Feedback control of enduring rotary motion of devil stick

Devil stick is a kind of juggling as to control a floating stick (the center stick) by hitting or pushing with other sticks (the hand sticks). Various stick motions are performed by jugglers and some of them are studied as control problem. The purpose of this paper is to rotate the center stick by only pushing with one hand stick. In other word, control objective is to maintain a position and an angular velocity of the center stick. The control strategy in this paper is as follows. To begin with, some states are controlled by output zeroing control. The output function is derived from observing the motion of good jugglers. But there are two problems in this strategy. First, since an attitude angle of the center stick is in unobservable subspace, the angular velocity can't be controlled. Secondly, a direction of the input force isn't considered. For the first problem, by solving equations of the zero dynamics, it turns out that it is possible to stabilize the angular velocity by changing a contact point of the center stick and the hand stick. For the second one, the analytical result shows that the required force depends on the angular velocity. Some numerical simulations show the usefulness of the proposed strategy.