The Geometry of Controlled Mechanical Systems

Abstract

The control of mechanical systems has become a principal application focus of nonlinear control theory. This development was to a very large extent foreseen in the early work of R.W. Brockett, and a number of his papers from the mid-1970s have been highly influential in establishing the links between nonlinear control theory and geometric mechanics. Taking some of Brockett’s early work as a starting point, we study intrinsically second-order nonlinear control systems. The theory is developed using the language of differential geometry and affine connections. Velocity- and acceleration-controlled Lagrangian systems provide a rich class of examples of second-order control systems. These arise in modeling the dynamics of superarticulated mechanical systems, in which only some of the configuration variables (or degrees of freedom) are controlled directly, with the remaining variables evolving under the dynamic influence of the actuated degrees of freedom. Our goal is to develop a nonlinear control theory that characterizes the way in which the unactuated degrees of freedom in these systems are influenced by the degrees of freedom which can be controlled directly. Central to the theory is a natural Riemannian structure and certain curvaturelike quantities which provide a measure of the lack of integrability in the mapping of input trajectories to configuration space trajectories. The chapter concludes with results connecting the geometry of second-order systems with the theory of averaged Lagrangian and Hamiltonian systems.