Motion planning in R3 for multiple tethered robots

The problem of motion planning in three dimensions for n tethered robots is considered. Motivation for this problem comes from the need to coordinate the motion of a group of tethered underwater vehicles. The motion plan must be such that it can be executed without the robots´ tethers becoming tangled. The simultaneous-motion plan is generated in three steps. First, an ordering of the robots is produced that maximizes the number of robots that can move along straight lines to their targets. Then paths for the robots are computed assuming they move sequentially in the given order. Two methods of computing the sequential-motion plan for the robots are presented. The first method is computationally simple but guarantees no bound on the path length with respect to the optimal length; the second method guarantees nearly optimal paths for the given ordering at the expense of additional computation. Finally, trajectories are determined that allow the robots to move simultaneously. The motion plan generated is guaranteed not to result in tangled tethers. The algorithms we present are shown to run in O(n⁴) time in total in the worst case, which is less than the additional computation needed to produce the nearly optimal paths using existing approximation algorithms