análise da força de impacto durante corrida com um modelo 5DOF

Prévia do material em texto

A 5-degree-of-freedom model to evaluate the ground reaction force 
during running, sentsitive to the effect of the velocity and lower 
joints torsional stiffness. 
 
 
Andrade, G. F., Zanetti, L. R., Ghoneim, H. 
 
 
Mechanical engineering department, Rochester Institute of Technology, 1 Lomb Memorial Drive, Rochester NY- USA 
 
13th December, 2015 
 
Abstract 
 
It is known that during running, the human body is subject to impact forces, and those can lead to injuries. Many studies have 
been developed in this area in order to understand and to improve the human body behavior during running. Different and more 
complex models have been proposed to simulate the ground reaction forces (GRF) based on obtained experimental data, but the 
simulation and experimental data still have divergences. This study proposed a 5 degree-of-freedom model with the purpose of 
obtaining the ground reaction force profile. The simulation verified the influence of running velocity, ankle stiffness and knee stiffness 
considering soft and hard sole conditions. The GRF was calculated using a non-linear curve fit based on experimental data. The 
model’s results were compared to a similar 4 degree-of-freedom model in order to validate it. The results showed that both ankle and 
knee stiffness have little effect over the GRF when being varied in small ranges. The increase of the running velocity led to a higher 
GRF impact (first) peak, a smaller active (second) peak and to a shorter stance time. The GRF for the soft sole condition had a smaller 
first peak when compared to the hard sole condition, as expected based on other studies. The proposed model reaction force curves 
had a bigger first peak and a smaller second peak when compared to the 4 degree-of-freedom model, probably due the direct effect of 
the running velocity in the proposed model. 
 
Keywords: Ground reaction force; Running; Velocity; Simulation; Joint stiffness 
 
 
1. Introduction 
 
 Running is a common activity for people who desire to 
maintain a healthy lifestyle. However, since running is a 
physical activity, related injuries can occur. Thus, it became 
convenient to study the running mechanisms in order to 
prevent or decrease injury frequency. Many running related 
studies have been developed in the past decades, as examples, 
Ker et al. (1989) proposed a relation between the foot strike 
in the human running and the mechanical properties of the 
heel pad; Tongen and Wunderlich (2010) studied the 
mathematics related to the forces produced by each foot in 
contact with the ground; Clark, Ryan and Weyand (2014) 
studied the effects of foot speed, foot-strike and footwear in 
the impact forces between the foot and the ground. A big part 
of these studies focus on analyzing the impact force between 
the ground and the human foot. This impact force during 
running occurs when the foot hits the ground with a certain 
velocity, the reaction force on the human body is called 
ground reaction force (GRF). 
 Several studies used simple mass-spring-damper models 
to simulate the human body behavior and to obtain the 
reaction force profile during the touchdown moment. Nigg 
and Liu (1999) used a 4-DOF mass-spring-damper model to 
simulate the impact force peaks during human running and 
study the effects of muscle stiffness and damping on this 
simulation; Yukawa, Tokizawa and Kawamura (2011) 
proposed a two dimensional runner mass-spring-damper 
model to generate ground reaction forces in steady pace 
running. Like Nigg and Liu (1999), several studies assumed 
that the body can be modeled as 4 lumped masses connected 
through springs and dampers. The masses represent the 
wobbling and rigid masses of our upper body and legs. The 
springs and dampers represent the muscles, bones and 
viscoelastic elements of our body and legs. The GRF profile 
resulting from the simulation with this type of model are 
consistent with the experimental results found during 
pendulum tests performed by Aerts and De Clercq (1992). 
Nevertheless, this study uses a more complex model which 
also includes the rotation around the ankle, with the intend of 
getting a more accurate GRF profile. 
 The ground reaction is affected by several different 
parameters. For example, Khajah and Hou (2015) used a 
nonlinear GRF relation and tried several different coefficients 
values that represents the hard and soft sole conditions, and as 
result the GRF peaks varies significantly. This study uses a 
range of parameters, found on similar studies, that simulates 
soft and hard sole conditions. The ankle and knee stiffness 
values here tested were also based on other related studies, as 
well as the values for the running velocity. These different 
parameters are simulated in order to check their effect in the 
GRF profile. 
 Most of the running related studies, which analyze the 
vertical reaction force, use a non-linear relation to evaluate it. 
This non-linear relation is based on a curve fit designed to 
match the results produced by the pendulum test, Aerts and 
De Clercq (1992). The non-linear force relation was used to 
calculate the GRF in this study, for many different parameters. 
 Finally, the purpose of this analysis is to simulate a 
human running using a 5 DOF mass-spring-damper model, 
verifying the GRF when varying sole stiffness, sole damping, 
ankle angular stiffness, knee angular stiffness and running 
velocity and compare the results produced with several 
published data, either analytical or experimental. 
 
 
2. Methods 
 
The human body was simplified as a 5-DOF spring-
damper-mass system, which include non-rigid “wobbling” 
masses, which is necessary since this is an impact motion 
model (Denoth et al., 1984). 
 
 
Fig 1. 5-degree-of-freedom spring-damper-mass model of the human 
body to study vertical impact force during the touch down phase of 
running. 
 
 
Fig 2. Vector components of the 5-DOF model. From left to right the diagrams represent the acceleration, velocity and position 
respectively. 
The model consists of four masses and one rotational DOF, 
θ. The masses m3 and m4 are the rigid and wobbling masses of 
the upper body respectively, m2 is the non-rigid mass of the 
shank and m1 represents the mass of the foot. The variable θ 
represents the rotation angle in the ankle and knee. This model 
is a modification of the 4-DOF representation from Nigg and 
Liu (1999), which is a particular case of the five-degrees-of-
freedom variant. The mathematical model associated with the 
spring-damper-mass system is described by: 
 
�̈�1 = 
𝐹𝑔
𝑚1
+
𝑐1
𝑚1
�̇�3 cos 𝜃 + 
𝑘1𝑟3 + 𝑐2�̇�2 + 𝑘2𝑟2
𝑚1
cos 𝜃 − 𝑔 
�̈�2 = −�̈�1 cos 𝜃 − 
𝑐2
𝑚2
 �̇�2 + (�̇�
2 −
𝑘2 + 𝑘3
𝑚2
) 𝑟2 +
𝑘3
𝑚2
 𝑟3 − 𝑔 
�̈�3 =
𝑐4
𝑚3
(�̇�4 − �̇�3) +
𝑘4 + 𝑘5
𝑚3
(𝑥4 − 𝑥3) 
−
𝑘1𝑟3 + 𝑘3(𝑟3 − 𝑟2)
𝑚3
cos 𝜃 −
𝑐1
𝑚3
�̇�3 cos 𝜃 − 𝑔 
�̈�4 = −
𝑐4
𝑚4
(�̇�4 − �̇�3) −
𝑘4 + 𝑘5
𝑚4
(𝑥4 − 𝑥3) − 𝑔 
 
�̈� = 
𝑚2𝑅2 + 𝑚3𝑅3
𝐼𝐺
 �̈�1 sin 𝜃 − 2 (
𝑚2𝑅2�̇�2 + 𝑚3𝑅3�̇�3
𝐼𝐺
) �̇�
−
𝑘𝑎 + 𝑘𝑘
𝐼𝐺
𝜃 +
𝑚2𝑅2𝑔
𝐼𝐺
sin 𝜃 
Where 𝐼𝐺 = 𝐼𝑠ℎ𝑎𝑛𝑘 + 𝑚2𝑅2
2 + 𝑚3𝑅3
2. 
 
The outcome variables of this study are the main 
parameters that can be changed and will influence directly the 
GRF profile: ankle angular stiffness (ka), knee angular 
stiffness (kk), running velocity (v0) and the sole properties. 
Several values of ka, kk and v0 will be tested and their influence 
over the GRF will be verified through the non-linear force 
relation. The non-linear GRF relation was proposed by Cole(1995): 
 
𝑉𝐺𝑅𝐹: {
 𝐴𝐶[𝑎(−𝑥1)
𝑏 + 𝑐(−𝑥1)
𝑑(−�̇�1)
𝑒] 𝑥1 < 0 
0 𝑥1 ≥ 0
(3) 
 
The 𝐴𝐶 coefficient is assumed to be 2, since the average 
contact area is assumed to be two times the area reported in the 
Aerts and de Clercq (1992) pendulum tests. The coefficients 
a, b, c, d and e represent the sole conditions and can be 
determined performing a curve fit on the force-deformation 
curves of the same pendulum tests performed by Aerts and de 
Clercq (1992). Table 1 shows the a, b, c, d, e coefficients 
obtained by Nikooyan and Zadpoor (2006) representing the 
curve fit for soft and hard shoe conditions. This study makes 
the use of these for estimating the GRF using the non-linear 
force relation. Note that the 𝑥1 and �̇�1 have negative signs 
because the coordinate system of reference assumes that the 
positive direction is upwards. 
 
Dimensionless 
Coefficient 
Soft Shoe Hard Shoe 
Ac 2.0 2.0 
a 0.6 x 106 0.6 x 106 
b 1.56 1.38 
c 2.0 x 104 2.0 x 104 
d 0.73 0.75 
e 1.0 1.0 
Table 1. Curve fit parameters for non-linear GRF evaluation. 
 
The values for masses, stiffness and damping listed on 
Table 2 were extracted from Nigg and Liu (1999) with an 
exception for the torsional stiffness, ka and kk, which were 
evaluated by Mauroy, Schepens and Willems (2014). 𝐼𝑠ℎ𝑎𝑛𝑘 
was approximated based on the body dimensions of a fit man 
of similar weight. 
 
 
Parameter Unit 
Mass / Mom. of Inertia 
m1 6.15 kg 
m2 6 kg 
m3 12.58 kg 
m4 50.34 kg 
Ishank 0.2488 kg.m² 
Linear and Torsional Stiffness 
k1 6,000 N/m 
k2 10,000 N/m 
k3 10,000 N/m 
k4 18,000 N.m 
Damping 
c1 300 kg/s 
c2 650 kg/s 
c4 1900 kg/s 
 Table 2. 5-DOF model parameters 
 
The initial conditions for position and velocity were 
obtained from Zadpoor et al. (2006) except for the angular 
parameters, which are new features. The initial angle at 
touchdown was estimated as 15º while the initial angular speed 
was estimated to be negative 150º/s. 
 
 
 
 
Initial Condition Value Unit 
v1 (0) -0.96 m/s 
v2 (0) -0.96 m/s 
v3 (0) -2.0 m/s 
v4 (0) -2.0 m/s 
θ0 15 deg 
𝜔0 -150 deg/s 
The initial position for all the masses are 0 
Table 3. Initial conditions for the masses position, velocity and initial 
angle θ and angular speed 𝜔. 
 
The range for v0 was defined based on the study 
conducted by Yukawa et al. (2011), where v0 varies between 
2.56 and 5.68 m/s. In this study, the velocity range used 
includes 8, 12, 16, 20 and 24 km/h (2.22, 3.33, 4.44, 5.56 and 
6.67 m/s, respectively). The range for kk and ka was defined 
based on Mauroy et al. (2014), where the tested values for kk 
are 6.06, 8.46 and 13.86 Nm rad-1kg-1 and for ka are 7.03, 11.54 
and 18.66 Nm rad-1kg-1.
 
In order to check the validity of the results, the GRF 
curves were plotted along with the result from the 4-DOF 
model, using the initial conditions parameters proposed by 
Zadpoor et al. (2007) where the upper and lower body have 
different touchdown velocities. 
 
 
3. Results 
 
Figure 3 and figure 4 present the plots for the GRF during 
the stance phase using the proposed 5-DOF model, comparing 
the results obtained by the 4-DOF model, and shows the effect 
of running velocity, knee stiffness, ankle stiffness and sole 
properties over the GRF profile. 
The GRF peaks are greatly affected by the running 
velocity. As the velocity increases, the first force peak 
increases and the second decreases. 
 The time in which the second peak occurs decreases, 
mainly because the stance phase gets shorter as the running 
velocity increases. For the ankle and knee stiffness, their
Fig 3: effect of ankle stiffness, knee stiffness and running velocity over the GRF for soft and hard soles and comparison with the 4-DOF 
model proposed by Nigg and Liu 
 
Fig 4: GRF profile for the proposed 5-DOF model compared with the 
GRF profile obtained with the 4-DOF model proposed by Nigg and 
Liu for soft and hard sole 
variation within the proposed range almost does not affect the 
GRF profile. 
The sole properties also have a great effect over the GRF. 
For the soft sole condition, the first force is smaller, as 
expected due to the fact that the soft sole helps to absorb the 
impact. For the hard sole condition, the first force peak is about 
0.5BW bigger. The second peak is not much affected by the 
changes in the sole properties. 
For all the simulations, the main GRF peak (second peak) 
is smaller for the proposed 5-DOF model when compared with 
the 4-DOF model, as for the first peak the 4 DOF model 
presents a GRF profile with a smaller value when compared 
with the proposed 5-DOF model. 
 
 
4. Discussion 
 
In this section, the results from the 5-DOF system, 
proposed as a modification of the 4-DOF from Nigg and Liu 
(1999), which includes the same rigid and wobbling masses as 
well as a rotational degree-of-freedom for the ankle and knee, 
are discussed and later compared to other experimental 
bibliography. The weaknesses and strengths of this model are 
analyzed along with the main problems faced during its 
simulation and suggestions and improvements for further 
research are also proposed. Adding the rotational DOF made 
possible to include the linear running velocity in the analysis, 
the single parameter that yielded the biggest difference 
between the two models. 
McMahon and Chang (1990), using a spring-mass model, 
proposed that the leg and joint stiffness change or adapt for 
different running scenarios. Later, using experimental support, 
Derrick, Hamill and Caldwell (1998) and Arampatzis, 
Brueggemann and Metzler (1999) demonstrated that the 
running speed affect the leg stiffness mostly influenced by a 
change in the rotational joint stiffness of the knee then 
followed by the ankle. Based on a joint torsional stiffness 
defined by Mauroy et al. (2014) the effect of the variation of 
both ka and kk were tested alone for both soft and hard shoe 
conditions. However, their effect on the GRF showed to be 
insignificant for this given range, and the major impacts on the 
vertical ground velocity were obtained by changing the linear 
velocity rather than joint stiffness, although more research is 
needed. 
Several studies linked the relation between the running 
speed and GRF (e.g. Payne, 1978; Cavanagh and Lafortune, 
1980; Roy, 1981; Munro, Miller and Fuglevand, 1987). 
Cavanagh and Lafortune results showed a huge variation in 
GRF curves shape and amplitude among the samples and 
further research was suggested. Later, Munro et al. (1987) 
performed similar experiments which demonstrated a positive 
relation between the impact force and running speed, 1.57 BW 
at 3m/s to 2.32 BW at 5m/s, and also concluded that more 
research was needed due to high variability in the results. 
Keller et al. (1996) compared ground reaction force between 
walking, slow jogging and running, the experimental data 
from this study shows GRF plots with 2 well distinct peaks 
with forces around 1.2 BW to walking to a single peak with 
more than 2 BW for running. Tongen and Wunderlich (2010) 
used a mathematical model, backed by experimental data, to 
predict the GRF with results qualitatively similar to those 
shown by Keller et al. (1996). Yukawa et al. (2011) used a 
different 4-DOF model, 3 masses and one rotational DOF, 
along with experimental testing with good concordance in the 
results. This study also showed a slight higher impact peak as 
the speed grows, generating a peak forceof approximately 
1500N (subject mass was not given) at 2.53 m/s to over 2000N 
at 5.16 m/s, an increase of roughly 33%. 
The curves for the presented 5-DOF model with different 
linear velocities resulted in shorter length of time when the 
GRF acts as well as an increased impact peak and reduced 
active peak, which tends to disappear while the speed 
increases. However, the net result of the linear velocity 
variations on the GRF was negligible; the maximum forces 
were only transferred from the active to the impact peak. More 
research is needed to understand this behavior and using the 
k1, k2 and k3 (linear stiffness of the leg) as variable parameters 
with running speed may yield more accurate outcomes. 
The comparison between the proposed model and the 
4-DOF shows a slight shift in the GRF curve peaks. In the 5-
DOF model, the impact (first) peak is bigger and the active 
(second) is smaller, when compared to the 4-DOF model. The 
main difference between these variants is that the 5-DOF 
model considers the effect of the leg angle. So, for this case, 
the kinematic relations show how running velocity affects 
directly the forces in all the obtained equations of movement. 
Therefore, the 5-DOF model GRF curves will be directly 
affected for the running velocity, while the 4-DOF model GRF 
does not show any direct relation with the running velocity. 
Thus, for this study, only two sets of sole experimental fit 
parameters (a, b, c, d and e) were tested, the same used for 
many related studies like Nigg and Liu (1999) and Nikooyan 
and Zadpoor (2006). Khajah and Hou (2015) showed that 
these parameters can be optimized in order to get a more 
realistic impact force profile, but that was not the goal of this 
study. The GRF plot for both 4 and 5-DOF models shows that 
the sole affects only the impact force peak. Analyzing the 5-
DOF reaction force curves, for the soft sole condition, the first 
peak is about 0.5BW smaller when compared to the hard sole 
shoe. The GRF profiles obtained by Khajah and Hou (2015) 
also show a smaller first peak for the soft sole condition and 
an almost irrelevant influence of the sole over the active peak. 
 
5. Conclusion 
 
The modification of the Nigg 4-DOF model, adding one 
rotational degree-of-freedom, allows checking the effect of the 
rotational mass of the leg, ankle and knee torsional stiffness 
and running velocity. The results showed a reasonable 
concordance between the 5-DOF curves and its precursor’s - 
which had good concordance with experimental data - as well 
as similar behavior when using soft or hard shoe soles. 
However, the shape and amplitude of the GRF curves for 
different speeds somewhat disagrees with published 
experimental data but this may be due the fact that same initial 
conditions were used for all simulations as well as some other 
factor which might have not been considered while developing 
the model. Since adding the speed to the model is a useful tool 
for the simulation and development of new models or even 
products, further research is desired once this approach is a 
promising method for checking the effects of forces during 
running with good accuracy. 
 
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