Parametric and Non-parametric Jacobian Motion Planning for Non-holonomic Robotic Systems

This paper addresses computational aspects of the Jacobian motion planning algorithms for non-holonomic robotic systems. The motion planning problem is formulated in terms of a control problem in the control affine system representing the system’s kinematics. Jacobian motion planning algorithms are derived by means of the continuation (homotopy) method applied to the inverse kinematics problem in the space of control functions. The solution of the motion planning problem is obtained as the limit solution of a functional differential equation involving the control function. Two methods of representing the control functions are studied: parametric and non-parametric. The parametric method parametrizes the control functions by truncated orthogonal series. The non-parametric method can manage without the parametrization. The functional differential equation can be solved using either the Euler method of integration or higher order methods. The paper focuses on the non-parametric Jacobian pseudo inverse motion planning algorithms incorporating a higher order integration method. Performance of this algorithm is illustrated by the numeric solution of a motion planning problem for the rolling ball kinematics.