Departmental Papers (ESE)


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.

Document Type

Book Chapter

Subject Area

GRASP, Kodlab

Date of this Version

January 2001


Reprinted from Algorithmic and Computational Robotics: New Directions 2000 WAFR, edited by Bruce Donald, Kevin Lynch, and Daniela Rus (Wellesley: A.K. Peters, 2001), pages 205-220.

NOTE: At the time of publication, author Daniel Koditschek was affiliated with the University of Michigan. Currently, he is a faculty member in the Department of Electrical and Systems Engineering at the University of Pennsylvania.



Date Posted: 02 September 2008