On simultaneous control of the energy and actuated variables of underactuated mechanical systems – example of the acrobot with counterweight

Abstract

The energy-based control approach aiming to control both the total mechanical energy and actuated variables of underactuated mechanical systems has generated renewed interest in recent years. Different from the reports of successful applications of this approach, we investigate whether there exists an underactuated mechanical system for which we fail to control both the total mechanical energy and actuated variable(s) to some given desired values by studying a CWA (Counter-Weighted Acrobot), which is a modified Acrobot with its first link having a counterweight and only its second link being actuated. By analyzing globally the solution of the closed-loop system consisting of the CWA and the controller designed via the energy-based control approach, we show that unless the CWA is linearly controllable at the up–up equilibrium, where links 1 and 2 are in the upright position, the controller fails to achieve the goal of controlling the energy to the potential energy at the equilibrium and controlling the actuated joint variable to zero. We also provide corresponding results for the up–down, down–up, and down–down equilibriums of the CWA, where up and down denote that the link is in the upright and downward positions, respectively. We present numerical simulation results to validate the theoretical results.