Computation of Regions of Attraction for Hybrid Limit Cycles Using Reachability: An Application to Walking Robots

TL;DR

This paper introduces a Hamilton-Jacobi reachability method to compute regions of attraction for hybrid systems like walking robots, effectively handling discontinuities, nonlinearities, and disturbances, and providing stabilizing controllers.

Contribution

It generalizes HJ reachability to include hybrid systems with state resets, enabling accurate region-of-attraction computation for complex robotic systems.

Findings

Larger regions of attraction compared to existing methods

Effective handling of state resets and disturbances

Provides stabilizing controllers leveraging resets

Abstract

Contact-rich robotic systems, such as legged robots and manipulators, are often represented as hybrid systems. However, the stability analysis and region-of-attraction computation for these systems are often challenging because of the discontinuous state changes upon contact (also referred to as state resets). In this work, we cast the computation of region-ofattraction as a Hamilton-Jacobi (HJ) reachability problem. This enables us to leverage HJ reachability tools that are compatible with general nonlinear system dynamics, and can formally deal with state and input constraints as well as bounded disturbances. Our main contribution is the generalization of HJ reachability framework to account for the discontinuous state changes originating from state resets, which has remained a challenge until now. We apply our approach for computing region-of-attractions for several underactuated walking robots and demonstrate that the proposed approach can (a) recover a bigger region-of-attraction than state-of-the-art approaches, (b) handle state resets, nonlinear dynamics, external disturbances, and input constraints, and (c) also provides a stabilizing controller for the system that can leverage the state resets for enhancing system stability.