On the energy based control for underactuated mechanical systems

The energy based control approach aiming to controlling the total 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 Pendubot (2-link planar robot with passive first joint), the Acrobot (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. In this paper, we attempt to answer this question by studying the problem of swinging up the Acrobot like robot with a counterweight connected to its first link, called CWA (Counter-Weighted Acrobot). 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 driving the energy to the desired value and stabilizing the actuated variable. Numerical simulation results are provided to validate the theoretical results.