Dynamically consistent Jacobian inverse for non-holonomic robotic systems

Abstract

This paper presents an extension of the concept of dynamically consistent Jacobian inverse from robotic manipulators (holonomic systems) to non-holonomic robotic systems, like mobile robots. This new inverse is derived within the framework of the endogenous configuration space approach, following a strict analogy with the original derivation of the dynamically consistent Jacobian inverse for holonomic systems. The analogy is founded on replacing a finite-dimensional configuration space of the manipulation robot by the space of control functions steering the non-holonomic system. Consequently, a curve in the space of control functions corresponds to the manipulator’s trajectory in the configuration space, whereas endogenous velocities and forces are defined as elements of the tangent and cotangent spaces to the control space. Three ways of introducing the dynamically consistent Jacobian inverse are proposed, referred to as the geometric method, the force method, and the optimization method. A crucial concept underlying all these methods is a Riemannian metric in the space of control functions of the non-holonomic system as well as in its operational space. It has been shown that, similarly as for holonomic systems, the dynamically consistent Jacobian inverse obtained prevents the transmission of certain internal forces acting in the system from the endogenous configuration space to the operational space. This property is illustrated with the example of the Pioneer 2DX mobile platform. Performance of the new Jacobian inverse is demonstrated in the context of motion planning of the rolling ball.