Forward Trajectory Steering for Hamilton-Jacobi Reachability Analysis

Emergency collision avoidance under extreme driving conditions demands safety-critical control that accounts for both obstacle proximity and vehicle dynamic stability over a future time horizon, yet existing methods often rely on instantaneous or local safety evaluations. This paper proposes a safe reinforcement learning framework guided by a Hamilton-Jacobi (HJ) reachability based motion safety set that provides forward-looking safety supervision for constrained policy optimization. Specifically, a unified signed safety function is formulated by combining geometric collision margins and chassis stability limits, and is then extended through reachability analysis into a finite-horizon motion safety set that characterizes whether safety can be maintained under future vehicle state evolution. To enable practical computation, the motion safety set is approximated from offline extreme driving data, mitigating the computational burden of grid-based HJ solvers. The learned motion safety set is then embedded as a continuous safety cost into a constrained Markov decision process, and a PID-Lagrangian policy optimization scheme is employed to adaptively regulate the Lagrange multiplier for safety constraint enforcement. Simulation and real-vehicle experiments on low-adhesion obstacle-avoidance scenarios demonstrate that the proposed method achieves higher goal-reaching rates, produces smoother avoidance maneuvers, and maintains larger unified safety margins than baseline methods.

Autonomous navigation requires planning to reach a goal safely and efficiently in complex and potentially dynamic environments. Graph search-based algorithms are widely adopted due to their generality and theoretical guarantees when equipped with admissible heuristics. However, the computational complexity of graph search grows rapidly with the dimensionality of the search space, often making real-time planning in dynamic environments intractable. In this paper, we combine offline Hamilton-Jacobi (HJ) reachability with online graph search to leverage the complementary strengths of both. Precomputed HJ value functions, used as informative heuristics and proactive safety constraints, amortize online computation of the graph search process. At the same time, graph search enables reachability-based reasoning to be incorporated into online planning, overcoming the long-standing challenge of HJ reachability requiring full knowledge of the environment. Extensive simulation studies and real-world experiments demonstrate that the proposed approach consistently outperforms baseline methods in terms of planning efficiency and navigation safety, in environments with and without human presence.

Neural network (NN) dynamics models and control policies achieve strong performance in robotics, but providing sound guarantees under uncertainty remains difficult, especially for closed-loop NN systems. Existing reachability tools provide formal over-approximations, yet are often non-differentiable, overly conservative, or too slow for modern learning and online planning pipelines. To address this, we present a parallelizable, differentiable reachability framework in JAX for continuous- and discrete-time systems with analytical and NN-based dynamics and controllers. Our framework combines Taylor-model flowpipe construction with CROWN-style linear bound propagation through a unified representation that preserves affine dependencies while supporting GPU-batched computation and automatic differentiation. Building on this reachability primitive, we develop (i) a certified training method that encourages reachability-friendly dynamics models and controllers, and (ii) a reachability-aware sampling-based MPC scheme with gradient-based refinement. Experiments on non-prehensile manipulation and quadrotor tasks, including hardware and higher-dimensional evaluations (up to 72D), demonstrate practical online planning while maintaining certified reachable-set over-approximations under bounded uncertainty.