Control of systems subject to Pfaffian constraints: motivation for Kane's formulation

Abstract

This paper provides motivation for the use of Kane's formulation of the equations of motion for systems with configuration coordinates that are subject to Pfaffian constraints. It is shown that the matrix-vector form of Kane's formulation, as outlined in this paper, results in an equation structure that is ideally suited to the backstepping approach to control system design that is described by Fierro and Lewis for the dynamical control of differentially steered mobile robots. The present paper casts this control system design approach into a form that coincides with the equations of motion derived via Kane's formulation. This unified modeling and control procedure is illustrated with two examples of systems with configuration coordinates that are subject to Pfaffian constraints. These examples are a rolling sphere that is subject to a non-holonomic constraint and a two-link robotic manipulator described in terms of unit-quaternions that is subject to two holonomic constraints.