Design and Analysis of the Swinging up Control for a 3 DOF Robot with Passive First Joint

Abstract

This paper addresses a swinging up control problem for a 3-link planar robot in the vertical plane, whose first joint is passive (unactuated) and the rest are actuated. By considering links 2 and 3 as a virtually composite link and performing a coordinate transformation of the joint variables of joints 2 and 3, this paper constructs a novel Lyapunov function based on the transformation, proposes an energy based swinging up control law, and analyzes the motion of the robot. For any initial state of the robot, this paper provides a necessary and sufficient condition for non-existence of any singular point in the control law for all future time, and shows how to choose the control parameters such that the state of the robot will eventually approach either any arbitrarily small neighborhood of the upright equilibrium point, or the downward equilibrium point. Moreover, this paper shows that the downward equilibrium point is unstable for the closed-loop system. To validate the theoretical results, this paper provides simulation results for the 3-link robot whose parameters are obtained from a human gymnast.