Braid-based entanglement-free trajectory planning for multiple tethered robots

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. This work presents a new approach to address this problem using topological braids. The dual algebraic and geometric representations of braids allow us to abstract the topological patterns from the complex spatiotemporal interactions among robots. By establishing a topological equivalence between the physical cables and the spatiotemporal 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 the trajectories corresponding to these interaction patterns. Based on this finding, we present a topology planning algorithm that computes a desired multi-robot path topology by searching in a graph of robot permutations and rejecting the entangling braid patterns. Then, the desired path topology serves as the high-level reference for each robot to generate non-entangling trajectories in a decentralized manner. We demonstrate that the proposed algorithm can achieve 100% goal-reaching capability without entanglement for 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 drones in indoor flight experiments.