Enforcing Temporal and Spatial Separation Constraints in Multi-Vehicle Trajectory Generation Problems Using a Bernstein Relaxation and Refinement Method

Satisfying collision-avoidance constraints at all time instances along vehicles' trajectories is crucial to the success of a multi-vehicle mission. A common approach to handling collision-avoidance constraints in a trajectory generation problem is to check the constraints on some discrete points in time (or space). This approach, while being straightforward, cannot always guarantee that the generated trajectories are collision-free in between the points. On the other hand, most approaches for ensuring collision avoidance at all times can get overly conservative or computationally expensive. Furthermore, with these approaches, spatial deconfliction between trajectories can be very difficult, if not impossible, to enforce. In this letter, we parameterize trajectories with Bézier curves and leverage the unique properties of these curves to propose a Bernstein relaxation and refinement method for evaluating temporal and spatial separation constraints in multi-vehicle trajectory generation problems. The proposed method can guarantee inter-vehicle collision avoidance at all times, while allowing for a flexible trade-off between the conservatism and the computational complexity of generating trajectories.