Swing-up control for a two-link underactuated robot with a flexible elbow joint: New results beyond the passive elbow joint

This paper concerns the swing-up control problem for a two-link underactuated robot moving in the vertical plane with a single actuator at the base joint and a spring between the two links (flexible elbow joint). First, we present two new properties of such a flexible robot about the linear controllability at the UEP (upright equilibrium point, where two links are in the upright position) and the limitation of the PD control on the angle of the base joint. Second, for the robot which can be not locally stabilized about the UEP by the PD control, we study how to extend the energy-based control approach, which aims to control the total mechanical energy and the angle and angular velocity of the base joint of the robot, to design a swing-up controller. We provide a necessary and sufficient condition for avoiding the singularity in the controller. Third, we analyze the motion of the robot under the presented controller by studying the convergence of the total mechanical energy and clarifying the structure and stability of the closed-loop equilibrium points. We validate the presented theoretical results via numerical investigation. This paper not only unifies some previous results for the Pendubot (a two-link robot with a passive elbow joint), but also provides insight into the control and analysis of the underactuated robots with flexible joints.