Control of Mechanical Systems with Cyclic Coordinates using Higher Order Averaging

The control and dynamics of complex mechanical systems with unactuated cyclic coordinates, using only internal controls, is treated here. The goal is to achieve full control-liability of the reduced dynamics obtained by eliminating the cyclic coordinates using standard Routh reduction. The reduced system is also underactuated. We use high frequency, high amplitude periodic inputs and the framework of chronological calculus and averaging theory, for this purpose. A feedback scheme based on this approach is applied to the example of a dumbbell body in planar motion with an attitude control input in a central gravitational field. From our earlier work on this model, based on linearization, we know that the system is controllable at its relative equilibria. This work supplements earlier research on the possible use of internal controls for orbital maneuvers of underactuated spacecraft.