Mechanical work for step-to-step transitions is a major determinant of the metabolic cost of human walking
1 Department of Integrative Biology, University of California, Berkeley, CA
94720-3140, USA
2 Department of Kinesiology and Applied Physiology, University of Colorado,
Boulder, CO 80309-0354, USA
3 Department of Mechanical Engineering, University of Michigan, Ann Arbor,
MI 48109-2125, USA
* Author for correspondence (e-mail: artkuo{at}umich.edu).
Accepted 13 August 2002
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Summary |
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Key words: biomechanics, biped, energetics, locomotion, oxygen consumption, human
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Introduction |
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For walking on level ground, however, it is unclear why mechanical work is
required. There is no dissipative load external to the body as for rowing or
cycling, nor is net work performed against gravity as for slope walking.
Perhaps the body and limbs themselves act as a mechanical load. There are a
variety of methods to quantify the mechanical work performed on the body and
limbs (Burdett et al., 1983;
Cavagna and Kaneko, 1977
;
Willems et al., 1995
), but
these neither predict nor explain where and why mechanical energy is
dissipated.
Several inverted pendulum models of walking
(Fig. 1) predict that work is
not needed within each step, but rather between steps
(McGeer, 1990;
Alexander, 1995
;
Garcia et al., 1998
;
Kuo, 2002
). In bipeds, single
support can be modeled as an inverted pendulum, with the center of mass moving
along an arc dictated by the stance limb
(Fig. 2A). A pendulum conserves
mechanical energy and requires no work to move along an arc, but the
transition from one stance limb to the next does require work. Negative work
is performed in the collision that redirects the center of mass velocity from
one arc to the next (Fig. 2B),
and positive work is required to restore the energy lost. These step-to-step
transition costs will exact a proportional metabolic cost if muscle efficiency
is constant.
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|
These model predictions have been supported by previous experiments
measuring mechanical and metabolic cost as a function of increasing step width
(Donelan et al., 2001). A
simple model (Kuo, 1999
)
predicted that collision costs would increase with the square of step width.
Subsequent experimental measurements of mechanical work performed during
step-to-step transitions (Donelan et al.,
2001
) showed a similar dependence (r2=0.91),
with a proportional increase in metabolic cost (r2=0.83).
This is, however, only a small part of the metabolic cost of normal walking
because humans prefer to walk with a relatively narrow step width.
Step-to-step transition costs associated with step length may comprise a
much greater fraction of the metabolic cost of normal walking. Our models
predict two important components to the cost of normal walking: a cost to
increasing step length due to step-to-step transitions, and a cost to
increasing step frequency due to moving the legs relative to the body
(Kuo, 2001). The rate of
mechanical work for step-to-step transitions is predicted to increase sharply
with the fourth power of step length when walking speed increases
proportionally with step length (Fig.
3A). The metabolic cost of moving the legs is predicted to depend
more heavily on step frequency (Kuo,
2001
) and to be isolated from the cost of step-to-step transitions
by keeping step frequency fixed. Several previous studies (e.g.
Atzler and Herbst, 1927
;
Zarrugh et al., 1974
;
Elftman, 1966
) indicate an
increase in metabolic cost with step length, but because these studies provide
few data points that specifically fix step frequency for a range of step
lengths, we embarked on a new study designed for this purpose.
|
In the present study, we tested predictions regarding step-to-step transition costs in walking by measuring mechanical and metabolic costs in humans as a function of step length. A fixed step frequency was used to control for other potential metabolic costs such as for moving the legs. Based on our model's predictions (Fig. 3), and assuming constant muscular efficiency, we hypothesized that both the mechanical and metabolic power associated with step-to-step transitions would increase with the fourth power of step length.
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Materials and methods |
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We applied a previously developed model, the `simplest two-dimensional
passive dynamic walking model' (Fig.
1; briefly reviewed in Appendix), to predict how collision costs
increase with step length (Kuo,
2002). During single support phases, the model behaves as an
inverted pendulum (Fig. 2A).
Each transition to a new stance limb (Fig.
2B) involves a collision, where the negative work per step,
denoted Wtrans(-), is performed by the leading
limb on the center of mass, according to:
![]() | (1) |
![]() | (2) |
Humans redirect the center of mass velocity during step-to-step transitions
not with instantaneous collisions, but with negative work performed by the
leading leg over a finite period of time (Donelan et al.,
2001,
2002
). The step-to-step
transition costs are the negative external work performed to redirect the
center of mass velocity from one inverted pendulum to the next, and the equal
amount of positive external work performed to restore the energy lost.
Equation 2 predicts that both of these quantities increase with step length
raised to the fourth power.
In addition to these step-to-step transition costs, motion of the legs back and forth relative to the body contributes to external mechanical work, whether or not work is performed on the legs (see Appendix). Keeping step frequency fixed, this motion contributes a term increasing with the square of step length (Fig. 3B). Even though leg motion is not related to step-to-step transition costs, it nevertheless affects the average external mechanical work rate.
Combining contributions from step-to-step transitions and limb motion,
simple bipedal models predict that when walking faster by increasing only step
length, the rate of external mechanical work
mech will be:
![]() | (3) |
Assuming constant muscular efficiency, the rate of mechanical work for
step-to-step transitions (Equation 3) translates directly into
met, a predicted metabolic rate:
![]() | (4) |
Experimental procedures
We measured the mechanical and metabolic costs of walking as a function of
step length in human adult subjects (N=9). All subjects (four male,
five female, body mass 66.0±8.4 kg; leg length 0.93±0.05 m;
means ± S.D.). were healthy and exhibited no clinical gait
abnormalities. Before the experiments began, volunteers gave their informed
consent to participate, in accordance with university policy.
We first measured each subject's preferred step length,
l*, and step frequency, f*, for
walking at 1.25 m s-1 on a treadmill. After allowing each subject
to acclimate to the treadmill for 10 min, we timed at least 100 steps at each
speed to find the average step period, which is the reciprocal of
f*. We then found l* by dividing speed
by f*. Average preferred step length was
l*=0.70±0.03 m and average preferred step frequency
was f*=1.81±0.07 Hz. Average step width, measured
in the same manner as by Donelan et al.
(2001), was 0.12±0.03 m
and did not change significantly with step length (P=0.44, ANOVA).
For all remaining trials, subjects walked at their own f*
by stepping to a metronome.
We measured ground reaction forces for subjects walking overground at six
different step lengths, keeping step frequency fixed. Subjects walked over two
ground-embedded force platforms mounted in series (described in detail in
Donelan et al., 2002), at
target speeds within the range 0.75-2.00 m s-1, presented in random
order. These speeds were chosen so as to produce multiples of each subject's
preferred step length: 0.6, 0.8, 1.0, 1.2, 1.4 and 1.6 l*.
The minimum step length was large enough to ensure that subjects could step on
two separate force platforms, and the maximum was close to the largest that
subjects could comfortably achieve. We discarded trials if the walking speed,
measured with photocells, was not within 0.05 m s-1 of the target
speed or if the individual feet did not fall cleanly on separate force
platforms. We analyzed data for three acceptable trials from each subject at
each of the step lengths. Reported values are averages from a single step,
beginning and ending with successive heel strikes, from each of the three
trials for each subject and condition.
In addition to the overground walking trials, we also conducted treadmill
trials to measure the metabolic cost of walking at six step lengths. Metabolic
cost was measured by indirect calorimetry using an open circuit respirometry
system (Physio-Dyne Instrument Co., Quogue, NY, USA). After first measuring
each subject's resting metabolic rate while standing, we then repeated the
same walking trials as above, with the exception of a
1.5l* condition (1.90 m s-1) in place of
1.6l* because subjects had difficulty maintaining the
longer step length for a sufficient duration without switching into a run.
Treadmill speed and metronome frequency were used to enforce step length and
frequency. Following a 3 min period to allow subjects to reach steady state,
we measured the average rates of oxygen consumption and carbon dioxide
production over 3 min, and calculated metabolic rate
(Brockway, 1987; described in
detail in Donelan et al.,
2001
). We subtracted the metabolic rate for standing from all
walking values and then divided by body mass to derive normalized net
metabolic rate,
met (W kg-1).
Data analysis
We calculated mechanical step-to-step transition costs using the individual
limbs method for quantifying external mechanical work
(Donelan et al., 2001;
reviewed in Appendix). Briefly, the external mechanical power generated by a
limb was found from the dot product of the limb's ground reaction force and
the velocity of the center of mass (see
Fig. 4 for intermediate results
showing these quantities). The magnitude of negative external mechanical work
per step was found from the time-integral of the negative portions of external
mechanical power generated by the limb
(Donelan et al., 2002
). We
determined the average normalized rate of negative external mechanical work,
Wtrans(-) (W kg-1), by dividing the
negative work for both limbs by body mass and step period.
|
The measures of step-to-step transition costs used here differ slightly
from our previous estimates, described by Donelan et al.
(2002). We previously
estimated transition costs as a function of step width using the negative
external mechanical work performed by the leading leg during double support
alone. While some negative work continued beyond double support, it probably
did not adversely affect our conclusions, as its magnitude was small. In the
present study, however, the leading limb performed substantial negative work
after double support during the longer step length conditions
(Fig. 4D). Integrating negative
power over the entire step therefore better quantifies step-to-step transition
costs for the conditions presented here.
A trade-off to quantifying external work over an entire step, rather than the double support phase alone, is that motion of the legs can affect our measurements. We expect that including the term Clegl2 in Equation 3 will underestimate Ctrans because external work from leg motion mathematically cancels some of the step-to-step transition costs (see Appendix). However, this estimate of the negative work of step-to-step transitions is sufficient to test the predicted relationship between step-to-step transition costs and step length given by Equation 3.
We used these data to test our predictions regarding step-to-step
transitions. We first tested whether the measured rate of mechanical work
increased with step length as predicted by Equation 3, and then tested whether
measured metabolic rate increased as predicted by Equation 4. These tests were
performed with a nonlinear regression to both equations, with
r2 and 95% confidence intervals (c.i.) indicating the
degree and significance of fit. Because the offsets D and D'
are purely empirical constants not predicted by the model, we performed the
regressions with an individualized offset subtracted from each subject's data.
To compare with previously reported data, we also calculated traditional
combined limbs measures of external mechanical work
(Cavagna, 1975) and percentage
recovery (Cavagna et al.,
1976
).
Finally, we tested whether metabolic rate increased in proportion to mechanical work rate, as would be expected if muscle performed this work at constant efficiency. We used a linear regression for this comparison, with r2 and 95% c.i. indicating the degree and significance of fit. The linear constant of proportionality was also used to estimate an efficiency, defined as negative external mechanical power divided by net metabolic power. We first estimated efficiency by performing a linear regression between total negative external mechanical power and net metabolic power. The result is probably an overestimate due to cancellation of swing leg work (see Appendix). To also estimate a lower bound on efficiency, we subtracted our estimated contribution of leg motion, Clegl2, from negative external mechanical power data (yielding a lower bound on negative step-to-step transition power), and then performed a linear regression between this transition power and net metabolic power. Using these methods, we were restricted to step lengths for which we collected both mechanical and metabolic data, i.e. excluding the longest step length condition.
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Results |
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Also in support of our hypothesis, metabolic rate associated with step-to-step transitions increased with the fourth power of step length (Fig. 5B). A nonlinear regression to Equation 4 yielded the coefficients Ctrans=0.877±0.060 W kg-1 m-4 (mean ± 95% c.i.) and D'=1.543±0.363 W kg-1 m-4 (mean ± S.D.) (r2=0.95). The metabolic step-to-step transition rate therefore increased from 0.11 W kg-1 to 5.75 W kg-1 over the range of step lengths we used.
Our estimates of the efficiency of step-to-step transitions ranged from 10%-25% (Fig. 6). A linear regression between mechanical costs, correcting for swing leg work, and metabolic costs yielded a slope of 0.10±0.02 (mean±95% c.i.) (r2=0.79), a lower bound on efficiency. Another regression, without the correction for swing leg work, yielded a slope of 0.25±0.03 (mean ± 95% c.i.) (r2=0.89), likely to be an overestimate of efficiency.
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Traditional combined-limbs measures of external mechanical work were on
average 31% less than individual-limbs measures
(Fig. 5A). The net metabolic
cost observed here was substantially higher than that for unconstrained normal
walking at the same speeds (e.g. by 87 W at 1.75 m s-1;
Tolani and Kram, 1999; ANOVA,
P=0.0011) but percentage recovery was not statistically different
(ANOVA, P=0.36).
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Discussion |
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Step-to-step transitions may also account for much of the overall metabolic
cost of freely selected gait. The rate of work for these transitions increases
sharply with step length, and to a lesser extent, step frequency (Equation 2).
Humans typically walk faster by increasing step length and step frequency in
almost equal proportion (for a review, see
Kuo, 2001), rather than by
increasing step length alone as in the present work. The preferred combination
of step length and step frequency minimizes metabolic cost of transport
(metabolic rate divided by speed, or energy per distance) at a given speed
(e.g. Elftman, 1966
), and is
expected to result in slightly lower, but still substantial, step-to-step
transition costs than were observed here.
In addition to step-to-step transitions, there appears to be another
substantial component to the metabolic cost of walking that depends more
heavily on step frequency (Atzler and
Herbst, 1927; Zarrugh et al.,
1974
). If step-to-step transitions alone determined the metabolic
cost of walking, they could be minimized by walking at high step frequencies
and short step lengths. The preferred combination of step length and frequency
(Elftman, 1966
) may be a
result of a trade-off between step-to-step transitions and a cost to
increasing step frequency, such as for moving the legs back and forth. Indeed,
our model of this trade-off predicts the preferred combination
(Kuo, 2001
). We aim to test
the cost of moving the legs, its trade-off against step-to-step transitions,
and the contribution of step frequency to step-to-step transitions (Equation
2) in future experiments.
The metabolic cost of high step frequencies does not, however, appear to be
proportional to work performed on the legs. The external work originating from
leg motion increases with the square of step length, corresponding to the term
Clegl2 in our mechanical cost
regression (Equation 3). But as predicted by our model, this work appears to
contribute negligibly to metabolic cost; addition of a similar term to the
metabolic cost regression (Equation 4) does not substantially improve the
degree of fit (r2 increases from 0.955 to 0.957). When
walking faster by increasing only step length, metabolic costs associated with
leg motion appear not to increase substantially. One possible explanation is
that metabolic cost depends more on the cost of producing force, rather than
work, to move the legs (Kuo,
2001). This would yield a large cost to high step frequencies that
would be nearly constant when step frequency is kept fixed.
Step-to-step transition costs depend not only on step length but also on
step width (Fig. 7A). We
previously studied transition costs as a function of step width while keeping
step length and frequency fixed and found that, as predicted, they increased
with the square of step width (Fig.
7B) (Donelan et al.,
2001). The present study examined the effect of step length while
keeping step width and frequency fixed, and found that transition costs
increased with step length to the fourth power
(Fig. 7C). These different
relationships are predicted by a single model of redirecting of the center of
mass between steps.
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There are other costs of walking that are not explicitly represented in our
models, as indicated by the y-intercept of the mechanical and
metabolic power curves (Fig.
5). These offsets (D and D') are important
in determining the magnitude of the minimum metabolic cost of transport and
the speed at which it occurs
(Schmidt-Nielsen, 1990).
Metabolic cost of transport is the metabolic energy required to move a unit
body weight or mass a unit distance, and animals prefer to move at speeds that
minimize this cost (Alexander,
1989
). A small part of the mechanical offset, D, may be
explained by step-to-step transition costs due to the non-zero step width.
There may be other mechanical work required of step-to-step transitions that
is not accounted for in our rigid body model, such as to restore energy
dissipated from flexible body deformations. It is also probable that there are
additional metabolic costs not attributable to step-to-step transitions or
external mechanical work, such as for supporting body weight, moving the legs,
moving other limbs, or controlling stability. However, our present data are
insufficient to resolve their contributions.
Another limitation is that even though our experimental data are consistent
with the proposed model, they also cannot preclude other possible
explanations. Our tests were based on a power law relationship predicted by a
simple model, in fact the simplest possible model based on mechanics
(Garcia et al., 1998). The
data fit of Equation 4 contains two coefficients, equivalent to a linear fit,
treating l4 as the independent variable. A linear fit can
confirm the statistical significance of the linear coefficient, but cannot
prove linearity. Our present results therefore do not prove that the
l4 term is exclusively superior to other possible terms.
In addition, polynomials with additional statistical degrees of freedom would
almost surely provide better fits. But a model capable of predicting such a
polynomial would also probably be more complex than the simple model
(Fig. 1A) proposed here. In
fact, the predictions of the more complex anthropomorphic model
(Fig. 1B) and our experimental
data are fitted nearly as well with l5, rather than the
l4 of the simple model. We feel that the present analysis
is a reasonable compromise between model simplicity (which facilitates
predictions made a priori) and goodness of fit. Not only is the model
simple, but its physical manifestation
(McGeer, 1990
) can also walk
down a slope with the same scaling of step-to-step transition costs as found
here.
Our results are based on a measure of the external mechanical work
performed by individual limbs. Traditional combined-limbs measures of external
mechanical work (Cavagna et al.,
1976) are prone to underestimating step-to-step transition costs
because they ignore simultaneous positive and negative mechanical work by the
trailing and leading legs (Donelan et al.,
2001
,
2002
). Measures of the total
mechanical work performed on the body and limbs
(Burdett et al., 1983
;
Cavagna and Kaneko, 1977
;
Willems et al., 1995
) include
work performed both for step-to-step transitions and to swing the leg, of
which the latter appears not to contribute to metabolic cost in a proportional
relationship. Our measure of external work by individual limbs appears to
better quantify step-to-step transitions than combined-limbs measures of
external work. It cannot differentiate the effects of swing leg motion, but is
less affected by this motion than measures of the overall mechanical work
performed on the body and limbs. Still better estimates might result from a
more complete separation of step-to-step transitions from swing leg motion,
perhaps through a joint power approach to estimating the mechanical work
performed by individual limbs (e.g.
Winter, 1990
) or through
measuring muscle mechanical work directly (e.g.
Prilutsky et al., 1996
;
Biewener and Roberts, 2000
).
The latter would assist in quantifying the degree to which transition work is
apportioned between work performed by muscle fibers, elastic energy stored and
returned by tendon, and energy dissipated in other structures (see
Appendix).
Step-to-step transition costs help to relate the observed metabolic cost of
walking with the inverted pendulum paradigm. An inverted pendulum by itself
conserves energy while the center of mass moves in a pendular arc. Yet one of
the enduring hypotheses of human walking is that it costs energy to produce
vertical excursions of the center of mass
(Saunders et al., 1953). The
present model of step-to-step transition costs predicts that larger vertical
excursions of the center of mass will indeed be correlated with, but do not
themselves cause, increasing metabolic cost. The vertical motion of an
inverted pendulum motion need not consume energy, but the transitions between
steps require mechanical work, and it is this work that consumes metabolic
energy. Longer steps result in greater vertical excursions of the inverted
pendulum, but more importantly, they incur higher step-to-step transition
costs. This theory is expected to apply not only to humans, but to any other
animals whose walking can be likened to an inverted pendulum.
In summary, humans perform substantial mechanical work to redirect the center of mass velocity during step-to-step transitions. This work exacts a proportional metabolic cost, consistent with studies on slope walking, rowing, cycling and isolated muscle. Legged animals vary widely in size, shape, and number of legs, but most walk with long steps and some walk with wide steps. The associated cost of step-to-step transitions may be a general and major determinant of the metabolic energy required for walking in all animals that make use of an inverted pendulum mechanism.
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Appendix |
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![]() | (A1) |
![]() | (A2) |
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Experimental details
We used the individual-limbs method to calculate external mechanical work
(Donelan et al., 2001,
2002
). The external mechanical
power (Fig. 4D,E), generated by
a limb is equal to the dot product of the limb's ground reaction force,
F, and the velocity of the center of mass, vcm. The
magnitude of negative external mechanical work per step,
W(-), performed by a limb is found from the time-integral
of the external mechanical power generated by the limb, restricted to the
intervals within each step over which the power is negative (denoted by the
domain NEG). Total negative individual limb external mechanical work per step,
WILM(-), is the summed magnitude of negative
external mechanical work from each limb. For a biped,
![]() | (A4) |
Analysis details
We use negative external mechanical power, averaged over an entire step, to
estimate the mechanical costs of step-to-step transitions. However, average
external mechanical work includes not only the work performed during
step-to-step transitions, but also work performed to move the legs and perhaps
energy fluctuations due to storage and return of elastic energy. Here we
discuss briefly how these separate contributions may affect total external
mechanical power and metabolic cost.
A large fraction of external mechanical work is due to the work required of step-to-step transitions (Fig. 4E,F). Most of this work occurs during the double support phase. But at longer step lengths, the stance leg performs some of the negative work extending beyond double support, into the beginning of single support (Fig. 4E; collision in Fig. A1). In addition, the stance leg performs some of the positive work prior to double support, at the end of single support (Fig. 4E; propulsion in Fig. A1).
|
While limb motion involves mostly internal work, it also contributes to the
work done on the center of mass because movement of the legs also results in
movement of the center of mass. This is true even if the legs move passively
or otherwise add no net mechanical energy over a step. In an anthropomorphic
two-dimensional model with a hip spring acting between the legs
(Kuo, 2001), motion of the
legs generates positive and negative external mechanical power at the
beginning and end of single support, respectively
(Fig. A1). The internal
mechanical power is equal and opposite, so that there is no net change in
total mechanical energy. The magnitude of negative external power increases
with the square of step length and the square of step frequency
(Fig. 3B). Keeping step
frequency fixed, moving the legs therefore contributes a term of the form
Clegl2 to our external mechanical
power regression model (Equation 3).
External mechanical power for moving the legs partially cancels power generated or dissipated during step-to-step transitions, making it impossible to separate the two contributions from force plate data alone (Fig. A1). This is a mathematical cancellation that is not representative of a physical cancellation, which would require a transfer of energy from one limb to the other. Inclusion of Clegl2 in a regression fit (Equation 3) will therefore underestimate Ctrans (and Cleg), making our estimate a lower bound on step-to-step transition costs. An alternative method is to exclude the swing leg from the regression. But exclusion of Clegl2 from Equation 3 will attribute all increases in total external mechanical power to Ctrans and none to Cleg. Though not a strict upper bound, the result is likely an overestimate of Ctrans. The results of such a regression are coefficients Ctrans=0.200±0.012 W kg-1 m-4 (95% c.i.) and D=0.314± 0.087 Wkg-1 m-4 (r2=0.96).
External mechanical work measured within a step may have contributions from
elastic energy storage and return. We consider three potential cases. First,
the external mechanical work for moving the legs
(Fig. A1; leg motion) could be
due to storage and return of elastic energy by hip tendons rather than work
performed by hip muscles (Alexander,
1990; Bennett,
1989
). This is supported by previous theoretical work
(Kuo, 2001
) and by our current
finding that metabolic cost is not proportional to work performed on the leg
(see Discussion). Second, positive external mechanical work as the leg extends
just prior to mid-stance (Fig.
A1; rebound) may be due to stored elastic energy during the
collision with ground (Fig. A1;
collision). Third, negative external mechanical work by the stance limb just
after mid-stance may reflect elastic energy being stored in tendon
(Fig. A1; preload). The
subsequent release of this stored energy would contribute to the positive
external mechanical work performed to redirect the center of mass velocity
(Fig. A1; propulsion). These
potential uses for storage and return of elastic energy represent
opportunities to save on work performed by muscle fibers and therefore to
reduce metabolic cost. If this reduction is substantial, the measured
metabolic cost could potentially differ from the prediction of Equation 4. The
present study, however, is insufficient to quantify elastic energy storage,
which is best measured in vivo
(Prilutsky et al., 1996
;
Biewener and Roberts,
2000
).
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Acknowledgments |
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