It is apparent that antero–posterior 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 fore–aft 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 fore–aft 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 fore–aft 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 b–p bias and relative contact time, would be expected during experimental loading of the limb girdles.

### Subjects and data collection

Five adult dogs (Canis familiaris L.) of various breeds were used in this experiment. Subjects ranged in body mass from 23.5 to 34.5 kg and from 2 to 9 years of age (Table 1). The dogs were borrowed from private owners for a period of 8 h or less and were never kept overnight. During data collection sessions, water was provided ad libitum and periodic rest breaks were given. The dogs trotted as they were led on a leash across a level, hard-packed soil runway in which two force platforms were positioned in series(Fig. 1A). This allowed independent measurement of simultaneous fore- and hindlimb ground reaction forces (Fig. 1B).

Table 1.

Subject descriptions

DogMass (kg)Hip height (m)Age (years)BreedSex
29.5 0.545 Pointer/scent hound
34.5 0.573 German shepherd
23.5 0.525 Coonhound
33.1 0.556 German shepherd mix
31.6 0.498 Golden retriever
DogMass (kg)Hip height (m)Age (years)BreedSex
29.5 0.545 Pointer/scent hound
34.5 0.573 German shepherd
23.5 0.525 Coonhound
33.1 0.556 German shepherd mix
31.6 0.498 Golden retriever
Fig. 1.

(A) Simultaneous fore- and hindlimb supports were recorded by two separate force platforms (thick lines) during trotting. (B) Ground reaction forces were measured independently by the two force platforms. Solid lines indicate vertical force and broken lines fore–aft force. BW, body weight.

Fig. 1.

(A) Simultaneous fore- and hindlimb supports were recorded by two separate force platforms (thick lines) during trotting. (B) Ground reaction forces were measured independently by the two force platforms. Solid lines indicate vertical force and broken lines fore–aft force. BW, body weight.

### 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 LabView™software and a National Instruments™ (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 fore–aft ground reaction force (GRF) with separate double-cantilever transducers at each corner post. Vertical GRF acting upward and fore–aft GRF acting in the direction of travel were considered positive. Vertical impulse jz and fore–aft 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:
$\ F_{\mathrm{z}}=(j_{\mathrm{z},\mathrm{total}}{/}t_{\mathrm{c},\mathrm{total}}){/}m\mathbf{g},$
1
where m is body mass (or 110% of body mass under loaded conditions)and g is gravitational acceleration. Likewise, normalized mean fore–aft acceleration of the center of mass, which is equivalent to force in dimensionless terms, was determined by:
$\ A_{\mathrm{y}}=(j_{\mathrm{y},\mathrm{total}}{/}t_{\mathrm{c},\mathrm{total}}){/}m\mathbf{g}.$
2
Normalized mean fore–aft force exerted separately on the forelimb and hindlimb during paired diagonal supports were determined by:
$\ F_{\mathrm{y},\mathrm{fore}}=(j_{\mathrm{y},\mathrm{fore}}{/}t_{\mathrm{c},\mathrm{total}}){/}m\mathbf{g},$
3
and
$\ F_{\mathrm{y},\mathrm{hind}}=(j_{\mathrm{y},\mathrm{hind}}{/}t_{\mathrm{c},\mathrm{total}}){/}m\mathbf{g}.$
4
The mean angle of the resultant force vector during fore- or hindlimb contact was determined by:
$\ {\theta}=\mathrm{tan}^{-1}(j_{\mathrm{y}}{/}j_{\mathrm{z}}).$
5
Forelimb vertical impulse was expressed as a fraction of total vertical impulse during paired diagonal supports. This quantity is referred to as the vertical impulse ratio:
$\ R=j_{\mathrm{z},\mathrm{fore}}{/}j_{\mathrm{z},\mathrm{total}},$
6
(Jayes and Alexander, 1978; Lee et al., 1999).

In order to determine a limb's functional bias toward braking or propulsion, a Fourier method (Hamming,1973) adapted to the analysis of force–time curves by Alexander and Jayes (1980) was used to quantify fore–aft 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 fore–aft force curve shape, which is referred to as the braking–propulsive (b–p) bias. Force–curve shapes defined by negative and positive values of a1/b2are 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 b–p bias(a1/b2) was computed from individual limb fore–aft force curves.

Fig. 2.

Illustration of braking–propulsive biases defined by different values of the Fourier shape ratio a1/b2. a1/b2 = –0.5 indicates a braking bias, a1/b2 = 0 indicates no b–p bias, and a1/b2 = 0.5 indicates a propulsive bias.

Fig. 2.

Illustration of braking–propulsive biases defined by different values of the Fourier shape ratio a1/b2. a1/b2 = –0.5 indicates a braking bias, a1/b2 = 0 indicates no b–p bias, and a1/b2 = 0.5 indicates a propulsive bias.

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 pbetween 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
$${\nu}_{\mathrm{y},\mathrm{step}}$$
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
$${\nu}_{\mathrm{y},\mathrm{step}}$$
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
$${\nu}_{\mathrm{y},\mathrm{step}}$$
, 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:
$\ \mathrm{Froude}={\nu}_{\mathrm{y},\mathrm{step}}{/}\sqrt{\mathbf{g}h},$
7
where g is gravitational acceleration and h is hip height as defined in the following section. The aforementioned step period tstep is one half of the stride period during trotting. For consistency with existing literature, the stride period(2tstep) was used to normalize individual foot and paired diagonal contact times. The ratio of foot contact time to stride period is termed the duty factor DF and was computed as:
$\ DF_{\mathrm{total}}=t_{\mathrm{c},\mathrm{total}}{/}2t_{\mathrm{step}},$
8
$\ DF_{\mathrm{fore}}=t_{\mathrm{c},\mathrm{fore}}{/}2t_{\mathrm{step}},$
9
and
$\ DF_{\mathrm{hind}}=t_{\mathrm{c},\mathrm{hind}}{/}2t_{\mathrm{step}}.$
10
In addition, the ratio of fore- to hindlimb contact time tc,f/tc,h was computed. The time of initial hindfoot contact with respect to initial forefoot contact was also expressed as a fraction of the stride period. This quantity, referred to as the hindlimb phase shift αhind, was computed as:
$\ {\alpha}_{\mathrm{hind}}=(t_{\mathrm{i},\mathrm{hind}}-t_{\mathrm{i},\mathrm{fore}}){/}2t_{\mathrm{step}},$
11
where ti,hind and ti,fore are initial contact times. A negative value of αhind indicates that hindlimb contact occurred before forelimb contact.

### Videographic measurements

A video camera with VCR (PEAK™ Performance Technologies Inc.,Centennial, CO, USA) acquired 120 images s–1 in lateral view. Force and video acquisition were synchronized by connection of a manual switch to a PEAK™ 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 Tukey–Kramer multiple comparison tests (InStat™ 2.0). The latter method was applied to mean vertical force, mean fore–aft 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 fore–aft force and b–p bias, which were functions of mean fore–aft 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

$$F_{\mathrm{y},\mathrm{fore}}$$
and
$$F_{\mathrm{y},\mathrm{hind}}$$
on
$$A_{\mathrm{y}}$$
were linear, as was the regression of a1/b2,fore on
$$A_{\mathrm{y}}$$
. 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 fore–aft 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.

During paired diagonal supports, mean vertical ground reaction force

$$F_{\mathrm{z}}$$
was greatest in the unloaded condition(Table 2A). The unloaded mean vertical force of 1.04 body weights (BW) was significantly(P<0.05) higher than that of the three loaded conditions, which were between 0.98 and 1.0 BW (Table 2B). Only the unloaded
$$F_{z}$$
was significantly different from one (P<0.05). Mean forward velocity
$${\nu}_{\mathrm{y},\mathrm{step}}$$
was, on average, higher in the unloaded than in the loaded conditions (U, 3.09 m s–1; M,2.81 m s–1; F, 2.70 m s–1; H, 2.77 m s–1). These differences were significant (P<0.05)only in the F condition. When
$${\nu}_{\mathrm{y},\mathrm{step}}$$
was normalized as a Froude number (yielding treatment means between 1.17 and 1.35), the statistical comparisons between loading conditions were the same as before normalization. Velocity differences between unloaded and loaded conditions were taken into account by predicting locomotor parameters at an intermediate
$${\nu}_{\mathrm{y},\mathrm{step}}$$
of 2.86 m s–1 whenever a parameter was significantly(P<0.10) related to
$${\nu}_{\mathrm{y},\mathrm{step}}$$
(Table 3). This analysis produced statistically similar values of distance between subsequent footfalls d as well as step period tstep across all four loading conditions (U, M, F and H) (Table 4). Ground reaction forces reconstructed from locomotor parameters reported in this section and in Appendix B illustrate steady speed GRF patterns at 2.86 m s–1 for U, M, F and H conditions(Fig. 3).

Table 2.

Mean vertical force, mean fore-aft acceleration and vertical impulse ratio

(A
1.037 0.0037 0.663 0.665
0.987 0.0535 0.640 0.678
1.075 0.0248 0.658 0.676
1.055 -0.0155 0.626 0.615
1.027 0.0142 0.596 0.606
U  1.036±0.015 0.0161±0.0115 0.637±0.012 0.648±0.016
0.998 -0.0330 0.693 0.670
0.927 0.0097 0.671 0.678
1.010 0.0200 0.684 0.698
1.004 0.0093 0.637 0.644
0.995 0.0042 0.617 0.619
M  0.987±0.015 0.0020±0.0091 0.660±0.015 0.662±0.014
0.976 -0.0323 0.712 0.689
0.957 0.0050 0.682 0.685
0.999 0.0211 0.700 0.715
1.009 -0.0392 0.682 0.654
0.976 0.0023 0.629 0.631
F  0.983±0.009 -0.0086±0.0116 0.681±0.014 0.675±0.015
1.013 0.0056 0.644 0.648
0.974 -0.0034 0.628 0.626
1.017 0.0158 0.629 0.640
1.004 -0.0101 0.591 0.584
1.005 0.0329 0.566 0.589
H  1.002±0.008 0.0082±0.0076 0.612±0.015 0.617±0.013
(B
U vs. -0.049*** -0.0141ns 0.024** 0.014ns
U vs. -0.053*** -0.0248ns 0.044*** 0.027**
U vs. -0.034** -0.0080ns -0.025*** -0.031**
M vs. -0.004ns -0.0107ns 0.021** 0.013ns
M vs. 0.016ns 0.0061ns -0.049*** -0.044***
F vs. 0.019ns 0.0168ns -0.069*** -0.057***
(A
1.037 0.0037 0.663 0.665
0.987 0.0535 0.640 0.678
1.075 0.0248 0.658 0.676
1.055 -0.0155 0.626 0.615
1.027 0.0142 0.596 0.606
U  1.036±0.015 0.0161±0.0115 0.637±0.012 0.648±0.016
0.998 -0.0330 0.693 0.670
0.927 0.0097 0.671 0.678
1.010 0.0200 0.684 0.698
1.004 0.0093 0.637 0.644
0.995 0.0042 0.617 0.619
M  0.987±0.015 0.0020±0.0091 0.660±0.015 0.662±0.014
0.976 -0.0323 0.712 0.689
0.957 0.0050 0.682 0.685
0.999 0.0211 0.700 0.715
1.009 -0.0392 0.682 0.654
0.976 0.0023 0.629 0.631
F  0.983±0.009 -0.0086±0.0116 0.681±0.014 0.675±0.015
1.013 0.0056 0.644 0.648
0.974 -0.0034 0.628 0.626
1.017 0.0158 0.629 0.640
1.004 -0.0101 0.591 0.584
1.005 0.0329 0.566 0.589
H  1.002±0.008 0.0082±0.0076 0.612±0.015 0.617±0.013
(B
U vs. -0.049*** -0.0141ns 0.024** 0.014ns
U vs. -0.053*** -0.0248ns 0.044*** 0.027**
U vs. -0.034** -0.0080ns -0.025*** -0.031**
M vs. -0.004ns -0.0107ns 0.021** 0.013ns
M vs. 0.016ns 0.0061ns -0.049*** -0.044***
F vs. 0.019ns 0.0168ns -0.069*** -0.057***

z, mean vertical force; y, mean fore-aft acceleration; R, vertical impulse ratio; R0, R at steady speed.

See the text for parameter definitions.

(A)

values are means for each dog and group means ± s.e.m.for each loading condition

(B)

Differences between means of each loading condition (e.g. U vs.M' was computed as M̄-Ū). Asterisks indicate the results of Tukey-Kramer tests comparing each loading condition to the other

***

P<0.001

**

P<0.01, *P<0.05

ns

P>0.05)

Table 3.

Regression coefficients (±95% C.I.) for various parametersversus mean fore-aft acceleration and mean forward velocity

Parametervs. Āy (BW)vs. v̄y (m s-1)r2
d (m)
U ns 0.111±0.021 0.58
M ns 0.103±0.028 0.54
F ns 0.095±0.032 0.35
H -0.495±0.424 0.114±0.048 0.32
tstep (s)
U ns -0.033±0.007 0.54
M ns -0.045±0.010 0.66
F ns -0.046±0.011 0.50
H -0.163±0.146 -0.040±0.017 0.52
αhind
U ns ns
M ns -0.046±0.032 0.16
F ns -0.042±0.023 0.17
H ns -0.042±0.034 0.11
DFtotal
U ns ns
M ns -0.034±0.024 0.16
F ns -0.031±0.013 0.25
H ns -0.020±0.019 0.08
DFfore
U ns -0.027±0.022 0.07
M ns -0.049±0.025 0.26
F ns -0.047±0.016 0.35
H ns -0.052±0.023 0.29
DFhind
U ns -0.037±0.011 0.34
M ns -0.019±0.019 0.08
F -0.224±0.144 ns 0.13
H ns -0.015±0.018 0.05
tc,f/tc,h
U ns ns
M ns ns
F 0.871±0.723 -0.103±0.071 0.13
H ns -0.083±0.074 0.09
θfore (deg.)
U 49.9±3.43 1.34±0.577 0.92
M 47.3±8.21 ns 0.75
F 50.4±8.94 ns 0.66
H 40.0±10.1 ns 0.56
θhind (deg.)
U 56.6±6.79 -1.23±1.14 0.77
M 59.7±13.0 1.21±1.28 0.73
F 63.8±13.6 ns 0.57
H 71.7±14.3 ns 0.68
Parametervs. Āy (BW)vs. v̄y (m s-1)r2
d (m)
U ns 0.111±0.021 0.58
M ns 0.103±0.028 0.54
F ns 0.095±0.032 0.35
H -0.495±0.424 0.114±0.048 0.32
tstep (s)
U ns -0.033±0.007 0.54
M ns -0.045±0.010 0.66
F ns -0.046±0.011 0.50
H -0.163±0.146 -0.040±0.017 0.52
αhind
U ns ns
M ns -0.046±0.032 0.16
F ns -0.042±0.023 0.17
H ns -0.042±0.034 0.11
DFtotal
U ns ns
M ns -0.034±0.024 0.16
F ns -0.031±0.013 0.25
H ns -0.020±0.019 0.08
DFfore
U ns -0.027±0.022 0.07
M ns -0.049±0.025 0.26
F ns -0.047±0.016 0.35
H ns -0.052±0.023 0.29
DFhind
U ns -0.037±0.011 0.34
M ns -0.019±0.019 0.08
F -0.224±0.144 ns 0.13
H ns -0.015±0.018 0.05
tc,f/tc,h
U ns ns
M ns ns
F 0.871±0.723 -0.103±0.071 0.13
H ns -0.083±0.074 0.09
θfore (deg.)
U 49.9±3.43 1.34±0.577 0.92
M 47.3±8.21 ns 0.75
F 50.4±8.94 ns 0.66
H 40.0±10.1 ns 0.56
θhind (deg.)
U 56.6±6.79 -1.23±1.14 0.77
M 59.7±13.0 1.21±1.28 0.73
F 63.8±13.6 ns 0.57
H 71.7±14.3 ns 0.68

y, mean fore-aft acceleration; y, mean forward velocity.

Only significant (P<0.10) regression coefficients are shown(ns, not significant.)

See the text for parameter definitions.

Table 4.

ParameterUMFH
d (m) 0.624±0.008 0.615±0.011 0.628±0.012 0.626±0.014
tstep (s) 0.219±0.003 0.217±0.004 0.222±0.004 0.220±0.005
αhind 0.001±0.009 0.012±0.012 0.016±0.009* —0.010±0.010*
DFtotal 0.461±0.005 0.487±0.009* 0.485±0.005* 0.478±0.006*
DFfore 0.451±0.009 0.476±0.009* 0.477±0.006* 0.455±0.007
DFhind 0.383±0.004 0.405±0.007* 0.389±0.005* 0.418±0.006*
tc,f/tc,h 1.19±0.017 1.18±0.032 1.23±0.024* 1.09±0.022*
Θfore (deg.) —3.32±0.221 —3.43±0.307 —2.92±0.302* —3.30±0.362
Θhind (deg.) 5.84±0.438 6.46±0.454* 5.70±0.459 5.14±0.512*
ParameterUMFH
d (m) 0.624±0.008 0.615±0.011 0.628±0.012 0.626±0.014
tstep (s) 0.219±0.003 0.217±0.004 0.222±0.004 0.220±0.005
αhind 0.001±0.009 0.012±0.012 0.016±0.009* —0.010±0.010*
DFtotal 0.461±0.005 0.487±0.009* 0.485±0.005* 0.478±0.006*
DFfore 0.451±0.009 0.476±0.009* 0.477±0.006* 0.455±0.007
DFhind 0.383±0.004 0.405±0.007* 0.389±0.005* 0.418±0.006*
tc,f/tc,h 1.19±0.017 1.18±0.032 1.23±0.024* 1.09±0.022*
Θfore (deg.) —3.32±0.221 —3.43±0.307 —2.92±0.302* —3.30±0.362
Θhind (deg.) 5.84±0.438 6.46±0.454* 5.70±0.459 5.14±0.512*

Values are ±95% C.I.

Significant differences (P<0.05) from the unloaded condition are indicated by asterisks.

See Appendix A and the text for parameter definitions.

Appendix B

Steady speed Fourier coefficients during level trotting at 2.86 m s-1

Vertical force curves

a1,fore (BW)b2,fore (BW)a3,fore (BW)
1.06±0.02 -0.071±0.014 0.051±0.008
1.05±0.03 -0.025±0.015* 0.061±0.013
1.08±0.02 -0.041±0.011* 0.064±0.012*
1.02±0.03* -0.039±0.015* 0.062±0.011
1.05 -0.044 0.060
a1,fore (BW)b2,fore (BW)a3,fore (BW)
1.06±0.02 -0.071±0.014 0.051±0.008
1.05±0.03 -0.025±0.015* 0.061±0.013
1.08±0.02 -0.041±0.011* 0.064±0.012*
1.02±0.03* -0.039±0.015* 0.062±0.011
1.05 -0.044 0.060
a1,hind (BW)b2,hind (BW)a3,hind (BW)
0.708±0.016 -0.079±0.008 0.019±0.003
0.640±0.010* -0.060±0.011* 0.010±0.006*
0.636±0.015* -0.058±0.010* 0.013±0.004*
0.706±0.021 -0.060±0.010* 0.011±0.006*
0.673 0.064 0.013
a1,hind (BW)b2,hind (BW)a3,hind (BW)
0.708±0.016 -0.079±0.008 0.019±0.003
0.640±0.010* -0.060±0.011* 0.010±0.006*
0.636±0.015* -0.058±0.010* 0.013±0.004*
0.706±0.021 -0.060±0.010* 0.011±0.006*
0.673 0.064 0.013
Fore-aft force curves
a1,foreb2,forea3,fore
-0.052±0.004 0.149±0.005 -0.002±0.005
-0.055±0.006 0.143±0.006 -0.012±0.004*
-0.045±0.005* 0.153±0.005 -0.007±0.004
-0.050±0.007 0.139±0.006* -0.004±0.005
-0.051 0.146 -0.006
Fore-aft force curves
a1,foreb2,forea3,fore
-0.052±0.004 0.149±0.005 -0.002±0.005
-0.055±0.006 0.143±0.006 -0.012±0.004*
-0.045±0.005* 0.153±0.005 -0.007±0.004
-0.050±0.007 0.139±0.006* -0.004±0.005
-0.051 0.146 -0.006
a1,hindb2,hinda3,hind
0.087±0.005 0.075±0.004 0.004±0.003
0.085±0.005 0.069±0.005* 0.002±0.004
0.073±0.006* 0.070±0.003* 0.004±0.003
0.076±0.007* 0.083±0.006* 0.005±0.005
0.080 0.074 0.004
a1,hindb2,hinda3,hind
0.087±0.005 0.075±0.004 0.004±0.003
0.085±0.005 0.069±0.005* 0.002±0.004
0.073±0.006* 0.070±0.003* 0.004±0.003
0.076±0.007* 0.083±0.006* 0.005±0.005
0.080 0.074 0.004

Values are ± 95% C.I. for each loading condition and overall means are shown.

Significant differences (P<0.05) from the unloaded condition are indicated by asterisks.

For an explanation of the Fourier coefficients a1, a2 and b2, see Materials and methods.

Fig. 3.

Fore- and hindlimb force curves reconstructed from steady speed Fourier coefficients predicted at 2.86 m s–1 (see Materials and methods and Appendix B) for each of the loading conditions (U, unloaded; M, mid-load; F, fore-load; H,hind-load). Solid lines indicate vertical force and broken lines fore–aft force.

Fig. 3.

Fore- and hindlimb force curves reconstructed from steady speed Fourier coefficients predicted at 2.86 m s–1 (see Materials and methods and Appendix B) for each of the loading conditions (U, unloaded; M, mid-load; F, fore-load; H,hind-load). Solid lines indicate vertical force and broken lines fore–aft force.

Mean fore–aft acceleration
$$A_{\mathrm{y}}$$
was not significantly different between U, M, F and H conditions(Table 2). Nevertheless,substantial inter-subject variation was observed, with
$$A_{\mathrm{y}}$$
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
$$A_{\mathrm{y}}$$
, 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
$$A_{\mathrm{y}}$$
,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
$$A_{\mathrm{y}}$$
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:
$\ R_{0}=R+A_{\mathrm{y}}(-0.71).$
12
Mean values of R and R0 are reported for each subject in Table 2. R0 was not significantly different between U and M, but fore-loading increased R0 by 0.027 (P<0.01)and hind-loading decreased R0 by 0.031(P<0.01) with respect to U(Table 2B). In agreement with Hypothesis 1, F and H conditions significantly altered the fore–hind impulse distribution, while M maintained the natural impulse distribution.

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.

Fig. 4.

Steady speed values (±95% CI) of mean fore–aft force (filled bars) and braking–propulsive bias a1/b2 (open bars) for the loading conditions U, M, F and H (see text). Significant difference(P<0.05) from the unloaded condition U is indicated by an asterisk.

Fig. 4.

Steady speed values (±95% CI) of mean fore–aft force (filled bars) and braking–propulsive bias a1/b2 (open bars) for the loading conditions U, M, F and H (see text). Significant difference(P<0.05) from the unloaded condition U is indicated by an asterisk.

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 fore–hind difference in b–p bias became much less pronounced in response to hind-loading, such that hindlimb propulsive bias was only twice the forelimb braking bias(Fig. 4).

$$F_{\mathrm{y},\mathrm{fore}}$$
were negative (braking) and steady speed values of
$$F_{\mathrm{y},\mathrm{hind}}$$
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 αhind was approximately zero, indicating simultaneous initial contacts of the forelimb and hindlimb. Values of αhind 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.

Fig. 5.

(A) The kinematic mid-point between the eye and the base of the tail is represented by the vertical line posterior to the elbow. The distance p between hindlimb and forelimb centers of pressure was measured from force platform data and was partitioned according to this mid-point. (B)Distances from the mid-point to the hindlimb (open bars) and forelimb (filled bars) centers of pressure for the loading conditions U, M, F and H (see text). Error bars indicate 95% confidence intervals. Significant difference(P<0.05) from the unloaded condition U is indicated by an asterisk.

Fig. 5.

(A) The kinematic mid-point between the eye and the base of the tail is represented by the vertical line posterior to the elbow. The distance p between hindlimb and forelimb centers of pressure was measured from force platform data and was partitioned according to this mid-point. (B)Distances from the mid-point to the hindlimb (open bars) and forelimb (filled bars) centers of pressure for the loading conditions U, M, F and H (see text). Error bars indicate 95% confidence intervals. Significant difference(P<0.05) from the unloaded condition U is indicated by an asterisk.

### Effect of adding mass near the center of mass

Opposing shoulder and hip moments reveal important functional patterns when whole-body mechanics are considered. Although exerting opposing moments and fore–aft 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.

Fig. 6.

Trunk mechanics during trotting with a mid-trunk load. The load tends to ventroflex the trunk. Hindlimb retractors and forelimb protractors exert moments that tend to dorsiflex the trunk. Hypaxial muscle tension also tends to dorsiflex the trunk.

Fig. 6.

Trunk mechanics during trotting with a mid-trunk load. The load tends to ventroflex the trunk. Hindlimb retractors and forelimb protractors exert moments that tend to dorsiflex the trunk. Hypaxial muscle tension also tends to dorsiflex the trunk.

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 fore–aft force and b–p bias.

### Effect of adding mass at the pectoral girdle

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 antero–posterior 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 antero–posterior 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 trot–gallop 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

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.

The axial and appendicular musculoskeletal systems of quadrupeds are designed to accommodate specific antero–posterior 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 antero–posterior 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 antero–posterior 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, R0≈0.52; Kruger, 1943), which have more symmetric fore–hind weight distributions, are predicted to show approximately equal fore- and hindlimb b–p 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 fore–hind 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 b–p 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 b–p bias. These duty factor predictions are also consistent with the tendency of more heavily loaded limbs to be relatively longer.

Fig. 7.

(A) Fore–aft GRF of trotting dog with R0=0.65,based upon the unloaded b–p biases(a1/b2) from Fig. 4 and duty factors(DF) from Table 4. (B)A hypothetical quadruped with R0=0.50, showing equal fore-and hindlimb magnitudes of b–p bias and equal duty factors. (C) A hypothetical quadruped with R0=0.35, showing a reversal of the fore- and hindlimb patterns observed in trotting dogs.

Fig. 7.

(A) Fore–aft GRF of trotting dog with R0=0.65,based upon the unloaded b–p biases(a1/b2) from Fig. 4 and duty factors(DF) from Table 4. (B)A hypothetical quadruped with R0=0.50, showing equal fore-and hindlimb magnitudes of b–p bias and equal duty factors. (C) A hypothetical quadruped with R0=0.35, showing a reversal of the fore- and hindlimb patterns observed in trotting dogs.

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 b–p 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 b–p 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 fore–aft 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 b–p 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 antero–posterior 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 antero–posterior mass distributions might favor galloping and bounding over trotting and pacing.

Although forelimb and hindlimb mechanics should generally accommodate a particular antero–posterior 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 20–30% 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

Appendix A

• $$F_{\mathrm{z}}$$

Dimensionless mean vertical force

•
• $$F_{\mathrm{y}}$$

Dimensionless mean fore–aft force

•
• $$A_{\mathrm{y}}$$

Dimensionless mean fore–aft acceleration

•
• Θ

Angle of the resultant force vector (degrees)

•
• R

Vertical impulse ratio

•
• R0

•
• p

Distance between diagonal foot centers of pressure (m)

•
• d

Distance between fore- and hindlimb centers of pressure of adjacent steps (m)

•
• tc

Time of contact (s)

•
• tstep

Time between fore- and hindlimb vertical force peaks of adjacent steps(s)

•
• $${\nu}_{\mathrm{y},\mathrm{step}}$$

Mean forward velocity (m s–1)

•
• DF

Duty factor

•
• αhind

Hindlimb phase shift

We wish to thank dog owners M. Floyd, G. Mercer and D. Ugan. We are indebted to J. E. A. Bertram, D. M. Bramble, S. M. Deban and J. S. Markley for their helpful suggestions. Two anonymous reviewers made substantial contributions to the final manuscript. This research was supported by NSF grant IBN-0212141 and done in accordance with University of Utah IACUC Protocol Number 99-06008.

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