Swing-up control based on virtual composite links for n-link underactuated robot with passive first joint

This paper concerns the swing-up control of an n-link revolute robot moving in the vertical plane with the first joint being passive and the others being active. The goal of this study 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 objective while preventing the robot from becoming stuck at an undesired closed-loop equilibrium point, we first address 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, we devise an energy-based swing-up controller that uses a new Lyapunov function based on that transformation. Third, we analyze 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, we determine the relationship between the closed-loop equilibrium points and a control parameter. Finally, we verify the theoretical results by means of simulations on a 4-link model of a gymnast on the high bar. This study not only unifies some previous results for acrobots and three-link robots with a passive first joint, but also provides insight into the energy- and passivity-based control of underactuated multiple-degree-of-freedom systems. (C) 2009 Elsevier Ltd. All rights reserved.