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In this note we study the generation of attractive oscillations of a class ofmechanical systems with underactuation one. The proposed design consists of twoterms, i.e., a partial linearizing state feedback, and an immersion andinvariance orbital stabilization controller. The first step is adopted tosimplify analysis and design, however, bringing an additional difficulty thatthe model loses its Euler-Lagrange structure after the collocated pre-feedback.To address this, we propose a constructive solution to the orbitalstabilization problem via a smooth controller in an analytic form, and themodel class identified in the paper is characterized via some easily aprioriverifiable assumptions on the inertia matrix and the potential energy function.

Immersion and invariance orbital stabilization of underactuated mechanical systems with collocated pre-feedback