A Set-Point Control for a Two-link Underactuated Robot With a Flexible Elbow Joint

This paper concerns a set-point control problem for a two-link underactuated robot moving in the vertical plane with a single actuator at the first 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 upright equilibrium point (UEP, where two links are in the upright position) and about its equilibrium configuration. Second, we show that for the robot with a certain range of spring constant, the proportional derivative (PD) control on the angle of the first joint can globally stabilize the robot at the UEP. Third, for the robot not satisfying the above range of spring constant, 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 first joint of the robot, to design a swing-up controller. We provide a necessary and sufficient condition for avoiding the singularity in the controller, and 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 present the results of numerical investigation, which support our theoretical conclusions.