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.
Figures & tables
Fig. 1 : A visualization of the conditions supposed in Theorem 1 . While the initial set can start with points in {h<0} , we assume there exists a time T∈[0,T) where the normotope is contained in {h>0} . Before T , trajectories may intersect {h=0} as long as {h˙>0} , avoiding impact with the switching surface S={h=0}∩{h˙<0} . By assumption, any intersection with {h=0} from T to T have {h˙<0} , guaranteeing they belong to the guard surface S . Finally, we continue flowing the normotope embedding until the normotope is contained in {h<0} to ensure every point resets.
Fig. 2 : The slicing and bounding procedure used in Lemma 1 and Theorem 2 is pictured. Left : the ℓ2 -normotope at time t is sliced with the subspace {h=0}={Bz+x˚(T)} to obtain a (n−1) -dimensional ℓ2 -normotope expressed in the basis B (Lemma 1 ). Right : The gain γ is bounded using (i) the distance from x∗ to the reset center of the sliced normotope ΔB(z˚(t)) measured in the initial α0 norm, and (ii) the maximum induced matrix norm of a linear inclusion bounding ΔB(z)−ΔB(z˚(t)) to obtain a radius in the α0 norm.
Fig. 3 : Simplified Bipedal Walker illustrated for the (a) continuous domain, (b) pre-impact, and (c) post-impact
Fig. 4 : Stable periodic limit cycle for the walker obtained from trajectory optimization.
Algorithm 1 Reachability-based Control Design Procedure
Fig. 5 : Slices of the computed Hybrid Invariant Tube with no tracking control (above) and tracking control (below). The blue dotted line represents the guard surface S . The green portions of the tube are the ones for which the tube intersects the guard surface.