Set-point control for folded posture of 3-link underactuated gymnastic planar robot: Beyond the swing-up control

This paper concerns a set-point control for a folded posture 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 posture of the kip motion of a gymnast on the high bar, the control objective is to drive the robot from any arbitrary initial state to any arbitrarily small neighborhood of the up-down-down equilibrium point and then balance the robot about that point, where the first link is in the upright position and the second and third links 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 robot. This paper shows that the control toward such a folded posture is different from the swing-up control problem, for which all three links are fully stretched out in the upright position at the goal posture. A new result of this paper is that in addition to some conditions on the control parameters, a constraint on the mechanical parameters of the robot is needed for achieving the set-point control of the folded posture. The simulation results for two 3-link robots are provided to validate the effectiveness of the proposed control law and the necessity of the constraint.