  Contact-rich robotic systems, such as legged robots and manipulators, are
often represented as hybrid systems. However, the stability analysis and
region-of-attraction computation for these systems are often challenging
because of the discontinuous state changes upon contact (also referred to as
state resets). In this work, we cast the computation of region-ofattraction as
a Hamilton-Jacobi (HJ) reachability problem. This enables us to leverage HJ
reachability tools that are compatible with general nonlinear system dynamics,
and can formally deal with state and input constraints as well as bounded
disturbances. Our main contribution is the generalization of HJ reachability
framework to account for the discontinuous state changes originating from state
resets, which has remained a challenge until now. We apply our approach for
computing region-of-attractions for several underactuated walking robots and
demonstrate that the proposed approach can (a) recover a bigger
region-of-attraction than state-of-the-art approaches, (b) handle state resets,
nonlinear dynamics, external disturbances, and input constraints, and (c) also
provides a stabilizing controller for the system that can leverage the state
resets for enhancing system stability.
