Hamilton-Jacobi Reachability-Based Safe Reinforcement Learning for Emergency Collision Avoidance

Robots deployed in safety-critical environments must execute motions that are simultaneously robust to disturbances and provably safe from collisions. Dynamic Movement Primitives (DMPs) offer inherent stability, temporal flexibility, and efficient trajectory generalization from single demonstrations, but they lack formal safety certificates. Conversely, Hamilton-Jacobi (HJ) Reachability analysis provides a principled framework for computing worst-case safety margins and forward-invariant safe sets, but classical grid-based methods suffer from the curse of dimensionality and are impractical for real-time control. This paper introduces HJ-SafeDMP, a framework that integrates DMPs with learned HJ Reachability-based safety value functions to achieve provably safe, robust, and computationally efficient robot motion. We learn a Control Barrier Value Function (CBVF) from offline demonstration data using a model-free, finite-difference HJ recursion and deploy it as a real-time safety filter via a closed-form control law that modulates the DMP output. Unlike optimization-based CBF-QP approaches, our method achieves safety filtering without online quadratic program solves, preserving the computational efficiency of DMPs. We further incorporate an expectile-based offline learning objective that avoids querying out-of-distribution actions, and a conformal prediction calibration step that provides finite-sample probabilistic safety coverage. Experimental evaluation on a 7-DOF robot manipulator demonstrates that HJ-SafeDMP achieves formal safety guarantees with orders-of-magnitude faster execution than optimization-based baselines, while maintaining the robustness and adaptability of DMPs for human-robot interaction.

We introduce $λ$-Reachability, a scalable approach to Hamilton--Jacobi safety analysis for high-dimensional robotic systems. Unlike prior discounted formulations that rely on fixed one-step Bellman updates, $λ$-Reachability employs a stochastic multi-step estimator of the safety value, using a geometrically distributed rollout horizon together with a randomly absorbed terminal. Conceptually analogous to TD($λ$), $λ$-Reachability interpolates between local self-consistency updates and long-horizon max-over-trajectory safety targets via an interpretable horizon-control parameter. Unlike TD($λ$), where the terminal value is always incorporated in learning targets, the terminal safety value in $λ$-Reachability is only used at a probability controlled by parameter $δ$. We formally show that for $δ<1$, the update induces a contraction mapping that allows temporal-difference learning; as $λ\to 1$, the estimator recovers the undiscounted reachability objective. We apply $λ$-Reachability to high-dimensional safety learning problems with both simulated and real humanoid robots under balance and collision avoidance constraints. Experimental results demonstrate that $λ$-Reachability significantly improves both safe-set boundary classification and safety margin estimation compared to single-step temporal-difference baselines.

Hamilton-Jacobi (HJ) reachability provides a mathematically rigorous framework for safe control of dynamical systems, but its practical application is bottlenecked by the computational complexity of solving Hamilton-Jacobi-Isaacs variational inequality PDEs in high dimensions. Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to classical mesh-based solvers, yet their performance is highly sensitive to the choice of collocation sampling. In order to learn accurate safety value functions, existing PINNs-based HJ reachability solvers must rely on complex training pipelines and auxiliary supervision. In this work, we propose STEER2REACH (S2R), a PINNs-based HJ reachability solver that requires minimal modification on top of standard PINNs training. S2R's key contribution is a lightweight, low-overhead adaptive collocation sampling distribution constructed by steering forward trajectories using a combination of the optimal control and disturbance signals induced by the current value function, with injected stochastic exploration noise. We demonstrate that despite its simplicity, S2R achieves competitive--and in some cases improved--performance on safety metrics while reducing relative L2 error across a range of reachability benchmarks compared with SoTA MPC-guided HJ reachability solvers, all without requiring multi-stage training or MPC-based supervision.