Pinned equilibria provide robustly stable multilegged locomotion

I. MOTIVATION Models of terrestrial legged locomotion are almost invariably hybrid, containing a discontinuity in the equations of motion when feet land on the ground. While some animals are bipedal, the great majority of legged locomotion on land is done with 4, 6 or more legs. Poly-pedal gaits for moderate or high-speed locomotion usually include multiple legs striking the ground that once (or nearly at once, on average). Familiar examples include trotting and pacing in mammalian quadrupeds, and the ubiquitous alternating tripod gaits in the insect world. We will present several arguments for why such simultaneous touchdowns should not be observed in nature, leading to the conclusion that observing them requires that a unique form of robust stability is associated with such behaviors. We proceed to present a mathematical formulation of such a novel form of stability, demonstrate sufficient conditions for its existence, and show that it can appear in fairly pedestrian models of locomotion.