Orbital Stabilization of Point-to-Point Maneuvers in Underactuated Mechanical Systems

The task of inducing, via continuous static state-feedback, an asymptotically stable heteroclinic orbit in a nonlinear control system is considered in this paper. The main motivation comes from the problem of ensuring convergence to a so-called point-to-point maneuver in an underactuated mechanical system. Namely, to a smooth curve in its state--control space, which is consistent with the system dynamics and connects two (linearly) stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver as to combine two linearization techniques for this purpose: the Jacobian linearization at the equilibria on the boundaries and a transverse linearization along the orbit. This allows for the computation of stabilizing control gains offline by solving a semidefinite programming problem. The resulting nonlinear controller, which simultaneously asymptotically stabilizes both the orbit and the final equilibrium, is time-invariant, locally Lipschitz continuous, requires no switching, and has a familiar feedforward plus feedback--like structure. The method is also complemented by synchronization function--based arguments for planning such maneuvers for mechanical systems with one degree of underactuation. Numerical simulations of the non-prehensile manipulation task of a ball rolling between two points upon the "butterfly" robot demonstrates the efficacy of the synthesis.