Periodic motion planning and control for underactuated mechanical systems

We consider the problem of periodic motion planning and of designing stabilising feedback control laws for such motions in underactuated mechanical systems. A novel periodic motion planning method is proposed. Each state is parametrised by a truncated Fourier series. Then we use numerical optimisation to search for the parameters of the trigonometric polynomial exploiting the measure of discrepancy in satisfying the passive dynamics equations as a performance index. Thus an almost feasible periodic motion is found. Then a linear controller is designed and stability analysis is given to verify that solutions of the closed-loop system stay inside a tube around the planned approximately feasible periodic trajectory. Experimental results for a double rotary pendulum are shown, while numerical simulations are given for models of a spacecraft with liquid sloshing and of a chain of mass spring system.