Differentiable Invariant Sets for Hybrid Limit Cycles with Application to Legged Robots

TL;DR

This paper introduces a method to compute and verify invariant sets around hybrid limit cycles in legged robots, enhancing robustness analysis and controller design for contact-rich systems.

Contribution

It extends parametric reachable set methods to hybrid systems, enabling formal verification of invariant sets and improved control strategies for legged robots.

Findings

Successfully verified invariant sets for a bipedal robot model

Maximized invariant set size through bi-level optimization

Implemented the approach using a JAX-based library for efficient computation

Abstract

For hybrid systems exhibiting periodic behavior, analyzing the invariant set containing the limit cycle is a natural way to study the robustness of the closed-loop system. However, computing these sets can be computationally expensive, especially when applied to contact-rich cyber-physical systems such as legged robots. In this work, we extend existing methods for overapproximating reachable sets of continuous systems using parametric embeddings to compute a forward-invariant set around the nominal trajectory of a simplified model of a bipedal robot. Our three-step approach (i) computes an overapproximating reachable set around the nominal continuous flow, (ii) catalogs intersections with the guard surface, and (iii) passes these intersections through the reset map. If the overapproximated reachable set after one step is a strict subset of the initial set, we formally verify a forward invariant set for this hybrid periodic orbit. We verify this condition on the bipedal walker model numerically using immrax, a JAX-based library for parametric reachable set computation, and use it within a bi-level optimization framework to design a tracking controller that maximizes the size of the invariant set.