A Constructive Procedure for Orbital Stabilization of a Class of Underactuated Mechanical Systems

In this note, we present a constructive procedure to solve the orbital stabilization problem of a class of nonlinear underactuated mechanical systems with n degrees of freedom and underactuation degree one using the Immersion and Invariance technique. We define sufficient conditions to solve explicitly the partial differential equations arising in the Immersion and Invariance methodology, so that mechanical systems with gyroscopic terms can be considered. At the end, we use two practical systems as examples to illustrate the design procedure, as well as validating performance via simulations and experiments.