Tight formation flying and sphere packing

In this paper, we introduce a method for finding a tight formation of unmanned air vehicles (UAVs) via the classical sphere packing scheme. We first translate the tight formation finding problem to the problem of maximizing the second smallest eigenvalue lambda₂(G) of the graph Laplacian L_G. We then show how close the formation Gs obtained from the sphere packing scheme is to the optimal formation G* that maximizes lambda₂(G). We show that lambda₂(G*)/lambda₂(G_s) is relatively small when the communication strength between two UAVs decays slowly with the distance between the two UAVs. This result implies that G_s can serve as a certificate that allows every graph to be quantitatively compared to G*. In the light of this tight formation result, a modelling technique is given for the optimal airborne refuelling of multiple UAVs.