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

In this note we study the generation of non-trivial oscillations of a class of mechanical systems with underactuation one. The proposed design consists of two terms, that is, a partial linearizing state feedback, and an immersion and invariance orbital stabilization controller. The first step is adopted to simplify analysis and design, however, bringing an additional difficulty that the model loses its Euler-Lagrange structure after the collocated pre-feedback. To address this, we propose a constructive solution to the orbital stabilization problem via a smooth controller in an analytic form, and the model class identified in the article is characterized via some easily a priori verifiable assumptions on the inertia matrix and the potential energy function.