Aircraft Proximity Maps Based on Data-Driven Flow Modeling

A = airspace A = exit coordinate along axis ai A = entry coordinate along axis ai ACi = any aircraft from flow F i AC i = qth aircraft from flow F i ai , b k i , c k i = vectors of the local orthonormal frame atW k i .b k i points towardW 1 i , c k i is vertical (usual up direction) B = exit coordinate along axis bi Bi = box defined by the window of convex hull of two consecutive windows, Wi and W k 1 i B = entry coordinate along axis bi C = exit coordinate along axis ci C = entry coordinate along axis ci Ci = ith cluster of “similar” trajectories F i = ith flow; three-dimensional tube that contains the trajectories of cluster Ci fO P = spatial distribution of the outliers f Di di = probability density function of the interaircraft distance fak i , fck i = lateral (along ai ) and vertical (along c k i ) spatial distributions of aircraft for window Wi , respectively fVi (vi) = probability distribution for the aircraft speeds in flow F i fRq i r i = probability density function of the residual distance r i of the aircraft AC q i I = event interarrival time associated to N N = Poisson counting process P = point of the airspace P1 P = probability that at least one aircraft flies in the proximity of a point P P2 P = probability that at least two aircraft from different flows fly in the proximity of a point P PO P = probability that one aircraft from the flows and one outlier are simultaneously in the same neighborhood of a point P Pc P = proximity volume centered at point P Pi P = proximity volume when considering box Bi R0 = east–north–up frame Ri = ai ;bi ; ci reference frame attached to box Bi r i = residual distance of AC q i T i = centroid of cluster Ci Vi P = intersection volume between Pi P and Bi vi = speed of aircraft ACi Wi = window of cluster Ci located at the kth point along centroid T i x; y; z = Cartesian coordinates di = distance between two consecutive aircraft from F i ti = time between two consecutive aircraft from F i = Poisson counting process parameter i t = aircraft arrival rate parameter for cluster Ci j = Poisson counting process parameter for the jth time interval t = aircraft arrival rate parameter considering all clusters Received 31 January 2011; revision received 22 May 2011; accepted for publication 31 July 2011. Copyright © 2011 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/12 and $10.00 in correspondence with the CCC. Postdoctoral Fellow, School of Aerospace Engineering, 270 Ferst Drive; erwan.salaun@gatech.edu. Member AIAA. Ph.D. Candidate, School of Aerospace Engineering. Currently, Postdoctoral Associate, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139; mgariel@mit.edu. Member AIAA. Ph.D. Candidate, School of Mechanical Engineering, 801 Ferst Drive; aevela@gatech.edu. Professor of Aeronautics and Astronautics, School of Aerospace Engineering, 270 Ferst Drive; feron@gatech.edu. Associate Fellow AIAA. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 35, No. 2, March–April 2012