Trajectory Optimization and Orbital Stabilization of Underactuated Euler-Lagrange Systems with Impacts

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.