Generalized polar polynomials for vehicle path generation with dynamic constraints

The paths generated by a high-level planner consist of sequences of positions and headings, i.e. configurations, that need a proper interpolation procedure for providing continuous controls of the vehicles. In the case of motions at constant speed, Nelson (1989) has proposed continuous-curvature paths using polar polynomials for describing the turns between two straight line segments. The corresponding curves have closed-form equations and are easy to compute. In this paper, we address the general case of interpolation between two consecutive segments travelled at different speeds, while taking into account the kinematic and dynamic limitations on the break angle, speed and acceleration. The generalized polar polynomials are thus obtained, which allow the planning of fast motions. Important properties of the polynomials are demonstrated and their use provides a method for computing the best motion consistent with the constraints. Examples are presented to illustrate the theoretical results.