On smooth and safe trajectory planning in 2D environments

A novel approach to generating optimal smooth piecewise trajectories based on a new energy measure is proposed. Given the configurations (position and direction) of two points in the plane, we search for the minimal energy trajectory that minimizes the integral of the squared acceleration opposed to curvature, which has been the predominant energy measure studied in the literature. The smoothness of the optimal trajectory depends on how the tangential and normal components of acceleration vary over an interval of time. A numerical iterative procedure is devised for computing the optimal piecewise trajectory as a solution of a constrained boundary value problem. The resulting trajectories are not only smooth but also safe with optimal velocity (acceleration) profiles and therefore suitable for robot motion planning applications. The feasibility of the proposed approach is illustrated by several simulation examples. Besides motion planning, the resulting trajectories may be useful in computer graphics and geometric design