Feedback stabilization for a special class of underactuated mechanical systems

We study feedback controllers that can stabilize the equilibrium of a special class of underactuated mechanical systems, namely mechanical systems with several directly actuated degrees of freedom and a single unactuated degree of freedom that must be controlled through the system coupling. Theoretical controllability and stabilizability results for a large class of such underactuated mechanical systems have been developed in our previous work. Here we propose feedback controllers so that a closed loop equilibrium is globally attractive. The proposed construction procedure depends on selection of a scalar configuration variable corresponding to a directly actuated degree of freedom so that the dynamics of the unactuated degree of freedom, after a rational coordinate transformation, has a special form. The construction procedure is introduced, and a number of possible modifications are suggested. This construction procedure unifies controllers that we have developed for particular examples, including an underactuated ship, a planar rigid body containing an unactuated internal degree of freedom, a planar manipulator with an unactuated elastically mounted end effector, and an aerospace vehicle with an unactuated internal degree of freedom