We consider a class of mechanical systems with an arbitrary number of passive (nonactuated) degrees of freedom, which are subject to a set of nonholonomic constraints. We assume that the challenging problem of motion planning is solved giving rise to a feasible desired periodic trajectory. Our goal is either to analyze orbital stability of this trajectory with a given time-independent feedback control law or to design a stabilizing controller. We extend our previous work done for mechanical systems without nonholonomic constraints. The main contribution is an analytical method for computing coefficients of a linear reduced-order control system, solutions of which approximate dynamics that is transversal to the preplanned trajectory. This linear system is shown to be useful for stability analysis and for design of feedback controllers orbitally, exponentially stabilizing forced periodic motions in nonholonomic mechanical systems.We illustrate our approach on a standard benchmark example.
There is no strife, no prejudice, no national conflict in outer space as yet. Its hazards are hostile to us all. Its conquest deserves the best of all mankind, and its opportunity for peaceful cooperation many never come again. But why, some say, the moon? Why choose this as our goal? And they may well ask why climb the highest mountain?
We choose to go to the moon. We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills, because that challenge is one that we are willing to accept, one we are unwilling to postpone, and one which we intend to win, and the others, too.
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.