Simultaneous control of energy and actuated variable of 3-link planar robot with underactuation degree two and its application

This paper concerns a control problem for a 3-link planar robot moving in the vertical plane with only the first joint being actuated by using the energy-based control approach. This robot is of underactuation degree two since the difference of the number of degrees of freedom and that of control input is two which is usually very difficult to control. The control objective is to control simultaneously the total mechanical energy, the angular velocity and the angle of link 1 of the robot to their values corresponding to those at the upright equilibrium point, where all three links are in the upright position. The main contribution of this paper is to provide a global analysis of the convergence of the energy and the motion of the robot under the derived control law in a systematic way. By presenting a new property with its strict proof about the motion of the robot, without any condition on the mechanical parameters of the robot, we prove that the control objective is achieved for all initial states with the exception of a set of Lebesgue measure zero provided that two conditions on control gains are satisfied. Although there is no theoretical guarantee that there exists time such that the robot can be swung up close to the upright equilibrium point, the numerical simulation shows that this simultaneous control can be applied successfully to the swing-up and stabilizing control for a 3-link robot. This paper reveals the difficulty of the motion analysis of mechanical systems with underactuation degree greater than one.