A unified solution to swing-up control for n-link planar robot with single passive joint based on virtual composite links and passivity

This paper concerns the swing-up control of an n-link revolute planar robot with any one of the joints being passive. The goal is to design and analyze a swing-up controller that can bring the robot into any arbitrarily small neighborhood of the upright equilibrium point, at which all the links are in the upright position. We present a unified solution based on the notion of virtual composite link (VCL), which is a virtual link made up of one or more active links. By using the angles of two series of VCLs separated by the passive joint and using the total mechanical energy of the robot, we design a swing-up controller and analyze the global motion of the robot under the controller. The main new results of this paper are: (1) we obtain a lower bound for each control gain related to the angle of each VCL such that the closed-loop system has only one undesired equilibrium point in addition to the upright equilibrium point, and we present an original proof of the conditions on the control gains for a class of n-link underactuation-degree-one planar robots with an active first joint; (2) we provide a bigger control gain region for achieving the control objective than those of previous results on three- and n-link robots with a passive first joint; (3) we validate the theoretical results via numerical simulations on a 4-link robot with the passive joint in each of the four positions. This paper gains an insight into the passivity-based control of underactuated multiple-DOF systems.