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Orbital stabilization of mechanical systems through semidefinite Lyapunov functions

In this paper we address the problem of generating asymptotically stable limit cycles for a fully actuated multibody mechanical system through a feedback control law. Using the concept of conditional stability the limit cycle can be designed for a lower dimensional dynamical system describing how the original one evolves on a chosen submanifold and the corresponding velocity space. Moreover, the controller can be split up in two parts that can be independently designed and analyzed in order to reach the constraint submanifold and then produce the oscillation. Even if designed assuming a lower dimensional system, the limit cycle implies a periodic motion for the whole system.