Efficient Parametric Synthesis of Optimal Mobile Robot Trajectories

The problem of computing a single continuous trajectory which connects an arbitrary initial posture (position, heading, curvature) with a final posture remains unsolved. Previous approaches have either failed to achieve curvature continuity for the general case or generated sampled solutions which have not been efficient enough for real time use. The polynomial spiral is a generalization of the clothoid which is general enough to represent any feasible vehicle motion with a few coefficients. Starting from an optimal control formulation, the problem of generating a polynomial spiral between two arbitrary postures is transformed naturally into a nonlinear programming problem. This formulation yields to standard numerical approaches which produce trajectories throughout the set of feasible motions in real time.