Autonomous Vehicle Overtaking Trajectory Based on Cubic Bézier Spirals: Analysis Under Multiple Physical Constraints

This paper is concerned with the overtaking problem for two autonomous vehicles. Note that the existing related works mainly focus on providing trajectory planning methods for overtaking vehicle. However, to achieve overtaking along the trajectory, what initial conditions does the vehicle need to satisfy has not been fully investigated. The challenge is that the initial conditions for the vehicle depend on multiple physical constraints in the trajectory planning, which are often difficult to be analytically solved. To overcome this difficulty, the proposed overtaking trajectory is decomposed into several cubic Bézier spiral segments firstly. Then, according to the decomposition, the multiple constraints with respect to the overtaking trajectory are simplified, based on which the solvability of the constraints is analyzed. Specifically, from the analysis, it is shown that if the initial linear velocity of the vehicle is less than $5/6$ times of the maximum linear velocity, then the multiple constraints can be solved, which implies that the vehicle can achieve overtaking along the proposed trajectory. Moreover, to avoid collision of two vehicles, a safety initial distance between two vehicles is determined. In addition, a control algorithm for the overtaking vehicle is given, which enables the overtaking vehicle to drive along the proposed overtaking trajectory. Finally, the theoretical results are verified through the simulations and experiments.