Energetics and mechanics of terrestrial locomotion. III. Energy changes of the centre of mass as a function of speed and body size in birds and mammals

The first paper in this series demonstrates two very general relationships about energetic cost of terrestrial locomotion in birds and mammals: (1) metabolic power increases nearly linearly with speed over a wide range of speeds; and (2) the cost to move a gram of body mass a given distance decreases as a regular function of increasing body mass (e.g. a 30 g quail uses approximately 13 times as much energy to move each gram of its body a given distance as a 100 kg ostrich or pony). The second paper quantifies the kinetic energy changes of the limbs and body relative to the centre of mass as a function of speed and body mass. It shows that the mechanical power required to maintain these changes in kinetic energy increases as the 1-55 power of speed and is independent of body size.

This third paper considers a second component of the mechanical work required to sustain a constant average speed along the ground: the work required to lift and reaccelerate the centre of mass within a step, E_CM,_tot Locomotion at a constant average speed consists of a series of cycles (steps) during which the potential and kinetic energy of the centre of mass oscillates as the centre of mass rises, falls, accelerates and decelerates. These oscillations in energy have been measured over a wide range of speeds in man (Fenn, 1930; Elftman, 1940; Cavagna, Saibene & Margaria, 1963, 1964; Cavagna, Thys & Zamboni, 1976); in one step of a cat (Manter, 1938); in one step of a quail (Clark & Alexander, 1975); and in two hops of a wallaby (Alexander & Vernon, 1975). They have been found to constitute an important part of the mechanical work of locomotion in all these studies.

More than five years ago we began studies designed to find out how Ė_CM,_tot varied as a function of speed and body mass. Our investigation had to be broken into two parts because the tool for measuring Ė_CMtot force platform, can only be used for a limited size range of animals. The first part of the study was carried out using a force platform that had originally been built for humans in Milan, Italy. It was suitable for studies of animals ranging in body mass from 3 to too kg. We studied Ė_CM,_tot as a function of speed for a diversity of bipeds, quadrupeds and hoppers that fell within this size range (Cavagna, Heglund & Taylor, 1977). Then we designed and built a force platform that was suitable for small animals ranging in body mass from 30 g to 3 kg. This paper reports the experiments relating Ė_CM,_tot and speed for small bipeds, quadrupeds and hoppers. We then utilize the data for Ė_CM,_tot as a function of speed for both the small animals and the large animals to find out how it varies as a function of body mass.

We utilized two force platforms (one for animals greater than 3 kg and one for animals less than 3 kg) to quantify the vertical displacement and the horizontal and vertical speed changes of the animal’s centre of mass as it moved along the ground at a constant average speed. The force platform measured the force exerted on the ground and resolved it into vertical and horizontal components. These forces were integrated to obtain horizontal and vertical velocities. The forces and velocities were recorded on a strip chart recorder. The velocity records were used to decide whether a particular experiment was acceptable for analysis of the energy changes of the centre of mass; we included only experiments where the animals moved at a constant average speed across the platform. Our criteria for inclusion of an experiment were: (1) records included one or more complete strides; (2) the sum of the increases in velocity (as measured by the integrators) was within 25 % of the sum of the decreases in velocity in both the horizontal and vertical directions for an integral number of strides; and (3) no drift in the integrators during the period of analysis. For a typical chipmunk stride at 2·1 m s⁻¹, our 25 % limit amounted to a forward speed change of less than 1·5 % of the average forward speed.

For experiments that met our criteria, we carried out a second integration of the vertical forces to give the vertical displacement of the centre of mass. Then kinetic and gravitational energy changes of the centre of mass within a stride were calculated from the velocities and the displacement. Cavagna (1975) has described this technique in detail.

Theoretically, it might be possible to calculate the displacements and the speed changes of the centre of mass using the film analysis technique described in the previous paper. Practically, however, these displacements and speed changes were too small to be resolved accurately with the filming technique. For example, the centr.e of mass of a 170 g quail running at 2·6 m s⁻¹ typically went up and down only 7 mm and decelerated (and reaccelerated) only 0·07 m s⁻¹ within each step.

The mechanical work required to lift and reaccelerate the centre of mass was first measured as a function of speed of locomotion for individual animals. Then we used the equations relating the work necessary to accelerate and lift the centre of mass and speed to develop an equation which described how this work changed with body mass.. Finally, we compared the equations for metabolic energy consumed with the work required to sustain a constant average speed of the centre of mass.

Two species of small bipedal runners, two species of small quadrupedal runners and two species of small bipedal hoppers were trained to run across the small force platform while we measured mechanical energy changes of their centre of mass. Measurements from two large bipedal runners, three large quadrupedal runners and two large bipedal hoppers had already been obtained on the large force plate, and these data have been reported (Cavagna et al. 1977).

We selected species in this study for which metabolic rate had been measured as a function of speed (see the first paper of this series) and which extended the range of body mass as much as was feasible. The bipedal runners included two 42–44 g Chinese painted quail (Excalfactoria chinensis) and three 150–180 g bobwhite quail (Colinus virgimamis). Measurements had been made on the large plate for turkeys, rhea and humans, giving us a 2000-fold range in body mass for bipedal runners. The quadrupedal runners included two 80–100 g chipmunks (Tamias striatus) and one 190 g ground squirrel (Spermophilus tridecemlineatus). Measurements had been made on the large force plate for monkeys, dogs and ram, giving us a 1600-fold range in body mass for quadrupedal runners. The bipedal hoppers included one 37 g kangaroo rat (Dipodomys merriami) and three 100–140 g kangaroo rats (Dipodomys spectabiHs). Measurements had been made on the large force plate for spring hares and kangaroos, giving us a size range 600-fold for bipedal hoppers.

In order to obtain measurements from small animals we constructed a smaller force platform suitable for measurements from animals ranging in body mass from 30 g to 3 kg. The small platform consisted of twelve mechanically distinct plates placed end to end in the middle of an 11 m runway. Each plate consisted of an aluminium honeycomb-panel surface (25 × 25 cm) with a sensing element at each corner. Each sensing element consisted of adjacent spring blades, one horizontal and one vertical, that were instrumented with metal-foil strain-gauges. The horizontally oriented spring blade was sensitive only to the vertical forces and the vertically oriented spring blade was sensitive only to the horizontal forces. Cross talk between the vertical and horizontal outputs of the force plate was less than 5 % in the worst case. The output of any particular plate was independent of where on the plate surface the force was exerted to within 3 %. The output of the platform was linear to within 1·5 % over the range of forces measured in these experiments. The natural frequency of oscillation of an unloaded plate was 170 Hz. The design of this force platform has been described in detail elsewhere (Heglund, 1979, 1981).

The horizontal force, and the vertical force minus the body weight, were each integrated (using an LM 208 op-amp with a 0·3 s R-C constant) to obtain continuous recordings of the velocity changes of the centre of mass. These recordings were entered directly into a microcomputer at 2 ms intervals using a 12-bit analog-digital converter. The remainder of these procedures were carried out by the microcomputer ; complete schematics of the electronics and listings of the programs utilized in this analysis have been given elsewhere (Heglund, 1979).

In order to calculate the absolute vertical and horizontal velocity of the centre of mass, the constants of integration have to be evaluated. The integration constant for the vertical velocity was taken to be zero over an integral number of strides, that is, we assume that the height of the centre of mass was the same at the beginning and end of the strides that were analysed. The integration constant for the horizontal velocity is the average running speed during the period of integration. The average speed was measured by placing two photocells along the path of the force platform; the first photocell turned the integrators on and the second photocell turned the integrators off. The computer then calculated the integration constant (average speed) from the distance between the photocells and the time the integrators were on. The system was calibrated daily for each animal.

The kinetic energy due to the horizontal component of the velocity of the centre of mass (E_H) was calculated as a function of time (⁠⁠.v², where v is the horizontal velocity of the centre of mass). The vertical velocity of the centre of mass was integrated to obtain the vertical displacement of the centre of mass as a function of time. Multiplying the vertical displacement by the animal’s body mass and the acceleration of gravity (ΔPE = M_bgΔh) gave the gravitational energy changes of the centre of mass as a function of time. The instantaneous sum of the changes in potential energy and the kinetic energy due to the vertical component of the velocity of the centre of mass gives the total changes in energy due to vertical position or movements of the centre of mass (ΔE_V).

The total energy of the centre of mass, E_CM,tot, was calculated as a function of time by summing the kinetic and gravitational potential energies of the centre of mass at the 2 ms intervals.

The average rate of increase in the total energy of the centre of mass, Ė_CM,tot, was calculated by summing the increments in the E_CM,tot curve over an integral number of strides and dividing by the time interval of those strides. This power had to be supplied by the muscles and tendons of the animal.

The procedure outlined above was repeated for 7–38 speeds in each animal. The function relating Ė_CM,tot to speed was then calculated by linear regression analysis.

We used the equations relating Ė_CM,tot to speed for the individual animals from this and the previous study (Cavagna et al. 1977) to develop an equation relating Ė_CM,tot to body mass.

The small quails (30 and 200 g) utilized the same walking mechanism as we had observed in larger animals (Cavagna et al. 1977) and humans (Cavagna et al. 1976). Fig. 1 shows force, velocity and energy records for a typical walking step of the quail. The changes in gravitational potential energy and kinetic energy due to the forward velocity of the animal are out of phase. Thus the decrease in kinetic energy that occurs as the animal slows during one part of the step is stored in gravitational potential energy as the centre of mass rises. This stored potential energy is recovered subsequently in the step as the animal reaccelerates and the centre of mass falls. This energy-saving mechanism is similar to an inverted pendulum or an egg rolling end over end.

Fig. 2 gives a quantitative measure of the energy savings resulting from this pendulum mechanism. As much as 75 % of the energy changes that would have occurred had there been no transfer were recovered by this pendulum mechanism. Percentage recovery was calculated using the following equation :

The small bipeds (quails) and hoppers (kangaroo rats) utilized a run or hop gait at speeds above 1 m s⁻¹ similar to the gaits observed in larger animals (Cavagna et al. 1977) and humans (Cavagna et al. 1976). Fig. 3 gives force, velocity and energy records for a typical run or hop step for a 35 g kangaroo rat, a 43 g quail, a 99 g kangaroo rat and a 176 g quail. The changes in gravitational potential energy and kinetic energy due to the forward velocity of the animal are in phase. Thus the decrease in kinetic energy as the animal slows within a stride occurs almost simultaneously with the decrease in gravitational potential energy as the animal’s centre of mass falls and little exchange can occur (Fig. 2).

The shape of mechanical energy curves are similar for a run, trot and hop, regardless of size of the animal or its mode of locomotion. The magnitude of the energy changes and the stride frequency, however, do change with body size. For example, E_CM,tot during the step of a human running at a moderate speed is 80 J, and about 2· 5 steps are taken each second. E_CM,tot for a step of a bobwhite quail running at a moderate speed, by contrast, is only 4·5 J × 10⁻² and the quail takes 9·5 steps each second. We were unable to train the chipmunk and ground squirrel to move across the platform slowly enough to obtain good records for the trotting gait.

The small quadrupeds (chipmunk and ground squirrel) galloped across the force platform over a wide range of speeds (1–3 m s⁻¹). The force, velocity and energy tracings obtained from these animals during a gallop (Fig. 4) are different from those we obtained from large animals (Cavagna et al. 1977). In each stride, the front legs decelerate the animal causing its kinetic energy to fall, and the rear legs reaccelerate the animal, causing its kinetic energy to increase. In the larger animals, both front and back legs decelerated and immediately reaccelerated the animal. Also, there was a significant transfer between gravitational potential energy and kinetic energy during the low-speed gallops of the larger animals, but not in the chipmunk and ground squirrel (Fig. 2).

Mass specific powers (W kg⁻¹) are plotted as a function of average speed in Fig. 5. Mass specific energy changes per unit time obtained from the vertical forces, Ė_v/M_b, and horizontal forces, Ė_H/M_b, and the total energy changes of the centre of mass, Ė_CM,tot/M_b^are plotted separately in Fig. 5. The divisions into vertical and horizontal power terms are useful in evaluating the relative amount of energy required to account for the height and speed changes of the centre of mass.

During a walk (walks were obtained only for the two quails), the vertical and horizontal power increased with increasing walking speed and are approximately equal in magnitude. This allows the relatively large transfer between kinetic and gravitational potential energy observed in Fig. 2. This is similar to what was observed in large animals and man during a walk (Cavagna et al. 1977; Cavagna et al. 1976). Because the details of the transfer have been discussed in these papers, we will not repeat them here.

During a run or hop and a gallop, the vertical power remained nearly constant over the entire range of speed, while the horizontal power increased. The magnitude of the vertical power was much greater in both the large and small hopping animals than in the running and galloping animals (Cavagna et al. 1977).

There are two components of the mechanical power expended to lift and reaccelerate the centre of mass (as there were with oxygen consumption) : an extrapolated zero speed power (the Y -intercept) and an incremental power (the slope) (see equation 2). Both terms are constant for an individual animal because the relationship between Ė_CM,tot/M_b and speed is linear. Both terms are independent of body mass because the slope of the function relating each term of the equation to body mass is not significantly different from zero. Both the metabolic energy consumed (Ė_metab/M_b) and are plotted as a function of speed in Fig. 6. This figure clearly demonstrates that the relationship between Ė_CM,tot/M_b and speed changes dramatically with body size, while the relationship between Ė_CM,tot/M_b and speed does not.

Our data show that quail, like some large animals, utilize a pendulum-like energy conservation mechanism during a walk. Up to 70% of the energy is exchanged between kinetic and gravitational potential energy within a step. This is the same magnitude of exchange that was found in humans (Cavagna et al. 1976) and large animals (Cavagna et al. 1977). However, it was extremely difficult to obtain good walking records from small animals, and we were never able to obtain them from the chipmunks and ground squirrels. At slow speeds, small animals normally moved in a series of short bursts, alternating with stops, rather than at a constant speed.

During a run or hop small animals exhibit force and velocity patterns similar to those we observed for large animals. However, one major difference exists. At high speeds dogs, kangaroos and humans stored energy as elastic strain energy in the muscles and tendons when they landed and recovered some of this energy when they took off. This was demonstrated because the magnitude of energy changes of the centre of mass was greater than the metabolic energy consumed by the muscles (assuming muscles convert energy stored in carbohydrates, fats and proteins into work at a 25 % efficiency). In the small animals, however, one could account for all of Ė_CM,tot/M_b with muscular efficiencies of less than 25 %. Recent studies by Biewener, Alexander & Heglund (1981) show that the tendons of small kangaroo rats are relatively thicker than those of large kangaroos, and are too stiff to store large amounts of elastic energy. It seems possible, therefore, that large animals are able to utilize an elastic storage mechanism during a run or hop, but that small animals are not. This matter certainly merits more investigation.

The gallop of the small quadrupeds was quite different from the gallop of the large quadrupeds (Cavagna et al. 1977). The small animals landed on their front legs, decelerating the body, and then, after an interval (aerial phase), reaccelerated their body with their hind legs. In the larger animals, both front and hind limbs alternately decelerated and then reaccelerated the body during a stride. This means that elastic storage and recovery within the tendons and muscles of the limb would be possible for large animals but not for small animals. However, small animals did exhibit enormous spinal flexion during a gallop, and it might be possible for them to store energy elastically in the muscles and tendons of the back as the animal landed on its front limbs which could be recovered as it pushed off from its hind limbs. In addition, the large animals were capable of alternately storing and recovering significant) amounts of forward kinetic energy in gravitational potential energy at slow galloping speeds. Small animals do not appear to be able to utilize this energy-saving mechanism in a gallop.

Ė_CM,tot like Ė_metab increases nearly linearly with speed. Thus, Ė_CM,tot might provide an explanation of the linear increase in Ė_metab for individual animals if it were the major component of mechanical work performed by the animal’s muscles. However, at the highest speeds achieved by the large animals, the rate at which muscles and tendons must supply energy to accelerate the limbs and body relative to the centre of mass, Ė′_{KE. tot}, becomes equal to or greater than Ė_CM,tot and cannot be ignored (see Fedak, Heglund & Taylor, 1982). For this reason, consideration of the explanation of the linear increase in Ė_metab is left to the following paper when total mechanical energy is calculated. The relationship between Ė_CM,tot as a function of speed is nearly independent of body size as predicted by Alexander’s mathematical models of running (Alexander, 1977; Alexander, Jayes & Ker, 1980). The increase in amplitude of the oscillations in energy of the centre of mass during a step with increasing body size appears to be nearly exactly compensated by a decrease in step frequency. Thus Ė_{CM, tot}/M_b does not help to explain the 10 to 15-fold changes in Ė_metab/M_b with body size observed in the first paper of this series.

This work was supported by NSF grants PCM 75-22684 and PCM 78-23319, NRS training grant 5 T 32 GM 07117 and NIH post-doctoral fellowship 1 F32 AM 06022.