Effects of grade and mass distribution on the mechanics of trotting in dogs

From pronghorn antelope (Antilocapra) on North American plains to goral (Naemorhedus) on Himalayan precipices, quadrupeds occupy virtually every terrestrial habitat on Earth. Pronghorn antelope, which rely on unrivaled speed for escape, prefer flat or rolling terrain of less than 5% grade (Kindschy et al., 1978) whereas goral readily traverse steep rock cliffs and boulder fields (Lekagul and McNeely, 1977). Most quadrupeds, including gray wolves (Canis lupus), which occupy an impressive variety of habitats in the Northern Hemisphere (Nowak and Walker, 1991), are generally capable of traversing all but the most precipitous terrain. Although some excellent kinematic data are available (e.g. Carlson-Kuhta et al., 1998; Hoyt et al., 2002; van Oldruitenborgh-Ooste et al., 1997; Smith et al., 1998; Nyakatura et al., 2008; Nyakatura and Heymann, 2010), relatively few studies have reported individual limb forces of quadrupeds during downhill and uphill locomotion. These studies include data from trotting horses (Dutto et al., 2004), trotting opossums (Lammers et al., 2006; Lammers, 2007), trotting and galloping goats (Lee et al., 2008), and geckos running on vertical surfaces (Autumn et al., 1999; Autumn and Peattie, 2002).

Because most quadrupeds encounter moderate to steep terrain in their habitats, locomotor systems designed largely for economical locomotion on flat terrain must be capable of adjusting to variable grades. The net work and power requirements of ascending and descending necessitate increased actuation by muscles. It is well known that skeletal muscles can produce greater force when actively lengthening and less force when shortening compared with isometric contraction (Close, 1972; Katz, 1939). Given that braking can employ active lengthening and propulsion cannot, it follows that propulsive force should require recruitment of a greater muscle cross-section than braking force of the same magnitude. Although its effect could be partially compensated for by altering musculoskeletal transmissions, this fundamental difference in muscle contractile mechanics has unavoidable consequences. As is typical of cursorial mammals, most of a dog's limb muscle is composed of its proximal retractors and, to a lesser extent, protractors (Pasi and Carrier, 2003; Williams et al., 2009a; Williams et al., 2009b) that actuate the limbs as levers. Hence, it is predicted that the limbs will function more as levers during uphill than downhill trotting (prediction 1).

The bias of forelimbs toward braking and hindlimbs toward propulsion during level quadrupedal locomotion has been known for several decades (Cavagna, 1977; Heglund et al., 1982; Jayes and Alexander, 1978). It is predicted that this functional differentiation will be maintained or amplified during grade running. Specifically, forelimbs will brake more strongly during downhill trotting, and hindlimbs will be more strongly propulsive during uphill trotting (prediction 2). This is most clearly expressed by individual limb impulse angles, determined from the quotient of shear and normal impulse. Impulse angles reported for level trotting of dogs are –3.3 deg in the forelimb and 5.8 deg in the hindlimb (Lee et al., 2004). Published values of shear and vertical impulse in other quadrupeds suggest that hindlimb impulse angle may be more propulsive during 6 deg incline than during level trotting in horses (Dutto et al., 2004), whereas no such pattern is evident in opossums ascending a 30 deg grade, nor is the predicted braking impulse angle evident during descent (Lammers et al., 2006).

In the present study, experimental center of mass (COM) manipulations were designed to provide insight into the functional consequences of quadrupedal body form variation during grade running. The effects of experimental COM manipulations on level trotting have been described previously in dogs (Lee et al., 2004). On the basis of these results, two effects of load distribution during downhill and uphill trotting are predicted here. By shifting the COM position upslope, hind-loading during downhill trotting and fore-loading during uphill trotting should result in a distribution of normal impulse more similar to that of level trotting (prediction 3) and a more equal distribution of shear impulse between the forelimbs and hindlimbs (prediction 4).

Although typically not measured from the same stride by multiple force platforms, an expected consequence of quadrupedal grade running is that force is redistributed from the upslope to the downslope limb or limbs. This redistribution can be expressed as the fraction of normal impulse exerted by the forelimb (i.e. the forelimb component of normal impulse divided by the sum of forelimb and hindlimb normal impulse). In agreement with this prediction, horses trotting up a 6 deg incline show a decreased forelimb fraction of mean vertical force of 0.52 compared with 0.57 on the level (Dutto et al., 2004). Likewise, in opossums running up and down a 30 deg grade, the forelimb fraction of mean vertical force decreased to 0.51 uphill and increased to 0.82 downhill compared with 0.67 on the level (Lammers et al., 2006). These investigators compared vertical rather than normal force distributions between level and grade locomotion. Yet, on a grade, vertical force includes a shear force component in the direction of travel, so vertical impulse ratios represent something mechanically distinct in grade versus level locomotion. The disparity between vertical and normal impulse distributions might reasonably be ignored in the case of slight grades, where the sine of the grade angle is approximately equal to the grade angle, but should generally be considered in comparisons of grade and level locomotion. Mean normal force distributions have been reported in goats running up and down a 15 deg grade, where the forelimb fraction of mean normal force decreased to 0.53 uphill and increased to 0.72 downhill compared with 0.65 on the level (Lee et al., 2008). The consensus of previous quadrupedal grade running studies is a substantial decrease in the fraction of forelimb support during uphill running and an increase during downhill running.

Although running up or down a grade at a steady speed requires no more force than level running (i.e. a vertical resultant force equal to body weight), a component of this force (i.e. shear force) acts in the direction of travel. During steady-speed, level trotting (Fig. 1A), the resultant ground reaction force is normal to the substrate and is distributed between the forelimbs and hindlimbs according to the craniocaudal position of the feet with respect to the COM. This determines the common center of pressure (COP) between the limbs. During steady-speed downhill or uphill running, the resultant ground reaction force has both a normal and shear component; however, the normal force distribution still defines the COP, which lies directly beneath the COM, on average, during steady-speed locomotion (Fig. 1B,C).

Shear force is required to maintain forward speed against the acceleration of gravity and is proportional to the sine of substrate angle in any steady-speed stride, whether downhill or uphill. During grade running, shear force may be provided by the angle of the limbs with respect to the substrate (Fig. 1B), such that the limbs only need to exert axial force (i.e. in line with the hip or shoulder). Hence, the limbs act purely as struts (Gray, 1968) and their normal force distribution and COP are unchanged from the level. Alternatively, shear force may be provided by hip and/or shoulder torques that tend to protract or retract the limbs (Fig. 1C). This is referred to as action of the limbs as levers (Gray, 1944; Gray, 1968) and results in an altered distribution of normal force between the limbs [i.e. a redistribution to the down-slope limb(s)]. A further consequence of lever function is that shear impulse is distributed to the forelimbs and hindlimbs according to hip and shoulder torques. Relative hip and shoulder torques drawn in Fig. 1C are not an a priori prediction of this model; that is, any torque distribution providing the requisite net torque is possible, as long as it does not exceed the allowable shear force between the foot and ground.

Previous work shows that telescoping strut versus lever mechanics largely explain the different strategies used by dogs and turkeys during net accelerations on the level. Dogs tend to use their limbs as levers and redistribute normal force between the forelimbs and hindlimbs during moderate accelerations (Lee et al., 1999) whereas turkeys alter their limb angles as telescoping struts during extreme accelerations (Roberts and Scales, 2002). Unlike bipeds, which are generally constrained to use single limb support during running, quadrupeds are free to use lever mechanics without exerting a net pitching moment about the COM – provided that the CoP is kept between the supporting forefeet and hindfeet. These mechanisms are not mutually exclusive, so telescoping strut and lever mechanisms can be combined in the function of individual limbs. Furthermore, multi-legged runners can allocate telescoping strut and lever functions differentially to individual limbs during a stride. This, combined with the mechanism of resisting pitch moments, yields a greater scope of functional differentiation in the forelimbs and hindlimbs of quadrupeds compared with the limbs of bipedal striders or hoppers.

Three adult dogs (a pointer, a Weimaraner and a Labrador retriever) were used in this experiment. Subjects ranged in body mass from 27 to 34 kg and from 3 to 10 years of age. 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 up and down a 15 deg grade, hard-packed soil runway in which two force platforms were positioned in series. Both downhill and uphill trotting were recorded. Force data were collected for 2 s as the dogs crossed the platforms. A video camera with VCR (PEAK™ Performance Technologies Inc.) acquired 120 images s^–1 in lateral view. Only data from uninterrupted trotting were saved. Data were collected while the dogs carried no load (unloaded, U) and under three loading conditions, in which tandem saddle bag packs were worn. Two small bags of lead shot totaling 10% body mass were inserted bilaterally in the anterior (fore-loaded, F), middle (mid-loaded, M) or posterior compartments (hind-loaded, H).

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™ data acquisition system (DAQCard AI-16-E4, SCXI 1000 chassis, SCXI 1121 strain/bridge modules and SCXI 1321 terminal blocks). 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. Each force platform was 0.6 m long by 0.4 m wide. Using platforms of this length increased the likelihood that diagonal forefeet 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 normal ground reaction force (GRF) and shear GRF along the line of travel with separate double-cantilever transducers at each corner post. Normal GRF acting upward and shear GRF acting in the direction of travel were considered positive.

During level locomotion, R_n is equivalent to the vertical impulse ratio (Jayes and Alexander, 1978) in that it defines the relative position of the common COP between the feet. Nonetheless, level locomotion is a special case whereas R_n defines the COP on any non-level substrate plane.

The COP on each force platform was determined by comparing the normal force from independent transducer elements at the fore and aft of the force platform. The two platforms were calibrated on a continuous scale to facilitate the computation of distance between footfalls on separate platforms (Bertram et al., 1997). The COP for each foot was determined by force-averaging over the duration of foot contact. In other words, instantaneous COPs were weighted according to the instantaneous normal force values, summed and then divided by the summation of normal force over t_c. This avoided the confounding effect of an extreme COP measurement during toe-off, for example, when the normal force is quite small. During paired diagonal contacts, the forefeet and hindfeet struck separate platforms allowing calculation of the distance between their mean COPs along the line of travel (p).

A general linear model (GLM) with three categorical effects [slope (downhill or uphill), loading condition (U, M, F or H) and subject (1, 2 or 3)] and two continuous effects (impulse angle θ and mean velocity ) was used to evaluate locomotor parameters in JMP 8.0™ software. Considering the effect of subject in the GLM permitted multiple runs to be considered from each subject, while assessing the significance of inter-subject differences. For certain parameters, one or more non-significant (P>0.05) effect was dropped from the model, as noted in the Results. GLM predictions with 95% confidence intervals were determined at θ≈0 (i.e. steady speed) and , except where one of these effects was non-significant. All of the parameters presented in the Results were assessed by the GLM, except Fourier coefficients and DF (supplementary material Tables S1 and S2), which were predicted by multiple regression at steady-speed velocities of 2.59 m s^–1 (downhill) and 2.18 m s^–1 (uphill).

Steady-speed Fourier coefficients of normal and shear force–time curves were predicted at 2.59 m s^–1 (downhill) and 2.18 m s^–1 (uphill) (supplementary material Table S1). These coefficients were used to reconstruct steady-speed normal and shear force–time curves for downhill and uphill trotting under each loading condition (Fig. 3). As expected, normal GRF was always positive (i.e. pushing against the foot) during downhill and uphill trotting. With the exception of very slight hindlimb propulsion late in the step, shear GRF was entirely braking during downhill trotting and entirely propulsive during uphill trotting. Hindlimb phase (Fig. 4B), duty factors (supplementary material Table S2) and step periods (Table 2) were also used in these force–time reconstructions.

Before proceeding with a description of step parameters, it is important to note that in the direction of travel was significantly greater in downhill than uphill trotting (Table 1). Hence, an intermediate velocity of 2.38 m s^–1 was chosen to make both downhill and uphill predictions in the general linear model (GLM). Step length and step period were thus predicted at the same velocity for downhill and uphill trotting (Table 2), revealing significant differences (d, P=0.0021; t_step, P<0.0001) between downhill and uphill locomotion. Because velocity is the quotient of d and t_step, the same velocity (i.e. 2.38 m s^–1) is achieved by long, slow steps during uphill trotting as by shorter, quicker steps during downhill trotting. Effects of load and θ were non-significant and were, therefore, dropped from the GLM for these two parameters.

Values of p can be expressed in dimensionless terms as a fraction of d. The ratio p/d, predicted by the GLM, was significantly greater during uphill trotting (P<0.0001; Fig. 4A). Given that d was also greater during uphill trotting (Table 2), this indicates a greater absolute and relative fore–aft distance between paired diagonal footfalls. Although individual limb angles were not measured here, this difference is likely due to a more retracted hindlimb angle during uphill trotting. The effect of θ was non-significant and was dropped from the GLM for this parameter.

Hindlimb phase (α_hind) quantifies the time of initial hindfoot contact with respect to initial forefoot contact. Downhill α_hind was significantly greater (i.e. later contact of the hindfoot; P<0.0001) than uphill α_hind. Fore-loading elevated α_hind significantly above unloaded values (Fig. 4B).

The ratio of forelimb to hindlimb contact time (t_c,fore/t_c,hind) was significantly greater (P<0.0001) during downhill than uphill trotting (Fig. 5A). The effect of subject was non-significant and was dropped from the GLM for this parameter. During grade trotting, fore-loading significantly increased t_c,fore/t_c,hind whereas mid- and hind-loading significantly decreased t_c,fore/t_c,hind with respect to the unloaded condition (Fig. 5A).

Mean normal force (F_n) for the contact time of paired diagonal footfalls (t_c,total) ranged from 0.9 to 1.05 body weights with no significant difference between downhill and uphill trotting (Fig. 4C).

Steady-speed normal impulse ratio (R_n,0) indicates the fraction of normal impulse on the forelimb during steady-speed trotting (i.e. θ=0). The effect of velocity was non-significant and was dropped from the GLM for this parameter. R_n,0 was significantly reduced during uphill compared with downhill trotting (P<0.0001; Fig. 5B). Without lever function (Fig. 1A,B), R_n,0 would remain similar to level trotting values. On average, downhill values were 0.06 greater whereas uphill values were 0.11 less than level. Hence, the fraction of normal impulse on the forelimb was increased with respect to level during downhill trotting, but not to the extent that it was decreased during uphill trotting. As discussed later, this supports prediction 1, that the limbs will tend to act more as levers during uphill than during downhill trotting.

In all three loading conditions, R_n,0 was significantly different from unloaded (Fig. 5B). In agreement with prediction 3, loading the upslope limb resulted in normal impulse distributions more similar to level. During downhill trotting, hind-loading decreased R_n,0 to within 0.025 of the unloaded level value (grey horizontal line in Fig. 5B) and, during uphill trotting, fore-loading increased R_n,0 to within 0.07 of the unloaded level value.

Steady-speed shear impulse ratio (R_s,0) indicates the fraction of shear impulse at steady-speed. Presented in bar plots and also graphically by impulse vectors in Fig. 5C, R_s,0 was significantly greater during downhill than during uphill trotting (P<0.0001). Patterns of R_s,0 across loading conditions were similar to those of R_s,0, with the exception that mid-loaded condition was not significantly different from the unloaded condition. As discussed later, loading the upslope limb resulted in a more equal fore–hind distribution of shear impulse (prediction 4). Specifically, hind-loading during downhill trotting reduced R_s,0 whereas fore-loading during uphill trotting increased R_s,0 (Fig. 5C).

Impulse angles of the forelimb (θ_fore) and hindlimb (θ_hind) are dimensionless expressions of a limb's braking or propulsive function. Extreme and highly significant (P<0.0001) differences between downhill and uphill trotting were evident in both θ_fore (Fig. 6A) and θ_hind (Fig. 6B). In addition to bar plots, angles are illustrated for subject 1 and the level, unloaded angle is drawn from Lee et al. (Lee et al., 2004) (Fig. 6). The braking angle of forelimbs is greatly amplified during downhill (–17.8 deg) compared with level trotting (–3.3 deg). Conversely, the propulsive angle of hindlimbs is greatly amplified during uphill (19.4 deg) compared with level trotting (5.8 deg). Both of these results are consistent with prediction 2 – that the fore–hind functional differentiation of braking–propulsive function will be maintained or amplified during grade running – but this is only relevant in the downslope limbs because the usual braking or propulsive functions of the upslope limbs are reversed during grade running.

Compared with the unloaded condition, fore-loading significantly reduced forelimb braking angle during downhill trotting and significantly increased forelimb propulsion angle during uphill trotting (Fig. 6A). A corresponding effect was seen in the hindlimb: hind-loading significantly increased hindlimb braking angle during downhill trotting and significantly decreased hindlimb propulsion angle during uphill trotting (Fig. 6B). No other loading effects on θ_fore or θ_hind were seen in the forelimb or hindlimb.

A prominent characteristic of uphill trotting is a significant increase in t_step with respect to downhill trotting (Table 2). Step length (d) also increased significantly, indicating longer, slower steps during uphill trotting at a given velocity. Because was statistically similar between downhill and uphill trotting, the present data indicate that the diagonal t_c,total remained a constant proportion of t_step at the steady-speed velocity of 2.38 m s^–1. Increasing t_step with respect to level is a common response to uphill trotting in quadrupeds, although stride period (2t_step) is normally reported. Horses trotting on treadmills inclined by 3.4 or 5.7 deg showed increases in t_step with respect to level (Hoyt et al., 2000; van Oldruitenborgh-Ooste et al., 1997). Stride period has also been found to increase during uphill locomotion in other quadrupeds at various gaits (Gillis and Biewener, 2002; Pierotti et al., 1989; Roy et al., 1991). If duty factors are consistent, greater step periods provide a clear advantage of increased contact time in which to exert force. This is not always the case, however, as no consistent change in stride period or hindlimb contact time was observed in squirrel monkeys walking or running up a treadmill inclined by 8, 16 or 28 deg (Vilensky et al., 1994). Also, small lizards walking and running up a treadmill inclined by 30 deg showed decreased stride periods and contact times (Jayne and Irschick, 1999).

In stark contrast to most quadrupeds, humans decrease their stride periods during uphill running. For example, stride period decreased when running up treadmills inclined by 16.7 deg (Swanson and Caldwell, 2000) and 2.9–8.5 deg (Minetti et al., 1994). These results are surprising, given that shorter stride periods require higher rates of force generation, provided that the duty factor is unchanged. This, in turn, might require greater muscle recruitment because of the force–velocity relationship and/or the recruitment of faster muscle fibers, both of which increase metabolic cost (Close, 1972). In guinea fowl on a treadmill inclined by 16 deg, stride period decreased, as in humans, but contact time actually increased due to greater duty factors during uphill than during level running (M. A. Daley, personal communication) – a result consistent with an increased time for muscle force generation. The question of why humans, and perhaps bipeds in general, employ shorter stride periods on an incline remains open.

A notable feature of downhill trotting is delayed hindlimb contact, as indicated by a greater hindlimb phase (α_hind) than during level trotting (Fig. 4C). The common COP depends upon the position of the feet with respect to the COM and the distribution of normal force between the forelimb and the hindlimb (Fig. 1). During downhill trotting, the common COP is the same as the forelimb COP until the hindlimb is set down. By delaying hindlimb contact, the COP remains farther forward and the resultant force (which is applied at the COP) can have a greater braking component without exerting a nose-down pitching moment about the COM. Hence, later hindlimb contact permits greater braking force during the first 20% of forelimb contact when normal force is relatively low (Fig. 3). Applying the same initial braking force with the hindlimb and a sufficient delay of forelimb contact could send the dog somersaulting nose down. Nose-down pitching is exacerbated during fore-loading by moving the COM more cranially but a significant increase in α_hind counteracts this effect. As expected, the risk of nose-up pitching during uphill trotting is substantially less, given the cranial COM position of most quadrupeds.

Grade trotting is also characterized by an increase in the t_c,fore/t_c,hind ratio during downhill trotting and a decrease in t_c,fore/t_c,hind during uphill trotting compared with level trotting (Fig. 5A) (Lee et al., 2004). A decrease in hindlimb duty factor with respect to level has also been shown in cats descending 26.6, 36.9 and 45.0 deg grades; however, no significant effect was found at 14 deg (Smith et al., 1998). In another study, decreases in stride period and hindlimb contact time, but not forelimb contact time, suggest that t_c,fore/t_c,hind might increase in squirrel monkeys descending a range of grades from 4.6 to 15.6 deg (Vilensky et al., 1994). The present data suggest that an increase in t_c,fore/t_c,hind during downhill trotting and a decrease during uphill trotting are related to redistribution of normal and shear impulse to the downslope limb (Fig. 5).

Lever and telescoping strut actions of the limbs were detected by measuring the steady-speed normal impulse ratio R_n,0. If the legs acted purely as telescoping struts and the degree of protraction or retraction of the forelimbs and hindlimbs were similar, R_n,0 would be unchanged from that of level trotting (Fig. 1B) and, as lever action became more prominent, R_n,0 would increasingly diverge from that of level trotting (Fig. 1C). Using the previously reported relationship between R_n and mean fore–aft acceleration for dogs (r²=–0.71) (Lee et al., 1999), R_n,0 is predicted to increase by 0.18 during 15 deg downhill trotting and decrease by 0.18 during 15 deg uphill trotting if the limbs are assumed to function strictly as levers to achieve braking or propulsion. The present data show that, with respect to level trotting, R_n,0 increases by 0.06 during downhill trotting and R_n,0 decreases by 0.11 during uphill trotting (Fig. 5B). Hence, the legs did not act purely as telescoping struts to maintain steady-speed downhill or uphill trotting. Nonetheless, the action of the legs as levers (indicated by the divergence of R_n,0 from level) contributed only approximately half as much to the braking force during downhill trotting as to propulsive force during uphill trotting. In other words, more shear force was supplied by thrust in line with the legs (i.e. telescoping strut action) during downhill trotting whereas relatively more shear force was supplied by proximal joint torques (i.e. lever action) during uphill trotting. This result is in good agreement with prediction 1, which indicates greater lever function during uphill trotting. As explained in the Introduction, this prediction is based upon the fact that the majority of limb muscle mass in the proximal retractors, combined with the dogs' inability to exploit high-force lengthening (i.e. energy absorbing) contractions during uphill running.

Bolstering this explanation are a number of recent in vivo extrinsic muscle studies comparing level, downhill and uphill locomotion. In a report of hindlimb extrinsic muscle function of dogs (Schilling et al., 2009), hindlimb retractor recruitment increased dramatically (2 to 22 times level values) during uphill trotting at 10 and 14 deg. This significant increase in the mean rectified electromyogram is due to both increased magnitude and duration of muscle recruitment. In a study by Gillis and Biewener (Gillis and Biewener, 2002), greater stance-phase recruitment and shortening of the biceps femoris muscle were reported during uphill walking and running of rats on a treadmill inclined by 15 deg. These results provide evidence of greater hindlimb lever action during uphill running because the hip is extended through a greater angle by the shortening biceps, a major hindlimb retractor. In vivo data from forelimbs of dogs provide evidence of a less-pronounced lever function: retractor recruitment increases moderately during uphill (10 and 14 deg) trotting and protractor recruitment increases moderately during downhill (10 and 14 deg) trotting (Carrier et al., 2006). Although the present analysis shows greater net lever function uphill than downhill, an analysis of hip and shoulder joint dynamics would be required to ascribe more or less of this function to the forelimb or the hindlimb. Furthermore, kinematic data showing individual limb angles are needed to determine more precisely the relative contributions of telescoping strut and lever action to downhill braking and uphill propulsion. The distance between diagonal footfalls represented by p/d (Fig. 4A) shows an ∼5% difference between downhill and uphill running, which could be explained by divergence of individual limb angles.

As expected from loading responses during level trotting (Lee et al., 2004), hind-loading during downhill trotting and fore-loading during uphill trotting yielded values of R_n,0 closer to those of steady-speed level trotting (prediction 3). Loading of the upslope limbs also yielded values of R_s,0 closer to 0.5, indicating a more symmetrical fore–hind distribution of shear impulse (prediction 4). It is reasonable, therefore, to suggest that quadrupeds with COM positions closer to the forelimbs would distribute propulsive force more equally between the forelimb and the hindlimb during uphill trotting. Conversely, quadrupeds with COM positions closer to the hindlimbs would distribute braking force more equally between the forelimb and the hindlimb during downhill trotting. This raises a basic question of quadrupedal design: what COM position allows adequate performance for both downhill and uphill locomotion? Here again, the explanation may be that muscle force during active lengthening (useful in braking but not propulsion) is much greater than that during active shortening (Close, 1972; Katz, 1939). Because less muscle is needed to supply a braking force of a given magnitude, it is possible for the forelimb to provide a larger fraction of the required braking impulse during downhill trotting than the fraction of propulsive impulse the hindlimb can provide during uphill trotting (∼0.84 versus ∼0.60; Fig. 5C). Given that the COM is typically closer to the forelimb (Rollinson and Martin, 1981), terrestrial quadrupeds likely benefit from a more equal distribution of propulsive impulse between forelimbs and hindlimbs during uphill running, whereas the forelimbs easily provide most of the braking impulse during downhill running.

The tendency of forelimbs to supply more braking, and hindlimbs more propulsion, is typical of running quadrupeds (Cavagna, 1977; Heglund et al., 1982; Jayes and Alexander, 1978; Lee et al., 1999; Lee et al., 2004) and hexapods (Full et al., 1991). For comparisons across size, species and even between different locomotor behaviors within subjects, braking and propulsive functions are best described as dimensionless impulse angles. Here, the forelimb impulse angle (θ_fore) was predicted to be more strongly braking during downhill trotting, and the hindlimb impulse angle (θ_hind) more strongly propulsive during uphill trotting (prediction 2). Compared with level impulse angles (Lee et al., 2004), θ_fore increased ∼ deg in magnitude to –17.8 deg (Fig. 6A) and θ_hind increased ∼14 deg in magnitude to 19.4 deg (Fig. 6B). Hence, the affinity of the forelimb to braking and the hindlimb to propulsion are not only maintained but also amplified during downhill and uphill trotting. Although the ultimate cause of these fore–hind functional differences may be the physical relationship between shear and normal force (e.g. front brakes are more effective than rear brakes and rear wheel propulsion is more effective than front wheel propulsion) (Gray, 1944; Gray, 1968; Lee et al., 1999), the proximate cause is likely the resulting specialization of forelimb and, to a greater extent, hindlimb structure. Evidence of such anatomical specialization is provided by the substantial fraction of muscle mass represented by hindlimb (propulsive) retractors (Pasi and Carrier, 2003; Payne et al., 2005a; Williams et al., 2007a) and, to a lesser extent, forelimb (braking) protractors (Payne et al., 2005b; Williams et al., 2007b). In addition to anatomical observations, the propulsive bias of hindlimbs and braking bias of forelimbs is reproduced by simple trotting simulations with knee-forward/elbow-back geometry (Lee and Meek, 2005). Together, pitch mechanics, limb geometry and limb anatomy explain the differential function of the forelimbs and hindlimbs of quadrupeds.

Principles of telescoping strut and lever function are applicable to both grade and level running, with the caveat that net lever function (Fig. 1C) can produce steady-speed grade running but would indicate acceleration on the level. Several recent studies have investigated net propulsive or braking accelerations of quadrupeds on the level. Walter and Carrier reported forelimb and hindlimb kinetics and kinematics during extreme propulsive accelerations of dogs, finding the fraction of forelimb shear impulse to be 0.43 (Walter and Carrier, 2009) – nearly equal to those reported here for steady-speed uphill trotting (Fig. 5C). The fraction of forelimb normal impulse in the present study, however, is considerably lower (0.43) compared with steady-speed uphill values between 0.5 and 0.6 (Fig. 5B). This discrepancy is likely associated with nose-up pitching during the accelerating strides (Walter and Carrier, 2009) due to greater propulsive forces and the use of a half-bound gait, which decouples forelimb and hindlimb footfalls. The authors conclude that lever function due to hip and shoulder retraction plays a primary role in propulsive acceleration and that telescoping strut function, indicated by a more retracted angle of forelimbs and hindlimbs, plays a secondary roll (Walter and Carrier, 2009). Another recent study of acceleration in dogs focused on hindlimb mechanics (Williams et al., 2009a). These authors show that hindlimb propulsion is primarily due to hip extension (suggesting lever function) and ankle extension (suggesting telescoping strut function). Assuming only lever function, the same authors applied Gray's (Gray, 1944; Gray, 1968) model to help explain maximum propulsion and braking in greyhounds and polo ponies (Williams et al., 2009b). The data from the present study indicate a combination of lever (Fig. 1B) and telescoping strut (Fig. 1C) function to achieve steady-speed downhill and uphill trotting, with significantly greater lever function uphill. This also seems to be the consensus for quadrupedal propulsion on the level, although level braking mechanics are less studied.

As evidenced by steady-speed normal impulse ratios, the limbs act collectively more like levers during uphill than downhill trotting (prediction 1). Hence, uphill trotting requires greater proximal limb torques, recruiting the large retractor muscles of the hip and shoulder. These results are consistent with a well-known contractile property of skeletal muscle – a greater cross-section needs to be recruited to exert a given force during active shortening (uphill propulsion) than during active lengthening (downhill braking). As shown by individual limb impulse angles, differential fore–hind function during level trotting is amplified in forelimb braking during downhill trotting and in hindlimb propulsion during uphill trotting (prediction 2). During both dowhnhill and uphill trotting, the normal impulse distribution between the forelimbs and hindlimbs is closer to level values when the upslope limb is loaded (prediction 3). The forelimb exerts ∼84% of the total braking impulse during downhill trotting whereas the hindlimb exerts only ∼59% of the total propulsive impulse during uphill trotting. During both downhill and uphill trotting, loading of the upslope limb resulted in a more equal distribution of shear impulse between the forelimb and the hindlimb (prediction 4). This last result suggests a functional trade-off in quadruped design: a COM closer to the hindlimbs would distribute downhill braking force more equally whereas a COM closer to the forelimbs would distribute uphill propulsive force more equally. Considering the reduced force capacity of skeletal muscle during shortening versus lengthening contractions, the anterior COM position of most terrestrial quadrupeds provides a functional advantage during uphill running without substantial detriment to downhill running. The advantage of this mechanical design seems to extend level propulsion (Walter and Carrier, 2009; Williams et al., 2009b) and jumping (Alexander, 1974).

I thank dog owners E. Bo, D. R. Carrier and L. Vernon, as well as E. F. Stakebake, J. S. Markley, R. M. Walter and S. M. Deban for invaluable help with data collection. I am indebted to D. R. Carrier for discussions that inspired these experiments and D. M. Bramble for discussions of data analysis and interpretation.

 
• COM

  center of mass

 
• COP

  center of pressure

 
• d

  distance between forelimb and hindlimb COPs of adjacent steps (m)

 
• DF

  duty factor

 
• 

  mean normal force (body weights, BW)

 
• g

  gravitational acceleration

 
• j_n

  normal impulse (BW s^–1)

 
• j_s

  shear impulse (BW s^–1)

 
• m

  body mass

 
• p

  distance between diagonal foot COPs (m)

 
• R_n

  normal impulse ratio

 
• R_s

  shear impulse ratio

 
• t_c

  time of contact (s)

 
• t_step

  time between forelimb and hindlimb normal force peaks of adjacent steps (s) (step period)

 
• 

  mean velocity in the direction of travel (i.e. shear) (m s^–1)

 
• α_hind

  hindlimb phase-shift with respect to forelimb initial contact

 
• θ

  impulse angle with respect to normal (deg)