Exploration of Vibrational Control of Two Underactuated Mechanical Systems

Control of underactuated mechanical systems is of interest as it allows for control authority over all of a system's degrees of freedom without requiring actuation of the full system. In addition to this, open-loop control of a system provides the advantage of applying to systems with unmeasurable states or where sensor integration is not feasible. Vibrational control is an open-loop control strategy that uses high-frequency, high-amplitude forcing to control underactuated mechanical systems. This thesis is concerned with exploring two underactuated mechanical systems that are controlled using vibrational inputs. The first system, a 3 degrees of freedom (DOFs) 2-link mechanism with 1 actuated DOF which is an example of a vibrational control system with 1 input and 2 unactuated DOFs, is used to review analytical results of stability analysis using the averaged potential. Theoretical and numerical results are presented for the achievable stable configurations of the system and the effects of changing the physical parameters on the achievable stable configurations are studied. The primary contribution of this effort is the development of an experimental apparatus where vibrational control is implemented. The second system is a 4DOF system composed of a 2DOF spherical pendulum supported by an actuated 2DOF cart used to study the effects of multiple vibrational inputs acting on a system. Theoretical and numerical analysis results are presented for three variants of harmonic forcing applied to the two actuated degrees of freedom: 1) identical input waveforms, except for the amplitudes, 2) identical input waveforms, except for the amplitudes and a phase shift, and 3) identical input waveforms, but at different frequencies and amplitudes. The equilibrium sets under open-loop vibrational forcing are determined for all three cases. A general closed-loop vibrational control scheme is presented using proportional feedback of the unactuated coordinates superposed with the zero-mean, $T$-periodic vibrational input.