Energetics and mechanics of human running on surfaces of different stiffnesses

Abstract

Mammals use the elastic components in their legs (principally tendons, ligaments, and muscles) to run economically, while maintaining consistent support mechanics across various surfaces. To examine how leg stiffness and metabolic cost are affected by changes in substrate stiffness, we built experimental platforms with adjustable stiffness to fit on a force-plate-fitted treadmill. Eight male subjects [mean body mass: 74.4 +/- 7.1 (SD) kg; leg length: 0.96 +/- 0.05 m] ran at 3.7 m/s over five different surface stiffnesses (75.4, 97.5, 216.8, 454.2, and 945.7 kN/m). Metabolic, ground-reaction force, and kinematic data were collected. The 12.5-fold decrease in surface stiffness resulted in a 12% decrease in the runner's metabolic rate and a 29% increase in their leg stiffness. The runner's support mechanics remained essentially unchanged. These results indicate that surface stiffness affects running economy without affecting running support mechanics. We postulate that an increased energy rebound from the compliant surfaces studied contributes to the enhanced running economy.

Full text

Energetics and mechanics of human running on surfaces
of different stiffnesses
AMY E. KERDOK,
1,2
ANDREW A. BIEWENER,
3
THOMAS A. MCMAHON,
1,2
†
PETER G. WEYAND,
3,4
AND HUGH M. HERR
1,5,6
1
Harvard Division of Health Sciences and Technology, and
5
Artificial Intelligence Laboratory,
Massachusetts Institute of Technology, Cambridge 02138;
3
Concord Field Station, Museum of
Comparative Zoology, Harvard University, Bedford 01730;
2
Division of Engineering and Applied
Sciences, Harvard University, Cambridge 02138;
4
United States Army Research Institute for
Environmental Medicine, Natick 01760; and
6
Department of Physical Medicine and Rehabilitation,
Harvard Medical School, Spaulding Rehabilitation Hospital, Boston, Massachusetts 02114
Received 12 December 2000; accepted in final form 24 September 2001
Kerdok, Amy E., Andrew A. Biewener, Thomas A.
McMahon, Peter G. Weyand, and Hugh M. Herr. Ener-
getics and mechanics of human running on surfaces of dif-
ferent stiffnesses. J Appl Physiol 92: 469–478, 2002; 10.1152/
japplphysiol.01164.2000.—Mammals use the elastic compo-
nents in their legs (principally tendons, ligaments, and mus-
cles) to run economically, while maintaining consistent sup-
port mechanics across various surfaces. To examine how leg
stiffness and metabolic cost are affected by changes in sub-
strate stiffness, we built experimental platforms with adjust-
able stiffness to fit on a force-plate-fitted treadmill. Eight
male subjects [mean body mass: 74.4 ⫾ 7.1 (SD) kg; leg
length: 0.96 ⫾ 0.05 m] ran at 3.7 m/s over five different
surface stiffnesses (75.4, 97.5, 216.8, 454.2, and 945.7 kN/m).
Metabolic, ground-reaction force, and kinematic data were
collected. The 12.5-fold decrease in surface stiffness resulted
in a 12% decrease in the runner’s metabolic rate and a 29%
increase in their leg stiffness. The runner’s support mechan-
ics remained essentially unchanged. These results indicate
that surface stiffness affects running economy without affect-
ing running support mechanics. We postulate that an in-
creased energy rebound from the compliant surfaces studied
contributes to the enhanced running economy.
biomechanics; locomotion; leg stiffness; metabolic rate
IN THEIR GROUNDBREAKING work, McMahon and Greene
(29) investigated the effects of surface stiffness (k
surf
)
on running mechanics. Their study sought to deter-
mine whether it was possible to build a track surface
that would enhance performance and decrease injury.
Their work showed that a range of k
surf
values existed
over which a runner’s performance was enhanced by
decreasing foot-ground contact time (t
c
), decreasing the
initial spike in peak vertical ground reaction force
(f
peak
), and increasing stride length. Tracks built
within this enhanced performance range at Harvard
University, Yale University, and Madison Square Gar-
den have been shown to increase running speeds by
2–3% and to decrease running injuries by 50% (29).
Despite the success of these “tuned tracks,” the mech-
anisms underlying the performance enhancement are
not clearly understood.
A major assumption of McMahon and Greene’s (29)
was that the running leg and surface could be repre-
sented as a simple spring and mass (Fig. 1). McMahon
and Cheng (27) subsequently described the leg spring
as having two stiffnesses: k
leg
and k
vert
. The k
leg
is the
actual leg stiffness describing the mechanical behavior
of the leg’s musculoskeletal system during the support
phase and is calculated from the ratio of f
peak
to the
compression of the leg spring (⌬l, defined in Eq. B4)
k
leg
⫽
f
peak
⌬l
(1)
In distinction, k
vert
is the effective vertical stiffness of
the runner. This stiffness serves as the mechanism by
which the direction of the downward velocity of the
body is reversed during limb contact (27, 30). There-
fore, k
vert
describes the vertical motions of the center of
mass during the ground contact phase (27, 30). The
k
vert
can be calculated from the ratio of f
peak
to the
maximum vertical displacement of the center of mass
during contact (⌬y
total
), measured from the onset of
limb contact (heel-strike) to midstep
k
vert
⫽
f
peak
⌬y
total
(2)
Farley et al. (12) and He et al. (19) determined that, for
a given ground stiffness, k
leg
changes little with speed.
Later experiments showed that k
leg
changes when an-
imals run on surfaces of different stiffnesses (16, 18). It
was specifically shown that human hoppers’ k
leg
ad-
justments were mainly due to changes in both ankle
joint stiffness and leg posture (14). However, Aram-
patzis et al. (3) provided evidence that the knee joint is
†Deceased 14 February 1999.
Address for reprint requests and other correspondence: A. E.
Kerdok, Harvard University, 29 Oxford St., Pierce Hall G-8, Cam-
bridge, MA 02138 (E-mail: kerdok@fas.harvard.edu).
The costs of publication of this article were defrayed in part by the
payment of page charges. The article must therefore be hereby
marked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734
solely to indicate this fact.
J Appl Physiol 92: 469–478, 2002;
10.1152/japplphysiol.01164.2000.
8750-7587/02 $5.00 Copyright
©
2002 the American Physiological Societyhttp://www.jap.org 469

the main determinant of k
leg
as a function of speed in
human running. In addition, modeling efforts and ex-
periments on humans have shown that a runner’s
center of mass deflections (⌬y
total
) remain nearly con
-
stant, independent of k
surf
, and that this may be a
general principle of running mechanics (16, 29). In
other words, by adjusting their “leg spring” stiffness to
adapt to different k
surf
values, a runner may be able to
maintain apparently uniform support mechanics.
Representations of the running leg as a simple
spring have described the mechanics of a running leg
remarkably well (2, 11, 12, 15, 16, 18, 19, 25–27). It has
been shown that the physical musculoskeletal elastic
components of the leg (tendons, ligaments, and mus-
cles) are used to minimize metabolic cost while running
(1, 2, 8–10). However, no one, to date, has related the
performance enhancements of running on surfaces of
different stiffnesses to metabolic cost. In this paper, we
assume that the leg can be represented by an un-
damped, linear spring and examine how the energetics
and mechanics of running vary on surfaces of different
stiffnesses.
The goal of this study is to relate human running
biomechanics to energetics on surfaces of different
stiffnesses. We expect that differences in the metabolic
cost of running on various surfaces are likely related to
the k
leg
variations observed by Farley et al. and Ferris
et al. (13, 17, 18). Specifically, we expect a less flexed
knee to account for a reduction in metabolic cost (30),
as well as an increase in k
leg
(3, 18).
In this study, we investigate the energetics and me-
chanics of running on surfaces having a stiffness range
from 75 to 945 kN/m. This range of stiffnesses was
selected to incorporate the range of McMahon and
Greene’s “tuned track” (29) and to extend the work of
similar recent studies (16–18). We hypothesize that
the metabolic cost of forward human running reaches a
minimum when the k
leg
of the runner is maximized on
surfaces of decreased stiffness. We expect a cost reduc-
tion to result from a change in leg posture, whereby the
knee is less flexed or straighter during stance (30).
Running with a straighter leg should improve the
limb’s mechanical advantage, thereby reducing the
amount of muscle force and muscle volume recruited to
support body weight (5). We also anticipate that a
reduction in metabolic cost could result from an in-
creased energy return to the runner from the more
compliant surfaces (14). Last, we expect that the run-
ner’s support mechanics will remain virtually unaf-
fected across the above-defined range of k
surf
.
EXPERIMENTAL METHODS
General procedures. Eight healthy male subjects [body
mass: 74.4 ⫾ 7.1 (SD) kg; leg length: 0.96 ⫾ 0.05 m] ran at 3.7
m/s on a level treadmill, fitted with track platforms of five
different stiffnesses (see descriptions below). All subjects
wore the same flat-soled running shoes. Approval was
granted from Harvard University’s Committee on the Use of
Human Subjects in Research, and subjects provided signed,
informed consent before participation. Subjects ran for 5 min
on each track platform stiffness in a mirrored fashion (run-
ning on stiffest to least stiff and then least stiff to stiffest).
Beaded strings hung from the ceiling to give the runner a
tactile sign as to where he needed to run so that his midstep
corresponded with the fore-aft center of the track platform.
Video was also used to ensure that the runner was both
centered and lateral enough not to be stepping on both sides
of the track simultaneously. If a runner was unable to avoid
the seam between tracks, he was asked to move laterally and
run on one track or the other. We recorded ground reaction
force (1,000 Hz) using a force plate (model OR6–5-1, Ad-
vanced Medical Technology, Newton, MA) mounted within
the treadmill (22). Kinematic data were collected at 60 Hz
using an infrared motion analysis system (MacReflex by
Qualysis), and oxygen uptake was measured using a closed
gas-collection Douglas bag setup. Oxygen and carbon dioxide
contents of the collected gas samples were analyzed using
Ametek (Pittsburgh, PA) S-3A O
2
and CD-3A CO
2
analyzers
equipped with an Ametek CO
2
sensor (P-61B) and flow con
-
troller (R-2). The analyzers were calibrated before each run
with gas by pumping several balloons of known gas mixture
(16.23% O
2
and 4.00% CO
2
medical gas mixture; AGA Gas,
Billerica, MA) through them. Force-plate and kinematic data
were obtained simultaneously, and oxygen consumption
(V
˙
O
2
) data were sampled during the fourth and fifth minutes
of running to ensure that the subject was at a steady state.
Subjects participated in two separate trials so that they ran
on each platform stiffness four times. Averages were taken
on each day and then averaged together for all variables
measured.
Experimental platform design. We built platforms with an
adjustable stiffness for our running surface. Because the
experiments were conducted on a treadmill, the running
surface was limited to platforms that would fit within the size
limitations of the treadmill. We used a treadmill fitted with
an AMTI force plate (22) that was accessible to the Douglas
bag oxygen analysis setup.
We tested five k
surf
based on ranges found in the literature
(14, 16, 18, 29). The McMahon and Greene (29) tuned track
stiffness range is between 50 and 100 kN/m. Because of size
limitations of the existing treadmill and earlier work done by
Farley and Morgenroth (15) and Ferris et al. (17), we de-
signed our variable stiffness track platforms to span from
Fig. 1. Spring-mass model representing a runner’s leg in contact
with a compliant surface. l
o
, Uncompressed leg length; m
r
, mass of
the runner represented as a point mass located at the hip; ⌬y
total
,
maximum vertical displacement of the center of mass; ⌬l, maximum
compression of the leg spring; ␪, angle of the leg spring at first
ground contact; k
leg
, spring constant of the runner’s leg; m
track
,
effective mass of the running surface; d
surf
, amount the running
surface deflects; k
surf
, spring constant of the running surface; and f
grf
,
vertical ground reaction force.
470 RUNNING ON SURFACES OF DIFFERENT STIFFNESSES
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75.4 kN/m to stiffnesses of 97.5, 216.8, 454.2, and 945.7
kN/m. The indoor track at Harvard University has a k
surf
of
⬃190 kN/m, allowing for a 9-mm deflection for a 75-kg
runner (assuming a runner exerts roughly 2.3 times body
weight at midstance). For a similar runner, our track would
result in 22.4-, 17.4-, 7.8-, 3.7-, and 1.8-mm deflections [sur-
face deflections (d
surf
)], respectively, according to the follow
-
ing equation
d
surf
⫽
2.3 䡠 m
r
䡠 g
k
surf
(3)
where m
r
is the mass of the runner, g is the gravitational
constant, and k
surf
is the stiffness of the track. The factor 2.3
is an estimate of how much the f
peak
exceeds body weight
during a running step.
The platform design is shown in Fig. 2. Garrolite (G-10,
Current, East Haven, CT) was chosen as the material for the
track platforms because it met all of the design criteria
described below and in
APPENDIX A and could be easily ma-
chined. The design consisted of two G-10 planks (1.22 ⫻
0.254 ⫻ 0.014 m) rigidly supported in the front and simply
supported in the rear by 0.016-m-thick acrylic. By moving the
treadmill rollers in at either end of the belt surface, enough
slack was provided to fit the platforms under the belt directly
on top of the force plate. Rollers were added to the existing
treadmill to reroute the treadmill belt over the platforms,
and a frame was built (not shown) to hold the platform in
place on top of the force plate during testing. The rear
support was movable so that, by simply adjusting it in closer
to or farther away from the front support, the stiffness of the
running surface was increased or decreased, respectively.
Once installed, the stiffness of each platform was cali-
brated by applying static loads to a person and measuring
force (f
peak
, from the force plate) and deflection (d
surf
; from an
LVDT cable extender ⫾ 0.25%, Celesco Transducer Products)
(Eq. 3).
As described in
APPENDIX A, the inertial effects of the run-
ning platform motion compared with the forces exerted by
the runner’s leg can be considered negligible if the effective
mass of the platform is ⬍17% of the m
r
(11.43 kg for the
smallest runner studied). The effective masses (6.88, 8.88,
4.94, 2.65, and 5.39 kg) gave inertial forces of ⫺41.43,
⫺30.41, ⫺10.04, ⫺3.33, and ⫺0.42 N, respectively. These
forces were ⬍2.5% of the peak forces exerted by the runner
and so were ignored.
Given that the running surface was a compliant surface
having the potential to return energy to the runner, we also
calculated the energy return of our variable-stiffness track
platform. We did this by using the track deflection to derive
the potential energy at each track stiffness (E
track
)
E
track
⫽
1
⁄
2
k
surf
d
surf
2
(4)
Multiplying this energy by two times the stride frequency
results in the mechanical power delivered from the track to
the runner. This was then related to a measurement of the
metabolic power (E
˙
metab
) consumed by the runner at each
track stiffness.
Force-plate measurements. A runner’s support mechanics,
defined as f
peak
, t
c
, duty factor, stride frequency, step length,
one-half of the angle swept by a runner’s leg during ground
contact (␪), and the total vertical displacement of the center
of mass, can be calculated from the force-plate data and the
assumption that the leg can be represented by an undamped,
linear spring (22). These parameters can then be indirectly
used to calculate the mass-spring characteristics of the run-
ner’s leg. Custom LabVIEW (version 4.0.1) software was used
to acquire the force-plate data. The force plate was calibrated
by applying known loads to the plate before and after each
set of running trials and sampling its output using the same
software. The derivation of all of the above parameters is
described in
APPENDIX B.
Kinematic measurements. To obtain information on the
posture of the limb in contact with the ground, we used an
infrared camera system (MacReflex; Qualysis) to follow
markers that were specifically placed on the subjects. Mark-
ers were positioned on the skin overlying the greater trochan-
ter, the lateral epicondyle of the femur, and the lateral
malleolus, so that the angle that the lower leg made with the
upper leg (knee angle) could be determined.
Kinematic data were collected simultaneously and syn-
chronized with the force-plate data (using an infrared light-
emitting diode in the camera’s field of view that gave a
voltage pulse that was recorded when the light-emitting
diode was switched on). Kinematic data were analyzed using
the Maxdos software from MacReflex (Qualysis) and incorpo-
rated into a Matlab (version 4.0) program to calculate the
knee angle at midstep. The program also calculated the
series minimum height points of the greater trochanter
marker for several strides over the 10-s collection period.
This marker was used to estimate the position of a runner’s
center of mass, and its minimum trajectory was used to
define the midpoint of each step when force application
reached its peak (13, 19, 26).
Fig. 2. Side view (A) and top view (B) of a schematic of
the experimental compliant track treadmill. The run-
ner’s foot strikes the treadmill belt (7) (note that this is
cut away on top view to show underlying structures and
also that the belt is longer than depicted) and exerts a
vertical force (F) on the compliant running platforms
(6) below. The vertical force is transmitted from the
platforms via the supports (2 and 8) to the force plate
(1). The stiffness of the running surface is adjusted by
moving the movable support (8) in and out. The force
plate (1) and entire treadmill apparatus are supported
by the treadmill base (5). By moving the treadmill
rollers (3) closer together, enough slack is recovered in
the belt (7) to insert the track platforms (6). The belt is
then redirected over the track platforms with the redi-
rection rollers (4).
471RUNNING ON SURFACES OF DIFFERENT STIFFNESSES
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Measuring metabolic cost. To quantify the metabolic cost of
human running, we used the indirect calorimetry method, as
described previously. After the runners ran for 3 min, we
collected the expired air for 2 min using two Douglas bags (1
per minute), a mouthpiece, and a nose clip, which were
attached to the runner via a special headpiece equipped with
a one-way valve. The rate of V
˙
O
2
(ml/min) was then calcu
-
lated using the volume of the expired air (from a dry-gas
volume meter; Parkinson-Cowan), room and vapor pressure
corrections, and the percentage of CO
2
and O
2
values. We
converted the rate of V
˙
O
2
into energy consumption using an
energy equivalent of 20.1 J/ml O
2
(6) and divided by 60 s/min
to obtain E
˙
metab
in watts.
Kram and Taylor (23) define the rate of metabolic con-
sumption (E
˙
metab
) in terms of a cost coefficient, C
0
E
˙
metab
⫽ C
0
冉
1
t
c
冊
共F
bw
兲 (5)
where C
0
is an empirical measure of the metabolic cost of
applying ground force to support the body’s weight (F
bw
) (4,
31). For this investigation, the C
0
is computed to determine
the effect of k
surf
on the energetics of supporting the runner’s
body weight.
Statistical methods. A1⫻ 5 ANOVA with a Scheffe´ post
hoc test of condition means was used to assess the effect of
k
surf
on the parameters of interest: t
c
, peak vertical force,
stride time, stride frequency, step length, ␪, ⌬y
total
, displace
-
ment of the limb with respect to the track displacement,
E
˙
metab
, C
0
, k
leg
, k
vert
, ⌬l, overall system stiffness, and knee
angle. P values ⬍0.05 were considered significant for all
tests.
RESULTS
The runner’s support mechanics were nearly invari-
ant across the 12.5-fold change in k
surf
of the experi
-
mental treadmill platform (Fig. 3), whereas their met-
abolic rate dropped dramatically with k
surf
(Fig. 6).
The results of the Scheffe´ post hoc test revealed that,
in virtually every case, the support mechanics re-
mained essentially unchanged over the four stiffest
surfaces tested. The basis for a significant difference in
the ANOVA results reported below was found to be due
to the data recorded for the lowest stiffness track
surface.
As shown in Fig. 3, the effect of k
surf
on t
c
(P ⫽
0.0001, F ⫽ 8.7), duty factor (P ⫽ 0.0001, F ⫽ 18.45),
step length (P ⫽ 0.0001, F ⫽ 8.51), stride frequency
(P ⫽ 0.0001, F ⫽ 15.35), ␪ (P ⫽ 0.0001, F ⫽ 8.44), and
f
peak
(P ⫽ 0.009, F ⫽ 4.19) were significant. However,
the data across these support mechanics showed only a
small difference between the two stiffness extremes.
The source of the difference occurred at the lowest
stiffness, with the remaining four stiffnesses being
essentially the same.
Fig. 3. When subjects ran at a con-
stant speed (3.7 m/s) over 5 different
surface stiffnesses (74–945 kN/m),
their support mechanics remained es-
sentially unchanged. As confirmed by
the Scheffe´ post hoc test, the differ-
ences observed were due to the lowest
surface stiffness only. The time that
the foot is in contact with the ground
[y ⫽ 0.0017 ln(x) ⫹ 0.204, R
2
⫽ 0.21]
(A); the duty factor [y ⫽ 0.0063 ln(x) ⫹
0.265, R
2
⫽ 0.58] (B); the step length
(right foot to left foot) [y ⫽ 0.0065
ln(x) ⫹ 0.76, R
2
⫽ 0.22] (C); the stride
(right foot to right foot) frequency [y ⫽
0.019 ln(x) ⫹ 1.31, R
2
⫽ 0.87] (D); the
angle swept by the runner’s leg [y ⫽
0.221 ln(x) ⫹ 23.44, R
2
⫽ 0.24] (E), and
the peak vertical ground reaction force
[y ⫽⫺41.4 ln(x) ⫹ 2163.2, R
2
⫽ 0.94]
(F) were all virtually constant. Values
are means ⫾ SD. T
c
, period of foot-
ground contact; T
a
, period of foot in air.
472 RUNNING ON SURFACES OF DIFFERENT STIFFNESSES
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In particular, when the support mechanics means
are compared, there was a 4% decrease in t
c
, step
length, and ␪ between the stiffest and least stiff sur-
faces studied. A 7% decrease in duty factor, 3% de-
crease in stride frequency, and a 5% increase in f
peak
were also observed between these stiffness extremes.
The post hoc test revealed that the runners also
maintained a nearly constant total leg plus track plat-
form stiffness (k
totL
) over the observed range of condi
-
tions (P ⫽ 0.0207, F ⫽ 3.45) (Fig. 4C). To achieve this,
the runner’s k
leg
increased by 29% with decreasing k
surf
(P ⫽ 0.0001, F ⫽ 23.76) (Fig. 4A). Given that the
runner’sf
peak
did not change greatly over the substrate
stiffness range (Fig. 3F), the observed increase in the
runner’s k
leg
most likely resulted from a decrease in the
amount that his leg spring was compressed (P ⫽
0.0001, F ⫽ 33.93) (Eq. 1, Figs. 1 and 4D). The ⌬l is a
function of leg length, ␪, and vertical displacement of
the runner relative to the displacement of the track
surface (y
limb
)(Eq. B4). Because leg length and ␪ re
-
mained essentially constant, the decrease in the ⌬l was
likely due to the observed decrease in y
limb
(P ⫽ 0.0001,
F ⫽ 94.05) (Fig. 5B, Eq. B5).
We achieved the five different k
surf
by allowing the
simply supported track to displace beneath the runner.
Therefore, d
surf
increased 12.5-fold from the stiffest to
the least stiff surface (Fig. 5A). This substantial in-
crease in surface displacement was mostly offset by
y
limb
so that y
total
was minimally changed between the
k
surf
extremes (⬃0.8 cm) (P ⫽ 0.0001, F ⫽ 16.28).
Again, the change in y
total
was only significant at the
lowest k
surf
studied (Fig. 5C). Our finding that the k
vert
(Fig. 4B) remained virtually constant (P ⫽ 0.0001, F ⫽
11.77) over the four stiffest surfaces further supports
the fact that the runners’ ⌬y
total
changed minimally, as
k
vert
is a function of f
peak
and y
total
.(Eq. 2).
We also found a 12% decrease in the runner’s rate of
E
˙
metab
as k
surf
decreased (P ⫽ 0.0001, F ⫽ 71.95) (Fig.
6A). The runner’s mean E
˙
metab
decreased from 896 to
792 W as k
surf
decreased from 945 to 75 kN/m. Refer
-
ring to Eq. 5 and recalling that t
c
remained essentially
unchanged (Fig. 3A), the observed decrease in meta-
bolic rate suggests that the C
0
defined by Kram and
Taylor (23) also decreased with decreasing k
surf
(P ⫽
0.0001, F ⫽ 32.54) (Fig. 6B).
In an attempt to evaluate limb mechanical advan-
tage, we used the kinematic data, together with the
vertical ground reaction force, to calculate the limb’s
knee angle at midstep for running over all surfaces
(Fig. 7). Knee angle increased 2.5% as k
surf
decreased
(P ⫽ 0.0001, F ⫽ 16.35). Thus only a slight straight-
ening of the leg was observed.
Last, to test our hypothesis that the track itself may
return significant energy to the runner, we calculated
the track platform’s mechanical power (E
track
, Eq. 4)at
each k
surf
and compared this with the reduction in
E
˙
metab
of the runner (Eq. 5) (Fig.
8). The results show
that, for every watt of mechanical power returned from
the track platform, there exists the possibility of a
1.8-W E
˙
metab
savings to the runner (R
2
⫽ 0.99).
DISCUSSION
Our results support the hypothesis that the meta-
bolic cost of running at an intermediate speed is pro-
gressively reduced and that the spring stiffness of the
leg is progressively increased as k
surf
is decreased from
945.7 to 75.4 kN/m. However, in contrast to our hy-
pothesis that a change in limb posture is the principal
factor underlying a change in both k
leg
and metabolic
cost, we found that only small changes in knee angle
were associated with the observed 29% increase in k
leg
and 12% decrease in E
˙
metab
. Our data do not provide
Fig. 4. When subjects ran at a con-
stant speed (3.7 m/s) over 5 different
surface stiffnesses (74–945 kN/m),
there was a 29% change in their leg
stiffness [y ⫽⫺1.37 ln(x) ⫹ 22.4, R
2
⫽
0.87] (A) and a 16% decrease in leg
compression [y ⫽ 0.008 ln(x) ⫹ 0.091,
R
2
⫽ 0.86] with decreasing surface
stiffness (D). The overall stiffness of
the system [y ⫽⫺0.175 ln(x) ⫹ 14.46,
R
2
⫽ 0.4] (C) and the effective vertical
stiffness of the runner [y ⫽ 0.81 ln(x) ⫹
29.31, R
2
⫽ 0.66] (B) remained essen
-
tially unchanged. Values are means ⫾
SD.
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any additional insight into the mechanism for k
leg
adjustment but do suggest that a reduction in meta-
bolic cost occurs as the elastic rebound provided by a
more compliant surface replaces that otherwise pro-
vided by a runner’s leg.
Previous work indicated that runners adjust the
stiffness of their limbs to maintain virtually constant
support mechanics on surfaces of different stiffnesses
(3, 14, 16, 18, 28). Although these studies provide
insight into the mechanics of human running, they did
not specifically examine the metabolic cost of running
on compliant surfaces. One study (30) looked at deep-
knee-flexed running and its effect on k
vert
and V
˙
O
2
but
did not incorporate k
leg
or compliant surfaces. Our goal
was to expand on these earlier studies and examine
how changes in limb-substrate stiffness interactions
affect the metabolic cost of running. Consequently, we
Fig. 5. A: as designed, the vertical displacement of the compliant
running surface increased with decreasing surface stiffness [y ⫽
⫺0.01 ln(x) ⫹ 0.064, R
2
⫽ 0.91]. B: the vertical displacement of the
runner’s center of mass relative to the track’s vertical displacement
decreases with surface stiffness [y ⫽ 0.0065 ln(x) ⫹ 0.011, R
2
⫽ 0.93].
This limb displacement is used to define the leg compression (Fig. 4,
Eq. B4), which in turn defines the leg stiffness of the runner (Fig. 4,
Eq. 1). C: the total vertical displacement of the runner’s center of
mass (com) measured from midstep to take-off using the vertical
displacement of the hip marker is essentially unchanged over the
surface stiffnesses [y ⫽⫺0.003 ln(x) ⫹ 0.076, R
2
⫽ 0.84]. This value
is used to define the effective vertical stiffness (Fig. 4, Eq. 2). Values
are means ⫾ SD.
Fig. 6. A: the runner’s metabolic rate decreased with surface stiff-
ness [y ⫽ 41.17 ln(x) ⫹ 622.31, R
2
⫽ 0.97] over the 12.5-fold change
in surface stiffness (74–945 kN/m) when running at a constant speed
(3.7 m/s). B: because the contact time remained essentially constant
(Fig. 3), the cost coefficient must also decrease with surface stiffness
[y ⫽ 0.0142 ln(x) ⫹ 0.171, R
2
⫽ 0.91] per Eq. 5 [E
˙
metab
⫽ C
0
⫻ (1/t
c
) ⫻
F
bw
, where E
˙
metab
is the rate of metabolic consumption, C
0
is an
empirical measure of the metabolic cost of applying ground force to
support the body’s weight (F
bw
), and t
c
is the period of foot-ground
contact]. Values are means ⫾ SD.
Fig. 7. The angle formed between the runner’s upper and lower legs
was defined from markers placed at the greater trochanter, lateral
epicondyle of the femur, and lateral malleolus. This knee angle
increased 2.5% as surface stiffness decreased [y ⫽⫺1.25 ln(x) ⫹
137.29, R
2
⫽ 0.97], giving rise to a slightly straighter leg on the softer
surfaces studied. Values are means ⫾ SD.
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adopted a similar mechanical and experimental ap-
proach to that of these studies, focusing on the knee
joint and assuming that the leg behaves as a massless,
undamped linear spring.
Using this simple model of the human leg, our find-
ings generally support those of Farley et al. (11, 13–
15), indicating that human runners alter their leg
spring stiffness to compensate for changes in k
surf
with
-
out altering their overall support mechanics. McMahon
and Greene (29), Farley et al. (14), and Ferris et al. (17)
have commented that an observer looking only at the
upper body of a runner would be unable to discern
when the runner experienced a change in ground stiff-
ness. This suggests that runners compensate for vari-
able ground stiffness without affecting the fluctuations
in the motion of their center of mass. This is consistent
with our findings that d
surf
is offset by y
limb
, thus
resulting in the minimal 0.8-cm change observed in
y
total
. Hence, utilizing preferred support mechanics
might represent a general principle of running.
Kram and Taylor’s (23) analysis suggests that the
mass-specific metabolic rates of running animals are
determined by the rate of ground-force application (1/
t
c
), regardless of the speed and size of the animal. Their
analysis assumes that animals maintain a uniform
limb mechanical advantage over a range of running
speeds and gaits. This assumption is supported by
previous studies of animals moving at steady speeds
over a constant (high) stiffness substrate (5). As a
result, the cost of force generation and the volume of
muscle that must be activated to support a given unit
body weight also appear to remain constant (21). How-
ever, we found a reduction in metabolic rate with
virtually no change in the t
c
(Figs. 3A and 6A). Thus
the energetic cost of applying a ground force to support
the runner’s body weight can be reduced at a given rate
of ground force application (1/t
c
) when running on more
compliant surfaces.
The close relationship between the reductions in
metabolic rates and the increased mechanical power
returned by the track to the runner in the latter por-
tion of foot-ground contact (Fig. 8) offers a straightfor-
ward explanation. This close relationship strongly sug-
gests that, when a greater share of the elastic rebound
elevating the center of mass in the latter portion of the
contact phase is provided by the elastic recoil of the
running surface rather than the biological springs in
the runner’s leg, the metabolic cost of running is re-
duced. We believe that these reductions in the meta-
bolic cost of operating leg springs are probably ex-
plained by decreases in the mechanical work and
shortening velocity performed by the muscles active
during foot-ground contact.
Although we had hypothesized that reductions in
metabolic cost and increases in k
leg
would be achieved
predominantly via changes in knee angle, it seems
evident that this mechanism cannot fully account for
these changes. The change in k
leg
is likely due to a
combination of local joint stiffness variation and over-
all limb posture adjustment (15). Whereas our study
provided some indication that the leg becomes
straighter at midstep on less stiff surfaces (Fig. 7), the
change at the knee was small and would require a
large sensitivity to have an effect on externally devel-
oped knee torque. Therefore, this small change in knee
angle could only account for a minority of the reduc-
tions in metabolic cost and increases in k
leg
that we
observed on more compliant surfaces.
Our hypothesis also anticipated that the decrease in
E
˙
metab
might well be explained by an enhanced energy
return from the more compliant track platforms. The
elastic surface could actually be assisting the runner
by assuming some of the cost necessary to operate the
leg spring, reducing the amount of mechanical work
required, and thereby allowing the leg muscles to op-
erate more isometrically. Reductions in relative short-
ening velocities would reduce metabolic cost in two
ways. First, the increased force per unit area of active
muscle would reduce the volume of muscle required to
support the body’s weight. Second, the E
˙
metab
con
-
sumed per unit of active muscle is also reduced when
the muscles shorten through a lesser distance (20).
To lend support for these ideas, experiments were
conducted to characterize the track-runner interaction.
The dynamic calibration of the four most compliant
experimental track platforms showed a linear relation-
ship between force and displacement (R
2
⫽ 0.96, 0.97,
0.95, and 0.94 from least to most stiff) with little
hysteresis (damping ratio ⬍0.1). Hence, the track can
indeed be considered an elastic substrate capable of
storing and returning mechanical energy. Also, by cal-
culating the resonant period of the track-plus-runner
system at the least stiff surface (⬃0.2 s) and comparing
the result to the contact times of the runners at this
same stiffness (0.21 ⫾ 0.02 s), we conclude that the
track has sufficient time to return its stored energy to
the runner. Last, our results show a consistent linear
Fig. 8. Change in the runner’s metabolic power (E
˙
metab
) as a function
of the change in mechanical power delivered from the track plat-
forms (E
track
) from the stiffest surface (K
5
). The power delivered from
the compliant track is derived from the mechanical energy due to the
track spring (Eq. 4) multiplied by the runner’s stride frequency. For
every watt delivered from the track platforms, there exists a poten-
tial 1.8-W reduction in metabolic power (y ⫽ 1.80x, R
2
⫽ 0.99).
Values are means ⫾ SD. K
n
, surface stiffness where n ⫽ 1, 2, 3, 4.
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relationship between the reduction in E
˙
metab
and track
mechanical power output across all surfaces studied
(Fig. 8). These results suggest that the track has the
capacity to save the runner 1.8 W of E
˙
metab
for every
watt of mechanical power that it returns.
Although our results support the fact that running
on a decreased k
surf
results in a reduction of metabolic
cost and an increase in k
leg
without affecting support
mechanics, future studies need to be done to find a true
metabolic minimum. Our measurements were de-
signed to examine surfaces that were within a stiffness
range that had already demonstrated an enhanced
running performance (29). However, support mechan-
ics are progressively altered to accommodate extreme
decreases in k
surf
. As mentioned above, our results
support our hypothesis that these support mechanics
would remain fairly constant over the 12.5-fold change
in k
surf
but also show a significant change in these
variables at the lowest k
surf
studied. This raises the
possibility of a trend in data as k
surf
goes even lower.
McMahon and Greene’s (29) work supports this spec-
ulation. We also anticipate that, as k
surf
decreases even
further and the virtual consistency of the support me-
chanics seen at the higher stiffnesses is lost, there
would exist a true metabolic minimum. Studies that
looked at running on surfaces with extremely low stiff-
ness, such as a trampoline and pillows (30) or sand
(24), which also have high damping ratios, indicate
that runners likely increase the amount of center-of-
mass work that they perform and thus substantially
increase their cost of locomotion (24). We propose that
a study be done to examine lower k
surf
values than
were studied here to determine at what substrate stiff-
ness a true metabolic minimum exists as a relation of
speed. We believe that there exists an optimal ratio of
t
c
to surface resonant period that can be used for the
future design of tracks and even running shoes to
minimize the cost of running.
Summary. Our study sought to link the mechanics
and energetics of human running on surfaces of differ-
ent stiffnesses. The results show that both metabolic
cost and k
leg
change when k
surf
is manipulated. The
metabolic reduction is largely due to the track’s elastic
energy return assisting the runner’s leg spring. Al-
though the mechanism for k
leg
adjustment still re
-
mains unclear, our results support the hypothesis that
human runners adjust k
leg
to maintain consistent sup
-
port mechanics across different surfaces.
This study has served to link previous studies on
animal locomotion and to open the door to future in-
vestigations on locomotory mechanics and energetics.
Understanding how metabolism, speed, and k
leg
relate
to substrate mechanics will not only lead to advances
in running shoe technology and track design, but may
also motivate the development of highly adaptive or-
thotic and prosthetic leg devices that change stiffness
in response to speed and ground surface variations,
enabling the physically challenged to move with
greater ease and comfort.
APPENDIX A
Experimental Track Platform Design
The design of the variable-stiffness track platform was
based on simply supported, two-point bending beam theory.
Pilot studies showed that this configuration would work well
within the size limitations of the treadmill (0.102-m maxi-
mum height from the force plate to beneath the belt, 1.22-m
long ⫻ 0.457-m wide force plate, and 0.5 ⫻ 2.64-m overall belt
surface). Materials and dimensions were chosen based on the
maximum deflection (y
max
) of the center of the beam accord
-
ing to the factor of safety (FS) associated with the loads that
would be applied in running (F) or
y
max
⫽
⫺ FL
3
48EI
(A1)
FS ⫽
␴
u
␴
max
(A2)
where L is the length of the beam, E is Young’s modulus, I is
the area moment of inertia, ␴
u
is the ultimate stress of the
material, and ␴
max
is the maximum allowable stress of the
material.
Another design criterion was that the track platform mass
needed to be small enough so that the inertial forces due to
the movement of the platform would be negligible compared
with the forces exerted by the runner’s leg. By modeling the
leg and platform surface as a two-mass and two-spring sys-
tem with a damper, we found that the effective mass of the
platform had to be ⬍12 kg (or 17% of the m
r
) in order for the
platform’s inertia to represent ⬍10% of the peak force devel-
oped by a 72-kg runner. Therefore, given that the masses of
the actual runners were from 67.3 to 81.5 kg, the effective
mass of the track platforms (m
track
) had to be ⬍11.4–13.9 kg
to meet this criterion.
The inertial effects of the track platforms on measure-
ments obtained from the force plate could be obtained by
calculating the effective mass of the platforms. The m
track
was estimated by treating the track as a harmonic oscillator
and finding the damped frequency (␻
d
). The ␻
d
was measured
by striking the platform and plotting the displacement vs.
time for the free vibration of the track surface (14). This was
accomplished by mounting the LVDT cable extender at the
center edge of the platform for each stiffness configuration,
with the platform resting in position on top of the AMTI force
plate and under the treadmill belt. The ␻
d
was computed
from the period of vibration (T
d
)or
␻
d
⫽
2␲
T
d
(A3)
The equation describing the envelope of the free vibration
curve can be used together with the damped natural fre-
quency of the track to obtain the natural frequency and the
damping ratio of the track surfaces
x ⫽ ⫾Ae
⫺ ␰␻
n
t
(A4)
␻
d
⫽ ␻
n
冑
1 ⫺ ␰
2
(A5)
where x defines the envelope of free vibration, A is the
amplitude of free vibration, ␻
n
is the natural frequency, and
t is time. With the use of Eqs. A4 and A5, the natural
frequency of the platform was calculated to be 105 rad/s, thus
resulting in a negligible damping ratio of ␰⬇0.07. Hence the
m
track
was estimated from the k
surf
and the ␻
d
,or
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m
track
⬇
k
surf
␻
d
2
(A6)
The m
track
was then used, together with the second deriva
-
tive of the displacement curves (to obtain acceleration), to
estimate the inertial force (F
inertial
) of the track platforms
using
F
inertial
⫽ m
track
d
2
x
dt
2
(A7)
APPENDIX B
Derivation of the Force-Plate Parameters
LabVIEW (version 4.0.1) was used to acquire the force-
plate data and output the parameters of the runner’s support
mechanics (f
peak
, t
c
, stride time, stride frequency, step length,
␪, and the vertical displacement of the center of mass).
Because of the vibrational noise from the treadmill belt,
motor, and track (22), we filtered the force data using a
low-pass, third-order Butterworth double-reverse filter. The
smoothed curve for the ground reaction force was used for
analysis. The f
peak
is the force at midstep and was taken to be
the maximum value of this curve. The duration of the force
provided a measure of the t
c
as well as total stride (right foot
to right foot) time (t
c
⫹ t
a
⫽ dur
tot
, where t
a
is the period the
foot is in the air and dur
tot
is total duration) that were then
used to calculate the stride frequency (freq) and step length
(SL) (distance traveled by the center of mass during one t
c
)
freq ⫽
Hz
dur
tot
(B1)
SL ⫽ t
c
u៮
x
(B2)
where u៮
x
is the horizontal (forward) velocity. With the addi
-
tional input of the runner’s leg length (l
o
) measured from the
runner’s greater trochanter to the floor while standing
straight legged, we calculated ␪ (see Fig. 1) from
␪ ⫽ sin
⫺ 1
冉
SL
2l
o
冊
(B3)
Because the ground reaction force is equal to the runner’s m
r
times his acceleration, we were able to calculate the ⌬y
total
of
the runner by twice integrating the vertical acceleration of
the center of mass over time (7).
To account for the displacement of the variable-stiffness
surfaces (d
surf
) in relation to the runner’s ⌬y
total
, we calcu
-
lated d
surf
from the calibrated values obtained for k
surf
and
the forces obtained from the force plate (Eq. 3, where 2.3 ⴱ
m
r
⫽ f
peak
from force plate).
The above variables were then used to calculate the mass-
spring characteristics of the runner’s leg. The maximum ⌬l
was calculated by using the runner’s l
o
, ␪, and the actual
⌬y
limb
(18, 26)
⌬l ⫽ ⌬y
limb
⫹ l
o
共1 ⫺ cos ␪兲 (B4)
⌬y
limb
⫽ ⌬y
total
⫺ d
surf
(B5)
Because the f
peak
occurs at the same time that the center of
mass is at its lowest height, the stiffness of the leg spring
(k
leg
) was calculated using the ratio of f
peak
to maximum leg
compression (13, 19, 26) (Eq. 1). Similarly, the effective
vertical spring stiffness was calculated using the peak force
and the total displacement of the center of mass of the system
(Eq. 2).
The total displacement is used in calculating k
vert
rather
than the actual displacement of the runner alone, because, on
less stiff surfaces, k
vert
is affected by the displacement of the
surface (18). If the actual (or relative) displacement is used,
the possibility exists that vertical stiffness could assume a
negative value (the runner moves in the opposite direction at
midstep in an effort to maintain a constant displacement of
the system’s center of mass), which is nonsensical.
Overall stiffness of the system (k
totL
) was calculated as the
sum of the k
leg
and the track platform stiffness (k
surf
)in
series
1
k
totL
⫽
1
k
leg
⫹
1
k
surf
(B6)
The authors thank Claire Farley and Roger Kram from the Uni-
versity of California at Berkeley for helpful discussions, as well as
Robert Wallace from the United States Army Research Institute for
Environmental Medicine for statistical analysis.
This research was supported in part by a graduate fellowship from
the Whitaker Foundation and the Division of Engineering and Ap-
plied Sciences, Harvard University.
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