Unfinished Business: Impulsive Models of Quadrupedal Running Gaits

Abstract

Ken Waldron has significantly advanced the understanding of the dynamics of high-speed quadrupedal locomotion through his research work in both modeling and experimentation. This paper revisits an impulsive model of quadrupedal running gaits that Waldron developed and seeks to find all feasible steady-state gait solutions for it. Prior work had reported only single solutions to the nonlinear systems of equations defining each gait. Using the Bertini software to implement a homotopy continuation method, all solutions were found for the trot, pace, bound, half-bound, and canter gaits of a biologically sized quadruped moving with a fixed stride period in the presence of drag. New solutions were identified for the trot and pace, and differences from previously reported solutions for the bound and halfbound were found. The approach has not yet been successful in comprehensively solving the transverse and rotary gallop systems of equations, so that remains a topic of ongoing research. In general, however, surprisingly few physically meaningful solutions were found for any gait despite the large numbers of possible solutions to consider.