3-Link Planar Robot with Passive First Joint

Abstract

This chapter concerns the swing-up control of an n-link revolute planar robot moving in the vertical plane with the first joint being passive and the others being active. The goal of this chapter is to design and analyze a swing-up controller that can bring the robot into any arbitrarily small neighborhood of the upright equilibrium point with all links in the upright position. To achieve this challenging control objective while preventing the robot from becoming stuck at an undesired closed-loop equilibrium point, first, this chapter addresses the problem of how to iteratively devise a series of virtual composite links to be used for designing a coordinate transformation on the angles of all the active joints. Second, it devises an energy-based swing-up controller that uses a new Lyapunov function based on that transformation. Third, it analyzes the global motion of the robot under the controller and establish conditions on the control parameters that ensure attainment of the swing-up control objective; specifically, it determines the relationship between the closed-loop equilibrium points and a control parameter. Finally, it verifies the theoretical results by means of simulations on a 4-link model of a human gymnast on the high bar. This chapter unifies some previous results for 2- and 3-link planar robots with a passive first joint.