Isolation of Unactuation: An Energetic Approach to Derive Static Equilibria for Underactuated Mechanical Systems

This paper revisits previous work on nonlinear mechanical control coupling for underactuated mechanical systems (UMS) to study how the coupling can be exploited in system dynamics. When decomposing a velocity field for an UMS into actuated and unactuated directions, the unactuated part can be isolated geometrically, and hence a "programmable" energetic approach is developed to directly obtain static equilibria for the system, instead of using Newton's second law and then eliminating variables. This approach is also interpreted with virtual work analysis. It is applicable to UMS, with any degree of unactuation, or any linkage type. It is also independent of choice of coordinates. A three-link model is adopted to demonstrate the results. Furthermore, a study between coupling and system response among a range of equilibria for an acrobot shows that larger coupling can instantly give more control authority to the unactuated dynamics.