Page - 388 - in Differential Geometrical Theory of Statistics

Entropy2016,9, 337 trajectoriesandoutputsasimplifiedonethatcanbeusedinanoperationalcontext. Pleasenote that theseparationnormconstraintswerenot taken intoaccount in thiswork. Inouralgorithm,wecannot enforce theregulatoryseparationnorms, justconstructclusterswith lowinteractions.Accordingto theapplications, theresultsofouralgorithmmaybeusedasaninitial solutionofapost-processing algorithmbasedonoptimal control inorder tokeep in linewith the regulatoryconstraints. Using entropyassociatedwith a curves system, agradientdescent is performed inorder to reduce it so as tostraightentrajectorieswhileavoidingareaswith lowaircraftdensity, thusenforcingroute-like behavior. Thiseffect is relatedto the fact thatentropy-minimizingdistributions favorconcentration. 2. Entropy-MinimizingCurves 2.1.Motivation Aspreviouslymentioned, air trafficmanagement of the futurewillmake an intensiveuseof 4Dtrajectoriesasabasicobject. Full automation isa far-reachingconcept thatwillprobablynotbe implementedbefore2040–2050,andeveninsuchasituation, itwillbenecessary tokeephumans in the loop soas togain awide societal acceptanceof the concept. Starting fromSESARorNextgen initialdeploymentandaimingtowards thisultimateobjective,a transitionphasewithhuman-system cooperationwill takeplace. SinceATCcontrollersareused toawell-structurednetworkof routes, it isadvisabletopost-processthe4Dtrajectories issuedbyautomatedsystemsinordertomakethemas closeaspossible to linesegmentsconnectingbeacons. Toperformthis task, inanautomaticway,flight pathswillbedeformedsoas tominimizeanentropycriterionthatenforcesavoidanceof lowdensity areasandat thesametimepenalizes length.Comparedtoalreadyavailablebundlingalgorithms[3] that tendtomovecurves tohighdensityareas, thisnewproceduregeneratesgeometrically-correct curves,withoutexcesscurvature. Letasetγ1, . . . ,γN ofsmoothcurvesbegiventhatwillbeaircraftflightpaths for theair traffic application. It will be assumed in the sequel that all curves are smoothmappings from [0,1] to adomainΩofRqwitheverywherenon-vanishingderivatives in ]0,1[. This last conditionallowsone toviewthemassmooth immersionswithboundariesandissoundfromtheapplicationpointofview, asaircraftvelocitiesareboundedbelowbytheefficiencyconsiderationandultimatelybythestall and, therefore, cannotvanishexpectat theendpoints. Inair trafficapplications, thedimensionof thestate space isgenerally twoandsometimes threewhen theevolutionof theaircraft in theverticalplane isof interest. Theapproachtakeninthisworkisfirst togetasounddefinitionofspatialdensityassociatedwith acurvesystem, thentoderive fromitanentropythatwillbeminimized. 2.2. SpatialDensityof aSystemofCurves Due to the fact that aircraft positions are acquired through radarmeasurements, a trajectory is knownonly at discrete sampling times. In theoperational context, the samplingperiod ranges from4 to 10 s,which corresponds roughly to a 100–250-m travelingdistance. Derived from that, aclassicalperformance indicatorusedinATMis theaircraftdensity [4],obtainedfromthesampled positionsγi(tj), j= 1,. . . ,ni oneachflightpathγi, i= 1,. . . ,N. It is constructed fromapartition Uk, k=1,. . . ,PofΩbycountingthenumberofsamplesoccurring inagivenUk, thendividingout by the totalnumberof samplesn=∑Ni=1ni. More formally, thedensitydk in the subsetUk ofΩ is expressedas: dk=n−1 N ∑ i=1 ni ∑ j=1 1Uk ( γi(tj) ) (1) 388