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How tight is sphere-packed formation flying?

Abstract In this article, a method for finding a tight formation of unmanned air vehicles (UAVs) via the classical sphere packing scheme is introduced. First, the tight formation finding problem is translated to the problem of maximizing the second smallest eigenvalue λ2(G) of the graph Laplacian L G . It is then shown how close the formation Gs, obtained from the sphere packing scheme, is to the optimal formation G∗ that maximizes λ2(G). λ2(G∗)/λ2(Gs) is shown to be relatively small (close to 1) when the communication strength between two UAVs decays slowly with the distance between the two UAVs. This result is also useful in that Gs 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.