Optimal maneuver for multiple aircraft conflict resolution: a braid point of view

We study conflict resolution for multiple aircraft encounter situations on a plane. First, the homotopy types of resolution maneuvers for n aircraft encounters are classified according to their images in the joint space-time coordinates, which are shown to bear a one-to-one correspondence with the well-known pure braid group PB/sub n/. Energy is then proposed as the cost function for choosing among all conflict-free maneuvers the optimal one. For two aircraft encounters, analytic expressions of the optimal resolution maneuvers are obtained, and for the multiple aircraft case, a convex optimization technique is used to find the optimal two-legged resolution maneuver within each type. The introduced solution, however, becomes computationally intractable as the number of aircraft increases. The use of the probabilistic resolution algorithm in Hu (1999) as "random type chooser" is then suggested as a randomized solution to the combinatorial optimization problem. Finally, simulation results are presented for some typical encounters.