Singularities, Normal Forms, and Motion Planning for Non-holonmic Robotic Systems

Abstract

We address the motion planning problem in non-holonomic robotic systems using the tools of control theory. To this objective we associate with the robotic system a control-affine system. The derivative of the end-point map of this control system defines the Jacobian of the robotic system. Control functions at which the Jacobian is not surjective are referred to as singular configurations of the robotic system. As a description of these singularities we propose normal forms under feedback of the associated control system. On the basis of the Jacobian we introduce Jacobian motion planning algorithms. Special attention is paid to Lagrangian Jacobian algorithms. As an illustration of theoretical concepts we analyse normal forms, singularities, and motion planning for a free-floating space manipulator. A motion planning problem is solved by a Lagrangian Jacobian motion planning algorithm and, alternatively, by a sinusoidal control applied to the normal form. Results of computations show the performance of the two algorithms.