2935
The Journal of Experimental Biology 201, 2935–2944 (1998)
Printed in Great Britain © The Company of Biologists Limited 1998
JEB1517
DETERMINANTS OF THE CENTER OF MASS TRAJECTORY IN HUMAN WALKING
AND RUNNING
CYNTHIA R. LEE AND CLAIRE T. FARLEY*
Locomotion Laboratory, Department of Integrative Biology, 3060 Valley Life Sciences Building, University of
California, Berkeley, CA 94720-3140, USA
*Author for correspondence (e-mail: [email protected])
Accepted 13 August; published on WWW 8 October 1998
Summary
Walking is often modeled as an inverted pendulum
than for running (0.123 m) at speeds near the gait transition
system in which the center of mass vaults over the rigid
speed. In spite of this relatively small difference, the center
stance limb. Running is modeled as a simple spring-mass
of mass moved upwards by 0.031 m during the first half of
the stance phase during walking and moved downwards by
system in which the center of mass bounces along on the
0.073 m during the first half of the stance phase during
compliant stance limb. In these models, differences in
running. The most important reason for this difference was
stance-limb behavior lead to nearly opposite patterns of
vertical movements of the center of mass in the two gaits.
that the stance limb swept through a larger angle during
Our goal was to quantify the importance of stance-limb
walking (30.4 °) than during running (19.2 °). We conclude
that stance-limb touchdown angle and virtual stance-limb
behavior and other factors in determining the trajectory of
compression both play important roles in determining the
the center of mass during walking and running. We
trajectory of the center of mass and whether a gait is a walk
collected kinematic and force platform data during human
or a run.
walking and running. Virtual stance-limb compression (i.e.
reduction in the distance between the point of foot–ground
Key words: locomotion, biomechanics, spring-mass model, inverted
contact and the center of mass during the first half of the
pendulum model, gait, walking, running, human.
stance phase) was only 26 % lower for walking (0.091 m)
Introduction
Humans walk at low speeds and run at high speeds. The
transition from walking to running is obvious because the two
gaits are distinctly different from each other. As a human
switches from a walking gait to a running gait, the kinematics
and kinetics of locomotion change abruptly. For example, when
a person switches from a walk to a run, the time of foot–ground
contact decreases by 35 % and the peak ground reaction force
increases by approximately 50 % (Nilsson et al. 1985; Hreljac,
1993; Minetti et al. 1994). In addition, the trajectory of the center
of mass is completely different for walking and running gaits in
humans and other animals (Cavagna et al. 1976, 1977; Heglund
et al. 1982; Blickhan and Full, 1987; Farley and Ko, 1997).
During walking, the center of mass reaches its highest point at
the middle of the stance phase. In contrast, during running, it
reaches its lowest point at the middle of the stance phase. This
difference in the pattern of vertical movement of the center of
mass has been proposed as the defining difference between a
walking gait and a running gait (McMahon et al. 1987).
This distinction between running and walking is reflected in
the simplest mechanical models used to describe the gaits
(Farley and Ferris, 1998). Running animals are often modeled
as simple spring-mass systems (Blickhan, 1989; McMahon and
Cheng, 1990; Blickhan and Full, 1993b; Farley et al. 1993;
Alexander, 1995; Farley and Gonzalez, 1996). The springmass system consists of a point mass equal to the runner’s body
mass and a compliant spring that connects the mass to the point
of ground contact (‘leg spring’). As the foot hits the ground
during running, the leg spring compresses as a result of joint
flexion, and the mass moves downwards. At the middle of the
stance phase, the leg is maximally compressed, and the mass
reaches its lowest point. In contrast, walking is often modeled
as an inverted pendulum system (Alexander, 1995). This model
consists of a point mass equal to the walker’s body mass and
a rigid strut that connects the mass to the point of ground
contact. During the stance phase of walking, the mass vaults
over the rigid strut, reaching its highest point at the middle of
the stance phase (Cavagna et al. 1976, 1977; Margaria, 1976).
The inverted pendulum model accurately predicts the general
pattern of mechanical energy fluctuations of the body during
walking (Cavagna et al. 1963, 1976, 1977; Margaria, 1976). In
moderate-speed walking, the kinetic energy and gravitational
potential energy of the center of mass are nearly 180 ° out of
phase. Between touchdown and mid-stance, the forward velocity
of the center of mass decreases as the trunk arcs upwards over
the stance foot. In this phase, kinetic energy is converted to
gravitational potential energy. During the second half of the
2936 C. R. LEE AND C. T. FARLEY
stance phase, the center of mass moves downwards as the
forward velocity of the center of mass increases. In this phase,
gravitational potential energy is converted back into kinetic
energy. This exchange of kinetic energy and gravitational
potential energy during walking is similar to the energy
exchange of an oscillating pendulum (Cavagna et al. 1976, 1977;
Margaria, 1976). An ideal inverted pendulum system has perfect
exchange between gravitational potential energy and kinetic
energy. In a human walker, energy exchange by the inverted
pendulum mechanism reduces the mechanical work required
from the muscular system by a maximum of 70 % (Cavagna et
al. 1976). One reason why a human walker does not achieve
100 % energy exchange is that the fluctuations in gravitational
potential energy and kinetic energy are not matched in
magnitude. For example, in high-speed walking, the fluctuations
in gravitational potential energy are smaller than the fluctuations
in kinetic energy (Cavagna et al. 1976).
The inverted pendulum model of walking emphasizes the
advantages of a stiff-legged gait in which the center of mass
arcs over the stance limb (a ‘compass gait’). An alternative
view of walking emphasizes the advantages of minimizing the
vertical movements of the center of mass (Inman, 1966). Inman
and colleagues carefully examined the kinematics of walking
and identified several mechanisms that are involved in
flattening the trajectory of the trunk (Saunders et al. 1953;
Inman, 1966; Inman et al. 1994). For example, the knee and
ankle flex and extend in the sagittal plane during the stance
phase, and the pelvis ‘lists’ or rotates in the frontal plane. They
pointed out that these movements appear to reduce the vertical
movements of the trunk. These observations strongly suggest
that the distance between the point of foot–ground contact and
the center of mass (‘virtual stance-limb length’) does not
remain constant during walking as it does in the inverted
pendulum model. Rather, the virtual stance limb shortens (i.e.
‘compresses’) and lengthens during the course of the stance
phase. Although it is well established that compression of the
stance limb plays an important role in determining the
dynamics of running (McMahon and Cheng, 1990; He et al.
1991; Farley et al. 1993; Farley and Gonzalez, 1996), little is
known about the magnitude and role of virtual stance-limb
compression during walking. Intuitively, it seems likely that
the virtual stance limb compresses more at higher speeds
during walking and that there is a gradual transition between
walking and running in terms of stance-limb behavior.
Inman and his colleagues compared human walking to a
compass gait in which the rigid stance limb contacts the ground
at a single point that remains fixed throughout the stance phase.
In an actual human walker, the heel of the foot hits the ground
first, and the toe of the foot leaves the ground last. In the
inverted pendulum model, this can be modeled as forward
translation of the point of ground contact during the stance
phase (Fig. 1A). If there is forward translation of the point of
ground contact, then the virtual stance limb will sweep through
a smaller angle because the movement of the point of ground
contact accounts for part of the forward movement of the
center of mass during the stance phase. Because of the smaller
angle swept by the stance limb, the vertical trajectory of the
mass is flattened if the point of ground contact is allowed to
translate forward during the stance phase (Fig. 1A). Thus, it is
possible that the trajectory of the center of mass is flattened in
a walking human because the point of ground contact rolls
forward from the heel to the toe.
The goal of our study was to examine the effects of
translation of the point of foot–ground contact and virtual
stance-limb compression on the vertical movements of the
center of mass in human walking. We began by comparing the
predicted trajectories of the center of mass for a model with a
rigid leg and a fixed point of foot–ground contact (‘compass
gait’) and for a model with a rigid leg and forward translation
of the point of ground contact (Fig. 1A). From this comparison,
we quantified the reduction in the vertical displacement of the
center of mass that occurred as a result of the movement of the
point of force application (∆PFA) under the foot in our
subjects. Subsequently, we quantified the magnitude of virtual
stance limb compression and its effect on the vertical
displacement of the center of mass (Fig. 1B). We defined
‘virtual stance-limb compression’ as a reduction in the distance
between the point of ground contact of the stance limb and the
center of mass. Throughout the study, we compared these
factors for walking and running to gain insight into the
biomechanical determinants of the extremely different
trajectories of the center of mass for the two gaits.
Materials and methods
Measurements
Five healthy human subjects (three females and two males)
agreed to participate in this study. The mean body mass of the
subjects was 56.3±9.9 kg (mean ± S.D.), mean leg length (the
distance from the ground to the greater trochanter during quiet
standing, L0) was 0.88±0.03 m, and the mean age was 23±2.1
years. Approval was obtained from the university committee
for the protection of human subjects and informed consent was
given by all subjects.
Subjects walked at five speeds (0.5, 1.0, 1.5, 2.0 and
2.5 m s−1) and ran at six speeds (1.5, 2.0, 2.5, 3.0, 4.0 and
5.0 m s−1) along a runway that had force platforms built into it.
The speed range corresponded to Froude numbers [u(gL0)−0.5,
where u is forward velocity and g is the gravitational
acceleration] of 0.17–0.85 for walking and 0.51–1.70 for
running. The only instruction given to each subject was to walk
or run normally. We measured the mean speed over the 3 m
section containing the force platforms using infrared photocells
placed at the beginning and end of the section. Subjects were
allowed to start as far back as necessary to accelerate to a
constant speed before the 3 m section and were given plenty of
room after the 3 m section to decelerate safely. We accepted
trials in which the measured speed was within 5 % of the
prescribed speed. We collected and analyzed three acceptable
trials from each subject under each condition. Thus, whenever
a mean value for all the subjects is stated for a given condition
(e.g. 2.5 m s−1 walk), it is the mean of 15 values.
Human walking and running 2937
Two AMTI force platforms (AMTI model LG6-4-1, Newton,
MA, USA), placed in series, were used to measure the vertical
and horizontal (i.e. fore–aft) components of the ground reaction
force. In addition, we used the force platform to measure the
moment about the medio-lateral axis of the force platform (Mx)
so that we could calculate the position of force application in the
fore–aft direction at each instant during the stride. Collection of
force data began when the subject passed the first photocell and
ended when the subject passed the second photocell. Signals
from the force platforms were sampled at 1000 Hz using
Labview Software and a computer A/D board (National
Instruments, Austin, TX, USA). The signals from the two force
platforms were summed using software before further analysis.
The horizontal ground reaction force signal was integrated with
respect to time in order to calculate the horizontal impulse and
the change in forward velocity while the subjects were on the
force platforms. We only accepted trials in which the net change
in forward velocity was less than 0.15 m s−1.
Video data were recorded in the sagittal plane at
200 fields s−1 (JC Labs, Mountain View, CA, USA). Video and
force platform data were synchronized using a circuit that
illuminated a light-emitting diode in the video field and
simultaneously sent a voltage signal to the A/D board when the
subject passed each photocell. Strips of reflective tape were
placed on the heel and toe of the subjects’ shoes to facilitate
identification of heel-strike and toe-off from the video
recordings. The time interval during which a foot was in
contact with the ground was referred to as its ‘stance phase’.
The period between successive heel-strikes (i.e. the time from
touchdown of one foot to touchdown of the contralateral foot)
was referred to as a ‘step’.
Calculation of the vertical displacement of the center of mass
in walking and running subjects
We calculated the vertical displacement of the center of
mass during the stance phase (∆y) for each trial of walking and
running. This calculation involved determining the vertical
acceleration of the center of mass from the vertical ground
reaction force data. Subsequently, we calculated the vertical
displacement of the center of mass by double integration of the
vertical acceleration over an integral number of steps as
described in detail elsewhere (Cavagna, 1975; Blickhan and
Full, 1993a). Sample data for the vertical displacement of the
center of mass as a function of time during the stance phases
of walking and running are shown in Fig. 2.
Calculation of the vertical displacement of the center of mass
for the inverted pendulum model with a fixed point of ground
contact
We used an inverted pendulum model to determine the
vertical movements of the center of mass that would have
occurred if the stance limb had behaved like a rigid strut with
a fixed point of ground contact (Fig. 1A). In the inverted
pendulum model, the virtual stance limb remained at a constant
length (L0) throughout the stance phase, and the rotation of the
virtual stance limb over the point of ground contact was
symmetrical during the first and second halves of the stance
phase (Fig. 1A). To predict the vertical displacement of the
center of mass that would have occurred if the virtual stance
limb had remained a constant length with a fixed point of
foot–ground contact, we began by calculating the angle that the
virtual stance limb would have had to sweep. The predicted
angle of the leg relative to the vertical (θcompass) for each
instant during the stance phase was calculated from:
θcompass(t) = sin−1[u(tc/2 − t)/L0] ,
(1)
where u is the forward velocity, t is time, tc is the stance
duration and L0 is the subject’s leg length (the distance from
the greater trochanter to the ground). This calculation was
performed for each instant from the beginning of stance (t=0)
to the end of stance (t=tc) for each trial. This calculation
assumed that the angle of the virtual stance limb changed
linearly with time during the stance phase. This assumption is
supported by the observation that the relationship between the
forward displacement of the center of mass (calculated by
double integration of the horizontal ground reaction force)
changed approximately linearly with time during the stance
phase. For all trials, linear regressions of forward displacement
versus time yielded r2 values greater than 0.98.
The θcompass values were used only for the theoretical
prediction of the vertical displacement of the center of mass
for a compass gait. Because the θcompass values do not account
for the movement of the point of foot–ground contact during
stance, we do not expect them to predict accurately the virtual
stance-limb angle during human locomotion. The θcompass
values were used in equation 2 to predict the vertical
displacement of the center of mass (relative to the beginning
of stance) for each instant during stance assuming a constant
leg length equal to L0 and a fixed point of ground contact:
∆ycompass = L0(cosθcompass − cosθ0,compass) .
(2)
The predicted angle of the leg at touchdown for a fixed point
of ground contact (θ0,compass) was calculated from equation 1
at t=0.
Calculation of the vertical displacement of the center of mass
for the inverted pendulum model with a translating point of
ground contact
The next step was to determine the predicted vertical
displacement of the center of mass for a constant-length virtual
stance limb but a forward-translating point of force application.
For this step, we began by calculating the angle swept by the
virtual stance limb while accounting for the forward translation
of the point of force application:
θPFAmodel(t) = sin−1{[u(tc/2 − t) − ∆PFA]/L0} .
(3)
This equation is very similar to equation 1, except that it
accounts for the forward translation of the point of foot–ground
contact. We used the forward displacement of the point of force
application (∆PFA) on the ground as an indicator of the
forward translation of the point of foot–ground contact. At
each instant from the beginning of the stance phase (t=0) to the
2938 C. R. LEE AND C. T. FARLEY
end of stance (t=tc), we used our actual ∆PFA data in equation
3, calculated from the force platform data. The forward
displacement of the point of force application (∆PFA) was
defined relative to the point of force application at the
beginning of the stance phase. Subsequently, we used the
θPFAmodel values to calculate the predicted vertical
displacement of the center of mass during the stance phase for
a virtual stance limb with a fixed length and a moving point of
force application (∆yPFAmodel):
∆yPFAmodel = L0(cosθPFAmodel − cosθ0,PFAmodel) .
A
utc/2
∆yPFAmodel,max
∆ycompass,max
θ0,compass
(4)
Calculation of virtual stance-limb compression
The actual trajectory of the center of mass for the subjects
differed from the prediction of the inverted pendulum model
with a translating point of ground contact because the length of
the virtual stance limb did not remain constant during stance in
each subject (Fig. 1B). Thus, the second major determinant of
the center of mass trajectory that we examined was virtual
stance-limb compression. We calculated the extent of virtual
stance-limb compression (∆L) for each instant of the stance
phase of walking and running. We began by using leg length
(L0) and the angle of the leg at touchdown (θ0,PFAmodel) to
calculate the horizontal (L0sinθ0,PFAmodel) and vertical
(L0cosθ0,PFAmodel) positions of the center of mass at the instant
that the foot hit the ground. After twice integrating the center of
mass acceleration, as described above, these values were used
as the integration constants to determine the instantaneous
horizontal and vertical positions of the center of mass throughout
the stance phase. The length of the virtual stance limb (L) at each
instant of the stance phase was then calculated from the
instantaneous position of the center of mass using the distance
formula. We calculated virtual stance-limb compression from
the difference between the leg length at heel-strike (L0) and the
leg length at each instant of the stance phase (L).
To compare the magnitude of virtual stance-limb
compression among the walking and running trials, we used
the virtual stance-limb compression value at the instant when
the center of mass reached its highest point for walking and its
lowest point for running (∆Lmidstance; Fig. 1B). The magnitude
of virtual stance-limb compression varied throughout the
B
∆yPFAmodel,max
∆Lmidstance
COM actual
path
∆y
al b
rtu lim
Vi nce
sta
The predicted angle of the virtual stance limb at touchdown for
the rigid leg model with a forward translating point of force
application (θ0,PFAmodel) was calculated from equation 3 at t=0.
As expected, θ0,PFAmodel was less than θ0,compass because of the
forward movement of the point of force application. The
θPFAmodel values should more closely approximate the virtual
stance-limb angle in a walking human than the θcompass values
because the θPFAmodel values account for the movement of the
point of force application under the stance foot.
We calculated the reduction in the vertical displacement of
the center of mass due to the movement of the point of force
application under the foot at each speed in each gait by taking
the difference between ∆ycompass and ∆yPFAmodel at the instant
at mid-stance when the center of mass reached its highest
position (Fig. 1A; see Fig. 4A).
∆PFA
∆PFA
Fig. 1. (A) A model with a constant-length virtual stance limb and
forward translation of the point of ground contact (∆PFA, dashed
lines) has a smaller vertical displacement of its mass than a similar
model with a fixed point of ground contact (compass gait, solid
lines). In both models, the circle represents the center of mass of the
whole body, and the line connecting the center of mass to the ground
represents the virtual stance limb. The length of the virtual stance
limb (L0) is the same for both models and remains constant
throughout the stance phase. Note that the touchdown angle of the
stance limb is reduced when the point of ground contact translates
forward during stance. As a result, the vertical displacement of the
mass during the first half of stance is smaller for the model with
forward translation of the point of ground contact (∆yPFAmodel, max)
than for a compass gait (∆ycompass,max). The horizontal distance
traveled by the mass during the first half of stance is the same for
both models and is denoted by utc/2, where u is the forward velocity
and tc/2 is half the stance time. θ0,compass, touchdown angle of the
stance limb for the compass gait. (B) The actual path of the center of
mass (COM) of a walking subject (solid arc) differed from the path
of the center of mass for an inverted pendulum model with forward
translation of the point of ground contact (dashed arc) because of
compression of the virtual stance limb. The maximum vertical
displacement of the center of mass of a walking human (∆y) was
much smaller than the maximum vertical displacement of the mass in
the model (∆yPFAmodel,max) because the virtual stance limb
compressed in a walking human by a distance ∆Lmidstance between
the beginning of the stance phase and the instant when the center of
mass reached its highest position.
stance phase for both walking and running (Fig. 2). For
running, maximum virtual stance-limb compression occurred
at nearly exactly the same time as the center of mass reached
its lowest point (Fig. 2C,D). For walking, maximum virtual
stance-limb compression often occurred after the center of
mass reached its highest point. Thus, by calculating virtual
Human walking and running 2939
stance-limb compression at the instant that the center of mass
had reached its highest point, we underestimated the full extent
of virtual stance-limb compression for walking (Fig. 2A,B).
However, one of our primary goals was to understand the role
of virtual stance-limb compression in determining the
maximum vertical displacement of the center of mass, and this
technique was most appropriate for achieving this goal.
The actual vertical displacement of the center of mass between
touchdown and mid-stance was smaller than that predicted by a
rigid leg model with a fixed point of ground contact at all speeds
of walking and running (Fig. 3; P<0.0001). The center of mass
moved upwards (positive vertical displacement) during the first
half of stance during walking at all speeds. The maximum vertical
displacement increased from 0.013±0.001 m (mean ± S.E.M.) at
0.5 m s−1 to 0.031±0.007 m at 2.5 m s−1 (P=0.037). If the virtual
stance limb had behaved as a rigid strut with a fixed point of
ground contact, the vertical excursion of the center of mass would
have increased more steeply with speed, reaching a value of
0.191 m during walking at 2.5 m s−1 (Fig. 3A).
The first factor that substantially reduced the vertical
displacement of the center of mass was the heel-to-toe
movement of the point of force application under the stance
foot. The point of force application moved forward by an
average of 0.201±0.008 m (mean ± S.E.M.) between the
beginning and the end of the stance phase of walking for all
subjects. During the stance phase of running, the point of force
application moved forward by 0.157±0.006 m (mean ± S.E.M.).
The forward displacement of the point of force application
during the stance phase was independent of speed in both gaits
(P>0.92). The movement of the point of force application
substantially flattened the trajectory of the center of mass during
Results
The actual path of the center of mass during the stance phase
of walking differed substantially from that predicted by the
inverted pendulum model with a rigid leg and a fixed point of
ground contact (compass gait; Fig. 2A,B). For both the subjects
and the pendulum model, the center of mass reached its highest
vertical position at approximately the middle of the stance phase.
However, the trajectory of the center of mass was much flatter
for the subjects than for the rigid leg model with a fixed point of
ground contact. In the typical examples illustrated in Fig. 2A,B,
the net upward movement of the center of mass of the subject
was less than 0.035 m between the beginning and middle of
stance during walking at 1.5 m s−1 and 2.5 m s−1. In contrast, the
rigid leg model with a fixed point of ground contact predicted an
upward movement greater than 0.12 m for both walking speeds.
WALK (1.5 m s-1)
0.18
A
Displacement (m)
0.12
Compass gait
PFA translation
0.06
0.06
Actual
0
PFA translation
0
∆L
-0.06
Actual
-0.06
∆L
-0.12
-0.12
0
25
50
75
0
100
25
50
WALK (2.5 m s-1)
0.18
Displacement (m)
C
Compass gait
0.12
Fig. 2. Typical data for the vertical
displacement of the center of mass
of a human subject (denoted by
actual), the predicted vertical
displacement of the center of mass
for the inverted pendulum model
with a constant leg length (0.87 m)
and a fixed point of force
application on the ground (compass
gait),
the
predicted
vertical
displacement of the center of mass
for a constant leg length (0.87 m)
but a translating point of force
application (PFA translation), and
the magnitude of stance-limb
compression (∆L). In these typical
trials, the distance of PFA
translation was 0.196 m for walking
(A,B) and 0.155 m for running
(C,D). Data are shown for a single
stance phase for (A) walking at
1.5 m s−1, (B) walking at 2.5 m s−1,
(C) running at 2.5 m s−1 and (D)
running at 3.0 m s−1.
RUN (2.5 m s-1)
0.18
0.18
B
Compass gait
0.12
0.12
PFA translation
0.06
-0.06
RUN (3.0 m s-1)
D
Compass gait
PFA translation
0
∆L
Actual
-0.06
-0.12
∆L
-0.12
0
25
50
100
0.06
Actual
0
75
75
Stance time (%)
100
0
25
50
75
Stance time (%)
100
2940 C. R. LEE AND C. T. FARLEY
A
0.20
WALK
Compass gait
Rigid limb with PFA translation
0.10
Actual
Vertical displacement (m)
0
-0.10
B
0.20
RUN
Compass gait
0.10
Rigid limb with PFA translation
0
Actual
-0.10
0
1
2
3
4
Speed (m s-1)
5
6
Fig. 3. Vertical displacement ∆y of the center of mass for walking and
running. For both walking (A) and running (B), the actual vertical
displacement of the center of mass (actual) during stance was smaller
than the vertical displacement for an inverted pendulum model with a
rigid leg and a fixed point of force application (compass gait) or for a
model with a rigid limb and translation of the point of force
application on the ground (rigid limb with PFA translation). During
walking, the center of mass moved upwards during the first half of
stance (open circles, ∆y>0), and the vertical displacement increased at
faster speeds (∆y=0.012+0.010u, r2=0.682, P=0.037, N=75, where u
is speed). During running, the center of mass moved downward
during the first half of stance (open squares, ∆y<0), and the magnitude
of the vertical displacement decreased at faster speeds
(∆y=−0.080+0.004u, r2=0.444, P=0.034, N=90). Symbols represent
the means and errors bars represent the standard error of the means. In
cases where the error bars cannot be seen, they are contained within
the symbols. The lines are the linear least-squares regressions.
the stance phase. The rigid leg model incorporating forward
translation of the point of ground contact (PFA translation) had
a smaller stance-limb touchdown angle (Fig. 1A, equations 1,
3). As a result, it also had a smaller vertical displacement of the
center of mass (Figs 2, 3). During walking, the forward
translation of the point of force application reduced the vertical
displacement of the center of mass by 0.035±0.002 m at the
lowest speed and by 0.069±0.002 m (means ± S.E.M.) at the
highest speed (Fig. 4A). During running, the forward
translation of the point of force application reduced the vertical
displacement by 0.030±0.002 m at the lowest speed and by
0.050±0.006 m at the highest speed (Fig. 4A).
The second factor that reduced the vertical displacement of the
center of mass was that the virtual stance limb compressed
(Fig. 1B). During walking, the virtual stance limb compressed
substantially at the beginning of the stance phase (Fig. 2A,B).
Later in the stance phase, the virtual stance limb lengthened,
reaching its maximum length at the end of the stance phase. We
quantified virtual stance-limb compression at the instant when the
center of mass reached its highest point for comparison among
different speeds and gaits. Because the virtual stance limb was
vertically oriented at that instant in the stance phase, the
magnitude of virtual stance-limb compression was equivalent to
the reduction in vertical displacement of the center of mass due
to virtual stance-limb compression. The magnitude of virtual
stance compression at the instant when the center of mass reached
its highest point (∆Lmidstance) during walking increased with
speed from 0.021±0.005 m at the lowest speed to 0.091±0.005 m
at 2.5 m s−1 (means ± S.E.M.; P<0.0001) (Fig. 4B).
During running at all speeds, the virtual stance limb reached
maximum compression at approximately the same time as the
center of mass reached its lowest position near the middle of the
stance phase (Fig. 2C,D). During running, the virtual stancelimb compression at the instant that the center of mass reached
its lowest point (∆Lmidstance) increased from 0.106±0.012 at
1.5 m s−1 to 0.141±0.016 m at 5 m s−1 (means ± S.E.M.; Fig. 4B).
The virtual stance limb compressed less during moderatespeed walking than during running at any speed (Figs 2, 4B). At
a moderate walking speed (1.5 m s−1), virtual stance-limb
compression reduced the vertical displacement of the center of
mass by 0.050±0.005 m. At a moderate running speed (3.0 m s−1),
virtual stance-limb compression reduced the vertical
displacement by 0.134±0.009 m. Thus, when moderate speeds for
each gait were compared, the difference in virtual stance-limb
compression was a major reason for the nearly opposite patterns
of vertical movement of the center of mass during the stance
phase. However, it is important to realize that this comparison
was not for walking and running at the same absolute speed.
At the highest walking speed, the virtual stance limb
compressed by 26 % less during walking (0.091 m) than during
running (0.123 m; P<0.0001; Figs 2B,C, 4B). In spite of this
relatively small difference in virtual stance-limb compression,
the pattern of movement of the center of mass during the stance
phase was very different for the two gaits. The center of mass
reached its highest point at mid-stance in walking but reached
its lowest point at mid-stance in running (Figs 2B,C, 3). The
major reason for this difference in the movement of the center
of mass was the difference in the stance-limb touchdown angle.
At a given absolute speed, stance-limb touchdown angle was
greater for walking than for running (Fig. 5). For example, at
2.5 m s−1, the leg touchdown angle was 30.4±1.4 ° for walking
and 19.2±0.9 ° (means ± S.E.M.) for running.
Human walking and running 2941
In the absence of compression of the virtual stance limb, a
greater stance-limb touchdown angle will result in a larger
upward vertical displacement of the center of mass during the
A
0.10
Walk
Run
0.05
Reduction in vertical displacement
due to virtual stance-limb compression (m)
0
B
0.15
Run
0.10
0.05
0
Discussion
The simplest models for walking and running depict the stance
limb as behaving very differently in the two gaits. The inverted
pendulum model for walking includes a rigid strut connecting the
Walk
0
1
2
3
4
Speed (m s-1)
5
6
Fig. 4. (A) Reduction in the maximum vertical displacement of the
center of mass due to translation of the point of force application (PFA)
as a function of speed (u) for walking (circles;
reduction=0.028+0.017u, r2=0.882, P<0.0001, N=75) and running
(squares; reduction=0.020+0.0065u, r2=0.433, P=0.0001, N=90). Note
that these values are for the instant near the middle of the stance phase
when the center of mass reached its extreme vertical position. (B)
Reduction in the maximum vertical displacement of the center of mass
due to virtual stance-limb compression as a function of speed for
walking (circles) and running (squares). The reduction in the maximum
vertical displacement of the center of mass was equal to ∆Lmidstance (see
Fig. 1B) because the virtual stance limb was oriented vertically at the
time when the center of mass reached its extreme vertical position.
Virtual stance-limb compression increased at faster speeds in both
walking (∆Lmidstance=0.0004+0.036u, r2=0.843, P<0.0001, N=75) and
running (∆Lmidstance=0.096+0.010u, r2=0.297, P=0.0019, N=90). Note
that these values are for the instant near the middle of the stance phase
when the center of mass reached its extreme vertical position. In both A
and B, the symbols represent the mean values for all of the subjects,
and the error bars are the standard errors of the means. In some cases,
the error bars cannot be seen because they are smaller than the symbols.
The lines are the linear least-squares regressions.
Touchdown angle, θ0,PFAmodel (degrees)
Reduction in vertical displacement
due to PFA translation (m)
0.15
stance phase. Because the stance-limb touchdown angle was
larger during walking than during running, stance-limb rotation
alone would have caused a greater upward movement of the
center of mass during the first half of the stance phase during
walking than during running. Without virtual stance-limb
compression, the rotation of the leg would have caused a
0.122 m upward movement of the center of mass during the
first half of the stance phase of walking at 2.5 m s−1 (Figs 2B,
3A). However, the 0.091 m virtual stance-limb compression
(Fig. 4B) reduced the upward movement of the center of mass
to 0.031 m. In contrast, during running at the same speed,
stance-limb rotation would have caused only a 0.050 m upward
movement of the center of mass during the first half of stance
if there had been no virtual stance-limb compression (Figs 2C,
3B). However, because the virtual stance limb compressed by
0.123 m, the center of mass moved downwards by 0.073 m
during the first half of the stance phase. It is interesting to note
that, even if the virtual stance limb had only compressed by
the same amount during running as during walking at 2.5 m s−1
(0.091 m), the center of mass still would have moved
downwards by 0.041 m during the first half of the stance phase
during running. This observation makes it clear that the stance
touchdown angle is the primary determinant of the trajectory
of the center of mass at speeds near the gait transition speed.
Walk
30
Run
20
10
0
0
1
2
3
4
Speed (m s-1)
5
6
Fig. 5. Touchdown angle of the virtual stance limb (θ0,PFAmodel)
during walking (circles) and running (squares) as a function of speed
u. Leg touchdown angle increased at higher speeds in walking
(θ0,PFAmodel=12.86+7.51u, r2=0.837, P<0.0001, N=75) and running
(θ0,PFAmodel=13.05+2.59u, r2=0.652, P<0.0001, N=90). The symbols
represent the mean values for all of the subjects, and the error bars
are the standard errors of the means. The lines are the linear leastsquares regressions.
2942 C. R. LEE AND C. T. FARLEY
center of mass to the point of foot–ground contact (‘virtual stance
limb’). The spring-mass model for running includes a compliant
spring representing the virtual stance limb. A comparison of these
models suggests that the dramatic differences in the trajectories
of the center of mass between the two gaits are due primarily to
differences in the amount of compression of the virtual stance
limb. However, our findings show that the virtual stance limb
compresses substantially during walking. Indeed, during fast
walking, the virtual stance limb compresses by an amount that is
only 26 % less than during running at the same speed.
Our findings demonstrate that the difference in the trajectory
of the center of mass between walking and running depends as
much on virtual stance-limb touchdown angle as on virtual
stance-limb compression. This is most obvious when we
compare walking and running at the same speed. Our findings
show that the virtual stance limb compresses by only 26 % less
when a person walks than when a person runs at 2.5 m s−1. In
spite of this relatively small difference in virtual stance-limb
compression, the center of mass reaches its highest point at
mid-stance in walking and reaches its lowest point at midstance in running. The reason is that the stance limb touches
down at a substantially larger angle relative to vertical during
walking than during running. Thus, at speeds near the gait
transition speed, the difference in the trajectory of the center
of mass between walking and running is caused primarily by
a difference in virtual stance-limb touchdown angle.
Previous studies have demonstrated the importance of
stance-limb touchdown angle in determining the dynamics of
running (McMahon and Cheng, 1990; He et al. 1991; Farley et
al. 1993; Alexander, 1995). Running, hopping and trotting
animals adjust their spring-mass systems for different speeds by
increasing the angle swept by the stance limb while keeping leg
stiffness nearly constant (He et al. 1991; Farley et al. 1993).
This simple adjustment reduces the vertical displacement of the
center of mass and the ground contact time at higher speeds (He
et al. 1991; Farley et al. 1993). Similarly, running, hopping and
trotting robots that have spring-based legs change their forward
speed by adjusting the angle swept by the stance limb while
keeping leg stiffness the same (Raibert et al. 1993). These
examples demonstrate the important role of stance-limb
touchdown angle in determining the kinematics and kinetics of
bouncing gaits in animals and robots. Our findings show that
stance-limb touchdown angle is also an important determinant
of whether the center of mass reaches its highest or lowest point
at mid-stance and thus of whether the gait is a walk or a run.
In a walking or running human, the heel usually strikes the
ground first and the toe leaves the ground last. This translation
of the point of ground contact reduces the stance-limb
touchdown angle (Fig. 1A). The center of mass moves forward
by a given distance during a single stance phase (‘utc’ in Fig. 1).
When forward translation of the point of ground contact
contributes part of the forward movement of the center of mass
during stance, the stance limb touches down with a smaller angle
relative to the vertical and rotates through a smaller angle during
stance. A smaller stance-limb touchdown angle results in a flatter
trajectory of the center of mass. The translation of the point of
ground contact under the human foot can be modeled as
translation of the point of ground contact of the stance limb in
the inverted pendulum model (Fig. 1A). In a human walking at
a moderate speed, this movement of the point of ground contact
reduces the vertical movements of the center of mass by
approximately 40 %. Interestingly, in the passive dynamic
bipedal walkers developed by McGeer (1990), a long semicircular foot is included. In the passive dynamic walker, having
a semi-circular foot is more stable than having a point foot and
more economical than having a flat foot. Humans are plantigrade
but most other mammals are not. A combination of modeling
and experiments might provide insight into the effects of foot
size and shape on bipedal and quadrupedal walking in animals.
In the present study, virtual stance-limb compression refers to
the change in distance between the center of mass and the point
of ground contact as shown in Fig. 1. We did not quantify the
specific contributors to virtual stance-limb compression. It could
be affected by several factors, including (1) flexion of limb joints
in the sagittal plane, (2) three-dimensional movements of the
pelvis and trunk, and (3) movement of the center of mass within
the body. The contributions of individual joints to virtual stancelimb compression have been quantified for humans hopping in
place (on the spot) (Farley et al. 1998; Farley and Morgenroth,
1998). As in forward running, the stance limb behaves like a
spring during hopping in place. Thus, hopping in place serves
as a relatively simple experimental system for gaining insight
into the link between the behavior of the individual joints and
the virtual stance limb during running. During hopping in place,
the stiffness and compression of the virtual stance limb can be
nearly fully explained by the angular displacements of the
stance-limb joints in the sagittal plane. However, this type of
analysis has not been performed for walking.
Kinematic studies of human walking offer insight into the
contributors to virtual stance-limb compression. Both the knee
and the ankle undergo flexion during the first half of the stance
phase and are likely to contribute to leg compression (Saunders
et al. 1953; Inman et al. 1994; Sutherland et al. 1994). The
contribution of knee flexion to leg compression is reflected in
small fluctuations in the distance between the greater trochanter
and the lateral malleolus of the ankle during the stance phase
of walking (Borghese et al. 1996). These fluctuations are
substantially smaller than the magnitude of virtual stance-limb
compression measured in our study, probably because they
depend only on knee flexion. In contrast, the distance between
the greater trochanter and the point of force application on the
ground, a distance that depends on ankle angle as well as knee
angle, yields much larger length changes (Siegler et al. 1982).
These observations suggest that ankle flexion (i.e. dorsiflexion)
during the first half of the stance phase contributes to virtual
stance-limb compression. In addition, movement of the center
of mass within the body (approximately 0.006 m; Whittle,
1997) and the actions of the contralateral limb during the brief
period of double support (Saunders et al. 1953; Inman, 1966;
Inman et al. 1994) are likely to contribute to virtual stance
compression as defined in our study. It has long been believed
that pelvic list plays an important role in flattening the trajectory
Human walking and running 2943
of the center of mass (Saunders et al. 1953; Inman, 1966; Inman
et al. 1994). However, recent research has shown that pelvic list
has very little effect on the vertical movements of the trunk
during walking (Gard and Childress, 1997).
Virtual stance-limb compression during walking has
kinematic and energetic consequences. If the stance limb
behaved like a rigid strut, the vertical movements of the center
of mass during walking would be much larger and would
increase steeply at higher speeds (Fig. 3A). However, because
virtual stance-limb compression increases at higher speeds, the
vertical movements of the center of mass and the fluctuations
in gravitational potential energy are relatively small even at the
highest walking speeds (Fig. 3A). At high walking speeds, the
fluctuations in kinetic energy exceed the fluctuations in
gravitational potential energy (Cavagna et al. 1976). As a
result, only part of the kinetic energy that is lost by the center
of mass during the first half of the stance phase can be
converted to gravitational potential energy. Clearly, at high
walking speeds, virtual stance-limb compression reduces the
upward movement of the center of mass during the first half of
the stance phase and, thus, reduces the pendulum-like
exchange of kinetic energy and gravitational potential energy.
Although virtual stance-limb compression has the
disadvantage of reducing pendulum-like energy exchange, it
may have the advantage of enhancing elastic energy storage.
Because the stance limb compresses during the first half of the
stance phase, the loss in kinetic energy exceeds the gain in
gravitational potential energy of the center of mass, thus
inhibiting inverted pendulum energy exchange. However, some
of the kinetic energy that is lost during the first half of the stance
phase may be converted to elastic energy. Elastic energy could
be stored in muscles, tendons and ligaments as the ankle and
knee flex, contributing to leg compression, during the first half
of the stance phase. This elastic energy could be utilized to help
increase the velocity of the center of mass in the second half of
the stance phase. Although elastic energy storage is certainly
more important during running than during walking, empirical
evidence supports the idea that elastic energy storage in the
ankle extensor muscle–tendon units plays an important role in
human walking (Hof et al. 1983). It seems likely that elastic
energy storage during walking becomes increasingly important
at higher speeds where virtual stance-limb compression limits
the pendulum-like exchange of kinetic energy and gravitational
potential energy of the center of mass.
An advantage of allowing the stance-limb joints to flex and
extend is a reduction in the peak musculoskeletal forces
associated with foot–ground impact. When knee flexion is
restricted during walking, there is a substantial increase in the
peak vertical ground reaction force and in the rate of rise of
the ground reaction force (Yaguramaki et al. 1995; Cook et al.
1997). Conversely, exaggerated limb joint flexion leads to a
decrease in the peak vertical ground reaction force (Li et al.
1996). These observations suggest that there are trade-offs
between maximizing energy exchange by the inverted
pendulum mechanism and reducing impact forces. However,
the loss in inverted pendulum exchange incurred by allowing
the virtual stance limb to compress may be offset by increases
in elastic energy storage as discussed above.
Springs and damping elements have been incorporated into
the legs of some models of walking in order to match the
ground reaction force patterns observed in human walking and
to predict correctly the relationship between speed and stride
length (Siegler et al. 1982; Pandy and Berme, 1988, 1989;
Alexander, 1992). The stiffness (k) of these springs is generally
somewhat higher (k=12–34.5 kN m−1) than the leg stiffness
values reported for normal human running (k≈11 kN m−1)
(Pandy and Berme, 1988; He et al. 1991; Alexander, 1992;
Farley and Gonzalez, 1996). It is important to realize that,
although a simple spring-mass model (McMahon, 1990;
McMahon and Cheng, 1990) simulates running reasonably
accurately, it does not include the pendulum-like energy
exchange that is a major factor in walking.
Most bipedal animals are birds, and an earlier study
demonstrated that the kinematics of locomotion change less
dramatically at the walk–run transition in birds than in humans
(Gatesy and Biewener, 1991). However, birds do have the
same pattern of vertical movement of the center of mass during
walking and running as humans and other mammals (Cavagna
et al. 1977; Heglund et al. 1982). In birds, the center of mass
reaches its highest point at mid-stance during walking and its
lowest point at mid-stance during running. Our findings on
human locomotion show that the trajectory of the center of
mass changes dramatically at the walk–run transition because
of an abrupt decrease in the touchdown angle of the stance
limb. In birds, the stance-limb touchdown angle does not
decrease abruptly at the transition from walking to running
(Gatesy and Biewener, 1991). We are not aware of published
information about virtual stance-limb compression in walking
and running birds, making a full comparison with humans
impossible. However, Gatesy and Biewener (1991) point out
that the posture of the stance limb of birds is very different
from that of humans during walking and running. Some limb
segments (e.g. the femur) that are nearly horizontal in birds are
nearly vertical in humans. This postural difference could
greatly affect limb compliance and the kinematics of the
walk–run transition (Gatesy and Biewener, 1991).
In conclusion, the inverted pendulum model with a rigid leg
qualitatively describes the mechanism of mechanical energy
exchange during walking in humans and other animals. However,
our findings show that virtual stance-limb compression is
surprisingly large during walking, reaching similar values during
high-speed walking to those during running. At these speeds, the
difference in the trajectory of the center of mass between walking
and running is due primarily to the difference in stance-limb
touchdown angle. The unexpectedly large virtual stance-limb
compression during walking could have implications for
understanding the energetic cost of walking and the determinants
of the gait transition speed. In addition, this finding could give
insight into the design of prosthetic limbs and robotic legs that
work effectively for both walking and running.
This research was supported by a National Institutes of
2944 C. R. LEE AND C. T. FARLEY
Health grant (R29 AR44008) to C.T.F. and by a UC Berkeley
Presidential Undergraduate Fellowship to C.R.L.
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