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First published online January 3, 2006
Journal of Experimental Biology 209, 249-259 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.01979
Differential leg function in a sprawled-posture quadrupedal trotter
1 Department of Integrative Biology, University of California Berkeley,
Berkeley, CA 94720-3140, USA
2 Department of Biology, Lewis and Clark College, Portland, OR 97219-7899,
USA
* Author for correspondence (e-mail: rjfull{at}berkeley.edu)
Accepted 9 November 2005
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Summary |
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Key words: biomechanics, locomotion, dynamics, leg function, mechanical stability, running, quadruped, gecko, Hemidactylus garnotii
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Introduction |
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;
Cavagna et al., 1977;
Full and Koditschek, 1999;
Herr et al., 2002;
McMahon and Cheng, 1990).
Initial decelerations of the body in the first half of a step are followed by
comparable accelerations in the second half. Remarkably, two-, four-, six- and
eight-legged runners produce similar whole body ground reaction force (GRF)
patterns when bouncing at a constant average-speed (Blickhan and Full,
1987,
1993;
Cavagna et al., 1977;
Farley et al., 1993). Both
horizontal and sagittal plane templates demonstrate the extraordinary
capability to self-stabilize (Kubow and
Full, 1999; Altendorfer et al.,
2002; Schmitt et al.,
2002; Schmitt and Holmes,
2000a,b;
Seyfarth et al., 2002). To
test hypotheses of control and energy management as they relate to diversity
in form and function, we must anchor these simple templates in sufficiently
elaborate models. Leg number and posture represent two of the most obvious
differences in a runner's morphology. How leg number and posture interact to
produce the common dynamics of the COM observed in diverse species remains a
challenge.
A simple way to operate legs during running is to have them all function as
passive springs that first decelerate and then accelerate the COM in the
fore–aft direction (Fig.
1A). Similar spring-like leg function has been discovered in
upright-trotting quadrupeds (Cavagna et
al., 1977; Jayes and
Alexander, 1978) and built into bouncing robots
(Raibert et al., 1986). Peak
force magnitudes vary depending on mass distribution and asymmetries, but
decelerations followed by accelerations remain the general pattern in
upright-posture, trotting quadrupeds
(Alexander, 2003;
Biewener, 2003;
Lee et al., 1999;
Witte et al., 2002). Full et
al. (1991) discovered
differential leg function in trotting, sprawled posture runners
(Fig. 1B). Cockroaches use an
alternating tripod as they bounce forward with individual legs producing large
lateral and opposing GRFs. Forelegs only decelerate the COM during a step
period, whereas hindlegs only accelerate the body. Middle legs first
decelerate and then accelerate the COM, much like the pattern in
upright-posture quadrupeds. Differential leg function in sprawled posture
runners can minimize joint moments, but still enable maneuvers such as rapid
turning (Jindrich and Full,
1999). Lateral forces generated as insects bounce from side to
side can contribute to passive, dynamic self-stabilization in the horizontal
plane (Kubow and Full, 1999;
Schmitt et al., 2002; Schmitt
and Holmes,
2000a,b;
Seipel et al., 2004).
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; Gregersen et al.,
1998; Heglund et al.,
1982) and small therian mammals
(Heglund et al., 1982;
Witte et al., 2002) produce
primarily decelerating forces, whereas the hindlegs accelerate the body. When
speeding up, the hindlegs of trotting dogs generate more acceleratory force
than the forelegs (Lee et al.,
1999). When slowing down, forelegs generate the majority of the
deceleration. During constant average speed trotting, both fore- and hindlegs
of dogs generate decelerations and accelerations, but the mean leg force can
be deceleratory for the foreleg and acceleratory for the hindleg,
demonstrating differential leg function in the sagittal plane
(Lee et al., 2004).
Simulations suggest that directionally compliant legs result in a
fore–aft force bias that passively reduces pitch
(Lee and Meek, 2005).
To determine the mechanical implications of posture and leg number, we measured leg function during trotting in a gecko, a sprawled-posture quadruped. We tested the hypothesis that sprawled-posture quadrupeds use fore- and hindlegs similarly to decelerate and subsequently accelerate the COM in the fore–aft direction during each step. Alternatively, sprawled-posture quadrupeds could exhibit differential leg function with large lateral and opposing leg GRFs, like arthropods. If legs function differently, then we predict that forelegs will generate decelerations of the COM, hindlegs will produce accelerations and all legs will create large, opposing lateral forces during a step.
Sprawled-posture, trotting lizards produce spring-mass dynamics of the COM
typical of other legged runners (Farley
and Ko, 1997; Ritter,
1996). However, whole body GRF data are insufficient to make
conclusions about single leg function if more than one leg is in contact with
the ground simultaneously. The hindleg of lizards has been hypothesized to be
the primary propulsor (Irschick and Jayne,
1999; Jayne and Irschick,
1999; Reilly and Delancey,
1997a,b).
However, single leg GRF data for the hindleg of iguanas show decelerations
followed by accelerations, supporting the hypothesis of similarity in leg
function (Blob and Biewener,
2001). To examine leg function in four-legged, sprawled-posture
runners, we selected the gecko Hemidactylus garnotii. These geckos
are rapid runners with masses similar to those of the hexapods studied to
determine sprawled-posture leg function
(Full et al., 1991). The
geckos we selected here are also extraordinary at rapid vertical climbing
(Autumn et al., 2005;
Irschick et al., 2003). Our
choice of experimental animal provides an opportunity to compare running
directly with climbing, further elucidating the advantages and disadvantages
of differential leg function, posture and leg number.
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Kinematics
Three high-speed digital cameras (Redlake Motionscope, Tucson, AZ, USA)
operating at a rate of 1000 frames s-1 captured the geckos' motion.
One camera recorded the dorsal view, and the other two cameras recorded the
left and right dorso-ventral views. An external trigger simultaneously
generated a pulse on the data acquisition system and lit LEDs in the three
camera views to synchronize the video with the measured forces.
Twenty landmarks painted on each gecko with correction fluid to facilitate digitization. Eight of these points were on the dorsal midline of the gecko: one in the middle of the head (snout), one between the shoulders (pectoral), three on the trunk, one between the hips (pelvic) and two on the tail. Each leg contained three points: one on the shoulder/hip, one on the elbow/knee and one on the foot. Video frames were transferred and the coordinates of various positions on the body at each frame were digitized into a computer (Gateway 2000, Irvine, CA, USA) using a motion analysis system (Motus, Peak Performance Technologies, Inc., Lake Forest, CA, USA). The data were filtered using a low-pass Butterworth filter with a cut-off frequency of 30 Hz.
To calculate the average forward velocity of the animal, we digitized the pectoral or pelvic point from the dorsal camera view as the gecko ran across the force platform. Video recordings were used to gather footfall data. Relative phase was calculated as the time elapsed between when a forelimb or hindlimb contacted the ground relative to the stride period. A stride period was defined as the time between consecutive footfalls of either the fore- or hindfoot when the animal was on the force platform. Legs contacting the ground at the same time have an equal relative phase, whereas legs in antiphase have relative phase values equal to 0.50.
Axial or lateral bending was calculated by using three of the points along the trunk of the gecko, the pectoral, the second point on the trunk and the pelvic point. We calculated joint moments using both kinematic and force platform data. The angles between the shoulder/hip and foot with respect to the horizontal plane were calculated in the anterior, lateral and dorsal planes. This was compared to the vector angle formed by GRFs in each plane.
Ground reaction force measurement
Geckos ran from a darkened box 0.55 m across a force platform into another
darkened box 0.15 m away. The floor of the track was made out of balsa wood
and walls were constructed from Plexiglas and polished with Brillianize
(Chemical Products Co. Inc., Omaha, NE, USA) to prevent geckos from clinging
to the side of the track. The track was long enough to allow the animals to
achieve and maintain a constant average velocity for at least one stride
across the length of a force platform. Trials were selected for analysis if
they contained one complete stride in which the sum of the increases and
decreases in velocity were less than 15% of the average forward velocity of
the animal. The degree of divergence was determined by integrating
fore–aft force at the time of data collection.
Vertical, fore–aft and lateral GRFs were measured using a force
platform (0.11 mx0.06 m). The design and performance of the force
platform have been described previously
(Full et al., 1991;
Full and Tu, 1991). Forces
were measured from the signals produced by semi-conductor strain gauges bonded
to spring blades at each corner of the force platform. Loads in the range
0.0001–0.1 N produced a linear response, and peak GRFs never exceeded
0.1 N. Signals from each force platform channel were amplified (Vishay,
Measurements Group, Malvern, PA, USA) and collected by a 16-bit data
acquisition system (National Instruments, Austin, TX, USA) on a computer
(Apple Power Macintosh) at a frequency of 5000 Hz. Force signals were filtered
using a Butterworth low-pass filter at a cut-off frequency of 150 Hz (unloaded
natural frequency of the plate was greater than 400 Hz). Crosstalk between
vertical, horizontal and lateral outputs was less than 5%.
Whole body dynamics
Velocity and position changes of the center of mass
Whole body GRFs were used to determine the velocities and position and
energy changes of the center of mass (COM) over an integral number of strides
(Biewener and Full, 1992;
Cavagna, 1985). Integrating
the fore–aft GRFs when the whole body of the animal was positioned on
the force platform yielded the fluctuations of the fore–aft velocity of
the COM. The integration constant for the fore–aft velocity was the
average speed over the force platform determined from the kinematic analysis.
Fluctuations in the vertical velocity of the COM were found by integrating the
vertical force minus the body weight. We only accepted trials for which the
integration of vertical force over a stride equaled body weight as measured by
a separate scale. Integrating the vertical velocity again yielded the vertical
displacement of the COM, assuming the average height of the COM did not vary
over an integral number of strides. Integrating the lateral forces gave the
fluctuations of lateral velocity of the COM. Integrating the lateral velocity
again yielded the lateral displacement of the COM, assuming the average
position of the COM did not move left or right over an integral number of
strides.
Energy changes of the center of mass
The kinetic and gravitational potential energy of the COM were calculated
from the fore–aft velocity and vertical displacement of the COM,
respectively (Blickhan and Full,
1992). The total external mechanical power used to lift and
accelerate the COM over one stride was found by summing the positive
increments in total mechanical energy for a stride and dividing by the stride
period. The lateral kinetic energy of the COM was calculated from the lateral
velocity by summing the positive increments in mechanical energy, assuming
zero energy storage and return.
Single leg ground reaction forces
Single leg force data were collected from trials that began or ended with a
single foot on the force platform. Single leg forces were included when the
stride before or after was determined to be constant average speed. These
forces were used to determine the alignment of the GRF relative to the
position of the joint centers of the leg. Angles in the anterior view were
determined at mid-stance, a phase of 0.5 within a step. Angles in the lateral
and dorsal view were determined at a phase of 0.20 within a step for the
forelegs and 0.70 for the hindlegs.
Statistical analysis
We used a nested analysis of variance model (NANOVA) to account for the
effects of measuring individuals more than once. We used NANOVA instead of
repeated-measures ANOVA since the number of measurements within individuals
varied from 1 to 6. For all statistical analyses, we used a commercial
statistical program (StatView 5.1, SAS, Cary, NC, USA) on a computer (Apple,
Macintosh). Values are means ± standard deviations (s.d.) unless
otherwise noted.
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Geckos used a trotting gait characterized by pairs of diagonal limbs hitting the ground synchronously (Fig. 3). We measured phase shift as a value between 0 and 1, with 1 indicating an in-phase relationship and 0.5 indicating antiphase. Diagonal limb phase averaged 0.92±0.06 (N=11) for the right hindleg–left foreleg pair and 0.94±0.03 (N=12) for the left hindleg–right foreleg pair. The contralateral limb pairs moved in antiphase with its diagonal pair (0.48±0.07, N=51). Speed had no significant effect on limb phase (ANCOVA, F=0.122; d.f.=1,72; P=0.73; power=0.06).
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Lateral flexion was greatest during the initial footfall at phase 0.03±0.04; the magnitude of bending was 20.1±6.32° (Fig. 4A). Lateral trunk bending became rectilinear through midstance and continued flexing attaining a maximum at phase 0.81±0.08 with 19.4±5.60°.
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Fore–aft. GRFs in the fore–aft direction showed a decelerating force driving the animal's COM rearwards followed by an accelerating force propelling the animal's COM forward (Fig. 4C; blue line). Peak fore–aft forces occurred at phases 0.10±0.04, 0.33±0.04, 0.60±0.07 and 0.83±0.03 (Fig. 5B). Peak magnitudes were one-third the normal force.
Lateral. Fore- and hindlegs summed to accelerate the COM in the
lateral direction. Left foreleg and right hindleg couplets summed to produce a
lateral GRF acting on the COM directed to the left (Figs
1D,
5C). Right foreleg and left
hindleg couplets summed to produce a lateral GRF directed to the right
(Fig. 5C). The lateral GRF
pattern was more variable than the vertical and fore–aft force pattern.
A single maximum per step occurred in the six trials. Nine trials with more
variable footfall phases showed more than one lateral GRF peak per step. To
make a general conclusion about the lateral GRF pattern, we examined
additional data serving as a control for a separate study but using the same
animals, equipment and experimental conditions
(Irschick et al., 2003).
Thirteen of a total of 15 trials displayed a lateral GRF pattern with a single
maximum per step, as depicted in Fig.
5C. Peak magnitudes of the summed lateral force were one-half the
peak normal force.
Velocity and energy changes of the center of mass
Velocity in the fore–aft direction fluctuated as the COM decelerated
and accelerated (Fig. 4D).
Fore–aft kinetic energy and gravitational potential energy of the COM
fluctuated in phase (Figs 4E,
6). Total mechanical energy of
the COM fluctuated as the sum of fore–aft kinetic energy and
gravitational potential energy (Fig.
4F). The mass-specific, mechanical power (Ecm)
used to lift and accelerate the COM increased linearly with speed and was not
significantly affected by individual (ANOVA; Ecm=
0.24+1.89v; r2=0.50). Mass-specific, lateral
mechanical power increased linearly with speed
(Elat=0.042+2.36v; r2=0.26)
and was on average 85±14% of Ecm.
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Single leg forces
Measuring single leg forces at the beginning and end of a run when only one
leg was in contact with the force platform demonstrated that geckos have
differential leg functions. Right and left, and fore- and hindlegs generated
different GRF patterns (Fig. 7; Table 1).
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Normal. Peak normal GRFs were evenly distributed among fore- and hindlegs during trotting (Table 1; Fig. 7). Individual legs did not generate measurable detachment forces, shown by the absence of negative normal forces.
Fore–aft. Fore- and hindlegs summed to produce the COM pattern where a deceleration was followed by an acceleration in the fore–aft direction, but each leg functioned differently. Forelegs only produced decelerating forces early in the step at phase 0.20±0.01 (Table 1; Fig. 7A,C). Hindlegs produced both decelerating and accelerating forces in the fore–aft direction. Accelerating forces of the hindleg occurred late in the step (0.69±0.02) and were twice the magnitude of the decelerating forces.
Lateral. Whole body lateral forces showed that the gecko's COM oscillated from side to side each step as it ran forward. Single leg forces revealed that lateral GRFs always pointed toward the midline of the body (Table 1). Hindleg lateral forces exceeded those measured in the forelegs. The peak magnitude of single leg lateral forces was equal to or greater than the peak fore–aft forces.
GRF vector orientation. All peak single leg GRFs were directed axially along the leg toward the COM (Table 2). In Table 2, we compare the single leg force vector orientation with the position of the joints for fore- and hindlegs. The angles of the force vectors represented in Table 2 are shown in Fig. 8. There were no differences for left vs right sides.
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;
Blickhan, 1989;
Schwind and Koditschek, 1997;
Blickhan and Full, 1993;
Farley et al., 1993;
McMahon and Cheng, 1990;
Raibert et al., 1986; Figs
4,
5,
8A). Forelegs only decelerated
the COM in the fore–aft direction
(Fig. 7; Table 1). Hindlegs provided all
the acceleration in the latter half of the step. Lateral GRFs were always
directed toward the midline and exceeded the magnitude of fore–aft
forces. Thus, the differential leg function of sprawled-posture geckos
resembled sprawled-posture hexapods more than upright-posture quadrupeds
(Fig. 1A,B,D). The advantages
and disadvantages of differential leg function as it relates to whole body
stability, maneuverability and energetics remain to be explained.
Sprawled posture increases pitch and roll stability
Pitch
Differential leg function in upright-posture quadrupeds has been
hypothesized to assist in the control of a body's pitch
(Herr et al., 2002;
Lee et al., 1999). The body
axis tends to pitch nose-down in the sagittal plane when the COM is
decelerated and nose-up during accelerations. Trotting labradors and
greyhounds redistribute vertical impulses of their fore- and hindlegs to
stabilize the pitch moments created by fore–aft accelerating and
decelerating forces (Lee et al.,
1999). Both fore- and hindlegs generate decelerations followed by
accelerations during trotting. However, the mean individual leg forces do not
sum to zero. When slowing the body down, the dogs' forelegs produce the
majority of the decelerating force. When speeding up, hindlegs generate the
bulk of the accelerating force. Even during constant average-speed trotting,
the mean leg force is deceleratory for the foreleg and acceleratory for the
hindleg (Lee et al., 2004).
Similar fore–hind patterns have been reported during trotting in sheep
(Jayes and Alexander, 1978).
Lee et al. (2004) successfully
manipulated antero-posterior mass distribution in dogs by adding loads to the
limb girdles. Added load to the pelvic girdle resulted in a greater foreleg
braking bias and a lesser hindleg propulsive bias, in addition to an increase
in the relative contact time of the experimentally loaded limb. Lee et al.
(2004) predicted that lizards,
which support more weight on their hindlegs, would show smaller hindleg
propulsion biases and greater hindleg duty factors relative to the forelegs.
In fact, this appears to be exactly what we observe in the present study on
geckos.
Sprawled-posture locomotors can actively reduce pitching. Sideways-moving
rock crabs increase their stance length in water relative to air, thereby
increasing stability against overturning by hydrodynamic forces
(Martinez et al., 1998). As
velocity increases, trotting cockroaches reduce the destabilizing effects of
increased momentum by moving their COM posteriorly within the triangle of
support (Ting et al., 1994).
This shift makes cockroaches less likely to tumble by being carried forwards
and out of the base of support by inertial forces, should they stop abruptly.
Although active adjustments have been observed, sprawled-posture animals
appear to be far more passively stable than upright posture animals with
respect to overturning moments in vertical planes. The low height of the COM
of sprawled-posture runners increases stabilizing moments, while their long
and wide support base decreases overturning moments
(Alexander, 1971;
Martinez et al., 1998;
Ting et al., 1994). Because
geckos such as H. garnotii have a very low COM, long bodies and a
tail, overturning in the sagittal plane is less likely than in upright-posture
quadrupeds (Walter and Carrier,
2002). In gecko locomotion, deceleration of the COM by the foreleg
early in the step that resulted in nose-down pitch appears to be easily
corrected by the hindleg acceleration later in the step
(Fig. 8A).
Roll
Stability in roll or lateral stability has received less attention than
pitch, except in human posture and locomotion focusing on aging and disease
(Zettel et al., 2002). Long
bodies and a tail provide little resistance to roll. If a pedestrian's COM is
low, then at least two options are available for rapid running. Short legs
must operate at high frequencies or legs can produce longer stride lengths by
projecting laterally. Laterally projecting legs decrease roll overturning
moments. In cockroaches, peak rolling moments during unperturbed running can
spin the body by more than 360° in the time it takes to complete one step
(Ting et al., 1994). These
moments cause the cockroach's body to roll toward the middle leg of the tripod
at the end of each step. Moments generating roll to one side in one step are
balanced by the opposite moment in the next step, maintaining dynamic
stability over each stride.
Trotting quadrupeds do not have the advantage of a hexapod's statically
stable tripod, nor can they roll to a side where two legs have an opportunity
for a foothold (Ting et al.,
1994). Still, we found that on ideal terrain, the laterally
projecting legs in geckos dynamically balanced roll moments over the stride.
Normal fore- and hindleg GRFs were equal to the animal's body weight
(Fig. 7; Table 1). Should the gecko
mis-step (i.e. significantly reduce foot GRF), substantial normal forces could
produce significant roll moments if unbalanced. However, as in
sprawled-posture cockroaches, geckos also generated large lateral forces
directed toward their midline as they bounced from side to side
(Table 1). If legs function as
springs rather than struts (Fig.
8B), unrecoverable roll from a missed foot contact may be delayed
or avoided. If geckos are rolling toward the side maintaining foot contact,
then that leg spring will take time to compress and return energy to reverse
the roll.
Coupling lateral and fore–aft GRFs can stabilize running in the horizontal plane
To better elucidate differential leg function in sprawled-posture runners,
we must look beyond the reference frame that humans are most familiar with.
Sprawled-posture animals operate primarily in the horizontal plane
(Kubow and Full, 1999). The
negative consequences of falling in a sprawled-posture runner already so close
to the substratum may be minor compared to the disruption of movement in the
horizontal plane.
Assume that a sprawled-posture runner uses legs that function similarly, as in upright-posture quadrupeds, where each leg generates a fore–aft deceleration followed by an equal acceleration (Fig. 1A). If a sprawled-posture trotter generates lateral forces because its legs operate as springs to minimize roll instability (Fig. 8B), then these same GRFs will also alter moments in the horizontal plane. If we add the midline-directed lateral GRFs measured for geckos (Table 1) to the typical upright posture pattern (Fig. 1A) without changing fore–aft forces, then significant yaw moments result (Fig. 1C). Moreover, these yaw moments would produce unbalanced rotations in a single direction during a step. Instead, our data show an increased fore–aft decelerating force generated by the foreleg early in the step and a greater accelerating force produced by the hindlimb later in the step. Both fore–aft force alterations minimize yaw by directing these GRF vectors towards the COM (Fig. 1D).
The lateral GRFs measured in the gecko
(Fig. 5), but absent in
upright-posture trotters, can be critical for stability in the horizontal
plane (Kubow and Full, 1999;
Schmitt et al., 2002). Kubow
and Full (1999) input the
lateral and fore–aft leg forces measured in cockroaches into a
six-legged anchored, dynamic model. Surprisingly, the model passively
self-stabilized to perturbations without the aid of neural feedback.
Perturbations to fore–aft, lateral and rotational velocity altered the
translation and/or rotation of the body that consequently provided mechanical
feedback by altering leg moment arms. Schmitt and Holmes
(2000a,b)
represented the synergistic behavior of the insect's legs by a simple
horizontal plane template, termed the lateral leg spring, that is similar to
the sagittal plane spring loaded inverted pendulum. The lateral leg spring
model consists of a single leg spring (representing the summed force of
individual legs) attached to a body free to yaw
(Fig. 8C). Perturbation
experiments showed that the model self-stabilizes as it runs forward bouncing
side to side. Stability results from the gains and losses of angular moment in
leg-to-leg transitions that couple lateral and fore–aft forces
(Schmitt et al., 2002).
Jindrich and Full (2002)
tested the passive, dynamic stability hypothesis on running cockroaches by
producing a rapid lateral impulse to the COM. Insects began recovering with
spring-like behavior in less than 10 ms, challenging all but the fastest
reflexes (Holtje and Hustert,
2003) and thereby simplifying control.
Geckos produced the step-to-step GRF patterns consistent with the operation
of a lateral leg spring in the horizontal plane (Figs
5,
8C). This hypothesis requires
testing to determine if passive stability results following perturbations. Our
preliminary efforts suggest that a more anchored model may be required.
Instead of using a single mass to represent the body, two or more segments
attached by a passive rotational spring could generate novel hypotheses of
lateral back bending (Ritter,
1996).
Differential leg function can increase maneuverability in the horizontal plane
The low COM and wide support base, which stabilize sprawled-posture runners
against over-turning moments, place legs in positions to facilitate maneuvers
such as rapid turning. Jindrich and Full
(1999) measured single leg
GRFs in cockroaches executing rapid turns. The force production necessary to
turn required only minor alterations in the force hexapods generate during
constant average velocity, straight-ahead locomotion. The lateral leg spring
template predicts that just such minor alterations of the center of pressure
could be effective in initiating maneuvers
(Schmitt and Holmes, 2000b).
Legs farther from the center of rotation on the outside of the turn
contributed the majority of force and torque impulse. Forelegs contribute most
effectively to turning by producing small forces nearly perpendicular to the
heading, whereas middle and hindlegs must produce additional force parallel to
the heading. The lateral and decelerating forces produced by the gecko's
forelegs may facilitate rapid turning by requiring only small changes in force
production. Turning dynamics in reptiles needs further investigation, since
the role of body bending and tails is unknown. Walter and Carrier
(2002) have shown that iguanas
and monitor lizards have 3.0- to 4.9-fold greater rotational inertia than
similarly sized murine rodents, but at least smaller lizards such as
Hemidactylus garnotii appear to be highly maneuverable.
Differential leg function with joint moment minimization
A low COM with a broad base of support in all directions favors increased
stability to overturning moments in pitch and roll caused by the animal's
momentum or external perturbations to the body. Sprawled-posture pedestrians
bounce when they locomote at high speeds. The initial deceleration at the
beginning of a step in crabs (Blickhan and
Full, 1987), insects (Full,
1993; Full et al.,
1991) and now geckos (Table
1; Fig. 8A) is
primarily produced by middle or forelegs, whereas accelerations later in the
step are generated by middle and hindlegs. In insects
(Ting et al., 1994), roll is
balanced by coupling normal forces supporting weight with lateral forces
directed toward the animal's midline. Sprawled-posture geckos appear to use
the same strategy (Fig.
8B).Insects attain passive, dynamic stability in the yaw or
horizontal plane by coupling initial fore–aft decelerations of middle
and forelegs and later middle and hindleg accelerations with lateral forces
directed toward the midline. The horizontal plane GRFs of sprawled-posture
geckos followed a similar pattern (Fig.
8B).
Remarkably, the GRF patterns generated by the individual legs of insects
and geckos can be modeled by simple spring-mass systems characterized by
axially aligned forces. The original dichotomy proposed by Gray
(1968) of legs as levers or
struts should be expanded to explicitly include springs. The same spring-like
GRF patterns that stabilize sprawled posture runners to pitch, roll and yaw
tend to minimize joint moments (Full,
1993; Full et al.,
1991). GRFs tend to be directed axially along legs, much as they
are found in upright-posture runners
(Alexander, 2003;
Full, 1993).
The tendency to minimize joint moments does not imply that all GRFs are
directed through joint centers of rotation. In fact, multi-segment bent legs
make this nearly impossible. Full et al.
(1991) tested the hypothesis
that deviations from the minimum moments in running cockroaches could be
explained by considering the minimization of the summed muscle forces in
several legs. The measured GRFs were within only 12% of that which would
minimize muscle forces. In general, individual leg GRFs in running geckos
tended to pass above shoulder/elbow and hip/knee joints
(Table 2). Lateral and
fore–aft forces were smaller than required to minimize joint
moments.
Spring-mass models as templates vs an anchored model
Simply fitting a spring-mass model to data from an animal's COM is not
conclusive proof of energy storage and return
(Alexander, 1988). Full and
Koditschek (1999) make a
distinction between a template and a more anchored model. A template is the
simplest model representing the system behavior and serving as a target of
control. Templates are a necessary abstraction if we are to integrate
mechanics with control. To determine how legs, joints and muscles function to
produce the COM behavior, a more anchored model is needed. It is entirely
possible that spring loaded inverted pendulum behavior results from one set of
muscles absorbing energy and another generating it anew
(Alexander, 1988). Although the
geckos in this and another study (Farley
and Ko, 1997) produce whole body GRFs consistent with a
spring-mass template, examination of the individual leg forces make it more
difficult to explain how the energy absorbed at the beginning of the step is
stored and returned later by the hindlimb
(Fig. 8A). Perhaps spring-mass
templates will continue to show their value in understanding control rather
than with respect to metabolic energetics. Clearly, more anchored models are
required to test hypotheses of mechanism. For example, it remains to be
discovered if energy storage and return can explain how the metabolic cost of
transport in geckos can be one-fourth that predicted for an animal of its mass
(Autumn, 1999; Autumn et al.,
1999,
2002;
Farley and Emshwiller, 1996).
Integrating limb dynamics (e.g. Blob and
Biewener, 2001; Reilly and
Blob, 2003) and muscle force development with whole-body dynamics
is required to resolve the trade-offs, if any, that result from stable
sprawled-posture running with differential leg function.
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TOPSummaryIntroductionMaterials and methodsResultsDiscussionReferences |
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