Nonlinear control of mechanical systems with one degree of underactuation

Numerous robotic tasks associated with underactuation have been studied in the literature. For a large number of these in the plane, the mechanical models have a cyclic variable, the cyclic variable is unactuated, and all shape variables are independently actuated. This paper formulates and solves two control problems for this class of models. If the generalized momentum conjugate to the cyclic variable is conserved, a set of that outputs is defined. If the generalized momentum conjugate to the cyclic variable is not conserved, a feedback that asymptotically stabilizes an equilibrium is given. The results are illustrated on a ballistic nip motion and on a balancing task.