COLLISIONAL ACCOUNTING OF ENERGY IN HORSE GALLOPING ...

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Manoj Srinivasan, Andrew Dressel, Andy Ruina†, and John E.A. Bertram‡,. †The Biorobotics Laboratory, Cornell University, Ithaca, NY, USA,.
COLLISIONAL ACCOUNTING OF ENERGY IN HORSE GALLOPING Manoj Srinivasan, Andrew Dressel, Andy Ruina†, and John E.A. Bertram‡, †The Biorobotics Laboratory, Cornell University, Ithaca, NY, USA, http://www.tam.cornell.edu/~ruina/hplab, [email protected] ‡Dept. of Nutrition, Food & Exercise Sciences, Florida State University, Tallahassee, FL, USA, [email protected] INTRODUCTION Terrestrial legged locomotion requires repeated, generally dissipative, collisions with the ground. Any dissipation is necessarily an energetic cost. One mechanism useful for reducing collisional dissipation is through elasticity in the tendons, muscles and ligaments. We have found another. Whether or not the legs have some elasticity, sequencing leg collisions, as in the three beat (Lone Ranger) "pa-dadump" of a slow horse gallop, can substantially reduce collisional losses over a one-beat gait with the same speed and period. METHODS Start by considering a seemingly unrelated springless locomotion system, brachiation. The gaits used by some arm swinging apes (Fig. a; 1) allows the making and breaking of contact without collisional loss by a smooth matching of the flight velocity with the swing velocity. The second inspiration is purely mechanical, although based on an apparently a new mechanical discovery (5). A rolling ellipse (or ellipsoid) that rolls too fast will leap in the air. If the speed is just right it will leap and flip in a manner so that the landing is symmetric with the take-off and the contact is at zero relative velocity (Fig. b). Thus even with no elasticity there is no dissipation and the ellipse can passively ‘bounce’, recovering all potential energy in each cycle, but without the requirement of a spring. Numerical simulation of this model

confirms this motion. We have found that a horse seems to employ mechanics similar to these models in order to bounce better with incompletely elastic legs. In more detail, first consider a particle collision mediated by a single massless leg. Conservation of momentum orthogonal to the leg yields v0 = vi cos α, where vi and v0 are incoming and outgoing speeds and α is the deflection angle caused by the contact. The post-collisional energy is E0 = Ei cos2α and the fractional energy loss is E/Ei = sin2α α2 (for small α). By the same reasoning, a collision at a shallower angle (α/2) has a fractional energy loss of (α/2)2, a quarter of that for α. Thus with a given incoming velocity, two successive collisions of deflection α/2 together accomplish a net deflection of α but with only half the total energy loss of a single collision. More generally, taking the net interaction of an animal with the ground as a sequence of n such inelastic collisions, each collision has a deflection angle of α/n and v0 = vi cosn(α/n) so E0 = Ei cos2n(α/n) and fractional energy lost E/Ei α2/n which goes to zero as n → . That is, an infinite sequence of plastic collisions makes up a fully elastic collision. Taking 3 as an approximation of , we get the 3-collision model of horse galloping in Fig. c. A similar analysis can be performed using elastic legs but keeping track of the work in restitution. The net restitution cost goes to zero as the number of legs goes to infinity.

RESULTS AND DISCUSSION Following through this mechanics and using g=10m/s2, T=0.5s, v=7m/s2 from (6) we predict a (dimensionless) specific cost of transport of cs = (power)/mgv=gT/8v=0.1 for a pronk and cs = gT/8v=0.03 for a 3 beat canter. Assuming 25% muscular efficiency the VO2 data in (6) shows cs=0.05. Thus our one-beat pronk is too costly (0.1>0.05) and our model for a three beat canter accounts for 60% of the measured cost of locomotion (0.03/0.05=60%), leaving some energy to swing the legs around and so on. With the given oxygen consumption even more energy is available to the horse for various "internal work" if the collision has some elasticity. Note also that ground force orientations predicted from this model agree qualitatively with the observations of a cantering horse (Fig. c, d, 3) and differ markedly from previous models of smaller animal galloping where the landing of the forelegs cock a spring that is released for the later push-off of the legs (4). Note also that both this model and real horses have a maximum horizontal speed at mid-stance, the opposite of that predicted by a "pogostick" model where the minimum horizontal speed is mid-stance. These arguments are made more precise by numerical optimization of a simple model of a horse consisting of a point mass with elastic legs and a metabolic cost proportional to the restitution work done by the legs at each collision. Fitting this model to kinematic data of a horse gives a good prediction of the measured metabolic cost. Given that the horse can do a maximum of 3 footfalls per stride, the strategy that minimizes this metabolic cost is the one that uses 3 footfalls per stride as opposed to one (pronk) or two (trot). This timing of the

footfalls and the metabolic cost at the optimum are well correlated to experimental data (6).

a) brachiation, b) bouncing ellipse, c) the model, d) horse. REFERENCES and FOOTNOTES 1. Bertram, J.E.A., Ruina, A., Cannon, C.E., Chang, Y-H., Coleman, M.J. (1999) Journal of Exp. Biol., 202, 2609-17. 2. Minetti, A.E. (1998) Proc. R. Soc. B 265, 1227-35. 3. Merkens, H.W., Schamhardt, H.C., van Osch, G.J., Hartman, W. (1993) Am. J. Vet. Res. 54, 670-74. 4. Alexander, R.McN. (1993) Am. Zool. 28, 237-45. 5. Haggerty, P, (2001) PhD thesis, U of Michigan independently constructed a similar model. 6. Minetti, A.E., Ardigo, L. P., Reinach, E. Saibene, F. (1999), J Exp Bio, 202, 23292338,