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Stability Analysis of Legged Locomotion Models by Symmetry-Factored Return Maps

Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA

Daniel E. Koditschek

Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109, USA

Philip Holmes

Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA

We present a new stability analysis for hybrid legged locomotion systems based on the "symmetric" factorization of return maps. We apply this analysis to two-degrees-of-freedom (2DoF) and three-degrees-of-freedom (3DoF) 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 characterize analytically the sensory cost and stabilizing benefit of various feedback schemes previously proposed for the 2DoF SLIP model, posited as a low-dimensional representation of running. We apply the result as well to a 3DoF SLIP model that will be treated at greater length in a companion paper as a descriptive model for the robot RHex.

Key Words: legged locomotion • hybrid system • return map • spring loaded inverted pendulum • stability • time-reversal symmetry

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