Survey of maneuvering target tracking: part III - Measurement Models

24 Pages·0.234 MB·English

by  Rong Li X., Vesselin P.Jilkov.

ProceedingsofSPIEConferenceonSignalandDataProcessing
ofSmallTargets,SanDiego,CA,USA,July-August2001.(4473-41)
A Survey of Maneuvering Target Tracking—Part III: Measurement Models
(cid:3)
X.RongLi and VesselinP.Jilkov
DepartmentofElectricalEngineering
UniversityofNewOrleans
NewOrleans,LA70148,USA
504-280-7416(phone),504-280-3950(fax),[email protected],[email protected]
Abstract
Thisisthethirdpartofaseriesofpapersthatprovideacomprehensivesurveyofthetechniquesfortrackingmaneuvering
targetswithoutaddressingtheso-calledmeasurement-originuncertainty.PartI[1]andPartII[2]dealwithgeneraltargetmo-
tionmodelsandballistictargetmotionmodels,respectively. Thispartsurveysmeasurementmodels,includingmeasurement
model-based techniques, used intargettracking. ModelsinCartesian,sensormeasurement, theirmixed, andothercoordi-
natesarecovered.Thestressisonmorerecentadvances—topicsthathavereceivedmoreattentionrecentlyarediscussedin
greaterdetails.
KeyWords: TargetTracking,MeasurementModel,Survey
1 Introduction
Thispaperisthethirdpartofaseriesthatprovidesacomprehensivesurveyoftechniquesformaneuveringtargettracking
withoutaddressingtheso-calledmeasurement-originuncertainty.
Mostmaneuveringtargettrackingtechniquesaremodelbased; thatis, theyrelyonexplicitlytwo descriptions: onefor
thebehaviorsofthe target,usuallyintheformofamotion(ordynamics)model,andtheotherforourobservationsofthe
target, knownas anobservationmodel. A surveyoftargetmotionmodelsin generaland ballistic targetmotionmodelsin
particular has been reported in Part I [1] and Part II [2], respectively. This part surveys the measurement models and the
relevantmodelingtechniques.
Moreprecisely,thispapersurveysthemodelsofmeasurementscharacterizedbythefollowing: theyaretrulyoriginated
fromthe“pointtarget”undertrack(i.e.,thereisnooriginuncertainty);andtheyaremeasurements,ratherthanobservations
inamoregeneralsense, whichmaycontainotherinformation,includingtargetfeaturesasprovidedbyanimagingsensor.
Also,thissurveyisconcernedwiththemathematicalmodelsasabasisformaneuveringtargettracking. Forexample,their
other applications are not addressed and the actual sensor models are not of concern. Further, this surveyincludes some
aspectsofestimationandfilteringtechniquesthatarehighlydependentonandthushardlyseparablefromthemeasurement
models.
As in our survey of motion models [1, 2], we highlight underlying ideas and try to clarify both explicit and implicit
assumptionsinvolvedineachmodel,inanattempttorevealprosandconsofthemodels.Aconsiderableamountofdiscussion
isgiventowardsthisend,muchofwhichcannotbefoundelsewhere. Weremindthereader,however,thatthesediscussions,
althoughintendedtobeaccurateandbalanced,areobviouslynotnecessarilyfreeofourpersonalpreferenceandbias. Also,
wefocusonmorerecentadvancesinmeasurementmodels—topicsthathavereceivedmoreattentionrecentlyarediscussed
ingreaterdetails.Furthermore,somemodelsandtechniquespresentedinthispaperhavenotyetappearedelsewhere.
AsstatedinPartI,weappreciatereceivingcommentsandanymissingmaterialthatshouldbeincludedinthispart.
Therestofthepaperisorganizedasfollows.Sec.2describesmeasurementmodelsintheoriginalsensorcoordinates.Sec.
3providesageneralviewoftherolesofcoordinatesystemsinmaneuveringtargettracking.Linearizedmeasurementmodels
in(Cartesian-sensor)mixedcoordinatesarepresentedinSec.4.MeasurementmodelsconvertedtoCartesiancoordinatesare
coveredinSec.5.PseudomeasurementmodelingtechniquesaresurveyedinSec.6.Finally,concludingremarksaregivenin
Sec.7.
ResearchsupportedbyONRGrantN00014-00-1-0677andNSFGrantECS-9734285.
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1
2 ModelsinSensorCoordinates
Sensorsusedfortargettrackingprovidemeasurementsofa targetina naturalsensorcoordinatesystem(CS) orframe.
In many cases (e.g., with a dish radar), this CS is spherical in 3D or polar in 2D with range , bearing (or azimuth) ,
elevation (Fig. 1)1, andpossiblyrangerate(orDoppler) . (Wedonotexplicitlyconsiderdirerctmeasurementsoftargebt
height,aseprovidedby,e.g.,ModeC.Suchmeasurementsarer_usuallynotavailablefornon-cooperativetargets.)Notallthese
measurementcomponentsare available fromall sensors. For example, some active sensors may not providerangerate or
elevationangle,whilepassivesensorsprovideonlyangles(althoughpassiverangingispossible). Weconsidergenerallythe
3Dcase—therespective2Dcasefollowsinastraightforwardmanner.
Inthesensorcoordinates,thesemeasurementsaregenerallymodeledinthefollowingformofadditivenoise
(1)
r = r+vr
(2)
b = b+vb
(3)
e = e+ve
(4)
r_ = r_+vr_
where denotestheerror-freetruetargetpositioninthesensorsphericalcoordinates,and aretherespective
random(rm;be;aes)urementerrors. Weassumethesemeasurementsaremadeattime (or forshvorrt;)vbbu;vtew;evr_willomitthetime
index wheneverpossiblewithoutambiguities. ItisnormallyassumedthatthetskemeaksurementerrorsinthesensorCSare
zero-mkean,Gaussiandistributed,anduncorrelated:
with cov diag (5)
2 2 2 2
vk (0;Rk) Rk = (vk)= ((cid:27)r; (cid:27)b; (cid:27)e; (cid:27)r_)
where isth(cid:24)eeNrrorvector2attime and isawhitenoisesequence.
0
vk =[vr;vb;ve;vr_]k k vk
f g
6
z
r
(cid:0)1
(cid:0)1 cos v
cos u e y -
b
x
=
Fig.1: Sensorcoordinatesystems.
Theabovemeasurementmodelinsphericalcoordinatesismostcommonfortrack-while-systems,e.g.,rotatingsurveil-
lanceradars[3,4,5].Forscan-while-tracksurveillancesystems[6],suchasphasedarrayradars[7,8,9],thesensorprovides
measurementsintermsofthedirectioncosines and —insteadofthebearing andelevation —ofthetargetposition
(Fig.1)relativetoreferenceaxes.TheRUVmeausuremventmodelisasgivenabovebwith(2)–(3)repelacedby
(6)
u = u+vu
(7)
v = v+vv
where and denote the error-free target position direction cosines, and are the respective measurement errors.
Sometiumesatvhirddirectioncosinemeasurement isusedforconvevnui;evnvce,albeitredundant( ).
w =w+vw w = p1 u2 v2
1Otherconventionsarealsoused.Forexample,thebearingmaybedefinedastheanglefromthe axis,ratherthanfromthe axis. (cid:0) (cid:0)
2WeuseSansSerifletters(e.g., )todenoteidealerror-freequantitiesandbold-faceletterstodenyotevectors. x
z
2
Itisalsonormallyassumedthattheerrorsarezero-mean,Gaussiandistributed,anduncorrelated:
with cov diag (8)
2 2 2 2
vk (0;Rk) Rk = (vk)= ((cid:27)r; (cid:27)r_; (cid:27)u; (cid:27)v)
(cid:24)N
with .
0
Nvokte=th[avtrt;hveuR;vUvV;-vCr_]Skisnotorthogonal. Nevertheless,theaboveuncorrelatednessassumptioncov diag ,
2 2
iswelljustifiedbythefactthat , ,and aremeasuredbythreephysicallyindependentsystem(vs.k) = ((cid:27)r; (cid:27)u
2 2
(cid:27)v;T(cid:27)hr_e)abovetwomodelsarisenaturallryfuromthvemeasurementprocess. TheyarelinearandGaussianandcanbewritten
compactlyinvector-matrixnotation
(9)
z=Hx+v; v (0;R)
where or , (cid:24)oNr , or
0 0 0 0 0
z = [r;,b; e; r_] ,zan=d [ri;suan; vid;er_n]tityxm=atr[irx;abn;de; r_i;s:a::z]eromxat=rix.[r; u; v; r_; :::] v = [vr;vb;ve;vr_] v =
0
[vr;Rvuan;vgve;avnr_d] anHgl=em[Ie;aOsu]remenItsmayhavevastlydifferenOtaccuracies. Forexample,aphasedarrayradarhasrangemea-
surementsmuchmoreaccuratethananglemeasurements—itserrorellipsoidlookslikeapancakenormaltotherangevector;
thesituationis reversedfora continuous-waveradar,whichoftenhas a cigar-shapederrorellipsoidalongtherangevector
[8].
Thislinearmodeliscompletelyuncoupledacrossdifferentcoordinates. Thisishighlydesirableforestimationandfil-
teringinanumberofaspects. Forexample,efficientparallelprocessingmaybeaccomplishedwithlittleornoperformance
degradation. More important, coordinate-decoupledfilters may be implementedthat mitigate the possible ill conditioning
arisingfromthevastlydifferentaccuraciesinmeasuringrangeandangles[8]. Thesedecoupledfiltersmaypossiblyoutper-
formthetheoreticallysuperiorfull-blown“optimal”filtersinthepresenceofreallyillconditioning.
3 TrackinginVariousCoordinates
Various coordinatesystems (CS) have been used in target tracking, including the Earth-centeredinertial (ECI), Earth-
centered (Earth) fixed (ECF, ECEF, or ECR), East-North-Up (ENU), and radar face (RF) coordinate systems. A concise
description of these coordinate systems has been given in Part II [2]. Many factors affect the selection of a coordinate
systems[8,10,11,9]. TheENU-CSisacommonchoicefortacticalsystemswithrelativelylimitedsensormotion,suchas
inaplatform-centricsystem. TheECI-CS,alongwithitsvariantECF-CS,isatypicalchoiceforastrategicsysteminvolving
multipleplatforms.
As far as tracking accuracy is concerned, the probably best choice of a coordinate system in principle is to align its
coordinatesto the principalaxes of the trackingerror ellipsoid [8]. This will avoid the corruptionof an accurate estimate
componentbyinaccurateones.Italsoprovidesagoodframeworkagainstillconditioning.Sincetheseprincipalaxesusually
vary with respect to time in a complex way, a sensible strategy is to align the coordinates to the principal axes of either
the measurement error ellipsoid or dynamic error ellipsoid, depending on which error has more important directionality
properties. Following this principle, several coordinate systems were discussed in [8] in the context of a single sensor,
includingradar-oriented,target-oriented,andtheircombinations.
A measurementis oftendescribedin a sensorreferenceframe, whichis usuallystabilized relativeto the motionof the
sensor,evenifadifferentCSisselectedfortrackingpurposes.Itisingeneraldifferentfromaplatform(orsite)CS(e.g.,the
ENU-CS)whenmultiplesensorsareinvolvedintheplatform(e.g.,aship). WhilethegeographicalENUframecenteredata
sensorisconvenientforarotatingtrack-while-scanradar,thesensor-specificradarfaceCSismoreoftenusedwithaphased
arrayradar,where and axesareintheradarfaceplaneand axisalongtheboresightdirection.
Fordetailedcoxnsideryationsofthecoordinatessystemsandztherespectivetransformations,thereaderisreferredto[7,8,
10,12,11,9].
In the sequel, by a Cartesian coordinate system, we mean a generic one unless otherwise is stated explicitly; and by
a sensor frame (or CS), we mean non-Cartesian (spherical or RUV) CS in which the measurements are available directly
withoutcoordinatetransformation.
TargetmotionisbestdescribedinaCartesianCS,butmeasurementsareavailablephysicallyinasensorCS.Assuch,there
arebasicallyfourpossibilitiesofdotracking:trackinginmixedcoordinates,inCartesiancoordinates,insensorcoordinates,
andinothercoordinates.Thesearedescribednext.
3
3.1 TrackinginMixedCoordinates
Thisisthemostpopularapproach.Thetargetdynamicsandmeasurementsaremodeledby
(10)
z = h(x)+v
wherethetargetstate andprocessnoiseareintheCartesiancoordinates,butmeasurement anditsadditivenoise arein
thesensorcoordinatesx3. Let bethetruepositionofthetargetintheCartesiancoordiznates.Forthecaseofspvherical
measurements,wehave (x;y;z) and with
0 0 0
z=[r; b; e; r_] h(x)=[r;b;e;r_] =[hr;hb;he;hr_]
(11)
hr = r= x2+y2+z2
(12)
hb = b=ptan(cid:0)1 y
x
(13)
he = e=tan(cid:0)1 z
2 2
x +y
(14)
xx_ +yy_+zz_
hr_ = r_ = p
2 2 2
x +y +z
ForRUVmeasurements,wehave and with
z=[r; u; v; r_]0 h(x)p=[r;u;v;r_]0 =[hr;hu;hv;hr_]0
(15)
x
hu = u=
2 2 2
x +y +z
(16)
y
hv = v= p
2 2 2
x +y +z
(and ). Clearly the measurement models are nonlinear and coupled across Cartesian coordinates,
hw = w = 2 z2 2 p
although the measuprexm+eynt+nzoise remains zero mean, Gaussian, and uncorrelated because measurements are in the sensor
coordinates.
Most nonlinearestimation and filtering techniques, such as the extendedKalman filters (EKF), formaneuveringtarget
trackinghavebeenappliedinthisframework. Thosebasedonmeasurementmodelsareaddressedinsubsequentsections.
Manyothertechniquesarecoveredinsubsequentpartsofthissurveyseries. AtypicalimplementationoftheEKFinmixed
coordinates(Cartesianstateandsphericalmeasurements)canbefoundin[13].
3.2 TrackinginCartesianCoordinates
Inthisapproach,themeasurementsinthesensorcoordinatesareconvertedtotheCartesiancoordinatesfortracking.Clearly,
anymeasurementexpressedinthesensorcoordinateshasanexactandequivalentrepresentationintheCartesiancoordinates.
Let betheequivalentrepresentationintheCartesiancoordinatesoftheerror-freesensormeasurement
0
xp =or[x;y;z] ,=wHithxtarget state and some , for example, if . Clearly, is in fact
0
(thr;ebt;reu)e pos(irt;ioun;vo)f the target in the Cxartesian cooHrdinates, not knoHwn=to[Iu;s.O]Oncxe=the[xn;oyis;yz;m:e:a:s]urements4 xofpthe target
positionare convertedto the Cartesian coordinates(i.e., the noisymeasurementsoriginallyavailablein sensor coordinates
are expressed in the Cartesian coordinates), the measurement equation takes the following “linear” form in the Cartesian
coordinates: ,thatis,
zc =xp+vc (17)
zc =Hx+vc
Thismeasurementissometimesreferredtoasapseudolinearmeasurement. Thismodelapparently“eliminates”theneedto
handlenonlinearmeasurements,incontrasttotheaboveapproachoftrackinginmixedcoordinates. Themajoradvantageof
thisapproachisthatalinearKalmanfilterthencanbeappliedifthedynamicsislinear.
Priorto[14]themeasurementnoise wascrudelytreatedtohavezeromeanandacovariancedeterminedbyafirst-order
Taylorseriesexpansion.Sincethensevevracltechniqueshavebeendevelopedtocomputeoraccountforthenonzeromeanand
thecovariancemoreaccurately(see,e.g.,[14,15,16,17,18,19]).ThesetechniquesaresurveyedinSec.5.
Weemphasizethatthemeasurementnoise isingeneralnotonlycoupledacrosscoordinates,non-Gaussian,butalso
state dependent. This state dependency is provbacbly more important but is largely ignored or overlooked in the literature
3Thismeasurementmodelwithadditivenoisecorrespondsto(9). Thenoise isnotnecessarilyinthesensorframeifthemoregeneralmodel
isused. v z =
h(x4;Tvh)atis,theactualobservedvalues(i.e.,realizations),nottheobservablesas(random)variables.
4
beyonditsimplicituseinthecomputationofthefirsttwomomentsof .Duetothenonlineardependencyof onthestate
,thismeasurementmodelisinfactnonlinear. Asaresult,eveniftvhecmeasurementconversionisdoneideavllcywithexact
xknpowledgeofthe(state-dependent)firsttwomomentsof ,itisstillanillusionthattheapplicationoftheKalmanfilterhere
inthecaseoflineardynamicsyieldsoptimalresults. Nevvecrtheless,sincethestatedependence(i.e.,nonlinearity)existsonly
inthemeasurementnoise ,ratherthaninthemeasurementfunction ,itseemsreasonabletoexpectthatitsimpacton
trackingperformanceisrevlactivelysmaller,ascomparedtothenonlineahri(ty)in whenhandledbymostpopularnonlinear
(cid:1)
filteringtechniques. Ontheotherhand,amajordrawbackofthisapproachstemhs(fr)omalackofavailabletechniquestohandle
(cid:1)
measurementswithstate-dependent,non-Gaussianerrors,whereasabundanttechniquesareavailableformeasurementswith
nonlinear . WehavedevelopedanextensionoftheKalmanfilterforsuchproblemsthatexplicitlyaccountsforthestate
dependenche(o)fthemeasurementnoisemoreeffectively,asreportedin[20].
(cid:1)
Another weakness of this approach is that the conversion from sensor to Cartesian coordinates requires knowledge of
range.Forangle-onlymeasurements,anestimatedrangecanbeused.However,theconvertedmeasurementshaveadegraded
accuracywhenaninaccuraterangeisused,suchasforpassivesensorsorrange-denialcountermeasures. Indeed,angle-only
measurementsarerarelyconvertedtotheCartesiancoordinateswithfewexceptions,oneofwhichisgivenin[21].Inaddition,
itisdifficulttodevelopcoordinate-decoupledfiltersinpureCartesiancoordinatesformitigatingthepossibleillconditioning
duetothelargedifferenceintheaccuraciesofrangeandanglemeasurements,aswellasforhighefficiency.
Themeasurementconversionasdescribedabovedoesnotdealwithrangeratemeasurements . Itissubstantiallymore
complexwhentherangeratemeasurementsareinvolved. Inthiscase,theconvertedmeasuremenr_tsarenonlinear(noteven
pseudolinear). The use of as a measurementof position and velocity for
(cid:1)
trackingintheCartesiancodor=dinrar_te=swxaxs_s+ugygye_s+tedzzi_n+[2v2d].Notethatthismeasurementisq(xua;dyr;azt)icinthestate,w(xh_;icy_h;zi_s)not
highlynonlinear. Thisisclearlysuperiortoconvertingtherangeratemeasurements directly,whichis
xx_+yy_+zz_
highlynonlinear. Foruncorrelatedrangeandrangerateerrors,however,themeasurer_m=entpxh2+asy2a+nze2rr+orvr_ with
zeromeanbutvariance ,whichcanbequitelargeforlong-rangetargetsd. vd = rr_ rr_
r2(cid:27)2r_ +r_2(cid:27)2r+(cid:27)2r(cid:27)2r_ (cid:0)
3.3 TrackinginSensorCoordinates
Alternatively,targetdynamicscanbeconvertedfromtheCartesiantothesensorcoordinatessothatthedesirablemeasurement
structureisunaltered,incontrasttoconvertingmeasurementsfromsensortoCartesiancoordinates.However,expressingtyp-
ical targetmotionsin sensorcoordinates(sphericalorRUV) leadsto highlynonlinear,coordinate-coupled,andsometimes
cumbersomemodels. For example, a constant-velocity(CV) motion has a simple Cartesian description with two or three
independenttwo-state one-dimensionalCV models. The same motion in the spherical coordinatesis rather nonlinearand
complicated,anexplicitmodelofwhichcanbefoundin,e.g.,[3]. Nontrivial,variableaccelerations(knownaspseudoaccel-
erations)[10,4,11]areinducedinthesensorcoordinatesbysuchaconversion,evenforaperfectCVmotion,andthusastate
vectorincludingaccelerationcomponentsisneeded.Inshort,itisimpossibletodescribetypicaltargetmotionsinthesensor
coordinatesinasimple,coordinate-uncoupledway.Further,theconvertedprocessnoiseisnon-Gaussianandstatedependent
evenifitisGaussianandcoordinate-uncorrelatedintheoriginalCartesiancoordinates.
Nevertheless, this approachhas certain advantages. The foremost oneis that the linear, uncoupled,Gaussian structure
ofthemeasurementmodelis maintained. Alargenumberoftrackingfiltersthatoperatepurelyinsensorcoordinateshave
appeared in the literature. Their common feature is the use of the above linear-Gaussian measurement model. Their key
differenceliesinhowthetargetdynamicsaremodeled.Adetailedcoverageofthesemodelsisbeyondthescopeofthispart,
whichissupposedtocovermeasurementmodels.Forcompleteness,however,wementionbrieflybelowthesetechniquesand
directthereadertothespecificreferences.
Asimplisticapproachistodirectlyemploysomedecoupled1Dtargetdynamicsmodels,suchastheCV,CA,andSinger
models(see Part I),forrange(rangerate)and othermeasurements(angles ordirectioncosines) separately. This approach
accountsforthetargetdynamicsinthesensorcoordinatesinacrudeway; itdoesnotreallyconvertthetargetdynamicsto
thesensorcoordinates.Asexplainedabove,however,high-ordermodelsthatincludeaccelerationsareneededto“cover”the
actualhighlynonlineardynamicsinthesensorcoordinatesevenforatrulyCVmotion. Thisleadstoaccuracydegradation.
The geometry-inducedpseudoaccelerationsare clearly not accountedfor if a two-state “CV” modelis used in each of the
sensorcoordinatesindependently.Anengineeringfixistocompensatetheresultingbias[23,24].
Amoreeffectiveapproachistousethetargetdynamicsmodelactuallyconvertedinthesensorcoordinates.Thisleadsto
afilteringproblemwithlinearuncoupledmeasurementsinGaussiannoisebutnonlineardynamicsandnon-Gaussian,state-
dependentprocess noise. Albeit theoreticallyand computationallychallenging, this approachis beneficial in a number of
cases[7,25,26,27,5]. Decoupledfirst-orderMarkovmotionmodelsinpolarcoordinatesformaneuveringaircrafttracking
wereusedin[28,29,30]. Asimilar,decoupledmotionmodelinsphericalcoordinateswasproposedin[31]. Itscompletely
5
coupled version was derived from the Cartesian version in [32]. For the developments in the context of ballistic target
tracking, the reader is referredto Part II. For example, [5] reportedthe developmentof a tracking filter where the reentry
vehicledynamicsaremodeleddirectlyinthesphericalcoordinates. Thesefiltersoperateentirelyinthesensorcoordinates.
Sec.4.2.1ofPartIIcontainsamoredetaileddiscussionofcomparisonbetweenreentry-vehicletrackinginCartesianandin
sensorcoordinates.
3.4 TrackinginOtherCoordinates
AlthoughtargetmotionandmeasurementsarebestdescribedinCartesianandsensorcoordinates,respectively,itisclearly
notnecessaryto dotrackingentirelyin oneorbothofthese coordinates. ThemodifiedCartesian coordinates[33,34] and
the better-knownmodifiedpolar coordinates[35] for angle-onlytrackingare goodexamples. Also, it is fairly commonto
propagatethe targetstate in a Cartesian frameand then convertthe predictedstate, alongwith the errorcovariance,to the
sensor coordinates for state update there (see, e.g., [7, 36, 37, 12, 27]). As such, while state update is decoupled across
coordinates, state predictionis in generalcoupled. This approachrelies heavily on coordinatetransformation: In addition
to the conversionof the predictedstate, the updatedstate andits errorcovariancemust be convertedbackto theCartesian
coordinates. The covariance conversions usually rely on linearization of the error models and is possibly biased, not to
emphasizethestatedependencyinherentintheapproach.
Alternatively,theuseoftheso-calledradarprincipalCartesiancoordinateswassuggestedin[8],whichisanintegration
of the Cartesian and the original sensor coordinates in that the range vector in the original sensor frame is retained as an
axisinthisorthogonalCartesianframe. (Theothertwoaxesquantifyangularcomponents,oneparalleltotheradarface,the
otherintheplanenormaltotheradarfaceandcontainingtherangevector.) Inthesamespirit,aschemebasedonrangeand
angularmodelswasdevelopedin[10,11]intheorthogonalrange-horizontal-“vertical”(RHV)frame5,whichinvolverange
(andrangerate)and, in Cartesian (H andV) coordinates, angularvelocityandacceleration. Similar approacheswere also
takenin[38,22,39,40,41,42]. Thankstotheweakcouplingbetweentherange(rangerate)andthenon-rangecoordinates,
a merit of this approachis that range and non-rangecoordinatesare processed in a quasi-independentmanner, capable of
alleviatingillconditioningandhavinghighefficiency. However,themeasurementmodelsinthenon-rangecoordinatesare
nolongerlinear.Inessencethisapproachcombines,inasensibleway,theframeworksinpurelysensorcoordinatesdescribed
abovebyusingthe rangecoordinate,andin mixedcoordinatesas describedin Sec. 3.1forthenon-rangecoordinates. An
additionaladvantageofusingrangeasacoordinateisthatrangeratemeasurementscanbeincorporatednicelyandeasily.
4 Linearized ModelsinMixedCoordinates
Throughoutthispaper,weconsideronlyagenericfilteringcyclefrom to (i.e.,from to )andwrite for
thepredictedstate and fortheupdatedstate . Theassociatedtke(cid:0)rr1orcotvkariancematkrices1aredkenotedby x(cid:22)and
,respectively. x^kjk(cid:0)1 x^ x^kjk (cid:0) P(cid:22)
P The“standard”techniqueforhandlingthenonlinearmeasurementmodel(10)istheextendedKalmanfilters(EKF)(see,
e.g.,[43,44,45])6.Ingeneral,itreliesonapproximatingthenonlinearmeasurementbythefirstfewtermsofitsTaylorseries
expansion.Specifically,thecornerstoneofitsfirst-orderversion,whichismostwidelyused,islinearizationofthenonlinear
model,resultinginaderivative-basedlinearizedmodel.Otherlinearizedmodelshavealsobeenproposedtohandlenonlinear
measurements.Wedescribetheselinearizedmodelsnext.
4.1 Derivative-Based
Themostwidelyusedtechniqueforlinearizinganonlinearmeasurementmodelintheformof(10)istoexpandthemeasure-
mentfunction atthepredictedstate andignoreallnonlinearterms7:
h(x) x(cid:22)
(18)
@h
h(x) h(x(cid:22))+ (x x(cid:22))
(cid:25) @x x=x(cid:22) (cid:0)
(cid:12)
5TheHaxisisreallyhorizontal,buttheVaxisisnotreallyvertical.Theyareb(cid:12)othperpendiculartotherangevector.
6ThefirstapplicationoftheKalmanfiltertoareal-lifeproblemwasinfactint(cid:12)heformofanEKF(see,e.g.,[46]).
(cid:12)
7Moregenerally,ifacompletelygeneralnonlinearmodel isconsidered,wewouldhave
z=h(x;v)
h(x;v)(cid:25)h(x(cid:22);v(cid:22))+ @@hx(cid:12)(cid:12)(cid:12)(cid:12)x=x(cid:22)(x(cid:0)x(cid:22))+ @@hv(cid:12)(cid:12)(cid:12)(cid:12)v=v(cid:22)(v(cid:0)v(cid:22))
Suchamodeldoesnotnecessarilyhaveadditivenoiseinthesensorcoordinates;forexample,itmayhaveadditivenoiseintheCartesiancoordinates.
6
Thisamountstoapproximatingthenonlinearmodel(10)bythelinearmodel
(19)
z=H(x(cid:22))x+d(x(cid:22))+v
where istheJacobianof and .
@h
ForHth(ixs(cid:22))m=ode@lxthxe=px(cid:22)redictedstateandithsc(oxv)ariancde(ax(cid:22)r)e=updha(txe(cid:22)d)usiHng(tx(cid:22)h)ex(cid:22)linearKalmanfilterequations
(cid:0)
(cid:12)
(cid:12) (20)
K = P(cid:22)H0 HP(cid:22)H0+R (cid:0)1
(21)
x^ = x(cid:22)+K(z (cid:22)z)
(cid:0) (cid:0) (cid:1) (22)
P = (I KH)P(cid:22)
= (I(cid:0)KH)P(cid:22)(I KH)+KRK0 (23)
(cid:0) (cid:0)
where and . Notethat isusedonlyinthecovarianceupdateand
filtergaHin=coHmp(ux(cid:22)t)ationaz(cid:22)n=dtHhat(x(cid:22)(2)3x(cid:22))+isvda(lix(cid:22)d)f+orEar[bvi]tr=aryhg(ax(cid:22)in)+Ean[vd] . ThesefHactsareusedinsometechniquesdiscussed
later. Althoughappearsalmosteverywhere,covarianceupdateKby(22H)shouldbeavoidedforatleasttworeasons: Itinvites
horriblenumericalproblemsandit is theoreticallyvalidonlywhenthe gain is trulyoptimal, whichis rarelythecasein
practice. Ithas beenlargelyoverlookedthatthegaingivenby(20)is nolonKgeroptimalandthuscanbeimprovedsinceit
ignoresthelinearizationerrors.
Thislinearizedmodelisadequateonlywhen issufficientlysmall,whichcanrarelybeguaranteedsincetheaccuracy
of reliesonthatoftargetstatepropagxatiox(cid:22)n(i.e.,dynamicsmodel)andthepreviousstateestimate .This
(cid:0)
inax(cid:22)cc=urax^ckyjkm(cid:0)a1ybuildupandresultinfilteringdivergence,asreportedinnumerousexamples(see,e.g.,[43,47x^])k.(cid:0)T1ejkc(cid:0)hn1iques
aimedatreducinglinearizationerrorsarediscussedinSec. 4.4.
4.2 Difference-Based
We present now a new linearized model, proposed in [48], that is not only expectably more accurate but also potentially
simplerthantheabovewidelyusedderivative-basedmodel.
h((cid:1))
x(cid:22) x(cid:3) x h(cid:0)1(z)
Fig.2: Variouslinearizations.
Considerfirstascalarnonlinearmeasurement forsimplicity.Let
z =h(x)+v
(24)
h(x) h(x(cid:22))
H(x;x(cid:22))= (cid:0) ; x=x(cid:22)
x x(cid:22) 8 6
Clearly, istheslopeofthestraightlineconnecting (cid:0) and (seeFig.2).Forconvenience,wedenote
(cid:3) (cid:3)
H(x ;x(cid:22)) h(x ) h(x(cid:22))
(25)
h(x) h(x(cid:22)) @h
H(x(cid:22);x(cid:22))= lim (cid:0) =H(x(cid:22))=
x!x(cid:22) x x(cid:22) @x x=x(cid:22)
whichistheslopeofthetangentof at .If isabett(cid:0)erestimatethan ,itis(cid:12)(cid:12)reasonabletoexpectthat
h(x) x(cid:22) x(cid:3) x(cid:22) (cid:12)
(cid:12)
(26)
(cid:3)
z =h(x(cid:22))+H(x ;x(cid:22))(x x(cid:22))+v
(cid:0)
7
isabetterlinearizedmodelthanthederivative-basedmodel .
Inthevectorcase,thisdifference-basedlinearizedmodelzo=f(h10(x(cid:22)))is+H(x(cid:22);x(cid:22))(x x(cid:22))+v
(cid:0)
(27)
(cid:3)
z=h(x(cid:22))+H(x ;x(cid:22))(x x(cid:22))+v
where (cid:0)
H(x(cid:3);x(cid:22))=[Hij]; Hij = hi(x(cid:3)j;(cid:3)x(cid:22))(cid:0)hi(x(cid:22)); hi =ithrowofh; hi(x(cid:3)j;x(cid:22))=hi(x)x=[x(cid:22)1;:::;x(cid:22)j(cid:0)1;x(cid:3)j;x(cid:22)j+1;:::;x(cid:22)n]0
xj x(cid:22)j j (28)
(cid:0)
Clearlyitisextremelyeasytoimplementthislinearizedmodel. ItdoesnotinvolvecomputationofanyJacobian,which
could be theoretically and/or computationally challenging for a complicated nonlinear function . It is expectably more
accuratein generalthan the derivative-basedlinearizedmodel, as widelyused in the EKF, providhed is a moreaccurate
(cid:3)
estimateof than . x
Severalxways ox(cid:22)f determining are possible. First, without loss of generality for tracking applications, assume that
(cid:3)
, where is invertxible. Let . In the case of a 3D measurement of the target position, for
0 0 0 (cid:0)1
hex=am[phl1e;,h2]wouldbeht1he3Dtargetpositionx.1W=e cahn1th(ezn)choose . Forthecomponents
(cid:3) 0 0 0 (cid:0)1 0 0
corresponxdi1ngto ,thederivatives canbeuxse=d.[Axl1te;rx(cid:22)n2a]ti=ve[lhy,1we(zc)a;nx(cid:22)fi2]rstupdatethestateestimHatiej
(cid:3) @hi
from to asinxajn=EKx(cid:22)Fj (noneedforcovHairjia=nce@uxpdxa(cid:3)jt=ex(cid:22)hjerethough)andthenuse intheabovelinearizedmodel.
(cid:3)
Theux(cid:22)seofx^this modelwill leadtoat least anexpecta(cid:12)blymoreaccuratecovarianceuxpdat=efx^ora state updateinthe formof
(cid:12)
[48]:
x^ =x(cid:22)+K(z z(cid:22)) (29)
(cid:0) P =[I KH(x(cid:3);x(cid:22))]P(cid:22)[I KH(x(cid:3);x(cid:22))]0+KRK0
(cid:0) (cid:0)
4.3 OptimallyLinearizedModel
Theabovederivative-basedanddifference-basedlinearizationmodelsingeneralhavenooptimalityandcanbequitebadin
manycases. We nowoutlinealinearizedmodelthatisoptimalinthemean-squareerror(MSE)sense[44],aspresentedin
[48]fortracking.
Anonlinearfunction canbeapproximatedoptimallyaround byalinearone:
h(x) x(cid:22)
(30)
h(x) a+H(x x(cid:22))
inthesenseofhavingtheminimumMSE,denoting (cid:25) , (cid:0)
x~ =x x(cid:22)
(cid:0) (31)
0
J =E[(h(x) a Hx~)(h(x) a Hx~)]
Itcanbeshownthat[48] (cid:0) (cid:0) (cid:0) (cid:0)
(32)
a = E[h(x)] E[h(x)x~0]P(cid:22)(cid:0)1E[x~] =(1 E[x~]0P(cid:22)(cid:0)1E[x~])
f (cid:0) g (cid:0) (33)
H = E[h(x)x~0] E[h(x)]E[x~0] P(cid:22)(cid:0)1(I E[x~]E[x~]0P(cid:22)(cid:0)1)(cid:0)1
where . Inthecaseof f ,itred(cid:0)ucesto g (cid:0)
P(cid:22) =E[x~x~0] E[x~]=0
(34)
a=E[h(x)]; H =E[h(x)x~0]P(cid:22)(cid:0)1
Consideran exampleofa scalar nonlinearmeasurement . Assume that . Thenthe optimally
linearizedmodelis z = x3 +v x (x(cid:22);P(cid:22))
(cid:24) N
(35)
z =x(cid:22)3+3P(cid:22)x(cid:22)+(3x(cid:22)2+3P(cid:22))(x x(cid:22))+v
since and . Com(cid:0)paredwiththederivative-basedlinearizedmodel
a = E[x3] = x(cid:22)3 +3,Pw(cid:22)x(cid:22)hichalHwa=ysEun[dxe3rx~e]sPt(cid:22)im(cid:0)1at=est3hx(cid:22)e2v+ari3aPt(cid:22)ion ,thismodelappearsmoreappealingfor
3 2
zma=nyx(cid:22)sit+ua3tix(cid:22)on(sx. x(cid:22))+v h(x) h(x(cid:22))
(cid:0) (cid:0)
Whilethederivative-basedlinearizationreliesontruncationoftheTaylorseries,whichwillincurlargeerrorsif isnot
small,thisoptimallylinearizedmodelaccountsforlargeerrorswithintheexpectationsbytheprobabilisticweightsanx~dthus
tendstogiveamoreconservativefiltergainandbetterperformanceforcasesinvolvinglarge . Anotherpossibleadvantage
ofthismodelisthat neednotbedifferentiable.Basically,ittradesintegrationwithdifferentiax~tion.Thismaybeparticularly
usefulinsuchcaseshwherehardlimiters(orsaturations)areinvolved.
In the calculationof the requiredexpectations, one may usually assume . It has a small tail probability,
whichisappealingbecausethegoalislocallinearization. Insomesituations,xonemay(x(cid:22)w;aP(cid:22)nt)touseadistributionflatterthan
(cid:24) N
theGaussian,buttheheavytailsshouldbetruncated;thatis, maybeassumedmoreevenlydistributedthanGaussian,but
onlyovera“small”neighborhoodof . Ingeneral,thelargertx~heneighborhood,themoreconservativethefiltergain.
x(cid:22)
8
4.4 Linearization-ErrorReductionTechniques
Sequential Processing. A well-known simple means to reduce linearization errors is sequential processing of the mea-
surementcomponents(see,e.g.,[49,50,51]). Itiswellknownthatthenonlinearmeasurementsshouldbeprocessedinthe
orderoftheir accuracy— moreaccuratefirst — (see, e.g., [51, 11]). A particularproblemwas consideredin [52], where
thesphericalmeasurementswereprocessedsequentiallyintheorderofdecreasingaccuracy: azimuth,elevation,andrange.
Performancecomparisonresultsbetweensequentialprocessingandtheconventionalvectorprocessingweregivenin[53].
Iterative EKF. Once the updated state is obtained, the nonlinear measurement model can be re-linearized at . This
willingeneralreducethelinearizationerrox^rcomparedwithlinearizationat . Thestateanditserrorcovariancecanx^thenbe
re-updatedbasedonthere-linearizedmodel. Thisprocesscanberepeated,x(cid:22)resultinginanalgorithmknownasaniterated
EKF(IEKF)inthecontextofKalmanfiltering[43]. FollowingTheorem8.2of[43]theiterationalgorithmis
(36)
0
x^ = x(cid:22)
(37)
i+1 i i i i
x^ = x(cid:22)+K(x^ ) z h(x^ ) H(x^ )(x x^ ) ; i=0;1; :::;L
(cid:0) (cid:0) (cid:0) (38)
L+1
x^ = x^ (cid:2) (cid:3)
(39)
P = I K(x^L)H(x^L) P(cid:22) I K(x^L)H(x^L) 0+K(x^L)RK(x^L)0
(cid:0) (cid:0)
where , , ifaKalmanfilteringisusedwitha
derivatHive(-x^bia)se=dl@@inhxeaxr=izx^eidKmo(xd^ei)l(=i(cid:2).eP.(cid:22),Hin(ax^ifi)r0s[Ht-o(rx^die)rP(cid:22)E(cid:3)HK(F(cid:2)x^).i)A0+simRi]l(cid:0)a1ritier=ati0o;(cid:3)n1;ca:n::b;eLwrittenifadifference-basedmodel
(Sec.4.2)isused.Incontrasttosomeotheriterationschemes(see,e.g.,[54,47]),thecomputationoftheupdatedcovariance
(cid:12)
bytheJoseph’sfo(cid:12)rm(39)isbettertobeoutsideoftheiterationloop.Thiswasusedin[43],andemphasizedanddiscussed
Pin[55]. A theoreticalconsiderationoftheEKF andIEKFmeasurementupdatesas Gauss-Newtoniterationschemesanda
demonstrationof the superiority of the IEKF were presented in [56]. The simulations of [7] (Table I) and [55] show that
suchre-linearizationiterationscanindeedimproveaccuracyatalevelthatisscenariodependent[55]. Itshouldbewarned,
however,thatanimprovementisnotguaranteed.Therearereportsthattherelinearizationiterationdegradestheperformance.
Thereaderisreferredto[57]forsomeinsightthatsubstantiatesthiswarning.
Higher-OrderPolynomialModels. Anotherstraightforwardideatoincreasetheaccuracyofapolynomialapproximation
ofanonlinearmeasurementmodelistousequadraticterm(andpossiblyhigher-orderterms)intheTaylorseriesexpansion.
IntheKalmanfilteringcontext,thisleadstowhatissometimesreferredtoasasecond-order(andhigher-order,respectively)
EKF[43,47,58]. Thesimulationresultsreportedin[7]showaconsiderableimprovementinperformanceofasecond-order
EKFoverafirst-orderEKF. However,second-orderEKFsarenotveryoftenusedinpracticemainlybecauseoftheirrather
burdensomecomputationandlimitedormarginalperformanceimprovement.
Followingthe sameideas it is alsopossibleto develophigher-orderversionsofthedifference-basedandoptimallylin-
earizedmodels.
Many other techniques are available for mitigating the performancedegradationdue to linearization, such as artificial
inflationoferrorcovariance[58].[8]includesashortlistofsuchtechniquesandabriefdiscussion.
5 ModelsinCartesianCoordinates
Sincetargetmotionis best describedinCartesian coordinatesbutmeasurementsareavailableinsensorcoordinates,as
explainedin Sec. 3.2, a commonlyused methodis to convertmeasurementsfrom sensor to Cartesian coordinates, and do
tracking entirely in the Cartesian coordinates. As before, we will assume a generic Cartesian frame since measurements
betweenCartesianframescanbeconvertedeasilyandexactly(assumingnosensorregistrationorgridlockerrors).
5.1 ConversionofMeasuredPositions
Thespherical-to-Cartesiantransformation with isgivenby
(cid:0)1 0
’=h h=[hr;hb;he]
xc rcosbcose (40)
zc = yc =’(z)=’(r; b;e)= rsinbcose
2 3 2 3
zc rsine
9
4 5 4 5
where and are one and the same noisy measurement, expressed in the original spherical
0 0
coordinzate=s a[rn;dbt;hee] convezrcted=C[axrcte;syica;nzcc]oordinates, respectively. The RUV-to-Cartesian transformation with
(cid:0)1
isgivenby (cid:30) = h
0
h=[hr;hu;hv;hw]
ru (41)
zc =(cid:30)(z)=(cid:30)(r; u;v;w)= rv
2 3
rw
Ifrangemeasurementsarenotavailable, asinthecaseofpassivesensorsandrange-denialcountermeasures,theabove
4 5
rangemeasurement canbereplacedbyanestimatedrange [21,11]andtherangemeasurementerror (anditsbias
andvariance)shouldrbereplacedbytherangeestimationerrorr^ (anditsbiasandvariance). r r
Inthesequel,wedescribetechniquesforconvertingmeasurr^emerntsfromsphericaltoCartesiancoordinat(cid:0)es.Thereaderis
(cid:0)
referredto[59,20]forconversionfromRUVtoCartesiancoordinates.
5.2 StandardModelofConvertedMeasurements
Afterconversion,themeasurementmodelinCartesiancoordinateshastheform
(42)
zc =Hx+vc
where isthepositionsubvectorofthestatevector and standsfortheresultingmeasurementerror.
ByxTpay:=lorHsexriesexpansionof aroundthenoisymeaxsuremevnct ,wehave
’(z) z
HOT (43)
xp =’(z)=’(z v)=’(z) J(z)v+ (v)
(cid:0) (cid:0)
where istheerror-freetruetargetpositioninsphericalcoordinatesandHOT standsforthehigherorder( )
0
terms,zan=d[trh;ebJ;aec]obian isevaluatedatthenoisymeasurement (v) 2
(cid:21)
J(z) z
cosbcose rsinbcose rcosbsine (44)
@’ (cid:0) (cid:0)
J(z)= = sinbcose rcosbcose rsinbsine
@z(cid:12)z=z 2 sine 0 (cid:0) rcose 3
(cid:12)
Thentheexactlyconvertedmeasurement(cid:12)model(442)canbewrittenas 5
(cid:12)
HOT (45)
zc =’(z)=xp+vc =xp+J(z)v (v)
(cid:0)
vc
Clearly,thisexpansionissuperiortoexpanding aroundtheerror-freemeasurement
’(z) | {z } z
HOT (46)
zc =’(z)=’(z+v)=xp+vc =xp+J(z)v+ (v)
vc
whichinvolvestheunknown .
z
Evidently,thetrueconvertedmeasurementerror ismeasurementdep|endent({ozrstated}ependent),non-Gaussian,cor-
relatedacrosscoordinates,andhasnonzeromean. Tvhucs,theconvertedmeasurement hasabias andaconditional
bias . zc E[vc]
CEle[avrlcy,z]conversion of the measured position values per se is straightforward and nothing is really subtle here. The
j
maintaskintheso-calledmeasurementconversion,whichisbettercalledmeasurementmodelconversion,reallyliesinthe
conversionoftheassociatednoisestatistics. AsexplainedinSec.3.2,themajoradvantageofexpressingmeasurementsin
Cartesiancoordinatesistheattractive“linear”structureofthecorrespondingmeasurementmodel(42). Inorderforalinear
filter(e.g.,theKalmanfilter)totakeadvantageofthis“linear”structure,thefirsttwomomentsof mustbedetermined.
vc
5.3 LinearizedConversion
The“standard”approachistotreat (approximately)aszero-meanwithcovariance[50,22,10,45,11]
vc
(47)
L 0
R =J(z)RJ(z)
10