Can the energy and actuated variables of underactuated mechanical systems be controlled? -Example of the Acrobot with counterweight

The energy-based control approach aiming to control the total mechanical energy and actuated variables of underactuated mechanical systems has generated renewed interests in studying these systems. Successful applications of the approach to the swing-up control for the Acrobot (2-link planar robot with passive first joint), the Pendubot (2-link planar robot with passive second joint), and an n-link revolute planar robot with a passive joint robot, have been reported. This paper investigates whether the objective of controlling the energy and actuated variables to some given desired values can be achieved for any underactuated mechanical system. We aim to answer this question by studying the problem of swinging up a modified Acrobot, which has a counterweight connected to its first link and is called CWA (Counter-Weighted Acrobot) below. By analyzing globally the behavior of the solution of the closed-loop system consisted of the CWA and the controller designed via the energy-based control approach, we show that unless the mechanical parameters of the CWA satisfying a constraint, the swing-up controller fails to achieve the goal of stabilizing the energy to the potential energy at the upright position and stabilizing the actuated joint variable to zero. Numerical simulation results are provided to validate the theoretical results.