Abstract

We present a new stability analysis for hybrid legged locomotion systems based on the “symmetric” factorization of return maps. We apply this analysis to 2 and 3 degree of freedom (DOF) models of the Spring Loaded Inverted Pendulum (SLIP) with different leg recirculation strategies. Despite the non-integrability of the SLIP dynamics, we obtain a necessary condition for asymptotic stability (and a sufficient condition for instability) at a fixed point, formulated as an exact algebraic expression in the physical parameters. We use this expression to study a variety of 2 DOF SLIP models that have previously been posited as low dimensional representations of running, focusing on the sensory “cost” required to achieve “fast ” transients as measured by the degree of singularity of the linearized dynamics. We introduce a new 3 DOF SLIP model with pitching dynamics whose stability properties, revealed by this analysis, provide for the first time the beginnings of a formal explanation for the surprisingly stable gaits of the open loop controlled robot, RHex.