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.
Poincare return maps are a fundamental tool for analyzing periodic orbits in hybrid dynamical systems, including legged locomotion, power electronics, and other cyber-physical systems with switching behavior. The Poincare return map captures the evolution of the hybrid system on a guard surface, reducing the stability analysis of a periodic orbit to that of a discrete-time system. While linearization provides local stability information, assessing robustness to disturbances requires identifying invariant sets of the state space under the return dynamics. However, computing such invariant sets is computationally difficult, especially when system dynamics are only available through forward simulation. In this work, we propose an algorithmic framework leveraging sampling-based optimization to compute a finite-step invariant ellipsoid around a nominal periodic orbit using sampled evaluations of the return map. The resulting solution is accompanied by probabilistic guarantees on finite-step invariance satisfying a user-defined accuracy threshold. We demonstrate the approach on two low-dimensional systems and a compass-gait walking model.
Varun Madabushi, Elizabeth Dietrich, Hanna Krasowski +1
Georgia Institute of Technology · University of California, Berkeley
Hybrid dynamical systems provide a powerful modeling framework for robotic systems, particularly in contact-rich environments. However, ensuring safety and performance in such systems remains challenging due to the intricate coupling between continuous dynamics and discrete mode transitions. In this work, we extend classical Hamilton-Jacobi (HJ) reachability analysis, a formal verification method for continuous-time nonlinear systems, to hybrid dynamical systems. Our framework characterizes safe sets for hybrid systems through a generalized value function defined over both discrete and continuous states while accounting for control constraints and model uncertainty. We additionally provide a numerical algorithm to compute this value function. Building on these safe sets, we propose two different mechanisms to integrate performance objectives. First, we introduce a hybrid least-restrictive safety filter that intervenes on both the discrete and continuous components of a nominal controller only when necessary to avoid unsafe states, thereby preserving nominal behavior whenever possible. Second, we formulate and compute hybrid backward reach-avoid tubes, enabling the simultaneous enforcement of safety and goal-reaching behavior, an extension not previously addressed within hybrid HJ reachability. This enables the synthesis of continuous and discrete control policies that guarantee both safety and task completion. We validate our framework through simulation studies and real-world experiments on a quadrupedal robot, demonstrating its effectiveness in hybrid mode planning and safety-critical applications.
Javier Borquez, Shuang Peng, Somil Bansal
Universidad de Santiago de Chile · University of Southern California · Stanford University
Controlling a hybrid system to a target set under parameter uncertainty can require informative actions that temporarily drive the system state away from the specified set. We propose a belief-informed dual-control algorithm that combines belief-space receding-horizon selection with an expected-decrease constraint on a nonnegative target-set progress function. To accommodate exploratory deviations, a scalar controller state bounds the constraint's cumulative slack. Our algorithm's two-step lookahead selection uses predicted observations to update the parameter belief before evaluating the subsequent action's admissibility. Under distance-comparison bounds, correct conditional prediction, and recursive feasibility, we prove almost-sure target-set convergence at decision times and bound both the sum of expected progress-function values and the expected neighborhood-entry time. Upper confidence bounds extend these convergence guarantees to bounded model samples with summable error probabilities. In planar regulation with an unknown control direction, we verify recursive feasibility: the proposed two-step selection reduces the Euclidean state norm below 0.01 within 11 decisions for either sign, whereas a myopic one-step selection loses admissibility immediately after zero input. In a simulated bimanual assembly task, our algorithm's one-step implementation uses clearance feedback to complete the assembly under nominal friction, yielding a 96.4% lower infinity-norm relative-position error than the reference-tracking baseline at the method's completion time. At lower friction, its posterior conditioning successfully detects an empty admissible set, whereas a fixed-prior alternative admits an action that violates the conditional decrease constraint. Project page: https://clintonenwerem.com/belief-hybrid-control/.
Clinton Enwerem, Saleh Kemal, John S. Baras +1
Institute for Systems Research, University of Maryland, College Park, MD, USA