Toward the Regulation and Composition of Cyclic Behaviors

Many tasks in robotics and automation require a cyclic exchange of energy between a machine and its envi ronment Since most environments are under actu ated that is there are more objects to be manip ulated 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 reg ulating coupled cyclic systems We prove for the rst time the local stability of a slight variant on a phase regulation strategy that we have been using with em pirical 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 rst time the analogies between juggling and running formally