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An inelastic quadrupedal model discovers four-beat walking, two-beat running, and pseudo-elastic actuation as energetically optimal

Fig 3

Empirical forelimb GRF (black) compared to model predictions for a range of force-rate penalty coefficients ().

(A) At lower , the solution becomes more impulsive, as expected from a work-minimizing “bang-coast-bang” solution [23]. As increases, the force peaks become more shallow, but in all cases the optimal solution maintains the double-hump profile characteristic of walking bipeds and quadrupeds under most situations. (B) The actuation is seemingly elastic, as indicated by the negative linear relationship between force and leg length change, for small changes in leg length from resting. However, there are no springs in the model. The “pseudo-elastic” actuation occurs because of the compromise between work and force rate. High force rate penalties lead to only small deviations from pseudo-elastic actuation and low apparent stiffness (k′), and small force rate penalties deviate substantially from pseudo-elasticity with high apparent stiffness. The “hook” in the force-length curve occurs when the leg shifts to stance, modulating the force to ∼0.5 mg. Morphological data for the simulations, and empirical ground reaction forces, are derived from [37] for a Dalmation (see also Fig 5B).

doi: https://doi.org/10.1371/journal.pcbi.1007444.g003