Resolution of Conflicts Involving Many Aircraft via Semidefinite Programming

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Abstract
Aircraft conèict detection and resolution is currently attracting the interest of many air transportation service providersand isconcernedwiththefollowingquestion: Given aset ofairborneaircraftand theirintended trajecto- ries,whatcontrolstrategyshould befollowedbythepilotsand theairtrafécserviceprovidertopreventtheaircraft from coming too close to each other? This paper addresses this problem by presenting a resolution methodology whereby each aircraft proposes its desired heading while a centralized air trafé c control authority resolves any conè ictarisingbetween aircraft, while minimizingthedeviation between desired and conè ict-freeheadingforeach aircraft. The resolution methodology relies on a combination of convex programming and randomized searches: It is shown that a version of the planar, multiaircraft conèict resolution problem, accounting for all possible cross- ing patterns among aircraft, might be recast as a nonconvex, quadratically constrained quadratic program. For this type of problem, there exist efé cient numerical relaxations, based on semideé nite programming, that provide lowerboundson thebestachievableobjective. Theserelaxationsalsoleadtoarandomsearch techniquetocompute feasible, locally optimal, and conè ict-free strategies. This approach is demonstrated on numerical examples and discussed.

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