Hamilton–Jacobi (HJ) reachability-based filtering provides a powerful framework to co-optimize performance and safety (or liveness) for autonomous systems. Under this filtering scheme, a nominal controller is minimally modified to ensure system safety or liveness. However, the resulting controllers can exhibit abrupt switching and bang-bang behavior, which is not suitable for applications of autonomous systems in the real world. This work presents a novel, unifying framework to design safety and liveness filters through reachability analysis. We explicitly characterize the maximal set of control inputs that ensures safety (or liveness) at a given state. Different safety filters can then be constructed using different subsets of this maximal set along with a projection operator to modify the nominal controller. We use the proposed framework to design three safety filters, each balancing performance, computation time, and smoothness differently. We highlight their relative strengths and limitations by applying these filters to autonomous navigation and rocket landing scenarios and on a physical robot testbed. We also discuss practical aspects associated with implementing these filters on real-world autonomous systems. Our research advances the understanding and potential application of reachability-based controllers on real-world autonomous systems.

Hamilton-Jacobi (HJ) reachability analysis is a powerful tool for analyzing the safety of autonomous systems. However, the provided safety assurances are often predicated on the assumption that once deployed, the system or its environment does not evolve. Online, however, an autonomous system might experience changes in system dynamics, control authority, external disturbances, and/or the surrounding environment, requiring updated safety assurances. Rather than restarting the safety analysis from scratch, which can be timeconsuming and often intractable to perform online, we propose to compute parameter-conditioned reachable sets. Assuming expected system and environment changes can be parameterized, we treat these parameters as virtual states in the system and leverage recent advances in high-dimensional reachability analysis to solve the corresponding reachability problem offline. This results in a family of reachable sets that is parameterized by the environment and system factors. Online, as these factors change, the system can simply query the corresponding safety function from this family to ensure system safety, enabling a real-time update of the safety assurances. Through various simulation studies, we demonstrate the capability of our approach in maintaining system safety despite the system and environment evolution.