Geometric Homogeneity and Configuration Controllability of Nonlinear Systems

This paper exploits notions of geometric homogeneity to show that (configuration) controllability results for a large class of mechanical systems with drift can be recovered by investigating a class of nonlinear dynamical systems satisfying certain homogeneity conditions. This broad class of mechanical systems, called 1-homogeneous systems, is defined to satisfy certain geometric homogeneous conditions. The properties of geometric homogeneity for vector bundles is developed for application to the analysis of systems with drift, followed by their control theoretic implications within this context. These theorems found in this paper generalize the configuration controllability results for simple mechanical control systems found in Lewis and Murray [34]. We also show how nonlinear controllability results for other classes of mechanical systems may be obtained with these methods.