Mechanical Control Systems and Kinematic Systems

The aim of this technical note is to analyze the equivalence between the second-order equations describing the dynamics of mechanical systems, and the associated kinematic system when dealing with nonholonomic systems with controls. If the system is fully actuated, both systems are equivalent. However, if it is underactuated an extra condition must be imposed to ensure that a weak equivalence holds. Furthermore, the notion of decoupling vector fields is generalized to a vector field distribution. This point of view may be used to obtain better solutions when a cost function is introduced to the controllability problem. The results are applied to some particular examples and to mechanical systems with symmetries.