Swing-up control for n-link planar robot with single passive joint using the notion of virtual composite links

In this paper, we concern a swing-up control problem for an n-link revolute planar robot with any one of the joint being a passive joint. 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, first, we address how to iteratively devise two series of virtual composite links separated by the passive joint to be used for designing a coordinate transformation on the angles of all 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 control parameters that ensure attainment of the swing-up control objective. The results obtained here unify some previous results for the Pendubot, the Acrobot, and three-link robots with a passive first joint. Finally, we validate the theoretical results via a numerical simulation investigation to a 4-link robot with a passive joint.