Differentiable Invariant Sets for Hybrid Limit Cycles with Application to Legged Robots
This work addresses robustness analysis for contact-rich cyber-physical systems like legged robots, but it is incremental as it builds on existing reachable set methods.
The authors tackled the problem of computationally expensive invariant set computation for hybrid systems with periodic behavior, specifically for legged robots, by extending parametric embedding methods to compute forward-invariant sets around nominal trajectories, resulting in a verified invariant set for a bipedal walker model and a controller that maximizes its size.
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.