Effects of mass distribution on the mechanics of level trotting in dogs
Department of Biology, University of Utah, 257 South 1400 East, Salt Lake City, UT 84112-0840, USA
* Author for correspondence at present address: Concord Field Station, Department of Organismic and Evolutionary Biology, Harvard University, Bedford, MA 01730, USA (e-mail: dlee{at}oeb.harvard.edu)
Accepted 16 February 2004
![]() |
Summary |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Key words: locomotion, running, dog, Canis, force, braking, propulsion, center of mass, limb, trunk
![]() |
Introduction |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
It is apparent that anteroposterior body mass distribution varies
substantially amongst quadrupeds, yet this characteristic has been measured
during standing in fewer than 20 species
(Rollinson and Martin, 1981).
In mammalian quadrupeds, the forelimbs typically support about 60% of body
mass during standing, while the forelimbs of primates, lizards and alligators
generally support less than 50%. Unfortunately, no data are available for
quadrupeds with more extraordinary body types, such as giraffes and spotted
hyenas, which appear to have extremely anterior center of mass positions, or
rabbits, which appear to have extremely posterior center of mass
positions.
We chose to study limb function during trotting because it is the gait most
commonly used by quadrupeds and it is a simple gait in which diagonal foot
pairs are set down alternately. This facilitates comparison of individual
forelimb and hindlimb function because diagonal fore- and hindfeet are set
down in the same functional step. Trotting is unique in that the forelimb and
hindlimb exert opposing foreaft forces while in-phase with one another
(Lee et al., 1999). The
forelimb exerts a net braking force and the hindlimb exerts a net propulsive
force during each trotting step. Although individual limb data from trotting
quadrupeds are limited to cats, macaques, dogs, goats, horses and alligators
(Demes et al., 1994
;
Kimura, 2000
;
Lee et al., 1999
;
Merkens et al., 1993
;
Pandy et al., 1988
;
Rumph et al., 1994
;
Willey et al., 2004
), this
phenomenon seems to be widespread in trotting animals. Opposing foreaft
force has also been reported in hexapedaly trotting cockroaches, which exhibit
primarily foreleg braking and hindleg propulsion
(Full et al., 1991
). This
pattern reflects the substantial anterior inclination of the forelegs and
posterior inclination of the hindlegs. In contrast, the mid-legs of
cockroaches show no bias in limb angle and, accordingly, exert equal braking
and propulsive force (Full et al.,
1991
).
In some cases, limbs act primarily as struts
(Gray, 1968), such that
minimal moments are exerted about their proximal joints. For example, the
foreleg resultant force of trotting cockroaches tends to align closely with
the leg axis (Full et al.,
1991
). On the other hand, strut action is sometimes accompanied by
substantial lever action (Gray,
1968
). For example, the strut action of the posteriorly inclined
cockroach hindleg would exert higher propulsive force if not for the opposing
lever action (i.e. proximal joint moment) tending to protract the hindleg
(Full et al., 1991
). During
steady speed trotting, it would be advantageous for animals to use their limbs
as springy struts, minimizing proximal joint moments and, thereby, increasing
the overall economy of locomotion
(Alexander, 1977
).
We predicted that body mass distribution would affect the individual limb force patterns, and tested this idea by adding 10% body mass near the center of mass, at the pectoral girdle, or at the pelvic girdle of trotting dogs. Assuming that the limbs act as struts, a disproportionate increase in loading of either the fore- or hindlimb would increase both the vertical and foreaft components of ground reaction force on that limb. Hence, in the absence of functional compensation, hindlimb propulsive force would tend to increase in response to pelvic girdle loading and forelimb braking force would tend to increase in response to pectoral girdle loading. Because braking and propulsion must be in equilibrium during steady speed trotting, compensatory changes in limb function, as evidenced by the bp bias and relative contact time, would be expected during experimental loading of the limb girdles.
![]() |
Materials and methods |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
|
Data were collected while the dogs carried no load (unloaded, U) and under three loading conditions, in which tandem saddle bag packs were worn (Fig. 1A). Two small bags of lead shot totaling 10% body mass were inserted bilaterally in the anterior compartments of the pectoral pack (fore-loaded, F), the posterior compartments of the pectoral pack (mid-loaded, M), or the posterior compartments of the pelvic pack (hind-loaded, H). We tested four hypotheses. Loading at the limb girdles (conditions F and H) will (1) change the forehind distribution of vertical impulse during trotting, (2) decrease the brakingpropulsive (bp) bias of the loaded limb and (3) increase the relative contact time of the loaded limb. (4) Loading (conditions M, F and H) will not affect the weight-normalized mean foreaft forces exerted individually by the fore- and hindlimb.
Force and center of pressure measurements
Force data were collected at 360 Hz from two strain gauge type force
platforms (made by N. T. Heglund) positioned in series, using LabViewTM
software and a National InstrumentsTM (Austin, TX, USA) data acquisition
system (DAQCard AI-16-E4, SCXI 1000 chassis, SCXI 1121 strain/bridge modules,
and SCXI 1321 terminal blocks). Data were collected for 2 s as the dogs
crossed the platforms. Only data from uninterrupted trotting were saved. Each
force platform was 0.6 m long by 0.4 m wide. Using platforms of this length
increased the likelihood that diagonal fore- and hindfeet would strike
separate force platforms simultaneously. Trials in which foot placements did
not meet this criterion were discarded. Furthermore, footfalls that struck the
platform edges as evidenced by negative vertical force (i.e. a moment tending
to lift the opposite end) were discarded. The force platforms measured
vertical and foreaft ground reaction force (GRF) with separate
double-cantilever transducers at each corner post. Vertical GRF acting upward
and foreaft GRF acting in the direction of travel were considered
positive. Vertical impulse jz and foreaft impulse
jy were determined by numerical integration of GRF from an
individual limb over the limb contact time (tc,fore or
tc,hind) or both diagonal limbs over the paired contact
time tc,total of the diagonal limbs. A key to the notation
used throughout this report is provided in Appendix A.
Normalized mean vertical force exerted on the center of mass during paired
diagonal supports was determined by:
![]() | (1) |
![]() | (2) |
![]() | (3) |
![]() | (4) |
![]() | (5) |
![]() | (6) |
In order to determine a limb's functional bias toward braking or
propulsion, a Fourier method (Hamming,
1973) adapted to the analysis of forcetime curves by
Alexander and Jayes (1980
) was
used to quantify foreaft force curve shape. This method decomposes a
complex waveform into five simple sinusoids of progressively higher frequency
(i.e. shape complexity). These sinusoids are known as Fourier terms and each
term has a coefficient, the magnitude of which indicates its influence on the
shape of the waveform. The five coefficients generated by this analysis are
a1, b2, a3,
b4 and a5, where a indicates
a cosine term and b, a sine term. The lower frequency terms
a1 and b2 define the basic waveform,
hence, by expressing a1 as a fraction of
b2 (or vice versa), the waveform shape can be
described in dimensionless terms (Alexander
and Jayes, 1980
). Here,
a1/b2 was used to quantify
foreaft force curve shape, which is referred to as the
brakingpropulsive (bp) bias. Forcecurve shapes defined by
negative and positive values of a1/b2
are shown in Fig. 2. Negative
values indicate a braking bias, positive values indicate a propulsive bias,
and zero indicates a symmetrical force curve with no bias toward braking or
propulsion. In this study, the bp bias
(a1/b2) was computed from individual
limb foreaft force curves.
|
Center of pressure was measured in a conventional manner by comparing the
vertical force from independent transducer elements at the fore and aft ends
of a force platform. The two platforms were calibrated on a continuous metric
scale to facilitate the computation of distance between supports on separate
platforms (Bertram et al.,
1997). Center of pressure position for each foot was determined by
force-averaging over the duration of foot contact. In other words,
instantaneous centers of pressure were weighted according to the instantaneous
vertical force values, summed, and then divided by the summation of vertical
force over the time of contact. This avoided the confounding effect of an
extreme anterior foot position during toe-off, for example, when the vertical
force is quite small. During paired diagonal contacts, the fore- and hindfeet
struck separate platforms allowing calculation of the distance p
between their mean center of pressure positions. The horizontal distance
between fore- or hindlimb centers of pressure and a kinematic mid-point
(described in the following section) was also determined.
Mean forward velocity was
determined directly from the force record by a method similar to that of Jayes
and Alexander (1978
), except
that time and distance parameters were computed from vertical force peaks
rather than initial foot contacts. The times tstep and
distances d between subsequent forelimb supports and subsequent
hindlimb supports were determined from the times of vertical force peaks and
the corresponding center of pressure positions of each foot. Mean forward
velocity
is the ratio of
d to tstep. Because the paired diagonal supports
of interest were preceded by a single forelimb support and followed by a
single hindlimb support, forelimb and hindlimb values of
, dstep, and
tstep were averaged to provide the best estimates of these
parameters. In order to account for size differences between dogs, mean
velocity was expressed as a Froude number:
![]() | (7) |
![]() | (8) |
![]() | (9) |
![]() | (10) |
![]() | (11) |
Videographic measurements
A video camera with VCR (PEAKTM Performance Technologies Inc.,
Centennial, CO, USA) acquired 120 images s1 in lateral view.
Force and video acquisition were synchronized by connection of a manual switch
to a PEAKTM Event Synchronization Unit, which simultaneously marked a
video frame and triggered force acquisition via a breakout connector
to the data acquisition card. The eye and the base of the tail were digitized
in every other frame during paired diagonal contacts (i.e.
tc,total) and their mean horizontal (y) positions
were computed. Then, the mean y-positions of the eye and the base of
the tail were averaged to define a kinematic mid-point. This mid-point was
used as a reference for the horizontal positions of the fore- and hindlimb
centers of pressure with respect to the trunk. Finally, the vertical position
of the base of the tail with respect to the substrate was used to approximate
the mean hip height h during trotting, because it is near the greater
trochanter (Table 1).
Videographic measurements were taken from four approximately steady speed
trials (i.e. 0.04
y
0.04) for each
dog under each loading condition.
Statistics
Comparisons between treatment conditions were made using either regression
analysis or repeated measures ANOVA and TukeyKramer multiple comparison
tests (InStatTM 2.0). The latter method was applied to mean vertical
force, mean foreaft acceleration, vertical impulse ratio, mean forward
velocity and Froude number, as well as the distance between diagonal
footfalls, p, and the distance between fore- or hind footfalls and
the kinematic midpoint between the eye and base of the tail. For individual
limb mean foreaft force and bp bias, which were functions of
mean foreaft acceleration, least-squares linear regression was used to
compute the y-intercept (i.e. the steady speed condition) and its 95%
confidence limits (Sokal and Rohlf,
1995). The regressions of
and
on
were linear, as was the regression of
a1/b2,fore on
. The log transformation of
a1/b2,hind allowed the use of a linear
regression to determine its steady speed value.
Values of other locomotor parameters were determined by simultaneously regressing the parameter of interest on mean foreaft acceleration and mean forward velocity. If both multiple regression coefficients were significantly different from zero (P<0.10), the multiple regression equation was used to predict the steady speed value of the locomotor parameter. If only one regression coefficient was significantly different from zero (P<0.10), the non-significant independent variable was dropped from the model and least-squares linear regression was used to predict either the steady speed value of the locomotion parameter or its value at the appropriate mean velocity. In cases where both multiple regression coefficients were not significantly different from zero (P>0.10), the sampled mean of the locomotor parameter was reported.
![]() |
Results |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
|
|
|
|
|
Mean foreaft acceleration was not
significantly different between U, M, F and H conditions
(Table 2). Nevertheless,
substantial inter-subject variation was observed, with
individual means ranging from 0.039 to
0.054 BW. Thus, some dogs tended to speed up and others tended to slow down as
they crossed the force platforms. Because the vertical impulse ratio
R is a linear function of
, individual
R values required adjustment to better predict the steady speed
vertical impulse ratio R0. This was done according to the
functional relationship between R and
,
reported to be 0.71 in separate analyses of Labrador retrievers and
greyhounds, which span a full range of R0 from 0.56 to
0.64 (Lee et al., 1999
). A
discrete slope of the regression of R on
was not available from the present analysis due
to disparity in body mass distribution associated with the use of various
breeds. Hence, the regression coefficient of 0.71 was applied in the
estimation of R0:
![]() | (12) |
In response to hind loading, a1/b2,hind decreased significantly (P<0.05; from 1.02 to 0.88) with respect to U (Fig. 4). This reduction in propulsive bias is consistent with Hypothesis 2. Nonetheless, fore-loading did not significantly reduce forelimb braking bias (Fig. 4). In response to mid-loading, a1/b2,fore decreased significantly (P<0.05) and a1/b2,hind increased significantly (P<0.05) with respect to U. These unexpected increases in forelimb braking bias and hindlimb propulsive bias are considered in the Discussion.
|
Across all four conditions, a1/b2,hind magnitude was, on average, 2.8 times that of a1/b2,fore (Fig. 4). In the unloaded condition, for example, a1/b2,fore was 0.31 and a1/b2,hind was 1.02. Hence, the hindlimb showed a large propulsive bias and the forelimb, a relatively small braking bias. Similar patterns were observed in conditions M and F. However, the forehind difference in bp bias became much less pronounced in response to hind-loading, such that hindlimb propulsive bias was only twice the forelimb braking bias (Fig. 4).
Like a1/b2,hind, the angle of the
hindlimb resultant force hind increased significantly
(P<0.05) during mid-loading and decreased significantly
(P<0.05) during hind-loading with respect to unloaded trotting,
while it was statistically unchanged during fore-loading
(Table 4). In contrast to
a1/b2,fore,
fore
increased significantly (P<0.05) during fore-loading but did not
decrease significantly during mid-loading with respect unloaded trotting
(Table 4).
In agreement with Hypothesis 3, the relative contact time of the forelimb tc,f/tc,h increased significantly (P<0.05) during fore-loading and decreased significantly (P<0.05) during hind-loading (Table 4). During mid-loading, relative contact time was statistically unchanged (P>0.05) from the unloaded condition. Duty factors increased significantly (P<0.05) under all loading conditions, with the exception of DFfore during hind-loading (Table 4).
Steady speed values of were
negative (braking) and steady speed values of
were positive (propulsive) under
all conditions (Fig. 4, filled
bars). By definition, braking and propulsion were of equal magnitude during
steady speed trotting. In the unloaded condition, for example, the magnitude
was 0.036 BW (i.e. 3.6% of body weight). In agreement with Hypothesis 4,
loading conditions F and H were not significantly different from U.
Nonetheless, mid-loading resulted in an unexpected, statistically significant
increase in braking and propulsive magnitude to 0.039 BW (P<0.05)
(Fig. 4).
During unloaded trotting, hindlimb phase shift hind was
approximately zero, indicating simultaneous initial contacts of the forelimb
and hindlimb. Values of
hind were significantly greater
(P<0.05) during fore-loading and significantly less
(P<0.05) during hind-loading
(Table 4). In other words,
hindlimb contact was relatively later during fore-loading and relatively
earlier during hind-loading. This pattern is illustrated in the GRF
reconstructions of Fig. 3.
The distance between the supporting diagonal limbs p was statistically unchanged (P>0.05) across loading conditions, with means of U, 0.569 m; M, 0.569 m; F, 0.562 m; H, 0.538 m. Hence, the spacing of diagonal supports was similar in U, M and F, while reduced, but not quite significantly, in H. Position of the diagonal supports with respect to the kinematic midpoint between the eye and base of the tail, was also conserved across U, M and F conditions; however, a statistically significant (P<0.05) anterior shift of the hindfoot position was observed during hind-loading (Fig. 5). The hindlimb reached about 0.03 m further forward in the hind-loaded than in the unloaded condition.
|
![]() |
Discussion |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
The dogs tended to trot faster without an added load, but the difference in
mean forward velocity was statistically significant only between the unloaded
and fore-loaded conditions. Multiple regression of locomotor parameters on
both mean foreaft acceleration and mean forward velocity (Tables
3 and
4) allowed the computation of
steady speed values at a common mean velocity of 2.86 m s1.
This analysis removed the potentially confounding effects of subtle velocity
differences between loading conditions. For example, the distance between
subsequent footfalls d was greater in the unloaded than in the loaded
conditions when simply comparing mean values, but multiple regression analysis
revealed that d was similar across the four loading conditions when
compared at an intermediate speed of 2.86 m s1
(Table 4). Furthermore, step
period tstep was similar to unloaded values across loading
conditions (Table 4). This
result agrees well with published stride periods of loaded and unloaded
trotting in horses carrying 14% body mass at 4.0 m s1
(Sloet van Oldruitenborgh-Ooste et al.,
1995) and in horses carrying 19% body mass at speeds above 3.0 m
s1 (Hoyt et al.,
2000
). In contrast to step period, duty factors were greater in
the loaded conditions than in the unloaded condition. This agrees
qualitatively with time of contact results from load-carrying experiments in
trotting horses (Hoyt et al.,
2000
; Sloet van
Oldruitenborgh-Ooste et al., 1995
).
Effect of adding mass near the center of mass
On the basis that limbs act primarily as struts, we predicted that adding
mass near the center of mass would not effect individual limb mean
foreaft force (normalized to body weight + added weight) (Hypothesis
4). Such a result was previously observed when mass was added near the center
of mass of running humans, in which absolute braking and propulsive impulse
components increased in direct proportion to the added weight
(Chang et al., 2000). We also
predicted that the individual limb bp bias (i.e. the Fourier ratio
a1/b2) would be affected by addition
of mass at the limb girdles, but not when mass was added near the
center of mass (Hypothesis 2). Nonetheless, both mean foreaft force and
bp bias showed increased magnitudes in the mid-loaded condition with
respect to the unloaded condition (Fig.
4). The magnitude of mean foreaft force increased from
0.036 to 0.039 BW, while forelimb bp bias decreased from 0.31 to
0.40 and hindlimb bp bias increased from 1.02 to 1.16. This
unexpected increase in both mean foreaft force and bp bias
magnitudes has two potential explanations. The first is a change in limb
excursions such that the forefoot would be positioned more anteriorly and the
hindfoot, more posteriorly on average during stance. This would alter the mean
foreaft force and bp bias due to the action of the limbs as
struts with steeper anteroposterior inclinations. Alternatively, the
mean foreaft force and bp bias could have been augmented by
moments exerted by the protractor muscles of the shoulder and retractor
muscles of the hip. Such a mechanism would constitute a classic example of
action of the limbs as levers, as described by Gray
(1968
). In our experiments, a
simple measurement of the horizontal distance p between simultaneous
fore and hind supports showed that this distance was the same in the unloaded
and mid-loaded conditions (Fig.
5). Hence, fore- and hindfoot positions could not have
shifted in opposite directions. Forelimb protracting and hindlimb retracting
moments must have been responsible for the observed changes in mean
foreaft force and bp bias during mid-loading. Assuming a mean
leg length of 0.54 m, both the shoulder and hip must have exerted additional
mean moments of ±0.53 N m to produce the additional ±0.003 BW of
mean foreaft force measured during mid-loading.
Opposing shoulder and hip moments reveal important functional patterns when
whole-body mechanics are considered. Although exerting opposing moments and
foreaft forces seems like a waste of metabolic energy, it may in fact
be necessary to stabilize the trunk under certain circumstances. Gray
(1968) hypothesized that
simultaneous forelimb protracting and hindlimb retracting moments would exert
an upward (dorsiflexing) bending moment on the trunk
(Fig. 6). He argued that this
mechanism could reduce the tension required in the hypaxial muscles that
resist the downward (ventroflexing) bending moment due to gravity. This effect
can be demonstrated easily with a flexible 15 cm ruler. Imagine holding the
ruler face up with thumb and forefinger of each hand at its ends. The ruler
will sag, or bend downward, under its own weight, putting the bottom surface
of the ruler in tension. Now twist your palms upward the resulting
bending moment causes the ruler to bend upward, putting the top surface of the
ruler in tension. The moments you applied to the ends of the ruler are
analogous to the moments exerted on the trunk by the muscles of the shoulder
and hip. If mass were added to the middle of the ruler, you would have to
exert larger moments to bend the ruler upward. When mass is added at the
center of mass of trotting dogs, our data show a compensatory increase of
±0.53 N m in shoulder and hip moments. Therefore, the shoulder plus hip
moment contributing to upward bending of the trunk would be1.06 N m.
|
By modeling the trunk as a beam rigidly fixed at its ends by the action of
shoulder and hip muscles, one can determine the hip and shoulder moments
required to maintain a net bending moment of zero across the length of the
trunk. Estimating that the distance between the shoulder and hip was
p (0.569 m) and that the added load of 0.1 BW (29.9 N) was applied at
a distance 0.66p anterior to the hip, the required shoulder plus hip
moment would be 3.8 N m. This suggests that the appendicular (i.e. shoulder
and hip) muscles contributed just 28% of the moment required to balance the
downward bending moment due to the added load. It is likely, however, that the
remaining upward bending moment was imparted to the trunk by hypaxial muscles
(Fig. 6). Because most
hypaxials, such as the rectus abdominis, act to bend the trunk upward, they
could provide the additional bending moment required to support a mid-trunk
load. In a previous study, such hypaxial action was evidenced by increased
electrical activity in the internal oblique muscle of dogs trotting with
mid-trunk loads (Fife et al.,
2001). The use of appendicular muscles to exert a bending moment
on the trunk is significant because it demonstrates a functional link between
axial and appendicular mechanics. Gray's hypothesis aptly explains the
observed effect of mid-loading on fore- and hindlimb mean foreaft force
and bp bias.
Effect of adding mass at the pectoral girdle
We hypothesized that the loaded limb (the forelimb in this case) would show
a decrease in bp bias (Hypothesis 2). However, no statistically
significant change in bp bias was observed in either of the limbs
during fore-loading. The forelimb braking bias was nearly identical in
fore-loaded and unloaded conditions and the hindlimb propulsive bias showed a
slight, insignificant decrease with respect to the unloaded value
(Fig. 4). The fore- and
hindlimb kinematics were similar in the fore-loaded and unloaded conditions,
as evidenced by the distance between simultaneous fore and hind supports
p and the distances of fore and hind supports from the mid-point
between the eye and base of the tail (Fig.
5). Hence, during steady speed trotting, the forelimb may act as a
strut with little or no anteroposterior inclination. This would explain
the absence of a reduced braking bias in response to increased forelimb
loading. As predicted (Hypothesis 4), mean foreaft force was
statistically unchanged from that of unloaded trotting.
In agreement with Hypothesis 3, the forelimb relative contact time
tc,f/tc,h increased from 1.19
(unloaded) to 1.23 during fore-loading
(Table 4). This follows the
same pattern observed in Labrador retrievers and greyhounds, which have
substantially different anteroposterior body mass distributions.
Although their hindlimb duty factors were statistically similar, Labrador
retrievers (R0=0.64) had forelimb duty factors of 0.505
and greyhounds (R0=0.56) had forelimb duty factors of
0.426, a statistically significant difference (P<0.05;
Bertram et al., 2000). Hence, a
relatively high forelimb duty factor is naturally associated with a higher
fraction of vertical impulse on the forelimb. We propose that, in general,
relative fore- and hindlimb duty factors reflect the anteroposterior
mass distributions of trotting quadrupeds. Biewener
(1983
) measured relative
contact time of the forelimb in trotting mammals across a size range of 0.01
to 270 kg. At the trotgallop transition speed, these ratios were less
than 1.0 in mammals likely to support a larger fraction of body weight on
their hindlimbs (i.e. pocket mouse, 0.88; mouse, 0.99; chipmunk, 0.96; ground
squirrel, 0.95), but greater than 1.0 in mammals known to support more body
weight on their forelimbs (i.e. dog, 1.07; pony, 1.04; horse, 1.01).
The fore-loaded condition also produced an unexpected change in the timing
of foot placement. The phase shift of hindlimb initial contact increased from
0.001 (unloaded) to 0.016 during fore-loading, indicating that hindlimb was
set down 1.5% of the stride period later in the fore-loaded condition
(Fig. 3,
Table 4). The same trend has
been observed in Labrador retrievers and greyhounds, with hindlimb initial
contact following forelimb initial contact in forelimb `heavy' Labradors, but
preceding forelimb initial contact in greyhounds
(Bertram et al., 2000). Whether
or not this pattern extends to other trotting quadrupeds is unknown.
Effect of adding mass at the pelvic girdle
Just as addition of mass at the pectoral girdle increased forelimb relative
contact time (tc,f/tc,h), addition of
mass at the pelvic girdle decreased forelimb relative contact time (i.e.,
increased hindlimb relative contact time). In agreement with Hypothesis 3,
tc,f/tc,h decreased from 1.19
(unloaded) to 1.09 during hind loading
(Table 4). As predicted
(Hypothesis 4), mean foreaft force was unchanged from that of unloaded
trotting (Fig. 4). The most
prominent and functionally important difference between unloaded and
hind-loaded trotting was the bp bias. We predicted that the propulsive
bias of the hindlimb would decrease in order to compensate for increased
vertical impulse due to hind-loading (Hypothesis 2). This would prevent
hindlimb propulsion from dominating forelimb braking. As expected, hindlimb
propulsive bias decreased significantly from the unloaded value of 1.02 to
0.88 in the hind-loaded condition (Fig.
4). This compensatory change could have been achieved by
decreasing the horizontal inclination of the hindlimb (strut action),
decreasing the hindlimb retracting moment (lever action), or a combination of
strut and lever action. From center of pressure data, it is clear that strut
action could account for the decrease in propulsive bias. During hind-loading,
the hindlimb center of pressure was significantly (P<0.05) more
anterior than during unloaded trotting
(Fig. 5). Given that this
anterior shift was, on average, 0.03 m and the mean hindlimb length was 0.54
m, we know that the mean hindlimb angle was approximately 3.2° retracted
with respect to the unloaded angle.
Without other changes in limb mechanics, a 3.2° change in mean limb angle would reduce propulsive force by 0.021 BW [i.e. 0.38 BW(tan3.2°)], which is seven times that required to compensate for the effect of hind-loading (i.e. 0.003 BW). The observed repositioning of the hindfoot would have reduced the mean foreaft force by more than half without a hindlimb retracting moment to maintain the observed mean foreaft force of 0.035 BW (Fig. 4). Given a foreaft force deficit of 0.018 BW (i.e. 0.0210.003) and a limb length of 0.54 m, the mean hindlimb retracting moment would need to be 2.9 N m, about three times that exerted by the shoulder plus hip during mid-loading. During hind-loading, the pelvis itself seems to have behaved much like a cantilever beam, supported primarily at the hip (posterior pelvis) while mass was added at the iliac crests (anterior pelvis). We conclude that the observed changes in the partitioning of hindlimb strut and lever action were largely due to local effects of loading the pelvis. If the added mass were applied 0.10 m anterior to the hip joint, a hindlimb retractor moment of 2.9 N m would be required to balance its effect. This is consistent with the actual placement of the added mass near the iliac crests. In contrast to mid-loading, the applied moment appears to have been resisted primarily by the hip muscles instead of being distributed between the shoulder, hip and, presumably, the trunk muscles.
Finally, the forelimb braking bias increased from the unloaded value of 0.31 to 0.41 in the hind-loaded condition (Fig. 4). The slight, statistically insignificant, anterior movement of the forefoot and/or an increase in forelimb protracting torque might have contributed to the observed increase in braking bias.
Implications for other quadrupeds
The axial and appendicular musculoskeletal systems of quadrupeds are
designed to accommodate specific anteroposterior mass distributions. In
this sense, the experimental addition of mass at discrete points seems
unnatural. A load concentrated at the center of mass, pectoral girdle, or
pelvic girdle is certainly not equivalent to the mass distributions inherent
to different structural designs. An artificial load is, however, equivalent in
one basic respect its effect on the center of mass position. Based
upon the assumption that limbs act primarily as struts, we hypothesized that a
shift in anteroposterior mass distribution would alter individual limb
mechanics. For example, the hindlimb propulsive bias decreased and its
relative contact time increased in the hind-loaded condition. We propose that
this relationship between anteroposterior mass distribution and limb
function represents a general trend in trotting quadrupeds. For example,
trotting dogs (R0=0.64) show a hindlimb propulsive bias
approximately three times greater than the forelimb braking bias and a
hindlimb duty factor 15% smaller than that of the forelimb. Cheetahs
(Acinonyx jubatus, R00.52;
Kruger, 1943
), which have more
symmetric forehind weight distributions, are predicted to show
approximately equal fore- and hindlimb bp biases and duty factors.
Lizards, which support more weight on their hindlimbs, are predicted to show
greater forelimb braking biases and reduced forelimb duty factors relative to
the hindlimb. This general pattern is illustrated in
Fig. 7, which summarizes our
results from trotting dogs and, based on these data, makes predictions for
hypothetical quadrupeds with different forehind mass distributions. As
shown in Fig. 7A,C, the limb
that supports a greater fraction of total vertical impulse is expected to have
a greater duty factor and a bp bias closer to zero. The limb that
supports a smaller fraction of total vertical impulse is expected to have a
smaller duty factor and, due to its reduced vertical force and contact time,
is expected to compensate with a greater bp bias. These duty factor
predictions are also consistent with the tendency of more heavily loaded limbs
to be relatively longer.
|
Individual limb forces have been reported for only a handful of trotting
quadrupeds. Although most of these data represent trotting with substantial
net braking, the results generally support the pattern of limb function
illustrated in Fig. 7. In Dutch
warmblood horses, Merkens and coworkers
(Merkens et al., 1993)
reported a vertical impulse ratio of 0.55 and a bp bias similar to that
of Fig. 7B. There was a small
mean braking force of 0.014 BW, on average, in their sample of trotting
steps. In cats, forelimb peak vertical force was found to be 1.69 times that
of the hindlimb (Demes et al.,
1994
), which suggests a vertical impulse ratio between 0.60 and
0.70 during trotting. The same study showed that forelimb braking impulse was
3.5 times forelimb propulsive impulse and hindlimb propulsive impulse was 2.7
times hindlimb braking impulse, indicating a larger bp bias in the
forelimb. This pattern seems unusual in light of
Fig. 7A, but is consistent with
the mean net braking acceleration of 0.037 BW in their sample of
trotting steps. The mean foreaft forces exerted by the forelimb
(0.068 BW) and hindlimb (0.031 BW) of trotting cats are quite similar
to those predicted for greyhounds trotting with a net braking acceleration of
0.037 BW (i.e. 0.070 and 0.033 BW, respectively;
Lee et al., 1999
).
Even fewer trotting data are available from quadrupeds with vertical
impulse ratios below 0.50. Although individual limb function has been
described in at least ten primate species
(Demes et al., 1994), most
primates use asymmetrical running gaits at higher speeds
(Hildebrand, 1967
).
Nonetheless, Kimura (2000
) has
collected fast trotting data from adult Japanese macaques Macaca
fuscata. He reported a vertical impulse ratio of 0.47, which is nearly
equal to his measurement of standing weight distribution (0.46;
Kimura et al., 1979
). This is
surprising, given that Kimura's sample of trotting steps represented a mean
net braking acceleration of more than 0.10 BW, on average
(Kimura, 2000
). In trotting
dogs, this degree of braking would cause a substantial increase in vertical
impulse ratio (e.g. from 0.64 to 0.71 in Labradors;
Lee et al., 1999
). It is
possible that macaques use a mechanism, such as forward foot placement
(Raibert, 1990
), to avoid a
nose-down pitching moment (and subsequent vertical impulse redistribution) due
to net braking. Because forelimb braking impulse was 11.1 times forelimb
propulsive impulse and hindlimb propulsive impulse was roughly equal to
hindlimb braking impulse, Kimura's data suggest a bp bias similar to
that of Fig. 7C. It is likely,
however, that steady speed trotting of macaques is more like
Fig. 7B.
Some mammals with extreme anteroposterior mass distributions, such
as giraffes, hyenas and rabbits, have abandoned trotting and pacing gaits in
favor of galloping and bounding gaits
(Estes, 1993;
Howell, 1944
;
Pennycuick, 1975
). This may
also be the case for some primates (Demes
et al., 1994
; Schmitt and
Lemelin, 2002
). Given that dogs (R0=0.64) have
a fairly extreme hindlimb propulsive bias, a more anterior center of mass
position, such as that of hyenas, may exceed a limit by which propulsive
impulse can be supplied by increasing hindlimb propulsive bias. Because
hindlimb propulsive and forelimb braking impulses must be equal in magnitude
during steady speed trotting steps, extreme anteroposterior mass
distributions might favor galloping and bounding over trotting and pacing.
Although forelimb and hindlimb mechanics should generally accommodate a
particular anteroposterior mass distribution, an individual's body mass
distribution often changes in response to behavioral or physiological factors.
Male cervids grow substantial masses at the end of a long neck every spring
and then discard this mass in the autumn. The `Irish elk' Megaloceros
giganteus, with 40 kg antlers estimated to be approximately 7% body mass,
provides a dramatic example (Geist,
1987). Carnivores can carry heavy prey in their jaws for some
distance before consuming it. For example, man-eating tigers have been
reported to run while carrying an intact human in their jaws
(Corbett, 1946
). Gray wolves
Canis lupus often gorge on meals as large as 2030% of their
body mass, yet they may subsequently travel a few miles to find a suitable
spot to rest (Mech, 1970
).
After making a kill and gorging as much as possible, African hunting dogs
Lycaon pictus may need to escape from large scavengers such as
spotted hyenas or lions or, in other cases, chase a lone hyena from their kill
at high speed (Creel and Creel,
2002
). The most ubiquitous natural increase in trunk loading
occurs in gravid females, which often carry substantial loads over periods of
weeks or months. For example, neonatal mass is approximately 15% of maternal
mass in the pronghorn and springbok
(Robbins and Robbins, 1979
).
This value neglects the masses of the amniotic fluid and placenta, however,
which are likely to be substantial. The distribution of fetal and placental
mass is generally posterior to the center of mass in quadrupeds.
Conclusions
Adding mass to the pectoral or pelvic girdles significantly altered the
forehind vertical impulse distribution of trotting dogs. Assuming that
the limbs act as struts, we predicted that these changes would lead to a
decrease in bp bias and an increase in relative contact time of the
experimentally loaded limb, while mean foreaft force would be
unaffected. All three of these predicted results were observed in the
hind-loaded condition. Only the latter two were observed in the fore-loaded
condition, perhaps due to a smaller initial bp bias and/or limb
inclination of the forelimb. We propose that the observed relationships
between anteroposterior mass distribution, bp bias, and relative
contact time will apply to other quadrupeds. Our data also show that the
mechanical effects of adding mass to the trunk are much more complex than
would be predicted from center of mass position alone. During mid-loading, the
shoulder and hip moments increased in order to resist the downward bending
moment applied to the trunk. During hind-loading, the hindlimb retractor
muscles exerted a large moment about the hip to resist the moment applied to
the pelvis. In accordance with the pioneering models proposed by Sir James
Gray (1968), both of these
results exemplify a link between appendicular and axial mechanics via
action of the limbs as levers.
Appendix A
![]() |
Acknowledgments |
---|
![]() |
References |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Alexander, R. M. (1977). Mechanics and scaling of terrestrial locomotion. In Scale Effects in Animal Locomotion: based on the proceedings of an international symposium held at Cambridge University, September, 1975 (ed. T. J. Pedley), pp.93 -110. London: Academic Press.
Alexander, R. M. and Goldspink, G. (1977). Mechanics and Energetics of Animal Locomotion. London, New York: Chapman and Hall. Distributed by Halsted Press.
Alexander, R. M. and Jayes, A. S. (1980). Fourier analysis of forces exerted in walking and running. J. Biomech. 13,383 -390.[Medline]
Bertram, J. E. A., Lee, D. V., Case, H. N. and Todhunter, R. J. (2000). Comparison of the trotting gaits of Labrador Retrievers and Greyhounds. Am. J. Vet. Res. 61,832 -838.[Medline]
Bertram, J. E. A., Lee, D. V., Todhunter, R. J., Foels, W. S., Williams, A. and Lust, G. (1997). Multiple force platform analysis of the canine trot: a new approach to assessing basic characteristics of locomotion. Vet. Comp. Orthoped. Traumatol. 10,160 -169.
Biewener, A. A. (1983). Allometry of quadrupedal locomotion: the scaling of duty factor, bone curvature and limb orientation to body size. J. Exp. Biol. 105,147 -171.[Abstract]
Chang, Y. H., Huang, H. W., Hamerski, C. M. and Kram, R.
(2000). The independent effects of gravity and inertia on running
mechanics. J. Exp. Biol.
203,229
-238.
Corbett, J. (1946). Man-eaters of Kumaon. New York: Oxford University Press.
Creel, S. and Creel, N. M. (2002). The African Wild Dog: Behavior, Ecology, and Conservation. Princeton, N.J.: Princeton University Press.
Demes, B., Larson, S. G., Stern, J. T. J., Jungers, W. L., Biknevicius, A. R. and Schmitt, D. (1994). The kinetics of primate quadrupedalism: `hindlimb drivex'reconsidered. J. Hum. Evol. 26,353 -374.[CrossRef]
Estes, R. (1993). The Safari Companion: A Guide to Watching African Mammals: Including Hoofed Mammals, Carnivores, and Primates. Post Mills, Vt.: Chelsea Green.
Fife, M. M., Bailey, C. L., Lee, D. V. and Carrier, D. R.
(2001). Function of the oblique hypaxial muscles in trotting
dogs. J. Exp. Biol. 204,2371
-2381.
Full, R. J., Blickhan, R. and Ting, L. H. (1991). Leg design in hexapedal runners. J. Exp. Biol. 158,369 -390.[Abstract]
Geist, V. (1987). On the evolution of optical signals in deer: a preliminary analysis. In Biology and Management of the Cervidae (ed. C. M. Wemmer), pp.235 -255. Washington: Smithsonian Institution Press.
Gray, J. (1968). Animal Locomotion. New York: Norton.
Hamming, R. W. (1973). Numerical Methods for Scientists and Engineers. New York: McGraw-Hill.
Hildebrand, M. (1967). Symmetrical gaits of primates. Am. J. Phys. Anthropol. 26,119 -130.
Howell, A. B. (1944). Speed In Animals; Their Specialization for Running and Leaping. Chicago: University of Chicago Press.
Hoyt, D. F., Wickler, S. J. and Cogger, E. A.
(2000). Time of contact and step length: the effect of limb
length, running speed, load carrying and incline. J. Exp.
Biol. 203,221
-227.
Jayes, A. S. and Alexander, R. M. (1978). Mechanics of locomotion of dogs (Canis familiaris) and sheep (Ovis aries). J. Zool. 185,289 -308.[Medline]
Kimura, T. (2000). Development of quadrupedal locomotion on level surfaces in Japanese macaques. Folia Primatol (Basel) 71,323 -333.[CrossRef][Medline]
Kimura, T., Okada, M. and Ishida, H. (1979). Kinesiological characteristics of primate walking: its significance in human walking. In Environment, Behavior and Morphology: Dynamic Interactions in Primates (ed. M. E. Morbeck, H. Preuschoft and N. Gomberg), pp. 297-311. New York: Gustav Fischer.
Kruger, W. (1943). Uber die beziehungen zwischen schwerpunktslage und starke der substantia compacta einzelner gliedmabenkochen bei vierfubigen saugetieren. Morphol. Jahrb. 88,377 -396.
Lee, D. V., Bertram, J. E. A. and Todhunter, R. J.
(1999). Acceleration and balance in trotting dogs. J.
Exp. Biol. 202,3565
-3573.
Mech, L. D. (1970). The Wolf: The Ecology and Behavior of an Endangered Species. Garden City, NY: Published for the American Museum of Natural History by the Natural History Press.
Merkens, H. W., Schamhardt, H. C., Van Osch, G. J. and Van den Bogert, A. J. (1993). Ground reaction force patterns of Dutch warmblood horses at normal trot. Equine Vet. J. 25,134 -137.[Medline]
Pandy, M. G., Kumar, V., Berme, N. and Waldron, K. J. (1988). The dynamics of quadrupedal locomotion. J. Biomech. Eng. 110,230 -237.[Medline]
Pennycuick, C. J. (1975). On the running of the gnu (Connochaetes taurinus) and other animals. J. Exp. Biol. 63,775 -799.
Raibert, M. H. (1990). Trotting, pacing and bounding by a quadruped robot. J. Biomech. 23 Suppl. 1,79 -98.[Medline]
Robbins, C. T. and Robbins, B. L. (1979). Fetal and neonatal growth patterns and maternal reproductive effort in ungulates and subungulates. Am. Nat. 114,101 -116.[CrossRef]
Rollinson, J. and Martin, R. D. (1981). Comparative aspects of primate locomotion with special reference to arboreal cercopithecines. Symp. Zool. Soc. Lond. 48,377 -427.
Rumph, P. F., Lander, J. E., Kincaid, S. A., Baird, D. K., Kammermann, J. R. and Visco, D. M. (1994). Ground reaction force profiles from force platform gait analyses of clinically normal mesomorphic dogs at the trot. Am. J. Vet. Res. 55,756 -761.[Medline]
Schmitt, D. and Lemelin, P. (2002). Origins of primate locomotion: gait mechanics of the woolly opossum. Am. J. Phys. Anthropol. 118,231 -238.[CrossRef][Medline]
Sloet van Oldruitenborgh-Ooste, M. M., Barneveld, A. and Schamhardt, H. C. (1995). Effects of weight and riding on work load and locomotion during treadmill exercise. Equine Vet. J. 18,413 -417.
Sokal, R. R. and Rohlf, F. J. (1995). Biometry: The Principles and Practice of Statistics in Biological Research. New York: Freeman.
Willey, J. S., Biknevicius, A. R., Reilly, S. M. and Earls, K.
D. (2004). The tale of the tail: Limb function and locomotor
mechanics in Alligator mississippiensis. J. Exp. Biol.
207,553
-563.