© Journal of Sports Science and Medicine ( 2003) 2, 139- 143
http:// www. jssm. org
Research article
A NEW APPROACH TO MODELING VERTICAL STIFFNESS
IN HEEL- TOE DISTANCE RUNNERS
Iain Hunter
Department of Physical Education, Brigham Young University, Provo, Utah, USA
Received: 07 June 2003 / Accepted: 10 October 2003 / Published ( online): 01 December 2003
ABSTRACT
Various models have been used to describe distance running technique. Among these, the mass- spring
model is fairly simple to use and apply, but when employed as a model, does not predict vertical force
accurately especially when a heel strike is exhibited. The purpose of this article is to demonstrate how the
mass- spring model can be modified to provide a simple, yet accurate prediction of ground reaction forces
in distance running. Sixteen subjects ran on a force instrumented treadmill. Vertical forces during
running at a self- selected pace were collected at 500 Hz. Vertical stiffness was calculated using the
conventional mass- spring model with a constant stiffness and then a high- low method where stiffness
was varied from a high to low value during the heel strike. Fishers z- test was used to compare
correlations between predicted and measured ground reaction forces for each method of calculating
stiffness. The high- low method of calculating stiffness provided a better fit of predicted to measured
ground reaction forces than the constant stiffness method ( p < 0.01). The high- low method of calculating
stiffness avoids the difficulties of applying multiple masses, springs, or dampers while simply, yet
accurately matching predicted to measured ground reaction forces.
KEY WORDS: Leg stiffness, modeling, leg spring
INTRODUCTION
The mass- spring model provides much useful
information when evaluating distance running
technique. The model describes the body as a mass
centered above a spring that is connected to the
ground during the stance phase of running ( Figure
1). It has been used to investigate energy cost of
running, aerobic demand, speed of running, stride
frequency, and how technique changes with various
surfaces ( McMahon and Cheng, 1990; Farley and
Gonzalez, 1996; Heise and Martin, 1998; Dalleau et
al., 1998; Arampatzis et al., 1999; Kerdok et al.,
2002).
However, there are limitations dealing with
the application of stiffness. When modeling a cycle
of running with a mass- spring system, prediction of
force versus time curves using a constant stiffness of
the spring are clearly imperfect, especially when the
runner exhibits a heel strike ( Figure 2). The mass-spring
model assumes a linear spring; however, in
running the body does not act linearly. Prior to heel
strike, runners’ bodies react more stiffly to the
ground than after heel strike.
Figure 1. The traditional mass- spring model. The
mass represents the center of mass of the body. The
stiffness of the spring is determined by how the body
reacts to the ground during the stance of running.
Variable stiffness 140
Figure 2. Force curve from one step of one subject demonstrating measured and modeled vertical ground
reaction forces using constant stiffness based upon maximum center of mass vertical displacement divided
by vertical force at the time of maximum vertical displacement.
Adding to the difficulty of matching predicted
versus measured ground reaction forces is the fact
that much variation exists from person to person,
with various speeds of running, stride rates, and
running surfaces. Other models have attempted to
more accurately reproduce the running motion ( Nigg
and Liu, 1999; Derrick et al., 2000). However, these
models are quite complex requiring a number of
variables to be defined and calculated. The large
quantity of variables in these models makes
statistical correlations with other aspects of running
difficult to investigate.
A modification to the traditional mass- spring
model was tested in this study to see whether it
could better match predicted ground reaction forces
to measured ground reaction forces. Since the body
does not react as a linear spring during stance,
application of a model with varying stiffness may be
more appropriate. The initial impact during stance
exhibits a relatively stiff reaction to the ground,
followed by a more pliant reaction following heel
strike. The model investigated in this study applies
an initially high stiffness which drops to a lower
stiffness, which it maintains following heel strike.
This article will show how applying a variable
stiffness to the region around heel strike provides a
much better fit of predicted to measured ground
reaction forces. This will allow for a mass- spring
model that will accurately model ground reaction
forces using only two variables, high and low
stiffnesses.
METHODS
Subjects all signed an informed consent approved by
the university’s internal review board. Nine men and
seven women ( age = 28 ± 6 yrs, height = 1.76 ± 0.10
m, mass = 70.1 ± 10.2 kg) ran at a self- selected pace
on a force instrumented treadmill for 10 minutes
allowing for sufficient warm- up as technique may
change slightly during the first few minutes of
running ( Candau et al., 1998). During the ninth
minute, vertical force data were collected at 500 Hz.
These data were smoothed using a Butterworth filter
set at 50 Hz. Average vertical stiffness was taken
from ten consecutive right steps.
Constant stiffness
Vertical stiffness can be calculated by solving for k
in the equation: F= ks.
Where F represents the ground reaction force
after accounting for body weight, k represents the
vertical stiffness, and s represents the vertical
displacement of the center of mass. In this study, we
divided vertical force by body weight to obtain
center of mass accelerations. A double- integration of
acceleration provided changes in center of mass
position. Assuming the person lands at the same
141 Hunter
Figure 3. Comparison of measured and modeled vertical ground reaction forces using a varying stiffness.
The varying stiffness, shown with the dark curve, begins high finishing low with a smooth curve connecting
the two with half a cosine wave.
height for each step, the initial vertical velocity
needed for the integration can be easily obtained
when consecutive steps are recorded. The stiffness is
then found by dividing peak vertical force by
vertical displacement of the center of mass during
stance ( Farley and Gonzalez, 1996).
High- low stiffness
Vertical stiffness is typically measured using a
constant k. However, this study applies a changing
stiffness in order to more accurately model what
occurs during stance. Initially, the body reacts
relatively stiffly to the ground. Following heel strike,
the body reacts more pliant. Four characteristics
were used to calculate stiffness for the high- low
method: initial or high stiffness, final or low
stiffness, and two points in time representing a
transition from the high to low stiffnesses. The
smooth transition from high to low stiffness was
created using half a cosine wave ( Figure 3). High
and low stiffnesses were determined by an optimal
fit of measured to predicted force using the initial
velocity and mass. The transition from high to low
stiffness begins at the peak of heel strike and ends at
the minimum of the valley immediately after heel
strike ( Figure 3). Outside of the transition, the
modeled stiffness remains constant.
Vertical forces throughout stance can be
predicted using the initial velocity, calculated
stiffness, and body weight. Correlations were made
between measured vertical force and predicted
vertical force for the constant and high- low stiffness
methods. Differences between correlations using the
two methods were investigated using Fisher’s z- test
for comparing correlations ( alpha = 0.05).
RESULTS
Fisher’s z- test for comparing correlations of
measured to predicted vertical force showed a
significant difference between the constant stiffness
method and the high- low method ( p < 0.01) ( Table
1). Visually, the high- low method was able to
closely match the heel strike, which the constant
stiffness method was unable to do at all ( Figure 3).
The high- low method also visually matched the
difference in slopes of force versus time up to heel
strike and following the active peak much better
than the constant stiffness method.
Table 1. Average correlations and stiffnesses with standard deviations for each stiffness calculation method.
Constant Stiffness High- low Stiffness
r Stiffness
( kN · m)
r High Stiffness
( kN · m)
Low Stiffness
( kN · m)
Average ( SD) .948 (. 026) 28.7 ( 5.0) .994 (. 004) 75.3 ( 15.2) 31.2 ( 4.1)
Variable stiffness 142
Figure 4. Force versus displacement curve showing hysteresis and heel strike.
DISCUSSION
For the mass- spring model to be useful in describing
running technique, the predicted force based upon
the model should match the measured force as
closely as possible. The heel strike and hysteresis of
a force versus center of mass displacement curve
shown in figure 4 are a major part of the problem
with the traditional mass- spring model ( Dutto and
Smith, 2002). The high- low stiffness mass- spring
model provides more accurate predictions of vertical
ground reaction forces than a constant stiffness
model. The varying stiffness even closely matches
the heel strike. This model provides a simple, but
relatively accurate representation of the body during
distance running.
The high- low stiffness model requires two
variables, creating a difficulty in making correlations
between vertical stiffness and other characteristics of
running such as stride length and rate, running
surface, footwear, and anthropometric variables.
However, other alternatives to the complete high-low
stiffness model are possible. The initial stiffness
in this model represents the high value of stiffness
immediately after ground contact. The final stiffness
represents the stiffness following heel strike. Using
these two characteristics individually may be useful
in determining correlations with other running
characteristics without using the entire model.
Vertical stiffness varies with running surface
( Ferris and Farley, 1997; Ferris et al., 1998; Farley
et al., 1998; Ferris et al., 1999). The variability
added by surface will likely add to inaccuracies of
the traditional mass- spring model. Vertical stiffness
also changes with fatigue, speed, and stride
frequency ( McMahon and Cheng, 1990; Farley et
al., 1991; Farley and Gonzalez, 1996; Dutto and
Smith, 2002). Since the high- low model adjusts for
variations in heel- strike and the hysteresis of the
force versus displacement curve, it should provide a
better fit of predicted versus measured ground
reaction forces for a variety of situations.
CONCLUSION
The high- low stiffness method of modeling the body
in distance running matches predicted ground
reaction forces to measured ground reaction forces
much better than the constant stiffness method.
Since the high- low method produces a better fit, it
may be more useful in correlating characteristics of
running with vertical stiffness. One disadvantage to
implementing the high- low method is that two
variables are required to describe stiffness during the
stance phase of running. However, there may be
useful application to using just one of these variables
for certain investigations.
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AUTHOR BIOGRAPHY
Iain HUNTER
Employment
Ass. Prof. at Brigham Young
University, USA
Degree
PhD
Research interest
Distance running economy and
technique of track and field
events.
E- mail: iain_ hunter@ byu. edu Iain Hunter, PhD
Brigham Young University, 120D Richards Building,
Provo, UT 84602, USA