A complete analytical characterization of optimal collision-free motion for two robots

Collision-free motion planning for multi-agent robotic systems remains an active research area because of the complexity of coordinating robot trajectories, with applications ranging from coordinated drone shows to autonomous operations on land, in the air, and on water. Importantly, the analytical difficulty of nonconvex collision-avoidance constraints has so far driven the development of exclusively numerical methods. We derive a complete analytical characterization of the optimal trajectories for two robots that start and end at rest, under double-integrator dynamics and a squared-acceleration objective. We prove that these trajectories belong to eight distinct contact families and determine which family applies for each configuration of initial and final positions. We analyze the geometrical properties of the optimal trajectories, provide a numerical algorithm for computing them, and discuss how these insights may improve the performance of existing numerical trajectory optimizers.