A novel approach to generating optimal smooth piecewise trajectories based o n a new energy memure i s 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 a n interval of time. A numerical iterative procedure is devised for computing the optimal piecewise trajectory as a solution of a constmined boundary value problem. The resulting trajectories are not only smooth but also safe wtth 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 wmputer graphics and geometric design.