Exploring the mechanical basis for acceleration: pelvic limb locomotor function during accelerations in racing greyhounds (Canis familiaris)

In a constantly varying environment, animals are confronted with the need to avoid obstacles, negotiate uneven terrain and rapidly change their speed and direction (for example during a predator–prey chase). Few studies have assessed the mechanics involved with such `non-steady state' locomotor behaviours in galloping quadrupeds. Whilst acceleration has been assessed in trotting dogs (Lee et al.,1999), running turkeys(Roberts and Scales, 2002; Roberts and Scales, 2004),wallabies (McGowan et al.,2005a), humans (Cavagna et al., 1971; Harland and Steele,1997; Johnson et al., 2001) and kangaroo rats(Biewener and Blickhan, 1988),to date understanding of acceleration in quadrupeds is minimal. It is difficult to consider factors such as muscle work and power output under steady-state conditions, as no net mechanical work is performed during running at steady speed on level ground. Some positive muscle work is performed, as is negative work during each step, but the net work output on the level is zero. In addition, passive elastic mechanisms are also present, which can store and recover mechanical work with each step, reducing the amount of mechanical work that is performed by muscular contractions [but to an uncertain extent(Alexander, 1991; Cavagna et al., 1977)]. During acceleration, however, work must be performed to increase the kinetic energy of the body. Passive elastic mechanisms cannot perform net work though they can modulate the time course of power output of a muscle(Lichtwark and Wilson, 2005; Lichtwark and Wilson, 2006),and thus work must be performed by the contraction of skeletal muscle.

In the racing greyhound, much of the locomotor musculature is located as hip extensors within the proximal pelvic limb(Williams et al., 2008). These muscles have long parallel fibres, an architecture which has been suggested as suitable for performing large amounts of muscle work; these muscles may therefore be able to produce substantial amounts of power. Those more distally located and many in the thoracic limb appear more adapted towards force production and elastic energy storage and return (due to shorter, pennate fascicles, which are in some cases associated with substantial tendons). Tendons in this portion of the limb may also play a role in amplifying the power produced by these distal muscles. Hence, a regional specialisation exists in terms of the locomotor musculature of the racing greyhound, as is seen in many other species (Lieber and Blevins, 1989; Payne et al.,2006b; Payne et al.,2005a; Payne et al.,2005b; Smith et al.,2006; Thorpe et al.,1999; Williams et al.,2007a; Williams et al.,2007b), suggesting that the functional roles of different limbs and joints within a limb may differ. Muscle architecture is not the only determinant of a muscle's ability to produce mechanical work at a joint:musculo-skeletal geometry, i.e. the moment arm of a muscle, is also important. Muscle moment arms have been shown to vary with joint angle [and therefore with limb posture (e.g. Payne et al.,2006a; Williams et al.,2008)]. It is likely therefore that the roles of muscles and joints may also change during different locomotor tasks.

That the hip extensor muscles in the racing greyhound appear to have so much potential for power production(Williams et al., 2008)suggests that much of the production of the mechanical work which is needed for acceleration of the centre of mass may occur via the pelvic limb,specifically at the hip joint, via hip extension. This may then be transferred by biarticular muscles to other joints. The presence of substantial distal limb tendons suggests that though the distal limb muscles possess less potential for producing work, their muscle power output could be amplified by tendons. The centre of mass acceleration of the animal may therefore also be affected by distal limb mechanics. The relative contribution of different hindlimb joints to acceleration thus requires investigation.

Much attention has been given to the mechanical significance of size-related changes in limb posture in relation to muscle and bone stress. A more upright posture aligns limb segments and joints more closely with the resultant ground reaction force vector (GRF_r) during stance. This reduces bending stresses to which the bones in the limb are subjected(Biewener, 1983) and also decreases the moments exerted about the limb joints(Biewener, 1989). This means that animals with a more upright posture are able to support their body mass with lower muscle forces. The `effective mechanical advantage' (EMA) of an animal's limb can be used as an index of limb posture, and here is defined as the ratio of the muscle moment arm of the extensor muscles (r) of a joint to the moment arm of the GRF vector (R) acting about that joint(see Fig. 1A). A scaling related change in limb EMA with body mass has been demonstrated, with larger animals having more extended limbs and hence higher limb EMAs(Biewener, 1989). This acts as a mechanism by which peak bone and muscle stresses are reduced in larger species, and thus similar magnitudes for these parameters are seen across the size range. This effect of changing EMA has mainly been considered in a quasi-static context. Far less is known about how limb/muscle EMA may change under different locomotor conditions – i.e. at different gaits or speeds. A large decrease in knee extensor muscle EMA occurs in humans across the gait transition from a walk to a run(Biewener et al., 2004). However, to date no studies have reported a change in EMA with speed,acceleration or other locomotor parameters in quadrupeds.

A muscle's moment arm (and thus leverage) can alter depending on joint angle (Thorpe et al., 1999; Payne et al., 2006b; Smith et al., 2006; Williams et al., 2007a; Williams et al., 2007b). Therefore, during the stride cycle the leverage of muscles must vary as joint angles are constantly changing. The GRF orientation relative to the limb and hence the GRF moment arm with respect to each joint centre of rotation also changes throughout a stance period, and so there is scope for the `gear ratio'or EMA to alter within and between stance phases during locomotion. This has been shown to occur in humans; trotting, galloping and jumping dogs; and accelerating turkeys (Carrier et al.,1998; Carrier et al.,1994; Gregersen et al.,1998; Gregersen and Carrier,2004; Roberts and Scales,2004). The changes in gearing that occur in dogs have been proposed as a mechanism by which performance can be enhanced, because, if muscles can be arranged to shorten at an appropriate velocity, power production can be maximised (Gregersen and Carrier, 2004). The gear ratio of a muscle that shortens at a constant rate would be predicted to increase with animal speed if the muscle is to make the maximum contribution to acceleration of the animal. This mechanism appears to occur at the knee joint in jumping and running dogs(Gregersen et al., 1998; Gregersen and Carrier,2004).

Here our aim was to determine patterns of pelvic limb kinematics, GRFs and impulses during varying sub-maximal accelerations. We assessed the production and modulation of mechanical work and power within the limb and considered the contributions of muscles around each hindlimb joint to the variations in work associated with acceleration. The EMA of the limb was also considered with respect to speed and acceleration to investigate the effect of limb posture on muscle power output and acceleration.

Six racing greyhounds (mass 28.5±5.0 kg, mean ± s.d.) were encouraged to accelerate across a purpose-built 20 m long and 0.6 m wide runway consisting of an upward sloping ramp (2 m long, 0.10 m rise) to 2 m of level runway, followed by five force platforms (AMTI, Watertown, MA, USA) in series. This was followed by another 2 m length of flat runway to a downward sloping ramp (2 m long, 0.10 m drop). The runway and individual force platforms were covered in thin-pile carpet to minimise slip. A six-camera 3-D motion analysis system (ProReflex, Qualisys, Gothenburg, Sweden) was positioned on both sides of the runway, and collected motion data at 240 Hz. Simultaneous force data in vertical (z), fore–aft (y)and medio-lateral (x) directions, were continuously logged in custom-made LabView software (500 Hz). Force and motion data were synchronised using a trigger, which recorded a pulse upon the start of motion analysis capture as a separate input channel of the force plate record. Retroreflective markers were placed over various anatomical landmarks representative of joint centres of rotation on both sides of the animal. These included the iliac crest, hip, stifle, hock, metatarsophalangeal (MTP) joints and the tip of the longest phalange (Fig. 2). Speed and acceleration were varied by increasing and decreasing the start distance of the dog from the first force plate, as necessary. Dogs were motivated to run by being called by a handler, by chasing a ball or toy, or in some cases with the use of a mechanical lure. Though GRFs were collected for all four feet (to allow calculation of whole-body centre of mass velocities and accelerations), only foot strikes that filled specific criteria were used for individual analysis. These conditions were: (1) only a clear single limb was in contact with the force plate (often, especially at low velocities, no aerial phase was present between forelimb and hindlimb contact resulting in two feet weight bearing on one force plate); (2) foot strikes had to be away from the edges of the plate; and (3) kinematics had to be of sufficient quality (i.e. no marker drop-outs).

Vertical, horizontal and medio-lateral GRFs were recorded for all four limbs of the animal. Medio-lateral forces were small and so were ignored for the purposes of this study. GRFs were used to calculate the magnitude and direction (angle from the vertical) of the resultant GRF vector(GRF_r). GRFs, joint angles and other continuous variables were normalised to per cent of stance (calculated from GRF traces) and averaged within groups. Stance time was greater in high acceleration trials(P<0.0001; Fig. 3)and, thus, showing GRFs normalised to stance time is not truly representative. Vertical and horizontal force impulses were therefore calculated (as the integrals of vertical and horizontal forces, respectively, over each actual stance time). These provide a clearer means by which to view differences in GRF data between conditions.

x- and y-coordinates of the GRF centre of pressure (CoP)were calculated. CoP_x was calculated as the negative of the force plate moment about the y-axis, divided by the vertical GRF. CoP_y was calculated as the plate moment about the x-axis divided by the vertical GRF. All CoP coordinates were checked manually against aligned kinematic data to ensure accurate values were used for further calculations (as joint moment calculations are sensitive to errors in the CoP value).

In this study, positive values for M represent an extensor muscle moment, and negative values a flexor muscle moment (i.e. positive values represent a moment that is being balanced by extensor musculature). Instantaneous power at each joint was calculated by multiplying the joint moment by the angular velocity at that joint. Total (single) limb power was then derived by summing the instantaneous powers from each joint. The net work delivered by each joint during stance was determined by integrating the power curve for that joint over the stance time. Absolute positive and negative work were calculated as the sum of the positive and negative integrated portions of the power curve, respectively. Total limb work is the sum of work performed at all of the joints in the one limb.

Velocity and acceleration in our data set were found to be related, given the absence of high accelerations at high initial velocities(P=0.002). Hence general linear model (GLM) one-way analyses of covariance (ANCOVA) were used to ascertain relationships between selected variables and both acceleration and speed, with speed and acceleration as covariates, and subject identity and lead/trailing limb as random factors. This accounted for any potentially confounding influence of speed on the dataset. Linear or multiple regression analysis (as appropriate) followed to determine the degree of the variance between dependent variables and acceleration (and speed if necessary). Prior to statistical analysis, data were checked to ensure that they did not violate the assumptions of normality and equality of variance of the statistical tests. Data were found not to violate these assumptions.

Forty strides were analysed, ranging from a minimum acceleration (per stride) of 0.3 m s^–2 to a maximum acceleration of 4.5 m s^–2. Accelerations were well spread across subjects, with all individuals performing low, medium and high accelerations (subject identity was not a significant influence on any of the variables analysed in the GLM). For the purposes of analysis, three groups were chosen to represent different levels of acceleration. These were termed `low', `medium' and `high' groups. The `low' acceleration category contained accelerations ranging from 0.5 to 1.3 m s^–2, with a mean acceleration of 0.9 m s^–2 (N=13). `Medium' accelerations ranged from 1.4 to 2.5 m s<sup>–2</sup> with a mean acceleration of 2.0 m s<sup>–2</sup> (N=15). `High' accelerations ranged from 2.51 to 4.5 m s^–2 (mean 3.2 m s^–2; N=12).

For graphical presentation of GRFs only, two groups were used as opposed to three, as (when normalised to per cent stance) visual differences between groupings were only apparent in this more extreme format. These groups are termed `lowest' accelerations, which range from 0.5 to 1.2 m s<sup>–2</sup> (mean 0.8 m s<sup>–2</sup>; N=11), and`highest' accelerations (range 2.8 to 4.4 m s^–2; mean 3.2 m s^–2; N=8). These groups, however, were only used for the purposes of figure presentation, and are not continued further in any analysis.

Patterns of GRFs are shown as means ± s.d. for two groups representing lowest and highest acceleration trials(Fig. 4), as discussed above. Even some of the low acceleration trials showed substantial imbalances between braking and propulsive components of the horizontal GRF, with larger braking forces to propulsive forces. During steady speed galloping the magnitudes of these braking and propulsive components are equal(Walter and Carrier, 2007). Larger accelerations had lesser braking forces than low acceleration trials,often showing almost no braking force at all. In addition, higher acceleration trials showed increased propulsive forces relative to low accelerations.

Hindlimb net horizontal impulses increased with acceleration(R²=0.65; P<0.0001), but did not change with speed (P>0.05). Vertical impulses also increased with acceleration(R²=0.40; P<0.0001; Fig. 5), not speed(P>0.05), but the increase with acceleration was of a lesser degree than that of horizontal impulses. The ratio of vertical to horizontal impulses therefore decreased non-linearly with acceleration, appearing to reach an asymptote between 2 and 4 (Fig. 6). Gallop lead/trailing limb was deemed a significant(P<0.0001) factor in the relationship between hindlimb horizontal impulse and acceleration, with the trailing limb showing a stronger association with acceleration (R²=0.96; P<0.0001) than the lead limb (R²=0.64; P<0.0001). For hindlimb vertical impulse, gallop lead was not significant (P>0.05).

Joint angles are shown as means and standard deviations of hindlimb joint angles for three acceleration categories(Fig. 7). In addition, net joint angle change at each joint with acceleration is shown in Fig. 8.

Patterns of hip angle were reasonably similar for all three acceleration categories (Fig. 7A). At low accelerations hip angle increased throughout most of stance, with a period of no change in hip angle occurring between around 40% and 50% stance. Net change in hip angle increased significantly (trailing P<0.0001, lead P=0.011) with acceleration whilst the period of no change in hip angle appeared earlier in medium accelerations (∼30–40% stance) and high accelerations (∼20–40% stance).

Differences were also apparent in the stifle joint angle between acceleration groups (Fig. 7B). In low acceleration trials, stifle angle was approximately 130 deg. at the beginning of stance, and underwent a fairly symmetrical pattern of flexion (to∼110 deg.) and extension, returning to a similar angle to that at the beginning of stance. Initial and minimum stifle angles were lower (∼110 and 100 deg.) in high accelerations, and the minima occurred earlier in the stance phase. Joint extension therefore began earlier, continuing to a similar maximum to that in lower acceleration conditions (meaning, however, that net change in joint angle was much increased at higher accelerations(Fig. 8; P<0.0001).

Greatest differences in joint kinematics were seen at the hock joint(Fig. 7C). Under low acceleration conditions, the hock angle was 140 deg. at initial foot contact,flexing to a minimum of ∼105 deg. at about 40% stance, and then extending to reach a maximum extension of 155 deg. as the foot leaves the ground. Under the highest acceleration conditions, hock angle was much more flexed at the start of stance (110 deg.) and underwent slightly less flexion to reach a minimum of 85 deg. at a slightly earlier part of stance (30–35%). The hock then extended to a similar maximum to that in low accelerations. As a result, net change in hock angle increased with acceleration(Fig. 8).

MTP joint angles (Fig. 7D)were measured on the extensor aspect of the joint (i.e. decreases in MTP joint angle represent functional extension of the joint). For clarity, the definitions used at each joint for flexion and extension are illustrated in Fig. 2. Joint angles here were more variable; however, kinematic trends were still similar across subjects. The general trend was for an initial extension of the MTP joint, with maximum(hyper) extension reached at 75% stance. The joint then flexed until the end of stance. As a result of the variation at this joint, differences in net MTP joint angle change were not significantly different between acceleration groups.

Hindlimb joint moments also appeared to differ depending on the acceleration condition (Fig. 9). Largest peak moments were seen at the hip and hock joints. At the hip, moments in low acceleration increased to a peak (22 Nm) during the first 25% stance, decreasing thereafter. The peak hip joint moment was greater in medium and high acceleration groups.

Stifle joint moment at low accelerations decreased during the first 25% of stance, then increased to a peak at 65% stance and decreased again. Similar joint moment patterns ensued in the higher acceleration trials. The initial decrease in moment was more apparent in the medium and high acceleration trials, with a more negative minimum moment reached in these two groups. In the highest acceleration category, peak stifle joint moments were at their greatest and this peak occurred later in stance, though more variation was apparent in these higher acceleration categories.

Hock joint moment increased, peaking just prior to 50% stance and then decreasing in a similar fashion. This pattern was similar for all three groups, with a trend towards greater peak moments being generated in the higher acceleration trials.

MTP joint moments were negative, reaching minima around mid-stance. Peak moments tended towards being more negative in higher accelerations; however,again, standard deviations were large in this group.

Typical patterns of hindlimb joint powers are displayed in Fig. 10 for an example low,medium and high acceleration trial. This example is for a lead limb where the increase in net work with acceleration is much greater than in the trailing limb (see later and Fig. 12). Plots for the non-lead limb were similar but less pronounced. Other than this difference between lead and trailing limbs, trends in the patterns of hindlimb power were similar within acceleration groups. Total hindlimb power generally decreased and was negative during the first 30% of stance in steady-state/low acceleration conditions. It then increased to a peak (∼8 W kg^–1; all power values are per kilogram total body mass) by 70% stance. In medium/high acceleration trials, the initial decrease in total hindlimb power was rarely seen, and peaks of a greater magnitude were observed(∼13 and 20 W kg^–1, respectively). Net hindlimb work increased with acceleration (R²=0.49; P<0.0001; Fig. 11A), but not speed (P=0.52). The majority of this increase in hindlimb work was associated with the lead limb(R²=0.60; P<0.0001), as when gallop leads were considered independently, no relationship was apparent for the trailing hindlimb (R²=0.08; P>0.05). Absolute positive limb work, however, was seen to increase significantly with acceleration in both lead (R²=0.30, P=0.009) and trailing(R²=0.39, P=0.002) limbs(Fig. 11B).

Under all conditions, hip power was substantial and positive during the first half of stance, falling to zero or slightly negative late in the second half of stance. Peak hip powers were not significantly greater in higher acceleration trials (Fig. 10),but hip power remained at or near its peak for a longer duration during higher accelerations. Net work at the hip joint had no significant association with acceleration for either lead (P=0.069) or trailing (P=0.29)limbs (Fig. 12); however,absolute positive hip work increased with acceleration in both lead(P=0.03) and trailing (P=0.003) limbs(Fig. 13). Absolute negative hip work was variable, as it was at all joints, and therefore showed no significant trend with acceleration at any joint.

Stifle joint power remained low throughout stance, reaching a maximum late in stance. Net stifle joint work increased with acceleration(R²=0.23; P=0.014; Fig. 12); however, as with total hindlimb work, the majority of this increase was associated with the lead (R²=0.28; P=0.007) and not trailing(R²=0.042; P>0.05) limb. Absolute positive stifle work also increased with acceleration in lead(R²=0.40; P=0.004) and trailing(R²=0.3; P=0.01) limbs(Fig. 13).

Hock power was consistently negative during the first 40% of stance,increasing to large positive powers for the remainder of stance. Under low acceleration conditions, the magnitudes of negative and positive work at the hock joint were roughly symmetrical, resulting in little/no net work. At higher accelerations the magnitude of the negative portion of the power curve appeared to be more variable; however, the magnitude of the positive peak increased with acceleration. Durations of positive and negative power peaks did not appear to differ between conditions. Net hock work did not show a significant relationship with acceleration (P=0.054; Fig. 12); however, again, when gallop lead was considered, lead hindlimb hock work was significantly associated with acceleration (R²=0.41; P=0.01). Absolute positive hock work increased with acceleration(Fig. 13), particularly so in the lead limb (R²=0.26; P=0.03).

Power at the MTP joint was roughly zero during the first 30% of stance, and remained small throughout. Sometimes, a peak was seen late in stance; however,patterns of MTP power appeared more variable than those at other joints. MTP joint work (both absolute positive and net) was independent of acceleration(P>0.05; Figs 12and 13).

EMA_vert decreased considerably with acceleration(R²=0.52; P=0.0001; Fig. 14A), and did not vary with speed (P>0.05). Both lead (R²=0.54; P<0.0001) and trailing (R²=0.53; P=0.01) hindlimbs showed significant changes in posture with acceleration, with the lead limb showing a slightly stronger relationship with acceleration. In addition, EMA_hz also decreased significantly with acceleration (Fig. 14B; lead R²=0.53, P=0.001; trailing R²=0.22, P=0.03). Patterns of gear ratio(EMA^–1) over stance are shown in Fig. 15 for two extreme trials– a low and a high acceleration trial. Under both conditions, hip muscle gear ratio decreased throughout stance, as did hock muscle gear ratio. At the stifle, gear ratio increased throughout stance in both steady-state locomotion and accelerations. Differences were only apparent in the magnitudes of the peaks in gear ratio, and not the overall patterns of change throughout stance. Higher maxima/minima of gear ratio were therefore seen at all joints in early stance during high accelerations. This resulted in a difference in the relative timing of when the stifle joint switched from a negative to positive gear ratio.

Peak hindlimb vertical GRFs here were similar to those described in dogs galloping at steady speed (Walter and Carrier, 2007); however, peak vertical force occurred later in the stance phase in the lead hindlimb of accelerating greyhounds(∼60–70% stance; Fig. 4). This `delay' in the development of the vertical GRF has previously been considered in bipeds as a mechanism by which the animal's centre of mass can be positioned more anteriorly by the time peak forces are reached (Harland and Steele,1997; Roberts and Scales,2002). This allows the GRF vector to be aligned more closely with the centre of mass (important for avoiding body pitching and therefore maintaining balance). Studies in dogs accelerating during a trot have suggested an alternative mechanism for aligning the GRF vector with the centre of mass (Lee et al., 1999). A redistribution of vertical force between forelimbs and hindlimbs was achieved by applying torques about the hip and shoulder, allowing the centre of pressure to originate more posteriorly. No such mechanism was clear here in dogs accelerating at a gallop, and in fact they appear to utilise the former strategy. In trotting, a front limb and hindlimb are in stance simultaneously allowing such a redistribution of body weight to take place; however, this will be more difficult in a canter or gallop when often only hindlimb or forelimb pairs are in stance at one time, and often this reduces to only single limbs. Thus in fast accelerations in greyhounds it appears that a similar mechanism is used to that in bipeds in aligning the body centre of mass with the GRF vector.

Hindlimb horizontal impulse increased with acceleration more than the increase in vertical impulse, such that the ratio of the two(vertical:horizontal) decreased with acceleration. It appears to reach an asymptote between 2 and 4 (Fig. 6). The ratio of horizontal to vertical impulse must be less than the coefficient of friction, as maximum shear force(F_y) is given by the product of the coefficient of static friction and the normal force (F_z). We do not know the coefficient of friction for the surface used in this study;however, many everyday substrates fall around the region of 0.5(Pardoe et al., 2001; Phillips and Morris, 2000; Phillips and Morris, 2001; Vos and Riebersma, 2007) (and thus the inverse of this is 2 – close to the beginning of the asymptote seen in this study). Given that the maximal accelerations we were able to obtain in the laboratory setting were reached close to this coefficient, this raises an interesting question: is acceleration grip limited? We were able to measure higher accelerations in racing greyhounds under field conditions, when they were maximally motivated. Another factor here, however, was that racing greyhounds run on a (watered) sand surface, potentially optimal for grip when compared with carpet used in the laboratory. Higher coefficients of friction than 1.0 could be achieved via inter-digitation between foot and substrate so a surface such as sand would be ideal. In addition, greyhounds have claws on their digits which will enhance grip on soft surfaces (like`in-built' running spikes). Further work comparing different surfaces under maximal field conditions might add insight to this question, particularly as an interesting trade-off is presented here: whilst soft surfaces may be advantageous for increasing the coefficient of friction, they will also absorb energy which would be detrimental for running performance. Greyhound racing surfaces are, however, relatively firm (a tractor can drive on them with minimal displacement) but the feet are still able to dig in.

Patterns of total limb power (Fig. 10) largely followed those of hock joint power. In addition, the hock joint showed a strong relationship between net joint work and acceleration (Fig. 12). Potential exists here, via the presence of a substantial tendon in the biarticular gastrocnemius muscle, for substantial power transfer and amplification (e.g. Lichtwark and Wilson,2005; Lichtwark and Wilson,2006; Peplowski and Marsh,1997; Aerts, 1998)– such a mechanism has been proposed to amplify power by up to 15 times(Aerts, 1998). Total hindlimb extensor muscle mass (one limb) equals around 2 kg(Williams et al., 2008), and therefore total limb peak powers in this study reach around 300 W kg^–1 of hindlimb extensor muscle mass. This approaches the top end of published values for potential maximum muscle power output in mammalian muscle (Weis-Fogh and Alexander,1977), and as accelerations in this study are only half-maximal,it would seem that there is scope for much higher peak powers during higher accelerations. This strongly suggests that power amplification in the tendon forms a substantial part of the hindlimb contribution to acceleration in this animal.

Unexpectedly, no significant relationship was seen between net hip joint work and acceleration (though that for lead hip joint was nearly significant, P=0.07). Previous studies have suggested that quadrupeds,particularly racing greyhounds, power locomotion by a torque about the hip(e.g. Usherwood and Wilson,2005) and so it was expected that the greatest contribution to total limb work would be made at the hip joint. An analysis of absolute positive and negative work at this joint (and others) was therefore conducted,because the initial period of positive work at the hip is often followed by a period of negative work. In this situation, negative work cannot `pay' for subsequent positive work in a physiological sense, unlike when negative work precedes the positive work (such as in a stretch–shorten cycle). Hence,during integration of power over stance to calculate net work, some of the positive work performed at the hip is effectively cancelled out. Absolute positive hip work increased significantly with acceleration(Fig. 13) in both hindlimbs,suggesting that the variable amount of negative work done (not associated with acceleration, P>0.05) in the latter part of the stance phase clouds the issue here. Muscles about the hip joint therefore do appear to be performing large amounts of additional work during accelerations (of around four times over the range of accelerations seen here).

In both stifle and hock joints, lead limb net work was significantly associated with acceleration, but not trailing limb net work. This is in contrast to absolute positive work, which increased significantly in both limbs (though still more so in the lead limb). This contrast indicates that lead and trailing limbs are likely to have different roles during acceleration. It is possible that this difference can be explained by the relative placement and timing of placement of the two limbs. The trailing hindlimb is the first to contact the ground, followed by the lead limb. A period of overlap exists when both limbs are on the ground, at which point the lead hindlimb is aligned with or in front of the hip joint. With this initial position, joints can undergo a wider range of angular excursions (thus increasing the potential for increasing work). In addition, placing the limb more anteriorly may result in the leg supporting a greater amount of body weight (higher GRFs are certainly apparent for the lead limb during high accelerations; Fig. 4) which will aid grip. This positioning thus may result in greater torques about the lead limb.

Caution must be applied when interpreting the results of any inverse dynamics analysis. Studies in jumping, sprinting and acceleration(Aerts, 1998; Dutto et al., 2004; Jacobs et al., 1996; McGowan et al., 2005a) have shown that whilst proximal muscles such as hip extensors produce most of the power required for the task, this power is delivered at the ankle joint. The power is transferred within the limb via biarticular muscles. Joint power analysis does not consider power transport across adjacent joints via two-joint muscles, nor that via muscle co-contraction. Thus power analysis at the joint level often leads to very different results from those at the muscle level (Neptune and van den Bogert, 1998). In this study several cases arise where power transfer may be substantial. Firstly, the back muscles of the greyhound are substantial, and the spinal column undergoes large amounts of flexion and extension, especially during high speed galloping(Alexander et al., 1985). The possibility exists that power is transferred from the musculature of the back to both forelimbs and hindlimbs during locomotion(Gambaryan, 1974). Back extension during hindlimb stance must aid in applying considerable propulsive forces. Quantifying the power produced in the spinal column presents a significant challenge and is beyond the scope of this study but it is likely that the contribution of the back to locomotion is substantial in the racing greyhound.

Secondly, within the hindlimb it is likely that much of the power produced by proximal muscles is transferred distally and delivered at the ankle joint. Anatomical studies (Williams et al.,2008) suggest that the maximum instantaneous power-generating potential of all racing greyhound hock extensor musculature amounts to only 36 W (approximately 1 W kg^–1 body mass). However mean hock power seen in the highest accelerations in this study reaches approximately 1.7 W kg^–1 body mass in a single limb (see Fig. 10) – nearly double the estimate for all ankle extensor muscle contraction alone. Given accelerations in this study were unexceptional (maximally motivated racing greyhounds reach accelerations of up to 10 m s^–2 (S.B.W.,J.R.U. and A.M.W., unpublished observations) this magnitude of difference will be even higher for more impressive accelerations. In addition, power profiles(Fig. 10) also show strong support for energy transfer. During the highest accelerations, the hip is generating a large amount of positive power for the first 20% of stance,whilst the ankle is absorbing energy. The result is that the total limb power during this period is effectively zero. It seems reasonable that some of this energy being developed at the hip is being stored at the ankle (and possibly elsewhere) and is returned at the ankle in late stance. Evidence for such a mechanism has been seen in cats (Prilutsky et al., 1996), and jumping and running humans(Prilutsky and Zatsiorsky,1994; Jacobs et al.,1996; Lichtwark and Wilson,2006); therefore, it should be considered when interpreting any results of an inverse dynamics analysis.

Other limitations to the inverse dynamics approach should also be considered. We assume fixed joint centres of rotation which may not be entirely accurate, particularly for the stifle joint. In addition, small measurement errors, caused by misalignment of markers with the joint rotation centres and by skin movement relative to underlying structures (particularly at proximal joints) may occur. Errors in this study were minimised by careful marker placement by a single person, using easily identifiable anatomical reference points. In addition, a small subset of data from the stifle joint were compared with mathematically derived values as per Dutto et al.(Dutto et al., 2004) and McGowan et al. (McGowan et al.,2005a). Errors were minimal, though this technique relies on reference markers which themselves may be influenced by soft tissue movement.

The EMA of the hindlimb was shown to decrease substantially with increased acceleration (Fig. 14). Thus the hindlimb assumed a more crouched posture at higher accelerations, such that the limb is generally required to recruit a larger volume of muscle, and the muscles experience more force relative to the GRF during periods of higher acceleration. A benefit of this may be to increase the summed length of the limb segments relative to hip or shoulder height(Biewener, 1983). This would allow an animal to exert ground forces over a longer contact period, thus increasing ground impulses for a given peak force. Finally, an additional advantage of a decreased EMA during high accelerations exists. Whereas an upright posture presents a much greater mechanical advantage in producing the vertical GRFs required to counteract gravity(Biewener, 1989) it decreases the mechanical advantage for horizontal ground forces(Fig. 1A). In contrast, the much reduced EMA of a greyhound undergoing accelerations will increase the mechanical advantage for applying horizontal forces and increasing work on the centre of mass (Fig. 1B)– as accelerating has a greater requirement for forces generated in the horizontal direction, this appears to be a simple strategy for increased acceleration performance. It remains to be seen whether the ability to decrease EMA is a prerequisite for superior acceleration capability. If this is the case, as larger animals seem to be constrained to a higher EMA in order to support their body weight (Biewener,1989), size may ultimately be detrimental to the ability to accelerate quickly or undergo other `unsteady' activities.

Changes in individual joint `gear ratio' throughout stance(Fig. 15) appeared to follow similar patterns to those during steady speed trotting, galloping and also jumping (Carrier et al., 1998; Gregersen and Carrier, 2004). The general patterns of gear ratio change under low and high acceleration conditions were similar; however, the magnitudes of the values were different. At the hock, gear ratio decreased during stance, a mechanism that may be beneficial in enhancing elastic energy storage and release, especially if limb forces are lower. This pattern of gear ratio change is typical for a system of a muscle in series with an elastic element(Roberts and Marsh, 2003) such as the gastrocnemius at the hock. A high gear ratio at the beginning of stance allows muscle force to rise to a significant level before shortening occurs. During this period, the series elastic element can be stretched by inertial effect prior to the muscle contracting against the spring. As muscle force is high throughout shortening and the muscle can contract slowly storing energy in the spring, muscle work is maximised. This can be released as high power when the muscle–tendon unit force declines.

At the knee joint, gear ratio increases throughout stance. This is consistent with the `dynamic gearing' hypothesis that has been proposed(Carrier et al., 1998; Carrier et al., 1994; Gregersen and Carrier, 2004)to occur during other activities in dogs and humans. This concept suggests that the total power that can be produced by a muscle during acceleration could be maximised if muscle fibres were to shorten continuously at their best possible shortening velocity [as an optimal shortening velocity exists at which power production is maximised (Hill,1950)]. With a tendon in the system maximum work in a stroke is performed when the muscle contracts as slowly as possible. This constant rate of shortening can only occur and contribute towards the acceleration of the animal if the gear ratio of the muscle in question increases with speed. Most interestingly, the magnitude of the increase in gear ratio at the knee in the accelerating dogs in this study appears greater for high accelerations than for low accelerations. Dogs undergoing greater changes in speed may hence be making greater use of this mechanism to enhance muscle power output.

Here we have explored the mechanical basis for acceleration in a galloping quadruped, the greyhound. Joint kinematics and kinetics were measured under differing acceleration conditions and inverse dynamics was undertaken to assess patterns of joint work and power in the hindlimb. Greatest increases in net joint work and power with acceleration appear at the hock joint,particularly in the lead limb. Large increases in positive hip work were,however, apparent with acceleration, suggesting that creating torque about this joint is essential for accelerating rapidly. Hindlimb EMA decreases substantially with increased acceleration – a potential strategy to increase stance time and thus ground impulses for a given peak force. This mechanism may also increase the mechanical advantage for applying the necessary horizontal forces for acceleration.

We opted to neglect moments generated by inertia and gravity in this study. However, there remains the possibility that errors may be generated by this approach, particularly during activities such as acceleration, when limb translational and joint angular accelerations are likely to be higher than during steady-state movement. The magnitude of these errors was assessed by comparing the total moment about each joint [including inertial and gravitational moments, calculated as per Winter(Winter, 1990)] with our approach.

A small subset of the four highest acceleration trials was used for this(where errors are likely to be the largest). As segment inertial properties were not available for our subjects, we used a geometric model(Crompton et al., 1996) which estimates these from external morphological segment measurements, assuming unvarying segment density. External segment measurements were taken from three greyhounds and the model was used to determine segment masses, centre of mass and moment of inertia. It has been shown that segment properties obtained via this method correlate very well with experimentally measured data(Isler et al., 2006). However,we also assessed the potential effect of any error in these inertial values on calculated joint moments by varying inertial values from 5% to 15% (above and below) of the model values. Inertial properties calculated as above are given in Table A1.

Errors associated with neglecting gravitational and inertial moments are shown in Fig. A1 for each hindlimb joint. Errors remained small during the majority of stance (below 6%of peak joint moments) even at the high accelerations we chose to analyse. This may be because the external joint moment dominated the total moment throughout most of stance, because the GRF_v is forwardly orientated in accelerations, resulting in large external moments (particularly about the proximal joints). Calculated errors were greatest at the proximal joints (hip and stifle) and were negligible at the distal joints (hock and MTP; Fig. A1). Largest errors occurred towards the beginning and end of stance, particularly at hip and stifle joints when the GRF is less influential in determining joint moment and segment angular accelerations are higher (highest angular accelerations are seen during the swing phase and so inertial moments will have a greater effect during this portion of the stride).

These findings are in agreement with other studies. For example McGowan and colleagues (McGowan et al.,2005a) found that in accelerating wallabies, internal moments were very small relative to external moments at all joints, except the hip. Even here, however, the major effect was in early stance, and inertial plus gravitational moments never exceeded 15% of the peak external moment. Furthermore, Dutto and colleagues (Dutto et al., 2004) showed that in jumping horses, inertial moments were small and clustered around zero for most of stance. We chose to base our analysis in this paper solely on `external' joint moments. However the (small)potential errors documented here should be considered when interpreting our results, particularly in early stance, and when designing future studies.

Varying segment inertial properties by 5% and 15% had negligible effects on calculated joint moments. These differences were consistently below 1% of the total joint moment from our model and therefore the use of modelled inertial properties in our error analysis appears justified in this instance. However,experimental measurements are clearly the ideal, especially for unusually shaped segments [e.g. hands and feet in primates(Isler et al., 2006; Crompton et al., 1996)].

LIST OF ABBREVIATIONS

 
• a_y

  horizontal acceleration

 
• a_z

  vertical acceleration

 
• CoM

  centre of mass

 
• CoP

  centre of pressure

 
• EMA

  effective mechanical advantage

 
• F_y

  horizontal force

 
• F_z

  vertical force

 
• g

  gravitational constant (–9.81 m s^–2)

 
• GRF

  ground reaction force

 
• GRF_r

  ground reaction force vector

 
• m

  mass

 
• M_ext

  external moment

 
• M_grav

  gravitational moment

 
• M_inert

  inertial moment

 
• M_tot

  total moment

 
• MTP

  metatarsophalangeal joint

 
• r

  muscle moment arm

 
• R

  ground reaction force moment arm

 
• s.d.

  standard deviation

 
• v_y

  horizontal velocity

 
• v_z

  vertical velocity

 
• V_init

  initial velocity

The authors would like to thank Simon Townsend, Melissa Upjohn and Matthew Tipper for providing their dogs for use in these experiments. We also extend our thanks to all members of the Structure and Motion Laboratory who assisted in data collection, and two anonymous reviewers, whose comments have improved this paper significantly. S.B.W. was funded by a Royal Veterinary College Studentship. A.M.W. is holder of a BBSRC Research Development Fellowship and a Royal Society Wolfson Research Merit Award. J.R.U. is funded by a Wellcome Trust Research Career Development Fellowship. Deposited in PMC for release after 6 months.