Biomechanics of quadrupedal walking: how do four-legged animals achieve inverted pendulum-like movements?
1 Orthopaedic Bioengineering Laboratory, Department of Surgery, Duke
University Medical Center, Durham, NC 27710, USA
2 Concord Field Station, Museum of Comparative Zoology, Department of
Organismic and Evolutionary Biology, Harvard University, Bedford, MA 01730,
USA
3 Locomotion Laboratory, Department of Integrative Physiology, University of
Colorado, Boulder, CO 80309, USA
* Author for correspondence (e-mail: tmgriff{at}duke.edu)
Accepted 7 July 2004
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Summary |
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Key words: locomotion, physiology, mechanical energy, work, ground force, gait, Canis familiaris
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Introduction |
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The inverted pendulum model of walking is characterized by a cyclic
exchange between gravitational potential energy and kinetic energy
(Cavagna et al., 1976). This
exchange process is best understood for bipedal animals. At the beginning of a
step, as the body's center of mass slows and gains height, kinetic energy
(Ek) is converted into gravitational potential energy
(Ep). During the second half of the step, as the body
falls forward and downward, Ep is converted back into
Ek. Energy recovery via this exchange is never
perfect (i.e. 100%) because the transition from one leg to the next inevitably
results in energy loss (Alexander,
1991
; Donelan et al.,
2002b
). However, due to effective energy exchange when the body is
supported by one limb, bipedal animals can substantially reduce the muscular
work of walking (Cavagna et al.,
1976
,
1977
).
The maximum values of mechanical energy recovery are lower for quadrupeds
(3065%) than for bipeds (7080%), suggesting that the inverted
pendulum mechanism for exchange of center of mass energy may be less effective
in quadrupedal animals (Cavagna et al.,
1977; Farley and Ko,
1997
; Griffin and Kram,
2000
; Heglund et al.,
1982
; Minetti et al.,
1999
). Differences in limb number make it more difficult to
understand how the limbs produce inverted pendulum-like dynamics in quadrupeds
than in bipeds. Bipeds behave like a single inverted pendulum, so the actions
of that inverted pendulum determine the vertical displacements and velocity
fluctuations of the center of mass. Walking quadrupeds, however, appear to
behave more like two inverted pendulums, with a `fore quarters pendulum'
located at the pectoral girdle and a `hind quarters pendulum' located at the
pelvic girdle (Fig. 1)
(Alexander and Jayes,
1978a
,b
).
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Unless a quadruped's fore and hind inverted pendulums move in-synch, such
as in a walking pace or a walking trot, the center of mass displacements and
velocity fluctuations will differ from those of either inverted pendulum. For
example, if weight is distributed equally between the fore and hind quarters
and the footfalls are evenly spaced in time, the fore and hind quarter
dynamics would completely offset each other
(Fig. 1). Thus, even when the
fore and hind quarters each behave individually as an inverted pendulum with
perfect energy exchange, the combined center of mass can undergo no energy
fluctuations and no inverted pendulum-like exchange. Although prior
observations indicate that the center of mass of quadrupedal animals does not
move in a perfectly smooth flat line
(Cavagna et al., 1977;
Farley and Ko, 1997
;
Minetti et al., 1999
), it is
not known how the movements of the fore and hind quarters are coordinated to
produce the observed inverted pendulum-like behavior.
Unlike the example in Fig.
1, animals use a wide range of footfall patterns (Hildebrand,
1968,
1976
), and some quadrupeds
support substantially more than half their body mass on either their fore
limbs or hind limbs (Demes et al.,
1994
; Schmitt and Lemelin,
2002
). These two factors, footfall pattern and body mass
distribution, may allow for much larger oscillations of the center of mass
than predicted from Fig. 1 by,
respectively: (1) synchronizing the movements of the fore and hind quarters,
even if only for brief time periods during the stride, and (2) allowing the
center of mass to track the movements of the heavier half of the body.
Although quadrupeds are likely to differ from the hypothetical example in
Fig. 1 in at least one of these
ways, the example provides a framework for investigating the determinants of
the center of mass motion in quadrupeds.
We hypothesized that quadrupeds achieve sufficient fluctuations in both
Ep and Ek to produce inverted
pendulum-like dynamics via two mechanisms: (1) by having footfalls
that are unevenly spaced in time (i.e. not 25% limb phase) and (2) by having
an unequal mass distribution between the fore and hind quarters. This
hypothesis was based on our hypothetical example in
Fig. 1 as well as previous
observations of limb phase relationships other than 25% and unequal
forehind limb loading in quadrupeds
(Budsburg et al., 1987;
Roush and McLaughlin,
1994
).
To test our hypothesis, we collected ground reaction force and high-speed video data from six dogs (Canis familiaris) walking over a range of speeds, and we calculated the mechanical energy fluctuations of the center of mass. Next, we compared the vertical displacements of the fore and hind quarters with those predicted if the legs functioned as incompressible struts during stance. Based on that comparison, we created a two-pendulum model to characterize the movements of the fore and hind quarters. This model provided insight into how limb phase and mass distribution collaborate to determine the center of mass movements in walking quadrupedal animals.
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Materials and methods |
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Measurements
Owners led their dogs along a runway that had two AMTI force platforms
(AMTI model LG6-4-1; Newton, MA, USA) built flush into it. We instructed the
owners to lead their dogs with a slack leash and to target four speeds (0.55,
0.80, 1.05 and 1.30 m s1). Owners and dogs were allowed as
many practice trials as needed to acclimate to this procedure. We measured the
speed of the owners walking past two infrared photocells placed 3 m apart on
either side of the force platforms. We later selected a random sample of
trials to compare these photocell speeds to the mean speed of the dog walking
through the 3 m section using our video data. The two speed measurements
produced nearly the same values (within 0.01 m s1 of each
other), so we used the photocell speeds in our analyses.
We collected the vertical (Fz), foreaft (Fy) and lateral (Fx) components of the ground reaction force at 1 kHz using Labview Software and a computer A/D board (National Instruments, Austin, TX, USA). Data were then filtered at 100 Hz with a fourth-order zero-lag Butterworth low-pass filter. We collected ground reaction forces when all the limbs were on the ground for a complete stride (i.e. whole-body forces) and then just for individual limbs. For whole-body forces, we summed the signals from the two force platforms and analyzed data when all the feet were on the force platforms for a complete stride. We used these data to calculate the mechanical energy fluctuations of the center of mass. Individual limb ground reaction force data were collected from separate left and right limb contacts with the force platforms. We then used these data to calculate the vertical displacements of the fore and hind quarters separately.
We recorded video data in the sagittal plane at 200 fields s1 and in the frontal plane at 60 fields s1 (JC Labs, Mountain View, CA, USA). Video and force platform data were synchronized using a circuit that illuminated a light-emitting diode in the video field and simultaneously sent a voltage signal to the A/D board. The video data were used to determine footground contact time, stride time and limb phase.
Kinematics, whole-body ground reaction forces and mechanical energies
For each dog, we analyzed two trials in which the mean speed was closest to
the target speed and the net change in speed was lowest. The mean speeds of
the analyzed strides were very close to the target speeds (0.57±0.01,
0.79±0.02, 1.06±0.03 and 1.31±0.02 m
s1; mean ± S.D.). The mean net
speed change of the analyzed strides was 2.1±1.6% (±
S.D.) of the mean trial speed.
Video recordings were used to calculate the stride time from the time between successive footfalls of the same limb. Duty factor was calculated as the footground contact time divided by stride time. Finally, we determined limb phase from the percentage of stride time that each limb first contacted the ground relative to the left hind limb; therefore, the left hind limb phase was always 0%. Because walking is a symmetrical gait (i.e. limb phase between each pair of left and right limbs is approximately 50%), the footfall pattern can be characterized as the average limb phase of the fore limbs relative to their ipsilateral hind limbs.
We calculated the velocity and displacement fluctuations of the center of
mass from the force platform measurements as described in detail elsewhere
(Blickhan and Full, 1993;
Cavagna, 1975
). The
Ek and Ep were calculated from the
velocity and vertical displacement of the center of mass, respectively
(Blickhan and Full, 1993
;
Cavagna, 1975
;
Willems et al., 1995
). The
instantaneous total mechanical energy of the center of mass
(Ecom) was calculated from the sum of the
Ek and the Ep at each instant. Percent
recovery, defined as the percent reduction in mechanical work required to lift
and accelerate the center of mass due to the inverted pendulum mechanism, was
calculated as follows (Blickhan and Full,
1987
; Cavagna et al.,
1976
,
1977
;
Farley and Ko, 1997
;
Heglund et al., 1982
;
Minetti et al., 1999
;
Willems et al., 1995
):
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A key parameter in determining the magnitude of percent recovery is the phase of the Ek and Ep fluctuations. We calculated the mechanical energy phase by determining the fraction of the stride time between the minimum Ek and the maximum Ep, multiplying it by 360° and adding 180°. The phase would be 180° if Ek and Ep fluctuated exactly out of phase. A phase value of >180° indicates that Ek reached its minimum after Ep reached its maximum.
Fore and hind quarter vertical displacements
To understand the link between fore and hind quarter vertical displacements
and center of mass dynamics, we collected individual limb ground reaction
forces for each dog at each target speed. We generally obtained three
acceptable force traces for each fore and hind limb per speed per dog. For
each component of the ground reaction force, we calculated an average fore and
hind limb force trace for each dog and then calculated an average force trace
for all the dogs. To do so, we normalized forces to body weight
(Wb) and expressed time as a percentage of contact time
before averaging.
We calculated the vertical displacements of the fore and hind quarters by
double integration of their vertical accelerations. These accelerations were
determined from the vertical ground reaction forces under the fore and hind
limbs and the effective mass of the fore and hind quarters, which was
estimated as the mass supported by either the fore or hind limbs during
standing (Jayes and Alexander,
1978). This approach was validated after data collection because
the distribution was independent of speed. The assumption of this approach,
that the vertical displacements of the fore quarters depend mainly on the fore
limb forces and that the hind quarter vertical displacements depend mainly on
the hind limb forces, is reasonable for two reasons
(Jayes and Alexander, 1978
).
First, the fore and hind quarters are connected by a flexible trunk. Second,
the trunk is long compared with the small vertical displacements of the fore
and hind quarters, so the trunk remains primarily horizontal. Thus, an axial
force transmitted through the trunk would have a negligible vertical
component. A similar approach, however, would not accurately determine the
individual foreaft movements of the fore and hind quarters because the
trunk is likely to transmit foreaft forces between them
(Alexander and Jayes, 1978b
).
This was confirmed after data collection: we found that the fore and hind
limbs generated net braking and propulsive forces, respectively. These net
forces must be offset by forces transmitted via the trunk to prevent
the hind quarters from overtaking the fore quarters.
To determine if the vertical center of mass displacements were more
influenced by the fore or hind limbs, we compared the times between the peak
center of mass displacement (tpeak,com) and the peak fore
(tpeak,fore) and hind (tpeak,hind)
quarters displacements. This temporal relationship () was calculated as a
percentage of the time interval between the peak fore and hind quarter
displacements:
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Compass gait displacements
We compared the empirical vertical displacements of the dogs with those
predicted if the fore and hind quarters vault up and over rigid limbs to
assess whether it was reasonable to compare a walking dog with two linked
bipeds with strut-like legs. In the theoretical rigid-leg gait, often referred
to as a `compass gait' (Rose and Gamble,
1994), each stance limb remained at a constant length and rotated
symmetrically over the point of contact during the first and second halves of
the stance phase as described by Lee and Farley
(1998
). To predict the
vertical displacement for the compass gait and compare it with our empirical
data, we incorporated experimentally derived values for limb phase, ground
contact time and standing leg length. The values for the vertical displacement
of the fore (zfore) and hind (zhind)
quarters were used to calculate the vertical displacement of the center of
mass (zcom) assuming a compass gait:
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Based on human walking data, in which the vertical position of the center
of mass is lowest at approximately mid-double support, we assumed that the
transition from left to right limb support occurred instantaneously at the
middle of the double support phase. Although this assumption could have a
significant effect on the absolute magnitude of the theoretical displacements
(2-fold range), the magnitude of the center of mass displacement relative
to the fore and hind quarters varied by <20% across the full range of
possible limb transition times within the leftright double support
phases. The relative timing of the center of mass displacement (
) was
unaffected by the limb transition assumption.
Two-pendulum model of quadrupedal walking
This model focuses on the link between the motions of two independent
pendulums and the motion of the system center of mass to address the question
of how the motions of quadrupeds' fore and hind quarters are coordinated to
produce inverted pendulum-like movement of the center of mass (see Appendix 1
for details). In the model, the vertical displacements of the two pendulums
were equal and the vertical displacement of the center of mass of the combined
two-pendulum system was expressed relative to that of a single pendulum. We
investigated the sensitivity of the center of mass vertical displacement to
the mass distribution and phase of the two pendulums. Although each individual
pendulum's motion does not depend on mass, the motion of the center of mass of
the whole two-pendulum system is affected by the mass distribution between the
pendulums. With this model, we hoped to gain insight into how body mass
distribution and limb phase affect center of mass movements in walking
quadrupedal animals. We did not examine the velocity and kinetic energy
fluctuations of the two-pendulum system center of mass due to the likely
forehind quarter interactions in the foreaft direction as
discussed earlier.
We calculated the magnitude (zcom) and timing
(
) of the center of mass vertical displacement for a full range of
pendulum phase relationships (
) and mass distributions
(Mf). We varied
from 0% (pendulums in-phase) to
25% (pendulums out-of-phase) and Mf from 0.5 (half of
total mass in fore pendulum) to 1.0 (total mass in the fore pendulum). Note
that Mf is dimensionless because it represents the
fraction of the total mass in the fore pendulum. The full range of possible
limb phase relationships for this model is 0 to 25% because the maximum
possible phase shift between successive peaks in the vertical displacements of
the fore and hind pendulums is 25%. This is because the model does not
distinguish between left and right limb movements. For example, in terms of
the two-pendulum model, a 50% limb phase (i.e. diagonal limb pairs move
synchronously) is the same as a 0% limb phase (i.e. ipsilateral limb pairs
move synchronously).
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Results |
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The recovery of mechanical energy by the dogs reached a maximum of 70% at
moderate speeds, a value similar to the maximum recovery in humans and other
bipeds (Fig.
3A)(Cavagna et al.,
1977). At moderate speeds (
0.8 m s1), the
muscular work required to lift and accelerate the center of mass per distance
walked was least (Fig. 3B) and
percent recovery was greatest. Recovery was maximized because the fluctuations
in Ep and Ek were nearly equal and
approximately out of phase at these speeds
(Fig. 3C,D). At all speeds, the
Ep reached its maximum value within 10% of the stride time
of when the Ek reached its minimum value.
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Vertical displacements
To understand the interaction between limb function and center of mass
dynamics, we calculated the fore quarter, hind quarter and center of mass
vertical displacements during a stride
(Fig. 4A). We focused on 0.80 m
s1 since the exchange of Ep and
Ek was greatest at this speed.
|
We found that the displacement patterns of the fore quarters, hind quarters
and center of mass for the dogs (empirical;
Fig. 4A) were remarkably
similar to the patterns calculated assuming that the legs behaved as rigid
struts (compass gait; Fig. 4B).
For the empirical data and the compass gait, the center of mass displacement
was 0.57 and 0.58, respectively, of the fore quarters displacement and 0.43
and 0.47, respectively, of the hind quarters displacement. Moreover, the
center of mass also reached its highest position at a similar moment in the
stride in the dogs (=66%) and the compass gait (
=62%). This
value indicates that the center of mass fluctuations followed the fore
quarters more closely than the hind quarters.
In both the dogs and the compass gait, the center of mass generally underwent two fluctuations in the vertical position per stride despite two fluctuations of the fore quarters and two fluctuations of the hind quarters. These four combined fluctuations of the fore and hind quarters produced two fluctuations of the center of mass because the displacements of the fore and hind quarters partially offset each other. The overall similarities between the empirical and compass gait data led us to further model walking quadrupeds as two pendulums: a fore quarters pendulum and a hind quarters pendulum.
Two-pendulum model of quadrupedal walking
Changing the pendulum phase and mass distribution dramatically altered the
magnitude of the zcom. When the pendulums fluctuated
exactly out of phase (=25%) and the mass was equally distributed
between them (Mf=0.5), the zcom was
zero (Fig. 5D), a result that
matches our hypothetical example in Fig.
1. Reducing the phase shift
(Fig. 5B) and/or redistributing
the mass (Fig. 5C) between the
pendulums increased the zcom. The two pendulums had equal
amplitudes in all cases.
|
When the two pendulums swung nearly synchronously (i.e. <5%, as
in a walking pace or trot), the zcom was nearly as large
as each individual pendulum displacement. In this case, mass distribution had
little effect on the zcom
(Fig. 6A). Alternatively, if
the pendulums swung out of phase (i.e.
25%), the
zcom increased as the distribution of mass became less
equal since the heavier pendulum had more influence on the center of mass
movement (Fig. 6A).
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The timing of the center of mass movements more closely followed the
movements of the heavier pendulum. Mass distribution primarily determined the
phase relationship between the peak zcom and the peak
displacements of the fore and hind pendulums () when the pendulums moved
nearly synchronously (Fig. 6B).
When the pendulums were more out of phase (i.e.
approached 25%), both
mass distribution and phase affected the relative timing of the center of mass
movements. For equal distribution of mass between the two pendulums (i.e.
Mf=0.5), the center of mass always reached its highest
position at a time exactly halfway between the hind and fore pendulum maximum
positions (
=50%; Figs 5B,
6B).
All combinations of pendulum phase and mass distribution, except for that shown in Fig. 5D, resulted in two fluctuations of the center of mass despite four total pendulum fluctuations (two by each pendulum). This observation is consistent with what was observed in the dogs and the compass gait predictions.
Comparison of dogs to model predictions
The two-pendulum model predicted that quadrupeds would walk with a flat
center of mass trajectory if they used a 25% limb phase and had equal body
mass distribution between the fore and hind quarters (Figs
1,
5D). The center of mass
trajectory of walking dogs, however, was not flat; the vertical displacement
was 53±6% (mean ± S.D., N=3) of
the mean fore and hind quarters displacement (e.g.
Fig. 4A). Dogs attained
significant fluctuations of the center of mass by deviating from the
flat-trajectory assumptions (i.e. 25% limb phase and equal mass distribution)
of the two-pendulum model.
We found that each fore limb lagged the hind limb on the same side of the body by, on average, 15% of stride time at all speeds in the dogs (Fig. 7; P=0.09 for limb phase vs speed, repeated-measures ANOVA). The two-pendulum model predicted that if limb phase decreased to 15% with equal mass distribution, the magnitude of the center of mass displacement would increase to 59% of each pendulum displacement (Figs 5B, 6A). This prediction was slightly greater than the observed 53% displacement in walking dogs.
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We also found that body mass was not distributed equally in dogs: the fore
limbs supported 63% of body mass during standing and at all walking speeds
(P=0.88, repeated-measures ANOVA). The 63:37 mass distribution
between the fore and hind quarters was similar to the differences in maximum
vertical ground reaction forces under the fore and hind limbs at 0.8 m
s1 (0.60x vs
0.35xWb, respectively;
Fig. 8). The two-pendulum model
predicted that altering mass distribution away from the equal distribution of
Fig. 5D and to the distribution
observed in dogs, without altering limb phase (i.e.
Mf=0.63 and =25%;
Fig. 5C), would increase the
vertical displacement of the center of mass to 26% of each pendulum
displacement. This center of mass displacement was still much less than that
observed in the dogs (53%). However, when both mass distribution and limb
phase were matched for the dog values (i.e. Mf=0.63 and
=15%; Fig. 5A), the
predicted center of mass displacement was 62% compared with 53% observed in
the dogs.
|
The relative timing of the center of mass fluctuations () was affected
by the unequal mass distribution of dogs as predicted by the two-pendulum
model. When more mass was concentrated in the fore pendulum to match the dog's
mass distribution, the center of mass of the two-pendulum model tracked the
fore pendulum more closely (
=68%). This observation suggests that the
center of mass followed the fore limbs more closely in the dogs (
=66%)
because they supported more weight than the hind limbs.
Although we focused on the vertical component of the ground reaction force
to determine the mass distribution between the fore and hind quarters, the
dog's unequal mass distribution was also evident in the foreaft and
lateral components of the ground reaction force. Both the fore limbs and hind
limbs generated a braking ground reaction force followed by a propulsive force
(Fig. 8). However, the fore
limbs spent more time braking than the hind limbs (55% vs 40% of
their respective ground contact times). Furthermore, the fore limbs generated
a greater peak braking force (0.11x vs
0.06xWb, fore vs hind limbs) and a
greater peak propulsive force (0.10x vs
0.07xWb) than the hind limbs. As a result, the fore
limbs generated 75% of the total braking impulse and 50% of the total
propulsive impulse. The fore and hind limbs contributed equally to the
propulsive impulse because the greater peak fore limb forces were offset by
the shorter time over which the fore limbs generated propulsive forces.
Similarly, the fore limbs generated 23 times greater peak lateral
forces than the hind limbs. For all limbs, the peak lateral ground reaction
force was directed medially (i.e. toward the body mid-line) and was less than
0.06x and 0.03xWb for the fore and hind limbs,
respectively. These small lateral forces help explain why the lateral
movements of the center of mass had only a small effect on the total
Ek.
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Discussion |
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Nearly all combinations of pendulum phase and mass distribution result in two fluctuations of the center of mass despite four total pendulum fluctuations (two by each pendulum). This finding makes sense mathematically since the addition of two sine waves of equal frequency results in a third wave of the same frequency. Thus, when applied to dogs or other walking quadrupeds, the center of mass will undergo two oscillations per stride as long as the fore and hind limbs each undergo two oscillations per stride. The one case when this does not occur is when the fore and hind quarters oscillate exactly out of phase and mass is distributed equally (Fig. 5D), acombination that results in no oscillation of the center of mass.
Determinants of the vertical displacements of the fore and hind quarters
We find that a dog's fore and hind quarters each reach their highest
position near mid-stance of their respective support limbs. This finding
assumes that the vertical movements of the fore and hind quarters are
mechanically independent of each other, as reasoned by us and other authors
(Alexander and Jayes, 1978a;
Jayes and Alexander, 1978
).
Although this assumption may not allow precise predictions of the pectoral and
pelvic girdle displacements, our goal was to understand the basis for the
pattern of center of mass movement. Consequently, our overall conclusions are
not likely to be affected by small deviations from this assumption.
The vertical displacement patterns of the fore quarters, hind quarters and
center of mass of a dog are remarkably similar to the patterns for a compass
gait. However, the displacement magnitudes in a dog are half of the compass
gait prediction (Fig. 4;
Jayes and Alexander, 1978).
This difference could be due to subtle non-strut-like limb behavior. For
example, the stance limb of walking humans does not actually behave like an
incompressible strut; joint flexion and the resulting limb compression reduces
the vertical displacement of the center of mass
(Lee and Farley, 1998
).
In dogs, the difference between the compass gait prediction and the
observed displacement could be reconciled if the fore and hind limbs
compressed by 3.4% and 4.6%, respectively, of limb length at mid-stance. For
comparison, dogs compress their limbs by 20% of limb length during
trotting (Farley et al.,
1993
). The limbs probably undergo some compression during walking
since the shoulder, elbow, knee and ankle joints flex by
20° during
the stance phase (Goslow et al.,
1981
). We could increase the accuracy of our predictions of the
absolute displacements of the fore quarters, hind quarters and center of mass
by adding a limb compression component to the compass gait model. However,
this refinement does not appear to be necessary to gain insight into the
determinants of the movement patterns of the center of mass since the
rigid-leg model accurately predicts the relative magnitude and timing of the
dog's center of mass movements.
If limb compression primarily affects absolute, but not relative,
displacements then it is unlikely to explain the greater relative center of
mass displacement predicted by the two-pendulum model (62%) compared with that
observed in dogs (53%). Instead, this difference may be due to our method of
calculating limb phase. Hildebrand
(1976) proposed that when the
fore and hind foot contact times are unequal, as in dogs, it may be more
functionally relevant to calculate limb phase based on the intervals between
mid-stance times of the limbs rather than touchdown times. This alternative
may indeed be functionally important for understanding the center of mass
vertical displacement pattern in walking dogs since the fore and hind quarters
each reach their highest position at mid-stance of their respective support
limbs. With this alternative method, limb phase is 17% rather than 15%. In the
two-pendulum model, this limb phase value and the observed mass distribution
(i.e.
=17% and Mf=0.63), leads to a center of mass
displacement that is 53% of the pendulum displacement the same value
observed in dogs.
Determinants of the foreaft movements of the fore and hind quarters
For effective inverted pendulum-like exchange, the magnitude of the
Ep and Ek fluctuations must be closely
matched. To what extent do limb phase and mass distribution factors
that affect the Ep fluctuations determine the
Eky fluctuations of the center of mass? We cannot answer
this question using the two-pendulum model because it assumes independent
pendulum movement but, in dogs, the trunk probably transmits foreaft
forces between the fore and hind quarters. We can, however, gain insight into
the factors that affect the foreaft movements of the center of mass by
examining the interaction between the fore and hind quarters.
The fore and hind limbs of dogs generate braking and propulsive forces simultaneously throughout the entire stride (Fig. 9A). Consequently, the net braking and propulsive forces acting on the center of mass are smaller than those generated by the individual limbs. Another consequence is that the amplitude of the Eky fluctuations is smaller for the center of mass than for both the fore and hind quarters (Fig. 9B).Because limb phase affects the relative timing of the Eky fluctuations of the fore and hind quarters, it is likely that limb phase has a large impact on the amplitude of the Eky fluctuations of the center of mass.
|
Indeed, limb phase appears to affect the Eky fluctuations to a similar extent as it affects the vertical displacement (Fig. 10). As limb phase approaches 0% (e.g. walking pace or trot), the Eky fluctuations of the center of mass increase since the fore and hind quarter fluctuations are nearly in phase with each other. These results, however, are subject to the assumptions discussed in Fig. 9B and they also assume that limb phase does not affect the foreaft ground reaction force pattern. It is difficult to test this last assumption since the dogs used the same limb phase across speed; a broader comparative study of animals that naturally vary in limb phase may be needed to evaluate this assumption. Overall, the data suggest that limb phase modulates the magnitude of Ek and Ep fluctuations to allow for inverted pendulum-like energy exchange across a range of limb phases.
|
By modulating the magnitude of mechanical energy fluctuations, limb phase
appears to affect the mechanical work of walking; the relative mechanical
energy fluctuations of the center of mass at a 25% limb phase are less than
half those at a 0% limb phase (Fig.
10). However, the magnitude of the total mechanical energy
increments of the center of mass does not account for all sources of limb
mechanical work. For example, two limbs perform mechanical work against each
other if one limb performs positive work while another limb simultaneously
performs negative work on the center of mass
(Alexander and Jayes, 1978b;
Donelan et al., 2002a
). Our
analysis suggests that, as limb phase approaches 25%, the periods of
simultaneous braking and propulsive force generation increase and likely lead
to greater amounts of inter-limb work. Therefore, the smaller fluctuations in
center of mass total mechanical energy as limb phase approaches 25% may be
offset by an increase in inter-limb work. In general, limbs probably work
against each other to a much greater extent in quadrupeds than bipeds because
limbs work against each other for 100% of the stride in dogs but only
30%
of the stride in humans.
Predicting center of mass movements: effect of morphology and limb phase
Based on the results of the two-pendulum model, the limb pair (e.g. fore or
hind) that supports more weight and generates the largest forces will
primarily determine the movements of the center of mass. Because the force
distribution is similar for standing as for walking, it is possible to make
predictions about which limbs most influence the center of mass movements for
most walking quadrupedal animals by simply measuring the body weight supported
by the fore and hind limbs during standing. However, this approach and the
results of the two-pendulum model may not apply to animals with heavy heads or
tails that do not move in synchrony with the fore or hind quarters,
respectively.
Animals use a wide range of limb phases and, according to our two-pendulum
model, these different phase values can lead to profound changes in the
displacement of the center of mass. If ipsilateral fore and hind limbs or
contralateral fore and hind limbs strike the ground together, such as in a
walking pace or walking trot, the center of mass vertical displacement equals
the displacement of the fore and hind quarters since their movements do not
offset each other. Conversely, the center of mass displacement decreases
dramatically if the limbs strike the ground at more evenly spaced time
intervals (i.e. approaching 25% limb phase). Given the great number of limb
phase measurements of walking animals
(Hildebrand, 1976), it may be
possible to make broad predictions about the relative center of mass movements
of diverse animals. These predictions, however, do not appear to correlate
with the extent to which animals utilize inverted pendulum-like energy
exchange. Diverse animals such as lizards
(Farley and Ko, 1997
) and
horses (Minetti et al., 1999
)
use vastly different limb phases 50% (equivalent to 0% in our model)
and 22%, respectively and they recover similar percentages of energy
via the inverted pendulum mechanism (
50%).
For most animals, each species' fore and hind limb lengths and duty factors
are similar (Hildebrand,
1976), which led us to assume equal fore and hind pendulum
displacements and frequencies in our model. Yet even if the limb lengths were
slightly different, as they probably are in most animals, it would have a
negligible effect on our conclusions since the natural frequency of a swinging
pendulum varies with L0.5, where L is leg
length. Although vertical displacement is proportional to limb length, the
fore limb length would have to be >1.8 times the hind limb length (for an
equal mass distribution) to cause the center of mass displacement to follow
the fore limb displacement to the same extent as when the mass distribution is
65:35 between the fore and hind quarters.
Conclusions
The inverted pendulum-like behavior of walking is observed in many
phylogenetically and morphologically diverse animals, and our study provides
some insight into the mechanical factors responsible for this convergent
behavior. The center of mass movements of a walking biped are primarily
determined by the mechanical behavior of the limb. Our study demonstrates that
changing limb phase or the distribution of weight supported among a
quadruped's limbs can alter the center of mass dynamics without changing the
behavior of individual limbs. Thus, a quadruped has more options for altering
the dynamics of walking than a biped.
Previous models of quadrupedal walking
(Alexander, 1980;
Alexander and Jayes, 1978b
)
suggest that animals can minimize the work performed by each limb by
generating ground force patterns that cause the fore and hind quarters to
vault over their respective stance limbs, like inverted pendulums. These
predicted force patterns are similar for a wide range of limb phases and mass
distributions. The results from these previous models, when combined with our
findings, suggest that animals can modulate their center of mass movements
over a wide range without deviating from the strut-like limb behavior that is
predicted to be most economical.
Unlike our hypothetical example with equal mass distribution and footfalls
evenly spaced through a stride, we found that the center of mass of a walking
dog does not maintain a flat trajectory because (1) the fore limbs lag the
hind limbs by less than 25% of the stride time and (2) the fore limbs support
more than half of body weight. In fact, our model demonstrates that many
combinations of limb phase and/or unequal fore:hind quarter mass distribution
will produce two fluctuations of the center of mass per stride if the fore and
hind quarters vault over their stance limbs like inverted pendulums. This
insensitivity to changes in limb phase and mass distribution may help explain
how animals as diverse as lizards and dogs achieve similar center of mass
dynamics (i.e. two fluctuations of the center of mass per stride) despite
vastly different limb postures (sprawled vs upright) and limb phases
(50% vs 15%) (Farley and Ko,
1997). Future studies of other species with different combinations
of limb phase and fore:hind mass distribution will provide further insight
into how gait pattern, morphology and limb mechanical behavior determine the
center of mass dynamics in walking.
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List of symbols |
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Appendix 1. Details of the two-pendulum model |
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![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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![]() | (A1) |
![]() | (A2) |
Accounting for the mass distribution between the fore and hind pendulums,
we calculated the displacement of the system center of mass, which was
expressed as a fraction of the maximum pendulum vertical displacement
(zpend):
![]() | (A3) |
![]() | (A4) |
![]() | (A5) |
![]() | (A6) |
To define ' in the same manner as equation 2, i.e. the
percentage time that the peak center of mass vertical position lags the peak
hind pendulum vertical position, we made the following modification:
![]() | (A7) |
To directly compare the limb phase values in the dogs and the phase shift
values between the fore and hind pendulums in the model, we expressed the
model's phase shift as a percentage of stride time:
![]() | (A8) |
![]() |
Acknowledgments |
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References |
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