A new model predicting locomotor cost from limb length via force production
Harvard University, Department of Anthropology, 11 Divinity Avenue, Cambridge, MA 02138, USA
e-mail: pontzer{at}fas.harvard.edu
Accepted 15 February 2005
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Summary |
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Key words: biomechanics, locomotion, cost of locomotion, energetics, force production, humans, running, walking
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Introduction |
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Physiological studies have demonstrated empirically that the metabolic cost
of locomotion primarily derives from the muscular force required to accelerate
the body's center of mass as it oscillates through the stride cycle. Notably,
the rate of force production, as proposed by Kram and Taylor
(1990), predicts the rate of
oxygen consumption during locomotion more accurately than other parameters,
including the work done in moving the center of mass and limbs through a
stride cycle (Heglund et al.,
1982
; Cavagna and Kaneko,
1977
). Added-mass studies of quadrupeds and bipeds
(Taylor et al., 1980
;
Kram, 1991
;
Wickler et al., 2001
;
Griffin et al., 2003
) and
gravity-manipulation studies of running humans
(Farley and McMahon, 1992
)
have shown that the muscular force needed to generate the vertical component
of ground reaction force, acting in opposition to gravitational acceleration,
accounts for the majority of locomotor cost (see
Taylor, 1994
). In addition,
the muscular force generated during braking and propulsion, associated with
the horizontal component of ground force generation, also contributes as much
as one-third of locomotor cost (Chang and
Kram, 1999
; Gottschall and
Kram, 2003
). Comparisons of bipeds and quadrupeds (Roberts et al.,
1998a
,b
)
suggest these determinants of locomotor cost work similarly for both, although
differences in muscle fiber length and effective mechanical advantage of the
limb joints may lead to higher costs for bipeds.
The relationship between limb length and locomotor energy cost is less
clear. Alexander and Jayes
(1983) initially proposed that
various gait parameters, including locomotor cost, are dynamically similar and
would scale by Froude number, a dimensionless constant that corrects for size
between pendular systems. Thus, the cost of locomotion for a given animal at a
given speed could be predicted by calculating the Froude number,
U2(Lg)1, where
U is travel speed and L is limb length. While there is some
support for this proposal from studies of human walking
(Alexander, 1984
;
Minetti et al., 1994
), Froude
numbers do not predict the scaling of cost or kinematic parameters during
running (Minetti et al., 1994
;
Donelan and Kram, 2000
).
Furthermore, recent studies that manipulate gravity during walking have shown
that stride length (Donelan and Kram,
1997
) and energy cost (Farley
and McMahon, 1992
) are not constant at a given Froude number,
suggesting dynamic similarity as proposed by Alexander and Jayes
(1983
) may not adequately
describe walking mechanics.
An inverse relationship between locomotor cost and limb length has also
been proposed for running gaits, which are typically considered to act as
mass-spring systems rather than pendular systems. Kram and Taylor
(1990) suggested that `larger
animals with longer limbs and step lengths will have lower transport costs,'
as the magnitude of the vertical impulse decreases with longer stance periods.
However, while stance phase duration, or contact time, tc,
has been shown to correlate with limb length
(Hoyt et al., 2000
), numerous
within- and between-species studies have found no relationship between limb
length and the cost of locomotion (walking humans,
Censi et al., 1998
; running
humans, Ferretti et al., 1991; Cavanaugh
and Kram, 1989
; Brisswalter et
al., 1996
; interspecific studies,
Steudel and Beattie, 1995
). In
the best study to date comparing short- and long-legged humans (mean limb
length 79 cm and 95 cm, short- and long-legged groups, respectively), Minetti
et al. (1994
) found locomotor
cost was lower for long-legged individuals during walking but higher during
running. This difference between walking and running gaits, and the lack of a
clear relationship between limb length and locomotor cost, suggests a simple
relationship between limb length and locomotor cost is unlikely.
One complication in predicting the effect of limb length on locomotor
efficiency is the cost of accelerating the limb during swing phase. While
initial studies suggested swing cost is negligible (Taylor et al.,
1974,
1980
;
Mochon and McMahon, 1980
),
more-recent studies have demonstrated that the muscular force required to
swing the limb can constitute a significant portion of total locomotor cost.
Several studies have measured the increase in energy cost when mass is added
to the limb and have shown that energy costs increase directly with increased
moment of inertia (Martin,
1985
; Myers and Steudel,
1985
; Steudel,
1990
). More recently, Marsh et al.
(2004
) in a study of guinea
fowl, measured energy consumption using blood flow in the limb muscles of
guinea fowl and found limb swing contributed over 20% of total locomotor cost
over a range of speeds. Thus, it appears a tradeoff may exist between the
force required to support the body and the force required to swing the limbs.
As limb length increases contact time increases and a lower rate of force
production is necessary to support the body, but more force is required to
swing the longer limb.
The model presented here predicts both the force required to support
bodyweight and the force required to swing the limb as functions of limb
length and proportion. Following Kram and Taylor
(1990), the predicted rate of
force production (i.e. the mean muscular tension required per step multiplied
by step frequency) is then used to predict the rate of oxygen consumption.
Predicted oxygen consumption is then tested against observed oxygen
consumption in a sample of human recreational runners over a range of running
and walking speeds. These results show that the model provides a useful
framework for understanding the link between limb length and the cost of
locomotion for both walking and running, incorporating both the cost of
supporting the bodyweight and swinging the limb.
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Materials and methods |
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The primary assumption of the LiMb model follows directly from the force
production hypothesis (Kram and Taylor,
1990; Taylor,
1994
): the mass-specific rate of energy consumption is a linear
function of the rate of muscular force production. This assumption requires
that the relative shortening velocities of muscles and the force exerted on
the ground per unit of active muscle are independent of body size and speed.
Empirical studies suggest these criteria are met. In vivo studies
have shown that most muscles supporting bodyweight during locomotion contract
isometrically during steady walking and running, and therefore at a constant
relative shortening velocity. This has been demonstrated in turkeys
(Roberts et al., 1997
),
walking humans (Fukunaga et al.,
2001
), and wallabies (Biewener
et al., 1998
), although other studies suggest shortening
contractions may be employed as well
(Gillis and Biewener, 2001
;
Daley and Biewener, 2003
).
Nevertheless, as long as the relative amount of muscle shortening work does
not change with speed or size, this assumption still holds. Additionally,
interspecific comparisons over a range of animal size have demonstrated that
the effective mechanical advantage, EMA, of extensor muscles scales inversely
with muscle fiber length (EMA
Mb0.26,
Biewener, 1989
; fiber length
Mb0.26,
Alexander et al., 1981
)
suggesting that a given volume of muscle should exert the same force on the
ground independent of body size. Indeed, one important result of Kram and
Taylor (1990
) was that the
ratio of metabolic energy expenditure to the rate of muscular force production
was constant across running speed, body mass, and species. Thus the available
evidence suggests that energy expenditure during locomotion may be predicted
by the rate of muscular force production without including major complexities
of muscle physiology.
Total predicted force production for the LiMb model is considered to depend on three components: vertical force, horizontal force and limb swing. Estimated force production for each component is derived separately.
Vertical forces
Vertical force production for both walking and running is estimated from
the change in vertical momentum of the body's COM. While `passive' mechanisms,
such as energy storage and release via tendons or the exchange of
potential and kinetic energy, may reduce the mechanical work done by the
muscles thereby improving energy economy
(Roberts et al., 1997), such
mechanisms still require muscular force to prevent the limb from collapsing
(i.e. to support bodyweight) and, to the extent the muscles perform true
mechanical work, to lift the COM. Therefore, in the LiMb model, positive
(upward) accelerations are viewed as a product of muscular force production
while negative (downward) accelerations of the COM are a product of gravity.
`Passive' mechanisms to reduce mechanical work, not considered explicitly in
the LiMb model, can be viewed as the efficiency with which oxygen consumption
is translated into force production. Accelerations of the COM require an
equivalent muscular force; passive mechanisms mediate the cost of producing
this force. Because the COM experiences free-fall during running but not
during walking, vertical accelerations are linked to kinematic and anatomical
variables differently for each gait and are, therefore, derived separately for
walking and running.
1. Running
Vertical force production during running is derived as follows (see
Fig. 1): during steady running
on level surface, the positive (upward) vertical acceleration,
ay, produced by muscular force must be equal in magnitude
to that of gravity, g. Furthermore, average positive vertical
acceleration during contact time, tc, must equal that of
gravity during step period, Tstep, or
tcay=Tstepg.
For a simple two-dimensional model in which the limbs are treated as simple
cylinders with no feet, the knees are modeled as telescoping (prismatic)
joints, and protraction of the hind limb is equal to retraction, contact time
during one step is a function of hind-limb length L, excursion angle
, and running speed U, such that:
![]() | (1) |
![]() | (2) |
![]() | (3) |
![]() | (4) |
|
Eq. 4 is similar to the prediction for locomotor cost proposed by Kram and
Taylor (1990; their equation
1: COLrun=kgtc1), with
the exception that tc1 has been replaced
with the equivalent expression
(U[2Lsin(
/2)]1). However, by
incorporating hind-limb length, running speed and excursion angle as
independent variables, this model has greater utility for comparative studies
investigating the different effects of these variables on locomotor costs.
2. Walking
Vertical force production during walking is predicted as follows (see
Fig. 2). In a walking gait the
COM follows a sinusoidal trajectory, alternately accelerating upward
via muscular force production and downward via gravity. The
rate of muscular force production necessary to achieve this change in momentum
(i.e. to prevent the limb from collapsing) is a function of the vertical
change in position of the COM through stance phase (i.e. the amplitude of
oscillation) and the duration of each step. In a simple `stick-figure' model,
the amplitude equals L[1cos(/2)], and step duration
equals U1[2Lsin(
/2)]. Given equal
periods of upward and downward acceleration of duration
U1[Lsin(
/2)], the average vertical
velocity of the COM during the first half of stance phase equals
+L[1cos(
/2)](U[Lsin(
/2)])1,
while the average vertical velocity during the second half equals
L[1cos(
/2)](U[Lsin(
/2)])1.
Note that in this case the vertical velocity of the COM is a function of
walking speed, which determines both the horizontal and vertical velocity of
the COM as is traverses its sinusoidal trajectory with gravity acting as a
restoring force. The movement of the COM in this case is analogous to that of
a ball rolling along a sinusoidal track; the forward speed of the ball also
determines its vertical velocity.
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Assuming maximum vertical velocity is equal to twice the average vertical
velocity (the mathematically simplest case), the change in velocity between
steps (i.e. during the trough of the COM trajectory) must equal
4L[1cos(/2)]. This change in velocity occurs over a
period of time equivalent to
U1[Lsin(
/2)], and therefore average
vertical acceleration for one step is given by:
![]() | (5) |
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In addition to the force needed to perform this change in momentum (Eq. 6), there is the constant acceleration of gravity, g. Here, I make the simplifying assumption that the force required to resist gravity and maintain an upright posture is equivalent to the metabolic cost of standing quietly, although these costs probably differ. As the cost of standing is subtracted from the `net' cost of locomotion, it is not included in the model prediction of walking cost.
Note that the form of Eq. 6 is similar to a Froude number in that the rate of energy expenditure scales with U2 and L1. However, Eq. 6 differs in that gravitational acceleration is not included: vertical force production is a function of inertia, not weight. Furthermore, stride frequency and excursion angle are included. The implications of these differences are discussed below.
Horizontal forces
To estimate horizontal forces, I make the simplifying assumption that the
combined vertical and horizontal ground reaction forces, GRF, produce a
resultant vector that passes through the COM throughout stance phase.
Empirical studies of ground reaction forces suggest this assumption is valid
(Chang et al., 2000;
Lee et al., 2004
), and that
horizontal GRF covaries with vertical GRF
(Breit and Wahlen, 1997
;
Chang et al., 2000
).
Furthermore, it is assumed that horizontal deceleration during the first
portion of stance phase and horizontal acceleration during the second portion
are generated via muscular contraction.
Instantaneous horizontal force (see Fig.
3) during stance phase must, therefore, equal
Fitani, where Fi is
the instantaneous vertical GRF and
i is the instantaneous
protraction or retraction angle of the limb. The summed horizontal force for
one stance phase, Fx, is therefore:
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|
The time course of both vertical ground force production and excursion
angle are, therefore, necessary to compute Fx. To
approximate this value, I assume the average force for both braking and
propulsive impulses, , is
equivalent to the product of the mean value for
i and the
mean vertical mass-specific GRF during stance phase,
. The mean value
for
i is equivalent to 0.5(
/2),
for the braking
or propulsive GRF is estimated as
,
and the combined horizontal force production (braking + propulsion) for one
step is estimated as:
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Limb swing
The work done to swing the limb can be calculated using the equation for
work done on a pendulum (Hildebrand,
1985):
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![]() | (16) |
Dividing both sides of this equation by body mass Mb
produces the mass-specific rate of force production necessary to swing the
limb. This rate of force production is the predicted cost of limb swing,
Climb:
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Total mass-specific force production (vertical + horizontal + limb swing)
for walking and running is, therefore, predicted via the LiMb model
as:
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![]() | (19) |
Testing the LiMb model
To test the LiMb model, nine human subjects (five male, four female; body
mass range: 53.394.3 kg) volunteered to perform a set of walking and
running trials at a range of speeds on a custom-built treadmill (tread
dimensions: 2x0.6 m) at the Concord Field Station in Bedford, MA, USA.
Subjects were recruited to maximize variation in limb length (range:
79112 cm); all were healthy, fit, recreational runners (self reported
miles/week running: 625, median N=15) ages 2035 with no
history of running-related injury or illness. Subjects wore their personal
running shoes for all trials. IRB (Human Subjects Committee) approval was
obtained from Harvard University prior to the study, and written informed
consent was obtained from each subject prior to participation. Subjects were
paid for their participation in accordance with Harvard University IRB
guidelines.
Limb length was measured as the vertical distance from the greater trochanter, determined by palpation, to the floor while shod. Subjects walked at four speeds, ranging from 1.02.5 m s1, or their fastest sustainable walking speed, and ran at three speeds ranging from 1.753.5 m s1. The range of running speeds was tailored to each subject such that the slowest running speed was slower than the subject's volitional walkrun transition speed. Subjects performed 610 min trials at each speed while wearing a loose-fitting mask. Air was pulled through the mask at 200300 l min1; this air was sampled and oxygen concentration monitored at 5 Hz using a paramagnetic analyzer (Sable Systems PA-1B; Las Vegas, NV, USA). The system was checked for leaks for each trial by bleeding N2 into the mask at a known rate and plotting (N2 rate/mass-flow rate) against the observed decrease in O2 content of the sub-sampled air; this relationship was consistent across trials (N=9, r2=0.98, P<0.001, second-order polynomial regression, no outliers).
Oxygen consumption was monitored during the trial in real time to ensure
that steady-state aerobic metabolism was achieved, and the rate of oxygen
consumption VO2 (s1) was measured
following Fedak et al. (1981)
using data from the last minute of each trial. The resting rate of oxygen
consumption, measured while standing for 6 min prior to the start of locomotor
trials, was subtracted from the rate of oxygen consumption for each trial, and
the difference divided by body mass to calculate the COL
(VO2 kg1 s1) for each
trial, where VO2 is in ml.
Kinematic data was also collected during each trial using a high-speed infrared camera system (Qualysis®; Qualysis Motion Capture Systems, Gothenburg, Sweden) operating at 240 Hz. Reflective markers were adhered to the skin overlying the greater trochanter and to the subject's shoe over the calcaneal tuberosity and distal fifth phalange. Qualysis® data analysis software was then used to measure: protraction angle at heelstrike (heeltrochanterfloor), retraction angle at toe-off (toetrochanterfloor), contact time duration (heel strike frametoe-off frame) and stride frequency. Swing period was estimated as (stride frequency)1. These variables, with speed, were used to calculate the rate of force production for each trial using equations from the model.
Estimating limb swing cost
To determine limb swing costs it was necessary to estimate the radius of
gyration, mass and resonant frequency of the limb for each subject. Limb mass
was estimated as 16% of body mass, following Dempster
(1955). The radius of gyration
D=hL, where h is a measure of mass distribution; a
value of 0.56 was used for h following Plagenhoef
(1966
).
The resonant period T is also a function of limb length L
and shape h, as
T0=2[I(MgL)1]0.5,
and
I=h2L2ML.
Thus:
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Predictions
To test the utility of the LiMb model, predicted COL was plotted against
observed COL. This was done using the equations for vertical force production
(Eq. 4, 6), vertical plus horizontal force production (Eq. 13, 14), and total
force production (Eq. 18, 19) to determine the contribution of each component
in predicting COL. It was predicted that each additional component would
improve the correlation between predicted and observed COL.
Next, the LiMb model (Eq. 18, 19) was compared with tc1 and Froude number as alternative predictors of COLrun and COLwalk, respectively. The LiMb model was predicted to outperform contact time and Froude number in predicting locomotor cost, as the model incorporates horizontal force production and limb swing costs.
Finally, I tested the prediction that k, the constant relating
oxygen consumption to force production, was the same for walking and running
gaits. This was predicted because walking and running employ isometric
contractions in the same muscle groups over similar ranges of limb excursion.
Therefore, muscle length, effective mechanical advantage and relative
shortening velocity and, therefore, k, should be independent of gait.
In fact, Kram and Taylor
(1990) found k was
nearly constant across a large range of body sizes and limb design (e.g.
rabbit to horse); it seems likely, therefore, that k should be
similar between gaits within a species.
Least squares regression was employed in each of the above tests to
determine the percentage of observed variation explained by a given predictor
with each trial treated as an independent data point. To test for differences
in k between gaits, k was determined as the slope of the LSR
for predicted versus observed COL for walking and running, and these
slopes were compared following Zar
(1984, pp. 292).
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Results |
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For walking, vertical force production (Eq. 6) and vertical + horizontal force production (Eq. 14) each explained 92% of the variance in observed COLwalk (N=34, r2=0.92, P<0.001 for both conditions), while total force production (Eq. 19) explained 94% (N=34, r2=0.94, P<0.001) (Fig. 5). Thus, while predicted force production explained a much higher percentage of the variance in observed COLwalk, the horizontal-force and limb-swing components did not improve the correlation between predicted and observed COLwalk significantly.
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The model performed as well or better than other predictors of cost. While
predicted COLrun (Eq. 18) explained over 40% of the variance in
observed COLrun, the inverse of contact time predicted less than
30% (N=27, r2=0.29, P<0.01). For
walking, predicted COLwalk (Eq. 19) predicted over 90% of the
variance in observed COLwalk, as did Froude number (N=34,
r2=0.91, P<0.001). While the model
outperformed (i.e. produced greater correlation coefficients) contact time and
Froude number as predictors of observed COL, comparisons of correlation
coefficients (Zar, 1984, pp.
313) revealed these differences were not significant (P>0.05).
The percentage of estimated total force production contributed from vertical, horizontal, and limb-swing components was similar for walking and running. For running, vertical force production accounted for 63.7% (S.D. ±7%) of estimated force production, while horizontal forces accounted for 19.8% (±1%) and limb-swing 16.5% (±6%). For walking, vertical forces accounted for 49.7% (±11%), horizontal forces for 21.3% (±5%), and limb-swing for 29% (±15%) of total estimated forces. However, limb-swing estimates are considerably lower when only normal walking speed (1.5 m s1) is considered; vertical forces at this speed, rated as the most `comfortable' speed by subjects, account for 59.8% (±5%), horizontal for 25.0% (±3%), and limb-swing for 15.2% (±8%) of total estimated force production. The contribution of each force component changes markedly with speed, particularly for walking, as shown for a representative subject in Fig. 6. Clearly, the proportion of total estimated force production was not correlated with the proportion of variance in COL explained by a given component.
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The force/oxygen constant, k, was not significantly different between gaits. The slope of the LSR for predicted versus observed COLrun was 0.0044, which was not significantly different than the slope for walking (slope=0.0045; P>0.05). Similarly, the y-intercept of the LSR equations for walking (0.021) and running (0.012) were not significantly different (P>0.05).
Values for k from this dataset are lower than that reported by
Kram and Taylor (1990) (mean
k=0.0092, S.D.=0.0022). This is probably a
result of estimated forces being greater when incorporating vertical,
horizontal and limb swing costs as in the LiMb model, rather than only
considering vertical forces as in Kram and Taylor
(1990
). When only vertical
force production is used to predict COL (Eq. 4, 6), the value for k
given by LSR is 0.0061, which is near the 95% confidence interval calculated
from Kram and Taylor (1990
)
(95% CI=0.00630.0119).
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Discussion |
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Vertical force production accounted for 50% or more of total estimated force production across speeds for both walking and running gaits, more than horizontal force and limb-swing combined. However, while vertical force production alone is a good predictor of COL for walking, it is a poor predictor in running in this dataset, with horizontal force and limb-swing contributing markedly to the predictive power of the LiMb model. Thus, the proportion of total force accounted for by a given component does not necessarily reflect the power of that component in predicting COL. This might be expected, as the regularity with which a given force component increases with total cost, and therefore the predictive power of that component, need not necessarily correspond with the magnitude of the force. Furthermore, force components that are highly correlated will not improve the fit of the model when combined. For example, estimated vertical and horizontal forces are highly correlated for walking (N=34, r2=0.99, P<0.01) but less so for running (N=27, r2=0.48, P<0.01); consequently, combining horizontal and vertical force production improves the fit of the LiMb model for running, but has no effect for walking (Table 1). This distinction between the magnitude of a given force component and its reliability as an index of COL may be relevant to studies seeking to identify discrete components of locomotor cost.
If muscular force production is the primary determinant of energy
expenditure during locomotion, predicted rate of force production should
relate to observed COL similarly across gaits in which similar muscle groups
and shortening velocities are employed. Indeed, this appears to be the case;
the relationship between predicted force production and observed energy
expenditure, as determined by LSR, is similar for walking and running
(Fig. 7). Thus, while no single
model may successfully describe the mechanics of mass-spring running gaits and
pendular walking gaits (Donelan and Kram,
2000), results of this study suggest models using a common
paradigm of force production to predict energy cost may be successful across
different gaits and activities. If so, it may be possible to compare directly
force production and cost across widely different activities (e.g. walking
versus climbing), providing a new means of comparing anatomical
specialization and locomotor performance.
|
Walking
The LiMb model was particularly effective in predicting COLwalk.
Predictions of the model fit observed COLwalk
(r2=0.94) as well or better than other proposed models for
walking cost, including Froude number (r2=0.91) and the
collision model proposed by Donelan et al.
(2002,
fig. 6;
r2=0.89). The success of the model for walking suggests
the cost of walking is primarily a function of the change in momentum inherent
in the sinusoidal trajectory of the COM through a stride: the upward
acceleration of the COM through the troughs of this trajectory requires
muscular force. In addition, limb swing contributes to cost, particularly at
higher walking speeds in which swing periods are far shorter than the natural
period of the lower limb. The `determinants of gait' described by Saunders et
al. (1953
) and others serve to
minimize walking cost by lowering the amplitude of the COM trajectory and by
increasing the duration (thereby decreasing the magnitude) of upward
acceleration.
Because walking cost is predicted as a function of inertia rather than
weight, the LiMb model for walking may explain deviations from dynamic
similarity reported previously (Farley and
McMahon, 1992; Donelan and
Kram, 1997
). The LiMb model predicts relative stride length,
Srel (stride length/limb length), will be a function of
excursion angle (Srel
4sin
/2) independent of
gravity. Similarly, COLwalk is predicted to be dependent on speed
but independent of gravity, as the positive (upward) change in momentum of the
COM during walking is a function of speed, not gravity (see Eq. 5, 6). Both of
these predictions run counter to dynamic similarity, which predicts relative
stride length to be inversely proportional to g, and
COLwalk to be a function of Froude number
(UL1 g1). Results from reduced
gravity experiments, in which gravitational acceleration, g,
is manipulated via a harness, fit LiMb model predictions better than
those of dynamic similarity: stride length
(Donelan and Kram, 1997
) and
COLwalk (Farley and McMahon,
1992
) were found to be largely independent of g
but not walking speed. Thus, while the LiMb model did not explain
significantly more of the variance in observed COLwalk than Froude
number, it does appear to outperform predictions of dynamic similarity in
reduced-gravity conditions.
Preferred step length, step frequency and speed relationships noted
previously (see Bertram and Ruina,
2001) for walking humans may also be explicable via the
LiMb model. Speed is equivalent to the product of step frequency,
0.5f, and step length, 2Lsin
/2 and, therefore, at any
given speed a range of step frequencies and step lengths are possible.
However, COLwalk for a given speed is predicted (Eqn 14) to
increase more steeply with frequency than with excursion angle. Thus the LiMb
model predicts long step lengths and low frequencies to be preferred, which
may explain why humans do not minimize step length during walking as predicted
by collision models of walking mechanics
(Donelan et al., 2002
).
Indeed, an interesting tradeoff may exist: high frequency and short steps
impose high costs as predicted by the present model while greater step length
increases collision costs, resulting in a U-shaped cost/step-length curve for
a given speed. If so, at any given speed, there will be one frequency/step
length combination that minimizes combined cost. Support for this hypothesis
is offered by Bertram and Ruina
(2001
), who investigated
walking speeds, step frequencies and step lengths chosen when one of these
variables was constrained. The frequency/speed relationships chosen under the
three constraint conditions were consistent with an energy-minimizing
strategy.
Running
The relationship between predicted and observed COLrun shows
considerably more variation (r2=0.43) than in walking
(Fig. 4). While the LiMb model
outperforms contact time (r2=0.29) as a predictor of cost,
the variation between predicted and observed cost begs explanation. One likely
source of increased variance in COLrun versus
COLwalk is between-subjects differences in the force/oxygen
constant, k. The LiMb model (Eqns 18, 19) predicts COL assuming that
k is constant across subjects, but this is unlikely. Differences in
the k (measured as COL/tc) have been noted
previously (Weyand et al.,
2001) and might be expected, as differences in variables, such as
muscle-fiber type, running mechanics and limb-proportion, will likely lead to
differences in the efficiency with which oxygen consumption is translated into
force production. Because k determines the slope of the
predictedobserved COL regression, differences in k will lead
to greater variance in running trials, in which predicted force production is
greater. To examine whether differences in k explain the variance in
COLrun, k was determined for each subject empirically as
the slope of the LSR between predicted and observed COL for all trials
(walking and running). While the fit of the LSR for each individual was
excellent (N=7 trials per subject, mean r2=0.98,
range: 0.930.99, P<0.001 for all subjects), differences
were observed in estimates of k (mean=0.0043, range
0.00280.0054, N=9 individuals, see
Fig. 8A). Using these estimates
of k to predict oxygen consumption via Eq. 18 and 19 reduced
the amount of unexplained variance considerably, and the fit of the LiMb model
was similar for walking (N=34, r2=0.95,
P<0.001) and running (N=27, r2=0.87,
P<0.001) (Fig. 8B).
This suggests between-subjects differences in k explain most of the
variance from predicted COL, but further work is necessary to test this
hypothesis.
|
As in walking, the LiMb model for running agrees with previous results from
reduced gravity experiments. COLrun is expected to increase
proportionally with gravity (Eq. 19), but be independent of body mass, because
COLrun predicted by the LiMb model is mass-specific. Farley and
McMahon (1992) reported
COLrun increases in direct proportion to gravity. Furthermore,
Chang et al. (2000
), in an
investigation of the separate effects of gravity and inertia on running
mechanics, found vertical and horizontal force impulses (a measure of
predicted cost via the present model) changed in direct proportion to
gravity but were independent of body mass. Another study, examining the effect
of gravity on walkrun transition speeds
(Kram et al., 1997
), found
preferred transition speeds decrease in proportion to gravity. Because
COLrun changes in proportion to gravity while COLwalk is
independent, the model predicts walkrun transition speeds, approximated
as the speed that COLrun=COLwalk, to decrease with
decreased gravity, in agreement with Kram et al.
(1997
).
Limb swing
Limb swing costs as predicted by the LiMb model are consistent with
previous studies in that these costs are low at normal walking speeds but
considerably greater at fast walking and running
(Fig. 6). However, predicting
limb swing cost is complicated by the necessity of estimating an oxygen/force
constant, b, and by the necessary constraint that hind-limb inertial
properties are estimated. This dependence on b is especially critical
for running, in which estimates of COLrun have a greater impact on
the fit of the model. At low values of b (e.g. b<10),
estimated swing costs relative to the cost of accelerating the COM are so low
as to be negligible, and the fit of the model for running does not exceed that
for Eq. 13. Similarly, at high values for b (e.g. b>100),
predicted limb swing cost dominates predicted COLrun, and the fit
of the model is diminished. Using an estimate of b that produces
swing costs similar to those reported by Marsh et al.
(2004) produces a good fit,
but further work is necessary to determine if this value (b=30) is
reasonable. For example, it is clear that limb proportion and therefore
inertial properties differ between guinea fowl and humans. Similarly, while
the method used here to estimate hind-limb inertial properties produces a
reasonable fit to the data, future work needs to improve these estimates by
incorporating anatomical data from individual subjects.
The effect of limb length on COL
The LiMb model predicts a somewhat complicated relationship between limb
length and COL in which longer limbs decrease the cost of accelerating the COM
(Eq. 13, 14) but increase the cost of limb swing (Eq. 18, 19). Data from this
study as well as others strongly suggests longer limbs do in fact decrease the
magnitude of vertical ground forces (i.e. the change in vertical momentum of
the COM) at a given speed. In the present human sample, vertical ground force
at a given speed, estimated as tc1
U1, was significantly greater for subjects with
shorter legs (Fig. 9), a result
predicted by the LiMb model (Eq. 4, 6). Similarly, Hoyt et al.
(2000) found contact time was
strongly correlated with limb length in comparisons between species. However,
while longer limbs decrease the force necessary to support bodyweight, the
increased cost of swinging longer, heavier limbs apparently eliminates a
simple univariate relationship between limb length and locomotor cost. As a
result, COL in this study was negatively correlated with limb length only at
moderately fast walking speeds (2.0 m s1:
r=0.87, P<0.01, N=9) where the cost of
accelerating the COM is high but swing cost is low. The trade-off between the
force needed to accelerate the COM and that needed to swing the limb obviates
a simple relationship between limb length and COL, at least for within-species
comparisons in which swing costs explain much of the variance in COL.
|
This trade-off may be less salient for comparisons of COL between species,
resulting in a simple inverse relationship between limb length and COL. Kram
and Taylor (1990) and other
studies (Taylor et al., 1974
,
1980
;
Taylor, 1994
) have suggested
the force produced to support bodyweight determines the scaling of COL with
body size between species. Swing cost, in contrast, may be less important in
between-species comparisons as decreases in stride frequency offset increases
in limb length with body size (Hildebrand,
1985
; Heglund and Taylor,
1988
). If the force produced to accelerate the COM does in fact
determine the scaling of COL, the LiMb model predicts COL to scale inversely
with limb length: vertical and horizontal ground forces are a product of
L1 (Eq. 13, 14). This prediction is supported by
interspecific comparisons of COL. Because limb length scales as
Mb0.33
(Alexander et al., 1979
), the
LiMb model predicts COL to scale as
Mb0.33. This exponent (0.33) is
similar to the scaling relationship reported by Taylor et al.
(1982
; exponent: 0.32,
95% CI 0.29 to 0.34). This agreement between predicted and
observed scaling suggests the LiMb model may be useful for between- as well as
within-species investigations of locomotor cost. Moreover, it suggests limb
length may drive the interspecific scaling of COL, as suggested by Kram and
Taylor (1990
) and others.
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Abbreviations |
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Acknowledgments |
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References |
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