arXiv:1104.1190v1  [cs.RO]  6 Apr 2011
A novel approach for determining fatigue resistances of different muscle groups in static
cases
Liang MAa,c,∗, Damien CHABLATb, Fouad BENNISb,c, Wei ZHANGa, Bo HUa, Franc¸ois GUILLAUMEd
aDepartment of Industrial Engineering, Tsinghua University, 100084, Beijing, P.R.China
b Institut de Recherche en Communications et en Cybern´etique de Nantes, UMR 6597
IRCCyN - 1, rue de la No¨e, BP 92 101 - 44321 Nantes CEDEX 03, France
c ´Ecole Centrale de Nantes, 1, rue de la No¨e, BP 92 101 - 44321 Nantes CEDEX 03, France
dEADS Innovation Works, 12, rue Pasteur - BP 76, 92152 Suresnes Cedex - France
Abstract
In ergonomics and biomechanics, muscle fatigue models based on maximum endurance time (MET) models are often used to
integrate fatigue effect into ergonomic and biomechanical application. However, due to the empirical principle of those MET
models, the disadvantages of this method are: 1) the MET models cannot reveal the muscle physiology background very well; 2)
there is no general formation for those MET models to predict MET. In this paper, a theoretical MET model is extended from a
simple muscle fatigue model with consideration of the external load and maximum voluntary contraction in passive static exertion
cases. The universal availability of the extended MET model is analyzed in comparison to 24 existing empirical MET models. Using
mathematical regression method, 21 of the 24 MET models have intraclass correlations over 0.9, which means the extended MET
model could replace the existing MET models in a general and computationally efficient way. In addition, an important parameter,
fatigability (or fatigue resistance) of different muscle groups, could be calculated via the mathematical regression approach. Its
mean value and its standard deviation are useful for predicting MET values of a given population during static operations. The
possible reasons influencing the fatigue resistance were classified and discussed, and it is still a very challenging work to find out
the quantitative relationship between the fatigue resistance and the influencing factors.
Relevance to industry :
MSD risks can be reduced by correct evaluation of static muscular work. Different muscle groups have different properties, and
a generalized MET model is useful to simplify the fatigue analysis and fatigue modeling, especially for digital human techniques
and virtual human simulation tools.
Key words: muscle fatigue, biomechanical muscle modeling, fatigue resistance, maximum endurance time, muscle groups
1. Introduction
Muscle fatigue is defined as “any reduction in the ability
to exert force in response to voluntary effort” (Chaffin et al.,
1999), and it is believed that the muscle fatigue is one of poten-
tial reasons leading to musculoskeletal disorders (MSDs) in the
literature (Westgaard, 2000; Kumar and Kumar, 2008; Kim et al.,
2008). Great effort has been contributed to integrate fatigue
into different biomechanical models, especially in virtual hu-
man simulation for ergonomic application, in order to analyze
the fatigue in muscles and joints and further to decrease the
MSD risks (Badler et al., 1993; Vignes, 2004; Hou et al., 2007).
In general, mainly two approaches have been adopted to
represent muscle fatigue, either in theoretical methods or in
empirical methods (Xia and Frey Law, 2008). One or more de-
cay terms were introduced into existing muscle force models in
theoretical fatigue models, and those decay terms were mainly
∗Corresponding author: Tel:+86-10-62792665; Fax:+86-10-62794399
Email addresses: liangma@tsinghua.edu.cn;
liang.ma@irccyn.ec-nantes.fr (Liang MA)
based on physiological performance of muscles in fatigue con-
traction. For example, a fatigue model based on the intracel-
lular pH (Giat et al., 1993) was incorporated into Hill’s muscle
mechanical model (Hill, 1938). This fatigue model was also
applied by Komura et al. (1999) to demonstrate the fatigue of
different individual muscles.
Another muscle fatigue model
(Ding et al., 2003) based on physiological mechanism has been
included into the Virtual Solider Research Program (Vignes,
2004), and in this model, dozens of parameters have to be fit
for model identification only for a single muscle. As stated in
Xia and Frey Law (2008), “these theoretical models are rela-
tively complex but useful at the single muscle level. However,
they do not readily handle task-related biomechanical factors
such as joint angle and velocity.” Meanwhile, several muscles
around a joint are engaged in order to realize an action or a
movement around the joint, and mathematically this results in
an underdetermined equation while determining the force of
each engaged muscle due to muscle redundancy and complex
muscle force moment arm-joint angle relationships. Although
different optimization methods have been used to face this load
sharing problem (Ren et al., 2007), it is still very difficult to val-
Preprint submitted to International Journal of Industrial Ergonomics
October 25, 2018
idate the optimization result and further the fatigue effect, due
to the complexity of anatomical structure and the physiological
coordination mechanism of the muscles.
Muscle fatigue is often modeled and extended based on
Maximum Endurance Time (MET) models at joint level in em-
pirical methods. These models are often used in ergonomic ap-
plications to handle task-related external parameters, such as
intensity of the external load, frequency of the external load,
duration, and posture. In these models, the MET of a mus-
cle group around a joint was often measured under static con-
traction conditions until exhaustion. Using this method can
avoid complex modeling of individual muscles, and net joint
strengths already exist in the literature for determining the rela-
tive load (Anderson et al., 2007; Xia and Frey Law, 2008). The
most famous one of these MET models is Rohmert’s curve
(Rohmert, 1960) which was usually used as guideline for de-
signing the static contraction task. Besides Rohmert’s MET
model, there are several other empirical MET models in the lit-
erature (El ahrache et al., 2006). These MET models are very
useful to evaluate physical fatigue in static operations and to de-
termine work-rest allowances (El ahrache and Imbeau, 2009),
and they were often employed in biomechanical models in order
to minimize fatigue as well. For example, Freund and Takala
(2001) proposed a dynamic model for forearm in which the fa-
tigue component was modeled for each single muscle by fit-
ting Rohmert’s curve in Chaffin et al. (1999). Rodr´ıguez et al.
(2003) proposed a half-joint fatigue model, more exactly a fa-
tigue index, based on mechanical properties of muscle groups.
The holding time over maximum endurance time is used as an
indicator to evaluate joint fatigue. In Niemi et al. (1996), dif-
ferent fiber type composition was taken into account with en-
durance model to locate the muscle fatigue into single mus-
cle level. However, in MET models, the main limitations are:
1) The physical relationship in these models cannot be inter-
preted directly by muscle physiology, and there is no universal-
ity among these models. 2) All the MET models were achieved
by fitting experimental results using different formulation of
equation. It has been found that muscle fatigability can vary
across muscles and joints. However, there is no general formu-
lation for those models. 3) Differences have been found among
those MET models for different muscle groups, for different
postures, and even for different models for the same muscle
group. Due to the limitation from the empirical principle, the
differences cannot be interpreted by those MET models. Thus,
it is necessary to develop a general MET model which is able
to replace all the experimental MET models and explain all the
differences cross these models.
Xia and Frey Law (2008) proposed a new muscle fatigue
model based on motor units (MU) recruitment (Liu et al., 2002)
to combine the theoretical models and the task-related muscle
fatigue factors. In this model, properties of different muscle
fiber types have been assumed to predict the muscle fatigue at
joint level. However, in their research, the validation of their fa-
tigue model was not provided. Furthermore, the different fati-
gability of different muscle groups has not been analyzed in
details in this model. Fatigability (the reciprocal of endurance
capacity or the reciprocal of fatigue resistance) can be defined
by the endurance time or measured by the number of times of
an operation until exhaustion. This measure is an important pa-
rameter to measure physical fatigue process during manual han-
dling operations (Lynch et al., 1999; Clark and Manini, 2003;
Larivi`ere et al., 2006; Hunter et al., 2004).
In Ma et al. (2009), we constructed a new muscle fatigue
model in which the external task related parameters are taken
into consideration to describe physical fatigue process, and this
model has also been interpreted by the physiological mecha-
nism of muscle. The model has been compared to 24 existing
MET models, and great linear relationships have been found
between our model and the other MET models. Meanwhile,
this model has also been validated in comparison to three the-
oretical models. This model is a simpler, theoretical approach
to describe the fatigue process, especially in static contraction
cases.
In this paper, further analysis based on the fatigue model
is carried out using mathematical regression method to deter-
mine the fatigability of different muscle groups. We are going
to propose a mathematical parameter, defined as fatigability, de-
scribing the resistance to the decrease of the muscle capacity.
The fatigue resistance for different muscle groups is going to be
regressed from experimental MET models. The theoretical ap-
proach for calculating the fatigue resistance will be explained in
Section 2. The muscle fatigue model in Ma et al. (2009) is go-
ing to be presented briefly in Section 2.1. A general MET model
is extended from this fatigue model in Section 2.2. The mathe-
matical procedure for calculating the fatigability contributes to
the main content of Section 2.3. The results and discussion are
given in Section 3 and 4, respectively.
2. Method
2.1. A dynamic muscle fatigue model
A dynamic fatigue model based on muscle active motor
principle was proposed in Ma et al. (2009). This model was
able to integrate task parameters (load) and temporal parame-
ters into manual handling operation in industry. The differen-
tial equation for describing the reduction of the capacity is Eq.
(1). The descriptions of the parameters for Eq. (1) are listed in
Table 1.
dFcem(t)
dt
= −k Fcem(t)
MVC Fload(t)
(1)
The fatigue model in Eq. (1) can be explained by the MU-
based pattern of muscle (Liu et al., 2002; Vøllestad, 1997). Ac-
cording to muscle physiology, larger Fload means more type II
motor units are involved into the force generation. As a result,
the muscle becomes fatigued more rapidly, as expressed in Eq.
(1). Fcem represents the non-fatigue motor units of the muscle.
The decreased fatigability is expressed by term Fcem(t)/MVC
due to the composition change in the non-fatigue muscle fibers.
k is a parameter indicating the fatigue ratio, and it has different
constant values for different muscle groups. In a general case,
it is assigned as 1. This dynamic model has been mathemati-
cal validated in Ma et al. (2009) with static MET models and
2
Table 1: Parameters in Dynamic Fatigue Model
Item
Unit
Description
MVC
N
Maximum voluntary contraction, maximum capacity of muscle
Fcem(t)
N
Current exertable maximum force, current capacity of muscle
Fload(t)
N
External load of muscle, the force which the muscle needs to generate
k
min−1
Constant value, fatigue ratio, here k = 1
%MVC
Percentage of the voluntary maximum contraction
fMVC
%MVC/100
other existing dynamic theoretical models. Much more detailed
explanation can be found in Ma et al. (2009).
The integration result of Eq. (1) is Eq. (2), if Fcem(0) =
MVC. Equation (2) demonstrates the fatigue process under an
external load Fload(u). The external load can be either static
external load or dynamic one.
Fcem(t) = MVC e
R t
0 −k Fload(u)
MVC
du
(2)
2.2. The extended MET model
A general MET model can be extended by supposing that
Fload(t) is constant in static situation to predict the endurance
time of a muscle contraction. MET is the duration in which
Fcem falls down to the current Fload. Thus, MET can be deter-
mined by Eq. (3) and (4).
Fcem(t) = MVC e
R t
0 −kFload(u)
MVC
du
= Fload(t)
(3)
t = MET = −
ln
 Fload(t)
MVC
!
k Fload(t)
MVC
= −ln(fMVC)
k fMVC
(4)
In comparison to the empirical MET models, the extended
MET model is obtained from a theoretical muscle fatigue model
based on motor units recruitment mechanism. In the extended
MET model, only one parameter k remains variable. This pa-
rameter is defined as fatigability (fatigue ratio) in the theoretical
model, and this parameter could enable us to explain the differ-
ences found in empirical MET models.
2.3. Regression for determining the fatigability
In Ma et al. (2009), two correlation coefficients were se-
lected to analyze the relationship between the extended MET
model (k = 1) and the empirical MET models. One is Pearson’s
correlation r in for evaluating the linear relationship and the
other one is intraclass correlation (ICC) for evaluating the sim-
ilarity between two models. The calculation results are shown
in columns r and ICC1 of Table 2. There are still large differ-
ences from 1 in ICC1 column, while high linear correlations
have also been found between the extended MET model and
the empirical MET models. That means that the extended MET
model can fit the empirical MET models for different muscle
groups by adjusting the parameter k.
A mathmatical regression process is carried out to deter-
mine the parameter k as follows. From Table 2, it is observed
that almost all the static MET models have high linear relation-
ship with the extended MET model (for most models, r > 0.95),
which means that each static model can be described mathemat-
ically by a linear equation (Eq. (5)). In Eq. (5), x is used to re-
place fMVC and p(x) represents the extended MET model in Eq.
(4).
m and n are constants describing the linear relationship
between an existing MET model and the extended MET model,
and they are needed to be determined in linear regression. It
should be noticed that m = 1/k indicates the fatigue resistance
of the static model, and k is fatigue ratio or fatigability of dif-
ferent static model.
f(x) = m p(x) + n
(5)
Due to the asymptotic tendencies of the empirical MET
models mentioned in El ahrache et al. (2006), when x →1
(%MVC →100), f(x) →0 and p(x) →0 (MET →0), there-
fore, we can assume that n = 0. For this reason, only one pa-
rameter m needs to be determined. Since some empirical MET
models are not available for %MVC under 15%, the regression
was carried out in the interval from x = 0.16 to x = 0.99. With
an space 0.01, N = 84 empirical MET values were calculated
to determine the parameter m by minimizing the function in Eq.
(6).
M(x) =
N
X
i=1
(f(xi) −m p(xi))2 = a m2 + b m + c
(6)
In Eq. (6), a =
NP
i=1
p(xi)2 is always greater than 0, and
b = −2
NP
i=1
p(xi)f(xi) is always less than 0, therefore the fatigue
resistance m can be calculated by Eq. (7).
m = −b
2a =
NP
i=1
p(xi)f(xi)
NP
i=1
p(xi)2
> 0
(7)
After regression for each empirical MET model, new ICC
values were calculated by comparing f(x)/m and p(x), and they
are listed in the column ICC2 of Table 2. It is easy to prove that
the regression does not change the r correlation. For this reason,
only ICC is recalculated.
3
3. Results
3.1. Regression results
Both ICC correlations before regression and after regres-
sion are shown in Table 2. It should be noticed that the results
ICC1 before regression in Table 2 were slightly different from
the results presented in Ma et al. (2009), because the range of
fMVC varies from 0.15 to 0.99 in this paper while it varied from
0.20 to 0.99 in Ma et al. (2009) in order to validate the dynamic
fatigue model. Some models were sensitive for such a change,
e.g. Monod’s model. However, the little change of the valida-
tion result does not change the conclusion in Ma et al. (2009).
The ICC results for different muscle groups after regression
are graphically presented in log-log diagram from Fig. 1 to 4
as well. The corresponding graphical results before regression
can be found in Ma et al. (2009). The x-axis is the MET values
predicted using the extended MET model at different relative
force levels, while the y-axis is the MET values predicted using
other MET models at corresponding relative force levels. The
straight solid diagonal line is the reference line. For the other
models, the one which approaches more closely to the straight
line has a higher ICC. If a model coincides completely over the
reference line, which means it is identical to the extended MET
model.
3.2. Fatigue resistance results
The regression results (m) for each MET model are listed in
Table 3. The mean value ¯m and the standard deviation σm were
calculated for different muscle groups as well. The Monod’s
general model is eliminated from the calculation due to its poor
ICC value. The intergroup differences are represented by the
mean value of each muscle group. The Hip/back models have
a higher mean value ¯m = 1.9701, while the other human body
segments and the general models have relative lower fatigue
resistances ranging from 0.76 to 0.90, without big differences.
The fluctuation in each muscle group, namely the intra muscle
group difference, is presented by σm. The stability in the gen-
eral group is the best, and the hip/back model has the largest
variation. There is no big difference between elbow and shoul-
der models.
4. Discussion
4.1. Result analysis
ICC: Almost all the ICC2 are greater than 0.89, and only
one is an exception (Monond and Scherrer, 0.4736). This ex-
ception is because of its relative worse linear correlation r with
the extended MET model, while almost all the other ones have
r over 0.96, and the Monod’s model has only 0.6241. For the
Monod’s model, the linear error occurs mainly when the fMVC
approaches to 0.15. This error is mainly caused by the way
in which the Monod’s model is formulated. This exception is
eliminated in the following analysis and discussion.
There are larger differences between the extended MET model
and the empirical MET models, especially when the fMVC ap-
proaches to 0.15. Those differences can be explained by the in-
terindividual difference in MET, and these differences are greater
for the low %MVC (El ahrache et al., 2006). From the graph-
ical representation, it can be noticed that the MET errors are
mainly decreased in the range from 100 min to 101 min, which
means the extended MET model after regression can predict
MET with less error than using the extended MET model be-
fore regression(refer to Fig.2, Fig. 4, Fig. 6 and Fig. 8 in
Ma et al. (2009)).
The greatest improvement of the fitness between the ex-
tended MET model and the empirical MET models is the Hip/Back
model (Fig. 4). This approves that the extended MET model
with a suitable fatigue ratio can adapt itself well to the most
complex part of human body. The same improvement can be
found for shoulder models and most of the elbow models. It
should be noticed not all the MET models have been improved
after the regression. Little fall can be found for the MET mod-
els (hand model) with ICC over 0.98 in the ICC1 column. The
possible reason is that it has already relative high ICC correla-
tion, and the regression does not improve its fitness. However,
those models after regression still have high ICC (> 0.95). As
a summary, the regression approach achieves high ICC and im-
proves the similarity between the extended MET model and the
existing models. This proves that the extended MET model can
be adapted to fit different body parts, and the extended MET
model can predict the MET for static cases.
Fatigue resistance: The regression result of the fatigue re-
sistance of different muscle groups were tested with normplot
function in Matlab in order to graphically assess whether the
fatigue resistances could come from a normal distribution. The
test result shows fatigue resistances for general models and el-
bow models scatter near the diagonal line in the Fig. 5 and Fig.
6. Due to limitation of sample numbers in shoulder models and
the large variance in hip/back models, the distribution test did
not achieve satisfying result. Once there are enough sample
models, it can be extrapolated that the fatigue resistances for
different muscle groups for the overall population distributes in
normal probability, therefore, the mean value locates in ¯m ± σ
could predict the fatigue property of 50% population.
Therefore, the mean value of ¯m and its standard deviation
σm are used to redraw the relation between MET and fMVC,
and they are presented from Fig. 7 to 10. The black bold solid
line is the extended MET model adjusted by ¯m and it locates
in the range constrained by two slim solid lines adjusted by
¯m ± σm. After adjusting our fatigue model with ¯m ± σm, the ex-
tended MET fatigue model can cover most of the existing MET
models from 15% MVC to 80% MVC. Although there is an ex-
ception in Hip/back model due to the relative large variability
in hip muscle groups, it should also be admitted that the adjust-
ment makes the extended MET model suitable for most of the
static cases. In another word, the adjustment by mean and de-
viation makes the extended MET model suitable for evaluating
the fatigue for the overall population.
The prediction by the extended MET model cannot cover
the models for the %MVC over 80 as well as the interval un-
der 15%. However in the industrial cases, it is very rare that
the force demand can cross that limit 80% in static operations.
Even if the physical demand beyond 80%MVC, the prediction
difference in the extended MET model from the other MET
4
Table 2: Static validation results r and ICC between the extended MET model and the other existing MET models in El ahrache et al. (2006)
Model
MET equations (in minutes)
r
ICC1
ICC2
General models
Rohmert
MET = −1.5 +
2.1
fMVC −
0.6
f 2
MVC +
0.1
f 3
MVC
0.9717
0..9505
0.9707
Monod and Scherrer
MET = 0.4167 (fMVC −0.14)−2.4
0.6241
0.0465
0.4736
Huijgens
MET = 0.865
h
1−fMVC
fMVC−0.15
i1/1.4
0.9036
0.8947
0.8916
Sato et al.
MET = 0.3802 (fMVC −0.04)−1.44
0.9973
0.8765
0.9864
Manenica
MET = 14.88 exp(−4.48 fMVC)
0.9829
0.9357
0.9701
Sjogaard
MET = 0.2997 f −2.14
MVC
0.9902
0.9739
0.9898
Rose et al.
MET = 7.96 exp(−4.16 fMVC)
0.9783
0.6100
0.9573
Upper limbs models
Shoulder
Sato et al.
MET = 0.398 f −1.29
MVC
0.9988
0.5317
0.9349
Rohmert et al.
MET = 0.2955 f −1.658
MVC
0.9993
0.7358
0.8982
Mathiassen and Ahsberg
MET = 40.6092 exp(−9.7 fMVC)
0.9881
0.8673
0.9711
Garg
MET = 0.5618 f −1.7551
MVC
0.9968
0.9064
0.9947
Elbow
Hagberg
MET = 0.298 f −2.14
MVC
0.9902
0.9751
0.9898
Manenica
MET = 20.6972 exp(−4.5 fMVC)
0.9832
0.9582
0.9708
Sato et al.
MET = 0.195 f −2.52
MVC
0.9838
0.9008
0.9688
Rohmert et al.
MET = 0.2285 f −1.391
MVC
0.9997
0.2942
0.9570
Rose et al.2000
MET = 20.6 exp(−6.04 fMVC)
0.9958
0.9627
0.9708
Rose et al.1992
MET = 10.23 exp(−4.69 fMVC)
0.9855
0.7053
0.9766
Hand
Manenica
MET = 16.6099 exp(−4.5 fMVC)
0.9832
0.9840
0.9646
Back/hip models
Manenica (body pull)
MET = 27.6604 exp(−4.2 fMVC)
0.9789
0.7672
0.9591
Manenica (body torque)
MET = 12.4286 exp(−4.3 fMVC)
0.9804
0.8736
0.9634
Manenica (back muscles)
MET = 32.7859 exp(−4.9 fMVC)
0.9878
0.8091
0.9819
Rohmert (posture 3)
MET = 0.3001 f −2.803
MVC
09655
0.4056
0.9482
Rohmert (posture 4)
MET = 1.2301 f −1.308
MVC
0.9990
0.8356
0.9396
Rohmert (posture 5)
MET = 3.2613 f −1.256
MVC
0.9984
0.1253
0.9263
Table 3: Fatigue resistance m of different MET models
Segment
m1
m2
m3
m4
m5
m6
m7
¯m
σm
General
Rohm.
Mono.
Hijg.
Sato.
Mane.
Sjog.
Rose
0.8328
-
0.9514
0.6836
0.8019
1.1468
0.4647
0.8135
0.2320
Shoulder
Sato.
Rohme.
Math.
Garg
0.4274
0.545
0.698
1.3926
0.7562
0.4347
Elbow
Hagb.
Mane.
Sato.
Rohm.
Rose00
Rose92
1.1403
1.1099
1.3461
0.2842
0.7616
0.5234
0.8609
0.4079
Hand
Mane.
0.8907
0.8907
-
Hip
pull
torq.
back
pos3
pos4
pos5
1.5986
0.7005
1.5931
3.2379
1.356
3.3345
1.9701
1.1476
5
10
-1
10
0
10
1
10
-1
10
0
10
1
Intraclass Correlation between extended MET model and General models
Endurance Time in extended MET model [min]
Endurance Time in General models [min]
Rohmert
Huijgens
Sato
Manenica
Sjogaard
Rose
Reference
Figure 1: ICC diagram for MET general models after regression
10
-1
10
0
10
1
10
-1
10
0
10
1 Intraclass Correlation between extended MET model and Elbow models
Endurance Time in Extended model [min]
Endurance Time in Elbow models [min]
Hagberg
Manenica
Sato
Rohmert
Rose00
Rose92
Reference
Figure 2: ICC diagram for MET elbow models after regression
10
-1
10
0
10
1
10
-1
10
0
10
1
Intraclass Correlation between extended MET model and Shoulder models
Endurance Time in extended MET model [min]
Endurance Time in Shoulder models [min]
Rohmert
Sato
Mathiassen
Garg
Reference
Figure 3: ICC diagram for MET shoulder models after regression
10
-1
10
0
10
1
10
-1
10
0
10
1
Intraclass Correlation between extended MET model and Hip/back models
Endurance Time in extended MET model [min]
Endurance Time in Hip/Back models [min]
Manenica86-pull
Manenica86-torque
Manenica86-back
Rohmert86-3
Rohmert86-4
Rohmert86-5
Reference
Figure 4: ICC diagram for Hip/back shoulder models after regression
models is less than one minute.
4.2. The extended MET model versus previous MET models
There are several MET models available in the literature,
and they cover different body parts. These models are all ex-
perimental models regressed from experimental data, and each
model is only suitable for predicting MET of a specific group
of people, although the similar tendencies can be found among
these models. Furthermore, those MET model cannot reveal
individual differences in fatigue characteristic. However, it is
admitted that different people might have different fatigue re-
sistances for the same physical operation.
In comparison to conventional MET models, the general an-
alytical MET model was extended from a simple dynamic fa-
tigue model in a theoretical approach. The dynamic muscle fa-
tigue model is based on muscle physiological mechanism. It
takes account of task parameters (Fload or relative load) and
personal factors (MVC and fatigue ratio k), and it has been val-
idated in comparison to other theoretical models in Ma et al.
(2009). Different from the other MET models, in this extended
MET model, there is a parameter k representing individual fa-
tigue characteristic.
After mathematical regression, great similarities (ICC >
0.90) have been found between the extended MET model and
the previous MET models. This indicates that the new theoret-
ical MET model might replace the other MET models by ad-
justing the parameter k. Therefore, the extended MET model
generalizes the formation of MET models.
In addition, different fatigue resistances have been found
while fitting to different MET models, even for the same mus-
cle group. Therefore, it is interesting to find the influencing
factors on the parameter k and to analyze its statistical distribu-
tion for ergonomic application. In this paper, we tried to use the
mean value and standard deviation of the regressed fatigue re-
6
0.5
0.6
0.7
0.8
0.9
1
1.1
0.05
0.10
0.25
0.50
0.75
0.90
0.95
Probability
Normal Probability Plot
Fatigue resistance in general MET models
Figure 5: Normal distribution test for the general model
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
0.05
0.10
0.25
0.50
0.75
0.90
0.95
Probability
Normal Probability Plot
Fatigue resistance in elbow MET models
Figure 6: Normal distribution test for the elbow model
sistances. It has been found that the extended MET model with
adjustable parameter k could cover most of the MET prediction
using experimental MET models. If further experiments can be
carried out, it should be promising that the statistical distribu-
tion of the fatigue resistance for a given population could be
obtained. This kind of information might be useful to integrate
early ergonomic analysis into virtual human simulation tools to
evaluate fatigue at early work design stage.
4.3. Influencing factors on fatigue resistance
Although the MET models fitted from experiment data were
formulated in different forms, the m can still provide some use-
ful information for the fatigue resistance, especially for differ-
ent muscle groups. The differences in fatigue resistance result
is possible to be concluded by the mean value and the devia-
tion, but it is still interesting to know why and how the fatigue
resistance is different in different muscle groups, in the same
muscle group, and even in the same person at different period.
There is no doubt that there are several factors influencing on
the fatigue resistance of a muscle group, and it should be very
useful if the fatigue resistance of different muscle groups can be
mathematically modeled. In this section, the fatigue resistance
and its variability are going to be discussed in details based on
the fatigue resistance results from Table 3 and the previous liter-
ature about fatigability. Different influencing factors are going
to be discussed and classified in this section.
All the differences inter muscle groups and intra muscle
groups in MET models can be classified into four types: 1) Sys-
tematic bias, 2) Fatigue resistance inter individual for construct-
ing a MET model, 3) Fatigue resistance intra muscle group:
fatigue resistance differences for the same muscle group, and
4) Fatigue resistance inter muscle groups: fatigue resistance
differences for different muscle groups. Those differences can
be attributed to different physiological mechanisms involved in
different tasks, and influencing variables are subject motivation,
central command, intensity and duration of the activity, speed
and type of contraction, and intermittent or sustained activi-
ties (Enoka, 1995; Elfving and Dedering, 2007). In those MET
models, all the contractions were exerted under static condi-
tions until exhaustion of muscle groups, therefore, several task
related influencing factors can be neglected in the discussion,
e.g., speed and duration of contraction. The other influencing
factors might contribute to the fatigue resistance difference in
MET models.
Systematic bias : all the MET models were regressed or
reanalyzed based on experiment results. Due to the experimen-
tal background, there were several sources for systematic er-
ror. One possible source of the systematic bias comes from ex-
perimental methods and model construction (El ahrache et al.,
2006), especially for the methods with subjective scales to mea-
sure MET. The subjective feelings significantly influenced the
result. Furthermore, the construction of the MET model might
cause system differences for MET model, even in the models
which were constructed from the same experiment data (e.g.
Huijgens’ model and Sjogaard’s model in General models). The
estimation error was different while using different mathematic
models, and it generates systematic bias in the result analysis.
Fatigue resistance inter individual : besides the system-
atic error, another possible source for the endurance difference
is from individual characteristic. However, the individual char-
acteristic is too complex to be analyzed, and furthermore, the
individual characteristic is impossible to be separated from ex-
isting MET models, since the MET models already represent
the overall performance of the sample participants. In addition,
in ergonomic application, the overall performance of a popula-
tion is often concerned. Therefore, individual fatigue resistance
is not discussed in this part separately, but the differences in
population in fatigue resistance are going to be discussed and
presented in the following part.
Fatigue resistance intra muscle group : the inter indi-
vidual variability contributes to the errors in constructing MET
models and the errors between MET models for the same mus-
7
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
2
3
4
5
6
7
8
9
10
Prediction of MET using extended MET model in comparison with General static models
fMVC
Endurance Time in General models [min]
Rohmert
Huijgens
Sato
Manenica
Sjogaard
Rose
Extended(mean)
Extended(- σ)
Extended(+σ)
Figure 7: MET prediction using extended MET model in comparison with
general static models
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
2
3
4
5
6
7
8
9
10
Prediction of MET using extended MET model in comparison with Elbow static models
fMVC
Endurance Time in Elbow models [min]
Hagberg
Manenica
Sato
Rohmert
Rose00
Rose92
Extended(mean)
Extended(- σ)
Extended(+σ)
Figure 8: MET prediction using extended MET model in comparison with
elbow static models
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
2
3
4
5
6
7
8
9
10
Prediction of MET using extended MET model in comparison with Shoulder static models
fMVC
Endurance Time in Shoulder models [min]
Rohmert
Sato
Mathiassen
Garg
Extended(mean)
Extended(- σ)
Extended(+σ)
Figure 9: MET prediction using extended MET model in comparison with
shoulder static models
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
2
3
4
5
6
7
8
9
10
Prediction of MET using extended MET model in comparison with Hip/back static models
fMVC
Endurance Time in Hip/Back models [min]
Manenica86-pull
Manenica86-torque
Manenica86-back
Rohmert86-3
Rohmert86-4
Rohmert86-5
Extended(mean)
Extended(- σ)
Extended(+σ)
Figure 10: MET prediction using extended MET model in comparison
with hip/back static models
cle group. The influencing factors on the fatigue resistance can
be mainly classified into sample population characteristic (gen-
der, age, and job), personal muscle fiber composition, and pos-
ture.
As mentioned in Section 1, the influences on fatigability
from gender and age were observed in the literature. In the re-
search for gender influence, women were found with more fa-
tigue resistance than men. Based on muscle physiological prin-
ciple, four families of factors were adopted to explain the fati-
gability difference in gender in Hicks et al. (2001). They are:
1) muscle strength (muscle mass) and associated vascular oc-
clusion, 2) substrate utilization, 3) muscle composition and 4)
neuromuscular activation patterns. It concluded that although
the muscle composition differences between men and women
is relatively small (Staron et al., 2000), the muscle fiber type
area is probably one reason for fatigability difference in gender,
since the muscle fiber type I occupied significantly larger area
in women than in men (Larivi`ere et al., 2006). In spite of mus-
cle fiber composition, the motor unit recruitment pattern acts
influences on the fatigability as well. The gender difference
in neuromuscular activation pattern was found and discussed
in Larivi`ere et al. (2006), and it was observed significantly that
females showed more alternating activity between homolateral
and contralateral muscles than males.
Meanwhile, in Mademli and Arampatzis (2008) older men
were found with more endurance time then young men in cer-
tain fatigue test tasks charging with the same relative load. One
of the most common explanations is changes in muscle fiber
composition for fatigability change while aging. The shift to-
wards a higher proportion of muscle fiber type I leads old adults
having a higher fatigue resistance but smaller MVC. Gender
and age were also already taken into a regression model to
predict shoulder flexion endurance (Mathiassen and Åhsberg,
1999). In Nussbaum (2009), the effects of age, gender, and task
parameters on muscle fatigue during prolonged isokinetic torso
exercises were studied. It constated that older men had less ini-
tial strength. It was also found that effects of age and gender
on fatigue were marginal, while significant interactive effects
8
of age and gender with effort level were found at the same time.
Besides those two reasons, the muscle fiber composition of
muscle varies individually in the population, even in a same
age range and in the same gender (Staron et al., 2000), and this
could cause different performances in endurance tasks. Differ-
ent physical work history might change the endurance perfor-
mance. For example, it appeared that athletes with different
fiber composition had different advantages in different sports:
more type I muscle fiber, better in prolonged endurance events
(Wilmore et al., 2008). Meanwhile, the physical training could
also cause shift between different muscle fibers (Costill et al.,
1979). As a result, individual fatigue is very difficult to be de-
termined using MET measurement (Vøllestad, 1997), and the
individual variability might contribute to the differences among
MET models for the same muscle group due to selection of sub-
jects.
Back to the existing MET models, the sample population
was composed of either a single gender or mixed. At the same
time, the number of the subjects was sometimes relative small.
For example, only 5 female students (age range 21–33) were
measured (Garg et al., 2002), while 40 (20 males, age range
22–48 and 20 females, age range 20–55) were tested in shoulder
MET model (Mathiassen and Åhsberg, 1999). Meanwhile, the
characteristics of population (e.g., students, experiences work-
ers) could cause some differences in MET studies. Due to dif-
ferent population selection method, different gender composi-
tion, and different sample number of participant, fatigue resis-
tance for the same muscle group exists in different experiment
results and finally caused different MET models under the sim-
ilar postures.
In Hip/back models, even with the same sample partici-
pants, difference existed also in MET models for different pos-
tures. The variation is possible caused by the different MU re-
cruitment strategies and load sharing mechanism under differ-
ent postures. Kasprisin and Grabiner (2000) observed that the
activation of biceps brachi was significantly affected by joint
angle, and furthermore confirmed that joint angle and contrac-
tion type contributed to the distinction between the activation of
synergistic elbow flexor muscles. The lever of each individual
muscle changes along different postures which results different
intensity of load for each muscle and then causes different fa-
tigue process for different posture. Meanwhile, the contraction
type of each individual muscle might be changed under differ-
ent posture. Both contraction type change and lever differences
contribute to generate different fatigue resistance globally. In
addition, the activation difference was also found in antago-
nist and agonist (Karst and Hasan, 1987; Mottram et al., 2005)
muscles as well, and it is implied that in different posture, the
engagement of muscles in the action causes different muscle
activation strategy, and as a result the same muscle group could
have different performances. With these reasons, it is much dif-
ficult to indicate the contribution of posture in fatigue resistance
because it refers to the sensory-motor mechanism of human,
and how the human coordinates the muscles remains not clear
enough until yet.
Fatigue resistance inter muscle groups: As stated before,
the three different muscle fiber types have different fatigue re-
sistances, and different muscle is composed of types of mus-
cles with composition determining the function of each muscle
(Chaffin et al., 1999). The different fatigue resistance can be
explained by the muscle fiber composition in different human
muscle groups.
In the literature, muscle fiber composition was used mea-
sured by two terms: muscle fiber type percentages and per-
centage fiber type area (CSA: cross section area). Both terms
contribute to the fatigue resistance of the muscle groups. Type
I fibers occupied 74% of muscle fibers in the thoracic mus-
cles, and they amounted 63% in the deep muscles in lumbar
region (Sirca and Kostevc, 1985). On average type I muscle
fibers ranged from 23 to 56% for the muscles crossing the hu-
man shoulder and 12 of the 14 muscles had average SO pro-
portions ranging from 35 to 50% (Dahmane et al., 2005). In
paper (Staron et al., 2000; Shepstone et al., 2005), the muscle
fiber composition shows the similar composition for the muscle
around elbow and vastus lateralis muscle and the type I mus-
cle fibers have a proportion from 35 - 50% in average. Al-
though we cannot determine the relationship between the mus-
cle type composition and the fatigue resistance directly and the-
oretically, the composition distribution among different muscle
groups can interpret the MET differences between general, el-
bow models and back truck models. In addition, the fatigue
resistance of older adults is greater than young ones could also
be explained by a shift towards a higher proportion of type I
fiber composition with aging. These evidences meet the physi-
ological principle of the dynamic muscle fatigue model.
Another possible reason is the loading sharing mechanism
of muscles.
Hip and back muscle group has the maximum
joint moment strength (Chaffin et al., 1999) among the impor-
tant muscle groups. For example, the back extensors are com-
posed of numerous muscle slips having different moment arms
and show a particularly high resistance to fatigue relative to
other muscle groups (Jorgensen, 1997). This is partly attributed
to favorable muscle composition, and the variable loading shar-
ing within back muscle synergists might also contribute signif-
icantly to delay muscle fatigue.
In summary, individual characteristics, population charac-
teristics, and posture are external appearance of influencing fac-
tors for the fatigue resistance. Muscle fiber composition, mus-
cle fiber area, and sensory motor coordination mechanism are
the determinant factors inside the human body deciding the fa-
tigue resistance of muscle group. Therefore, how to construct a
bridge to connect the external factors and internal factors is the
most important way for modeling the fatigue resistance for dif-
ferent muscle groups. How to combine those factors to model
the fatigue resistance remains a challenging work. Despite the
difficulty of modeling the fatigue resistance, it is still applicable
to find the fatigue resistance for a specified population by MET
experiments in regression with the extended MET model due to
its simplicity and universal availability.
4.4. Limitations
In the previous discussion, the fatigue resistance of the ex-
isting MET models were quantified using m from regression.
The possible reasons for the different fatigue resistance were
9
analyzed and discussed. However, how to quantify the influ-
ence from different factors on the fatigue resistance remains
unknown due to the complexity of muscle physiology and the
correlation among different factors.
The availability of the extended MET model in the interval
under 15% MVC is not validated. The fatigue resistance is only
accounted from the 15% to 99% MVC due to the unavailability
of some MET models under 15% MVC. For the relative low
load, the individual variability under 15% could be much larger
than that over 15%. The recovery effect might play a much
more significant role within such a range.
5. Conclusions and perspectives
In this paper, fatigue resistance of different muscle groups
were calculated by linear regression from the new fatigue model
and the existing MET static models. High ICC has been ob-
tained by regression which proves that our fatigue model can be
generalized to predict MET for different muscle groups. Mean
and standard deviation in fatigue resistance for different muscle
groups were calculated, and it is possible to use both of them
together to predict the MET for the overall population. The pos-
sible reasons responsible for the variability of fatigue resistance
were discussed based on the muscle physiology.
Our fatigue model is relative simple and computation ef-
ficient. With the extended MET model it is possible to carry
out the fatigue evaluation in virtual human modeling and er-
gonomic application, especially for static and quasi-static cases.
The fatigue effect of different muscle groups can be evaluated
by fitting k from several simple static experiments for certain
population.
Acknowledgments
This research was supported by the EADS and the R´egion
des Pays de la Loire (France) in the context of collaboration
between the ´Ecole Centrale de Nantes (Nantes, France) and Ts-
inghua University (Beijing, PR China).
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11
arXiv:1104.1190v1  [cs.RO]  6 Apr 2011
A novel approach for determining fatigue resistances of different muscle groups in static
cases
Liang MAa,c,∗, Damien CHABLATb, Fouad BENNISb,c, Wei ZHANGa, Bo HUa, Franc¸ois GUILLAUMEd
aDepartment of Industrial Engineering, Tsinghua University, 100084, Beijing, P.R.China
b Institut de Recherche en Communications et en Cybern´etique de Nantes, UMR 6597
IRCCyN - 1, rue de la No¨e, BP 92 101 - 44321 Nantes CEDEX 03, France
c ´Ecole Centrale de Nantes, 1, rue de la No¨e, BP 92 101 - 44321 Nantes CEDEX 03, France
dEADS Innovation Works, 12, rue Pasteur - BP 76, 92152 Suresnes Cedex - France
Abstract
In ergonomics and biomechanics, muscle fatigue models based on maximum endurance time (MET) models are often used to
integrate fatigue effect into ergonomic and biomechanical application. However, due to the empirical principle of those MET
models, the disadvantages of this method are: 1) the MET models cannot reveal the muscle physiology background very well; 2)
there is no general formation for those MET models to predict MET. In this paper, a theoretical MET model is extended from a
simple muscle fatigue model with consideration of the external load and maximum voluntary contraction in passive static exertion
cases. The universal availability of the extended MET model is analyzed in comparison to 24 existing empirical MET models. Using
mathematical regression method, 21 of the 24 MET models have intraclass correlations over 0.9, which means the extended MET
model could replace the existing MET models in a general and computationally efficient way. In addition, an important parameter,
fatigability (or fatigue resistance) of different muscle groups, could be calculated via the mathematical regression approach. Its
mean value and its standard deviation are useful for predicting MET values of a given population during static operations. The
possible reasons influencing the fatigue resistance were classified and discussed, and it is still a very challenging work to find out
the quantitative relationship between the fatigue resistance and the influencing factors.
Relevance to industry :
MSD risks can be reduced by correct evaluation of static muscular work. Different muscle groups have different properties, and
a generalized MET model is useful to simplify the fatigue analysis and fatigue modeling, especially for digital human techniques
and virtual human simulation tools.
Key words: muscle fatigue, biomechanical muscle modeling, fatigue resistance, maximum endurance time, muscle groups
1. Introduction
Muscle fatigue is defined as “any reduction in the ability
to exert force in response to voluntary effort” (Chaffin et al.,
1999), and it is believed that the muscle fatigue is one of poten-
tial reasons leading to musculoskeletal disorders (MSDs) in the
literature (Westgaard, 2000; Kumar and Kumar, 2008; Kim et al.,
2008). Great effort has been contributed to integrate fatigue
into different biomechanical models, especially in virtual hu-
man simulation for ergonomic application, in order to analyze
the fatigue in muscles and joints and further to decrease the
MSD risks (Badler et al., 1993; Vignes, 2004; Hou et al., 2007).
In general, mainly two approaches have been adopted to
represent muscle fatigue, either in theoretical methods or in
empirical methods (Xia and Frey Law, 2008). One or more de-
cay terms were introduced into existing muscle force models in
theoretical fatigue models, and those decay terms were mainly
∗Corresponding author: Tel:+86-10-62792665; Fax:+86-10-62794399
Email addresses: liangma@tsinghua.edu.cn;
liang.ma@irccyn.ec-nantes.fr (Liang MA)
based on physiological performance of muscles in fatigue con-
traction. For example, a fatigue model based on the intracel-
lular pH (Giat et al., 1993) was incorporated into Hill’s muscle
mechanical model (Hill, 1938). This fatigue model was also
applied by Komura et al. (1999) to demonstrate the fatigue of
different individual muscles.
Another muscle fatigue model
(Ding et al., 2003) based on physiological mechanism has been
included into the Virtual Solider Research Program (Vignes,
2004), and in this model, dozens of parameters have to be fit
for model identification only for a single muscle. As stated in
Xia and Frey Law (2008), “these theoretical models are rela-
tively complex but useful at the single muscle level. However,
they do not readily handle task-related biomechanical factors
such as joint angle and velocity.” Meanwhile, several muscles
around a joint are engaged in order to realize an action or a
movement around the joint, and mathematically this results in
an underdetermined equation while determining the force of
each engaged muscle due to muscle redundancy and complex
muscle force moment arm-joint angle relationships. Although
different optimization methods have been used to face this load
sharing problem (Ren et al., 2007), it is still very difficult to val-
Preprint submitted to International Journal of Industrial Ergonomics
October 25, 2018
idate the optimization result and further the fatigue effect, due
to the complexity of anatomical structure and the physiological
coordination mechanism of the muscles.
Muscle fatigue is often modeled and extended based on
Maximum Endurance Time (MET) models at joint level in em-
pirical methods. These models are often used in ergonomic ap-
plications to handle task-related external parameters, such as
intensity of the external load, frequency of the external load,
duration, and posture. In these models, the MET of a mus-
cle group around a joint was often measured under static con-
traction conditions until exhaustion. Using this method can
avoid complex modeling of individual muscles, and net joint
strengths already exist in the literature for determining the rela-
tive load (Anderson et al., 2007; Xia and Frey Law, 2008). The
most famous one of these MET models is Rohmert’s curve
(Rohmert, 1960) which was usually used as guideline for de-
signing the static contraction task. Besides Rohmert’s MET
model, there are several other empirical MET models in the lit-
erature (El ahrache et al., 2006). These MET models are very
useful to evaluate physical fatigue in static operations and to de-
termine work-rest allowances (El ahrache and Imbeau, 2009),
and they were often employed in biomechanical models in order
to minimize fatigue as well. For example, Freund and Takala
(2001) proposed a dynamic model for forearm in which the fa-
tigue component was modeled for each single muscle by fit-
ting Rohmert’s curve in Chaffin et al. (1999). Rodr´ıguez et al.
(2003) proposed a half-joint fatigue model, more exactly a fa-
tigue index, based on mechanical properties of muscle groups.
The holding time over maximum endurance time is used as an
indicator to evaluate joint fatigue. In Niemi et al. (1996), dif-
ferent fiber type composition was taken into account with en-
durance model to locate the muscle fatigue into single mus-
cle level. However, in MET models, the main limitations are:
1) The physical relationship in these models cannot be inter-
preted directly by muscle physiology, and there is no universal-
ity among these models. 2) All the MET models were achieved
by fitting experimental results using different formulation of
equation. It has been found that muscle fatigability can vary
across muscles and joints. However, there is no general formu-
lation for those models. 3) Differences have been found among
those MET models for different muscle groups, for different
postures, and even for different models for the same muscle
group. Due to the limitation from the empirical principle, the
differences cannot be interpreted by those MET models. Thus,
it is necessary to develop a general MET model which is able
to replace all the experimental MET models and explain all the
differences cross these models.
Xia and Frey Law (2008) proposed a new muscle fatigue
model based on motor units (MU) recruitment (Liu et al., 2002)
to combine the theoretical models and the task-related muscle
fatigue factors. In this model, properties of different muscle
fiber types have been assumed to predict the muscle fatigue at
joint level. However, in their research, the validation of their fa-
tigue model was not provided. Furthermore, the different fati-
gability of different muscle groups has not been analyzed in
details in this model. Fatigability (the reciprocal of endurance
capacity or the reciprocal of fatigue resistance) can be defined
by the endurance time or measured by the number of times of
an operation until exhaustion. This measure is an important pa-
rameter to measure physical fatigue process during manual han-
dling operations (Lynch et al., 1999; Clark and Manini, 2003;
Larivi`ere et al., 2006; Hunter et al., 2004).
In Ma et al. (2009), we constructed a new muscle fatigue
model in which the external task related parameters are taken
into consideration to describe physical fatigue process, and this
model has also been interpreted by the physiological mecha-
nism of muscle. The model has been compared to 24 existing
MET models, and great linear relationships have been found
between our model and the other MET models. Meanwhile,
this model has also been validated in comparison to three the-
oretical models. This model is a simpler, theoretical approach
to describe the fatigue process, especially in static contraction
cases.
In this paper, further analysis based on the fatigue model
is carried out using mathematical regression method to deter-
mine the fatigability of different muscle groups. We are going
to propose a mathematical parameter, defined as fatigability, de-
scribing the resistance to the decrease of the muscle capacity.
The fatigue resistance for different muscle groups is going to be
regressed from experimental MET models. The theoretical ap-
proach for calculating the fatigue resistance will be explained in
Section 2. The muscle fatigue model in Ma et al. (2009) is go-
ing to be presented briefly in Section 2.1. A general MET model
is extended from this fatigue model in Section 2.2. The mathe-
matical procedure for calculating the fatigability contributes to
the main content of Section 2.3. The results and discussion are
given in Section 3 and 4, respectively.
2. Method
2.1. A dynamic muscle fatigue model
A dynamic fatigue model based on muscle active motor
principle was proposed in Ma et al. (2009). This model was
able to integrate task parameters (load) and temporal parame-
ters into manual handling operation in industry. The differen-
tial equation for describing the reduction of the capacity is Eq.
(1). The descriptions of the parameters for Eq. (1) are listed in
Table 1.
dFcem(t)
dt
= −k Fcem(t)
MVC Fload(t)
(1)
The fatigue model in Eq. (1) can be explained by the MU-
based pattern of muscle (Liu et al., 2002; Vøllestad, 1997). Ac-
cording to muscle physiology, larger Fload means more type II
motor units are involved into the force generation. As a result,
the muscle becomes fatigued more rapidly, as expressed in Eq.
(1). Fcem represents the non-fatigue motor units of the muscle.
The decreased fatigability is expressed by term Fcem(t)/MVC
due to the composition change in the non-fatigue muscle fibers.
k is a parameter indicating the fatigue ratio, and it has different
constant values for different muscle groups. In a general case,
it is assigned as 1. This dynamic model has been mathemati-
cal validated in Ma et al. (2009) with static MET models and
2
Table 1: Parameters in Dynamic Fatigue Model
Item
Unit
Description
MVC
N
Maximum voluntary contraction, maximum capacity of muscle
Fcem(t)
N
Current exertable maximum force, current capacity of muscle
Fload(t)
N
External load of muscle, the force which the muscle needs to generate
k
min−1
Constant value, fatigue ratio, here k = 1
%MVC
Percentage of the voluntary maximum contraction
fMVC
%MVC/100
other existing dynamic theoretical models. Much more detailed
explanation can be found in Ma et al. (2009).
The integration result of Eq. (1) is Eq. (2), if Fcem(0) =
MVC. Equation (2) demonstrates the fatigue process under an
external load Fload(u). The external load can be either static
external load or dynamic one.
Fcem(t) = MVC e
R t
0 −k Fload(u)
MVC
du
(2)
2.2. The extended MET model
A general MET model can be extended by supposing that
Fload(t) is constant in static situation to predict the endurance
time of a muscle contraction. MET is the duration in which
Fcem falls down to the current Fload. Thus, MET can be deter-
mined by Eq. (3) and (4).
Fcem(t) = MVC e
R t
0 −kFload(u)
MVC
du
= Fload(t)
(3)
t = MET = −
ln
 Fload(t)
MVC
!
k Fload(t)
MVC
= −ln(fMVC)
k fMVC
(4)
In comparison to the empirical MET models, the extended
MET model is obtained from a theoretical muscle fatigue model
based on motor units recruitment mechanism. In the extended
MET model, only one parameter k remains variable. This pa-
rameter is defined as fatigability (fatigue ratio) in the theoretical
model, and this parameter could enable us to explain the differ-
ences found in empirical MET models.
2.3. Regression for determining the fatigability
In Ma et al. (2009), two correlation coefficients were se-
lected to analyze the relationship between the extended MET
model (k = 1) and the empirical MET models. One is Pearson’s
correlation r in for evaluating the linear relationship and the
other one is intraclass correlation (ICC) for evaluating the sim-
ilarity between two models. The calculation results are shown
in columns r and ICC1 of Table 2. There are still large differ-
ences from 1 in ICC1 column, while high linear correlations
have also been found between the extended MET model and
the empirical MET models. That means that the extended MET
model can fit the empirical MET models for different muscle
groups by adjusting the parameter k.
A mathmatical regression process is carried out to deter-
mine the parameter k as follows. From Table 2, it is observed
that almost all the static MET models have high linear relation-
ship with the extended MET model (for most models, r > 0.95),
which means that each static model can be described mathemat-
ically by a linear equation (Eq. (5)). In Eq. (5), x is used to re-
place fMVC and p(x) represents the extended MET model in Eq.
(4).
m and n are constants describing the linear relationship
between an existing MET model and the extended MET model,
and they are needed to be determined in linear regression. It
should be noticed that m = 1/k indicates the fatigue resistance
of the static model, and k is fatigue ratio or fatigability of dif-
ferent static model.
f(x) = m p(x) + n
(5)
Due to the asymptotic tendencies of the empirical MET
models mentioned in El ahrache et al. (2006), when x →1
(%MVC →100), f(x) →0 and p(x) →0 (MET →0), there-
fore, we can assume that n = 0. For this reason, only one pa-
rameter m needs to be determined. Since some empirical MET
models are not available for %MVC under 15%, the regression
was carried out in the interval from x = 0.16 to x = 0.99. With
an space 0.01, N = 84 empirical MET values were calculated
to determine the parameter m by minimizing the function in Eq.
(6).
M(x) =
N
X
i=1
(f(xi) −m p(xi))2 = a m2 + b m + c
(6)
In Eq. (6), a =
NP
i=1
p(xi)2 is always greater than 0, and
b = −2
NP
i=1
p(xi)f(xi) is always less than 0, therefore the fatigue
resistance m can be calculated by Eq. (7).
m = −b
2a =
NP
i=1
p(xi)f(xi)
NP
i=1
p(xi)2
> 0
(7)
After regression for each empirical MET model, new ICC
values were calculated by comparing f(x)/m and p(x), and they
are listed in the column ICC2 of Table 2. It is easy to prove that
the regression does not change the r correlation. For this reason,
only ICC is recalculated.
3
3. Results
3.1. Regression results
Both ICC correlations before regression and after regres-
sion are shown in Table 2. It should be noticed that the results
ICC1 before regression in Table 2 were slightly different from
the results presented in Ma et al. (2009), because the range of
fMVC varies from 0.15 to 0.99 in this paper while it varied from
0.20 to 0.99 in Ma et al. (2009) in order to validate the dynamic
fatigue model. Some models were sensitive for such a change,
e.g. Monod’s model. However, the little change of the valida-
tion result does not change the conclusion in Ma et al. (2009).
The ICC results for different muscle groups after regression
are graphically presented in log-log diagram from Fig. 1 to 4
as well. The corresponding graphical results before regression
can be found in Ma et al. (2009). The x-axis is the MET values
predicted using the extended MET model at different relative
force levels, while the y-axis is the MET values predicted using
other MET models at corresponding relative force levels. The
straight solid diagonal line is the reference line. For the other
models, the one which approaches more closely to the straight
line has a higher ICC. If a model coincides completely over the
reference line, which means it is identical to the extended MET
model.
3.2. Fatigue resistance results
The regression results (m) for each MET model are listed in
Table 3. The mean value ¯m and the standard deviation σm were
calculated for different muscle groups as well. The Monod’s
general model is eliminated from the calculation due to its poor
ICC value. The intergroup differences are represented by the
mean value of each muscle group. The Hip/back models have
a higher mean value ¯m = 1.9701, while the other human body
segments and the general models have relative lower fatigue
resistances ranging from 0.76 to 0.90, without big differences.
The fluctuation in each muscle group, namely the intra muscle
group difference, is presented by σm. The stability in the gen-
eral group is the best, and the hip/back model has the largest
variation. There is no big difference between elbow and shoul-
der models.
4. Discussion
4.1. Result analysis
ICC: Almost all the ICC2 are greater than 0.89, and only
one is an exception (Monond and Scherrer, 0.4736). This ex-
ception is because of its relative worse linear correlation r with
the extended MET model, while almost all the other ones have
r over 0.96, and the Monod’s model has only 0.6241. For the
Monod’s model, the linear error occurs mainly when the fMVC
approaches to 0.15. This error is mainly caused by the way
in which the Monod’s model is formulated. This exception is
eliminated in the following analysis and discussion.
There are larger differences between the extended MET model
and the empirical MET models, especially when the fMVC ap-
proaches to 0.15. Those differences can be explained by the in-
terindividual difference in MET, and these differences are greater
for the low %MVC (El ahrache et al., 2006). From the graph-
ical representation, it can be noticed that the MET errors are
mainly decreased in the range from 100 min to 101 min, which
means the extended MET model after regression can predict
MET with less error than using the extended MET model be-
fore regression(refer to Fig.2, Fig. 4, Fig. 6 and Fig. 8 in
Ma et al. (2009)).
The greatest improvement of the fitness between the ex-
tended MET model and the empirical MET models is the Hip/Back
model (Fig. 4). This approves that the extended MET model
with a suitable fatigue ratio can adapt itself well to the most
complex part of human body. The same improvement can be
found for shoulder models and most of the elbow models. It
should be noticed not all the MET models have been improved
after the regression. Little fall can be found for the MET mod-
els (hand model) with ICC over 0.98 in the ICC1 column. The
possible reason is that it has already relative high ICC correla-
tion, and the regression does not improve its fitness. However,
those models after regression still have high ICC (> 0.95). As
a summary, the regression approach achieves high ICC and im-
proves the similarity between the extended MET model and the
existing models. This proves that the extended MET model can
be adapted to fit different body parts, and the extended MET
model can predict the MET for static cases.
Fatigue resistance: The regression result of the fatigue re-
sistance of different muscle groups were tested with normplot
function in Matlab in order to graphically assess whether the
fatigue resistances could come from a normal distribution. The
test result shows fatigue resistances for general models and el-
bow models scatter near the diagonal line in the Fig. 5 and Fig.
6. Due to limitation of sample numbers in shoulder models and
the large variance in hip/back models, the distribution test did
not achieve satisfying result. Once there are enough sample
models, it can be extrapolated that the fatigue resistances for
different muscle groups for the overall population distributes in
normal probability, therefore, the mean value locates in ¯m ± σ
could predict the fatigue property of 50% population.
Therefore, the mean value of ¯m and its standard deviation
σm are used to redraw the relation between MET and fMVC,
and they are presented from Fig. 7 to 10. The black bold solid
line is the extended MET model adjusted by ¯m and it locates
in the range constrained by two slim solid lines adjusted by
¯m ± σm. After adjusting our fatigue model with ¯m ± σm, the ex-
tended MET fatigue model can cover most of the existing MET
models from 15% MVC to 80% MVC. Although there is an ex-
ception in Hip/back model due to the relative large variability
in hip muscle groups, it should also be admitted that the adjust-
ment makes the extended MET model suitable for most of the
static cases. In another word, the adjustment by mean and de-
viation makes the extended MET model suitable for evaluating
the fatigue for the overall population.
The prediction by the extended MET model cannot cover
the models for the %MVC over 80 as well as the interval un-
der 15%. However in the industrial cases, it is very rare that
the force demand can cross that limit 80% in static operations.
Even if the physical demand beyond 80%MVC, the prediction
difference in the extended MET model from the other MET
4
Table 2: Static validation results r and ICC between the extended MET model and the other existing MET models in El ahrache et al. (2006)
Model
MET equations (in minutes)
r
ICC1
ICC2
General models
Rohmert
MET = −1.5 +
2.1
fMVC −
0.6
f 2
MVC +
0.1
f 3
MVC
0.9717
0..9505
0.9707
Monod and Scherrer
MET = 0.4167 (fMVC −0.14)−2.4
0.6241
0.0465
0.4736
Huijgens
MET = 0.865
h
1−fMVC
fMVC−0.15
i1/1.4
0.9036
0.8947
0.8916
Sato et al.
MET = 0.3802 (fMVC −0.04)−1.44
0.9973
0.8765
0.9864
Manenica
MET = 14.88 exp(−4.48 fMVC)
0.9829
0.9357
0.9701
Sjogaard
MET = 0.2997 f −2.14
MVC
0.9902
0.9739
0.9898
Rose et al.
MET = 7.96 exp(−4.16 fMVC)
0.9783
0.6100
0.9573
Upper limbs models
Shoulder
Sato et al.
MET = 0.398 f −1.29
MVC
0.9988
0.5317
0.9349
Rohmert et al.
MET = 0.2955 f −1.658
MVC
0.9993
0.7358
0.8982
Mathiassen and Ahsberg
MET = 40.6092 exp(−9.7 fMVC)
0.9881
0.8673
0.9711
Garg
MET = 0.5618 f −1.7551
MVC
0.9968
0.9064
0.9947
Elbow
Hagberg
MET = 0.298 f −2.14
MVC
0.9902
0.9751
0.9898
Manenica
MET = 20.6972 exp(−4.5 fMVC)
0.9832
0.9582
0.9708
Sato et al.
MET = 0.195 f −2.52
MVC
0.9838
0.9008
0.9688
Rohmert et al.
MET = 0.2285 f −1.391
MVC
0.9997
0.2942
0.9570
Rose et al.2000
MET = 20.6 exp(−6.04 fMVC)
0.9958
0.9627
0.9708
Rose et al.1992
MET = 10.23 exp(−4.69 fMVC)
0.9855
0.7053
0.9766
Hand
Manenica
MET = 16.6099 exp(−4.5 fMVC)
0.9832
0.9840
0.9646
Back/hip models
Manenica (body pull)
MET = 27.6604 exp(−4.2 fMVC)
0.9789
0.7672
0.9591
Manenica (body torque)
MET = 12.4286 exp(−4.3 fMVC)
0.9804
0.8736
0.9634
Manenica (back muscles)
MET = 32.7859 exp(−4.9 fMVC)
0.9878
0.8091
0.9819
Rohmert (posture 3)
MET = 0.3001 f −2.803
MVC
09655
0.4056
0.9482
Rohmert (posture 4)
MET = 1.2301 f −1.308
MVC
0.9990
0.8356
0.9396
Rohmert (posture 5)
MET = 3.2613 f −1.256
MVC
0.9984
0.1253
0.9263
Table 3: Fatigue resistance m of different MET models
Segment
m1
m2
m3
m4
m5
m6
m7
¯m
σm
General
Rohm.
Mono.
Hijg.
Sato.
Mane.
Sjog.
Rose
0.8328
-
0.9514
0.6836
0.8019
1.1468
0.4647
0.8135
0.2320
Shoulder
Sato.
Rohme.
Math.
Garg
0.4274
0.545
0.698
1.3926
0.7562
0.4347
Elbow
Hagb.
Mane.
Sato.
Rohm.
Rose00
Rose92
1.1403
1.1099
1.3461
0.2842
0.7616
0.5234
0.8609
0.4079
Hand
Mane.
0.8907
0.8907
-
Hip
pull
torq.
back
pos3
pos4
pos5
1.5986
0.7005
1.5931
3.2379
1.356
3.3345
1.9701
1.1476
5
10
-1
10
0
10
1
10
-1
10
0
10
1
Intraclass Correlation between extended MET model and General models
Endurance Time in extended MET model [min]
Endurance Time in General models [min]
Rohmert
Huijgens
Sato
Manenica
Sjogaard
Rose
Reference
Figure 1: ICC diagram for MET general models after regression
10
-1
10
0
10
1
10
-1
10
0
10
1 Intraclass Correlation between extended MET model and Elbow models
Endurance Time in Extended model [min]
Endurance Time in Elbow models [min]
Hagberg
Manenica
Sato
Rohmert
Rose00
Rose92
Reference
Figure 2: ICC diagram for MET elbow models after regression
10
-1
10
0
10
1
10
-1
10
0
10
1
Intraclass Correlation between extended MET model and Shoulder models
Endurance Time in extended MET model [min]
Endurance Time in Shoulder models [min]
Rohmert
Sato
Mathiassen
Garg
Reference
Figure 3: ICC diagram for MET shoulder models after regression
10
-1
10
0
10
1
10
-1
10
0
10
1
Intraclass Correlation between extended MET model and Hip/back models
Endurance Time in extended MET model [min]
Endurance Time in Hip/Back models [min]
Manenica86-pull
Manenica86-torque
Manenica86-back
Rohmert86-3
Rohmert86-4
Rohmert86-5
Reference
Figure 4: ICC diagram for Hip/back shoulder models after regression
models is less than one minute.
4.2. The extended MET model versus previous MET models
There are several MET models available in the literature,
and they cover different body parts. These models are all ex-
perimental models regressed from experimental data, and each
model is only suitable for predicting MET of a specific group
of people, although the similar tendencies can be found among
these models. Furthermore, those MET model cannot reveal
individual differences in fatigue characteristic. However, it is
admitted that different people might have different fatigue re-
sistances for the same physical operation.
In comparison to conventional MET models, the general an-
alytical MET model was extended from a simple dynamic fa-
tigue model in a theoretical approach. The dynamic muscle fa-
tigue model is based on muscle physiological mechanism. It
takes account of task parameters (Fload or relative load) and
personal factors (MVC and fatigue ratio k), and it has been val-
idated in comparison to other theoretical models in Ma et al.
(2009). Different from the other MET models, in this extended
MET model, there is a parameter k representing individual fa-
tigue characteristic.
After mathematical regression, great similarities (ICC >
0.90) have been found between the extended MET model and
the previous MET models. This indicates that the new theoret-
ical MET model might replace the other MET models by ad-
justing the parameter k. Therefore, the extended MET model
generalizes the formation of MET models.
In addition, different fatigue resistances have been found
while fitting to different MET models, even for the same mus-
cle group. Therefore, it is interesting to find the influencing
factors on the parameter k and to analyze its statistical distribu-
tion for ergonomic application. In this paper, we tried to use the
mean value and standard deviation of the regressed fatigue re-
6
0.5
0.6
0.7
0.8
0.9
1
1.1
0.05
0.10
0.25
0.50
0.75
0.90
0.95
Probability
Normal Probability Plot
Fatigue resistance in general MET models
Figure 5: Normal distribution test for the general model
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
0.05
0.10
0.25
0.50
0.75
0.90
0.95
Probability
Normal Probability Plot
Fatigue resistance in elbow MET models
Figure 6: Normal distribution test for the elbow model
sistances. It has been found that the extended MET model with
adjustable parameter k could cover most of the MET prediction
using experimental MET models. If further experiments can be
carried out, it should be promising that the statistical distribu-
tion of the fatigue resistance for a given population could be
obtained. This kind of information might be useful to integrate
early ergonomic analysis into virtual human simulation tools to
evaluate fatigue at early work design stage.
4.3. Influencing factors on fatigue resistance
Although the MET models fitted from experiment data were
formulated in different forms, the m can still provide some use-
ful information for the fatigue resistance, especially for differ-
ent muscle groups. The differences in fatigue resistance result
is possible to be concluded by the mean value and the devia-
tion, but it is still interesting to know why and how the fatigue
resistance is different in different muscle groups, in the same
muscle group, and even in the same person at different period.
There is no doubt that there are several factors influencing on
the fatigue resistance of a muscle group, and it should be very
useful if the fatigue resistance of different muscle groups can be
mathematically modeled. In this section, the fatigue resistance
and its variability are going to be discussed in details based on
the fatigue resistance results from Table 3 and the previous liter-
ature about fatigability. Different influencing factors are going
to be discussed and classified in this section.
All the differences inter muscle groups and intra muscle
groups in MET models can be classified into four types: 1) Sys-
tematic bias, 2) Fatigue resistance inter individual for construct-
ing a MET model, 3) Fatigue resistance intra muscle group:
fatigue resistance differences for the same muscle group, and
4) Fatigue resistance inter muscle groups: fatigue resistance
differences for different muscle groups. Those differences can
be attributed to different physiological mechanisms involved in
different tasks, and influencing variables are subject motivation,
central command, intensity and duration of the activity, speed
and type of contraction, and intermittent or sustained activi-
ties (Enoka, 1995; Elfving and Dedering, 2007). In those MET
models, all the contractions were exerted under static condi-
tions until exhaustion of muscle groups, therefore, several task
related influencing factors can be neglected in the discussion,
e.g., speed and duration of contraction. The other influencing
factors might contribute to the fatigue resistance difference in
MET models.
Systematic bias : all the MET models were regressed or
reanalyzed based on experiment results. Due to the experimen-
tal background, there were several sources for systematic er-
ror. One possible source of the systematic bias comes from ex-
perimental methods and model construction (El ahrache et al.,
2006), especially for the methods with subjective scales to mea-
sure MET. The subjective feelings significantly influenced the
result. Furthermore, the construction of the MET model might
cause system differences for MET model, even in the models
which were constructed from the same experiment data (e.g.
Huijgens’ model and Sjogaard’s model in General models). The
estimation error was different while using different mathematic
models, and it generates systematic bias in the result analysis.
Fatigue resistance inter individual : besides the system-
atic error, another possible source for the endurance difference
is from individual characteristic. However, the individual char-
acteristic is too complex to be analyzed, and furthermore, the
individual characteristic is impossible to be separated from ex-
isting MET models, since the MET models already represent
the overall performance of the sample participants. In addition,
in ergonomic application, the overall performance of a popula-
tion is often concerned. Therefore, individual fatigue resistance
is not discussed in this part separately, but the differences in
population in fatigue resistance are going to be discussed and
presented in the following part.
Fatigue resistance intra muscle group : the inter indi-
vidual variability contributes to the errors in constructing MET
models and the errors between MET models for the same mus-
7
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
2
3
4
5
6
7
8
9
10
Prediction of MET using extended MET model in comparison with General static models
fMVC
Endurance Time in General models [min]
Rohmert
Huijgens
Sato
Manenica
Sjogaard
Rose
Extended(mean)
Extended(- σ)
Extended(+σ)
Figure 7: MET prediction using extended MET model in comparison with
general static models
0.2
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0.8
0.9
1
1
2
3
4
5
6
7
8
9
10
Prediction of MET using extended MET model in comparison with Elbow static models
fMVC
Endurance Time in Elbow models [min]
Hagberg
Manenica
Sato
Rohmert
Rose00
Rose92
Extended(mean)
Extended(- σ)
Extended(+σ)
Figure 8: MET prediction using extended MET model in comparison with
elbow static models
0.2
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0.7
0.8
0.9
1
1
2
3
4
5
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8
9
10
Prediction of MET using extended MET model in comparison with Shoulder static models
fMVC
Endurance Time in Shoulder models [min]
Rohmert
Sato
Mathiassen
Garg
Extended(mean)
Extended(- σ)
Extended(+σ)
Figure 9: MET prediction using extended MET model in comparison with
shoulder static models
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1
2
3
4
5
6
7
8
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10
Prediction of MET using extended MET model in comparison with Hip/back static models
fMVC
Endurance Time in Hip/Back models [min]
Manenica86-pull
Manenica86-torque
Manenica86-back
Rohmert86-3
Rohmert86-4
Rohmert86-5
Extended(mean)
Extended(- σ)
Extended(+σ)
Figure 10: MET prediction using extended MET model in comparison
with hip/back static models
cle group. The influencing factors on the fatigue resistance can
be mainly classified into sample population characteristic (gen-
der, age, and job), personal muscle fiber composition, and pos-
ture.
As mentioned in Section 1, the influences on fatigability
from gender and age were observed in the literature. In the re-
search for gender influence, women were found with more fa-
tigue resistance than men. Based on muscle physiological prin-
ciple, four families of factors were adopted to explain the fati-
gability difference in gender in Hicks et al. (2001). They are:
1) muscle strength (muscle mass) and associated vascular oc-
clusion, 2) substrate utilization, 3) muscle composition and 4)
neuromuscular activation patterns. It concluded that although
the muscle composition differences between men and women
is relatively small (Staron et al., 2000), the muscle fiber type
area is probably one reason for fatigability difference in gender,
since the muscle fiber type I occupied significantly larger area
in women than in men (Larivi`ere et al., 2006). In spite of mus-
cle fiber composition, the motor unit recruitment pattern acts
influences on the fatigability as well. The gender difference
in neuromuscular activation pattern was found and discussed
in Larivi`ere et al. (2006), and it was observed significantly that
females showed more alternating activity between homolateral
and contralateral muscles than males.
Meanwhile, in Mademli and Arampatzis (2008) older men
were found with more endurance time then young men in cer-
tain fatigue test tasks charging with the same relative load. One
of the most common explanations is changes in muscle fiber
composition for fatigability change while aging. The shift to-
wards a higher proportion of muscle fiber type I leads old adults
having a higher fatigue resistance but smaller MVC. Gender
and age were also already taken into a regression model to
predict shoulder flexion endurance (Mathiassen and Åhsberg,
1999). In Nussbaum (2009), the effects of age, gender, and task
parameters on muscle fatigue during prolonged isokinetic torso
exercises were studied. It constated that older men had less ini-
tial strength. It was also found that effects of age and gender
on fatigue were marginal, while significant interactive effects
8
of age and gender with effort level were found at the same time.
Besides those two reasons, the muscle fiber composition of
muscle varies individually in the population, even in a same
age range and in the same gender (Staron et al., 2000), and this
could cause different performances in endurance tasks. Differ-
ent physical work history might change the endurance perfor-
mance. For example, it appeared that athletes with different
fiber composition had different advantages in different sports:
more type I muscle fiber, better in prolonged endurance events
(Wilmore et al., 2008). Meanwhile, the physical training could
also cause shift between different muscle fibers (Costill et al.,
1979). As a result, individual fatigue is very difficult to be de-
termined using MET measurement (Vøllestad, 1997), and the
individual variability might contribute to the differences among
MET models for the same muscle group due to selection of sub-
jects.
Back to the existing MET models, the sample population
was composed of either a single gender or mixed. At the same
time, the number of the subjects was sometimes relative small.
For example, only 5 female students (age range 21–33) were
measured (Garg et al., 2002), while 40 (20 males, age range
22–48 and 20 females, age range 20–55) were tested in shoulder
MET model (Mathiassen and Åhsberg, 1999). Meanwhile, the
characteristics of population (e.g., students, experiences work-
ers) could cause some differences in MET studies. Due to dif-
ferent population selection method, different gender composi-
tion, and different sample number of participant, fatigue resis-
tance for the same muscle group exists in different experiment
results and finally caused different MET models under the sim-
ilar postures.
In Hip/back models, even with the same sample partici-
pants, difference existed also in MET models for different pos-
tures. The variation is possible caused by the different MU re-
cruitment strategies and load sharing mechanism under differ-
ent postures. Kasprisin and Grabiner (2000) observed that the
activation of biceps brachi was significantly affected by joint
angle, and furthermore confirmed that joint angle and contrac-
tion type contributed to the distinction between the activation of
synergistic elbow flexor muscles. The lever of each individual
muscle changes along different postures which results different
intensity of load for each muscle and then causes different fa-
tigue process for different posture. Meanwhile, the contraction
type of each individual muscle might be changed under differ-
ent posture. Both contraction type change and lever differences
contribute to generate different fatigue resistance globally. In
addition, the activation difference was also found in antago-
nist and agonist (Karst and Hasan, 1987; Mottram et al., 2005)
muscles as well, and it is implied that in different posture, the
engagement of muscles in the action causes different muscle
activation strategy, and as a result the same muscle group could
have different performances. With these reasons, it is much dif-
ficult to indicate the contribution of posture in fatigue resistance
because it refers to the sensory-motor mechanism of human,
and how the human coordinates the muscles remains not clear
enough until yet.
Fatigue resistance inter muscle groups: As stated before,
the three different muscle fiber types have different fatigue re-
sistances, and different muscle is composed of types of mus-
cles with composition determining the function of each muscle
(Chaffin et al., 1999). The different fatigue resistance can be
explained by the muscle fiber composition in different human
muscle groups.
In the literature, muscle fiber composition was used mea-
sured by two terms: muscle fiber type percentages and per-
centage fiber type area (CSA: cross section area). Both terms
contribute to the fatigue resistance of the muscle groups. Type
I fibers occupied 74% of muscle fibers in the thoracic mus-
cles, and they amounted 63% in the deep muscles in lumbar
region (Sirca and Kostevc, 1985). On average type I muscle
fibers ranged from 23 to 56% for the muscles crossing the hu-
man shoulder and 12 of the 14 muscles had average SO pro-
portions ranging from 35 to 50% (Dahmane et al., 2005). In
paper (Staron et al., 2000; Shepstone et al., 2005), the muscle
fiber composition shows the similar composition for the muscle
around elbow and vastus lateralis muscle and the type I mus-
cle fibers have a proportion from 35 - 50% in average. Al-
though we cannot determine the relationship between the mus-
cle type composition and the fatigue resistance directly and the-
oretically, the composition distribution among different muscle
groups can interpret the MET differences between general, el-
bow models and back truck models. In addition, the fatigue
resistance of older adults is greater than young ones could also
be explained by a shift towards a higher proportion of type I
fiber composition with aging. These evidences meet the physi-
ological principle of the dynamic muscle fatigue model.
Another possible reason is the loading sharing mechanism
of muscles.
Hip and back muscle group has the maximum
joint moment strength (Chaffin et al., 1999) among the impor-
tant muscle groups. For example, the back extensors are com-
posed of numerous muscle slips having different moment arms
and show a particularly high resistance to fatigue relative to
other muscle groups (Jorgensen, 1997). This is partly attributed
to favorable muscle composition, and the variable loading shar-
ing within back muscle synergists might also contribute signif-
icantly to delay muscle fatigue.
In summary, individual characteristics, population charac-
teristics, and posture are external appearance of influencing fac-
tors for the fatigue resistance. Muscle fiber composition, mus-
cle fiber area, and sensory motor coordination mechanism are
the determinant factors inside the human body deciding the fa-
tigue resistance of muscle group. Therefore, how to construct a
bridge to connect the external factors and internal factors is the
most important way for modeling the fatigue resistance for dif-
ferent muscle groups. How to combine those factors to model
the fatigue resistance remains a challenging work. Despite the
difficulty of modeling the fatigue resistance, it is still applicable
to find the fatigue resistance for a specified population by MET
experiments in regression with the extended MET model due to
its simplicity and universal availability.
4.4. Limitations
In the previous discussion, the fatigue resistance of the ex-
isting MET models were quantified using m from regression.
The possible reasons for the different fatigue resistance were
9
analyzed and discussed. However, how to quantify the influ-
ence from different factors on the fatigue resistance remains
unknown due to the complexity of muscle physiology and the
correlation among different factors.
The availability of the extended MET model in the interval
under 15% MVC is not validated. The fatigue resistance is only
accounted from the 15% to 99% MVC due to the unavailability
of some MET models under 15% MVC. For the relative low
load, the individual variability under 15% could be much larger
than that over 15%. The recovery effect might play a much
more significant role within such a range.
5. Conclusions and perspectives
In this paper, fatigue resistance of different muscle groups
were calculated by linear regression from the new fatigue model
and the existing MET static models. High ICC has been ob-
tained by regression which proves that our fatigue model can be
generalized to predict MET for different muscle groups. Mean
and standard deviation in fatigue resistance for different muscle
groups were calculated, and it is possible to use both of them
together to predict the MET for the overall population. The pos-
sible reasons responsible for the variability of fatigue resistance
were discussed based on the muscle physiology.
Our fatigue model is relative simple and computation ef-
ficient. With the extended MET model it is possible to carry
out the fatigue evaluation in virtual human modeling and er-
gonomic application, especially for static and quasi-static cases.
The fatigue effect of different muscle groups can be evaluated
by fitting k from several simple static experiments for certain
population.
Acknowledgments
This research was supported by the EADS and the R´egion
des Pays de la Loire (France) in the context of collaboration
between the ´Ecole Centrale de Nantes (Nantes, France) and Ts-
inghua University (Beijing, PR China).
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