Human-machine shared control in critical collision scenarios aims to aid drivers' accident avoidance through intervening only when necessary. Existing methods count on replanning collision-free trajectories and imposing human-machine tracking, which usually interrupts the driver's intent and increases the risk of conflict. Additionally, the lack of guaranteed trajectory feasibility under extreme conditions can compromise safety and reliability. This paper introduces a Reachability-Aware Reinforcement Learning framework for shared control, guided by Hamilton-Jacobi (HJ) reachability analysis. Machine intervention is activated only when the vehicle approaches the Collision Avoidance Reachable Set (CARS), which represents states where collision is unavoidable. First, we precompute the reachability distributions and the CARS by solving the Bellman equation using offline data. To reduce human-machine conflicts, we develop a driver model for sudden obstacles and propose an authority allocation strategy considering key collision avoidance features. Finally, we train a reinforcement learning agent to reduce human-machine conflicts while enforcing the hard constraint of avoiding entry into the CARS. The proposed method was tested on a real vehicle platform. Results show that the controller intervenes effectively near CARS to prevent collisions while maintaining improved original driving task performance. Robustness analysis further supports its flexibility across different driver attributes.

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