Planar Curve Routing for Tethered-Robot Motion Planning

The following problem appears in robotics. A number of small, circular robots live in a common planar workspace. Each is attached by a flexible cable of finite length to a point on the boundary of this workspace. Each robot has a target point in the workspace it must reach. When a robot has reached its target, its cable will have been dragged out into the workspace and possibly pushed and bent by other robots. The cables remain taut at all times and may not overlap, but may bend around other robots. When only the target points are specified for the robots, their motion can produce arbitrarily complex cable configurations. The more complex a cable configuration is, the more restrictive it is to the motion of the robots. To keep restrictions on the robots' motions at a minimum, it is necessary to specify, in addition to the target points, a configuration of the cables that is as simple as possible but allows all robots to reach their targets. The problem of finding the simplest cable configuration for a given set of target points is shown to reduce to the problem of finding a minimal set of nonintersecting routes in a Euclidean graph whose nodes are the robots' cable line. That no set of edges can intersect any other set of edges is an unusual characteristic of this graph problem. This consideration leads to interesting geometric analysis used to determine which relative placements of the graph edges represent overlapping cable lines. An algorithm is suggested that uses an exhaustive search method with pruning to find a set of nonintersecting routes in the graph that is minimal according to a chosen criterion. The algorithm has been implemented and tested; examples of its performance are given.