ContentslistsavailableatScienceDirect
Information
Fusion
journalhomepage:www.elsevier.com/locate/inffus
Distributed
multiple
model
joint
probabilistic
data
association
with
Gibbs
sampling-aided
implementation
Shaoming
He,
Hyo-Sang
Shin
∗,
Antonios
Tsourdos
School of Aerospace, Transport and Manufacturing, Cranfield University, Cranfield MK43 0AL, UK
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r
t
i
c
l
e
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f
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Keywords:Multiple target tracking Distributed information fusion Joint probabilistic data association Gibbs sampling
Average consensus
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This paper proposes a new distributed multiple model multiple manoeuvring target tracking algorithm. The proposed tracker is derived by combining joint probabilistic data association (JPDA) with consensus-based dis- tributed filtering. Exact implementation of the JPDA involves enumerating all possible joint association events and thus often becomes computationally intractable in practice. We propose a computationally tractable approxi- mation of calculating the marginal association probabilities for measurement-target mappings based on stochastic Gibbs sampling. In order to achieve scalability for a large number of sensors and high tolerance to sensor failure, a simple average consensus algorithm-based information JPDA filter is proposed for distributed tracking of mul- tiple manoeuvring targets. In the proposed framework, the state of each target is updated using consensus-based information fusion while the manoeuvre mode probability of each target is corrected with measurement prob- ability fusion. Simulations clearly demonstrate the effectiveness and characteristics of the proposed algorithm. The results reveal that the proposed formulation is scalable and much more efficient than classical JPDA without sacrificing tracking accuracy.
1. Introduction
Proliferationoflow-cost, lightweight,andpowerefficient sensors andadvancesinnetworkedsystemsenabletheemploymentofmultiple sensornodes,capableofcommunicatingwitheachother.Thesensorsin anetworkcooperativelyenablecomplicatedsensingandtrackingtasks, whichareotherwisedifficulttoaccomplish.Comparedtothesingle sen-sortargettracking,utilisingmultiplesensors,throughinformation fu-sion,cansignificantlyimprovethesensorcoverageandtheestimation accuracy[1].Thechallengeisthatthesesensorsarelikelytocontain somedegreeofuncertainties.Low-costsensorsaregenerallysubjectto highclutterrateandlowdetectionprobability.Combinedwiththe in-herentuncertaintiesandcomplexityof theproblem, thepoor perfor-mance issuewiththesesensors couldbe significantlyexacerbatedin targettracking,especiallyinmulti-targettracking[2–4].Whentargets aremanoeuvring,theproblembecomesevenmorechallenging. Practi-calapplicationsthatinvolvemanoeuvringtargetsinclude,butarenot limitedto,aircrafttracking,groundmovingvehicletracking,re-entry vehicletracking,andhumantracking.However,algorithmsformultiple manoeuvringtargetstrackinginasensornetworkarerare.Therefore,it ismeaningfultodevelopatractablemulti-sensormultiplemanoeuvring targetstrackingalgorithm.
The objective of this paper is, in fact, to address the problem ofdistributedmultiplemanoeuvringtargetstrackingin asensor
net-∗Corresponding author.
E-mailaddresses:shaoming.he@cranfield.ac.uk(S. He), h.shin@cranfield.ac.uk(H.-S. Shin), a.tsourdos@cranfield.ac.uk(A. Tsourdos).
work,subjecttoacertaindegreeofuncertainties.Generally,the multi-sensor multi-targettracking isdividedintotwostages:thefirststage is a local multi-target tracking phase andthe second is the estima-tion fusion among all sensors. Thefocus of this paper is the devel-opment ofefficient algorithmsforhandlingimportantissuesinboth stages.
Inthelocalestimationstage,eachsensornoderunsamulti-target tracking(MTT)algorithmtoobtainthelocaltracks.Asdiscussed,the keyissueisthatthemeasurementuncertaintycouldsignificantly de-grade the performance of MTT. Dataassociation is a plausible and widely-acceptedsolutioninmulti-targettrackingtoresolvetheproblem ofmeasurementuncertainty.Thistechniquediscernstarget-generated measurementsfromcluttersandfindsthemappingsbetweentargetsand measurements.Oneofthemostwell-knownassociationalgorithmsisthe multiplehypothesistracking(MHT)[5,6].MHTsolvestheproblemof associationambiguitybyadelayedlogic,whichmaintainsalldata as-sociationhypothesesinadecision-makingtreeunlessenough informa-tionisavailabletoremovetheimpossiblehypothesis.AlthoughMHT is provedtobeBayesianoptimalforMTT,findingtheexactsolution iscomputationallyintractableandhencerequiresapproximated imple-mentations[7–10].Anotherwidely-acceptedprobabilisticdata associa-tionapproach,jointprobabilisticdataassociation(JPDA),isknownasa suboptimalMTTestimatorthatcanachievereasonableresultsatlower computationalburden[11].
https://doi.org/10.1016/j.inffus.2020.04.007
Received 15 February 2019; Received in revised form 14 April 2020; Accepted 26 April 2020 Available online 15 May 2020
1566-2535/© 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license. ( http://creativecommons.org/licenses/by-nc-nd/4.0/)
Consideringthebalancebetweenthesensitivityissueandthe compu-tationalcost,thispaperadoptsJPDAastheunderlyingdataassociation approach.TheissuewithJPDAisthatitrequirestoenumerateall fea-siblejointeventstofindthemarginalassociationprobability.Notethat themarginalprobabilityisthenusedtoperformmomentmatchingto mergetheposteriorGaussianmixtureintoasingleGaussianform. Com-putingthemarginalprobabilitiesitselfis#P-complete,therebyleading tointractabilityforalargenumberoftargets.Toreducethe computa-tionalcostoftheJPDA,manyadhocformulationsandapproximations havebeendeveloped,[12–15].Unlikepreviousaccountable approxima-tions,thispaperproposesanewstochasticsampling-based implemen-tationofJPDAthatgreatlyimprovesthecomputationalefficiencyand maintainstherobustnessofstandardJPDAagainstoutliers.Notethat thesecharacteristicsareofgreatimportanceforsensornetworks,where thecomputationalpowerislimited.
Inthefusionstage,thesensornodescommunicatewitheachother toperformestimationfusionthroughanetworktopology.Unlikethe centralisedfilter, the distributedestimationis known toexhibit the advantagesof scalability forlarge-scale networks andstrong robust-nessagainstsensorfault [16–19].Amongtheexistingdistributed ap-proaches, consensus-based methods [19–21] arewidely-used due to theirglobalconvergenceandeasyimplementation.Byiteratively com-municatingwiththeadjacentsensorsateachtimeinstant,the estima-tionobtained byeach sensor asymptotically convergestotheglobal one.TheKalmanconsensusfilter(KCF)[16,19,22] isawell-known dis-tributedfilter,whichdirectlyappliestheaverageconsensusalgorithm tolocalstateestimations.However,itonlyworkswellinthesituation whereeachsensorcangetameasurementfromthetarget[23,24].In [25,26],adistributedmulti-sensormulti-targettrackingalgorithmwas proposedonthebasisofKCFandthuscannotaccountforthenaive sen-sors,1e.g.,targetsareoutofsensors’field-of-view[24].Inarecent no-tablecontribution[24],theauthorsproposedaninformationconsensus filter(ICF)thataddressestheinherentproblemsofKCFandguarantees convergencetothecentralisedone.BasedoneitherKCForICF,different distributedestimationalgorithmsforsinglemanoeuvringtargettracking wereproposedin[27–29].Notethatalthoughdistributedmulti-sensor usingconsensusalgorithmforsingletargettrackingiswell-established, directextensiontotheMTTscenarioisunreasonableduetothe mea-surementoriginuncertaintyandthereforerequirescarefuladjustment. Combiningtheprobabilisticdataassociation(PDA)filter[30]withthe ideaofICF,amulti-sensormulti-targettrackingfilterwasdevelopedin [31] forasensornetwork,butthisalgorithmwasshowntobe sensi-tivetoclutterrate.Byincorporatingconsensusalgorithmwith Proba-bilityHypothesisDensity(PHD)filter,theauthorsin[32,33]proposed novelmulti-sensormulti-targettrackingapproaches.Notethatthesetwo distributedPHDfiltersarebasedonclassicalgeometricaveragefusion rule.Theauthorsin[34] firstdevelopedanewconsensus-basedPHD filterbyutilisingthearithmeticaveragefusionrule.Ithasbeenlater demonstratedthatthesimplerarithmeticaveragefusionoutperforms thegeometricaverage fusion in somecases [35,36]. The consensus-basedPHDfilters,however,cannotpreservetrackcontinuity,e.g., can-notprovidetargetidentityinformation.Thisissuewaslaterresolvedby theconsensus-based[37,38]𝛿-GeneralisedLabeledMulti-Bernoulli( 𝛿-GLMB),whichsharessimilarconceptasclassicaldataassociation tech-niquesin theimplementation.Theauthorsin [39–41] proposed two multi-sensormulti-targettrackingfilters:parallelandsequential multi-sensorJPDAs.Theparallelversionwasshowntobeexponentially com-putationallycomplexasthetotalnumberofsensorsincreases[39].On theotherhand,thesequentialonehaslowersurvivability[40],i.e.,it requireseachsensor’sfield-of-viewtocovertheentiresurveillance re-gion.Theseapproaches,however,requiresequentiallyconnectedsensor networksandarenotreallydistributedtrackers.
1 Naive sensors mean the sensors that cannot detect the same target as other
sensors.
Motivatedbytheaboveobservations,thispaperaimstodevelopa tractable/practicalalgorithmthatissuitableformultiplemanoeuvring targetstrackingusingapartiallyconnectedsensornetwork.Themain contributionsofthispaperarehighlightedasfollows:
(1)AnefficientalgorithmfortheJPDAimplementationisproposed byutilisingstochasticGibbssampling.Eachpossiblejointeventis con-sideredasarandomvariablethatcanbegeneratedbystochasticGibbs samplingandhencethemarginalassociationprobabilitycanbeeasily approximatedbytheeventoccurrence.Thispolynomial-time approxi-mationmakesitfeasibletoapplyJPDAinasensornetworkfor multi-targettracking.Experimentsshowthattheproposedapproximationis scalableandofgreatefficiencywithignorableperformancesacrifice.
(2)Ageneralframeworkofdistributedmultiplesensorsmultiple ma-noeuvringtargetstrackingalgorithmisdevelopedbyincorporatingthe consensusalgorithmandinteractivemultiplemodel(IMM)approach intotheproposedGibbs-JPDAfilter.Morespecifically,weformulatethe stateestimationofeach manoeuvremodeintheformof information statefusion bydevelopingadistributedinformationconsensusJPDA filter.BasedonthejumpMarkovnonlinearsystem(JMNS)modelling, theproposedgeneralframeworkincorporatestheIMMfilterbyusinga distributedmeasurementprobabilityfusionschemetoprovidethe ca-pabilityofaccurateestimationformanoeuvringtargets.Duetothe dis-tributednature,theproposedalgorithmhasstrongrobustnessagainst sensorfailures.
Notethatthetrackingalgorithmdevelopedhasalreadybeenapplied andtestedintheEuroSwarm2project.Thecorrespondingindoordemo isalsoattachedintheSupplementaryfile.
The rest of thepaper is organised asfollows. Section 2 presents somepreliminariesandbackgrounds.Section3 providesthedetailsof theproposedGibbssampling-aidedmarginalisation.InSection4,the UKF-baseddistributedinformationJPDAfilterisderivedindetail, fol-lowedbytheproposedmultiplemodelUKF-baseddistributed informa-tionJPDAfiltershowninSection5.Finally,somesimulationresultsand conclusionsareoffered.
2. Backgroundsandpreliminaries
Thissectionfirstprovidessomenecessarybackgroundsofthebasics ofJMNS,tofacilitatetheanalysisinthefollowingsections.Then,the problemformulationofthepaperisstated.
2.1. Multiple-targetjumpmarkovnonlinearsystem
Let𝑋𝑘=
{
𝑥1
𝑘,…,𝑥𝑁𝑘𝑘
}
bethesetoftargetstatesatscank,where
Nkdenotesthenumberoftargetsatscank, 𝑥𝑖𝑘 theithtargetatscan k.Thetargetconsideredinthispapermanoeuvresaccordingtovarious kinematicmodels.Inthecaseofmanoeuvringtargetestimation,one keyissueishowtoconstructasuitablemodeltorepresentthesystem transitionmodel.Todate,themostwidely-acceptedideaistheJMNS modelling[42,43],whichassumesthatthetargetmotioncanbe quan-tifiedbyaweightedsumofseveralmanoeuvremodes.AJMNSconsists ofasetofdifferentnonlinearmodelsandeachmodelisquantifiedbyits modeprobability.Themodeprobabilityevolveswithtimeaccordingto afinitestateMarkovchainanddetermineshowprobablethe manoeu-vremodefollowstherealtargetmotionmodel.UndertheJMNS frame-work,theithtargetcanbemodeledbythefollowingdiscrete-timejump Markovnonlinearsystem
𝑥𝑖 𝑘=𝑓𝑖 ( 𝑥𝑖 𝑘−1,𝑟𝑖𝑘 ) +𝜔𝑖𝑘−1(𝑟𝑖𝑘) (1)
wherefidenotesthesystemdynamicstransitionfunctionoftheith
tar-get,𝑟𝑖
𝑘 the targetmanoeuvre mode,and𝜔𝑖𝑘−1 (
𝑟𝑖 𝑘
)
theprocessnoise. We assume that 𝑟𝑖𝑘 takes value from a finite set Ξ ={1,2,…,𝑟} 2Description can be accessed through
Fig.1. General information flow of the proposed algorithm.
withmodetransitionprobabilitymatrixΠ =[𝜋𝑚𝑠]
𝑟 ×𝑟 ,where𝜋𝑚𝑠 Δ = Pr { 𝑟𝑖 𝑘+1=𝑠|||𝑟𝑖𝑘=𝑚 }
forallm,s∈ Ξand∑𝑠=1𝑟 𝜋𝑚𝑠=1foranym∈ Ξ. TheMarkoviantransitionprobabilitymatrixdeterminesthepossibility ofacertainmotionmodelthatthetargetfollowsinthenextscan𝑘+1 givenamotionmodelatcurrentscank.
2.2. Problemformulation
Theaimofthispaperistodesignadistributedmultiplemanoeuvring targets(e.g.,multipleJMNSs)trackingalgorithmusingapartially con-nectedsensornetwork.Eachsensornoderunsalocalmulti-target track-ingalgorithmandthelocalestimationsarethenfusedinadistributed way.Notethatinapartiallyornotfullyconnectedsensornetwork,each sensorcanonlycommunicatewithitsneighbours.Themainchallenges oftheconsideredproblemaretwofold.Ononehand,mostmulti-target trackingalgorithmshavehighcomputationalburdenandthushave lim-itedapplicationsinmulti-sensornetworksystems.AlthoughtheJPDA canachievereasonableaccuracywithlesscomputationalpowerthanthe MHT,fullenumerationofallpossiblejointeventsisstillintractablefor practicalapplications.Ontheotherhand,theposteriorofmulti-target estimationscontainthemeasurementoriginuncertainty,meaningthat directextensionofdistributedsinglemanoeuvringtargetmulti-sensor fusionalgorithmstothemultiplemanoeuvringtargetscaseisnot feasi-bleandthusrequirescarefuladjustment.
Inordertotacklethesechallenges,wetailoraframework incorporat-ingtheinformation-formJPDAfilterwithIMMformulti-sensormultiple manoeuvringtargetstracking.Theproposedframeworkisillustratedin Fig.1.Inthelocalestimation,weconsidereachjointassociationeventin JPDAasarandomvariablethatsatisfiesadistributionandthenpropose toleveragethestochasticGibbssamplingtocalculatetheapproximated marginalprobabilityofJPDAfilter.Thisstochasticapproximationcan significantlyreducethecomputationalburdenandretaintheproperties oftheoriginalJPDA.Forestimationfusion,wereformulatetheJPDA
filterinaninformationformandthenutilisetheaverageconsensus al-gorithmforbothtargetstatefusionandmodelprobabilityfusionina distributedway.Insummary,theproposedtrackingalgorithmconsists ofthreemodules:(1)GibbssamplingbasedJPDA(Section3);(2) dis-tributedUKF-basedinformationJPDAfilterforeachmanoeuvremode (Section4);and(3)multiplemodelestimationfusion(Section5). 3. GibbssamplingbasedJPDAfilter
ThissectionproposesanewJPDAfilteralgorithmusingastochastic Gibbssamplingapproach. Forbrevity,weignorethesensor indexas wellasthemodeindexhere.
3.1. StandardJPDAfilter
LetusbrieflyreviewtheclassicalJPDAfilterforthecompleteness ofthepaper.Thesetofmeasurementsreceivedbyonesensoratscan
kisdefinedas𝑍𝑘=
{
𝑧0,𝑘,𝑧1,𝑘,…,𝑧𝑀𝑘 ,𝑘
}
,whereMkdenotesthe
num-berofmeasurementsreceivedatscank,zj,k(j≠ 0)thejthmeasurement receivedatscank,z0,kthedummymeasurementforconvenient
rep-resentationofmissdetectionandfalsealarm.Thenumberofclutters orfalsealarmsisassumedtobeaPoissondistributionand𝜆Fdenotes
theexpectednumberofcluttersperunitvolumeofthevalidationgate, knownasspatialdensityofclutters.
InthestandardJPDAfilteringapproach,each measurementis as-sumedtooriginatefromanumberofcandidatetargetsandthustracks areupdatedbyaweightedsumofthevalidatedmeasurementsfrom thecurrenttime.ThisisthereasonwhyJPDAisknownasa’soft de-cision’filter.Thisfeaturemakestheposteriorprobabilitydistributiona Gaussianmixtureform.PropagationoftheGaussianmixture distribu-tionovertimecontainsanexponentialnumberofmixturecomponents andisthusintractablewithoutapproximations.Inordertomaintainthe feasibility,JPDAusesasingleGaussianmodeltoapproximatethe Gaus-sianmixtureateachtimestep.Morespecifically,thestateestimationis obtainedbyusingaweightedinnovationterm
̃𝑧𝑖 𝑘= 𝑀𝑘 ∑ 𝑗=1𝛽 𝑖 𝑗̃𝑧𝑖𝑗,𝑘= 𝑀𝑘 ∑ 𝑗=1𝛽 𝑖 𝑗 ( 𝑧𝑗,𝑘−̂𝑧𝑖𝑘) (2) where ̂𝑧𝑖
𝑘denotesthepredictedmeasurementoftheithtarget,𝛽𝑗𝑖 the
marginalassociationprobabilitythatthejthmeasurementisassociated withtheithtarget.
Itisclearthatdeterminingthemarginalassociationprobabilities𝛽𝑖 𝑗
is thekeypartof JPDA.JPDAalgorithmcalculatesthemarginalised association probability basedonall possiblejoint associationevents. Afeasiblejointeventisdefinedasonepossiblemappingofthe mea-surementstothetrackssuch that:(1)eachmeasurement (exceptfor thedummyone)isassignedtoatmost onetarget;(2)eachtargetis uniquelyassignedtoameasurement.LetΘ𝑘={𝜃𝑖𝑘},𝑖∈{1,2,…,𝑁𝑘}, denotethejointassociationeventatscank,where𝜃𝑖
𝑘∈
{
0,1,…,𝑀𝑘} standsforthesingleassociationevent.Here,𝜃𝑖𝑘=𝑗 meansthatthejth measurementoriginatesfromtheithtarget.Theposteriordistribution ofthejointeventis
𝑝(Θ𝑘||𝑍𝑘)∝ (𝑁𝑘 ∏ 𝑖=1 𝜑𝑖(𝜃𝑖𝑘) )( ∏ (𝑖,𝑖′)∈𝐸 𝜑𝑐 ( 𝜃𝑖 𝑘,𝜃𝑖 ′ 𝑘 )) (3) where𝜑𝑖(𝜃𝑖𝑘 )
denotestheun-normalisedPDAprobability ofevent𝜃𝑖 𝑘 givenby[30] 𝜑𝑖(𝜃𝑘𝑖=𝑗 ) ∝ {( 1−𝑃𝐷𝑃𝐺)𝜆𝐹, 𝑗=0 (𝑧𝑗,𝑘;𝐻𝑘𝑖𝑥𝑖𝑘,𝑆𝑖𝑘 ) 𝑃𝐷, 𝑗≠ 0 (4)
where(𝑥;𝜇,Σ)denotestheGaussiandistributionofvariablexwith mean𝜇 andcovarianceΣ,𝐻𝑖
𝑘themeasurementmatrix,𝑆𝑖𝑘the
innova-tioncovariancematrix,PDtheprobabilityofdetection,PG thegating probability,and 𝜑𝑐 ( 𝜃𝑖 𝑘,𝜃𝑖 ′ 𝑘 ) = { 0, 𝜃𝑘𝑖=𝜃𝑖𝑘′>0 1, otherwise (5)
𝐸={(𝑖,𝑖′)|||∃𝑖∈[𝑁𝑘],𝑖′∈[𝑁𝑘],𝑖≠ 𝑖′} (6) whichensurethatonemeasurement(exceptforthedummy measure-ment)canonlybeallocatedtoonetarget.
Themarginalisedassociation probability𝛽𝑖
𝑗 indicatingthatthejth
measurementisassociatedwiththeithtargetcan beobtainedbythe lawoftotalprobabilityas
𝛽𝑖 𝑗=
∑ Θ𝑘 ∶𝜃𝑖 𝑘 =𝑗
𝑝(Θ𝑘||𝑍𝑘) (7)
whichimpliesthatexactsolutionofthemarginalassociationprobability requiresfullyenumeratingallpossiblejointevents.
Remark1. NotethatthestandardJPDAfilterrequirestheassumption thatthenumberoftargetsisknownapriori[11].Thesameassumption isalsoutilisedinthispaper.However,thisalgorithmcanbeadaptedto morepracticalscenarios,wherethenumberoftargetsisunknown,by eithersimpleheuristicM/Nlogic[6] orbirthmodelapproach[44].
3.2. Gibbssampling-aidedmarginalisation
Determiningthemarginaljointassociationprobabilities𝛽𝑖 𝑗between
measurementsandtargets, whichisawell-known#P-complete prob-lem.Totacklewiththecombinatorialnatureinobtainingthe associa-tionprobability,thispaperproposesasamplingbased,specificallyGibbs sampling-based,algorithm.
Thekeyideathatthispaperproposesistoconsidereachjoint asso-ciationeventΘkasarandomvariablethatsatisfiesadistribution𝜋(Θk).
Tothisend,weconstructaMarkovchainwhosestatespaceisthesetof allfeasiblejointeventswithstationarydistributionastheposteriorjoint eventdistribution.Toensurethatthejointeventswithhigher probabil-ityaremoreeasilytobesampled,itisnaturaltoconstructthesampling proposal𝜋(Θk)proportionaltoitscorrespondingjointposterior proba-bility,i.e., 𝜋(Θ𝑘)∝ (𝑁𝑘 ∏ 𝑖=1 𝜑𝑖(𝜃𝑘𝑖 ))( ∏ (𝑖,𝑖′)∈𝐸 𝜑𝑐 ( 𝜃𝑖 𝑘,𝜃𝑖 ′ 𝑘 )) (8) Usingthesamplingproposal𝜋(Θk),wecouldgeneratesufficiently
enoughsamplesof Θk.Fromthesamples,itisstraightforwardto
ap-proximatethemarginaljointassociationprobabilitybythesample oc-currence.
Theissueisthatdirectsamplingfrom(8)isverydifficultas enumer-atingallpossiblejointeventsisimpossibleforreal-timeapplications. Therefore,wedevelopasampling-basedmarginalisationalgorithm us-ingGibbssampling.GibbssamplingisastochasticmethodforBayesian inferencetoapproximatetheposteriormultivariateprobability distri-butioninapolynomialtime[45,46].Thissamplingapproachwasalso utilisedintheimplementationof𝛿-GLMBin[47].Themainadvantage isthatitissimplertorecursivelysamplefromaconditionaldistribution thantosampledirectlyfromthejointdistributionitself.More specifi-cally,thetransitionkernelfromonejointeventΘ𝑘=
(
𝜃1
𝑘,…,𝜃𝑁𝑘𝑘
) to anotherjointevent ̄Θ𝑘=(̄𝜃1
𝑘,…,̄𝜃𝑘𝑁𝑘 ) isgivenby 𝜋( ̄Θ𝑘||Θ𝑘)= 𝑁𝑘 ∏ 𝑚=1 𝜋𝑚 ( ̄𝜃𝑚 𝑘|||̄𝜃𝑘1,…,̄𝜃𝑘𝑚−1,𝜃𝑚𝑘+1,…,𝜃𝑁𝑘𝑘 ) (9) where𝜋mcanbeobtainedfrom(4)and(8) as
𝜋𝑚 ( ̄𝜃𝑚 𝑘|||̄𝜃1𝑘,…,̄𝜃𝑘𝑚−1,𝜃𝑘𝑚+1,…,𝜃𝑘𝑁𝑘 ) = 𝜋(̄𝜃1 𝑘,…,̄𝜃𝑘𝑚,𝜃𝑘𝑚+1,…,𝜃𝑁𝑘𝑘 ) 𝜋(̄𝜃1 𝑘,…,̄𝜃𝑘𝑚−1,𝜃𝑘𝑚+1,…,𝜃𝑘𝑁𝑘 ) ∝𝜋 ( ̄𝜃1 𝑘,…,̄𝜃𝑘𝑚,𝜃𝑘𝑚+1,…,𝜃𝑁𝑘𝑘 ) = ( 𝜑𝑚(𝜗𝑚𝑘 ) ∏ (𝑚,𝑖′)∈𝐸 𝜑𝑐 ( 𝜗𝑚 𝑘,𝜗𝑖 ′ 𝑘 )) × ( ∏ 𝑖≠𝑚 𝜑𝑖(𝜗𝑖𝑘) ∏ (𝑖,𝑖′)∈𝐸,𝑖≠𝑚 𝜑𝑐 ( 𝜗𝑖 𝑘,𝜗𝑖 ′ 𝑘 )) ∝𝜑𝑚(𝜗𝑚𝑘) ∏ (𝑚,𝑖′)∈𝐸 𝜑𝑐 ( 𝜗𝑚 𝑘,𝜗𝑖 ′ 𝑘 ) (10) where𝜗𝑖 𝑘= ̄𝜃𝑘𝑖,𝑖∈ {1,…,𝑚},𝜗𝑖𝑘=𝜃𝑖𝑘,𝑖∈ { 𝑚+1,…,𝑁𝑘}
‘Proportionto’inEq.(10) highlightsthedependenceofindividual conditionaldistributionon𝜗𝑚
𝑘,whilerestpartsareformedasthe
nor-malisationconstant.GiventhejointeventΘk,ajointevent ̄Θ𝑘canbe obtainedbyrecursivesamplingaccordingtothefollowingindividual conditionaldistributions ̄𝜃1 𝑘∼𝜋1 ( ̄𝜃1 𝑘|||𝜃2𝑘,…,𝜃𝑘𝑁𝑘 ) ⋮ ̄𝜃𝑚 𝑘 ∼𝜋𝑚 ( ̄𝜃𝑚 𝑘|||̄𝜃𝑘1,…,̄𝜃𝑘𝑚−1,𝜃𝑚𝑘+1,…,𝜃𝑁𝑘𝑘 ) ⋮ ̄𝜃𝑁𝑘 𝑘 ∼𝜋𝑁𝑘 ( ̄𝜃𝑁𝑘 𝑘 |||̄𝜃1𝑘,…,̄𝜃𝑁𝑘𝑘 −1 ) (11) Onceanenoughnumberofsamplesgenerated,themarginaljoint as-sociationprobabilityisapproximatedbyoccurrence.Afterconstructing theMarkovchain,itisnecessarytoprovethatthegeneratedMarkov chain asymptotically converges toits invariant distribution andthis propertyisformulatedinTheorem1.
Theorem1. Givenanyinitialfeasiblejointevent,thedistributionofGibbs samples(9) asymptoticallyconvergestothetargetdistribution(8) withan exponentialrateas
||
|𝜋𝑛( ̄Θ𝑘||Θ𝑘)−𝜋( ̄Θ𝑘)||| ≤(1−2𝛽)⌊𝑛∕2⌋ (12) where𝜋𝑛( ̄Θ
𝑘||Θ𝑘)denotesthenthpoweroftransitionkernel𝜋( ̄Θ𝑘||Θ𝑘),𝛽 =
minΘ𝑘 𝜋2( ̄Θ𝑘||Θ𝑘 )
∈ (0,0.5]theleastlikelytwo-steptransitionprobability.
Proof. Ingeneral,theconvergenceoffinite-stateMarkovchainis guar-anteedbyitsirreducibilityandregularity.TheirreducibilityofaMarkov chainisquantifiedintermsofthepossibilitythatonestatehas capabil-itytotransfertoanotherstatewithinfinitestep.Andtheregularityof aMarkovchaincanbecheckedbythepositivityoftheentriesofsome finitepowerofitstransitionmatrix.
Let 0n denote then dimensional zero vector. Since every target cansharethedummymeasurement,i.e.,𝜑𝑐(0,𝜃𝑖𝑘
) =1,itfollowsfrom (10)that 𝜋(0𝑁𝑘 ||Θ𝑘)∝ 𝑁𝑘 ∏ 𝑚=1 𝜑𝑚(0)>0 𝜋(Θ𝑘|| |0𝑁𝑘 ) ∝ 𝑁𝑘 ∏ 𝑚=1 𝜑𝑚(𝜃𝑘𝑚)>0 (13)
Then,thetwo-steptransitionkernelfromany𝜃 toany ̄𝜃 satisfies
𝜋2( ̄Θ 𝑘||Θ𝑘)= ∑ 𝜁 𝜋 ( ̄Θ𝑘|𝜁)𝜋(𝜁||Θ𝑘) >𝜋(̄Θ𝑘|||0𝑁𝑘 ) 𝜋(0𝑁𝑘 ||Θ𝑘)>0 (14) ThisimpliesthattheMarkovchain{Θ(𝑘𝑡)}
∞
𝑡=1generatedbytheGibbs samplerisirreducibleandrecurrent,andtherefore theMarkovchain willasymptoticallyconvergetoitsinvariantdistribution,e.g.,the pos-teriorofthejointevent,bytheergodictheorem[48].Next,applying Lemma2,presentedinAppendixA,to𝜋2( ̄Θ
𝑘||Θ𝑘)gives max Θ𝑘 𝜋 2𝑛( ̄Θ 𝑘||Θ𝑘)−minΘ𝑘 𝜋2𝑛( ̄Θ𝑘||Θ𝑘)≤(1−2𝛽)𝑛 lim 𝑛→∞maxΘ𝑘 𝜋 2𝑛( ̄Θ 𝑘||Θ𝑘)= lim 𝑛→∞minΘ𝑘 𝜋 2𝑛( ̄Θ 𝑘||Θ𝑘)≥𝛽 >0 (15)
Since Lemma 1, presented in Appendix A, states that maxΘ𝑘 𝜋𝑛( ̄Θ𝑘||Θ𝑘) is non-increasing and minΘ𝑘 𝜋𝑛( ̄Θ𝑘||Θ𝑘) is
non-decreasinginn,(15) canbereformulatedas max Θ𝑘 𝜋 𝑛( ̄Θ 𝑘||Θ𝑘)−minΘ𝑘 𝜋𝑛( ̄Θ𝑘||Θ𝑘)≤(1−2𝛽)⌊𝑛∕2⌋ lim 𝑛→∞maxΘ𝑘 𝜋 𝑛( ̄Θ 𝑘||Θ𝑘)=𝑛lim→∞minΘ 𝑘 𝜋 𝑛( ̄Θ 𝑘||Θ𝑘)>0 (16)
Due to the asymptotical convergence property of the proposed Markovchain,wehave
𝜋( ̄Θ𝑘)= lim 𝑛→∞maxΘ𝑘 𝜋 𝑛( ̄Θ 𝑘||Θ𝑘)=𝑛lim→∞minΘ 𝑘 𝜋 𝑛( ̄Θ 𝑘||Θ𝑘) (17)
Since𝜋( ̄Θ𝑘)liesbetweentheminimumandmaximum𝜋𝑛( ̄Θ 𝑘||Θ𝑘)
foranygivenstate𝜃,(12)canbedirectlyensured.QED. □
Remark2. Theorem1showsthat,givenanyfeasiblejointassociation eventastheinitialstateΘ(1)𝑘 ,thegeneratedMarkovchain{Θ(𝑘𝑡)}∞
𝑡=1 ex-ponentiallyconvergestotheposteriorof thejoint event.Duetothe convergenceproperty,onecaneasilyselectafeasiblejointeventasthe initialstateforGibbssampler.Forexample,onecanchoosethejoint associationeventthatalltargetsareassumedtobemiss-detectedasthe initialstateoftheGibbssampler,e.g.,Θ(1)𝑘 =0𝑁𝑘 .
Remark3.SinceGibbssamplerisinitialisedwithrandomvalues, sam-plesgeneratedatearlyiterations,knownastheburn-inphase,usually cannotrepresentthetargetdistributionandneedtobediscarded. Typi-cally,thereisnorule-of-thumboranalyticallywaytosetthenumberof burn-inphasesamples.However,duetotheexponentialconvergence rate,theburn-inphaseoftheproposed Gibbssampleris short.Even thoughthenumberofburn-insamplesisempiricallyset,itsinfluence onthemarginalisationisignorablesincethegeneratedGibbssamples arenotusedforinferencethestationarydistribution(3).Thiswillbe empiricallyanalysedinthesimulationpart.
Remark 4. SimilartoMetropolis-Hastingssampling, the Gibbs sam-pling mightbecomeinefficient in exploring thespace withhigh di-mensionality,e.g., extremelylarge numberof targets,becauseofthe random-walkbehaviour [49]. Underthiscondition,the Hamiltonian MonteCarlosamplingcould beutilisedasanalternativewayto ob-tainthesamplesinamoreefficientway.ComparedtoGibbssampling, theHamiltonianMonteCarlosamplinghasfasterconvergencespeedfor high-dimensionaltargetdistribution,althoughthepriceofsingle itera-tionishigher[49].
TheGibbssampling-basedmarginalisationalgorithmdevelopedis summarisedinAlgorithm1.
4. DistributedUKF-basedinformationJPDAfilter
ThissectionproposesadistributedUKF-basedinformationJPDA fil-ter.Wefirstbrieflyreviewthewell-knownaverageconsensusalgorithm andthenpresentthedetailedfilteringalgorithm.Asthissectiononly considersdistributedestimationforeachmanoeuvremode,weignore themodenotationhereforsimplicity.
4.1. Averageconsensus
SupposethatNssensorsparticipateinacooperativedistributed
es-timationmission.Forthismulti-sensor system,weuseanundirected graph=(,)torepresentthecommunicationtopology,where= {
𝜈1,𝜈2,…,𝜈𝑁𝑠 }
isasetofverticesthatrepresentNs sensorsand= {(𝑖,𝑗)∈×}isasetofedgesthatstandfortherelationshipbetween twoneighbouringsensorsinthistopology.Iftwosensors(iandj)are ad-jacent,namely,theycancommunicatewitheachother,then(𝜈𝑖,𝜈𝑗)∈
and(𝜈𝑖,𝜈𝑗)∈.Thegraphissaidtobeconnectedifthereexistsapath
betweenanytwosensors.Theadjacencymatrixofgraph,denotedby
𝐴=[𝑎𝑖𝑗]∈ℝ𝑁𝑠 ×𝑁𝑠 isdefinedas𝑎𝑖𝑗=1,if(𝜈𝑖,𝜈𝑗)∈,otherwise𝑎𝑖𝑗=0.
Algorithm1 MarginalisationbyGibbssampling.
Input:Previous target estimation,receivedmeasurements,allowable samples𝑁𝑔𝑖𝑏𝑏𝑠,burn-insamples𝑁𝑏𝑢𝑟𝑛−𝑖𝑛
Output:Marginalassociationprobability𝛽𝑖 𝑗
1: 𝑖𝑡𝑒𝑟←1{settheinitialiterationcounterasone} 2: 𝑁𝜃𝑗 𝑖 ←0{settheeventcounteraszero}
3: Θ(1)𝑘 ←0𝑁𝑘 {settheinitialstatefortheGibbssampler} 4: while𝑖𝑡𝑒𝑟<𝑁𝑔𝑖𝑏𝑏𝑠do
5: 𝑖𝑡𝑒𝑟←𝑖𝑡𝑒𝑟+1 6: for𝑚=1∶𝑁𝑘do
7: Recursivesamplingaccordingto(11) 8: endfor 9: Θ(𝑘𝑖𝑡𝑒𝑟)= ( 𝜃1 𝑘,…,𝜃𝑘𝑁𝑘 ) {
oneGibbssample} 10: if𝑖𝑡𝑒𝑟>𝑁𝑏𝑢𝑟𝑛−𝑖𝑛then 11: if𝜃𝑖 𝑘=𝑗then 12: 𝑁𝜃𝑖 𝑗 ←𝑁𝜃𝑗 𝑖 +1 13: endif 14: endif 15: endwhile 16: 𝛽𝑖𝑗=𝑁𝜃𝑖 𝑗 ∕ ( 𝑁𝑔𝑖𝑏𝑏𝑠−𝑁𝑏𝑢𝑟𝑛−𝑖𝑛 )
Toperformestimationfusionin adistributedway,theconceptof consensusisadoptedhere.Inparticular,averageconsensusamong net-workedsensorsisperformed.Theaverageconsensusalgorithmisused toobtainthemeanvalueoftheinformationofallsensorsinadistributed way.Denotealastheavailableinformationfromthelthsensorandal
isinitialisedasal(0).Then,thedistributedaverageconsensusalgorithm [18,19]atthemthiterationisdefinedas
𝑎𝑙,𝑚=𝑎𝑙,𝑚−1+𝜀 ∑ (𝜈𝑙 ,𝜈𝑙 ′)∈𝑙 ( 𝑎𝑙′,𝑚−1−𝑎𝑙,𝑚−1 ) (18)
where𝑙denotesthesetofsensorsthathaveconnectionswiththelth
sensorand𝜀istheconsensusgain,whichisdesignedtotunethe conver-gencespeed.Toguaranteethestabilityoftheconsensusphase,thegain
𝜀shouldsatisfy𝜀∈(0,1/Δmax),whereΔmaxisthemaximumdegreeof undirectedgraph.
Basedontheanalysisof[18,19],itcanbeconcludedthat lim 𝑚→∞𝑎𝑙,𝑚= 1 𝑁𝑠 𝑁𝑠 ∑ 𝑙=1 𝑎𝑙(0) (19)
which meansthat theinformationof allsensors asymptotically con-vergestotheaveragevalue.
4.2. UKF-baseddistributedestimation
Thispaperconsidersmanoeuvringtargettrackingwithanonlinear observationmodel. Inthisregard,thewell-knownUKFis utilisedto handletheissueofnonlinearity.Itisknownthattheinformation fil-terisasuitableformulatoaddressmulti-sensordatafusionproblemin adistributedmanner[23,50].However,directapplicationofthe well-establishedinformationfilterforsingletargettrackingtotheMTTis intractableduetothemeasurementoriginuncertainty.Tothisend,this paperdevelopsadistributedinformation-formoftheJPDAfilterthat onlyexploitstheexchangedinformationamongtheneighboursensors. Thisformofthefilterisexceptedtoenablelowcommunicationload, fastimplementationandmorerobustnessagainstsensor failuresthan thecentralisedimplementation.Sincethepredictionstepinmulti-sensor filteringissimilartostandardpredictionofJPDA,weonlyderivethe correctionstepinthissubsection.
ThestateestimationinJPDAfiltercan berewrittenusing matrix inversionlemmaas
̂𝑥𝑖 𝑘|𝑘 =̂𝑥𝑖𝑘|𝑘−1+ [ 𝑃𝑖 𝑘|𝑘−1 ( 𝐻𝑖 𝑘 )𝑇( 𝑆𝑖 𝑘 )−1] ̃𝑧𝑖 𝑘 =̂𝑥𝑖𝑘|𝑘−1+ [( 𝑃𝑖 𝑘|𝑘−1 )−1 +(𝐻𝑘𝑖)𝑇𝑅−1𝐻𝑘𝑖 ]−1 ×(𝐻𝑘𝑖)𝑇𝑅−1 (𝑀𝑘 ∑ 𝑗=1𝛽 𝑖 𝑗𝑧𝑖𝑗,𝑘− ( 1−𝛽0𝑖)𝐻𝑘𝑖̂𝑥𝑖𝑘|𝑘−1 ) =̂𝑥𝑖𝑘|𝑘−1+(𝑌𝑘𝑖|𝑘−1+I𝑖𝑘)−1[i𝑖𝑘−(1−𝛽0𝑖)I𝑖𝑘̂𝑥𝑖𝑘|𝑘−1] =(𝑌𝑘𝑖|𝑘−1+I𝑖𝑘)−1(𝑦𝑖𝑘|𝑘−1+i𝑖𝑘+𝛽0𝑖I𝑖𝑘̂𝑥𝑖𝑘|𝑘−1) (20) wheretheinformation-relatedtermsaredefinedas
𝑌𝑖 𝑘|𝑘−1= ( 𝑃𝑖 𝑘|𝑘−1 )−1 ,𝑦𝑖 𝑘|𝑘−1= ( 𝑃𝑖 𝑘|𝑘−1 )−1 ̂𝑥𝑖 𝑘|𝑘−1 I𝑖𝑘=(𝐻𝑘𝑖)𝑇𝑅−1𝐻𝑖 𝑘, i𝑖𝑘= ( 𝐻𝑖 𝑘 )𝑇𝑅−1 𝑀𝑘 ∑ 𝑗=1 𝛽𝑖 𝑗𝑧𝑖𝑗,𝑘 (21)
Define a new information contribution ̄i𝑖𝑘=i𝑖𝑘+𝛽0𝑖I𝑖𝑘̂𝑥𝑖𝑘|𝑘−1, then, (20) canbereducedto ̂𝑥𝑖 𝑘|𝑘 = ( 𝑌𝑖 𝑘|𝑘−1+I𝑖𝑘 )−1( 𝑦𝑖 𝑘|𝑘−1+̄i𝑖𝑘 ) (22) Basedonthematrixinversionlemma,thecorrectionofthe informa-tionmatrix𝑌𝑖 𝑘|𝑘−1isderivedas 𝑌𝑖 𝑘|𝑘 = { 𝑃𝑖 𝑘|𝑘−1−𝐾𝑘𝑖 [( 1−𝛽𝑖0)𝑆𝑘𝑖− ̄𝑃𝑘𝑖](𝐾𝑘𝑖)𝑇} −1 =𝑌𝑘𝑖|𝑘−1+̄I𝑖𝑘 (23)
where ̄𝑃𝑘𝑖isapositivesemi-definitematrixrepresentingthe measure-mentoriginuncertaintyandtakestheform
̄𝑃𝑖 𝑘= 𝑀𝑘 ∑ 𝑗=1𝛽 𝑖 𝑗̃𝑧𝑖𝑗,𝑘 ( ̃𝑧𝑖 𝑗,𝑘 )𝑇 −̃𝑧𝑖𝑘(̃𝑧𝑖𝑘)𝑇 (24) and ̄I𝑖 𝑘=𝑌𝑘𝑖|𝑘−1𝐾𝑘𝑖 {[( 1−𝛽0𝑖)𝑆𝑘𝑖− ̄𝑃𝑘𝑖]−1−(𝐾𝑘𝑖)𝑇𝑌𝑘𝑖|𝑘−1𝐾𝑘𝑖 }( 𝐾𝑖 𝑘 )𝑇𝑌𝑖 𝑘|𝑘−1 (25) isaninformationmatrixcontribution.
Eqs.(22) and(23) constitutetheinformationformoftheJPDA fil-ter.Differentfromclassicalinformationfilter,theinformationformof JPDAfilterconsistsoftwodifferentinformationstatecontributionsand twodifferentinformationmatrixcontributions.Thedifferencesbetween i𝑖𝑘 and̄i𝑖𝑘,I𝑖𝑘and̄I𝑖𝑘areresulted fromthemeasurementorigin uncer-tainty. Apparently,ifthere is nomeasurement uncertainty,we have
𝛽𝑖
0=0and ̄𝑃𝑘𝑖=0,whichmeansthattheproposedinformationJPDA
filterreducestotheclassicalinformationfilter[19].Furthermore,ifthe
ithtargetismissdetected,then,𝛽𝑖
0=1and ̄𝑃𝑘𝑖=0,whichimpliesthat
̄i𝑖 𝑘= ( 𝐻𝑖 𝑘 )𝑇𝑅−1𝐻𝑖
𝑘̂𝑥𝑖𝑘|𝑘−1and𝑌𝑘𝑖|𝑘 =𝑌𝑘𝑖|𝑘−1.Therefore,iftheithtargetis missdetectedbyonesensor,thatsensorcanonlyprovidethe informa-tionaboutthepredictionoftheithtarget.
Notethatimplementing(20) requiresthemeasurementmatrix𝐻𝑘𝑖, whichisnotexplicitlygivenbythenonlinearmeasurementmodel.In ordertoapplyUKFinnonlinearfilteringtotheJPDAfilter,weusethe pseudomeasurementmatrixthatcanbederivedfromthestatistical lin-earerrorpropagationapproach as𝐻𝑘𝑖≈
( 𝑃𝑖 𝑘|𝑘−1 )−1 𝑃𝑖 𝑘,𝑥𝑧, where𝑃𝑘,𝑥𝑧𝑖
isthecross-correlationcovariance,whichcanbeapproximatedby un-scentedtransformationas 𝑃𝑖 𝑘,𝑥𝑧= 2𝑛 ∑ 𝑠=0 𝑊𝑠 ( 𝜆𝑖,𝑠 𝑘|𝑘−1−̂𝑥𝑖𝑘|𝑘−1 )( 𝛾𝑖,𝑠 𝑘 −̂𝑧𝑖𝑘 ) (26) where𝜆𝑖,𝑠𝑘|𝑘−1denotesthemappedsigmapointsthroughsystem transfor-mationfunction,𝛾𝑖,𝑠𝑘 themappedsigmapointsthroughsystem observa-tionfunction,Wstheweightsofsigmapoints,and̂𝑧𝑖𝑘thepredicted
mea-surementoftheithtarget,whichisapproximatedbŷ𝑧𝑖
𝑘=∑2𝑠𝑛=0𝑊𝑠𝛾𝑘𝑖,𝑠.
Basedonthepropertyofestimatorswithinformationform, incorpo-ratingadditionalinformationfromothersensorscouldbeachievedby summationofthecorrespondinginformationterms.Thisimpliesthat theoptimalcentralisedimplementationofJPDAwithNssensorsisgiven
by 𝑌𝑖 𝑘|𝑘=𝑌𝑘𝑖|𝑘−1+ 𝑁𝑠 ∑ 𝑙=1 ̄I𝑖 𝑙,𝑘 ̂𝑥𝑖 𝑘|𝑘= ( 𝑌𝑖 𝑘|𝑘−1+ 𝑁𝑠 ∑ 𝑙=1 I𝑖𝑙,𝑘 )−1( 𝑦𝑖 𝑘|𝑘−1+ 𝑁𝑠 ∑ 𝑙=1 ̄i𝑖 𝑙,𝑘 ) (27) Itfollowsfrom(27) thatcentralisedestimationrequiresfull infor-mationof allsensors.Consideringeachsensorusuallycanonly com-municatewithitsneighboursduetocommunicationlimit,thispaper developsadistributedimplementationbasedonconsensusalgorithmto recovertheperformanceofthecentralisedestimation(27).Assumethat theinformationstatesandmatricesofallsensorsconvergetotheglobal onesatpreviousscan,e.g.,eachsensorhasanidenticalcopyofthe sys-temstateandthesameamountofinformationmatrixafterconsensusat previousscan,(27)canbereformulatedas
𝑌𝑖 𝑘|𝑘= 𝑁𝑠 ∑ 𝑙=1 (𝑌𝑖 𝑙,𝑘|𝑘−1 𝑁𝑠 +̄I𝑖𝑙,𝑘 ) ̂𝑥𝑖 𝑘|𝑘= [𝑁𝑠 ∑ 𝑙=1 (𝑌𝑖 𝑙,𝑘|𝑘−1 𝑁𝑠 +𝐼 𝑖 𝑙,𝑘 )]−1(𝑁 𝑠 ∑ 𝑙=1 (𝑦𝑖 𝑙,𝑘|𝑘−1 𝑁𝑠 +̄i 𝑖 𝑙,𝑘 )) (28) Defineconsensusvariables𝑣𝑖
𝑙,𝑘,𝑉𝑙,𝑘𝑖 ,𝐺𝑖𝑙,𝑘,whichareinitialisedas 𝑣𝑖 𝑙,𝑘(0)= 𝑦𝑖 𝑙,𝑘|𝑘−1 𝑁𝑠 +̄i 𝑖 𝑙,𝑘, 𝑉𝑙,𝑘𝑖(0)= 𝑌𝑖 𝑙,𝑘|𝑘−1 𝑁𝑠 +̄I 𝑖 𝑙,𝑘 𝐺𝑖 𝑙,𝑘(0)= 𝑌𝑖 𝑙,𝑘|𝑘−1 𝑁𝑠 +I 𝑖 𝑙,𝑘 (29)
Inapracticalimplementationscenarioofmulti-sensorestimation, notallsensorscangetthemeasurementinformationofeachtargetdue tolimitedsensorfield-of-viewandnon-unitydetectionprobability.In thecasewherenomeasurementinformationoftheithtargetis avail-ableatthelthsensor,thequalityoflocalestimation̂𝑥𝑖
𝑘|𝑘 willbevery poorandisfarfromtherealstate𝑥𝑖
𝑘|𝑘.Fusingthispoorinformation withotherrelativelygoodlocalestimationwillobviouslydeteriorate theperformanceofthefusedresultsandmightresultinestimation di-vergence.Toaccommodatethisissue,wesettheinitialvaluesofthree consensusvariablesaszeroifthelocalsensornodecannotgetthe mea-surementof aspecifictarget.Inthissituation,thenaivesensorswill notperforminformationfusionstepsandleveragetheinformationfrom othernon-naivesensorsforestimationupdate.Thissimplestrategyis demonstratedtobehelpfulinimprovingthestabilityofthefusion pro-cess.
Afterseveralaverageconsensusiterations,eachsensorobtainsthe distributedestimationofthesystemstateandinformationmatrixas
𝑌𝑖 𝑙,𝑘|𝑘=𝑁𝑠𝑉𝑙,𝑘𝑖 ̂𝑥𝑖 𝑙,𝑘|𝑘 = ( 𝑁𝑠𝐺𝑖𝑙,𝑘 )−1( 𝑁𝑠𝑣𝑖𝑙,𝑘 ) = ( 𝐺𝑖 𝑙,𝑘 )−1 𝑣𝑖 𝑙,𝑘 (30)
Basedontheabovederivations,thefollowingpointsoftheproposed distributedinformationJPDAfilterareimportant.
(1)Theconsensusvariables𝑣𝑖
𝑙,𝑘and𝑉𝑙,𝑘𝑖 containtheeffectof
mea-surementoriginuncertainty.Thisshowshowdataassociationistightly integratedinconsensus-baseddistributedfiltering,whichhasnotbeen exploredinpreviousworks.
(2)Itfollowsfrom(30) thattheproposeddistributedmulti-target trackingalgorithmrequiresthetotalnumberofsensorsNs for imple-mentation.Thisinformationcanbecalculatedinadistributedwayas shownin[51].Inthecaseofsensorfailure,however,onemaygetthe wrongestimationofNs.InSec.VI,wewillshowthat,evenunderthe
conditionofsensorfailure,theproposedalgorithmcangetcomparable performancetothecentralisedone.
(3)Theoretically,theaverageconsensusguaranteesasymptotic sta-bility.Inpractice, sinceonlyfinitenumberof iterationsis tractable, convergencewillnotbefullyachieved.However,wecansafelyassume thatboththeestimatedsystemstateandtheinformationmatrixofall sensorsaretheequalafterenoughfiniteiterations.Thisassumptionisa keypointthatwecanapplyaverageconsensusalgorithmindistributed filteringandisreasonableastheconsensuserrorcouldbemade arbi-trarilysmallaftersufficientiterations.
Remark5. InMTToverasensornetwork,eachsensornodeordersits estimatedtracksdifferentlyandthereforetrack-to-trackassociationis requiredtoassociatethetracksfromdifferent sensorsthat represent theithtarget. Typicaltrack-to-trackassociation utilisestheso-called multi-dimensionalassociation(MDA)formulation.Ifonlytwosensors areutilisedinfusion,theMDAproblemreducestoaclassical2D assign-mentproblem,whichcanbeefficientlysolvedbythewell-known Hun-garianalgorithm.Whenthenumberofsensornodesislargerthantwo, thetrack-to-trackassociationMDAproblem becomesNP-hard.There areanumberofelegantchoicesforsolvingtheMDAproblemin com-binatorialoptimisationbyreformulatingtheproblemasanetworkflow [52] orusingapproximateLagrangianrelaxation[53,54] and stochas-ticsamplingapproach[55].However,thediscussionoftrack-to-track associationisbeyondthescopeofthispaperandweassumelocal es-timateshaveperfectmatchingwhenevaluatingtheperformanceofthe proposedalgorithmforsimplicity.
5. DistributedUKF-basedmultiplemodelinformationJPDAfilter ThissectiondevelopsthedistributedUKF-basedmultiplemodel in-formationJPDAfiltertoprovidethecapabilityforaccuratelyestimating manoeuvringtargetsbasedontheJMNSmodellingusingIMMconcept [42,56].
Let𝑍𝑙
𝑘denotethemeasurementsetreceivedfromthelthsensorat
scankanddefine𝑍1∶𝑘𝑁𝑠 ={𝑍1
𝑘,…,𝑍𝑘𝑁𝑠 }.Ingeneral,onefilteringcycle
ofJMNSconsistsoffoursteps: (1)𝑝(𝑥𝑖 𝑘−1|||𝑟𝑖𝑘−1,𝑍 1∶𝑁𝑠 𝑘−1 )Mixing → 𝑝(𝑥𝑖 𝑘−1|||𝑟𝑖𝑘,𝑍 1∶𝑁𝑠 𝑘−1 ) (2)𝑝(𝑥𝑖 𝑘−1|||𝑟𝑖𝑘,𝑍 1∶𝑁𝑠 𝑘−1 )Prediction → 𝑝(𝑥𝑖 𝑘|||𝑟𝑖𝑘,𝑍 1∶𝑁𝑠 𝑘−1 ) (3)𝑝(𝑥𝑖𝑘|||𝑟𝑖𝑘,𝑍𝑘1∶−1𝑁𝑠 )Bayes→ 𝑝(𝑥𝑖𝑘|||𝑟𝑖𝑘,𝑍𝑘1∶𝑁𝑠 ) (4)𝑝(𝑟𝑖 𝑘|||𝑍 1∶𝑁𝑠 𝑘−1 )Bayes → 𝑝(𝑟𝑖 𝑘|||𝑍 1∶𝑁𝑠 𝑘 )
Sincesteps(2)and(3)areaccomplishedbytheproposeddistributed informationJPDAfilter,thissectiononlyfocusesonsteps(1)and(4). Accordingtothetotalprobabilitytheorem,themixedpriorcanbe de-rivedas 𝑝(𝑥𝑖 𝑘−1|||𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘−1 ) = 𝑟 ∑ 𝑚=1 𝑝(𝑥𝑖 𝑘−1|||𝑟𝑖𝑘−1=𝑚,𝑍 1∶𝑁𝑠 𝑘−1 ) × Pr(𝑟𝑖𝑘−1=𝑚|| |𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘−1 ) (31) andthemodemixprobabilityPr(𝑟𝑖𝑘−1=𝑚||
|𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘−1 ) isgivenby Pr(𝑟𝑖𝑘−1=𝑚|| |𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘−1 ) = 𝜋𝑚𝑟𝑝 ( 𝑟𝑖 𝑘−1=𝑚|||𝑍 1∶𝑁𝑠 𝑘−1 ) ∑𝑟 𝑚=1𝜋𝑚𝑟𝑝 ( 𝑟𝑖 𝑘−1=𝑚|||𝑍 1∶𝑁𝑠 𝑘−1 ) (32)
Itfollowsfrom(31) thattheexactsolutionof JMNSestimationis aGaussiansumwith(𝑟)𝑘termsatscank.PropagationofGaussian mixturesisnaturallyintractableinrealapplications.Inorderto main-tainthecomputationalefficiency,theconceptofIMMisadoptedhere byusingasingleGaussiantoapproximatethemixedprior(31)atevery timeinstant.Thisimpliesthatthemixedinitialconditionforeachfilter
isgivenby ̂𝑥0𝑖,𝑟 𝑘−1|𝑘−1 = 𝑟 ∑ 𝑚=1 𝜇𝑖,𝑟𝑘−1|𝑚̂𝑥 𝑖,𝑚 𝑘−1|𝑘−1 𝑃0𝑖,𝑟 𝑘−1|𝑘−1= 𝑟 ∑ 𝑚=1 𝜇𝑖,𝑟𝑘−1|𝑚 [ 𝑃𝑘𝑖,𝑚−1|𝑘−1+ ( ̂𝑥𝑖,𝑚𝑘−1|𝑘−1−̂𝑥 0𝑖,𝑟 𝑘−1|𝑘−1 ) × ( ̂𝑥𝑖,𝑚 𝑘−1|𝑘−1−̂𝑥 0𝑖,𝑟 𝑘−1|𝑘−1 )𝑇] (33) where𝜇𝑘𝑖,𝑟−1|𝑚 =Pr ( 𝑟𝑖 𝑘−1=𝑚|||𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘−1 ) .
Byfeedingthemixedprior(̂𝑥0𝑘𝑖,𝑟−1|𝑘−1,𝑃𝑘0−1𝑖,𝑟|𝑘−1)astheinitial condi-tiontoeachinformationJPDAfilter,only𝑟modesarekeptatone
estimationcycle.
Based on the mode-conditioned update 𝑝(𝑥𝑖 𝑘|||𝑟𝑖𝑘,𝑍
1∶𝑁𝑠
𝑘
) and the mode probabilityupdate 𝑝(𝑟𝑖𝑘|||𝑍𝑘1∶𝑁𝑠 ),theposteriorprobability den-sityfunctionofonetargetcanbe representedbyaGaussianmixture distributionusingthetotalprobabilitytheoremas
𝑝(𝑥𝑖 𝑘|||𝑍 1∶𝑁𝑠 𝑘 ) = 𝑟 ∑ 𝑟=1 𝑝(𝑥𝑖 𝑘|||𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘 ) ×𝑝(𝑟𝑖𝑘=𝑟|| |𝑍 1∶𝑁𝑠 𝑘 ) (34) Thestateestimate anderrorcovariancematrixofeach target, ex-tractedfrom(34) usingmoment-matching,areobtainedas
̂𝑥𝑖 𝑘|𝑘= 𝑟 ∑ 𝑟=1 𝜇𝑖,𝑟 𝑘|𝑘̂𝑥𝑖,𝑟𝑘|𝑘 𝑃𝑖 𝑘|𝑘 = 𝑟 ∑ 𝑟=1 𝜇𝑖,𝑟𝑘|𝑘 [ 𝑃𝑘𝑖,𝑟|𝑘+(̂𝑥𝑖,𝑟𝑘|𝑘−̂𝑥𝑘𝑖|𝑘)(̂𝑥𝑖,𝑟𝑘|𝑘−̂𝑥𝑖𝑘|𝑘)𝑇 ] (35) where𝜇𝑘𝑖,𝑟|𝑘 =𝑝(𝑟𝑖𝑘=𝑟|| |𝑍 1∶𝑁𝑠 𝑘 )
istheposteriormodeprobability,which canbecalculatedbyBayesianruleas
𝜇𝑘𝑖,𝑟|𝑘 = 𝑝(𝑟𝑖 𝑘=𝑟|||𝑍 1∶𝑁𝑠 𝑘−1 ) 𝑝(𝑍1∶𝑁𝑠 𝑘 |||𝑍 1∶𝑁𝑠 𝑘−1 ) 𝑝(𝑍1∶𝑁𝑠 𝑘 |||𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘−1 ) (36) where𝑝(𝑍1∶𝑁𝑠 𝑘 |||𝑍 1∶𝑁𝑠 𝑘−1 )
isthenormalisationconstant,andthepredicted modeprobability𝑝(𝑟𝑖 𝑘=𝑟|||𝑍 1∶𝑁𝑠 𝑘−1 ) canbeobtainedas 𝑝(𝑟𝑖 𝑘=𝑟|||𝑍 1∶𝑁𝑠 𝑘−1 ) = 𝑟 ∑ 𝑚=1 𝜋𝑚𝑟𝑝 ( 𝑟𝑖 𝑘−1=𝑚|||𝑍 1∶𝑁𝑠 𝑘−1 ) (37)
Since the measurements from different sensors are independent, (36)canbefurtherreducedto
𝜇𝑘𝑖,𝑟|𝑘 = 𝑝(𝑟𝑖 𝑘=𝑟|||𝑍 1∶𝑁𝑠 𝑘−1 ) 𝑝(𝑍1∶𝑁𝑠 𝑘 |||𝑍 1∶𝑁𝑠 𝑘−1 ) 𝑁𝑠 ∏ 𝑙=1𝑝 ( 𝑍𝑙 𝑘|||𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘−1 ) (38)
Eq.(38) providesthecentralisedform ofmode probability calcu-lationandisrequiredtobedistributedforsensornetworks.Sincethe productformin(38) causesanissueindirectlyapplyingtheaverage consensusalgorithmintheproposeddistributedfusion,wedefinean auxiliaryvariable𝛿𝑙,𝑘𝑖,𝑟=lnΛ𝑙,𝑘𝑖,𝑟.NotethatΛ𝑖,𝑟𝑙,𝑘=𝑝
( 𝑍𝑙 𝑘|||𝑟𝑖𝑘=𝑟,𝑍 1∶𝑁𝑠 𝑘−1 ) is themodeconditionedmeasurementlikelihoodofthelthsensor.Then, wedefineconsensusvariable𝑄𝑖,𝑟𝑙,𝑘,whichisinitialisedas𝑄𝑖,𝑟𝑙,𝑘(0)=𝛿𝑙,𝑘𝑖,𝑟. Afterrunningaverageconsensusalgorithmfor𝑄𝑖,𝑟𝑙,𝑘,thefusionof the posteriormodeprobabilityinadistributedwayisgivenby
𝜇𝑖,𝑟 𝑘|𝑘= 𝑝(𝑟𝑖 𝑘=𝑟|||𝑍 1∶𝑁𝑠 𝑘−1 ) 𝑝(𝑍1∶𝑁𝑠 𝑘 |||𝑍 1∶𝑁𝑠 𝑘−1 ) exp ( 𝑁𝑠 ( 1 𝑁𝑠 𝑁𝑠 ∑ 𝑙=1 𝛿𝑖,𝑟 𝑙,𝑘 )) = 𝑝(𝑟𝑖 𝑘=𝑟|||𝑍 1∶𝑁𝑠 𝑘−1 ) 𝑝(𝑍1∶𝑁𝑠 𝑘 |||𝑍 1∶𝑁𝑠 𝑘−1 ) exp(𝑁𝑠𝑄𝑖,𝑟𝑙,𝑘 ) (39)
Afterfindingtheposteriormodeprobability𝜇𝑖,𝑟𝑘|𝑘,thefinalstate es-timateandcovariancematrixofeachtargetisgivenby(35).
Remark6. Consensuson𝑄𝑖,𝑟𝑙,𝑘requiresΛ𝑖,𝑟𝑙,𝑘:thiscanbereadilyobtained bysummingupallthePDAprobability(4),i.e.,
Λ𝑖,𝑟𝑙,𝑘=(1−𝑃𝐷𝑃𝐺)𝜆𝐹+ 𝑀𝑙,𝑘 𝑖,𝑟 ∑ 𝑗≠0𝑝 ( 𝑧𝑙,𝑗,𝑘|||𝑥𝑖,𝑟𝑙,𝑘 ) 𝑃𝐷 (40)
where𝑀𝑙,𝑘𝑖,𝑟denotesthenumberofvalidatedmeasurementsfortheith targetwiththerthmodefromthelthsensor.
Remark7. Comparedwithsingle-modedistributedJPDAfilter, incor-poratingIMMwithmodeprobabilityfusionwillinevitablyincreasethe computationalburden.However,itiscleartoverifythatthe complex-ityoftheproposeddistributedUKF-basedmultiplemodelinformation JPDAfilterisproportionaltothenumberofmodes𝑟,e.g.,the
pro-posedalgorithmisscalable.
The proposed multiple model UKF-baseddistributed information JPDAfilterissummarisedinAlgorithm2.
6. Numericalsimulations
Inthissection,theeffectivenessoftheproposedmultiplemodel UKF-baseddistributedinformationJPDAfilteringalgorithmisdemonstrated throughnumericalsimulationsinaclutteredenvironment.Thetarget trackingproblemisbasedonsomegenericairtrafficcontrol(ATC) sce-narios.FortypicalATCcases,thebehaviourofcivilianaircraftmaybe modelledbytwodifferentmodes:constantvelocity(CV) and coordi-natedturning(CT).
6.1. Simulationsetup
Inoursimulations,7targets,switchingbetweenCVmodelandCT model,areconsidered.Thestatevectorcontains planarpositionand velocity.Morespecifically,forCVmodel,thestatetransitionis
𝑥𝑘=𝐹𝐶𝑉𝑥𝑘−1+𝐺𝑤𝑘−1 (41) with 𝐹𝐶𝑉 Δ =𝕀2×2⊗ [ 1 𝑇 0 1 ] , 𝐺=Δ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑇2∕2 0 𝑇 0 0 𝑇2∕2 0 𝑇 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (42)
where𝕀2×2denotesthe2×2identitymatrix,𝑇=1𝑠thesamplingperiod, and𝑤𝑘∼(⋅;0,𝜎2𝑣)theGaussianprocessnoisewith𝜎𝑣=5𝑚∕𝑠2.The
statetransitionofCTmodelis
𝑥𝑘=𝐹𝐶𝑇𝑥𝑘−1+𝐺𝑤𝑘−1 (43) with 𝐹𝐶𝑇 Δ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 sin(𝜔𝑘 𝑇) 𝜔𝑘 0 − 1−cos(𝜔𝑘 𝑇) 𝜔𝑘 0 cos(𝜔𝑘𝑇) 0 −sin(𝜔𝑘𝑇) 0 1−cos𝜔(𝜔𝑘 𝑇) 𝑘 1 sin(𝜔𝑘 𝑇) 𝜔𝑘 0 sin(𝜔𝑘𝑇) 0 cos(𝜔𝑘𝑇) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (44) 𝐺=Δ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 𝑇2∕2 0 𝑇 0 0 𝑇2∕2 0 𝑇 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (44)
Algorithm2 DistributedUKF-basedmultiplemodelinformationJPDA filter.
Input:Previoustargetestimation,receivedmeasurements Output:Currenttargetestimation
(1) Step1.ModeMixing
a.Calculatethemodemixprobability𝜇𝑘𝑖,𝑟−1|𝑚 using(32) b.Calculatethemixedpriorstateestimation̂𝑥0𝑖,𝑟
𝑘−1|𝑘−1and errorcovariance𝑃0𝑖,𝑟
𝑘−1|𝑘−1 using(33)
(2) Step2.Mode-conditioneddistributedJPDAestimation (Sec.4)
a.Predicttargetstateandcalculatetheerrorcovariance basedon ̂𝑥0𝑖,𝑟
𝑙,𝑘−1|𝑘−1and𝑃 0𝑖,𝑟 𝑙,𝑘−1|𝑘−1
b.Receivemeasurementsandperformgatingwith probability𝑃𝐺
c.ApplyGibbssamplingformarginalassociationprobability
𝛽𝑖
𝑗approximation(Sec.3)
d.Calculatetheinformationterms𝑌𝑙,𝑘𝑖|𝑘−1,
𝑦𝑖
𝑙,𝑘|𝑘−1,i𝑖𝑙,𝑘,I𝑖𝑙,𝑘,̄i𝑖𝑙,𝑘,̄I𝑖𝑙,𝑘
e.Broadcastmessagetoneighboursensorsandreceive neighbours′messageson𝑣𝑖,𝑟
𝑙,𝑘,𝑉𝑙,𝑘𝑖,𝑟,𝐺𝑖,𝑟𝑙,𝑘
f.Performaverageconsensusforeach𝑣𝑖,𝑟𝑙,𝑘,𝑉𝑙,𝑘𝑖,𝑟,
𝐺𝑖,𝑟𝑙,𝑘independently
g.Getfusedposteriortargetestimation̂𝑥𝑖,𝑟𝑘|𝑘
and𝑃𝑘𝑖,𝑟|𝑘foreachmode
(3) Step3.Modeprobabilityfusionandupdate (Sec.5)
a.Calculatethemodeconditionedmeasurementlikelihood (40)
b.Broadcastmessagetoneighboursensorsandreceive neighbours′messageson𝛿𝑖,𝑟
𝑙,𝑘
c.Performaverageconsensusfor𝛿𝑖,𝑟𝑙,𝑘
d.Calculatefusedposteriormodeprobability𝜇𝑖,𝑟𝑘|𝑘
accordingto(39) (4) Step4.Output
Givenallthemode-conditionedfusedestimates,thefinal stateestimateofeachtargetisobtained
asaweightedsumofindividualfusedestimatesbyusing(35)
where𝜔𝑘=6𝜋∕180𝑟𝑎𝑑∕𝑠istheturningrate.Thenonlinearrangeand bearingmeasurementmodelforstatecorrectionis
𝑧= ⎡ ⎢ ⎢ ⎣ √( 𝑥𝑇−𝑥𝑅)2+(𝑦𝑇−𝑦𝑅)2 arctan(𝑦𝑇 −𝑦𝑅 𝑥𝑇 −𝑥𝑅 ) ⎤⎥ ⎥ ⎦ +𝑣𝑘 (45)
where (xT, yT) is target position, (xR, yR) radar position, and𝑣𝑘∼
(⋅;0,𝑅𝑘) the Gaussian measurement noisewith 𝑅𝑘=𝑑𝑖𝑎𝑔(𝜎2
𝑟,𝜎2𝑎
)
, 𝜎𝑟=20𝑚,𝜎𝑎=2(𝜋∕180)𝑟𝑎𝑑.
Forallsensors, themeasurementsaregeneratedwithadetection probability 𝑃𝐷=0.85andtheclutterisassumedtobeuniformly dis-tributedinthesurveillanceregionwithitsnumberbeingPoissonwith 10averagereturnsateachscan.Gatingisperformedwithathreshold suchthatthegatingprobabilityis𝑃𝐺=0.999.Thefield-of-viewofall
radarsaresetas[0,3000m]×[0,90∘].Inordertofullycovertheentire surveillanceregion,weusetheminimumfourradars,whicharefixed at(−1500𝑚,−600𝑚),(−1500𝑚,2000𝑚),(1600𝑚,−600𝑚),(1600m,2000m). Thedefaultcommunicationtopologyamongthesefourradarsis repre-sentedbytheadjacentmatrix
𝐴= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (46)
Fig.2. Snapshots of the considered scenario with grey stars as measurements, black solid line ground truth, colour dashed line estimation, red triangle sensors. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Itfollowsfrom(46)thatthemaximumdegreeofthenetwork undi-rected graphis Δmax=2 andthen the consensus gain is set as 𝜀= 0.5∕2.Forinformationexchange,theconsensusiterationisselectedas
𝐼𝑡𝑒𝑟max=10,whichisshowntobeenoughtoreducetheconsensus er-ror.ThedesignparametersforGibbssampling-basedimplementation aresetas𝑛max=200,𝑛𝑏𝑢𝑟𝑛−𝑖𝑛=100.Allexperimentsareperformedon
Matlab2016bplatformusinganIntelCorei5-6500CPUandthenumber ofMonte-Carlorunsis50.
Insimulations,threedifferentmodelsareconsidered,i.e.CVmodel, leftCTmodel,rightCTmodel.Themodetransformationprobabilityis selectedas Π ={𝜋𝑚𝑟}3×3= ⎡ ⎢ ⎢ ⎣ 0.8 0.1 0.1 0.1 0.8 0.1 0.1 0.1 0.8 ⎤ ⎥ ⎥ ⎦ (47)
SnapshotsoftheconsideredscenarioaredepictedinFig.2.The well-knownunitlessoptimalsub-patternassignment(OSPA)distancemetric [57] forMTTproblemisconsideredhereforperformanceevaluation. LetXandYbethepositionestimationsetandtruetargetpositionset, respectively.Thecardinalityofthesetwosetsaremandn,respectively. DenoteΠnasthesetofallpermutationson{1,2,…,𝑛}foranypositive integern.𝑑𝑐(𝑥
𝑖,𝑦𝜋(𝑖) )
=min(𝑑(𝑥𝑖,𝑦𝜋(𝑖) )
,𝑐)withd(xi,y𝜋(i))is the cut-off Euclideandistancebetweentwovectorswithd(xi,y𝜋(i))beingthe Euclideandistance.Then,forc>0and1≤p<∞,theOSPAdistance
𝑑𝑐 𝑝(𝑋,𝑌)isdefinedas[57] 𝑑𝑐 𝑝(𝑋,𝑌) Δ = ⎧ ⎪ ⎨ ⎪ ⎩ [ 1 𝑛 ( min𝜋∈Π𝑛 ∑𝑚𝑖=1𝑑𝑐(𝑥𝑖,𝑦𝜋(𝑖) )𝑝+𝑐𝑝(𝑛−𝑚))]1∕𝑝 , 𝑚≤𝑛 𝑑𝑐 𝑝(𝑌,𝑋), 𝑚>𝑛 (48) wheretheorderparameterpdeterminesthesensitivityof𝑑𝑐
𝑝(𝑋,𝑌)in
penalizingestimationoutliers,whilethecut-off parametercdetermines therelativeweightingofthepenaltiesallocatedtocardinalityand local-izationerrors.Inallsimulations,thesetwoparametersaresetas𝑝=2,
𝑐=100.
6.2. Characteristicsoftheproposedalgorithm
Forconsensus-basedfiltering,multiplecommunicationsbetween dif-ferentsensorsarerequiredandtheperformanceisrelatedtothe
consen-susgainandthenumberofiterations.Inordertoinvestigatetheeffect ofthesetwoparametersonfilteringperformance,Monte-Carlo simula-tionsareperformedwithrespecttodifferentiterationsandconsensus gain.ThesimulationresultsoftheOSPAdistancearedepictedinFigs.3 (a)and(b).Whentestingtheeffectofoneparameter,theotheroneis settoitscorrespondingdefaultvaluepresentedinSec.VIA.Fig.3(a) showsthattheperformanceofdistributedestimationisimprovedwhen thenumberofiterationsincrease.DistributedJPDAfilteringwith iter-ationsgreaterthan5hascloseperformancewithitscentralisedJPDA counterpart.Thisimpliesthattheperformanceofcentralisedestimation canberecoveredwithenoughconsensusiterations.Fig.3(b)showsthat improvementinestimationcanbeobtainedbyincreasingtheconsensus gain.However,withlargeenoughgains,thereisnotmuchdifference fortheproposeddistributedJPDAwithdifferentconsensusgains.
Asmentionedbefore,thenumberoftheburn-insamplesis empir-icallyset.Fig.3 (c)studiestheimpactofnumberburn-insampleson theperformanceoftheproposedtrackingalgorithm.Fromthisfigure,it isclearthat𝑛𝑏𝑢𝑟𝑛−𝑖𝑛=60achievesthebestestimationaccuracy,butthe
trackingperformanceintermsoftheOSPAdistanceunderall consid-eredconditionsiscomparable.Thismeansthattheproposedalgorithm isrobustagainstthevariationofnumberofburn-insamples.
In distributed network estimation, the communication structure playsanimportantroleingoverningtheoverallfilteringperformance. Theperformanceoftheproposedalgorithmiscomparedbyusingfour differentsensornetworksA1,A2,A3andA4:
𝐴1= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 1 1 0 1 0 0 1 0 1 1 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐴2= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 𝐴3= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 0 1 0 0 1 0 0 1 0 1 1 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , 𝐴4= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 0 0 1 1 0 0 1 0 1 1 0 1 1 0 1 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (49)
Theresults,showninFig.3(d),revealthattheOSPAdiscrepancies between differentcommunication topologiesarequitesmall, demon-stratingthatproposedalgorithmisrobustagainstthevariationof com-municationstructures.
Now,letusinvestigatetheeffectofthetotalnumberofsensorson filteringperformance.Exceptforthefourdefaultsensorsmentionedin Fig.2,thepositionsofothersensorsarerandomlyplaced.Fig.3(e) pro-videstheOSPAdistanceoftheMonte-Carlosimulationswithrespectto differenttotalnumbersofsensors.Inthissimulation,thesensorrangeis setas3000m.OnecannotefromFig.3(e)thatlargernumberofsensors renderssmallerOSPAdistance,butthedifferenceintermsofestimation accuracyisignorablewhenthesensornumberislargerthan6.This re-sultrevealsthattheminimumnumberofsensorsthatcovertheentire surveillanceregionisenoughfortheproposedfiltertoobtaingood per-formance.Therecordedmeanrunningtimeofthesefourscenariosare 14.9137s,17.6921s,20.3121s,22.9994s,25.0001s,respectively,which revealsthattherunningtimeoftheproposedalgorithmgrowslinearly withtheincreasingofsensornumbers.
Asmentionedearlier,theproposeddistributedmulti-targettracking algorithmrequiresthetotalnumberofsensorsforimplementation. Al-thoughthisinformationcanbecalculatedinadistributedwayasshown in[51],onemaygetthewrongestimationofNsinthepresenceof
sen-sorfailure.Tothisend,wetesttherobustnessandsensitivityof the proposeddistributedfilteragainstthewrongestimationofNs.Inthis
regard,letthetotalnumberofsensorsbe𝑁𝑠+Δ𝑁𝑠withΔNsbeingthe
biasedterm.Fig.3(f)showstheOSPAdistanceoftheMonte-Carlo simu-lationswithrespecttodifferentbiasedterms.Theactualsensornumber forthistestis7.Fig.3(f)revealsthattheOSPAdiscrepanciesbetween differentbiasedtermsarequitesmall,demonstratingthatproposed al-gorithmishighlytoleranttothewrongestimationofNs.
Fig.3. Characteristics of the proposed algorithm: OSPA distance under different conditions.
(s)
Fig.4. OSPA distance of different algorithms.
6.3. Comparisonswithotheralgorithms
Inordertoverifytheeffectivenessoftheproposedalgorithm com-paredtootherPDA-typemulti-sensorfilters,wecomparetheproposed onewithJPDA-KCF[25,26] andPDA-ICF[31].Tomakefair compar-isons,alltestedalgorithmsareenhancedbytheIMMformanoeuvring targettracking.Fig.4showstheOSPAdistanceoftheMonte-Carlo sim-ulationsobtained bydifferent algorithms.As notall sensorscan the measurementofeachtargetduetosensorfield-of-viewlimitand none-unitydetectionprobabilityintheconsideredscenario,thereexistnaive sensorsforsometargetsatsometimeinstants.Thisfactresultsin per-formancedegradationofJPDA-KCF.Notsurprisingly,sincePDAonly considersonemeasurement-targetassociationsandneglecttheeffectof
otherpossiblesolutions,notrealisticforsomescenarios,theobtained meanOSPAdistanceofthePDA-ICFislargerthanthatoftheproposed algorithm.
TodemonstratetheefficiencyofGibbssampling-aided implementa-tion,Monte-Carlosimulationsarecarriedoutandthecomparison re-sultsamongexactJPDA,ENNJPDA[12],m-bestJPDA[15] andGibbs sampling-aidedimplementationarepresentedinFigs.5 and6,where Fig.5 isfordifferentnumberofclutterreturnsatonescanandFig.6is fordifferentnumberoftargets.InENNJPDA,onlythejointeventwith thehighestprobabilityispickedupformarginalisation.Asanextension ofthisidea,m-bestJPDAmaintainsm-bestjointeventsforthe marginali-sation.Inthisregard,theENNJPDAcanbeviewedasaspecialcaseofm -bestJPDAwith𝑚=1.In[15],itwasshownthatthem-bestjointevents canbeiterativelysolvedbylinearprogramming(LP).Inthesimulations, weleveragethecommercialGurobisolvertoderivebothENNJPDAand
m-bestJPDAwith𝑚=5.Additionally,inordertoreducethecomplexity ofLPproblemsform-bestJPDA,thebinarytreepartitionmethod[15]is alsoadoptedinsimulationsform-bestJPDAimplementation.Theresults inFigs.5 (a)and6 (a)revealthattheproposedGibbssampling-aided implementationrunsmuchfasterthanstandardJPDAaswellasm-best JPDA.However,asENNJPDAonlyneedstosolveoneLPproblemto ob-tainthebestjointevent,itrequiresslightlylessrunningtimethanthe proposedmethodinadenseclutterenvironmentorwithlargenumber oftargets.Notably,thezoomed-ingraphinFig.6(a)demonstratesthat theexecutiontimeoftheproposedGibbssampling-aidedJPDAgrows linearlywithrespecttothetargetnumbersandtherebytheproposed algorithmisscalable.Forlarge-scaleproblem,thescenariowith14 tar-getsforinstance,exactJPDA,m-bestJPDAtake1374.8034s,41.8488s, respectively,whileGibbssampling-aidedimplementationonlyrequires 25.9695s.ItisevidentfromFigs.5(b)and6(b)thattheestimation per-formanceofexactJPDA,m-bestJPDAandtheproposedoneintermsof OSPAdistanceaccuracyiscomparable.Fig.5 (b)alsoshowsthatthe meanOSPAdistanceofstandardJPDAisrelativelylowerthanthe pro-posedalgorithmandm-best JPDAwhenthenumberofaverage clut-terreturnsincreases.Thereasonofthisphenomenonliesinthatboth
