Departmental Papers (ESE)

Document Type

Working Paper

Subject Area


Date of this Version

October 2004


Reprinted from The International Journal of Robotics Research, Volume 23, Issue 10-11, October-November 2004, pages 979-999.
DOI: 10.1177/0278364904047389


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 threedegrees- 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.


legged locomotion, hybrid system, return map, spring loaded inverted pendulum, stability, time-reversal, symmetry



Date Posted: 30 June 2008

This document has been peer reviewed.