Winning a Won Game: Strict Reach-Avoid-Stay Control Barrier Functions for High-Dimensional Black-Box Systems

Abstract

Robots must complete their tasks and maintain the achieved outcomes while avoiding safety failures at all times. Strict reach-avoid-stay (sRAS) formalizes this requirement: safely reaching a target and remaining there indefinitely after first entry. We propose an sRAS Q-control barrier function (CBF) safety filter for high-dimensional black-box systems under bounded uncertainty. Our construction combines a stay value encoding safe permanent residence in a target subset with a reach-avoid value encoding safe reachability of this subset while avoiding target states from which safe permanent residence cannot be guaranteed. We prove that these values jointly yield a valid robust discrete-time CBF and lift them to state-action Q-functions for runtime intervention. For exact values and under a measure-zero condition, our filter preserves sRAS feasibility from almost every winnable initial state and keeps the system safely within the target after first entry, against all admissible uncertainty realizations. We adopt reachability-based adversarial reinforcement learning for scalable value approximation using only black-box interactions. Notably, neither synthesis nor deployment of our filter requires known dynamics, affine structure, value derivatives, or hand-designed barriers. We validate our framework in quadruped gap jumping in simulation and hardware, where the robot crosses the gap, lands safely, and remains safe afterward. Simulated F1TENTH races further demonstrate safe overtaking and lead retention.