Orbital Stabilization of Point-to-Point Maneuvers in Underactuated Mechanical Systems
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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, that is, to a smooth curve in its state–control space that is consistent with the system dynamics and which connects two stabilizable equilibrium points. The proposed method uses a particular parameterization, together with a state projection onto the maneuver's orbit 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. Numeric simulations of the non-prehensile manipulation task of a ball rolling between two points upon the "butterfly" robot demonstrates the efficacy of the full synthesis.
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