Polynomial Time Near-Time-Optimal Multi-Robot Path Planning in Three Dimensions with Applications to Large-Scale UAV Coordination

For enabling efficient, large-scale coordination of unmanned aerial vehicles (UAVs) under the labeled setting, in this work, we develop the first polynomial time algorithm for the reconfiguration of many moving bodies in three-dimensional spaces, with provable $1.x$ asymptotic makespan optimality guarantee under high robot density. More precisely, on an $m_1\times m_2 \times m_3$ grid, $m_1\ge m_2\ge m_3$, our method computes solutions for routing up to $\frac{m_1m_2m_3}{3}$ uniquely labeled robots with uniformly randomly distributed start and goal configurations within a makespan of $m_1 + 2m_2 +2m_3+o(m_1)$, with high probability. Because the makespan lower bound for such instances is $m_1 + m_2+m_3 - o(m_1)$, also with high probability, as $m_1 \to \infty$, $\frac{m_1+2m_2+2m_3}{m_1+m_2+m_3}$ optimality guarantee is achieved. $\frac{m_1+2m_2+2m_3}{m_1+m_2+m_3} \in (1, \frac{5}{3}]$, yielding $1.x$ optimality. In contrast, it is well-known that multi-robot path planning is NP-hard to optimally solve. In numerical evaluations, our method readily scales to support the motion planning of over $100,000$ robots in 3D while simultaneously achieving $1.x$ optimality. We demonstrate the application of our method in coordinating many quadcopters in both simulation and hardware experiments.