Set-point control for folded configuration of 3-link underactuated gymnastic planar robot: new results beyond the swing-up control

Abstract

This paper concerns a set-point control for a folded configuration of a 3-link gymnastic planar robot moving in the vertical plane with the first joint being passive and the others being active. To realize the goal configuration of the kip motion of a human gymnast on the high bar, the control objective is to drive the robot from any initial state to any small neighborhood of the up–down–down equilibrium point and then balance the 3-link robot about that point, where link 1 is in the upright position and links 2 and 3 are in the downward position. This paper uses the energy-based control approach and the notion of virtual composite link to design a controller and provides a global motion analysis of the 3-link robot. Different from the swing-up control problem for which three links are fully stretched out in the upright position at the goal configuration, a new result of this paper is that in addition to some conditions on control parameters, a constraint on the mechanical parameters of the 3-link robot is needed for achieving the set-point control of the folded configuration (links 1 and 2 are folded). The proposed constraint guarantees the linear controllability at the up–down equilibrium point of the Acrobot (degenerated from the 3-link robot) whose actuated second link is the merge of links 2 and 3 of 3-link robot with all possible relative angles. The simulation results for two 3-link robots are presented to validate the obtained theoretical results.