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1
Effects of Muscle Fatigue on the Ground
Reaction Force and Soft Tissue Vibrations
during Running: A Model Study
Ali Asadi Nikooyan and Amir Abbas Zadpoor

Abstract—A modeling approach is used in this paper to study
the effects of fatigue on the ground reaction force and the
vibrations of the lower extremity soft tissues. A recently
developed multiple-degrees-of-freedom mass-spring-damper
model of the human body during running is used for this
purpose. The model is capable of taking the muscle activity into
account by using a nonlinear controller that tunes the mechanical
properties of the soft-tissue package based on two physiological
hypotheses, namely “constant-force” and “constant-vibration”.
In this study, muscle fatigue is implemented in the model as the
gradual reduction of the ability of the controller to tune the
mechanical properties of the lower body soft-tissue package.
Simulations are carried out for various types of footwear in both
pre- and post- fatigue conditions. The simulation results show
that the vibration amplitude of the lower body soft-tissue
package may considerably increase (up to 20 %) with muscle
fatigue while the effects of fatigue on the ground reaction force
are negligible. The results of this modeling study are in line with
the experimental studies that found muscle fatigue does not
significantly change the GRF peaks but may increase the level of
soft tissue vibrations (particularly for hard shoes). A major
contribution of the current study is formulation of a hypothesis
about how the central nervous system tunes the muscle
properties after fatigue.
Index Terms—ground reaction force, mass-spring-damper
model, muscle fatigue, soft tissue vibrations
R
I. INTRODUCTION
running has become the preferred mode of
exercise for millions of people worldwide. One of the
consequences of this profound interest in running is a high
incidence rate of running injuries. For example, 24-67% of 30
million Americans who run recreationally suffer from some
type of injury that prevents them from running for at least one
ECREATIONAL
Manuscript received August 3, 2011; revised October 26, 2011; accepted
November 27, 2011. Asterisk indicates corresponding author.
*A. A. Nikooyan is with the Dep. Biomech. Eng., Delft University of
Technology, Mekelweg 2, Delft 2628 CD, The Netherlands (e-mail:
[email protected]).
A. A. Zadpoor is with the Dep. Biomech. Eng., Delft University of
Technology, Delft, The Netherlands (e-mail: [email protected]).
Both authors have equally contributed to this manuscript and should
therefore be considered as joint first authors.
Copyright (c) 2011 IEEE. Personal use of this material is permitted.
However, permission to use this material for any other purposes must be
obtained from the IEEE by sending an email to pubs‐[email protected].
week per year [1, 2]. Many researchers have presumed that
running injuries are linked to the transient force that is
developed in the human body after its collision with the
ground and is often called Ground Reaction Force (GRF), see
e.g. [3]. The temporal pattern of the GRF results from a
complex interaction between the neural and musculoskeletal
systems that follows the collision and aims to attenuate the
shock [4, 5]. As a result of the transient impact force, soft
tissues may start to vibrate. It has been, however, shown that
the vibrations of soft tissues are heavily damped [4-6]. It is
believed that not only regulation of the GRF but also damping
of soft tissue vibrations are linked to muscle activity (the
muscle tuning paradigm) [4].
Given the importance of muscle activity in the attenuation
of the collision shock and damping of the resulting vibrations,
it is important to understand how the GRF and vibrations of
soft tissues change when muscles fatigue. Any change in the
shock attenuation capability of the human body due to fatigue
is important both from fundamental and injury development
viewpoints. The effects of muscle fatigue on the GRF [7-12]
and tissue vibrations [13] during running are therefore studied
before.
As far as the GRF is concerned, there is confusion in the
literature as to whether the GRF increases or decreases with
muscle fatigue. While some researchers have found that the
GRF decreases with fatigue [9, 10, 12], others have not found
any substantially change [8], or have even observed increase
[7, 11]. Not many researchers have studied the effects of
fatigue on soft tissue vibrations. In a recent study,
Friesenbichler et al found that the level of soft tissue
vibrations increases with fatigue [13].
Beside these experimental researches, the modeling
approach has recently received attention. In some recent
studies, a mass-spring model was used to predict the changes
in the passive stiffness during sprint [14] and self-paced [9]
running. The single-degree-of-freedom passive mass-spring
model developed by McMahon and Cheng [15] was used in
those studies. The simulation results showed that both vertical
and leg stiffness significantly decrease with fatigue.
No experimental or modeling studies have so far
investigated the effects of muscle fatigue on both GRF and
vibration level. In the current study, a modeling approach is
used to understand the effects of muscle fatigue on the GRF
and tissue vibration during running. A recently developed
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multiple-degrees-of-freedom mass-spring-damper model that
is capable of taking the muscle activity into account is used for
this purpose. The model is modified to enable it to take
account of muscle fatigue. The model is then used to calculate
the changes in the GRF and the vibration level of soft tissue
package with fatigue. The simulation results are compared
with experimental observations.
The cost functions that are used with this controller are based
on two main physiological hypotheses. The first objective
function is designed based on the constant force hypothesis
according to which the human body adjusts the mechanical
properties of the lower-body soft tissues such that the changes
in the GRF are minimal [19-22]. For each set of shoe-hardness
parameters (bi, di), the force objective function (Jf) is
formulated as follows:
II. MATERIALS AND METHODS
A. The basic and active models
The mass-spring-damper model presented in references [16,
17] is called the basic model which, in fact, is a corrected
version of the original model which was previously developed
by Liu and Nigg [18]. The basic model has four degrees-offreedom and consists of four masses, five springs, and three
dampers. The model parameters are listed in Table 1. Two of
these masses represent the upper-body rigid (m3) and
wobbling (m4) masses and the other two represent the lowerbody rigid (m1) and wobbling (m2) masses. The springs and
dampers represent the stiffness and damping properties of the
human body hard and soft tissues.
An improved version of the basic model was recently
proposed by Zadpoor and Nikooyan [5]. In this new version
(active model, Fig. 1), the pre-landing muscle activity is taken
into account by considering the lower body wobbling mass
(LBWM) as an active element. The governing equations of the
motion can be written as [5]:
m1x1 = m1g - Fg - k1 ( x1 - x3 ) - k2 c
-c1 ( x1 - x3 ) - c2 c ( x1 - x2 ) ,
where p1 and p2 are the first and second peaks of the GRF as
functions of the shoe hardness parameters, and p1,0 (=1436.8
N) and p2,0 (=2026.4 N) are the reference values of the first
and second GRF peaks [17] calculated using some default
shoe parameters (Table 2).
k5
m4
k
x
m3
c
4
4
4
k
x
3
3
2
k
1
1
2
x
c
k
m
c
2
2
ground
reaction
model
1
x
(1)
1
Fig. 1. A schematic representation of the active model [5].
m3 x3 = m3 g + k1 ( x1 - x3 ) + k3 ( x2 - x3 )
TABLE I
THE DEFAULT VALUES OF THE MASSES, STIFFNESS, AND DAMPING
- ( k4 + k5 ) ( x3 - x4 ) + c1 ( x1 - x3 ) - c4 ( x3 - x4 ) ,
PARAMETERS OF THE MODEL
m4 x4 = m4 g + ( k4 + k5 ) ( x3 - x4 ) + c4 ( x3 - x4 )
where χ represents the excitation signal issued by the Central
Nervous System (CNS). The LBWM stiffness (k2) and
damping (c2) properties are both functions of the excitation
signal.
The model is connected to the ground via a GRF element.
The force acting on this element (Fg) is a function of the
displacement (x1) and velocity (v1) of the LBWM [18]:
( x1 > 0 )
( x1 £ 0 )
b
d e
ì
ï Ac éë ax1 + cx1 v1 ùû
ï
î0
(3)
+ p2,i bi ,di - p2,0 b0 ,d0
m
+c2 c ( x1 - x2 ) ,
Fg = í
J f = p1,i bi ,di - p1,0 b0 ,d0
( x1 - x2 )
m2 x2 = m2 g + k2 c ( x1 - x2 ) - k3 ( x2 - x3 )
2
m1 (kg)
m2 (kg)
m3 (kg)
m4 (kg)
6.15
k1 (kN/m)
6
k5 (kN/m)
18
6
k2 (kN/m)
6
c1 (kg/s)
300
12.58
k3 (kN/m)
10
c2 (kg/s)
650
50.34
k4 (kN/m)
10
c4 (kg/s)
1900
TABLE 2
THE DEFAULT VALUES FOR TOUCHDOWN VELOCITIES AND THE SHOE-GROUND
MODEL PARAMETERS
(2)
where, a, b, c, d, Ac, and e are the parameters of the shoeground model. Parameters a, c, and e are the same for all shoe
types. The hardness of the shoe is determined by two
parameters b and d. The default values of the shoe-ground
model are listed in Table 2.
In the active model, a controller adjusts the stiffness and
damping properties of the LBWM (k2 and c2). The controller is
a nonlinear optimizer that minimizes some cost functions (J).
touchdown velocities (m/s)
a
b
0.6e6 1.38
for m1 and m2
c
d
2.0e4
0.75
for m3 and m4
0.96
2.0
e
1.0
The second objective function is based on the vibration
hypothesis according which the human body adjusts the
mechanical properties of the lower-body soft tissues such that
the changes in the level of vibrations are minimal [23-27]. The
vibration level is quantified based on the amplitude (Λ) of the
displacements of the lower-body soft tissue package. For each
set of shoe hardness parameters (bi, di), the vibration
amplitude cost function (Jv1) can be calculated as:
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J v1 = Li bi ,di - L0 b0 ,d0
(4)
where Λ0 (=10.1 cm) is the reference value of the vibration
amplitude and is calculated using the default values of b and d
(Table 2) [28].
In a more recent study, Nikooyan and Zadpoor [29]
introduced a combined cost function that matches
experimental observations better. The improved cost function
(Jfv) is the normalized sum of the two previously-proposed
(force and vibration amplitude) cost functions:
J fv =
p1,i bi ,di - p1,0 b0 ,d0
p1,0 b0 ,d0
+
p2,i bi ,di - p2,0 b0 ,d0
p2,0 b0 ,d0
+
L i bi ,di - L 0 b0 ,d0
L 0 b0 ,d0
(5)
The simulation results using this improved cost function
showed that the new cost function can predict the GRF and
vibrations levels that are in agreement with experimental
observations.
To make comparison between the present and previous
studies [5, 29] easier, the same range of shoe hardness
parameters (1 ≤ b ≤ 2 and 0.6 ≤ d ≤ 0.9) and the same
optimization technique (i.e. Pattern Search Method for boundconstraint minimization [30]) were used in this study. In the
pattern search method, the objective function is optimized
while applying some bound constraints. A pattern is a set of
vectors defined as the scalar multiplication of the basis and the
generating matrices. The number of independent variables
determines the size of the generating matrix.
For every iteration, the pattern search algorithm uses the
pattern to search the points around the computed point in the
last iteration. If the new point lowers the value of the objective
function, it will be selected for comparison in the next
iteration. The process will continue till the final error is less
than a very small value (e.g. 0.0001). In our simulations, the
independent variables of the objective function are the
stiffness and damping coefficients (k2 and c2) of the lower
body wobbling mass.
B. Implementation of muscle fatigue in the active model
It is well known that the pre-landing muscle activity
prepares the musculoskeletal system for collision with the
ground [31]. In the active model, it is assumed that the prelanding muscle activity results in adjustment of the properties
of the LBWM [5]. The interval within which the stiffness and
damping properties of the LBWM may change, i.e. the
constraints of the controller, were determined [5] based on
experimental observations [26, 32]. In this study, it is assumed
that:
1. The adjustments in the properties of the LBWM are
caused by muscle activity. This assumption is in line with the
muscle tuning paradigm [4].
2. The intervals, within which the properties of the LBWM
can change, shrink as muscles fatigue. This assumption is
3
based on the reasoning that larger changes in the properties of
the LBWM need higher levels of muscle forces.
Based on these two assumptions, muscle fatigue is
implemented in the active model as time decay of the ability
of the human body to adjust the mechanical properties of the
LBWM (i.e. k2 and c2). The bound limits for pattern search
optimization were therefore assumed to narrow with time. For
the purpose of the current modeling study, the exact shape of
the time decay function is not important. That is because we
are primarily interested in the pre- and post-fatigue conditions
and not as much in the intermediate steps. However, it makes
sense to use a relatively generic decay function such as the
exponential decay function used by Franken et al [33].
Franken et al [33] observed that exponential functions can be
used to represent the changes in the measured values of the
quadriceps muscle force with time. A similar exponential
function as in [33] was used in the current study to narrow
down the bound limits that are used for pattern search
optimization:
é
æ
ù
tö
x (t) = xmax ê(1- xmin ) exp ç - ÷ + xmin ú
è tø
ë
û
(6)
where
t is the time passed after the start of running
ξ(t) is a coefficient that determines to what extent the
optimization bound limits shrink.
ξmax and ξmin are, respectively, the maximum and minimum
permissible values of the coefficient ξ. Similar to the previous
studies [5, 29], these two parameters were assumed to be
respectively 4 and ¼.
τ is the time-constant of fatigue.
At each time-step t, the upper (UB) and lower (LB)
optimization bounds of the stiffness (k2) and damping (c2)
parameters were defined as:
LBk (t) =
k2,0
, LB (t) =
c2,0
,
c2
x (t)
x (t)
UBk (t) = k2,0 .x (t), UBc (t) = c2,0 .x (t)
2
2
(7)
2
where k2,0 and c2,0 are the default values of k2 and c2 and are
given in Table 1.
C. Modeling considerations
Modeling the human body with a mass-spring-damper
model that can only move in the vertical direction is an
approach that has been used by several researchers to study
the loading of the body during running and hopping (for a
thorough review see reference [34]). The reader is referred to
those studies for a detailed discussion of why such simplified
models are useful for studying the loading of the body during
bouncing gait. The spring and dampers that are used in the
construction of the vertical mass-spring-dampers models
represent the stiffness and damping properties of the various
segments of the body. Certain characteristics of the
musculoskeletal system such as the joint angles chosen during
running cannot be directly represented in such models.
However, the net effect of the variations in the joint angles
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during running, i.e. changes in the stiffness of the body, can be
captured. The stiffness and damping properties that are used in
the model should therefore be considered as equivalent springs
and dampers that also represent the changes in the stiffness
caused by variations in the joint angles and other similar
mechanisms.
It should be noted that the way the muscle fatigue is
modeled in the current study is mostly based on biomechanical
reasoning and is therefore somewhat arbitrary. That is because
there is no consensus in the literature about the definition of
fatigue, the consequences of muscle fatigue for running [35],
and the protocols for inducing fatigue [36]. One of the benefits
of such a modeling study is that we can formulate certain
hypotheses and test them with the model to determine whether
or not the hypotheses result in meaningful and relevant
simulation results. In this study, we have proposed a simple
model and have used it to test a certain hypothesis. The way in
which muscle fatigue is implemented in the model is actually
a representation of the tested hypothesis. We postulate that
muscle fatigue occurs in the certain way implemented in the
model and run the simulations. We then try to understand and
validate the simulation results. In this way, the relationship
between the assumed fatigue mechanism and previous
experimental observations becomes clearer. This may
ultimately result in a better understanding of how lowerextremity muscle fatigue affects the performance of the human
body during running. A major contribution of this study using
such a simple model is proposing a certain hypothesis about
how the CNS tunes muscle properties after fatigue.
D. Simulations
Simulations were carried out for four different conditions:
1. the pre-fatigue condition: t/τ = 0,
2, 3. the intermediate conditions: t/τ = 1, and t/τ = 2,
4. the post-fatigue condition: t/τ = 3
For each condition and every set of shoe hardness
parameters b and d (varying in the ranges of 1 ≤ b ≤ 2 and
0.6 ≤ d ≤ 0.9), the k2 and c2 values that minimized the
objective function (i.e. Equation (5)) were calculated by using
the pattern search algorithm.
As stiffness may be related to activation, it could be argued
that the interval (within which the stiffness and damping
properties of the lower-body soft tissue package change)
shrinks mainly with regard to the upper bound, and not that
much for the lower bound. To investigate the possible effects
of this assumption, the stiffness variation mechanism
described by Equation (7) was updated such that only the
upper-bound (UB) changed with muscle fatigue. The
simulations were repeated for this updated stiffness variation
mechanism.
III.
RESULTS
Similar to the results of our previous studies [5, 29], there is
a particular area of the b-d plane (Fig. 2, the safe region)
within which the normalized error is very close to zero. The
safe region was defined in the same way as it was defined in
reference [29]: a part of the b-d plane (Fig. 2) within which the
4
normalized error is less than 0.05. The safe region can be
identified as the region that lies in between two dashed lines in
each subfigure of Fig. 2. One can see (Fig. 2) that the size of
the safe region does not considerably change as a result of
muscle fatigue. The updated assumption about stiffness
variation mechanism does not significantly change the
simulation results (Fig. 3).
Another important observation is that, inside the safe
region, the magnitude of the GRF as well as the vibration
amplitude is not much different between the pre-fatigue and
post-fatigue conditions (compare subfigures 2a and 2d, see
subfigures 4a and 4b). It can therefore be said that for a certain
range of shoe hardness parameters neither GRF nor vibration
amplitude change with fatigue.
Outside the safe region, the controller is not capable of
bringing the objective function close to zero. Therefore, the
magnitude of the GRF and the vibration amplitude that are
calculated for very soft or very hard shoes are far from their
reference values (subfigures 4a and 4b). Comparing pre- and
post-fatigue conditions (subfigures 2a-d), one can see that the
GRF and vibration amplitude exhibit different behaviors as the
level of muscle fatigue increases. While the GRF values do
not change with muscle fatigue even outside the safe region
(compare subfigures 3e-h), the vibration amplitudes deviate
from their pre-fatigue values as muscles fatigue (compare
subfigures 2i-l). The deviations from the pre-fatigue values are
larger for larger values of the shoe hardness parameter b.
It can therefore be concluded that the vibration amplitude
increases with fatigue for a certain range of shoe hardness
parameters (Fig. 4b, hard shoe). For b > 1.6, the vibration
amplitude increases by up to 20% (Fig. 2l and 2p) after longterm (t/τ = 3) running. For soft shoes, the vibration amplitude
does not change with fatigue, even though the peak
displacement occurs in different times before and after muscle
fatigue (Fig. 4b, soft shoe).
IV.
DISCUSSION
The simulation results obtained for both stiffness variations
mechanisms are almost the same (compare Fig. 2 and 3). This
is an important point because it shows that the main results of
the study are not dependent on the mechanism through which
the stiffness and damping properties of the model change.
There is a major difference between the effects of muscle
fatigue on the vertical GRF and on the level of soft tissue
vibrations. While the so-called safe region remains more or
less the same regardless of how much the muscles are
fatigued, the level of soft tissue vibrations may increase as
muscles fatigue. As for the GRF, the minimization error does
not significantly change with fatigue neither inside nor outside
the safe region (compare Fig. 2e-h).
These results have important implications for understanding
the effects of muscle fatigue on the GRF and vibration level.
When muscles are significantly fatigued (t/τ > 3), the intervals
within which the damping and stiffness properties of the
human body may be adjusted are very narrow. Nevertheless,
the controller is capable of adjusting the stiffness and damping
properties of the human body such that the GRF values remain
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5
constant. As Orishimo and Kremenic describe [37],
insignificant changes in the GRF with fatigue illustrate the
Winter’s assertion [38] that the human body adjusts the motor
patterns to produce a consistent GRF when landing. Moreover,
only minimal changes in the properties of the LBWM are
required for producing a consistent GRF before and after
fatigue. However, these minimal changes may not be
sufficient for producing a consistent level of vibrations and
that is why the vibration amplitude increases with fatigue for
hard shoes.
consistent levels of the GRF and vibration amplitude are
produced at contact with the ground. Both above-mentioned
lines of reasoning suggest that muscle fatigue ultimately
results in change in the stiffness and/or elastic storage
capability of the human body. Even though the proposed
model cannot capture such detailed mechanisms as change in
flexion angle, it can capture the net effect of those
mechanisms, i.e. the change in the stiffness of the human
body. That explains the agreement between the predictions of
the proposed model and experimental observations.
Comparison with experimental observations
It is important to compare the results of this modeling study
with experimental observations. As for the GRF, there is yet
no consensus in the literature on how the magnitude of the
vertical GRF changes with fatigue. Some researchers have
found that the vertical GRF decreases [9, 10, 12] with muscle
fatigue during running. However, some other researchers have
found no notable change [8], or have even observed increase
[7, 11]. The results of a recent meta-analysis study by the
current authors [35] showed that as far as running is
concerned, neither first nor the second GRF peak significantly
changes with fatigue. The results of the current modeling
study are in agreement with that conclusion.
We are only aware of one study that has investigated the
effects of muscle fatigue on soft tissue vibrations. The results
of that study [13] showed that the vibration amplitude of the
soft-tissue compartments (triceps-surae muscle) increase with
fatigue. The results of the current study coincide with the
findings of that study.
Limitations of the current study
The mass-spring-damper model that is proposed in this
study inherits all the limitations of mass-spring-damper
models. For example, it only works with vertical GRF and
cannot say anything about the horizontal GRF. Moreover, it
cannot distinguish between different muscles, as they are all
lumped into one single element with one single set of stiffness
and damping properties. Kellis et al [39] studied agonist vs.
antagonist muscle fatigue effects on the GRF during drop
landing. They found that these two different types of fatigues
might have different effects on the GRF. It is, however, not
clear to what extent their conclusions are generalizable to
running. In order to be able to study the fatigue effects of
individual muscles, more complex models such as large-scale
musculoskeletal models [44-46] are needed. However, study
of muscle fatigue and vibrations using musculoskeletal models
is not straightforward and involves modification of a large
number of modeling assumptions and parameters.
The other limitation of the model proposed here is that it
does not take the effects of upper-body vibrations into
account. The trunk-stabilizing (lower-back) musculature may
also play an important role during the shock absorption phase,
to minimize impact on the upper body and/or head. Such
effects are not included in the current study.
Why vibration/GRF changes/does not change with fatigue?
Researchers have tried to explain the effects of muscle
fatigue on the GRF and/or vibration using two major lines of
biomechanical reasoning:
The first line of reasoning speculates that the human body
possesses a protective mechanism that lowers the GRF and/or
vibration level when muscles fatigue in order to protect the
body from injuries. Decrease of the peak GRF values has been
explained based on such detailed mechanisms as change in
joint kinematics such as greater flexion angles at impact
during landing [8, 11, 39] or changes in the regulation of
muscle stiffness and reduced storage of elastic energy [10, 40].
According to the second line of reasoning, the capacity of
the human body’s protective mechanism decreases with
muscle fatigue. Based on this reasoning, one would expect that
the GRF and/or the vibration level should increase with
muscle fatigue. As James et al note [41], the increase of the
GRF has been explained by increased pre-activation of the
stabilizing musculature [42] that may result in increased joint
or system stiffness [40, 41, 43]. James et al [18] observed that
muscle activation increases with fatigue in the vastus medialis
and gastrocnemius, further supporting the explanation based
on increased stiffness. Friesenbichler et al [13] argue that the
vibration amplitude increases due to the weakening of the
protective mechanism with fatigue.
In the model proposed here, the controller tries to adjust the
stiffness and damping properties of the body such that
Implications of the simulation results
The major question we are asking in this paper is that ‘how
do the GRF and/or vibrations change due to the changes in the
stiffness of the lower body caused by muscle fatigue?’ The
simple model presented in this study provides an unexpected
answer to this question: the changes in the stiffness of the
lower body soft tissue package do not necessarily result in
significant changes in the GRF. The model shows that even
for such a simple model, there are certain ways for minimal
adjustment of the stiffness and damping properties of the
lower body tissue package such that the GRF remains almost
unchanged for a wide range of shoe hardness parameters. If
such a simple model could exhibit this feature, it may well be
the case that the more complex musculoskeletal system is also
capable of maintaining the same level of impact forces despite
muscle fatigue. A modeling study using dynamical rigid body
models is needed to clarify whether or not that actually is the
case.
Comparing the pre- and post-fatigue simulation results, one
can see that the controller converges to different values of the
stiffness and damping properties before and after fatigue.
However, for a wide range of shoe hardness parameters, these
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very different values of stiffness and damping properties result
t/τ = 0
(b)
(e)
(f)
(i)
t/τ = 2
t/τ = 3
(c)
(d)
(g)
(j)
(m)
in very similar values of the GRF.
t/τ = 1
(a)
(k)
(n)
6
(h)
(l)
(o)
(p)
(q)
(r)
(s)
(t)
(u)
(v)
(w)
(x)
(y)
(z)
(aa)
(bb)
(cc)
(dd)
(ee)
(ff)
Fig. 2. Simulation results at four different time-ratios (larger versions of the sub-figures of this figure are presented in the electronic supplement of the paper);
(a to d) the normalized error, (e to h) the absolute error of the force part, (i to l) the absolute error of the amplitude part, and (m to p) the vibration amplitude, (q
to t) the stiffness coefficients, (u to x) the damping coefficients, (y to bb) the first GRF peak, and (cc to ff) the second GRF peak. The shoe hardness changes
from “the softest” at the left top to “the hardest” at the right down. t/τ = 0 : pre-fatigue condition; t/τ = 3 : post-fatigue condition;
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t/τ = 3
Fig. 3. Simulation results for the post-fatigued condition (t/τ = 3) while only
the upper bound (UB) changes with time. For description of the labels see the
caption of Fig. 2.
3
(a)
2.5
VGRF (BW)
2
default model, pre-fatigue
default model, post-fatigue
safe region, pre-fatigue
safe region, post-fatigue
soft shoe, pre-fatigue
soft shoe, post-fatigue
hard shoe, pre-fatigue
hard shoe, post-fatigue
1.5
1
0.5
0
0
0.05
0.1
0.15
Stance Time (s)
0.2
0.25
(b)
0.1
X2 (m)
0.08
default model, pre-fatigue
default model, post-fatigue
safe region, pre-fatigue
safe region, post-fatigue
soft shoe, pre-fatigue
soft shoe, post-fatigue
hard shoe, pre-fatigue
hard shoe, post-fatigue
0.04
0.02
0
0
0.05
0.1
0.15
Stance Time (s)
0.2
The existence of several solutions for the same control
problem suggests the following hypothesis: the CNS is aware
of the existence of several solutions for the problem of
maintaining the level of the GRF and uses these various nonunique solutions to maintain similar levels of the GRF before
and after muscle fatigue. This is an important hypothesis that
needs to be experimentally examined.
The way of implementing fatigue in the proposed model is a
relatively practical way that is based only on biomechanical
reasoning and not detailed physiological observations. There
are two reasons for that. First, mass-spring-damper models are
not capable of implementing detailed physiological
observations. Second, there is no consensus in the literature as
to how exactly the running performance changes with muscle
fatigue [35, 36]. In particular, there are many different fatigue
protocols [36] each of which may affect the running
performance differently. This limitation is not only applicable
to this modeling study. Experimental studies of fatigue have
similar limitations, because they use very different fatigue
protocols [36] that may not be consistent and may affect the
running performance differently. We therefore need a
quantitative and universally accepted definition of fatigue
upon which a consistent fatigue protocol can be built. That
protocol may then be used for more consistent experimental
and modeling studies of how fatigue affects running.
This modeling study may contribute towards a better
definition of the fatigue protocol. The hypothesis of the
current modeling study is the way of implementing muscle
fatigue in the model. The fact that the simulation results are in
agreement with experimental observations may (at least
partially and subject to certain conditions) confirm the tested
hypothesis. This information may be useful for defining better
fatigue protocols in human running experiments.
V. CONCLUSIONS
0.12
0.06
7
0.25
Fig. 4. (a) Vertical GRF (VGRF) in terms of Body Weight (BW) and (b)
displacement of the LBWM vs. stance time for pre- and post-fatigue
conditions using selected pairs of (b, d) parameters; default values b=1.38,
d=0.75, inside the safe region: b=1.33, d=0.69, relatively soft shoe: b=1.24,
d=0.81, relatively hard shoe: b=1.96, d=0.85.
A mass-spring-damper model of fatigued running was
proposed in this study. In agreement with experimental
observations, the proposed model predicts that the vertical
GRF does not significantly change with fatigue. The level of
soft tissue vibrations may, however, increase. The results of
the current study may be ultimately useful for defining better
and more consistent fatigue protocols in human running
experiments. Moreover, a certain hypothesis is proposed based
on the obtained simulation results: ‘multiple solutions exist for
the problem of maintaining the same level of the GRF before
and after muscle fatigue. The CNS is aware of these
solutions.’
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING
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