  This paper initiates the use of vector fields to design, optimize, and
implement reactive schedules for safe cooperative robot patterns on planar
graphs. We consider Automated Guided Vehicles (AGV's) operating upon a
predefined network of pathways. In contrast to the case of locally Euclidean
configuration spaces, regularization of collisions is no longer a local
procedure, and issues concerning the global topology of configuration spaces
must be addressed. The focus of the present inquiry is the achievement of safe,
efficient, cooperative patterns in the simplest nontrivial example (a pair of
robots on a Y-network) by means of a state-event heirarchical controller.
