Biomechanical and energetic determinants of the walktrot transition in horses
1 Department of Integrative Biology, University of California, Berkeley, CA
94720-3140, USA
2 Department of Integrative Physiology, University of Colorado, Boulder, CO
80309, USA
3 Equine Research Center and Departments of Animal and Veterinary Sciences
and Biological Sciences, California State Polytechnic University, Pomona, CA
91768, USA
* Author for correspondence at present address: Orthopaedic Bioengineering Laboratory, Department of Surgery, Duke University Medical Center, Durham, NC 27710, USA (e-mail: tmgriff{at}duke.edu)
Accepted 3 September 2004
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Summary |
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Key words: equine, Froude number, gait, inverted pendulum, locomotion, oxygen consumption
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Introduction |
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Gait transitions occur as the result of information the body senses or
perceives. Transitions do not occur because some maximal limit is reached
animals can be trained to use a gait over a broader speed range than
they normally choose to use (Farley and
Taylor, 1991; Hoyt and Taylor,
1981
; Wickler et al.,
2003
). Rather, animals naturally switch gaits when they reach some
critical kinematic, kinetic or metabolic level that is relieved by changing
gaits. Most investigators seeking to identify these critical factors in
terrestrial locomotion have focused on the walkrun transition of bipeds
or the trotgallop transition of quadrupeds (e.g.
Biewener and Taylor, 1986
;
Diedrich and Warren, 1995
;
Farley and Taylor, 1991
;
Hreljac,
1993a
,b
,
1995a
,b
;
Kram et al., 1997
;
Mercier et al., 1994
;
Minetti et al., 1994
;
Prilutsky and Gregor, 2001
;
Raynor et al., 2002
;
Rubenson et al., 2004
;
Thorstensson and Roberthson,
1987
; Turvey et al.,
1999
). Our goal was to understand the determinants of the
walktrot transition in quadrupeds.
The walktrot transition is biomechanically analogous to the bipedal
walkrun transition (Cavagna et al.,
1977). Walking is traditionally defined as a gait in which there
is at least one leg always in contact with the ground
(Howell, 1944
). In contrast,
an animal that runs or trots generally has periods during a stride when all
the legs are off the ground. Biomechanical data also differentiate these two
gaits. For walking animals, the center of mass reaches its highest position
near the middle of the stance phase; whereas in running, the center of mass
attains its lowest position near mid-stance
(McMahon et al., 1987
). These
patterns reflect two different basic mechanisms of legged locomotion
the inverted-pendulum-like mechanics of walking and the spring-like mechanics
of running (Cavagna et al.,
1977
). The change from a walk to a trot is a critical transition
in biomechanics from inverted-pendulum to spring-like mechanics.
In a simple inverted-pendulum model of walking, maximal walking speed is
set by gravity and leg length (Alexander,
1977). This model simplifies the legs to `mass-less' struts and
body mass to a point mass located at the hip. During a step, body mass follows
a curved path and to do so, it must be accelerated (centripetally) towards the
ground contact point. Because legs do not stick to the ground, this
acceleration must be provided by gravity. Based on this inverted-pendulum
model, Alexander predicted that animals could not walk faster than the speed
at which the centripetal force required to keep the body moving along the
curved path exceeded the force provided by gravity
(Alexander, 1989
). This limit
occurs when the ratio of these forces exceeds 1.0:
![]() | (1) |
Central to the inverted-pendulum model of walking is the exchange of
kinetic energy and gravitational potential energy
(Cavagna et al., 1963). This
exchange conserves mechanical energy by allowing forward motion to help raise
the center of mass, thereby reducing muscular work. The extent to which this
exchange occurs is, in part, limited by the amount of kinetic energy
(xMbv2) available to convert
into gravitational potential energy
(xMbgL). The ratio of these energies
also reduces to the dimensionless Froude number. Therefore, the Froude number
may link mechanical and metabolic energy-saving aspects of the
inverted-pendulum mechanism across the walkrun and walktrot
transitions.
We hypothesized that quadrupeds switch from a walk to a trot at similar inverted-pendulum dynamics. We also hypothesized that quadrupeds would switch gaits at the metabolically optimal transition speed. To test these hypotheses, we studied nine adult horses spanning an eightfold range in mass and a twofold range in leg length. If horses switch gaits at the same Froude number, then given these size differences, we would expect the largest horse to switch from a walk to a trot at a speed that is 1.4 times faster than the smallest horse. We chose to study horses because they are experimentally tractable and they provided a large range of similarly shaped adult body sizes. Furthermore, by using an intra-specific size comparison, we hope to control for potential phylogenetic and morphological differences associated with inter-specific comparisons.
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Materials and methods |
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Morphological measurements
We performed an allometric analysis to determine whether the horses in our
study were geometrically similar to one another. If different sized animals
are geometrically similar, then their linear dimensions should be proportional
to body mass to the power
(Schmidt-Nielsen, 1984
). We
measured the trunk, fore limb and hind-limb lengths by palpation during quiet
standing. Trunk length was measured as the cranialcaudal distance
between the anterior aspect of the scapulohumoral joint and the tuber
ischii. Fore-limb length was measured in two ways: as the vertical distance
between (1) the ground and the shoulder and (2) the ground and the dorsal edge
of the scapula. The functional fore-limb length is probably intermediate to
these two measurements because the scapula rotates during stance
(Back et al., 1993
). Hind-limb
length was measured as the vertical distance between the ground and the
greater trochanter of the femur. To estimate limb posture across body size, we
calculated the relative hip (lh) and shoulder
(ls) heights by dividing the height of the hip and
shoulder joints above the ground by the sum of the segment lengths for the
hind and fore limbs, respectively (Gatesy
and Biewener, 1991
).
We used reduced major-axis regression (model II;
Sokal and Rohlf, 1995) to
determine the scaling coefficient (a) and exponent (b)
relating parameter length (Y) to body mass (Mb),
where Y=aM bb
(Schmidt-Nielsen, 1984
). The
regression coefficients and confidence intervals were determined after
logarithmic transformation of the data:
![]() | (2) |
The 95% confidence interval of the scaling coefficient (intercept, a) and exponent (slope, b) are presented along with the correlation coefficient, R. Horses were considered to be geometrically similar if the 95% confidence interval for the scaling exponent, b, included a value of 0.33.
Transition speed
The walktrot transition speed was determined by changing speed in
incremental steps and observing the horse's gait. Transition speeds were
measured prior to the measurement of oxygen consumption. Horses began each
treadmill session with a warm-up that consisted of moderate-speed walking for
2 min followed by another 2 min of moderate-speed trotting. We obtained a
preliminary estimate of the walktrot transition speed on an animal's
first session by beginning at a moderate walking speed and steadily increasing
the speed until the horse switched to a trot. To determine the transition
speed more precisely we then adjusted the treadmill speed to 0.5 m
s1 slower than the estimated transition speed. Speed was
increased in incremental steps (0.1 m s1 steps every 1 min),
and we observed the animal's gait. We defined the walktrot transition
speed as the slowest speed at which the horse trotted continuously for a
minute. The treadmill speed was increased by another two steps to be sure that
we identified the correct transition speed. We also noted the fastest speed at
which the horse walked continuously for one minute so we could consider how
alternative definitions of the transition speed would affect our conclusions.
We repeated this process over a period of about two weeks for a total of 15
observations per horse.
Some investigators of the human walkrun transition have noted that
the transition speed varies whether it is approached from faster (run to walk)
or slower (walk to run) speeds (Diedrich
and Warren, 1995; Hreljac,
1993b
; Li and Hamill,
2002
; Raynor et al.,
2002
). We measured the trotwalk transition speed in the
Arabian horses and found that the trotwalk transition speed
(1.96±0.12 m s1; mean ± S.D.) was
numerically greater, but not statistically different, from the walktrot
transition speed (1.91±0.14 m s1; P=0.14;
two-tailed paired Student's t-test). A post hoc analysis of
these data indicated that more than nine horses in each size class would be
needed to have at least an 80% chance of detecting this difference to a
significance level of P<0.05. Our sample size was limited for each
size group so we did not further examine the trotwalk transition
speed.
Oxygen consumption
The metabolically optimal walktrot transition speed was determined
by measuring the rate of oxygen consumption
(O2) over a
range of walking and trotting speeds, including trotting speeds that were
slower than normally used.
O2 was measured
with an open-flow system (Fedak et al.,
1981
). The system and methods used in this study have been
described previously (Wickler et al.,
2000
). Briefly, animals wore a loose-fitting facemask, and air was
drawn past the mask at
4004700 l min1 depending
on the animal's size and locomotor speed. Flow rates were calibrated at the
end of each experimental session using the nitrogen flow technique
(Fedak et al., 1981
). Oxygen
content from a sample of the gas exiting the mask was analyzed downstream of
the flowmeter after CO2 and water vapor were removed (FC-1B
O2 analyzer®; Sable Systems, Las Vegas, NV, USA). We used a
commercial data-acquisition program (Datacan®; Sable Systems) to
simultaneously record analog data from the O2 analyzer, flowmeter
and treadmill speed. All gases were corrected to STP and
O2 (ml
O2 kg1 s1) was calculated using
equation 4b of Withers (1977
):
![]() | (3) |
Animals were warmed-up on the treadmill prior to
O2 measurements
by walking and trotting for 810 min. Steady-rate
O2 was measured
during the last minute of 35 min bouts at the selected experimental
speeds. This time period was sufficient to obtain steady-rate
O2. The average
O2 during the
last 60 s in the first 3 min was not significantly different from the lowest 2
min mean over a 15 min time period
(Wickler et al., 2000
).
O2 was collected
at 45 speeds spaced between
0.5 m s1 and the
fastest normal walking speed, and another 46 speeds between the slowest
normal trotting speed and a fast trotting speed. We obtained additional
walking and trotting
O2 data near the
walktrot transition by training the horses to extend their gaits beyond
the speeds normally used for these gaits. No more than six trials were
collected on any one sampling day to minimize potential fatigue effects and
speeds were varied randomly. We collected two trials per speed condition for
each horse and we averaged the
O2 from these
trials.
We defined the optimal metabolic transition speed as the speed above which
the walking cost of transport was more than trotting and below which the
trotting cost of transport was more than walking. We calculated this speed for
each animal by fitting separate 2nd order polynomial equations
(Kaleidagraph® 3.0; Synergy Software, Reading, PA, USA) to the walking and
trotting cost of transport versus speed data, and solving for the
speed where the two curves intersected. However, for the miniature and draft
horses, we were not able to obtain walking and trotting
O2 data at the
same speeds in the transition zone, so we had to extrapolate the polynomial
curve fits beyond the data. For the miniature horses, this involved
extrapolating the walking and trotting curve fits by 0.13 m
s1 and 0.10 m s1, respectively (mean
values). For the draft horses, this only involved extrapolating the walking
curve fit (mean = 0.26 m s1).
To assess the potential errors of extrapolating the polynomial curves, we used the Arabian horse data to determine how excluding the data points for the extended walking speeds would affect the calculation of the optimal metabolic transition speed. We were able to train the Arabian horses to extend their walking gait to speeds faster than they would normally use for prolonged time periods by an average of 0.20 m s1. If we excluded these data and relied on an extrapolated curve fit, we would have underpredicted the optimal metabolic transition speed by an average of 0.06 m s1. This value is small relative to the range of speeds in which horses spontaneously switch between walking and trotting (0.28±0.07 m s1, mean ± S.D.). Therefore, it is likely that extrapolating the curve fits beyond the data do not lead to substantial errors because the curve fits were only extrapolated by a small amount and the R2 values of the walking curve fits were >0.96.
Statistical analysis
We tested for statistical differences among the miniature, Arabian and
draft horse values using one-way analysis of variance (ANOVA) (JMP®,
4.0.2; SAS Institute, Cary, NC, USA). Horse groups were treated as ordinal
categories, ordered by increasing body size. If a significant difference was
found (i.e. P<0.05), we then performed a TukeyKramer
Honestly Significant Difference (HSD) test to determine which groups were
significantly different from each other.
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Results |
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Walktrot transition speed
Larger horses switched from a walk to a trot at faster absolute speeds, but
all the horses switched gaits at approximately the same Froude number
(Table 2;
Fig. 1). The miniature horses
switched from a walk to a trot at 1.63±0.10 m s1
(mean ± S.D.), which corresponded to a Froude number
(v2/gL) of 0.37±0.02. The next size
group, the Arabian horses, switched gaits at 1.91±0.14 m
s1 and a Froude number of 0.30±0.04. The largest
horses, the draft horses, transitioned to a trotting gait at 2.27±0.03
m s1, corresponding to a Froude number of 0.37±0.02.
Froude number at the walktrot transition speed did not change across
size (P=0.87, ANOVA) because the transition speed increased with the
square root of limb length.
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Oxygen consumption
The mass-specific oxygen consumed to travel a given distance (i.e. the cost
of transport) was least at moderate speeds within each gait and was greater
for speeds near the walktrot transition
(Fig. 2). This pattern was
similar for all the horses in our study. The optimal metabolic transition
speed (i.e. the speed at which the cost of transport for walking intersected
that for trotting) occurred at faster absolute speeds in the larger horses
(miniature horses = 1.50±0.04 m s1, Arabian horses =
1.87±0.09 m s1 and draft horses = 2.16±0.07 m
s1; mean ± S.D.).
|
The optimal metabolic transition speeds corresponded closely to the observed transition speeds (Figs 2, 3). The slowest speed that the horses trotted at continuously for 1 min was slightly faster than the optimal metabolic transition speed, whereas the fastest speed that the horses walked at continuously for 1 min was slightly slower (Fig. 2). Horses spontaneously alternated between walking and trotting gaits within a small range of speeds (0.28±0.07 m s1; mean ± S.D. for all horses) the optimal metabolic transition speed fell within this range (Fig. 3).
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Discussion |
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Inverted-pendulum dynamics
Analysis of data from other studies supports the idea that the Froude
number is an appropriate means of comparing inverted-pendulum dynamics in
different-sized animals walking at different speeds. The traditional method of
quantifying inverted-pendulum-like dynamics in walking animals is to calculate
the percent of mechanical energy recovered via the exchange between
kinetic and gravitational potential energies of the center of mass
(Cavagna et al., 1977). In
humans, percent recovery is similar at a given Froude number over a twofold
variation in leg length (same as the horses in our study) and gravity levels
(Fig. 4A; see also
Saibene and Minetti, 2003
).
Although humans do not walk in a dynamically similar fashion at equal Froude
numbers in simulated reduced gravity
(Donelan and Kram, 1997
),
gravity does affect the speed at which humans prefer to change gaits. Kram et
al. (1997
) found that at lower
levels of gravity, people switch from a walk to a run at progressively slower
absolute speeds but at approximately the same Froude number.
|
Bipeds (i.e. humans and birds) transition from a walk to a run at a Froude
number of about 0.5 (Alexander and Jayes,
1983; Gatesy,
1999
; Hreljac,
1995b
; Kram et al.,
1997
; Rubenson et al.,
2004
). Our results, as well as those from prior studies
(Heglund and Taylor, 1988
;
Pennycuick, 1975
), indicate
that horses and other quadrupeds switch from a walk to a trot at lower Froude
numbers than bipeds (
0.35). This difference in the relative
gait-transition speeds between bipeds and quadrupeds may correspond to
differences in inverted-pendulum dynamics
(Griffin et al., 2004
). In
quadrupeds, percent recovery of mechanical energy decreases more precipitously
at faster speeds (i.e. increasing Froude numbers) compared with bipeds
(Fig. 4B). The traditional
measure of percent recovery does not take into account the mechanical energy
lost to the footground collisions
(Donelan et al., 2002
).
However, no alternative global measure of inverted-pendulum-like behavior has
been proposed.
Intra- versus inter-specific size comparisons
We chose to use an intra-specific size comparison to control for potential
phylogenetic and morphological differences associated with inter-specific
comparisons. Although different sized species only deviate slightly from
geometric similarity in terms of limb segment lengths
(Alexander and Jayes, 1983),
limb posture changes dramatically across size, with larger species using more
upright limb postures (Biewener,
1989
). However, by restricting our study to horses, we minimized
body shape and limb posture differences across an eightfold range in body
mass. This approach allowed us to better isolate the effects of leg length on
the walktrot transition speed.
Our intra-specific approach may also provide unique insight into the
determinants of the metabolic cost of locomotion. In different species, the
metabolic energy required to move a unit body mass a unit distance decreases
in larger animals (Full and Tu,
1991; Taylor et al.,
1982
). For example, an elephant uses
1/40 the energy to carry
1 g of its body weight for 1 mcompared with that of a mouse
(Langman et al., 1995
). This
difference in metabolic economy across size can be explained by the volume of
active muscle required to support body weight and the rate of activating this
volume (Kram and Taylor, 1990
;
Roberts et al., 1998
;
Taylor, 1994
). The volume of
active muscle required to support a unit of body weight appears to be similar
across size (Biewener, 1989
).
Therefore, the primary reason that larger animals are more economical is
because their longer legs allow them to generate muscular force more slowly,
which uses slower, less expensive muscle fibers
(Kram and Taylor, 1990
).
We find, however, that across an eightfold size range in horses, metabolic economy remains unchanged (Fig. 5). This finding may be due to the consistent limb posture among different-sized horses. If the muscular dimensions of horses are geometrically similar like their body dimensions, then muscle stress would increase with size (xMb0.33), where muscle stress = force (xMb 1.00)/area (xMb0.67), and larger horses would require a larger volume of active muscle to support a unit of body weight compared with smaller horses. This differs from the inter-specific case in which muscle stress is nearly independent of body size because the more upright limb posture of larger animals decreases the muscle force required to support a unit of body weight. Therefore, within horses, the metabolic savings associated with longer leg lengths and slower rates of generating force may simply act to offset the relatively greater volume of active muscle in larger horses.
|
We recognize that the large size range of horses obtained in this study is
the result of selective breeding rather than natural variation. However, the
ancestors of modern horses exhibited significant natural plasticity in body
size. Throughout the 57 million year history of horses, the body size of
fossil equids ranged from approximately 25500 kg, with a number of
species lineages evolving dwarfism
(MacFadden, 1986
). This
suggests that the size range of horses used in our study is not unnatural.
Given that the cost of transport does not change across body size within
modern horses, locomotor economy seems unlikely to have driven the selection
for greater or smaller body size in the ancestral lineage of modern horses.
However, other factors could have influenced the metabolic cost of locomotion.
For example,
12 million years ago, the first equid clade developed a
medial trochlear ridge of the femur
(Hermanson and MacFadden,
1996
). This ridge, which was retained in all subsequent clades,
facilitates passive locking of the knee joint while standing
(Hermanson and MacFadden,
1996
) and may also provide greater mechanical advantage for the
knee extensor muscles during locomotion
(Kappelman, 1988
).
Determinants of gait transitions
Seminal studies by Margaria
(1938), and Hoyt and Taylor
(1981
), showed that humans and
horses naturally use gaits that require the least amount of metabolic energy
to travel a given distance. These findings suggested that animals switch gaits
at speeds that minimize the metabolic cost of locomotion. Yet it is difficult
to imagine how the body senses these metabolic measures over a time course
that is as rapid as the observed gait transitions; rather, biomechanical
factors may provide cues that are more rapidly sensed. Farley and Taylor
(1991
) found that
musculoskeletal force level, not metabolic energy minimization, triggered the
trotgallop transition in ponies. Recent data, however, indicate that
the maximum sustained trotting speed is not statistically different from the
metabolically optimal transition speed in Arabian horses
(Wickler et al., 2003
). Within
humans, studies provide conflicting reports as to the role of metabolic and
mechanical factors in determining the walkrun transition
(Hreljac, 1993b
;
Kram et al., 1997
;
Mercier et al., 1994
;
Minetti et al., 1994
;
Raynor et al., 2002
). Some of
these differences appear to be due to different methods of determining the
gait transition.
Most gait transition studies search for a specific trigger, often metabolic
or biomechanical in nature, that controls gait choice. Diedrich and Warren
(1995,
1998a
,
1998b
) proposed an alternative
theoretical approach to analyze gait transitions based on dynamic systems
analysis. This approach views gaits as complex systems that self-organize
around the system's natural dynamics. Gait transitions occur when the
stability of the current system dynamics decreases so much that switching to a
new system (i.e. gait) improves stability. In this approach, metabolic energy
is used as a surrogate for the stability of the underlying dynamics, but it is
not considered a trigger per se.
Our results are consistent both with traditional `trigger' and with dynamic
systems approaches. The different-sized horses switched gaits at different
absolute speeds but at the same Froude number. This suggests that horses
switched gaits at speeds that produced similar pendular dynamics. Because
horses switched gaits at a Froude number less than that observed in bipeds
(0.35 versus 0.5) suggests that the Froude number is adjusted to the
particular system rather than being a preset critical value. Furthermore, the
walktrot and walkrun transitions represent a critical transition
between the stability of pendular versus spring-based systems
(Diedrich and Warren, 1998b).
Dynamic systems theory predicts that spending extended periods of time at
speeds in the transition region will decrease strong hysteresis effects and
result in spontaneous shifts between gaits. Our findings are consistent with
these predictions: the hysteresis for the walktrot and trotwalk
transition speed is minor, and horses spontaneously switch between gaits
around the metabolically optimal transition speed.
Conclusions
We examined the walktrot transition in horses spanning a twofold
range in leg length and eightfold range in body mass to test the hypotheses
that quadrupeds switch gaits at similar inverted-pendulum dynamics, and at the
speed that maximizes metabolic economy. Larger horses switched from a walk to
a trot at faster absolute speeds but at the same mechanically equivalent speed
(i.e. a Froude number of 0.35). In addition, horses spontaneously switched
between gaits in a small region of speeds that corresponded to the
metabolically optimal transition speed. These results indicate that
biomechanical and metabolic factors are tightly coupled at the walktrot
transition. These findings are consistent with the interpretation that the
walktrot transition is triggered by the dynamics of an
inverted-pendulum system. However, an alternative interpretation, that horses
switch gaits at a critical transition between inverted-pendulum- and
spring-based system dynamics, may better explain why horses switch gaits at a
particular dimensionless speed well below the theoretical limit of 1.0.
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Acknowledgments |
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