  Path planning for multiple tethered robots is a challenging problem due to
the complex interactions among the cables and the possibility of severe
entanglements. Previous works on this problem either consider idealistic cable
models or provide no guarantee for entanglement-free paths. In this work, we
present a new approach to address this problem using the theory of braids. By
establishing a topological equivalence between the physical cables and the
space-time trajectories of the robots, and identifying particular braid
patterns that emerge from the entangled trajectories, we obtain the key finding
that all complex entanglements stem from a finite number of interaction
patterns between 2 or 3 robots. Hence, non-entanglement can be guaranteed by
avoiding these interaction patterns in the trajectories of the robots. Based on
this finding, we present a graph search algorithm using the permutation grid to
efficiently search for a feasible topology of paths and reject braid patterns
that result in an entanglement. We demonstrate that the proposed algorithm can
achieve 100% goal-reaching capability without entanglement for up to 10 drones
with a slack cable model in a high-fidelity simulation platform. The
practicality of the proposed approach is verified using three small tethered
UAVs in indoor flight experiments.
