Orbital stabilization for linear mechanical systems via a speed gradient algorithm

In some situations, it is useful to regulate the total energy function of a mechanical system to a constant value with the purpose of generating periodic motions. That is the case, for example, when a periodic motion is used to the fulfilment of a repetitive task. However, for mechanical systems with n degrees-of-freedom, regulation of the total energy function may not be sufficient by itself. In such situations, the regulation of more functions, closed-loop invariants, may be needed. As a simple solution to this problem, we design a control law in the principal coordinates of a linear mechanical system providing a sufficient quantity of such invariants. The proposed control scheme is based on a speed gradient algorithm added to a proportional term. We show that our controller can control the amplitude and frequency of periodic solutions in principal coordinates. Numerical simulations on a two degrees-of-freedom linear mechanical system are presented.