Optimal Motion Planning for Differentially Flat Systems Using Bernstein Approximation

This letter presents a computational framework to efficiently generate feasible and optimal trajectories for differentially flat autonomous vehicle systems. We formulate the optimal motion planning problem as a continuous-time optimal control problem, and approximate it by a discrete-time formulation using Bernstein polynomials. These polynomials allow for efficient computation of various constraints along the entire trajectory, and are particularly convenient for generating trajectories for safe operation of multiple vehicles in complex environments. The advantages of the proposed method are investigated through theoretical analysis and numerical examples.