  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.
