Cyclic Nullspace Coordination: Perpetual Flight of Aerial Carriers for Static Suspension

This work demonstrates that the nonstop flights of three or more carriers are compatible with holding a constant pose of a cable-suspended load. It also presents an algorithm for generating the carriers’ coordinated nonstop trajectories. The proposed method builds upon two pillars: 1) the choice of$n$special linearly independent directions of internal forces within the$3n-6$-dimensional nullspace of the grasp matrix of the load, chosen as the edges of a Hamiltonian cycle on the graph that connects the cable attachment points on the load. Adjacent pairs of directions are used to generate$n$forces evolving on distinct 2-D affine subspaces, despite the attachment points being generically in 3-D and 2) the construction of elliptical trajectories within these subspaces by mapping, through appropriate graph coloring, each edge of the Hamiltonian cycle to a periodic coordinate while ensuring that no adjacent coordinates exhibit simultaneous zero derivatives. Combined with conditions for load statics and attachment point positions, these choices ensure that each of the$n$force trajectories projects onto the corresponding cable constraint sphere with nonzero tangential velocity, enabling perpetual motion of the carriers while the load is still. The work provides a scalable constructive design for any$n\ge 3$with tuning guidelines, quantifies sensitivity and single-carrier failures, and provides a fixed-wing–compatible planner that preserves load statistics under speed/bank/flight-path constraints. The theoretical findings are validated through simulations and laboratory experiments with quadrotor uncrewed aerial vehicles (UAVs).