Bernstein polynomial-based transcription method for solving optimal trajectory generation problems

This paper presents a method and an open-source implementation, Bernstein/B\'ezier Optimal Trajectories (BeBOT), for the generation of trajectories for autonomous system operations. The proposed method is based on infinite dimensional optimal control formulations of trajectory generation problems. By approximating the trajectories using Bernstein polynomials, these problems can be transcribed as nonlinear programming problems, which can then be solved using off-the-shelf solvers. Bernstein polynomials possess favorable geometric properties that enable the trajectory planner to efficiently evaluate and enforce constraints along the vehicles' trajectories, including maximum speed and angular rates, minimum distance between trajectories and between the vehicles and obstacles. By virtue of these properties, feasibility and safety constraints typically imposed in autonomous vehicle operations can be enforced and guaranteed independently on the order of the polynomials. Thus, the trajectory generation algorithm can efficiently generate feasible and collision-free trajectories, and can be deployed for real-time safety critical applications in complex environments and for multiple vehicle missions.