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Journal of Biomechanics 44 (2011) 984–987
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Journal of Biomechanics
journal homepage: www.elsevier.com/locate/jbiomech
www.JBiomech.com
Short communication
An improved cost function for modeling of muscle activity during running
Ali Asadi Nikooyan 1, Amir Abbas Zadpoor n,1
Department of Biomechanical Engineering, Faculty of Mechanical, Maritime, and Materials Engineering, Delft University of Technology, Mekelweg 2, Delft 2628 CD, The Netherlands
a r t i c l e i n f o
a b s t r a c t
Article history:
Accepted 23 November 2010
This paper tries to improve a recently developed mass-spring-damper model of the human body during
running. The previous model took the muscle activity into account using a nonlinear controller that tuned
the mechanical properties of the soft-tissue package based on two physiological hypotheses, namely
‘‘constant-force’’ and ‘‘constant-vibration’’. Three cost functions were used, out of which one was based on
the constant-force hypothesis and two others were based on the constant-vibration hypothesis. The
results of the study showed that the proposed cost functions are only partially successful in capturing the
experimentally observed trends of the ground reaction force and vibration. The current paper proposes an
improved cost function that combines both above-mentioned hypotheses. It is shown that the improved
cost function can capture all the trends that were observed in the measurements of the ground reaction
force and vibration level. It is therefore advised to use the new cost function in place of the previous ones.
& 2010 Elsevier Ltd. All rights reserved.
Keywords:
Mass-spring-damper model
Muscle tuning
Running
Ground reaction force
Vibrations
1. Introduction
According to the muscle-tuning paradigm, the human body tries
to regulate the muscle activity to minimize the vibrations of the
lower body’s soft-tissue package (Boyer and Nigg, 2006, 2007;
Wakeling et al., 2003, 2002, 2001; Wakeling and Nigg, 2001) or to
keep the ground reaction force (GRF) constant (Clarke et al., 1983;
Nigg et al., 1987, 1983; Snel et al., 1985) during running.
In a recent study (Zadpoor and Nikooyan, 2010), a previously
developed mass-spring-damper model of the human body (Liu and
Nigg, 2000; Zadpoor and Nikooyan, 2006; Zadpoor et al., 2007) was
improved by taking the pre-landing muscle activity into account.
For a full description of the model and the physiological questions it
tries to answer see Zadpoor and Nikooyan (2010) and the electronic
supplement of the current paper. In the so-called improved model,
the stiffness and damping properties of the lower body wobbling
mass (LBWM) were controlled by a mechanism that mimicked the
functionality of the central nervous system (CNS) in determining
and issuing the motor signals that are required for muscle tuning.
Based on experimental observations, three cost functions, namely
force, vibration amplitude, and vibration acceleration, were defined to
study the effects of shoe hardness on the loading of the human body
during running. The force criterion was based on the constant-force
hypothesis whereas the other two criteria were based on the
constant-vibration muscle-tuning hypothesis. The ‘‘vibration level’’
n
Corresponding author. Tel.: + 31 15 2786794; fax: + 31 15 2784717.
E-mail addresses: [email protected], [email protected],
[email protected] (A.A. Zadpoor).
1
Both authors have equally contributed to this article and should therefore be
considered as joint first authors.
0021-9290/$ - see front matter & 2010 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jbiomech.2010.11.032
was defined based on either the vibration amplitude or the half-sine
peak acceleration following impact.
The results of the previous study (Zadpoor and Nikooyan, 2010)
showed that by applying the force cost function, the controller can
keep the GRF constant in a specific range of shoe hardness
parameters (i.e. safe region). By applying the vibration cost functions, the controller was able to minimize the vibrations for the
whole range of shoe parameters. Although each control strategy
could capture a part of the experimentally observed trends, none
could cover all aspects. Therefore, an improved cost function
capable of simultaneously satisfying force and vibration hypotheses
is needed.
In this study, two cost functions (force and vibration amplitude)
are combined into a new criterion. The model is simulated using the
new cost function and the simulation results are compared with
experimental observations.
2. Materials and methods
2.1. Basic and improved models
The ‘‘basic model’’ presented in Zadpoor and Nikooyan (2006) and Zadpoor et al.
(2007) was later improved by considering the LBWM as an active element (Fig. 1;
Zadpoor and Nikooyan, 2010). The system is connected to the ground via an element
(Fg), whose force is a function of the displacement (x1) and velocity (v1) of the LBWM
(Liu and Nigg, 2000):
(
Ac ½axb1 þ cxd1 ve1 ðx1 4 0Þ
Fg ¼
ð1Þ
0
ðx1 r 0Þ
The hardness of the shoe is determined by two parameters b and d. The
governing equations of the motion can be written as follows:
m1 x€ 1 ¼ m1 gFg k1 ðx1 x3 Þk2 /wSðx1 x2 Þc1 ðx_ 1 x_ 3 Þc2 /wSðx_ 1 x_ 2 Þ
m2 x€ 2 ¼ m2 g þ k2 /wSðx1 x2 Þk3 ðx2 x3 Þ þ c2 /wSðx_ 1 x_ 2 Þ
A.A. Nikooyan, A.A. Zadpoor / Journal of Biomechanics 44 (2011) 984–987
985
Method for bound-constraint minimization (Lewis and Torczon, 1999)), and the
same optimization bound limits are used. The limits within which k2 and c2 could
change are as follows (Zadpoor and Nikooyan, 2010):
bound limits 1:14 k2,0 r k2,i r 4k2,0 ,
1
4c2,0 r c2,i r 4c2,0
2 :13 k2,0 r k2,i r 3k2,0 ,
3:12 k2,0 r k2,i r 2k2,0 ,
1
3c2,0 r c2,i r 3c2,0
1
2c2,0 r c2,i r 2c2,0
bound limits
bound limits
ð6Þ
For each pair of b and d, the k2 and c2 values that minimized the objective
functions were calculated using the controller. Simulations were carried out for each
bound limit, resulting in three sets of simulations.
3. Results
Fig. 1. Schematic drawing of the ‘improved model’.
m3 x€ 3 ¼ m3 g þ k1 ðx1 x3 Þ þ k3 ðx2 x3 Þðk4 þ k5 Þðx3 x4 Þ þ c1 ðx_ 1 x_ 3 Þc4 ðx_ 3 x_ 4 Þ
m4 x€ 4 ¼ m4 g þ ðk4 þ k5 Þðx3 x4 Þ þ c4 ðx_ 3 x_ 4 Þ
ð2Þ
The model and its parameters (Fig. 1) are fully described elsewhere (Liu and
Nigg, 2000; Zadpoor and Nikooyan, 2010; Zadpoor et al., 2007).
In the improved model, for each pair of the shoe hardness parameters (b and d), the
controller minimizes the cost function by adjusting the stiffness (k2) and damping (c2)
coefficients of the LBWM (Zadpoor and Nikooyan, 2010). The stiffness coefficient k2 and
damping coefficient c2 represent the muscle-tendon properties and are in the range of
experimentally measured values (Liu and Nigg, 2000). The bound limits used for
optimization of k2 and c2 are based on experimental observations and guarantee that
the optimized values are within the physiological range (Zadpoor and Nikooyan, 2010).
The parameters k2 and c2 are considered time-independent. This assumption is
thoroughly discussed in Zadpoor and Nikooyan (2010).
Similar to the simulations that used the constant-force hypothesis, there is a particular area of the b–d plane within which the
normalized optimization error (optimization residual) is very close
to zero (Fig. 2a–c). This region is called the ‘‘safe region’’ and is
defined as a part of the b–d plane within which the normalized
error is less than 0.05 (the region lies between two dashed lines in
each subfigure of Fig. 2). However, comparing the current simulation results with those of the previous study shows that the new
safe region has reduced in size by about 50%. This is because
satisfaction of the combined cost functions is stricter that satisfying
either force or vibration cost functions.
The maximum and minimum values of k2 and c2, when the error
is brought to zero by the controller, are listed in Table 1. One can see
in this table that the selected bound limits for the pattern search
optimization considerably affect the minimum value of c2, while
they have no sizable impact on the minimum value of k2 and
maximum value of c2. The maximum values of k2 are only slightly
(2–6%) affected.
The GRF plots for different pairs of shoe hardness parameters
(Fig. 3) confirm that only inside the safe region the peak forces
remain unchanged (compared to the default values). For soft shoes,
the pattern of the time response of the GRF (Fig. 3) when using
the improved cost function is similar to that of the force criterion.
For hard shoes, the pattern resembles the simulation results of
the improved model, when vibration amplitude cost function
was used.
4. Discussions
2.2. Combined force–vibration criterion
Two cost functions, namely force (Jf) and vibration amplitude (Jv1), were defined
as follows (Zadpoor and Nikooyan, 2010):
Jf ¼ 9p1 /bi ,di Sp1,0 /b0 ,d0 S9þ 9p2 /bi ,di Sp2,0 /b0 ,d0 S9
ð3Þ
Jv1 ¼ 9Li /bi ,di SL0 /b0 ,d0 S9
ð4Þ
where p1 and p2 are the first and second ground reaction force peaks. The default
force values (p1,0 ¼ 1436.8 N and p2,0 ¼ 2026.4 N) are the force peaks calculated using
the default values of the shoe hardness parameters b0 (¼ 1.38) and d0 (¼0.75)
(Zadpoor et al., 2007). L is the vibration amplitude (the vertical displacement of the
soft tissues) and L0 (¼ 10.1 cm) is the default value of the vibration amplitude that
were calculated using the default values of b0 and d0 (Nikooyan and Zadpoor, 2007).
The new cost function is defined as sum of the two above-mentioned cost
functions. The objective functions are first nondimensionalized by normalizing each
term with respect to its default value. The new criterion (Jfv) can be therefore written as
p1 /bi ,di Sp1,0 p2 /bi ,di Sp2,0 Li /bi ,di SL0 ð5Þ
Jfv ¼ þ
þ
p1,0
p2,0
L0
2.3. Simulations
To make comparison of simulation results between the current and the previous
study (Zadpoor and Nikooyan, 2010) as easy as possible, the same range of the shoe
hardness parameters (b and d), the same optimization technique (Pattern Search
In order to assess the usefulness of the combined cost function,
the simulation results are compared with three sets of experimental observations. The first comparison is between the GRF
values calculated here and the ones measured during actual
running exercises (Clarke et al., 1983) as well as the findings of
the study by Ferris and Farley (1997). Both in the simulation results
and in experimental observations, there is a range of hardness
values (safe region) for which the GRF remains constant. Outside
the safe region, the GRF deviates from default values. Many other
researchers have observed similar trends in their experiments.
Lieberman et al. (2010) observed that the mean impact transient
force is not dependent on the hardness of the platform nor is it
dependent on whether the runners are barefoot or shod. Other
researchers (Dixon et al., 2000; Kerdok et al., 2002) have observed
that by varying the stiffness of the surface and/or footwear, there is
a range of stiffness parameters for which the ground reaction force
does not significantly change.
Comparing the force cost function (Jf) with the improved cost
function (Jfv), one can see that the latter is less sensitive to the
selected bound limits (Table 1): when using Jfv, the minimum
values of the stiffness show no dependency on the bound limit,
while this is not the case for Jf.
986
A.A. Nikooyan, A.A. Zadpoor / Journal of Biomechanics 44 (2011) 984–987
Fig. 2. Simulation results: the normalized error (a)–(c), the absolute error of the force part (d)–(f), the absolute error of the amplitude part (g)–(i), stiffness coefficients (j)–(l),
damping coefficients (m)–(o), the first force peak (p)–(r), the second force peak (s)–(u), and vibration amplitude (v)–(x). Columns 1–3 of this figure relate to bound limits 1–3
defined by Eq. (6). The shoe hardness changes from ‘‘the softest’’ at the left top to ‘‘the hardest’’ at the right down.
A.A. Nikooyan, A.A. Zadpoor / Journal of Biomechanics 44 (2011) 984–987
Table 1
Values of stiffness (k2) and damping (c2) coefficients when the optimization error
(optimization residual) is close to zero for three different bound limits (UB, upper
bound; LB, lower bound). These values show that the involved muscles do not need
to be extremely stiff to keep the GRF and the level of vibrations constant.
k2 (N/m)
c2 (kg/s)
Bound limit 1
Bound limit 2
Bound limit 3
UB
Max.
24,000
8048
18,000
7536
12,000
7350
LB
Min.
1500
4976
2000
4976
3000
4976
UB
Max.
2600
907
1950
907
1300
907
LB
Min.
162.5
174
216.67
279
325
327
987
latest stages of contact with the ground, when the post-landing
muscle activity plays an important role.
Conflict of interest statement
We hereby state that we have not had any financial or personal
relationships with other people or organizations that could inappropriately influence (bias) our work.
Appendix A. Supplementary Materials
Supplementary data associated with this article can be found in
the online version at doi:10.1016/j.jbiomech.2010.11.032
References
Vertical Ground Reaction Force (N)
2500
2000
default model
Jf safe region
1500
Jv1 safe region
Jfv safe region
Jf soft shoe
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Jv1 soft shoe
Jfv soft shoe
Jf hard shoe
500
Jv1 hard shoe
Jfv hard shoe
0
0
0.05
0.1
0.15
0.2
0.25
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