A Simple Method for Measuring Stiffness during Running

Abstract

The spring-mass model, representing a runner as a point mass supported by a single linear leg spring, has been a widely used concept in studies on running and bouncing mechanics. However, the measurement of leg and vertical stiffness has previously required force platforms and high-speed kinematic measurement systems that are costly and difficult to handle in field conditions. We propose a new "sine-wave" method for measuring stiffness during running. Based on the modeling of the force-time curve by a sine function,this method allows leg and vertical stiffness to be estimated from just a few simple mechanical parameters: body mass, forward velocity, leg length, flight time, and contact time. We compared this method to force-platform-derived stiffness measurements for treadmill dynamometer and overground running conditions, at velocities ranging from 3.33 m.s-1 to maximal running velocity in both recreational and highly trained runners. Stiffness values calculated with the proposed method ranged from 0.67 % to 6.93 % less than the force platform method, and thus were judged to be acceptable. Furthermore, significant linear regressions (p < 0.01) close to the identity line were obtained between force platform and sine-wave model values of stiffness. Given the limits inherent in the use of the spring-mass model, it was concluded that this sine-wave method allows leg and stiffness estimates in running on the basis of a few mechanical parameters, and could be useful in further field measurements.

Full text

167
A Simple Method for Measuring
Stiffness During Running
Jean-Benoît Morin
1
, Georges Dalleau
2
, Heikki Kyröläinen
3
,
Thibault Jeannin
1
, and Alain Belli
1
1
University of Saint-Etienne;
2
University of La Réunion
3
University of Jyväskylä
The spring-mass model, representing a runner as a point mass supported by
a single linear leg spring, has been a widely used concept in studies on run-
ning and bouncing mechanics. However, the measurement of leg and vertical
stiffness has previously required force platforms and high-speed kinematic
measurement systems that are costly and difcult to handle in eld condi-
tions. We propose a new “sine-wave” method for measuring stiffness during
running. Based on the modeling of the force-time curve by a sine function,
this method allows leg and vertical stiffness to be estimated from just a few
simple mechanical parameters: body mass, forward velocity, leg length, ight
time, and contact time. We compared this method to force-platform-derived
stiffness measurements for treadmill dynamometer and overground running
conditions, at velocities ranging from 3.33 m·s
–1
to maximal running velocity
in both recreational and highly trained runners. Stiffness values calculated with
the proposed method ranged from 0.67% to 6.93% less than the force platform
method, and thus were judged to be acceptable. Furthermore, signicant linear
regressions (p < 0.01) close to the identity line were obtained between force
platform and sine-wave model values of stiffness. Given the limits inherent in
the use of the spring-mass model, it was concluded that this sine-wave method
allows leg and stiffness estimates in running on the basis of a few mechanical
parameters, and could be useful in further eld measurements.
Key Words: spring-mass, modeling, practical calculation
During running, the musculoskeletal structures of the legs alternately store
and return elastic energy, so the legs can be described as springs loaded by the
runner’s body mass, constituting the “spring-mass model” (Alexander, 1992; Blick-
han, 1989; Cavagna, Heglund, & Taylor, 1977; Cavagna, Saibene, & Margaria,
1
Physiology Laboratory, PPEH Res. Unit, Univ. of Saint-Etienne, CHU Bellevue,
42055 St-Etienne Cedex 02, France;
2
Sport Sciences Res. Center, Faculty of Science and
Technology, Univ. of La Réunion, 117 General Ailleret St., 97430 Le Tampon, France;
3
Neuromuscular Res. Center, Univ. of Jyväskylä, PO Box 35, 40014 Jyväskylä, Finland.
JOURNAL OF APPLIED BIOMECHANICS, 2005, 21, 167-180
©2005 Human Kinetics Publishers, Inc.

168 Morin, Dalleau, et al.
Measuring Stiffness During Running 169
1964; McMahon & Cheng, 1990). This model has been a widely used concept for
describing and studying the storage and return of elastic energy in the lower limbs
in humans and other animals during bouncing and running (Dalleau, Belli, Bourdin,
& Lacour, 1998; Farley, Blickhan, Saito, & Taylor, 1991; Farley & Gonzalez, 1996;
Ferris, Louie, & Farley, 1998; He, Kram, & McMahon, 1991; McMahon, Valiant,
& Fredrick, 1987; McMahon & Cheng, 1990). The model consists of a point mass
supported by a single massless linear “leg spring.”
The main mechanical parameter studied when using the spring-mass model
is the stiffness of the leg spring, dened as the ratio of the maximal force in the
spring to the maximum leg compression at the middle of the stance phase (Farley &
Gonzalez, 1996). Moreover, although it does not correspond to any physical spring,
the term vertical stiffness is used to describe the vertical motion of the center of
mass (CM) during contact (Farley & Gonzalez, 1996; McMahon & Cheng, 1990),
and is dened as the ratio of the maximal force to the vertical displacement of the
CM as it reaches its lowest point, i.e., the middle of the stance phase. Maximal
ground reaction force and CM displacement measurements during running are
needed to measure these parameters requiring dynamometers such as overground
or treadmill-mounted force platforms, or even video motion-analysis (e.g., Aram-
patzis, Brüggemann, & Metzler, 1999). Such equipment is costly and not practical
for eld measurements.
Considering the descriptive and predictive power of the spring-mass model
in running mechanics and the serious technical issues mentioned, the aim of this
study was to propose and validate a simple calculation method for assessing leg and
vertical stiffness during running. The computations inherent in the proposed method
are based on a few simple mechanical parameters: contact and ight times, forward
running speed, leg length, and body mass. We validated the method by comparing
leg and vertical stiffness values measured with a force platform to those obtained
using the proposed method during treadmill and overground running.
Methods
For practical reasons our study was divided into two different protocols, one aiming
to validate the presented method during treadmill running, the other during over-
ground running. However, in both cases the force platform and sine-wave method
stiffness values were each calculated in the same way. All participants gave their
informed consent to take part in those protocols.
Treadmill running: Eight young men (age 24 ± 2 yrs, height 1.78 ± 0.07 m,
body mass 76.0 ± 7.0 kg; mean ± SD) volunteered to participate in this study. They
were physical education students and experienced in treadmill running. After a 5-min
warm-up at 3.33 m·s
–1
, they performed 30-s running bouts at 3.33, 3.89, 4.44, 5, 5.56,
6.11, and 6.67 m·s
–1
at their preferred step frequency (separated by 2 min of rest)
on a treadmill dynamometer (HEF Techmachine, Andrézieux-Bouthéon, France).
Using the same calibration procedure as Belli, Bui, Berger, Geyssant, and Lacour
(2001), we determined the treadmill’s static nonlinearity to be less than 0.5% and 1%,
respectively, in the vertical and horizontal directions. Natural vibration frequency
(treadmill hit with a hammer) were 147 Hz in the vertical direction and 135 Hz in
the anterior-posterior and mediolateral directions. Vertical ground reaction forces,
belt velocity, and ight and contact times were measured at a sampling frequency
of 500 Hz. All values were averaged for 10 consecutive steps for each velocity.

168 Morin, Dalleau, et al.
Measuring Stiffness During Running 169
Overground running: 10 young men (age 23 ± 3 yrs, height 1.80 ± 0.05 m,
body mass 66.4 ± 5.3 kg) who were elite middle-distance runners volunteered to
run over a 10-m force platform (Kistler, Winterthur, Switzerland; natural frequency
higher than 150 Hz, horizontal to vertical cross-talk lower than 2%) at 4, 5, 6, and
7 m·s
–1
and at their maximal running velocity. Velocity was measured by two pairs
of photocells placed at each end of the force platform. Vertical ground reaction
forces and ight and contact times were measured for one step at each velocity at
a sampling rate of 1.8 kHz.
Reference method. Vertical stiffness: The vertical stiffness k
vert
(in kN·m
–1
)
was calculated as:
k
vert
= F
max
· ∆y
c
–1
(1)
with F
max
the maximal ground reaction force during contact (in kN) and ∆y
c
the
vertical displacement of the CM when it reaches its lowest point (in m), determined
by double integration of the vertical acceleration over time, as proposed by Cavagna
(1975). Figure 1 shows a typical example of force and vertical displacement of the
CM evolutions during contact.
Leg stiffness: The stiffness of the leg spring k
leg
(in kN·m
–1
) was calculated
as follows:
k
leg
= F
max
· ∆L
–1
(2)
with ∆L the peak displacement of the leg spring (in m) calculated from the initial
leg length L (great trochanter to ground distance in a standing position), running
velocity v (in m·s
–1
), and the contact time t
c
(in s), as per Farley and Gonzales
(1996; Eq. 2 and 3):
∆L = L – √ L
2
– (––)
2
+ ∆y
c
(3)
vt
c
2
Figure 1 — Typical measured (thick line) and modeled (dotted line) vertical force over
time during contact on treadmill running, and corresponding downward vertical CM
displacement (thin line) (velocity of 5 m·s
–1
, body mass of 68 kg).

170 Morin, Dalleau, et al.
Measuring Stiffness During Running 171
Sine-wave method. Model proposed: This method is based on a model used
by Dalleau et al. (2004) for vertical jumps, which considers the force as a func-
tion of time during contact to be a simple sine function (Alexander, 1989; Kram
& Dawson, 1998):
F(t) = F
max
sin(— t) (4)
The validity of this postulate was checked by comparing areas under the mea-
sured and modeled F(t) curves for all steps analyzed (Figure 1). The mean absolute
bias was calculated as (Modeled – Force platform)/Force platform×100.
Calculations: Complete calculations and assumptions are shown in the
Appendix section of this paper.
The modeled vertical stiffness k
vert
was calculated as the ratio of the modeled
maximal force F
max
over the modeled vertical CM displacement ∆y
c
:
k
vert
= F
max
· ∆y
c
–1
(5)
with F
max
= mg — (— +1) (6)
m being the participant’s body mass (in kg), and t
f
and t
c
, respectively, being the
ight and contact times (in s)
and ∆y
c
= —— + g — (7)
The modeled leg stiffness k
leg
was calculated as follows:
k
leg
= F
max
· ∆L
–1
(8)
with ∆L = L – √L
2
– (—)
2
+ ∆y
c
(9)
the modeled leg length variation (in m).
The participant’s leg length was also modeled as: L
mod
= 0.53h where h is the
participant’s height (in m), according to the anthropometric equations of Winter
(1979), in order to check the validity of the presented stiffness calculation method
using leg length values obtained from participant’s height data.
In order to quantify the force platform-model values difference, we calculated
an absolute mean error bias as follows: Bias = (Modeled – Force platform)/Force
platform×100. The force platform-model relationships were further described by
means of linear regressions and calculation of the determination coefcient R².
We then performed an ANOVA with the Scheffé post hoc test in order to
check the eventual effect of velocity on the above-mentioned force platform-model
bias. The statistically signicant level was set at 0.05.
Results
The bias between force platform and modeled F(t) areas was 5.33% (ranging from
11.7% at 3.33 m·s
–1
to 1.7% at 6.67 m·s
–1
) for treadmill running, 542 steps analyzed,
and 2.93% (ranging from 3.17% at 4 m·s
–1
to 2.33% at 7 m·s
–1
) for overground
running, 50 steps analyzed.
π
t
c
^
^
^
^
^
^
^
^
^
^
^
t
f
t
c
π
2
F
max
t
c
2
m π
2
t
c
2
8
^
^
^
^
vt
c
2

170 Morin, Dalleau, et al.
Measuring Stiffness During Running 171
The values of the different mechanical parameters measured and modeled and
the reference-model bias are shown in Tables 1 and 2 for treadmill and overground
running, respectively. For the vertical and leg stiffness, we obtained reference-model
error biases of 0.12% (ranging from 1.53% at 6.67 m·s
–1
to 0.07% at 6.11 m·s
–1
) and
of 6.05% (from 9.82% at 3.33 m·s
–1
to 3.88% at 6.67 m·s
–1
), respectively, during
treadmill running. During overground running the bias was 2.30% (from 3.64% at
5 m·s
–1
to 0.25% at 6 m·s
–1
) for the vertical stiffness, and 2.54% (from 3.71% at 5
m·s
–1
to 1.11% at maximal velocity) for leg stiffness.
Further, the reference-model linear regressions were signicant (p < 0.01; R²
= 0.89 to 0.98) for both vertical and leg stiffness either on the treadmill or during
overground running (Figures 2 and 3, respectively).
The changes in reference and modeled values of stiffness with running veloci-
ties are reported in Figure 4 for treadmill and overground running. The ANOVA
and the Scheffé post hoc test demonstrated for treadmill running a signicant dif-
Table 1 Main Mechanical Parameters Measured With Reference Method During
Treadmill Running (M ± SD)
Reference Modeled Bias (%)
∆y
c
(m) 0.05 ± 0.01 0.05 ± 0.02 3.28 ± 1.10
∆L (m) 0.20 ± 0.03 0.20 ± 0.03 0.93 ± 0.43
F
max
(kN) 2.05 ± 0.34 1.91 ± 0.32 6.93 ± 2.52
k
vert
(kN·m
–1
) 37.70 ± 8.84 37.74 ± 8.87 0.12 ± 0.53
k
leg
(kN·m
–1
) 10.37 ± 2.34 9.75 ± 2.19 6.05 ± 3.02
Note: Measured with reference method, corresponding values calculated with proposed
method and error bias values in between. Values presented are averaged for all partici-
pants and all velocities.
Table 2 Main Mechanical Parameters Measured With Reference Method During
Overground Running (M ± SD)
Reference Modeled Bias (%)
∆y
c
(m) 0.05 ± 0.01 0.05 ± 0.01 2.34 ± 2.42
∆L (m) 0.16 ± 0.02 0.16 ± 0.02 0.67 ± 1.09
F
max
(kN) 2.13 ± 0.21 2.06 ± 0.24 3.24 ± 2.08
k
vert
(kN·m
–1
) 51.39 ± 21.46 50.21 ± 20.40 2.30 ± 1.63
k
leg
(kN·m
–1
) 13.28 ± 1.85 12.95 ± 2.13 2.54 ± 1.16
Note: Measured with reference method, corresponding values calculated with proposed
method and error bias values in between. Values presented are averaged for all partici-
pants and all velocities.

172 Morin, Dalleau, et al.
Measuring Stiffness During Running 173
Figure 2 — Signicant reference-model linear regressions (p < 0.01) obtained during
treadmill running for vertical stiffness (upper panel) and leg stiffness (lower panel)
compared to the identity line. Each dot represents a mean value of stiffness for a par-
ticipant at the corresponding velocity.
Reference vertical stiffness (kN·m
–1
)
Modeled leg stiffness (kN·m
–1
)
Modeled vertical stiffness (kN·m
–1
)
Reference leg stiffness (kN·m
–1
)

172 Morin, Dalleau, et al.
Measuring Stiffness During Running 173
Figure 3 — Signicant reference-model linear regressions (p < 0.01) obtained during
overground running for vertical stiffness (upper panel) and leg stiffness (lower panel)
compared to the identity line. Each dot represents the value of stiffness of the step
analyzed for each participant at the corresponding velocity.
Modeled leg stiffness (kN·m
–1
)
Modeled vertical stiffness (kN·m
–1
)
Reference vertical stiffness (kN·m
–1
)
Reference leg stiffness (kN·m
–1
)

174 Morin, Dalleau, et al.
Measuring Stiffness During Running 175
Figure 4 — Changes in the modeled and reference vertical and leg stiffness with run-
ning velocity during treadmill running (upper panel) and overground running (lower
panel). *Signicant (p < 0.05) difference in reference-model bias obtained by ANOVA
and Scheffé post hoc test.
Velocity (m·s
–1
)
Stiffness (kN·m
–1
)
Velocity (m·s
–1
)
Stiffness (kN·m
–1
)

174 Morin, Dalleau, et al.
Measuring Stiffness During Running 175
ference (p < 0.05) between the bias in leg stiffness assessment at 3.33 m·s
–1
and
that obtained at the four highest velocities. For overground running, a signicant
difference was observed on the vertical stiffness bias (p < 0.05) obtained at 6 m·s
–1
and at the maximal velocity (Figure 4).
The leg length calculation using Winter’s equation led to a mean error bias of
1.94 ± 1.51%, and the linear regression between the measured leg length and that
obtained using Winter’s model was signicant (R² = 0.89; p < 0.01).
Discussion
The values of the different mechanical parameters obtained in this study—verti-
cal ground reaction forces, displacements, and stiffness—are in line with those of
studies about the spring-mass model in running (Arampatzis et al., 1999; Blickhan,
1989; Farley & Gonzales, 1996; Ferris et al., 1998; He et al., 1991; McMahon et al.,
1987; McMahon & Cheng, 1990). It should be noted that we obtained slight differ-
ences in displacements and vertical and leg stiffness between our two experimental
protocols (treadmill and overground running), due to the different running abilities
of the two populations in these two parts of the study.
Indeed, highly trained middle-distance runners who performed the force plat-
form protocol showed lower CM displacement and leg length change and higher
vertical and leg stiffness than the nonspecialists who volunteered to perform the
treadmill protocol (e.g., leg stiffness: 10.37 ± 2.34 kN·m
–1
for the treadmill protocol
group vs. 13.28 ± 1.85 kN·m
–1
for overground runners). This is in line with the study
of Mero and Komi (1986), who observed higher “apparent spring constants” of the
support leg and lower CM displacement during running in highly trained sprinters
compared to less-skilled ones (Luhtanen & Komi, 1980) or marathon runners (Ito,
Komi, Sjödin, Bosco, & Karlsson, 1983).
The stiffness vs. velocity patterns (Figure 4), i.e., the increase in vertical
stiffness and the constancy of leg stiffness with increasing velocity either on the
treadmill or in overground conditions, has been previously noted in the literature
(He et al., 1991; McMahon et al., 1987).
Validity of the proposed sine-wave method: The aim of this study was to
provide a calculation method based on a few simple parameters allowing vertical
and leg stiffness to be assessed during treadmill and overground running without
a force platform. The low bias obtained between force platform and model values
(from 0.12% to 6%), and the high determination coefcients (from 0.89 to 0.98, p
< 0.01), demonstrate the validity of this calculation method during both treadmill
and overground running. Furthermore, the validity of this method was tested for a
wide range of velocities (i.e., from 3.33 to 6.67 m·s
–1
on the treadmill and 4 m·s
–1
to maximal velocity on the force platform) with runners of different ability levels,
from nonspecialists to highly trained middle-distance runners, all giving acceptable
results (Figures 2 and 3). This may allow us to use the proposed method during
submaximal to maximal velocity running, with either nonspecialists or elite athletes,
in eld conditions. It should be also noted that the mechanical parameter input
to the model (maximal force, CM displacement, leg length change) also showed
acceptable reference-model bias (0.67 to 6.93%, Tables 1 and 2).
Basis postulate and assumptions: The basis postulate of this study was that
the F(t) curve can be tted by means of a simple sine function. The validity of this
sine modeling, recently used in a study aimed at validating a stiffness measuring

176 Morin, Dalleau, et al.
Measuring Stiffness During Running 177
method during hopping (Dalleau, Belli, Viale, Lacour, & Bourdin, 2004), was
checked by comparing areas under F(t) curves for all steps analyzed on the treadmill
and on the force plate, and we obtained low error bias values (5.33% and 2.93%,
respectively). On the treadmill the accuracy of this sine modeling was improved
at faster velocities, the bias ranging from 11.7% at 3.33 m·s
–1
to only 1.7% at 6.67
m·s
–1
. This was probably due to the alteration of the passive impact peak in the
vertical ground reaction force at low velocity and to an F(t) curve more closely
approximating a sine curve at faster velocities. This could explain the signicant
effect of velocity on reference-model bias in leg stiffness observed between that
obtained at 3.33 m·s
–1
and that at the four highest velocities (Figure 4, upper panel),
the accuracy of the basic sine-wave postulate increasing at faster velocities on the
treadmill. To the contrary, this velocity effect on the bias was not observed during
overground running.
The bias in F(t) curve tting by the sine function was rather constant, and the
previously mentioned velocity effect on reference-model bias was only signicant (p
= 0.043) in vertical stiffness between that obtained at 6 m·s
–1
and that at the maximal
velocity (Figure 4, lower panel). This smaller effect of velocity may be explained
by the fact that F(t) curves of elite level “forefoot striking” runners showed smaller
or no passive impact peaks, whatever the velocity, with shorter contact times and
F(t) plot shapes closer to a sine-wave (Nilsson & Thorstensson, 1989).
Sensitivity analysis: In order to further determine the inuence of the dif-
ferent mechanical parameters constituting the presented model on the vertical and
leg stiffness calculated, we performed a sensivity analysis (Figure 5). It was then
possible to observe the relative inuence of each mechanical parameter on the
calculated stiffness values. It should be noted that the most sensitive parameter, for
both vertical and leg stiffness estimates, is the contact time. Its variation inuences
the stiffness in a proportion of about 1 to 2, i.e., for instance a 10% reduction in
contact time leads, according to this model, to a 20% increase in vertical stiffness
or even 25% in leg stiffness. All the other parameters have a 1 : 1 weight or even
less, especially anthropometrical parameters of body mass and leg length.
Therefore, although the legs are not stiff either at landing or takeoff, resulting
in a slight overestimation of leg length (Arampatzis et al., 1999), the inuence of
such a phenomenon on the accuracy of the stiffness calculations was not important.
In addition, the results of the present study showed that the leg length value used
in this sine-wave model can be obtained using anthropometric equations, on the
basis of the individual’s height, without signicant changes in the stiffness values
obtained. Furthermore, a recent study did not nd any signicant variation in the
stiffness values obtained with the proposed method, using an estimated value of leg
length from the participant’s height according to Winter’s model (Winter, 1979),
showing that this parameter is not crucial for improving the accuracy of the sine-
wave model (Jeannin, 2003).
This model also assumes a constant point-of-force application on the ground
during the entire contact phase. However, the location of the point-of-force applica-
tion was shown to move forward by about 0.16 m (Lee & Farley, 1998), and this
constitutes another limitation of the simplest spring-mass model.
Finally, it is worth noting that the limits of the proposed sine-wave method are
also those inherent in the use of any theoretical spring-mass model, i.e., the human
lower limb is not a true linear spring in a physical sense. This point has been widely
discussed elsewhere (Blickhan, 1989; Farley & Gonzales, 1996; Ferris, Louie, &

176 Morin, Dalleau, et al.
Measuring Stiffness During Running 177
Figure 5 — Sensitivity analysis: Variations of the different mechanical parameters
constituting the equation’s model plotted against the corresponding leg stiffness (upper
panel) and vertical stiffness (lower panel) variations.
Farley, 1998; He et al., 1991; McMahon & Cheng, 1990). Even if one prefers the
term “quasi-stiffness” (Latash & Zatsiorsky, 1993), the simplicity and predictive
power of the spring-mass model in studies aimed at understanding and analyzing
human running performance outweigh the limitations.
The equations based on a force-time curve sine modeling allow these calcu-
lations from simple mechanical parameters of ight and contact times, leg length,
body mass, and running velocity. The reference-model biases on calculated stiff-
ness were acceptable for a wide range of velocities, different levels of runners, and
Vertical stiffness variation (%)
Leg stiffness variation (%)
Parameters variation (%)
Parameters variation (%)

178 Morin, Dalleau, et al.
Measuring Stiffness During Running 179
either on a treadmill dynamometer or overground. To conclude, this new method
may allow researchers to measure and to understand the inuence of leg stiffness
on performance in eld running conditions and also to improve track & eld coach-
ing and training.
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Measuring Stiffness During Running 179
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APPENDIX
Modeled vertical stiffness computations
The modeled vertical stiffness k
vert
was calculated as:
k
vert
= F
max
· ∆y
c
–1
(5)
with F
max
the modeled maximal force and ∆y
c
the modeled vertical peak displace-
ment of the center of mass (CM) during contact.
F
max
computations
The pattern of vertical ground reaction force over time was modeled using the
following equation:
F(t) = F
max
· sin(— · t) (4)
with F
max
the peak force value and t
c
the contact time.
From this equation, the momentum change during contact is:
∫
tc
[F(t) – mg] · dt = m∆u = mgt
f
(A10)
with m the participant’s body mass, u the vertical velocity, g the gravity accelera-
tion, and t
f
the mean ight time (mean of ight times before and after contact).
Substituting Eq. A4 in Eq. A10 gives:
∫
tc
[F
max
· sin(— · t) – mg] · dt = m∆u = mgt
f
(A11)
[–F
max
— cos (— · t)] –mgt
c
= mgt
f
(A12)
2 F
max
— mg (t
f
+ t
c
) (A13)
The modeled peak force during contact is then obtained as:
F
max
= mg — (— + 1) (6)
t0
t0
π
t
c
π
t
c
t
c
π
π
t
c
t
c
t0
t
c
π
π
2
t
f
t
c
^
^ ^
^
^
^
^
^

180 Morin, Dalleau, et al.
∆y
c
computations
Based on Eq. A4 and according to the fundamental law of dynamics, vertical velocity
is obtained by integrating the vertical acceleration of the CM during contact:
u(t) = ∫
tc
[—– – g] · dt + u(t0) (A14)
u(t0) being the downward vertical velocity at the beginning of contact.
u(t) = ∫
tc
[—–– sin (— · t) – g] · dt + u(t0) (A15)
u(t) = [– —– — cos (— · t)] – gt + u(t0) (A16)
Knowing that the vertical velocity is nil at the time of half-contact:
u(—) = —– — – g— + u(t0) = 0 (A17)
—– — + u(t0) = g— (A18)
The nal expression of vertical velocity during contact being:
u(t) = – —– — cos (— · t) – gt + g— (A19)
Integrating this vertical velocity over time, vertical displacement can be obtained:
y(t) = ∫
tc
u(t) · dt + y(t0) (A20)
with y(t
0
) the vertical position of the CM at the beginning of contact.
Assuming y(t
0
) = 0 and substituting Eq. A19 in Eq. A20:
y(t) = ∫
tc
[– —– — cos (— · t) – gt + g—] · dt (A21)
y(t) = [– —– — sin (— · t) – — gt
2
]
tc
+ g—t (A22)
y(t) = – —- — sin (— · t) – — gt
2
+ g—t (A23)
The total CM displacement at the time of half contact, i.e., for t = t
c
/ 2 is then:
∆y
c
= – —- — + g— (7)
Modeled leg stiffness computations
The modeled leg stiffness k
leg
was calculated as:
k
leg
= F
max
· ∆L
–1
(8)
with ∆L the modeled leg peak displacement during contact.
∆L computations
∆L was obtained on the basis of the spring-mass model’s typical equations and
assumptions (Farley & Gonzales, 1996; McMahon & Cheng, 1990):
∆L = L – √ L
2
– (—–)
2
+ ∆y
c
(9)
L being the participant’s leg length and v the constant horizontal velocity.
F(t)
m
F
max
m
F
max
m
π
t
c
t
c
π
π
t
c
t0
t
c
2
F
max
m
t
c
π
t
c
2
F
max
m
t
c
π
t
c
2
F
max
m
t
c
π
π
t
c
t
c
2
t0
t0
F
max
m
t
c
m
π
t
c
t
c
2
F
max
m
t
c
2
π
2
π
t
c
1
2
t
c
2
F
max
m
t
c
2
π
2
π
t
c
1
2
t
c
2
F
max
m
t
c
2
π
2
t
c
2
8
vt
c
2
t0
^
^
^
^
^
^
^
^
^
^
^
^
^ ^ ^
^
^
^
^
^
t
c
t0
t0