A model of scale effects in mammalian quadrupedal running
1
Harvard/MIT Division of Health Sciences and Technology, Physical Medicine
and Rehabilitation, Harvard Medical School, Artificial Intelligence
Laboratory, MIT, 200 Technology Square, Room 820, Cambridge, MA 02139,
USA
2
Division of Engineering and Applied Science, Harvard University,
Cambridge, MA 02138, USA
Deceased
* Author for correspondence (e-mail: huang{at}ai.mit.edu )
Accepted 23 January 2002
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Summary |
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Key words: biomechanics, locomotion, running, mammal, quadruped, body size, leg stiffness, metabolic cost of transport, computational model, limb control
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Introduction |
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To capture the diversity of animal design and locomotory performance, more
detailed models of morphology, musculoskeletal mechanisms and motor control
are needed (Full and Koditschek,
1999; Kubow and Full,
1999
). Here, we develop a model of running that is based on the
body structure and stride-to-stride dynamics of a variety of quadruped
species. This work builds upon recent computer simulations of a running horse
(Herr and McMahon, 2000
,
2001
) by extending the theory
to animals of different size.
The broad aim of this work is to explain how the extensive data on
locomotory scale effects are related. It is not fully understood, for example,
how the size-dependence of leg stiffness and limb excursion angle in trotting
and galloping quadrupeds may be related to the size-dependence of the
metabolic cost of transport. Here, we hypothesize that a single, integrative
model of mechanics, control and energetics can predict how important features
of running change with size in mammalian quadrupeds. To test the hypothesis,
we conduct computational experiments on six morphologically realistic animal
models (`virtual animals') ranging in size from a chipmunk (0.115 kg) to a
horse (676 kg). Each virtual animal trots and gallops in numerical simulations
using the following set of biologically plausible strategies, to be justified
below: (i) each stance limb acts as a linear spring of constant stiffness
(Cavagna et al., 1988;
Blickhan, 1989
;
McMahon and Cheng, 1990
); (ii)
forward motion is powered and controlled by active hip and shoulder torques;
and (iii) metabolic cost is predicted from the time course of supporting body
weight (Kram and Taylor,
1990
). With overall leg stiffness constrained by published
experimental data (Farley et al.,
1993
), we select model parameter values such that the virtual
animals remain upright and the running kinematics is smooth and periodic.
Finally, we test the model for internal consistency by comparing the
simulation results with available experimental data. We would reject the
model's set of assumptions if we were to find discrepancies between
predictions and data (e.g. in metabolic cost).
To summarize, we ask whether there exists a set of biologically plausible locomotory principles that, when specified in the formulation of our model, unifies well-known features of quadrupedal trotting and galloping across body size. It is our belief that answering this question will be a step towards identifying mechanisms of gait performance in a wide variety of terrestrial mammals and, ultimately, a more unified theory of locomotory mechanics, control and energetics.
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Materials and methods |
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Similar to the horse model of Herr and McMahon
(2000,
2001
), each limb was formed
with an upper and a lower segment connected by a prismatic (telescoping) joint
such that the limb could change length. The simplifying assumption of
prismatic joints for the elbows and knees was justified because the quadruped
limbs were lightweight (limb mass <7 % of total body mass)
(Fedak et al., 1982
), so that
hip and shoulder torques to accelerate each limb were small compared with
those required to sustain forward running. Consequently, errors in limb
moments of inertia from the prismatic assumption led to negligible errors in
total torque output.
For this study, we did not use an average quadrupedal form; rather, the
virtual animals were given mass distributions and shapes similar to those of
particular species. Joint locations and segment lengths were measured from the
animal photographs of Muybridge
(1957) or, in the case of the
chipmunk model, from video camera images (200 frames s-1) of a
running chipmunk. The back flexion point, hip-to-shoulder distance, neck and
head lengths, shoulder-to-elbow distance and hip-to-knee distance were all
measured from photographic or video images and normalized to leg length. The
back flexion point was measured by estimating the midway point between the
tail base and the caudal aspect of the rib cage. The shoulder-to-hip distance
was measured from a point midway between the greater tubercle and the dorsal
aspect of the scapula and the greater trochanter of the femur. The distance
from the elbow to the shoulder point and the distance from the knee to the hip
point were also measured from the animal images.
These dimensionless sagittal-plane lengths were then multiplied by the
animal's leg length. Leg lengths were taken from the literature
(Fedak et al., 1982;
Farley et al., 1993
), except
for the large horse's leg length, which was measured directly on a horse
specifically for the study. Each leg length was computed using the protocol of
Farley et al. (1993
) by taking
the average of the forelimb and hindlimb lengths at first contact in trotting.
The forelimb length was taken as the distance from the foot to a point midway
between the greater tubercle and the dorsal aspect of the scapula and the
hindlimb length as the distance from the foot to the greater trochanter of the
femur. Mass was distributed throughout each virtual animal in a realistic
manner using data from the literature
(Taylor et al., 1974
;
Fedak et al., 1982
). The
lateral thicknesses of the trunk, neck and limbs were computed using the mass
of each segment, the sagittal-plane lengths and the volume formula for each
segment shape.
Biological assumptions of the model
For the virtual animals to be viewed as plausible biological
representations, three main assumptions were made irrespective of size and
gait. First, each limb behaved as a linear spring of constant stiffness
throughout each ground-contact phase in running. During stance, a limb changed
length through a passive telescoping joint whose stiffness was linear and
invariant with time. The support for this assumption includes linear
measurements of force versus displacement in mammalian limbs and
agreement of bouncing spring-mass models with gait data
(Cavagna et al., 1988;
Blickhan, 1989
;
McMahon and Cheng, 1990
;
Blickhan and Full, 1993
).
Second, forward motion was controlled by active torques about the proximal leg
joints (hips and shoulders); these torques were the only energy input to the
model during stance. This assumption was made to achieve a simple control
scheme that was biologically realistic. Anatomical descriptions of limb
musculature and measurements of ground-reaction forces from trotting dogs are
consistent with this assumption (Gray,
1968
; Lee et al.,
1999
). Third, metabolic cost of running was predicted from the
cost of supporting body weight and the time course of generating that force.
This last assumption was based on evidence that, during running, the metabolic
rate is inversely proportional to the time per stride that a given foot
contacts the ground (Kram and Taylor,
1990
). It allowed metabolic costs to be calculated from the
kinematics of the virtual animal simulations. There are other approaches for
estimating the metabolic cost of locomotion on the basis of external
mechanical work (Taylor et al.,
1982
; Full, 1989
;
Donelan et al., 2002
); we used
the rule of Kram and Taylor
(1990
) because of its
simplicity and because it was tested on species similar to those in this
study.
Gaits
Two distinct quadrupedal gaits were modeled: trotting and galloping. During
trotting, diagonal pairs of limbs moved approximately in concert, with one
pair on the ground at a time. During galloping, which was used at higher
speeds, the four limbs touched the ground sequentially during each contact
period. The present model did not attempt to explain why a particular gait was
used at a given speed. Rather, for each speed, experimental observations were
used to select the relevant gait. Interspecies comparisons were made at
physiologically equivalent speeds (Heglund
et al., 1974) (Table
1). For trotting, model predictions and measurements were compared
near the midpoint of each animal's natural range of trotting speeds, with
similar Froude numbers and duty factors occurring among species
(Alexander, 1988
;
Heglund and Taylor, 1988
;
Farley et al., 1993
). The
Froude number was defined as
u/(gL0)1/2, where u is
forward speed (averaged over a stride), g is gravitational
acceleration and L0 is leg length. Duty factor was defined
as the percentage of a stride period during which a foot was on the ground.
For galloping, comparisons were made at each animal's lowest galloping speed,
i.e. its trotgallop transition speed
(Heglund et al., 1974
;
Heglund and Taylor, 1988
).
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Model control implementation
The dynamics of trotting and galloping were simulated by programming motor
control into the model structures, subject to the laws of Newtonian mechanics.
A commercially available software package (SD/Fast, Symbolic Dynamics Inc.)
was employed to generate and integrate the non-linear equations of motion
using a fourth-order Runge-Kutta method (0.4 ms time step). Local software
enabled communication between the control algorithms and SD/Fast to determine
the forces and torques commanded to the joints. All running simulations were
two-dimensional, operating within the sagittal plane. Yaw and roll degrees of
freedom were neglected.
Parameter definitions
For the virtual animals to trot and gallop in numerical simulation,
mechanical and kinematic control parameters needed to be defined (see
Table 1 for key parameters).
The mechanical parameters were the stiffnesses of the limbs (at the
telescoping joints), back and neck during ground contact. These passive
properties, which were fixed for the duration of ground contact (the first
assumption of the model), helped to determine the behavior of each virtual
animal as it rebounded from the ground during trotting and galloping. The
kinematic control parameters were target limb-retraction speeds and gains,
aerial position and velocity gains, target joint positions and limb-retraction
times. To control forward running speed in trotting or galloping, torques were
applied about the hip and shoulder (the second assumption of the model) such
that the tangential velocity component of each foot, measured relative to each
foot's proximal hip or shoulder joint, was sustained. The target
limb-retraction speeds were the desired tangential velocities of the feet
relative to the hip and shoulder during ground contact. There were two target
speeds, one corresponding to the forelimbs and a second corresponding to the
hindlimbs. Foot velocity was computed by multiplying the virtual animal's leg
length by the angular velocity of the hip or shoulder joint measured relative
to the trunk. The applied torque was set proportional to the difference
between a measured tangential velocity component and a target limb-retraction
speed. To control forward speed in trotting and galloping simulations, this
proportionality constant, or gain, was defined together with two
limb-retraction speeds corresponding to the fore- and hindlimbs (Herr and
McMahon, 2000,
2001
).
During each aerial phase, conventional proportional-plus-derivative (PD)
controllers (Nise, 1995) (pp.
460-469) were used to position the hip, shoulder, back and neck joints to
desired angular positions relative to the model trunk in preparation for
landing. PD controllers were also used to shorten the limbs for foot clearance
and then to lengthen the limbs for landing. Here, the force or torque applied
to a joint was proportional to errors in position and velocity.
After achieving a desired position, each limb retracted just before
striking the ground. Retraction times were set by an internal clock (analogous
to a neural pattern generator) that determined how the limbs were phase-locked
in a trot or gallop. Throughout this paper, limb retraction is defined as a
backward displacement of the foot by means of rotating a limb about the hip or
shoulder joint within the sagittal plane
(Gray, 1968). In trotting
simulations, a diagonal limb pair began to retract towards the ground after a
fixed time interval from the beginning of the aerial phase. In galloping
simulations, the retractions of the first hindlimb and the first forelimb to
strike the ground were separated by fixed time intervals; the second hindlimb
began to retract when the first hindlimb became perpendicular with the trunk,
and similarly, the second forelimb began to retract when the first forelimb
became perpendicular with the trunk.
Setting model parameters
The first constraint on the model parameters was that simulations were
required to be smooth and periodic. For each gait (trotting and galloping) and
for each virtual animal, we adjusted the model parameters to achieve this
condition. Periodicity was defined as no significant change
(P<0.05) in the animal's maximum aerial height, forward speed and
body pitch over 20 running cycles. Running simulations were started with the
animal off the ground. Initial conditions were defined for the position and
velocity of the center of mass and the individual body segments. When
periodicity was satisfied, we found that the simulation dynamics was
insensitive to the initial conditions. When periodicity was not satisfied,
however, simulation dynamics was found to be strongly dependent on the initial
conditions and, typically, a simulation run would become unstable within only
a few strides. For the small horse simulations, we demonstrated the model's
capacity to recover from an environmental disturbance: a sudden 20% reduction
in ground stiffness (Herr and McMahon,
2000,
2001
). These numerical
experiments suggested that periodicity may be related to dynamic stability,
but in no way served as proof of stability.
Simulation experiments showed that the model's cyclic behavior was
sensitive to variations (among simulation runs) in fore- and hindlimb
stiffnesses during stance, target limb-retraction speeds, target limb angle
and limb-retraction times. To set these parameters, genetic algorithms were
employed to search the parameter space for smooth and periodic behavior.
Specifically, we performed a genetic-algorithm search
(Goldberg, 1989) to find these
control values that minimized the variance in step-to-step maximum aerial
height, forward speed and body pitch. The resulting ranges of parameter values
demonstrated the mechanical correlates of the periodicity constraint. Most
notably, we found that stance-limb stiffnesses must exceed certain minimum
values to keep the body upright from stride to stride. Below this stiffness
threshold, each virtual animal could be stabilized, but only with active
limbs, i.e. when non-conservative forces were applied along the axis of the
limb (see Herr and McMahon,
2000
).
The second constraint on model parameters was that the overall leg
stiffness (kleg) for each animal should match published
experimental data. It is important to point out that kleg
represents the stiffness of the entire musculoskeletal system during stance
(McMahon and Cheng, 1990;
Farley et al., 1993
). The
method for computing kleg is reviewed in the next section.
From the sets of parameter values that led to smooth and periodic trotting, we
chose the particular set of values that led to the closest agreement between
the computed kleg and experimental
kleg values from the literature. That is, after
stabilizing the model, we tuned the fore- and hindlimb stiffnesses, target
limb-retraction speeds, target limb angle and retraction times to match the
kleg data. The data are well-fitted by the power law
kleg=0.715M0.67, where M is
the body mass of the animal in kg, and the units of kleg
are kN m-1 (Farley et al.,
1993
). The same fore- and hindlimb stiffness values were used in
galloping and in trotting simulations.
For trotting and galloping simulations, we also adjusted model parameters that had little effect on periodicity or overall leg stiffness, such as neck and back stiffness during stance and aerial-phase PD gains. Neck and back stiffnesses were selected to minimize the number of oscillations in the trunk per stride, and aerial PD gains (position and velocity) were set to position the joints such that each joint moved to its target position without overshoot. For all PD controllers, target velocities were set to zero.
The ground was modeled with linear springs and dampers in the vertical and horizontal directions to model the viscoelastic properties of a natural running surface. Ground stiffness was first set so that the limbs penetrated the ground by a small amount when running (0.3 cm for the small horse model). Increasing damping from zero then minimized oscillations between the ground and foot.
Model outputs
The model's outputs were computed from the dynamics of each set of animal
simulations as follows. For trotting animals, leg stiffness was defined as
kleg=F/l, where F is the
peak vertical ground-reaction force and
l is the compression
of a virtual leg spring, based on a spring-mass representation of the center
of mass as it rebounded from the ground
(McMahon and Cheng, 1990
;
Farley et al., 1993
). The
spring compression was given by
l=
y+L0(1-cos
),
where
y is the vertical displacement of the center of mass
during stance,
=sin-1(utc/2L0) is
the virtual-leg angle from vertical at touchdown (or half the angle swept by
the leg during stance), tc is the foot-contact time per
stride and L0 is leg length. Vertical stiffness was
defined as kvert=F/
y and
described the center-of-mass mechanics of the stance phase in the vertical
direction.
In the model, these properties of overall stiffness depended on the joint
stiffnesses, target limb positions and target retraction speeds. (The
stiffnesses kleg and kvert are defined
for symmetrical gaits such as trotting, hopping and bipedal running, but they
do not describe the mechanics of galloping.) Cost of transport (COT) was
defined as the metabolic energy required to move a unit mass over a unit
distance (Taylor et al., 1970;
Schmidt-Nielsen, 1984
). In the
model, COT was computed from the kinematics of the animal simulations (the
third assumption of the model) based on the empirical rule
COT=C0/utc, where u is
forward speed and C0 (C0=1.8 J
kg-1) was a size- and speed-independent cost coefficient
(Kram and Taylor, 1990
). In
this work, the empirical rule of Kram and Taylor served as a bridge between
model mechanics and energetics.
For galloping, we calculated stride frequency, hindlimb excursion angle, forelimb duty factor, peak vertical ground-reaction force and the cost of transport. These output variables depended primarily on limb stiffness (fore- and hindlimb) and the parameter values (target limb positions, target limb speeds, retraction times) resulting from the genetic-algorithm search for smooth and periodic solutions. We set the cost coefficient C0 to 1.8 J kg-1 (as in trotting), the forward speed u to the minimum galloping speed (trotgallop transition speed; Table 1) and tc to the mean contact time predicted from the galloping simulations.
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Results |
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Trotting performance
In this section, we compare the model predictions for trotting with
experimental data. The model's leg stiffness (kleg) was
constrained to match published data
(Farley et al., 1993), as
described in Materials and methods. This constraint, together with the
requirement for smooth and periodic trotting solutions, led to predictions of
vertical stiffness, limb angle, peak force and the cost of transport
versus body mass (Fig.
2). Least-squares regression lines were fitted to the simulation
results (filled circles, N=6) and compared with experimental data
(open circles). The linear regressions (with units of the variables as
plotted) are: vertical stiffness
(kvert=3.2M0.61,
r2=0.98); limb angle from the vertical at touchdown
(
=38M-0.081, r2=0.91); peak
vertical ground-reaction force (F=24M0.96,
r2=0.99); and cost of transport
(COT=12M-0.33, r2=0.99). Least-squares
regression lines fitted to the experimental data of Farley et al.
(1993
) and Taylor et al.
(1970
,
1982
) are
kvert=2.64M0.61±0.10,
=34.35M-0.034±0.092,
F=30.1M0.97±0.14; and
COT=10M-0.36±0.09, where the uncertainties are
standard errors on the slope. We conclude that the model predictions for
trotting are in quantitative agreement with published experimental data on
mechanical and energetic properties versus size.
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Galloping performance
In this section, we compare the model predictions for galloping with
experimental data. With the same values for fore- and hindlimb stiffness as
used in the trotting simulations, the constraint of stride-to-stride
smoothness and periodicity led to predictions of stride frequency, limb
excursion angle, duty factor and peak vertical ground-reaction force
(Fig. 3). Least-squares
regression lines were fitted to the simulation results (filled circles,
N=6) and compared with experimental data (open circles). The linear
regressions are: stride frequency (fS), in cycles per minute
(fS=282M-0.13, r2=0.96);
hindlimb excursion angle () (
=68M-0.06,
r2=0.77); forelimb duty factor (DF)
(DF=34M0.018, r2=0.45) and peak
vertical force (F) per body weight
(F/Mg=2.2M0.011, r2=0.43).
Least-squares regression lines fitted to the experimental data are
fS=269M-0.14±0.01,
=66M-0.07±0.01,
DF=34M-0.03±0.02 and
F/(Mg)=2M0.04±0.03, where the
uncertainties are standard errors on the slope. We conclude that the model
predictions of galloping mechanics versus size at the minimum
galloping speed are generally in quantitative agreement with published
experimental data.
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The energetic predictions for galloping are also in agreement with
experimental animal data. A least-squares regression line for the model's
cost-of-transport (COT) values versus mass plotted on double
logarithmic coordinates gives COT=14M-0.36
(r2=0.95, N=6). This model prediction is in
agreement with the cost of transport allometric equation
COT=14.6M-0.37±0.1, adapted with permission from
Heglund and Taylor (1988). The
uncertainties are 95% confidence limits for the slope.
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Discussion |
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Significance of model control
It is important to point out that the overall behavior of the model was
strongly dependent on the methods used for pitch and speed control. In
investigations of alternative control schemes, we found strategies that kept
the virtual animals upright from stride to stride but with features grossly
inconsistent with biological data. For example, when forward speed was
sustained by actively extending the contact limbs beyond their equilibrium
lengths at the end of stance (by applying non-conservative forces about the
knee and elbow), the vertical oscillations of the body increased dramatically
and the gait resembled a bound (see
Hildebrand, 1976) rather than
a trot or gallop, with a vertical stiffness much smaller than those measured
in animals (see Fig. 2A).
In developing the model, such alternative strategies were rejected in favor
of more biologically plausible control methods
(Herr and McMahon, 2000).
Given the realistic morphology of the virtual animals, the control features
responsible for their realistic dynamics were passive springy legs and active
wheel-like limb control. Here, the axial limb response (fore- and hindlimbs)
was passive and spring-like throughout stance, while active hip and shoulder
torques rotated each stance limb such that the tangential velocity component
of each foot was sustained like the rim of a steadily rolling wheel. The
model's requirements for sensory information (e.g. joint positions and
velocities) and active joint-torque magnitudes (e.g. 120Nm at the hip and
shoulder of the small horse) did not exceed the reported capabilities of
mammals (Eyzaguirre and Fidone,
1975
; Roberts,
1995
; Herr and McMahon,
2000
,
2001
). The model's control
thus represents a set of simple, plausible rules by which running quadrupeds
might operate.
Implications for running mechanics and motor control
For both trotting and galloping, the constraint of smooth and periodic
solutions led to the selection of hindlimb target speeds that were greater
than the forward speed of running and of forelimb target speeds that were less
than the forward speed (Table
1). This relationship between the target speeds caused the hip
generally to apply a thrusting torque (supplying mechanical energy) and the
shoulder a braking torque (dissipating energy). When the forelimb speed was
greater than the hindlimb speed, the model was unstable. This result suggests
that thrusting hip torques and braking shoulder torques may be crucial for
dynamic stability. This strategy is qualitatively consistent with ground-force
measurements in running dogs (Lee et al.,
1999; Herr and McMahon,
2000
).
The size-dependencies of stride frequency and excursion angle in galloping
(Fig. 3) have previously been
explained as mechanical constraints of natural frequencies of vibration and
muscle force acting across a joint, respectively
(McMahon, 1975). The present
model provides an alternative explanation. Our results suggest that biological
values of stride frequency and excursion angle may arise from interactions
between motor control and stride-to-stride dynamics. We found that the timing
of limb movements was crucial to stability and changed with the size of the
model, suggesting that neural pattern generators that control limb movements
might be tuned for stability as a function of body size. The emergent model
behaviors of duty factor and normalized peak force were relatively invariant
with size, which is consistent with experimental data
(Fig. 3C). The agreement
between model predictions and experimental data for both trotting and
galloping supports the idea that, in addition to mechanical considerations,
stride-to-stride periodicity constrains scale effects in running
quadrupeds.
Irrespective of size, animals must sustain their forward speed and remain
balanced while running. However, it is not known whether different control
strategies are required to stabilize animals of different size or whether
species have evolved with altogether different strategies for optimizing
running economy and/or speed. In this paper, a single control strategy was
used to stabilize six virtual animals spanning nearly three orders of
magnitude in body size. This control-independence suggests that scale effects
in quadrupedal running are attributable primarily to morphological differences
among animals, not to fundamental differences in how they remain balanced from
stride to stride. As animals get larger, the basic control scheme required to
maintain stability need not change, but the stiffness and timing of limb
movements change on the basis of the morphology of the limbs, trunk and head.
This result supports the idea that the natural dynamics of the body simplify
the control of locomotion (Raibert and
Hodgins, 1993; Kubow and Full,
1999
).
For each virtual animal, smooth and periodic trotting solutions were found
when we increased kleg or stride frequency above the
biological range, with the disadvantage of increasing the predicted cost of
transport (Herr and McMahon,
2000). However, when we decreased kleg or
stride frequency below the biological range, smooth and periodic solutions
could not be found. We speculate that the biological range of
kleg may represent a minimum-stiffness boundary for
stability. At these stiffness values, the predicted cost of transport, which
is in agreement with the data (Fig.
2D), may represent the lowest metabolic energy level within the
region of stiffness where stability can be achieved with passive, spring-like
legs. This explanation of biological stiffness values seems plausible, given
the evidence that energy consumption is minimized by gait transitions in
humans and horses (Margaria,
1976
; Hoyt and Taylor,
1981
), and warrants further investigation.
Concluding remarks
Perhaps the simplest summary of our findings is that the present model,
constrained by periodicity and stiffness and incorporating empirical
energetics, predicted a substantial data set across size on the basis of
simple mechanical and control features. Further development of the model may
include an investigation of the determinants of leg stiffness, the
incorporation of virtual muscle mechanics to derive energetic properties from
first principles, the representation of limb postures to test whether postural
variations with size (Biewener,
1983,
1989
) are constrained by
stability and extension to bipedal running. Before this work, it had not been
shown that the size-dependent properties of quadrupedal running could be
unified within a single theoretical framework. In the study of body size to
understand locomotory function, we believe that identifying mechanisms
critical to stability and metabolic economy can lead to simple ways to think
about how animals operate.
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Acknowledgments |
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