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This paper introduces a method for efficiently computing forward-invariant sets for hybrid systems with limit cycles, specifically applied to legged robots. They extend parametric embedding techniques for reachable set computation to handle hybrid dynamics, including guard surface intersections and reset maps. The method is demonstrated on a bipedal walker model, integrated with a bi-level optimization to design controllers that maximize the size of the invariant set, thereby improving robustness.
Guaranteeing robustness for legged robots just got easier: this work provides a computationally tractable method to compute invariant sets around periodic orbits, enabling controller design that explicitly maximizes stability margins.
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.