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

Abstract

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.

Excessive Transverse Coordinates for Orbital Stabilization of (Underactuated) Mechanical Systems

“…1 The same set of excessive coordinates we consider in this paper, together with the linearization of their dynamics, has previously been considered in [6] for stabilizing periodic motions of a fully actuated robot manipulator. Moreover, they were utilized in [7] for the stabilization of a hybrid walking cycle of a three-link biped robot with two degrees of underactuation. There, a particular choice of the parameterizing variable s allowed for one coordinate to be trivially omitted in order to obtain a minimal set of transverse coordinates.…”