Stochastic Modeling and Optimal Control for Automated Overtaking

This paper proposes a solution to the overtaking problem where an automated vehicle tries to overtake a human-driven vehicle, which may not be moving at a constant velocity. Using reachability theory, we first provide a robust time-optimal control algorithm to guarantee that there is no collision throughout the overtaking process. Following the robust formulation, we provide a stochastic reachability formulation that allows a trade-off between the conservative overtaking time and the allowance of a small collision probability. To capture the stochasticity of a human driver's behavior, we propose a new martingale-based model where we classify the human driver as aggressive or nonaggressive. We show that if the human driver is nonaggressive, our stochastic time-optimal control algorithm can provide a shorter overtaking time than our robust algorithm, whereas if the human driver is aggressive, the stochastic algorithm will act on a collision probability of zero, which will match the robust algorithm. Finally, we detail a simulated example that illustrates the effectiveness of the proposed algorithms.