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

Abstract

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 nonparametric 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.

Egalitarian versus prioritarian approach in multiple task motion planning for nonholonomic systems

“…Till now [9,13,19,20], the solution of motion planning algorithm with endogenous configuration space approach has been mostly obtained by control function parametrization (truncated series parametrization) and by the algorithm discretization (fixed-step-size Euler method). The recent research [15] shows that the computations could be made in a more effective way. In this work, the multiple task algorithms are implemented in the nonparametric version, assisted with higher order algorithms of differential equations integration.…”

“…A secondary contribution of this paper is the implementation of both multiple task algorithms by means of the nonparametric and higher order differential equation solver [15]. The paper [15] presents the vari-ous approaches to numerical computation of the single task motion planning algorithm. In this paper, this approaches are extended to the multiple task motion planning algorithms.…”

Lagrangian Jacobian Motion Planning: A Parametric Approach

“…Some of those features were presented in a preliminary way in [11] and then proven in [12]. Jacobian motion planning algorithms can be run in either parametric or non-parametric formulation [13]. For practical and computational reasons the parametric formulation is beneficial, based on transferring the Lagrangian Jacobian motion planning algorithm from the infinite dimensional space of control functions to a finite dimensional space of control parameters.…”

Design of a Non-Holonomic Spherical Wrist

“…Non-holonomic constraints have been long studied in past years and important theorems have been developed on the subject, such as the Frobenius' theorem that helps understanding whether a set of Pfaffian constraints can be integrated or not, therefore giving rise to a true non-holonomic system [4]. More interesting results have been obtained also in the field of motion planning of mobile robots, where the kinematics of nonholonomic systems meets control theory [5,6].…”