Control of underactuated mechanical systems using high frequency input

This paper describes an approach to closed-loop control of a class of underactuated mechanical systems using a high frequency periodic input whose amplitude is modulated through feedback. The approach makes use of the averaging theorem and the time scale separation between the high frequency forcing and the low frequency amplitude modulation. The paper begins with the special case of a two degree of freedom system with a single input. A closed-loop controller is developed that forces the time-averaged value of the unactuated coordinate to follow a desired (low frequency) trajectory. The method is then extended to systems with more degrees of freedom and multiple inputs and it is modified to enable low frequency trajectory tracking in the (averaged) actuated coordinates, as well as the unactuated coordinates. To illustrate, the approach is applied to the problem of position control for a three degree of freedom flapping plate in a uniform flow.