Free flight in 2000: games on Lie groups

With growth of air traffic worldwide, revolutionary new architectures for air traffic management called "free flight" and "user preferred routing" are being considered by the Federal Aviation Administration as possible ways to ease the increasing congestion in the skies. The basic analytical and control theoretic breakthrough needed to enable free flight is airborne conflict resolution, that is, decentralized algorithms for preventing aircraft from losing separation with one another. Motivated by this problem, we explore special features of the dynamical games solution when the underlying dynamics correspond to left-invariant control systems on a Lie group. We show that the motion of an aircraft may be modeled as a control system on the Lie groups SE(2) (horizontal motion only) and SE(3) (full 3-dimensional motion). In this formulation, some simplification in the derivation of saddle and Nash strategies follows from the use of Marsden-Weinstein reduction techniques: we give an outline for the solution of N-aircraft conflict resolution using Nash type strategies. This simplification allows us to efficiently compute optimal solutions to complex conflict resolution problems for more than 2 aircraft, using numerical techniques which could be programmed into the flight management computers on board each aircraft.