arXiv:1204.1756v1  [cs.RO]  8 Apr 2012
Human Muscle Fatigue Model in
Dynamic Motions
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
Abstract Human muscle fatigue is considered to be one of the main reasons
for Musculoskeletal Disorder (MSD). Recent models have been introduced
to define muscle fatigue for static postures. However, the main drawbacks of
these models are that the dynamic effect of the human and the external load
are not taken into account. In this paper, each human joint is assumed to
be controlled by two muscle groups to generate motions such as push/pull.
The joint torques are computed using Lagrange’s formulation to evaluate the
dynamic factors of the muscle fatigue model. An experiment is defined to
validate this assumption and the result for one person confirms its feasibility.
The evaluation of this model can predict the fatigue and MSD risk in industry
production quickly.
Key words: Muscle fatigue model, Dynamic motions, Human simulation
1 Introduction
Muscle fatigue is defined as “any reduction in the ability to exert force in
response to voluntary effort” [2] and is one of the main reasons leading to
MSD [7]. From Hill’s muscle model [4] to today’s muscle fatigue models, this
topic has been researched from different scientific field with special point
of views. In general, mainly two approaches have been adopted to evaluate
muscle fatigue [10], either in theoretical methods or in empirical methods.
In [11], Wexleret et al. proposed a new muscle fatigue model based on Ca2+
cross-bridge mechanism and verified the model with simulation experiments.
Although this model can be used to predict the muscle force fatigue under
different simulation frequencies, the large number of variables make it diffi-
Ruina Ma, Damien Chablat, Fouad Bennis,
Institut de Recherche en Communications et Cybern´etique de Nantes, UMR CNRS 6597
e-mail: {Ruina.Ma, Damien.Chablat, Fouad.Bennis}@irccyn.ec-nantes.fr
Liang MA,
Department of Industrial Engineering, Tsinghua University.
e-mail: liangma@tsinghua.edu.cn
1
2
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
cult to use compare with other models. In [8], Liu et al. proposed a fatigue
and recovery models based on motor units pattern. They demonstrated the
relationship among muscle activation, fatigue and recovery. This model is
available under maximum voluntary contraction situation; this condition is
rare in the manual working situation. Another muscle fatigue model was de-
veloped by Giat [3] based on force-pH relationship. This fatigue model was
obtained by curve fitting of the pH level with time in the course of stimu-
lation and recovery, but it cannot used in evaluating the muscle fatigue in
the whole working process. In [9], Ma et al. proposed a muscle fatigue model
from the macroscopic point of view. External physical factors and personal
factors were taken into consideration to construct the model. This model
can predict the muscle fatigue trend in static working posture (θelbow = 90◦,
θshoulder = 30◦), but in dynamic working situation this model was limited.
The purpose of this work is to extend muscle fatigue model to dynamic
working situations. The difference force generation between static working
posture and dynamic working motions is depended on the activation of dif-
ferent muscle types. There are three types of fibers of muscle: slow-twitch
fibers, fast-twitch A fibers and fast-twitch B fibers [5]. In every postures and
motions all of the three muscle fibers are used, but the percent of every
fibers in static and dynamic situation is different. In a static working posture
fast-twitch fibers is mostly used and this type of muscle fibers have a low
resistance to fatigue. In a low speed dynamic working motions slow-twitch
fibers are mainly used and this type of muscle fibers have a high resistance
to fatigue. Meanwhile, the blood circulation during dynamic motions is bet-
ter than in a static working posture. For these reasons, the behavior of the
muscle and its fatigue rate are different in the two types of situations. In
this paper, a muscle fatigue model in dynamic situation is proposed. A new
approach to identify the fatigue rate parameter k is used. An experimental
setup is defined to validate this assumption.
Firstly, some assumptions are given and a new dynamic muscle fatigue
model is proposed. Secondly, an experiment is designed to verify this model.
Thirdly, a case-study for one person is illustrated and the fatigue parameter
k is evaluated. Finally, some perspectives are presented.
2 Proposal of a new muscle fatigue model
Dynamic muscle fatigue model: Muscles in the human body have one most
important function, such as force generating devices. They can work only
in a single direction. Hence, for each single joint, at least two groups of
muscles (agonistic muscle and antagonistic muscle) are necessary to control
the motion. The co-contraction of the two groups of muscles provide stability
to joint and balance to the posture. From the articulation point of view,
it is assumed that a joint is controlled by two groups of muscles (one for
flexion and one for extension). These muscle groups create a torque on the
Human Muscle Fatigue Model in Dynamic Motions
3
joint. This torque drives the human movement and whether it is positive or
negative depends on the angle and direction of joint rotation. Based on the
previous model of Ma et al., we propose that:
1. the fatigue of muscle is proportional to the joint torque, i.e. in the same
period of time, the larger the torque of joint exerted, the more fatigue
people feel;
2. the fatigue of muscle is inversely proportional to the muscle torque capacity
i.e. the smaller the capacity is, the quicker the muscle becomes tired.
This can be mathematically described by the following equation.
dΓcem(t)
dt
= −k · Γcem(t)
ΓMVC
· Γjoint(t)
(1)
where the set of parameters are listed in Table 1.
Item
Unit
Description
ΓMVC
N.m
Maximum voluntary contraction of joint torque, i.e. Γmax
Γcem(t) N.m
Current exertable maximum joint torque
Γjoint
N.m
Joint torque, i.e. the torque which the joint needs to generate
k
min−1 Fatigue rate
Table 1 Parameters in dynamic fatigue model
If we assume that Γcem(0) = ΓMVC and k is a constant, the integration result
of the previous equation is given by
Γcem(t) = ΓMVC · e−
k
ΓMVC
R t
0 Γjoint(u)du
(2)
The value of ΓMVC is a fixed value determined by individual person. In the
first approximation, we assume that ΓMVC is a constant of a joint torque
during a limited period of time. According to robotic dynamic model [6],
Γjoint(u) can be modeled by a variable depending on the angle, the velocity,
the acceleration and the internal/external load.
Γjoint(u)
def
= Γ(u, θ, ˙θ, ¨θ)
(3)
This way, Equation (2) can be further simplified in the form.
Γcem(t) = ΓMVC · e−
k
ΓMVC
R t
0 Γ (u,θ, ˙θ,¨θ)du
(4)
Equation (4) defines our new dynamic muscle fatigue model. The model takes
consideration of the motion by the variations of the torque Γjoint from joint
level. This torque which is computed using robotic method is integrated to
obtain the current exertable maximum joint torque. At first stage, we do not
take into account the muscle co-contraction factor [1]. This work enlarges the
muscle fatigue model usefulness range.
The new dynamic fatigue model is in joint level. As mentioned in the
assumption, the motion of joint is driven by a pair of muscles. Obviously,
4
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
this model can be easily applicated in muscle level. For one cycle, the Γjoint is
negative or positive related with elbow rotation range. If Γjoint is positive, we
suppose it is the effect of agonistic muscle. Inversely, if Γjoint is negative, we
suppose it is the effect of antagonistic muscle. Based on this consideration,
there will be two fatigue rate parameters k (kagonist and kantagonist) for a
dynamic operation in the muscle level. Our new dynamic fatigue model can
applicate fatigue evaluation in muscle level.
3 Experiment design for validation
The aim of the experiment design is to
Fig. 1
Measurement device
evaluate the muscle fatigue model. We sup-
pose that in a push operation the agonis-
tic muscle is mainly used whereas in a pull
operation the antagonistic muscle is mainly
used. Based on this assumptions, we concen-
trate the study on the elbow joint and use
a push/pull operation to simulate dynamic
motions. This evaluation consists in measur-
ing the maximal push and pull strength after
a continuous movement of the lower arm.
Experiment materials (Figure 1):
1. A dynamometer. This device is used to measure the maximum push/pull
force after lengths of time’s movement of the lower arm.
2. A bar. This weight is grabbed by the hand of the participant and is used
to simulate the weight of an operation tool in industrial environment. For
our experiment the bar weight is 3 Kg.
3. A metronome. This tool is used to define the sample times of the motion.
For our experiment the frequency is 1Hz.
4. A self-made support. This support is used to maintain the elbow posture
during the motion and measure the torque after the operation.
Experiment procedure: The participant seats in a chair and puts his elbow
on the support. The procedure is to repeat a rotation of the elbow joint from
0 to 75 degrees and then from 75 degrees to 0 during ti unit of time. This
movement is done with the bar in hand. The experiment procedure is as
follows:
1. Measure ΓMVC before starting the operation (ΓMVC(0) = Γcem(0));
2. Perform the dynamic operation during ti unit of time;
3. Measure the remained maximum torque of elbow joint Γcem(ti);
4. Take a rest about 1-2 hours until complete recovery;
5. Repeat steps 2, 3 and 4 for different values of ti, ti ∈{0, 1, 2, 3, 4, 5}
minutes.
Human Muscle Fatigue Model in Dynamic Motions
5
4 Case-study of muscle fatigue for the elbow joint
Kinematics and dynamics of the arm: In this section we will illustrate the dy-
namic calculation in our dynamic muscle fatigue model using robotic method.
An example of the lower arm cyclic periodic movement during 2 seconds is
demonstrated in details.
Geometric modeling of arm: As presented in Figure 2, the arm model
is composed of an upper arm, a lower arm and a hand. Figure 2(b) gives
the simplified model of the arm used for the calculation. In order to deter-
mine the torque Γjoint, several parameters of the arm need to be obtained.
These parameters are the length of the lower arm (ℓf), the length of the
hand (ℓh), the radius of the lower arm (rf) and the mass of the lower arm
(mf). If the human has a height of H and a weight of M, according to the
anthropometry database [2], the related geometric human parameters are:
ℓf = 0.146H,
rf = 0.125ℓf,
ℓh = 0.108H,
mf = 0.023M.
M, I,G,L
i
i
i
i
O
g
Upper arm
Lower arm
i
M , I ,G ,L
o
o
o
o
O
y
O
x
Q
(a)
l
r
mf
O
lf
rf
O
y
x
(b)
Fig. 2 (a) Geometric and inertia parameters of one arm, (b) Simplified arm structure
Trajectory generation: We suppose that for one motion of lower arm up
and down movement both the initial and the final velocity and acceleration
are nul. We use a polynomial function to describe this movement. Accord-
ing to the hypothesis, the minimum degree of the polynomial satisfying the
constraints is at least five and has the following form.
P = a0 + a1t + a2t2 + a3t3 + a4t4 + a5t5
(5)
where the coefficients ai are determined from the boundary conditions:
θ(0) = θinitial, ˙θ(0) = 0, ¨θ(0) = 0
θ(tf) = θend,
˙θ(tf) = 0, ¨θ(tf) = 0
(6)
The trajectory between θinitial and θend is determined by
θ(t) = θinitial + r(t) ·
θend −θinitial

,
0 ≤t ≤tf
(7)
Solving Equation (7) with the above mentioned condition we can get the
following interpolation function
r(t) = 10 (t/tf)3 −15 (t/tf)4 + 6 (t/tf)5
(8)
6
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
Based on this interpolation function we can get the velocity and acceleration
of every moment in the joint trajectory. The Figure 3 represents the evolution
of θ, ˙θ and ¨θ for the considering experiment with the angle change between
0 to 5π/12.
0
0.01
0.02
0.03
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time(Minutes)
Angle(rad)
(a)
0
0.01
0.02
0.03
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Time(Minutes)
Vitess(rad/s)
(b)
0
0.01
0.02
0.03
-8
-6
-4
-2
0
2
4
6
8
Time(Minutes)
Acceleration(rad/s²)
(c)
Fig. 3 (a) Joint angle, (b) Joint angular velocity, (c) Joint angular acceleration evolution
during one cycle motion
Dynamic model and joint torque evaluation: The Lagrange method
is applied to compute the dynamic model [6]. Firstly, we calculate the joint
kinetic energy and the joint potential energy
E = Ejoint + Eobject,
U = Ujoint + Uobject
(9)
Then, the joint torque is given by
Γ = d
dt
∂L
∂˙θ

−
∂L
∂θ

(10)
where L = E −U. In our dynamic muscle fatigue model,
R t
0 Γ(u, θ, ˙θ, ¨θ)du is
0
0.01
0.02
0.03
-5
0
5
10
15
20
25
Time(Minutes)
Torque(N.m)
(a)
0
0.01
0.02
0.03
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time(Minutes)
Torque(N.s)
(b)
0
0.01
0.02
0.03
0
1
2
3
4
5
6
7x 10
-3
Time(Minutes)
Torque(N.s)
(c)
Fig. 4 (a) Elbow joint torque, (b) Momentum of agonistic muscle, (c) Momentum of
antagonistic muscle
the joint momentum. This is the most important difference between dynamic
muscle fatigue model and static muscle fatigue model. In static situation the
joint torque is a constant, and with time goes by the joint momentum is a
linear function. In dynamic muscle fatigue model the joint torque is changing
with joint angle and time. The joint momentum is a non-linear function. Fig-
ure 4(a) is the torque of elbow joint in a cyclic motion of 2 seconds and there
are a part of positive torque and a part of negative torque. We consider the
positive torque as a result of the effect of angonistic muscle and the negative
Human Muscle Fatigue Model in Dynamic Motions
7
torque as a result of antagonistic muscle. Figure 4 presented joint torque and
momentum evolution of two groups of muscles during 2 seconds.
Experimental results: The experiment part is an implementation and verifi-
cation of above mentioned experiment design. At first stage we just measure
one person to test its operability and feasibility. A large number of tests will
be carried out in the future stage.
Experiment result for one person: A male subject (H =188cm, M =80Kg)
took part in the presented experiment. Push and Pull torque of lower arm
are measured for the different operation times of t in {0, 1, 2, 3, 4, 5} minutes.
The experiment results are presented in Table 2.
Γcem[N · m] 0 min 1 min 2 min 3 min 4 min 5 min
Push
31.46 30.08 28.07 29.33 26.32 26.82
Pull
31.71 28.33 22.94 22.31 20.05 18.67
Table 2
Current exertable maximum joint torque for push and pull action
Fatigue rate parameter k evaluation: The parameter k represents fatigue
rate and it depends on individual person itself. To evaluate the parameter k of
our model, we suppose k is constant. The following Eq. (11) which is deduced
from Eq. (4) is used to calculate ki with the help of using the experiment
measurement of Γcemi for each operation.
ki = −ln
Γcem(t)
ΓMVC

/
Z ti
0
Γ(u, θ, ˙θ, ¨θ)du
(11)
For t = 1, 2, 3, 4, 5 minutes, the agonistic and antagonistic muscle group fa-
tigue rate were evaluated as follows:
kagonist = [0.13, 0.17, 0.07, 0.13, 0.09][min−1]
kantagonist = [19.56, 28.07, 20.32, 19.86, 18.36][min−1]
In Ma [9], the values of k obtained are
0
1
2
3
4
5
23
24
25
26
27
28
29
30
31
32
Time(Minutes)
Torque(N.m)
minimum k
average k
max k
Experiment data
Fig. 5 Theoretical evolution of Γcem
and experiment data using different
values of kagonist
around 0.87, so kagonist is a realistic value
due to that the fact the blood circulation
is better during dynamic motions. Con-
versely, kantagonist seems to be too high.
In fact, due to the co-contraction activ-
ities influence, the torque of the antag-
onistic muscle group is higher than the
results computed by the dynamic model.
To characterize kantagonist more precisely,
another experimental measurement is nec-
essary to make the same motion with a
pulley based system that inverse the grav-
ity force. Because of the measurement errors of forces, the calculated k is
not exactly the same for each time t. To evaluate the confidence of the fa-
tigue rate parameter k, with the minimum, average and maximum values of
8
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
kagonist, Γcem is evaluated seperately and compared with the experimental
measurements in Fig. 5. It seems that the first two experimental measure-
ments overestimate kagonist. This means that we have to wait three minutes
to have a good evaluation of the muscle fatigue properties. In fact, we can
consider the force capacity of one muscle group can increase in the beginning
of the activity as a warming-up period of the muscle. As only one person par-
ticipated the experiment, the conclusion cannot be generalized but we have
obtained interesting informations. The test will be done for a representivie
number of participants in the future works.
5 Conclusion and Perspectives
In this paper, a new muscle fatigue model for dynamic motions is presented.
Thanks to the robotic method, dynamic factors have been introduced to char-
acterize a new dynamic muscle fatigue model from the joint level. This model
can be explained theoretically. Meanwhile, an experiment has been designed
to validate it. This model could demonstrate the potential for predicting mus-
cle fatigue in dynamic motions. The limit of this work is that it still lacks
experimental validation for more participants. In the future, validations of
experiments for a number of participants will be carried out.
References
1. Ait-Haddou, R., Binding, P., Herzog, W.: Theoretical considerations on cocontraction
of sets of agonistic and antagonistic muscles. Journal of Biomechanics 33(9), 1105 –
1111 (2000)
2. Chaffin, D.B., Andersson, G.B.J., Martin, B.J.: Occupational Biomechanics, 3rd Edi-
tion. Wiley-Interscience (1999)
3. Giat, Y., Mizrahi, J., Levy, M.: A musculotendon model of the fatigue profiles of
paralyzed quadriceps muscle under fes. IEEE Transactions On Biomedical Engineering
40(7), 664–674 (1993)
4. Hill A, V.: The heat of shortening and dynamic constants of muscle. Biological Sciences
126, 136–195 (1983)
5. Karp, J.R.: Muscle fiber types and training. Strength and Conditioning Journal 23(5),
21 (2001)
6. Khalil, W.: Modeling and control of manipulators. Ecole Central de Nantes (2009-
2010)
7. Kumar, R., Kumar, S.: Musculoskeletal risk factors in cleaning occupation: a literature
review. International Journal of Industrial Ergonomics 38, 158–170 (2008)
8. Liu, J., Brown, R., Yue, G.: A dynamical model of muscle activation, fatigue, and
recovery. Biophysical Journal 82(5), 2344–2359 (2002)
9. MA, L.: Contributions pour lanalyse ergonomique de mannequins virtuels.
Ph.D.
thesis, Ecole Central de Nantes (2009)
10. Westgaard, R.H.: Work-related musculoskeletal complaints: some ergonomics chal-
lenges upon the start of a new century. Applied Ergonomics 31, 569–580 (2000)
11. Wexler, A.S., Ding, J., Binder-Macleod, S.A.: A mathematical model that predicts
skeletal muscle force. IEEE Transactions On Biomedical Engineering 44(5), 337–348
(1997)
arXiv:1204.1756v1  [cs.RO]  8 Apr 2012
Human Muscle Fatigue Model in
Dynamic Motions
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
Abstract Human muscle fatigue is considered to be one of the main reasons
for Musculoskeletal Disorder (MSD). Recent models have been introduced
to define muscle fatigue for static postures. However, the main drawbacks of
these models are that the dynamic effect of the human and the external load
are not taken into account. In this paper, each human joint is assumed to
be controlled by two muscle groups to generate motions such as push/pull.
The joint torques are computed using Lagrange’s formulation to evaluate the
dynamic factors of the muscle fatigue model. An experiment is defined to
validate this assumption and the result for one person confirms its feasibility.
The evaluation of this model can predict the fatigue and MSD risk in industry
production quickly.
Key words: Muscle fatigue model, Dynamic motions, Human simulation
1 Introduction
Muscle fatigue is defined as “any reduction in the ability to exert force in
response to voluntary effort” [2] and is one of the main reasons leading to
MSD [7]. From Hill’s muscle model [4] to today’s muscle fatigue models, this
topic has been researched from different scientific field with special point
of views. In general, mainly two approaches have been adopted to evaluate
muscle fatigue [10], either in theoretical methods or in empirical methods.
In [11], Wexleret et al. proposed a new muscle fatigue model based on Ca2+
cross-bridge mechanism and verified the model with simulation experiments.
Although this model can be used to predict the muscle force fatigue under
different simulation frequencies, the large number of variables make it diffi-
Ruina Ma, Damien Chablat, Fouad Bennis,
Institut de Recherche en Communications et Cybern´etique de Nantes, UMR CNRS 6597
e-mail: {Ruina.Ma, Damien.Chablat, Fouad.Bennis}@irccyn.ec-nantes.fr
Liang MA,
Department of Industrial Engineering, Tsinghua University.
e-mail: liangma@tsinghua.edu.cn
1
2
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
cult to use compare with other models. In [8], Liu et al. proposed a fatigue
and recovery models based on motor units pattern. They demonstrated the
relationship among muscle activation, fatigue and recovery. This model is
available under maximum voluntary contraction situation; this condition is
rare in the manual working situation. Another muscle fatigue model was de-
veloped by Giat [3] based on force-pH relationship. This fatigue model was
obtained by curve fitting of the pH level with time in the course of stimu-
lation and recovery, but it cannot used in evaluating the muscle fatigue in
the whole working process. In [9], Ma et al. proposed a muscle fatigue model
from the macroscopic point of view. External physical factors and personal
factors were taken into consideration to construct the model. This model
can predict the muscle fatigue trend in static working posture (θelbow = 90◦,
θshoulder = 30◦), but in dynamic working situation this model was limited.
The purpose of this work is to extend muscle fatigue model to dynamic
working situations. The difference force generation between static working
posture and dynamic working motions is depended on the activation of dif-
ferent muscle types. There are three types of fibers of muscle: slow-twitch
fibers, fast-twitch A fibers and fast-twitch B fibers [5]. In every postures and
motions all of the three muscle fibers are used, but the percent of every
fibers in static and dynamic situation is different. In a static working posture
fast-twitch fibers is mostly used and this type of muscle fibers have a low
resistance to fatigue. In a low speed dynamic working motions slow-twitch
fibers are mainly used and this type of muscle fibers have a high resistance
to fatigue. Meanwhile, the blood circulation during dynamic motions is bet-
ter than in a static working posture. For these reasons, the behavior of the
muscle and its fatigue rate are different in the two types of situations. In
this paper, a muscle fatigue model in dynamic situation is proposed. A new
approach to identify the fatigue rate parameter k is used. An experimental
setup is defined to validate this assumption.
Firstly, some assumptions are given and a new dynamic muscle fatigue
model is proposed. Secondly, an experiment is designed to verify this model.
Thirdly, a case-study for one person is illustrated and the fatigue parameter
k is evaluated. Finally, some perspectives are presented.
2 Proposal of a new muscle fatigue model
Dynamic muscle fatigue model: Muscles in the human body have one most
important function, such as force generating devices. They can work only
in a single direction. Hence, for each single joint, at least two groups of
muscles (agonistic muscle and antagonistic muscle) are necessary to control
the motion. The co-contraction of the two groups of muscles provide stability
to joint and balance to the posture. From the articulation point of view,
it is assumed that a joint is controlled by two groups of muscles (one for
flexion and one for extension). These muscle groups create a torque on the
Human Muscle Fatigue Model in Dynamic Motions
3
joint. This torque drives the human movement and whether it is positive or
negative depends on the angle and direction of joint rotation. Based on the
previous model of Ma et al., we propose that:
1. the fatigue of muscle is proportional to the joint torque, i.e. in the same
period of time, the larger the torque of joint exerted, the more fatigue
people feel;
2. the fatigue of muscle is inversely proportional to the muscle torque capacity
i.e. the smaller the capacity is, the quicker the muscle becomes tired.
This can be mathematically described by the following equation.
dΓcem(t)
dt
= −k · Γcem(t)
ΓMVC
· Γjoint(t)
(1)
where the set of parameters are listed in Table 1.
Item
Unit
Description
ΓMVC
N.m
Maximum voluntary contraction of joint torque, i.e. Γmax
Γcem(t) N.m
Current exertable maximum joint torque
Γjoint
N.m
Joint torque, i.e. the torque which the joint needs to generate
k
min−1 Fatigue rate
Table 1 Parameters in dynamic fatigue model
If we assume that Γcem(0) = ΓMVC and k is a constant, the integration result
of the previous equation is given by
Γcem(t) = ΓMVC · e−
k
ΓMVC
R t
0 Γjoint(u)du
(2)
The value of ΓMVC is a fixed value determined by individual person. In the
first approximation, we assume that ΓMVC is a constant of a joint torque
during a limited period of time. According to robotic dynamic model [6],
Γjoint(u) can be modeled by a variable depending on the angle, the velocity,
the acceleration and the internal/external load.
Γjoint(u)
def
= Γ(u, θ, ˙θ, ¨θ)
(3)
This way, Equation (2) can be further simplified in the form.
Γcem(t) = ΓMVC · e−
k
ΓMVC
R t
0 Γ (u,θ, ˙θ,¨θ)du
(4)
Equation (4) defines our new dynamic muscle fatigue model. The model takes
consideration of the motion by the variations of the torque Γjoint from joint
level. This torque which is computed using robotic method is integrated to
obtain the current exertable maximum joint torque. At first stage, we do not
take into account the muscle co-contraction factor [1]. This work enlarges the
muscle fatigue model usefulness range.
The new dynamic fatigue model is in joint level. As mentioned in the
assumption, the motion of joint is driven by a pair of muscles. Obviously,
4
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
this model can be easily applicated in muscle level. For one cycle, the Γjoint is
negative or positive related with elbow rotation range. If Γjoint is positive, we
suppose it is the effect of agonistic muscle. Inversely, if Γjoint is negative, we
suppose it is the effect of antagonistic muscle. Based on this consideration,
there will be two fatigue rate parameters k (kagonist and kantagonist) for a
dynamic operation in the muscle level. Our new dynamic fatigue model can
applicate fatigue evaluation in muscle level.
3 Experiment design for validation
The aim of the experiment design is to
Fig. 1
Measurement device
evaluate the muscle fatigue model. We sup-
pose that in a push operation the agonis-
tic muscle is mainly used whereas in a pull
operation the antagonistic muscle is mainly
used. Based on this assumptions, we concen-
trate the study on the elbow joint and use
a push/pull operation to simulate dynamic
motions. This evaluation consists in measur-
ing the maximal push and pull strength after
a continuous movement of the lower arm.
Experiment materials (Figure 1):
1. A dynamometer. This device is used to measure the maximum push/pull
force after lengths of time’s movement of the lower arm.
2. A bar. This weight is grabbed by the hand of the participant and is used
to simulate the weight of an operation tool in industrial environment. For
our experiment the bar weight is 3 Kg.
3. A metronome. This tool is used to define the sample times of the motion.
For our experiment the frequency is 1Hz.
4. A self-made support. This support is used to maintain the elbow posture
during the motion and measure the torque after the operation.
Experiment procedure: The participant seats in a chair and puts his elbow
on the support. The procedure is to repeat a rotation of the elbow joint from
0 to 75 degrees and then from 75 degrees to 0 during ti unit of time. This
movement is done with the bar in hand. The experiment procedure is as
follows:
1. Measure ΓMVC before starting the operation (ΓMVC(0) = Γcem(0));
2. Perform the dynamic operation during ti unit of time;
3. Measure the remained maximum torque of elbow joint Γcem(ti);
4. Take a rest about 1-2 hours until complete recovery;
5. Repeat steps 2, 3 and 4 for different values of ti, ti ∈{0, 1, 2, 3, 4, 5}
minutes.
Human Muscle Fatigue Model in Dynamic Motions
5
4 Case-study of muscle fatigue for the elbow joint
Kinematics and dynamics of the arm: In this section we will illustrate the dy-
namic calculation in our dynamic muscle fatigue model using robotic method.
An example of the lower arm cyclic periodic movement during 2 seconds is
demonstrated in details.
Geometric modeling of arm: As presented in Figure 2, the arm model
is composed of an upper arm, a lower arm and a hand. Figure 2(b) gives
the simplified model of the arm used for the calculation. In order to deter-
mine the torque Γjoint, several parameters of the arm need to be obtained.
These parameters are the length of the lower arm (ℓf), the length of the
hand (ℓh), the radius of the lower arm (rf) and the mass of the lower arm
(mf). If the human has a height of H and a weight of M, according to the
anthropometry database [2], the related geometric human parameters are:
ℓf = 0.146H,
rf = 0.125ℓf,
ℓh = 0.108H,
mf = 0.023M.
M, I,G,L
i
i
i
i
O
g
Upper arm
Lower arm
i
M , I ,G ,L
o
o
o
o
O
y
O
x
Q
(a)
l
r
mf
O
lf
rf
O
y
x
(b)
Fig. 2 (a) Geometric and inertia parameters of one arm, (b) Simplified arm structure
Trajectory generation: We suppose that for one motion of lower arm up
and down movement both the initial and the final velocity and acceleration
are nul. We use a polynomial function to describe this movement. Accord-
ing to the hypothesis, the minimum degree of the polynomial satisfying the
constraints is at least five and has the following form.
P = a0 + a1t + a2t2 + a3t3 + a4t4 + a5t5
(5)
where the coefficients ai are determined from the boundary conditions:
θ(0) = θinitial, ˙θ(0) = 0, ¨θ(0) = 0
θ(tf) = θend,
˙θ(tf) = 0, ¨θ(tf) = 0
(6)
The trajectory between θinitial and θend is determined by
θ(t) = θinitial + r(t) ·
θend −θinitial

,
0 ≤t ≤tf
(7)
Solving Equation (7) with the above mentioned condition we can get the
following interpolation function
r(t) = 10 (t/tf)3 −15 (t/tf)4 + 6 (t/tf)5
(8)
6
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
Based on this interpolation function we can get the velocity and acceleration
of every moment in the joint trajectory. The Figure 3 represents the evolution
of θ, ˙θ and ¨θ for the considering experiment with the angle change between
0 to 5π/12.
0
0.01
0.02
0.03
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Time(Minutes)
Angle(rad)
(a)
0
0.01
0.02
0.03
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Time(Minutes)
Vitess(rad/s)
(b)
0
0.01
0.02
0.03
-8
-6
-4
-2
0
2
4
6
8
Time(Minutes)
Acceleration(rad/s²)
(c)
Fig. 3 (a) Joint angle, (b) Joint angular velocity, (c) Joint angular acceleration evolution
during one cycle motion
Dynamic model and joint torque evaluation: The Lagrange method
is applied to compute the dynamic model [6]. Firstly, we calculate the joint
kinetic energy and the joint potential energy
E = Ejoint + Eobject,
U = Ujoint + Uobject
(9)
Then, the joint torque is given by
Γ = d
dt
∂L
∂˙θ

−
∂L
∂θ

(10)
where L = E −U. In our dynamic muscle fatigue model,
R t
0 Γ(u, θ, ˙θ, ¨θ)du is
0
0.01
0.02
0.03
-5
0
5
10
15
20
25
Time(Minutes)
Torque(N.m)
(a)
0
0.01
0.02
0.03
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time(Minutes)
Torque(N.s)
(b)
0
0.01
0.02
0.03
0
1
2
3
4
5
6
7x 10
-3
Time(Minutes)
Torque(N.s)
(c)
Fig. 4 (a) Elbow joint torque, (b) Momentum of agonistic muscle, (c) Momentum of
antagonistic muscle
the joint momentum. This is the most important difference between dynamic
muscle fatigue model and static muscle fatigue model. In static situation the
joint torque is a constant, and with time goes by the joint momentum is a
linear function. In dynamic muscle fatigue model the joint torque is changing
with joint angle and time. The joint momentum is a non-linear function. Fig-
ure 4(a) is the torque of elbow joint in a cyclic motion of 2 seconds and there
are a part of positive torque and a part of negative torque. We consider the
positive torque as a result of the effect of angonistic muscle and the negative
Human Muscle Fatigue Model in Dynamic Motions
7
torque as a result of antagonistic muscle. Figure 4 presented joint torque and
momentum evolution of two groups of muscles during 2 seconds.
Experimental results: The experiment part is an implementation and verifi-
cation of above mentioned experiment design. At first stage we just measure
one person to test its operability and feasibility. A large number of tests will
be carried out in the future stage.
Experiment result for one person: A male subject (H =188cm, M =80Kg)
took part in the presented experiment. Push and Pull torque of lower arm
are measured for the different operation times of t in {0, 1, 2, 3, 4, 5} minutes.
The experiment results are presented in Table 2.
Γcem[N · m] 0 min 1 min 2 min 3 min 4 min 5 min
Push
31.46 30.08 28.07 29.33 26.32 26.82
Pull
31.71 28.33 22.94 22.31 20.05 18.67
Table 2
Current exertable maximum joint torque for push and pull action
Fatigue rate parameter k evaluation: The parameter k represents fatigue
rate and it depends on individual person itself. To evaluate the parameter k of
our model, we suppose k is constant. The following Eq. (11) which is deduced
from Eq. (4) is used to calculate ki with the help of using the experiment
measurement of Γcemi for each operation.
ki = −ln
Γcem(t)
ΓMVC

/
Z ti
0
Γ(u, θ, ˙θ, ¨θ)du
(11)
For t = 1, 2, 3, 4, 5 minutes, the agonistic and antagonistic muscle group fa-
tigue rate were evaluated as follows:
kagonist = [0.13, 0.17, 0.07, 0.13, 0.09][min−1]
kantagonist = [19.56, 28.07, 20.32, 19.86, 18.36][min−1]
In Ma [9], the values of k obtained are
0
1
2
3
4
5
23
24
25
26
27
28
29
30
31
32
Time(Minutes)
Torque(N.m)
minimum k
average k
max k
Experiment data
Fig. 5 Theoretical evolution of Γcem
and experiment data using different
values of kagonist
around 0.87, so kagonist is a realistic value
due to that the fact the blood circulation
is better during dynamic motions. Con-
versely, kantagonist seems to be too high.
In fact, due to the co-contraction activ-
ities influence, the torque of the antag-
onistic muscle group is higher than the
results computed by the dynamic model.
To characterize kantagonist more precisely,
another experimental measurement is nec-
essary to make the same motion with a
pulley based system that inverse the grav-
ity force. Because of the measurement errors of forces, the calculated k is
not exactly the same for each time t. To evaluate the confidence of the fa-
tigue rate parameter k, with the minimum, average and maximum values of
8
Ruina Ma, Damien Chablat, Fouad Bennis, and Liang Ma
kagonist, Γcem is evaluated seperately and compared with the experimental
measurements in Fig. 5. It seems that the first two experimental measure-
ments overestimate kagonist. This means that we have to wait three minutes
to have a good evaluation of the muscle fatigue properties. In fact, we can
consider the force capacity of one muscle group can increase in the beginning
of the activity as a warming-up period of the muscle. As only one person par-
ticipated the experiment, the conclusion cannot be generalized but we have
obtained interesting informations. The test will be done for a representivie
number of participants in the future works.
5 Conclusion and Perspectives
In this paper, a new muscle fatigue model for dynamic motions is presented.
Thanks to the robotic method, dynamic factors have been introduced to char-
acterize a new dynamic muscle fatigue model from the joint level. This model
can be explained theoretically. Meanwhile, an experiment has been designed
to validate it. This model could demonstrate the potential for predicting mus-
cle fatigue in dynamic motions. The limit of this work is that it still lacks
experimental validation for more participants. In the future, validations of
experiments for a number of participants will be carried out.
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