  Hybrid dynamical systems with non-linear dynamics are one of the most general
modeling tools for representing robotic systems, especially contact-rich
systems. However, providing guarantees regarding the safety or performance of
such hybrid systems can still prove to be a challenging problem because it
requires simultaneous reasoning about continuous state evolution and discrete
mode switching. In this work, we address this problem by extending classical
Hamilton-Jacobi (HJ) reachability analysis, a formal verification method for
continuous non-linear dynamics in the presence of bounded inputs and
disturbances, to hybrid dynamical systems. Our framework can compute reachable
sets for hybrid systems consisting of multiple discrete modes, each with its
own set of non-linear continuous dynamics, discrete transitions that can be
directly commanded or forced by a discrete control input, while still
accounting for control bounds and adversarial disturbances in the state
evolution. Along with the reachable set, the proposed framework also provides
an optimal continuous and discrete controller to ensure system safety. We
demonstrate our framework in simulation on an aircraft collision avoidance
problem, as well as on a real-world testbed to solve the optimal mode planning
problem for a quadruped with multiple gaits.

Robotics;Systems and Control;Systems and Control