Controllability of Under-Actuated Planar Manipulators with One Unactuated Joint

This paper is concerned with analysis on controllability for a class of nonholonomic systems. We discuss controllability of underactuated planar manipulators with one unactuated joint. We show that if the first joint (in the base side) is actuated, these systems are completely controllable, namely, there exists an admissible trajectory from any given initial point to any given final point. In order to prove this, we use global stabilizing feedback control law to converge the state to a manifold, where the system is locally controllable. By this controller, we have two trajectories, one starting at the given initial position and the other starting at the given final position. Then we connect them using a kind of bi-directional approach to show the existence of the whole admissible trajectory. Finally, we give some simulation results to discuss controllability of more general cases, the first joint being actuated and all other joints being unactuated.