This paper considers control systems defined on Lie algebroids. After deriving basic controllability tests for general control systems, we specialize our discussion to the class of mechanical control systems on Lie algebroids. This class of systems includes mechanical systems subject to holonomic and nonholonomic constraints, mechanical systems with symmetry and mechanical systems evolving on semidirect products.

Abstract This papers applies a recently developed ”generalized averaging theory ” to construct stabilizing feedback control laws for underactuated driftless systems. These controls exponentialy stabilize in the average. Conditions relating the properties of the averaged and unaveraged systems are given. An example validates the theory, demonstrating its utility. 1

Abstract 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. 1

Abstract This paper uses a recently developed generalized averaging theory [18] to develop stabilizing control laws for a large class of nonlinear systems with drift. These control laws exponentially stabilize in the average. 1

Motion planning and control are key problems in a collection of robotic applications including the design of autonomous agile vehicles and of minimalist manipulators. These problems can be accurately formalized within the language of ane connections and of geometric control theory. In this paper we overview recent results on kinematic controllability and on oscillatory controls. Furthermore, we discuss theoretical and practical open problems as well as we suggest control theoretical approaches to them.