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

2026-04-06

AI Summary

This work addresses the high computational cost of traditional invariant set computation for hybrid systems with limit cycles, such as legged robots. The authors propose a three-step method that computes an upper bound of the continuous-flow reachable set around a nominal trajectory, records its intersection with switching surfaces, and propagates this set through the reset map to formally verify forward invariance by checking whether the one-step reachable set is strictly contained within the initial set. Innovatively, they extend parametric embedding—a differentiable invariant set computation technique—to hybrid limit-cycle systems for the first time and integrate it within a bilevel optimization framework to synthesize controllers that maximize the invariant set volume. Implemented in the JAX-based immrax library, the approach demonstrates significant improvements in closed-loop robustness on a simplified bipedal walking model.

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.

Problem

• hybrid systems
• limit cycles
• invariant sets
• legged robots
• robustness

Innovation

• differentiable invariant sets
• hybrid limit cycles
• parametric reachable sets
• legged robots
• forward-invariant verification