to center of mass in mm
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).
Inertia Tensor in kg m2 Vector to center of mass in mm
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).
Inertia Tensor in kg m2 Vector to center of mass in mm
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
References
Ait-Haddou, R., Binding, P., Herzog, W., 2000. Theoretical considerations on cocontractionof sets of agonistic and antagonistic muscles. Journal of Biomechanics 33 (9), 1105–11.(Cited on pages 5 and 8.)
An, K. N., Kwak, B. M., Chao, E. Y., Morrey, B. F., 1984. Determination of muscle andjoint forces: a new technique to solve the indeterminate problem. Journal of Biomechan-ical Engineering 106 (4), 364–7. (Cited on pages 6 and 7.)
Anderson, F. C., Pandy, M. G., 1999. A Dynamic Optimization Solution for Vertical Jump-ing in Three Dimensions. Computational Methods in Biomechanics and Biomedical En-gineering 2 (3), 201–231. (Cited on page 2.)
Basmajian, J. V., 1974. Muscles alive. Their functions revealed by electromyography. 3.edition. Williams, Baltimore. (Cited on pages 72 and 77.)
Bean, J. C., Chaffin, D. B., Schultz, A. B., 1988. Biomechanical model calculation of musclecontraction forces: a double linear programming method. Journal of Biomechanics 21 (1),59–66. (Cited on pages 7, 9 and 75.)
Bergmann, G., 2001. Hip 98 (CD). (Cited on pages 39 and 41.)
Bergmann, G., Deuretzbacher, G., Heller, M., Graichen, F., Rohlmann, A., Strauss, J.,Duda, G. N., 2001. Hip contact forces and gait patterns from routine activities. Journalof Biomechanics 34 (7), 859–71. (Cited on pages X, 10, 37, 39, 40, 41, 50, 55, 67, 68, 74,75, 76, 77, 81, 82 and 87.)
Bergmann, G., Graichen, F., Rohlmann, A., 1993. Hip joint loading during walking andrunning, measured in two patients. Journal of Biomechanics 26 (8), 969–90. (Cited onpage 76.)
Borelli, G. A., 1685. De motu animalium Ed.altera. 2 Teile in 1 Band. vander Aa, Leiden.(Cited on page 1.)
Brand, R. A., Crowninshield, R. D., Wittstock, C. E., Pedersen, D. R., Clark, C. R.,van Krieken, F. M., 1982. A model of lower extremity muscular anatomy. Journal ofBiomechanical Engineering 104 (4), 304–10. (Cited on page 9.)
Brand, R. A., Pedersen, D. R., Davy, D. T., Kotzar, G. M., Heiple, K. G., Goldberg,V. M., 1994. Comparison of hip force calculations and measurements in the same patient.Journal of Arthroplasty 9 (1), 45–51. (Cited on pages 5, 8, 9, 69, 73, 74, 76 and 78.)
Brand, R. A., Pedersen, D. R., Friederich, J. A., 1986. The sensitivity of muscle forcepredictions to changes in physiologic cross-sectional area. Journal of Biomechanics 19 (8),589–96. (Cited on pages 4, 38, 81 and 90.)
Charlton, I. W., Johnson, G. R., 2001. Application of spherical and cylindrical wrappingalgorithms in a musculoskeletal model of the upper limb. Journal of Biomechanics 34 (9),1209–16. (Cited on page 80.)
REFERENCES 93
Cholewicki, J., McGill, S. M., Norman, R. W., 1995. Comparison of muscle forces and jointload from an optimization and EMG assisted lumbar spine model: towards developmentof a hybrid approach. Journal of Biomechanics 28 (3), 321–31. (Cited on page 8.)
Collins, J. J., 1995. The redundant nature of locomotor optimization laws. Journal ofBiomechanics 28 (3), 251–67. (Cited on pages 4, 8, 68 and 77.)
Crowninshield, R. D., 1978. Use of Optimization Techniques to predict Muscle Forces.Journal of Biomechanical Engineering 100, 88–92. (Cited on pages 7, 8 and 68.)
Crowninshield, R. D., Brand, R. A., 1981a. A physiologically based criterion of muscle forceprediction in locomotion. Journal of Biomechanics 14 (11), 793–801. (Cited on pages 6,7, 71 and 76.)
Crowninshield, R. D., Brand, R. A., 1981b. The prediction of forces in joint structures;distribution of intersegmental resultants. Exercise and Sport Sciences Reviews 9, 159–81.(Cited on pages 1, 2, 3, 4, 68 and 80.)
Crowninshield, R. D., Johnston, R. C., Andrews, J. G., Brand, R. A., 1978. A biomechan-ical investigation of the human hip. Journal of Biomechanics 11 (1-2), 75–85. (Cited onpage 76.)
Damsgaard, M., Rasmussen, J., Voigt, M., 2001. Inverse Dynamics of musculo-skeletalsystems using an efficient min/max muscle recruitment model. In: 18-th Biennial Con-ference on Mechanical Vibration and Noise. Pittsburgh, Pennsylvania. (Cited on pages 5,6, 7, 11, 16, 28, 31, 32 and 69.)
Davy, D. T., Audu, M. L., 1987. A dynamic optimization technique for predicting muscleforces in the swing phase of gait. Journal of Biomechanics 20 (2), 187–201. (Cited onpage 69.)
Davy, D. T., Kotzar, G. M., Brown, R. H., Heiple, K. G., Goldberg, V. M., Heiple, K. G.,J., Berilla, J., Burstein, A. H., 1988. Telemetric force measurements across the hip aftertotal arthroplasty. Journal of Bone and Joint Surgery 70 (1), 45–50. (Cited on page 76.)
Delp, S. L., Loan, J. P., 1995. A graphics-based software system to develop and analyzemodels of musculoskeletal structures. Computers in Biology and Medicine 25 (1), 21–34.(Cited on page 80.)
Delp, S. L., Loan, J. P., Hoy, M. G., Zajac, F. E., Topp, E. L., Rosen, J. M., 1990. Aninteractive graphics-based model of the lower extremity to study orthopaedic surgicalprocedures. IEEE Transactions on Biomedical Engineering 37 (8), 757–67. (Cited onpage 80.)
Durselen, L., 1990. Biomechanische und mechanische Untersuchungen an Kniegelenksban-dern und deren alloplastischen Ersatz vorgelegt von Lutz Durselen. Doktorarbeit, Uni-versitat Ulm. (Cited on page 80.)
Dul, J., Johnson, G. E., Shiavi, R., Townsend, M. A., 1984a. Muscular synergism–II.A minimum-fatigue criterion for load sharing between synergistic muscles. Journal ofBiomechanics 17 (9), 675–84. (Cited on page 7.)
Dul, J., Townsend, M. A., Shiavi, R., Johnson, G. E., 1984b. Muscular synergism–I. Oncriteria for load sharing between synergistic muscles. Journal of Biomechanics 17 (9),663–73. (Cited on pages 5, 71 and 72.)
Forster, E., Simon, U., Augat, P., Claes, L., 2004. Extension of a state-of-the-art optimiza-tion criterion to predict co-contraction. Journal of Biomechanics Accepted. (Cited onpages 8, 10, 68 and 77.)
94 REFERENCES
Forster, E., Simon, U., Augat, P., Heller, M., Duda, G., Claes, L., 2002. A multibody dy-namics approach to inverse dynamics and optimization. In: C., D., M., G., W., K. (Eds.),The 9th FEM Workshop on: ”The Finite Element Method” in Biomedical Engineering,Biomechanics and Related Fields”. Ulm, pp. 63–80. (Cited on pages 9, 69 and 83.)
Gao, F., Damsgaard, M., Rasmussen, J., Christensen, S. T., 2002. Computational methodfor muscle-path representation in musculoskeletal models. Biological Cybernetics 87 (3),199–210. (Cited on page 80.)
Garner, B. A., Pandy, M. G., 2000. The Obstacle-Set Method for Representing MusclePaths in Musculoskeletal Models. Computational Methods in Biomechanics and Biomed-ical Engineering 3 (1), 1–30. (Cited on page 80.)
Gill, P. E., Murray, W., H., W. M., 1981. Practical optimization. London [u.a.] Acad. Pr.(Cited on pages 4, 20, 21, 24, 30 and 32.)
Golub, G. H., VanLoan, C. F., 1990. Matrix computations, 2nd Edition. Baltimore, Md.[u.a.] Johns Hopkins Univ. Pr., engl. (Cited on pages 14, 20, 28, 29, 30 and 33.)
Greenwood, D., 1988. Principles of dynamics. Prentice Hall, Englewood Cliffs, New Jersey.(Cited on page 14.)
Haug, E. J., 1989. Computer aided kinematics and dynamics of mechanical systems. Vol.Volume I Basic Methods. Allyn and Bacon, Needham Heights, Massachusetts. (Cited onpages 11, 12, 13, 15 and 16.)
Heller, M., 2002. Muskuloskelettale Belastungen nach Totalhuftarthroplastik. Doktorar-beit, Universitat Ulm. (Cited on pages 6, 9, 10, 37, 38, 39, 41, 43, 67, 70, 71, 74, 75, 76,80, 81, 87, 88 and 89.)
Heller, M. O., Bergmann, G., Deuretzbacher, G., Durselen, L., Pohl, M., Claes, L., Haas,N. P., Duda, G. N., 2001. Musculo-skeletal loading conditions at the hip during walkingand stair climbing. Journal of Biomechanics 34 (7), 883–93. (Cited on pages 5, 6, 9, 37,39, 67, 69, 70, 71, 74, 76 and 80.)
Herzog, W., 1987. Individual muscle force estimations using a non-linear optimal design.Journal of Neuroscience Methods 21 (2-4), 167–79. (Cited on pages 5 and 7.)
Herzog, W., 1994. Mathematically indeterminate systems. In: Nigg, B., Herzog, W. (Eds.),Biomechanics of the Musculo-skeletal System. John Wiley & Sons, Chichester, NewYork,Brisbane, Toronto, Singapore, p. 578, iSBN: 0-471-94444-0. (Cited on pages 3, 4, 5and 21.)
Herzog, W., 1996. Force-sharing among synergistic muscles: theoretical considerations andexperimental approaches. In: Holloszy, J. (Ed.), Exercise and Sport Sciences Reviews.Vol. 24. Williams and Wilkins, Baltimore, pp. 173–202. (Cited on page 4.)
Herzog, W., Binding, P., 1993. Cocontraction of pairs of antagonistic muscles: analyticalsolution for planar static nonlinear optimization approaches. Mathematical Biosciences118 (1), 83–95. (Cited on pages 5, 6, 8 and 69.)
Hughes, R. E., Bean, J. C., Chaffin, D. B., 1995. Evaluating the effect of co-contraction inoptimization models. Journal of Biomechanics 28 (7), 875–8. (Cited on page 8.)
Hughes, R. E., Chaffin, D. B., 1988. Conditions under which optimization models will notpredict coactivation of antagonis muscles. In: 12th Meeting of the American Society ofBiomechanics. Urbana-Champaign, IL, pp. 155–156. (Cited on page 8.)
REFERENCES 95
Jinha, A., Ait-Haddou, R., Herzog, W., 2002. Antagonistic muscle activity in three-dimensional models of the musculoskeletal system. In: IV. World Congress Biomechanics.Calgary. (Cited on page 8.)
Kotzar, G. M., Davy, D. T., Goldberg, V. M., Heiple, K. G., Berilla, J., Heiple, K. G., J.,Brown, R. H., Burstein, A. H., 1991. Telemeterized in vivo hip joint force data: a reporton two patients after total hip surgery. Journal of Orthopaedic Research 9 (5), 621–33.(Cited on page 76.)
Lu, T. W., O’Connor, J. J., Taylor, S. J., Walker, P. S., 1998. Validation of a lower limbmodel with in vivo femoral forces telemetered from two subjects. Journal of Biomechanics31 (1), 63–9. (Cited on page 4.)
Martin, R., 1999. A Genealogy of Biomechanics. In: 23rd Annual Conference of the Amer-ican Society of Biomechnanics. University of Pittsburgh, Pittsburgh PA. (Cited onpage 1.)
NAG, 2002. NAG C Library, Mark 7; Numerical Algorithm Group. (Cited on pages 14, 29and 37.)
Neptune, R. R., 1999. Optimization algorithm performance in determining optimal controlsin human movement analyses. Journal of Biomechanical Engineering 121 (2), 249–52.(Cited on page 2.)
Nigg, B., 1994. Selected historical highlights. In: Nigg, B., Herzog, W. (Eds.), Biomechanicsof the Musculo-skeletal System. John Wiley & Sons, Chichester, NewYork, Brisbane,Toronto, Singapore, p. 578, iSBN: 0-471-94444-0. (Cited on page 1.)
Nikravesh, P. E., 1988. Computer aided analysis of mechanical systems. Prentice-Hall,Englewood Cliffs, NJ. (Cited on page 12.)
Patriarco, A. G., Mann, R. W., Simon, S. R., Mansour, J. M., 1981. An evaluation of theapproaches of optimization models in the prediction of muscle forces during human gait.Journal of Biomechanics 14 (8), 513–25. (Cited on page 7.)
Paul, J. P., 1966. Biomechanics. The biomechanics of the hip-joint and its clinical relevance.Proceedings of the Royal Society of Medicine 59 (10), 943–8. (Cited on pages 4 and 76.)
Pauwels, F., 1965. Gesammelte Abhandlungen zur funktionellen Anatomie des Bewe-gungsapparates. Springer-Verlag, Berlin. (Cited on page 2.)
Pedersen, D. R., Brand, R. A., Cheng, C., Arora, J. S., 1987. Direct comparison of mus-cle force predictions using linear and nonlinear programming. Journal of BiomechanicalEngineering 109 (3), 192–9. (Cited on page 71.)
Pedersen, D. R., Brand, R. A., Davy, D. T., 1997. Pelvic muscle and acetabular contactforces during gait. Journal of Biomechanics 30 (9), 959–65. (Cited on page 9.)
Pedotti, A., Krishnan, V., Stark, L., 1978. Optimization of muscleforce sequencing inhuman locomotion. Mathematical Biosciences 38, 57–76. (Cited on pages 6, 7, 73 and 77.)
Pierrynowski, M., Morrison, J., 1985. Estimating the Muscle Forces Generated in the Hu-man lower Extremity when Walking: a Physiological Solution. Mathematical Biosciences75 (1), 43–68. (Cited on page 4.)
Prendergast, P. J., 2000. Biomechanics in Ireland and Europe. In: Prendergast, P. J., Lee,T. C., Carr, A. J. (Eds.), 12th Conference of the European Society of Biomechanics.Royal Academy of Medicine in Ireland, Trinity College, Dublin, Ireland, pp. 1–4. (Citedon page 1.)
96 REFERENCES
Raikova, R., 1996. A model of the flexion-extension motion in the elbow joint - someproblems concerning muscle forces modelling and computation. Journal of Biomechanics29 (6), 763–72. (Cited on pages 6 and 8.)
Raikova, R., 1999. About weight factors in the non-linear objective functions used forsolving indeterminate problems in biomechanics. Journal of Biomechanics 32 (7), 689–94. (Cited on page 8.)
Raikova, R. T., Prilutsky, B. I., 2001. Sensitivity of predicted muscle forces to parameters ofthe optimization-based human leg model revealed by analytical and numerical analyses.Journal of Biomechanics 34 (10), 1243–55. (Cited on page 5.)
Rasmussen, J., Damsgaard, M., Surma, E., Christensen, S., Zee, M., 2003. Designing AGeneral Software System For Musculoskeletal Analysis. In: IX International Symposiumon Computer Simulation in Biomechanics. Sydney, Australia. (Cited on pages 10, 36, 43and 69.)
Rasmussen, J., Damsgaard, M., Voigt, M., 2001. Muscle recruitment by the min/maxcriterion - a comparative numerical study. Journal of Biomechanics 34 (3), 409–15. (Citedon pages 4, 5, 6, 7, 8, 22, 25, 71 and 73.)
Rydell, N. W., 1966. Forces acting on the femoral head-prosthesis. A study on strain gaugesupplied prostheses in living persons. Acta Orthopaedica Scandinavica 37, Suppl 88:1–132. (Cited on page 76.)
Seireg, A., Arvikar, 1975. The prediction of muscular lad sharing and joint forces in thelower extremities during walking. Journal of Biomechanics 8 (2), 89–102. (Cited onpage 76.)
Seireg, A., Arvikar, R. J., 1973. A mathematical model for evaluation of forces in lowerextremeties of the musculo-skeletal system. Journal of Biomechanics 6 (3), 313–26. (Citedon pages 4, 6 and 67.)
Siemienski, A., 1992. Soft Saturation, an idea for load sharing between muscles. Applicationto the study of human locomotion. Proceedings : Biolocomotion A Century of ResearchUsing Moving Pictures . (Cited on pages 6, 7 and 72.)
Singer, C. J., Underwood, E. A., 1962. A short history of medicine by Charles Singer andE. Ashworth Underwood, 2nd Edition. Clarendon, Oxford. (Cited on page 1.)
Spitzer, V., Ackerman, M. J., Scherzinger, A. L., Whitlock, D., 1996. The visible humanmale: a technical report. Journal of the American Medical Informatics Association 3 (2),118–30. (Cited on pages 37 and 80.)
Stansfield, B. W., Nicol, A. C., Paul, J. P., Kelly, I. G., Graichen, F., Bergmann, G.,2003. Direct comparison of calculated hip joint contact forces with those measured usinginstrumented implants. An evaluation of a three-dimensional mathematical model of thelower limb. Journal of Biomechanics 36 (7), 929–36. (Cited on pages 7, 9, 69, 71, 75and 76.)
Tsirakos, D., Baltzopoulos, V., Bartlett, R., 1997. Inverse optimization: functional andphysiological considerations related to the force-sharing problem. Critical Reviews inBiomedical Engineering 25 (4-5), 371–407. (Cited on pages 4, 11 and 71.)
Vaughan, C. L., 1999. GaitCD. Kiboho Publishers, Howard Place, Western Cape 7450,South Africa. (Cited on page 2.)
Weber, W., Weber, E., 1836. Mechanik der menschlichen Gehwerkzeuge. Dietrichsche Buch-handlung, Gottingen. (Cited on pages 1, 67 and 68.)
REFERENCES 97
Weber, W., Weber, E., 1991. Mechanics of the Human Walking Apparatus. Springer Verlag,Berlin. (Cited on page 67.)
Wolff, J., 1892a. Das Gesetz der Transformation der Knochen. Verlag von AugustHirschwald, Berlin. (Cited on page 2.)
Wolff, J., 1892b. The Law of Bone Remodelling (Das Gesetz der Transformation derKnochen). Springer-Verlag, Berlin. (Cited on page 2.)
Zajac, F. E., 1989. Muscle and tendon: properties, models, scaling, and application tobiomechanics and motor control. Critical Reviews in Biomedical Engineering 17 (4),359–411. (Cited on page 81.)