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A numerical framework for finding and stabilizing periodic trajectories of underactuated mechanical systems with impacts is presented. By parameterizing a trajectory by a set of synchronization functions, whose parameters we search for, the dynamical constraints arising due to underactuation can be reduced to a single equation on integral form. This allows for the discretization of the planning problem into a parametric nonlinear programming problem by Gauss-Legendre quadratures. A convenient set of candidates for transverse coordinates are then introduced. The origin of these coordinates correspond to the target motion, along which their dynamics can be analytically linearized. This allows for the design of an orbitally stabilizing feedback controller, which is also applicable for degrees of underactuation higher than one.

The design of robust orbitally stabilizing feedback is considered. From a known orbitally stabilizing controller for a nominal, disturbance‐free system, a robustifying feedback extension is designed utilizing the sliding‐mode control (SMC) methodology. The main contribution of the article is to provide a constructive procedure for designing the time‐invariant switching function used in the SMC synthesis. More specifically, its zero‐level set (the sliding manifold) is designed using a real Floquet–Lyapunov transformation to locally correspond to an invariant subspace of the Monodromy matrix of a transverse linearization. This ensures asymptotic stability of the periodic orbit when the system is confined to the sliding manifold, despite any system uncertainties and external disturbances satisfying a matching condition. The challenging task of oscillation control of the underactuated cart–pendulum system subject to both matched‐ and unmatched disturbances/uncertainties demonstrates the efficacy of the proposed scheme.

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With the purpose of highlighting the concept of orbital stabilization as an alternative to the reference tracking control methodology, this paper considers simple, informative examples in relation to motion control of an onedegree-of-freedom, double integrator system. In this regard, the notions of (excessive) transverse coordinates, projections operators and the transverse linearization are introduced, and it is illustrated how these can be used both for the design and analysis of orbitally stabilizing feedback controllers.

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 functionbased 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.

Transverse linearization is a useful tool for the design of feedback controllers that orbitally stabilizes (periodic) motions of mechanical systems. Yet, in an n-dimensional statespace, this requires knowledge of a set of (n − 1) independent transverse coordinates, which at times can be difficult to find and whose definitions might vary for different motions (trajectories). Motivated by this, we present in this paper a generic choice of excessive transverse coordinates defined in terms of a particular parameterization of the motion and a projection operator recovering the "position" along the orbit. We present a constructive procedure for obtaining the corresponding excessive transverse linearization and state a sufficient condition for the existence of a feedback controller rendering the desired trajectory (locally) asymptotically orbitally stable. The approach is demonstrated through numerical simulation by stabilizing oscillations around the unstable upright position of the underactuated cart-pendulum system, in which a novel motion planning approach based on virtual constraints is utilized for trajectory generation.