Many tasks in robotics and automation require a cyclic exchange of energy between a machine and its environment. Since most environments are “under actuated”—that is, there are more objects to be manipulated than actuated degrees of freedom with which to manipulate them—the exchange must be punctuated by intermittent repeated contacts. In this paper, we develop the appropriate theoretical setting for framing these problems and propose a general method for regulating coupled cyclic systems. We prove for the first time the local stability of a (slight variant on a) phase regulation strategy that we have been using with empirical success in the lab for more than a decade. We apply these methods to three examples: juggling two balls, two legged synchronized hopping and two legged running—considering for the first time the analogies between juggling and running formally.
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