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

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.