New Form of Kane's Equations of Motion for Constrained Systems

Abstract

A differentiated form of the constraint equations, that is, in terms of accelerations, is incorporated into Kane's equations for nonholonomic systems, resulting in equations of motion that are both full order and separated in the generalized accelerations. This means that the time derivatives of all generalized speeds appear in the equations, but only one in each equation. Thus, one obtains a single set of consistent equations without reducing the dimensionality of the space of generalized speeds from the number of generalized coordinates to the number of degrees of freedom. Furthermore, this full dimensionality is maintained without employing Lagrange multipliers. Finally, a systematic method to obtain analytical expressions for the constraint forces is derived. The resulting formulation is believed to be simple and useful in performance analysis and control system design for constrained dynamical systems.