3-Link Planar Robot with Two Passive Joints

Abstract

This chapter concerns a control problem for a 3-link planar robot moving in the vertical plane with only the first joint being actuated (called the APP robot below) by using the energy-based control approach. The control objective is to control simultaneously the total mechanical energy, the angular velocity and the angular displacement of the 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. This chapter presents a global analysis of the convergence of the energy and the motion of the APP robot under the derived control law in a systematic way. Specifically, by presenting a new property with its strict proof about the motion of the APP robot, without any condition on the mechanical parameters of the robot, this chapter proves 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 chapter reveals the difficulty of the motion analysis of the above energy-based control for mechanical systems with underactuation degree greater than one.