The Work and Energetic Cost of Locomotion: II. Partitioning the Cost of Internal and External Work within a Species

The energetic cost of locomotion has been shown to vary in a strongly size dependent manner. Larger animals have systematically lower mass-specific costs of locomotion (see Taylor et al. 1982 for a summary of available data). A probable explanation for this pattern lay in differences in mechanical efficiency: larger animals had lower costs because they did less work in moving 1 g of their mass for a given time or distance. Experimental results, however, have contradicted this interpretation. This study looks at how this decoupling of work and cost can occur.

The mechanical work done per unit time in locomotion (in watts) is divided into external work (Ẇ_EXT, that done to raise and reaccelerate the center of mass of an animal) and internal work (Ẇ_INT, that of accelerating and decelerating individual body segments relative to the center of mass). Heglund et al. (1982) reported a linear relationship between external work per unit time and velocity within each of 14 species tested. However, they found no systematic decrease in mass-specific external work per unit time with increasing body size that would explain the scaling of cost. Since cost shows an approximately linear relationship to velocity within an individual (Taylor et al. 1982), as does Ẇ_EXT, there is considerable similarity between these two in their response to changes in velocity within an animal, perhaps even a causal linkage. Such a linkage may break down interspecifically owing to scaling of physiological and morphological determinants of locomotion.

This linkage is absent, however, for internal work and cost even within an individual. Fedak et al. (1982) found that the internal work done per unit time increased as a power function of speed within a species and showed no tendency to decrease in larger animals. This curvilinear relationship between Ẇ_INT and velocity (Cavagna and Kaneko, 1977; Fedak et al. 1982; Winter, 1979) coupled with the linear relationship between total cost and velocity (Taylor et al. 1982) could be interpreted in two ways. Taylor et al. (1980) argued that the results of their back-loading experiments were explicable only if the contribution of internal work to the total cost of locomotion was very small, possibly as a result of the use of elastic strain energy to power this aspect of locomotion. Alternatively, the cost of internal work may be substantial but increase with speed at a much slower rate than would be predicted from the increase in work itself; that is, the cost of internal work may be decoupled from the amount of internal work across a range of speeds.

These two possibilities can be discriminated through direct measurement of the cost of internal work within individual animals across a series of speeds. The present study develops a method of partitioning the metabolic cost of internal work from that of external work through the addition of artificial loads either at the center of mass or on the limbs. This technique is then used to estimate the relative magnitudes of the effects on cost of increases in internal and external work and to determine the pattern of variation with speed in the cost of internal work.

Three adult male dogs (body mass: 20,20·9 and 26·8 kg) were trained to run on a treadmill at four speeds between 1·07 and 2·68 ms⁻¹ under three different loading conditions. The speeds used differed somewhat among the three subjects. The lowest test speed was the highest speed at which each dog consistently used a walking gait; the highest test speed was the highest speed at which each dog could locomote comfortably while wearing the limb loads. For all dogs the latter was within the trot range. The remaining two test speeds were intermediate. The experimental conditions involved adding 0·77 kg of lead either to a harness or distributed equally to the four limbs. Details of the harness and loads and of the training are given in Steudel (1990).

Rates of oxygen consumption were measured using the open-circuit system described in Steudel (1990). values for the last three successful runs for each dog under each of the 12 test conditions were used for data analysis.

All values of in this study are given in mlO₂s⁻¹. To avoid the difficulties associated with the statistical treatment of ratios (see Atchley et al. 1976), I have not followed the widespread approach of dividing values by body mass to obtain mass-specific values. Instead, differences in body size between the subjects are treated as a component of the between-subject variation.

All trials were videotaped at 200 frames s⁻¹ using a NAC high-speed video camera. Stride frequency was measured for each trial by counting the number of fields per stride and averaging over 10 strides. Knowing the tape speed allows conversion to strides per second.

The position of the mass on the harness was very close to the center of mass of the animal and, therefore, affected external work (Ẇ_EXT) while leaving internal work (ẆINT) virtually unchanged. Consequently, the cost of the increased external work (Ċ_EXT) produced by a load of this mass can be determined by subtracting the control value for a given dog at a given speed from the value for the same dog at the same speed running with a back load.

The cost of increased internal work (Ċ_INT) can be closely approximated by subtracting the values for trials in which the 0.77 kg mass was added to the back from the values obtained when the same mass was added to the limbs, again separately for each dog and each running speed.

I do not contend that adding mass on the trunk affects external work with no effect whatsoever on internal work or that adding mass to the limbs affects internal work with no change in external work. Rather, I argue that the back loads produce a small increase in external work while having a negligible impact on internal work, and that, similarly, the limb loads produce a substantial increase in internal work, while having a very small effect on external work.

Because the position of the center of mass of an animal oscillates slightly during locomotion as a result of changes in the position of body segments, a stationary load cannot be fixed precisely at the center of mass. The consequent changes in the relative positions of the center of mass of the trunk segment and the total body center of mass will result in some change in internal work. Given, however, the small mass of the loads and the fact that the vertical oscillations of the center of mass over a stride are of the order of a few centimeters (Cavagna et al. 1977; Fukunaga et al. 1980), the magnitude of the effect on internal work seems likely to be extremely small.

Because the total mass of the loads was between 2·9 and 3·9 % of body mass, the effect of their addition to the limbs of a stationary animal on the position of the center of mass should be modest. Moreover, the oscillations of the limbs during locomotion will not greatly alter the position of the center of mass. Because the gaits used by the dogs in this study, the walk and the trot, are both symmetrical (Hildebrand, 1966, 1980), the footfalls of the two hindfeet and two forefeet are evenly spaced in time. Consequently, the forward shift in the center of mass that would be produced by the forward placement of one limb of a pair will be approximately balanced by the backward shift in the center of mass produced by the more posterior placement of the other limb. Thus, the increase in external work produced by limb loads over the external work produced by back loads should not be very great, nor should any such effect show a strong relationship with speed because the dogs used symmetrical gaits at all speeds tested. Thus, attributing the increases in cost observed in limb-loaded trials over the costs from back-loaded trials to increases in Ẇ_INT seems justified.

The minimal increments in Ẇ_INT due to the 0·77 kg limb loads that should be observed across the velocity range tested for each dog were calculated from the results of Fedak et al. (1982) based (i) on the relationship between Ẇ_INT and velocity observed for their 5 kg dog and (ii) on their summary equation (given above) based on data from seven species. The internal work done by each dog at each running speed was estimated by substituting the treadmill speed for that trial for v, calculating Ẇ_INT, and then multiplying the result by 0·77 kg to estimate the effect of a load of that size on internal work.

Calculating the change in Ẇ_1NT produced by the limb loads based on the equations of Fedak et al. (1982) will underestimate the change in Ẇ_INT actually produced by the loads. In unloaded animals, an increase in mass will be distributed across many body segments. This is the situation described by the equations of Fedak et al. (1982). With artificial loads, only the mass of the loaded segment changes. Martin (1985) has shown that it is only the mechanical work done to move the loaded segment that is altered as a result of added loads. Fedak et al. (1982) noted that the kinetic energy of the distal limb segments generally goes through greater oscillations than that of more proximal segments, because the distal segments move further during a stride and reach higher velocities. Consequently, the addition of mass to distal segments will tend to have a greater effect on total internal work than if the same mass was more dispersed.

More importantly for the arguments given here, using the equations of Fedak et al. (1982) should also underestimate the magnitude of the increase with speed in work due to the loads. Because the substantial contribution of the distal segments to internal work results from their velocities at a given speed of locomotion being greater than that of more proximal segments (Fedak et al. 1982), the enhancement of limb velocity that occurs with increased running speed will tend to produce a larger effect on the kinetic energy of distal segments as compared to more proximal segments that have higher masses but lower velocities. Consequently, the effect of the loads being localized on the distal extremities, rather than spread evenly across the body, should result in actual increases in Ẇ_INT with speed being larger than the baseline values estimated here.

In summary, using the equations of Fedak et al. (1982) to estimate the changes in the mechanical work of locomotion produced in these experiments by the limb loads biases the results against the conclusion reported below, that Ẇ_INT increases faster with speed than does Ċ_INT. This approach to the estimation of internal mechanical work is, therefore, a conservative one, given the conclusions that are drawn below.

To determine whether there is evidence for a curvilinear increase in Ċ_INT with velocity similar to that seen for Ẇ_1NT with velocity, I have compared the fit of the relationship between Ċ_INT and speed to the same relationship with logarithmic transformations to see if the exponential model produced by the transformation results in a better fit. Both relationships have also been tested to see if Ċ_INT shows a significant relationship to speed. In addition, I have made direct comparisons between values for limb-loaded trials and values from back-loaded and control trials using regression analysis and analysis of variance (ANOVA).

Another approach involves direct comparisons of the observed increases in Ċ_INT across a series of speeds with estimations of the increases in Ẇ_1NT across the same series of speeds. Taylor et al. (1980) compared the ratios of loaded and unloaded mass with the ratios of loaded and unloaded in a series of animals at a series of speeds. The analogous ratios in the present experiment are the ratio of Ċ_INT at the highest and lowest running speed for each animal to the ratio of estimates of Ẇ_INT at the highest and lowest running speed for each animal. If these values are similar -that is, if the proportional increase in Ċ_INT is similar to the proportional increase in Ẇ_INT -a clear association between internal work and its cost is indicated. A lack of proportionality between these ratios, however, would indicate a decoupling of internal work and its cost.

The rates of oxygen consumption for each of the three subjects under each of the three loading conditions (control, back loading and limb loading) at each of four test speeds are given in Table 1. Results from analysis of covariance (ANCOVA) indicate that the addition of 0·77 kg of mass to the limbs produces a consistent increase in at all four speeds that is statistically highly significant. ANCOVA with as the dependent variable, dog and loading condition as independent variables and velocity as a covariate showed that loading condition had a significant effect on (P<0·001). Least significant difference tests of the difference between mean values for each paired combination of the weighting conditions showed that the back loads did not produce a significant increase in over the control condition, whereas the limb loads produced a significant increase over both back loads and control.

Since there was no significant difference between the values produced under the back-loading and control conditions, in subsequent analyses the cost of internal work was measured as the ⁠, with limb loads minus the control ⁠. Since the values for limb-loaded and control dogs that are subtracted to produce each estimation of Ċ_INT were measured on the same day in the same test session, these values for Ċ_1NT contain less experimental error than those calculated by subtracting back-loaded values.

The cost of internal work does not increase with speed at nearly the rate at which internal work itself increases. Least-squares regression analyses of Ċ_INTversus speed are not significantly different from zero. This is true when the analysis is done on the untransformed data and also when the data have been logarithmically transformed to test the exponential model. Not only do the data fail to support the hypothesis that Ċ_INT increases curvilinearly with speed, they show no evidence of any increase with speed. The cost of increasing internal work is significant, but the magnitude of this effect appears to be independent of speed (Fig.1).

Other statistical approaches confirm this result. The observed rate of linear increase in the cost of locomotion with speed was not significantly greater in the limb-loaded trials than in the controls. Fig. 2 shows the values obtained for each speed and loading condition for one of the animals. The results of least squares regression analysis of against speed for each dog and each loading condition are given in Table 2. For dogs 1 and 2, the slopes for the limb-loaded trials are larger than for control or back-loaded trials, but in neither case was this difference significant at P<0·05. In dog 3, the slope for the limb-loaded trials was nearly identical to that for the control runs.

This same conclusion is indicated by the results of ANOVA with as dependent variable, and dog, weighting condition and speed as the independent variables. Treating speed as a discrete variable rather than as a covariate allows one to test for the significance of an interaction between weighting condition and speed. If the effects of the loads are different at different speeds, as would be expected if Ċ_INT follows Ẇ_INT, one should find a significant interaction. This interaction term is not significant (P=0·579).

Analysis of the stride frequency data showed that the dogs responded to the addition of limb loads with a small but significant (P<0·05) drop in the rate at which they cycled their limbs (Fig. 3). This result, however, does not affect the conclusions about the rate of increase in the cost of internal work with speed. When stride frequency was regressed on speed, separately for each dog and weighting condition, the resulting slopes were not significantly different from one another at P=0·05. Thus, while the dogs responded to the limb loads by reducing stride frequency compared to back-loaded or control trials, the rate of increase of stride frequency with speed was not significantly different.

A direct comparison of the expected increases in Ẇ_INT with the observed values for Ċ_INT resulted in the same conclusion. Values for the increase in Ẇ_INT for the velocity range tested in each dog predicted from data on a 5 kg dog and from the summary equation of pooled data (Fedak et al. 1982) can be seen in Table 3 along with the observed values for Ċ_INT across the same velocity ranges. In all three dogs, the increase in cost is approximately half the increase that would be predicted from the increase in work. These differences are significant at P<0·03 (two-tailed t-tests, N=3). Again, these data show no evidence of a curvilinear increase in the cost of internal work with increasing velocity. This result is shown graphically in Fig. 4.

These experiments were designed to determine how the variation of internal work with velocity within a species can be so different from the variation of cost with velocity within a species. As indicated above this dissimilarity could result either from the cost of internal work (Ċ_INT) being a very small proportion of total cost or from a decoupling of Ċ_INT from the amount of internal work ( ẆJNT) across velocities. The latter possibility is the only one consistent with the results reported here. The cost of increasing internal work is substantially greater than that of increasing external work at all speeds tested, but the enhancement of this effect that would be expected at higher speeds does not occur.

This mitigation of cost at higher speeds implies that at such speeds the ability of the organism to use non-metabolic means to power internal work is enhanced. This is exactly the pattern that would be expected if the storage and recovery of elastic strain energy were an important interface between internal work and cost. While many aspects of the role of elastic strain energy in enhancing locomotor economy are poorly understood, it is widely believed that elastic mechanisms become increasingly effective at higher speeds. Dawson and Taylor (1973) reported an actual decline in with increasing speed in hopping kangaroos, suggesting that elastic recoil paid for increasing amounts of work as speed increased. Heglund and Cavagna (1985), working on isolated muscle preparations, found that the effect of a prestretch on muscle efficiency increased with contraction rate. Gregor et al. (1988) have documented in vivo changes in time tq peak force at three speeds in the cat, showing the increased opportunity for elastic storage with increasing velocity. Furthermore, they were able to demonstrate enhanced generation of force and power output for a given shortening velocity in in vivo compared with in situ data at their higher speeds of locomotion, which they interpreted as the result of elastic storage in vivo.

Although most workers expect elastic mechanisms to become more important at higher speeds, a wide range of opinions exist concerning the likelihood that elastic mechanisms are important in mitigating Ċ_INT. Fedak et al. (1982) concluded that elastic recoil supplies a significant fraction of internal kinetic energy, especially at high speeds, based on comparisons of their estimates of internal work and published data on metabolic cost (Taylor et al. 1982). Alexander (1984) was persuaded by these arguments that elastic mechanisms should save internal work, but found that the possible mechanisms for these savings seemed ‘unlikely to be very effective’. Subsequently, Alexander et al. (1985) presented data that make a convincing case for the importance of elastic structures in the back as a means of powering internal work. These arguments, however, apply only to galloping. Goslow et al. (1981) found muscle activity patterns consistent with the storage and recovery of elastic strain energy in the limbs in both trotting and galloping dogs. This occurred both in the stance phase and at the end of the swing phase and into the recovery stroke. Furthermore, Bennett et al. (1989) estimated that elastic strain energy stored in the hind foot could account for about 12·5 % of the cost of half a stride in trotting monkeys. The extent to which this value might change with increasing velocity is not known.

The results obtained here unequivocally indicate that some mechanism for powering internal work that supplements the muscular contribution becomes increasingly important at higher speeds. Thus, it seems very likely that the storage and recovery of elastic strain energy in muscles and tendons is important in supplying energy for internal work. Energy stored and recovered in the muscular and associated tendon complexes of the limbs described by Goslow et al. (1981) and Bennett et al. (1989) is especially likely to be involved at these speeds.

Do these artificial loading experiments tell us anything about the role of non-metabolic mechanisms, such as elastic storage, in driving external work? Since the relatively small mass added in the present experiments did not produce a statistically significant effect when added at the center of mass, the data collected here do not indicate whether the cost of increasing external work (Ċ_EXT) keeps pace with the increase of external work itself (Ẇ_EXT)-Previous experiments, however, have measured the cost of locomotion in dogs running with and without back loads amounting to 21 and 27 % of body mass (Taylor et al. 1980). These data allow Ċ_EXT to be calculated in a manner analogous to the internal work calculations made above. In this case the relevant statistic is cost with the back loads minus cost at the same velocity with no load. The appropriate calculations were carried out for both the 21 and the 27 % increases in mass. The results can then be compared with the expected increases in external work due to loads of that size based on equations given in Heglund et al. (1982). Data from Heglund et al. (1982) were used to calculate expected increases in external work in two ways: (i) based on the relationship between work to move the center of mass and velocity observed in a 17 kg dog and (ii) based on their equation summarizing the results for that same relationship over all 14 species.

As can be seen from Table 4, ĊEXT increases with velocity to an extent very similar to that predicted from the external work equations. Thus, both Ċ_EXT and Ẇ_EXT are closely linked to velocity and apparently to one another within an individual. This empirical result is not at all surprising given the observation that both cost of locomotion and the work needed to move the center of mass have been shown to increase linearly with speed (Taylor et al. 1970,1982; Heglund et al. 1982).

Thus, there is no evidence of a mitigation of the cost of external work with increasing speed. If elastic savings mechanisms become increasingly important at higher speeds and if these savings could be applied to increases in external work, one would not expect to see such a direct response of cost to work across speeds. This argument, however, is not conclusive. The direct proportionality described above for Ẇ_EXT and Ċ_EXT will hold whenever two variables have linear relationships to a third variable with y-intercepts near zero. If the extent of elastic effects in saving external work showed a similar linear increase with speed and a zero intercept, this could result in a reduction of the observed slope for cost versus speed, a retention of the near-zero intercept and, hence, a retention of the direct proportion to cost. Thus, a role for elastic storage in mitigating external work can neither be confirmed nor ruled out by these results. If present, however, it must conform to the pattern just described -the effect must increase linearly with speed and be near zero at 0ms⁻¹.

As indicated above, Taylor et al. (1980) thought it necessary to assume that the direct proportionality of the change in cost to the change in mass produced by their back loads implied that most of the energetic cost of locomotion was allocated to external work. The Appendix contains a mathematical argument concluding that the change in cost should be proportional to the change in external work as a fraction of total work, making no assumptions about the relative proportions of internal and external work. The result described here, that increasing internal work produces a substantial impact on cost, thus becomes readily explicable. The addition of external loads to either the limbs or the back alone does not produce an estimate of the relative importance of external and internal work in determining cost. This assessment can only be made using comparisons of the relative costs of limb loads to back loads.

Transfers of energy between body segments or between the kinetic and potential energy of the center of mass are also thought to be an important energy saving mechanism (Cavagna et al. 1977; Pierrynowski et al. 1980; Robertson and Winter, 1980; Williams and Cavanagh, 1983). The efficiency of such transfers has been shown to vary with speed in walking humans (Cavagna et al. 1964). Current data on running speeds, however, are inadequate to determine the relationships between speed and the savings due to energy transfers. If the efficiency of such transfers increases with increased running speeds, it is possible that this could be another mechanism that preferentially mitigates internal work.

Although adding mass to the limbs of an animal results in substantial, statistically significant increases in across a range of speeds, the slope of the versus speed relationship -the cost of transport -is unaltered (see Table 2 and Fig. 2). Species differences in limb mass distribution should not, therefore, emerge in studies of the cost of transport. Rather, the effect of limb morphology becomes apparent in comparisons of the absolute cost of locomotion or the mass specific cost of locomotion at a given velocity.

The observation that the mechanical work of locomotion does not show a pattern of interspecific scaling that could be driving the scaling of the energetic cost of locomotion has caused interest to shift elsewhere for the determinants of cost (Heglund et al. 1982; Taylor, 1985; Steudel and Strang, 1987; Blickhan and Full, 1987; Heglund and Taylor, 1988; Strang and Steudel, 1990). Research on human locomotion, however, continues to search for a linkage between the mechanical work and the energetic cost of locomotion, attributing the lack of a perfect correspondence between the two to imprecise measurement and, especially, to imprecise understanding of the interface provided by elastic storage and energy transfers (Winter, 1979; Fukunaga et al. 1980; Williams and Cavanagh, 1983; Williams, 1985; Cavanagh and Kram, 1985). Perhaps what one sees is a substantial relationship between work and cost within a species, modulated by the storage and recovery of elastic strain energy and transfers of energy, that breaks down when interspecific comparisons are made.

A variety of physiological and morphological factors might contribute to such a situation. Allometric variation in the intrinsic contraction rate of muscles (Goldspink, 1977; Taylor et al. 1980) or in the stride frequencies necessary to achieve a given velocity are two possibilities (Steudel and Strang, 1987; Heglund and Taylor, 1988). A third possibility is that larger mammals run at higher speeds, at which internal work becomes large relative to external work. Even though the relationship between internal work and speed is independent of mass (Fedak et al. 1982), the contribution of internal work to total work will be dependent on mass if running speed is dependent on mass.

Mass-dependence of running speeds is demonstrated by the scaling of speed at the trot-gallop transition as M^0·24 (Heglund et al. 1974). This gait transition is often regarded as a ‘physiologically similar speed’, useful for comparisons across species (Heglund et al. 1974; Heglund and Taylor, 1988; Perry et al. 1988). Garland (1983) reported a less pronounced scaling of maximal running speed of M⁰·¹⁶⁵. The maximum speed at which has actually been measured for each species scales as M⁰·²⁰, which is similar to the scaling of maximal aerobic speed, M^0·19 (Garland et al. 1988).

The results described here indicate that the increased internal work produced at higher speeds does not exact a comparable increase in cost. This may be an important factor mitigating cost in larger animals.

In conclusion, the cost of increasing internal work does not increase as a power function of speed in a manner comparable to the amount of increase in internal work. Rather, the cost of increased limb mass is the same at all speeds tested. Consequently, differences in limb morphology will not be reflected in measurements of the cost of transport, but rather in measurements of the total cost of running at a given speed.

The decoupling of internal work from its energetic cost strongly implies the existence of some buffer between these parameters that becomes increasingly effective at higher speeds. The storage and recovery of elastic strain energy and the transfer of energy within and between body segments are two obvious candidates for such a buffer. The fact that larger animals more commonly run at the higher speeds at which the cost of internal work is mitigated might account, in part, for their lower energetic costs of locomotion.

I would like to thank Paula Kroma, Karla Pecelunas, Cynthia Dott, Patricia Breen, Phillip Gilson, Allen Caviles and Julie Cherney for help with the training and care of the dogs. I am especially grateful to two of my graduate students, Marcella Myers and Kevin Strang, for useful discussions about the determinants of the cost of locomotion and to Ted Garland, Tom Givnish, Kevin Strang and Marcella Myers for helpful criticisms of the manuscript. Carole Hom contributed important insights to the mathematical treatment given in the Appendix. Cheryl Hughes drew the figures.