Swing-up and stabilizing control for two-link underactuated robot with flexible elbow joint

This paper concerns the swing-up and stabilizing 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 in the presence of gravity 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 not be 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 singular points 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. When the robot is swung-up close to the UEP, a locally stabilizing controller is switched to balance the robot about the UEP. We validate the presented theoretical results via numerical investigation. This paper not only unifies some previous results for the Pendubot (a two-link underactuated robot with a passive elbow joint), but also suggests that the underactuated robots with flexible joints have richer nonlinear behaviors than those with only passive joints.