Coordinate-free Formulation for Dynamics and Control of a Chain Pendulum on a Cart

A new multibody dynamics model, that is referred to as a chain pendulum on a cart, is presented. It is composed of an arbitrarily number of rigid links that are serially connected by spherical joints, and a cart that is moving in a horizontal plane. This yields rich dynamic structures on the nonlinear configuration manifold that is comprised of the copies of the unit-sphere and the two dimensional Euclidean space. In this paper, we provide Lagrangian mechanics, controllability analysis, and controller design for a chain pendulum on a cart. In particular, it is shown that any equilibrium can be locally asymptotically stabilized by linear control analysis, and a nonlinear control system is also presented to stabilize the inverted equilibrium for the special case of a single link. The unique contribution is that all of these dynamic models and control analysis are developed in a coordinate-free fashion such that the corresponding results can be uniformly applied to a chain pendulum on a cart with an arbitrary number of links. The proposed coordinate-free formulation completely avoids singularities and complexities that are associated with local coordinates of the nonlinear configuration manifold. Copyright c © 0000 John Wiley & Sons, Ltd.