An Input-to-State Stability Perspective on Robust Locomotion

TL;DR

This paper introduces a new robustness measure for bipedal robot stability on uneven terrain, using input-to-state stability theory to certify and verify stable walking despite disturbances.

Contribution

It proposes a novel $\,\delta$-robustness definition and develops Lyapunov-based tools to certify stability of periodic orbits under terrain uncertainties.

Findings

Validated in simulation with a bipedal robot on uneven terrain

Formulated an optimization framework for robustness verification

Established Lyapunov functions certifying $\,\delta$-robustness

Abstract

Uneven terrain necessarily transforms periodic walking into a non-periodic motion. As such, traditional stability analysis tools no longer adequately capture the ability of a bipedal robot to locomote in the presence of such disturbances. This motivates the need for analytical tools aimed at generalized notions of stability -- robustness. Towards this, we propose a novel definition of robustness, termed \emph{$\delta$-robustness}, to characterize the domain on which a nominal periodic orbit remains stable despite uncertain terrain. This definition is derived by treating perturbations in ground height as disturbances in the context of the input-to-state-stability (ISS) of the extended Poincar\'{e} map associated with a periodic orbit. The main theoretic result is the formulation of robust Lyapunov functions that certify $\delta$-robustness of periodic orbits. This yields an optimization framework for verifying $\delta$-robustness, which is demonstrated in simulation with a bipedal robot walking on uneven terrain.