On closed-loop vibrational control of underactuated mechanical systems

This paper discusses vibrational stabilization of a class of single-input, two degree-of-freedom mechanical systems. Considering two different control formulations—position-input and force-input—and both open- and closed-loop control, we find that the sets of attainable equilibrium positions for the unactuated coordinate are identical in every case. The subset of positions that are stabilizable, however, depends on the formulation. In general, the set of equilibria that can be stabilized using open-loop force-input is larger than the set that can be stabilized using open-loop position-input. And the use of feedback expands this stabilizable set even further. As examples, this paper presents the dynamic analysis, open- and closed-loop vibrational control, and the mechanics behind the stability of two underactuated systems, the Kapitza pendulum and a one-link horizontal pendulum.