The Tethered Robot Problem

Robot vehicles tethered to fixed bases move about in a com mon region. To perform its task, each vehicle must visit a prespecified set of locations. This paper discusses the problem of controlling the vehicles so that they do not tangle their tethers. The problem arises, for example, in connection with small air-bearing supported vehicles used to fetch parts for mechanical assembly. The tethers in this case are "umbilical cords" carrying compressed air, power, and signals. The tethered robot problem can be seen as a special case of the general problem of scheduling the motions of several cooperating robots. In more conventional cases the robot's manipulator is connected to the base by a series ofjointed members whose movements in 3D space are quite compli cated. The tethered robot problem considered here is much simpler because the manipulators (vehicles) move in only two dimensions, and they are connected to their bases in the geometrically simplest way. Relatively large numbers of teth ered robots may be able to share a common space without undue complication. Such sharing can greatly increase the ef ficiency and flexibility with which an array ofprocessing equipment can be used. The technical problem outlined above leads in particular to the following mathematical problem: identify those rectilin ear drawings ( plane embeddings) of the complete bipartite graph K m,n whose edges can be colored with n colors so that no two edges of the same color have common points (includ ing endpoints). Solutions are presented for useful special cases. The general case of this apparently new problem re mains open.