the interacting multiple model algorithm for accurate ... · the interacting multiple model...

614 JOHNSHOPKINSAPLTECHNICALDIGEST,VOLUME22,NUMBER4(2001)

A. F. GENOVESE

A

TheInteractingMultipleModelAlgorithmforAccurateStateEstimationofManeuveringTargets

Anthony F. Genovese

ccurate state estimation of targets with changing dynamics can be achievedthroughtheuseofmultiplefiltermodels.The interactingmultiplemodel(IMM)algo-rithmprovidesastructuretoefficientlymanagemultiplefiltermodels.DesignofanIMMrequiresselectionofthenumberandtypeoffiltermodelsandselectionofeachoftheindi-vidualfilterparameters.Inthisarticletheresultsforfivefiltermodelson10targettrajec-torysegmentsarediscussedandcompared.Thecomplexityofthefiltermodelsincreasesfromasingleconstantvelocitymodeltoathree-modelIMMfilter.Theresultsshowthattheoverallperformanceofthestateestimates,formosttargets,improvesasthecomplexityofthefiltermodelsincreases.SelectionofIMMfilterparametersisaddressedandresultsareprovidedtoshowthatperformanceoftheIMMappearstoberelativelyinsensitivetolargechangesinfilterparameters.TheperformanceofanIMMisprimarilydeterminedbytheselectionofthecomponentfiltermodels.

INTRODUCTIONThe performance of a tracking system is governed

by the performance of the state estimation algorithmemployed. Accurate state estimation of targets in atrackingsystemisrequiredforreliabledataassociationand correlation. The states to be estimated are typi-callythekinematicquantitiesofposition,velocity,andacceleration.Filtersareusedonmeasurementstoreducethe uncertainty due to noise on the observation andtoestimatequantitiesnotdirectlyobserved.Thefilteruses a model of the state process that can be used toaccuratelypredictthebehavioroftheobservedtargettoestimatethedesiredkinematicquantities.

Stateestimationofpotentiallymaneuveringtargetsfrom sensor measurements often requires the use of

multiple filter models to account for varying targetbehavior. Efficient management of the multiple filtermodels is critical to limiting algorithm computationswhileachievingthedesiredtrackingperformance.Thisrequirement is achievedwith the interactingmultiplemodel(IMM)algorithm.1

TheIMMalgorithmisamethodforcombiningstatehypotheses from multiple filter models to get a betterstateestimateof targetswithchangingdynamics.ThefiltermodelsusedintheIMMforeachstatehypothesiscanbeselectedtomatchthebehavioroftargetsofinter-est.ModelmanagementfortheIMMalgorithmisgov-ernedbyanunderlyingMarkovchainthatcontrolstheswitchingbehavioramongthemultiplemodels.Forthe

JOHNSHOPKINSAPLTECHNICALDIGEST,VOLUME22,NUMBER4(2001) 615

INTERACTING MULTIPLE MODEL ALGORITHM

resultingalgorithm,logicdecisionsarenotrequiredforestimationofthemodelprobabilities.2,3

BACKGROUNDStateestimationfortrackingismosteffectivelydone

bymodelingthetargettrajectoryasalinearsystem.Thediscrete-time state representationofa linear system isgiveninthefollowingequation:

Xk+1=kXkwk ,

whereXk isthestateestimate,k isastatetransitionmatrix from time k to k+1, and wk is system processnoiseassumedtobeGaussian-distributedzeromeanandwhite.

Observationsforthisprocessareassumedtobelinearwithrespecttothestateestimate.Theobservationsarethengivenas

yk=HkXk+vk ,

whereHkisthematrixrelatingthestatetoobservationquantities and vk is observation noise assumed to beGaussian-distributed zero mean and having zero crosscorrelationwiththeprocessnoisewk.

The Kalman filter provides the minimum meansquared error solution to this linear system problemwhentheprocessunderobservationiscompletelyrep-resentedbythestatemodel.4Theequationsusedtopre-dictandupdatethestateandcovarianceforaKalmanfilteraregivenasfollows:

˜ ˆ

˜ ˆ

˜ ( ˜ )ˆ ˜ ( ˜)ˆ ( ) ˜ ,

X =

= Q

=

=

=

T

T

�

� �

X

P P

K PH HPH R

X X K y HX

P I KH P

T

+

+

+ −

−

−1

k

where X =thestateestimate, P =thecovariancematrix, ~,^=thepredictedandfilteredquantities,

respectively, =thediscretetimestatetransitionmatrix, Q =theprocessnoisematrix, K =theKalmangain, R =thecovarianceofthemeasurementquantity, I =anidentitymatrix, yk =themeasurementquantityusedtoupdatethe

stateestimate,and T =thematrixtransposeoperation.

Most tracking systems employ a single filter modelwithadaptivegainsforstateestimationofmaneuvering

targets.Thesesystemsrequirethedetectionofthetargetmaneuverviaasecondestimatoranddecision-directedlogic to change gains. The problems with this systemare that the decision to switch can be delayed as aresultoflagsinthemaneuverdetectionfilterandfalsealarmscangivefalsemaneuverindications.Inaddition,a single state estimator will exhibit biases when themodelisnotmatchedtothetargetmotion.

Multiple filter models enable a tracking system tobettermatchchangingtargetdynamics.Thiswillyieldthebestoverallperformanceonthemaneuveringandnonmaneuveringtimeintervalsoftargets.Theeffectiveapplication of multiple models requires an algorithmto manage the models. Desired performance must beweighedagainstsystemresources.TheIMMalgorithmhasbeenshowntobeaveryefficientimplementationofthemultiplemodelapproach.1

The IMM AlgorithmTheIMMalgorithmisamethodforcombiningstate

hypotheses frommultiple filtermodels to get a betterstateestimateoftargetswithchangingdynamics.Thefiltermodelsusedtoformeachstatehypothesiscanbederived to match the behavior of targets of interest.Figure1showstheflowdiagramforanIMMalgorithmwith two filter models. Superscripts in the state vari-ablesrepresentthemodelhypotheses(1and2),andthesymbols^and~areusedtorepresentfilteredandpre-dictedquantities,respectively.

Thestateestimates foreachmodel fromtheprevi-ouscycle,X1andX2,aremixedpriortostateupdateusingasetofconditionalmodel probabilities.Thecon-ditionalmodelprobabilities ( ˜ ) i j arecomputedusingthemodelprobabilitiesfromthepreviousupdateanda

Figure 1. A block diagram of the IMM algorithm with two filter models.

µState interaction

Filter Filter

Modelprobability

update

Stateestimate

combination

X1 X2

Z1 X01 X02

Z2

Λ2

Λ1

X1 X2 µ X

µ

Μ Μ1 2

~


A. F. GENOVESE

stateswitchingmatrixselecteda priori.Themixedstateestimatesareupdatedusingeachfiltermodel.Thelike-lihood (i) for each filter model is computed duringthe state update from the innovations (Zi) and inno-vationscovariancematrix.Thelikelihood,priormodelprobabilities,andstateswitchingmatrixarethenusedtoupdatethemodelprobabilities.Theestimates fromeachfiltermodelarecombinedasaweightedsumusingtheupdatedmodelprobabilities.

TheequationsgoverningtheIMMalgorithmforanarbitrarynumberoffiltermodels,N,areoutlinedinthefollowingsteps.Theprocessthenbeginswiththecom-putedquantitiesfromthepreviousfilteriteration.Ini-tialization procedures are required to obtain the stateestimate,covariance,and initialprobabilities foreachfiltermodel.

State InteractionPriortothefilterupdate,themodelstateestimates

andcovariancesaremixedusingcomputedconditionalmodelprobabilities.Themixedstateandcovarianceformodeljattimekiscomputedas

ˆ ˆ ˜

1X X0 j i i j

i =

N= ∑

and

ˆ ˜ ˆ ˆ ˆ ˆ ˆ ,

1P P X X X X

T0 0 0j i j i i j i j

i =

N= + −( ) −( )

∑

where

˜ ˆi jj

ij ip= 1

with

j ij i

i

Np= ∑

=1,

wherepijistheijelementofthestateswitchingmatrix() thatdefines thea priori probability for switchingfrom model i to model j, and c j is a normalizationvectorusedtomaintainatotalmodelprobabilityof1.AlsonotethatX0 jisthemixedstateestimateforeachfiltermodelandP0 jisthemixedstatecovariance.

Model Probability UpdateThe likelihood of each model is computed using

the innovationsZ jcomputedduring stateupdateandtheinnovationscovariancematrix S j computedintheKalman gain. This step is done after state predictionofeachmixedstateestimate.IfGaussianstatisticsareassumed,thelikelihoodofmodeljisgivenby

j

j

j j j= −[ ]−1

20 5 1

˜exp . ( ) ( ˜ ) ( ) ,

SZ S ZT

where

Z

S H P H RT

j j

j j j j

= −

= +

m mo ˜

˜ ˜ ( ) ,0

where mo is a vector of observations for the currentupdate and m j is the predicted track state for filtermodeljtransformedintotheframeoftheobservations.

The model probabilities are updated after all filtermodelshavebeenupdatedas

ˆ j j jc= 1

c

with

c i i

i =

N= ∑ � c

1,

where j istheupdatedmodelprobabilityformodel jandcisanormalizationconstant.Notethatinnovationscovariancematrix S j is computedusing thepredictedcovariancematrixP0 j.

State Estimate CombinationThecombinedstateestimateandcovarianceiscom-

putedfromtheupdatedfilteredstatesfromeachmodelweightedbytheupdatedmodelprobabilities:

ˆ ˆ ˆX X= ∑ i i

i

N�

=1

and

ˆ ˆ ˆ ( ˆ )( ˆ ) .P P X X X X T= + − −[ ]∑ � i

i

Ni i i

=1

FILTER MODEL DEFINITIONSThree filter models have been selected to test the

IMM algorithm with different configurations. Thesemodelsareaconstantvelocity(CV),aconstantaccel-eration(CA),andathree-dimensionalturnwithakine-maticconstraint(TURN).

CV Model ThestatevectorfortheCVfiltermodelisdefinedas

X = [ ˙ ˙ ˙ ,x x y y z z]T



where x corresponds to the east component, y corre-sponds to the north component, z corresponds to thezenith component, and x , y, and z are the corre-sponding rates. Target accelerations are modeled as acontinuous-time white noise process to ensure modelstability. This model will yield the best estimates ofpositionandvelocityonnonmaneuveringtargets.TheextendedKalmanfilterderivedinRef.5isusedasthebasisfortheCVfiltermodel.

The state transition matrix for the CV model isdefinedforalinearpredictionfromthetrackvalidtimetothetimeofthemeasurements

=

C

CV

CV

,V

0

0

0

0

0

02

2

2

2

2

2

where

�CV and ,=

=

10 1

00

002

�t0

andwhere∆tisthedifferenceofthemeasurementtimeandvalidtimeofthetrack.

The plant noise matrix for the CV filter model isderivedasthediscretetimerepresentationofthewhitenoiseacceleration.Thismatrixisgivenas

Q

q

0

0

0

q

0

0

0

q

=

CV

CV

CV

,2

2

2

2

2

2

where

qCV .=

q t

t

t

t2

�

�

�

�

3

2

2

1

Theparameterqisthefilterplantnoisespectralden-sityandhasunitsofm2/s3.Thisparameterisselectedtocontrolthesteady-stategainperformanceofthefilter.

CA Model The filter state vector for the CA filter model is

definedas

X T= [ ˙ ˙ ˙ ] ,..

x x x y y y z z z.. ..

wherethepositionandratetermsarethesameasthoseintheCVmodeland x

.., y

..,and z

..aretheacceleration

estimates.Thestatetransitionmatrixisdefinedforalinearpre-

dictioninallthreedimensionsusingallstateestimateterms

=

C

C

C

,A

A

A

0

0

0

0

0

03

3

3

3

3

3

where

C

/

and .=

=

100

10

2

1

000

000

000

3

� �

�

t tt

2

0

Theplantnoisematrixisdefinedas

Q

q

0

0

0

q

0

0

0

q

CA

CA

CA

=

3

3

3

3

3

3 ,

where

qCA =

q t�

000

000

001

.

The parameter q for this model has units of m2/s5.ThepredictionandprocessnoisemodelforthisfilterisderivedinRef.6.

TURN Model ThefilterstatevectorfortheTURNfiltermodelis

thesameasthatfortheCAmodel,

X T= [ ˙ ˙ ˙ ] ...

x x x y y y z z z.. ..

Thestatetransitionmatrixforthismodelisdefinedtoperformaconstant-speed turnmaneuveralong thetrajectorydefinedbythestateestimatesofvelocityandacceleration.Thismatrixisgivenas

=

TURN

TURN

TURN

0

0

0

0

0

03

3

3

3

3

3 ,

where

�TURN =−

−

− −

−100

11 2

1

sin( )

cos( )

sin( )

( cos( ))

sin( )

cos( )

,

and is the turning ratecalculated fromelementsofthefilteredtrackstateas

� = =+ +

+ +

�

�A

V

ˆ ˆ ˆ

ˆ ˆ ˆ.

x y z

x y z

2 2 2

2 2 2


A. F. GENOVESE

filtermodelcovariance,withtheresultthatthecovari-anceissmallerthantruemeasurementnoise-onlyerrors.However,lagerrorsaresignificantlyreducedduringtheturnperiodusingthisprocedure.ThefullderivationofthisprocedureisgiveninRef.7,andthefiltermodelisappliedwithinanIMMstructureinRef.8.

Theconstraint isappliedasapseudo-measurementupdatetotheTURNmodel.Thefilteringequationsforthisformationhavebeenderivedas

ˆ [ ˆ ˆ ˆ]

ˆˆ [ ˆˆˆ ]ˆ [ ˆ] ˆ

ˆ ( ) ˆ ,

T T

vV

K Pv vPv R

X I K v X

P I K v P

=

= +

= −

= −

−

10 0 0 0 0 0

1

� x y z

c c

c

c

wherethesuperscriptcineachequationdenotestermsrelated to the pseudo-measurement update. Rc is thevariance of the pseudo-measurement update selectedfor this application to achieve a gain of 0.5. In theIMMfiltermodelthisconstraintisappliedtwice:onceafterstateinteractionandonceafterthemeasurementupdatefortheTURNfiltermodel.

FILTER MODELS FOR PERFORMANCE COMPARISON

Astudyhasbeenconductedtocomparetheperfor-manceoffivefilteringmethods.Theapplicationcon-sideredistrackingofairbornetargets.Thefilteredrootmeansquare(RMS)positionandvelocityerrorshavebeen compared on a variety of maneuvering targets.The five filtering methods with operating parametersaregiveninTable1.

The five filter models have been selected to showfilterperformanceasafunctionofincreasingfiltercom-plexity.Thefirsttwomethodsaresinglefiltermethods,

Table 2. Sensor model parameters.

High Scan precisionUpdateperiod(s) 4 1Rangeaccuracy(m) 50 5Bearingaccuracy(Mrad) 5 1Elevationaccuracy(Mrad) 5 1

Table 1. Filtering methods for comparison study.

Method Filtermodels Filterparameters 1 CV qCV=400m2/s3 2 CA qCA=400m2/s5 3 CV-CVIMM qCV =1m2/s3 qCA =3600m2/s3

4 CV-CAIMM qCV=1m2/s3 qCA=400m2/s5

5 CV-CA-TURNIMM qCV=1m2/s3 qCA=400m2/s5 qTURN=25m2/s5

� =0 95 0 050 12 0 88

. .

. .

� =0 95 0 050 12 0 88

. .

. .

� =0 90 0 08 0 020 15 0 70 0 150 04 0 16 0 80

. . .

. . .

. . .

The plant noise matrix for thismodel is the same as for the CAmodel.

The TURN model also uses apseudo-measurementupdatederivedfromtheconstraintthatthetargetisundergoingaconstant-speedturningmaneuver. Using the pseudo-mea-surementderived fromthismaneu-ver assumption tends to influencethe state estimates to change tofitthe profile of this type of target.Namely, the vector representationof the acceleration estimate willchangetobenormaltothevelocityvector.Applicationof this pseudo-measurement will also affect the

andthelastthreeareIMMfilters.Methods1and2willshow filter model performance for nonadaptive singlemodel filters.TheCVfilter q value formethod1wasselected high in order to reasonably limit lags duringtargetmaneuvers.TheCAfilterqvaluewasselectedthesame forallfiltermethodswhereaCAmodel isused.ThethreeIMMmethodsrepresentanincreaseincom-plexitythatshouldbereflectedintheresults.Thefilterparameters were selected experimentally using generalguidelines.Filterparameterselectionanditsimpactonthe IMM filter performance will be addressed later inthisarticle.

Twosensormodelsareselectedtoprovideabroaderviewofthefilterperformanceasappliedunderdifferentoperatingconditions.Thefirst sensor is a surveillance(scan) radar that provides detections at a 4-s updateperiod.Thesecondisahigh-precision(HP)radarwithan update period of 1 s. Each sensor model providesthree-dimensional measurements of position that arezero-mean and Gaussian. The standard deviation ofmeasurement noise for each sensor model is given inTable2.

Thetwosensormodelsareusedincombinationwithfivemaneuveringtargetmodelstoprovideatestingsuiteforthefiltermodels.Thefivemaneuveringtargetmodelsareaweavingmaneuverwitha12-sweaveperiod,10-glinearspeedaccelerationfor10s,high-altitude6-gdive,1-gconstantspeedturn,and5.6-gconstantspeedturn.



RESULTSThe simulation results for each filter model are

obtainedfromMonteCarlosimulationswith100real-izations. The RMS errors for position and velocityare computed from thefiltered track state estimateofeachfiltermodel.Table3providesthepeakRMSposi-tionandvelocity errors for10 selectedperiodsof thetargetandsensorcombinations.Thetargetandsensorcombinationforeachperiodislistedinthefirsttwocol-umnsofTable3.

Thefirst two rowsofTable3 show thepeakRMSerrors of each filter model on the nonmaneuveringperiodofatargetmodel.Thescansensorwasthemea-surementsourceforperiod1,andtheHPsensorwasthemeasurement source for period 2. Examination of thetableentriesshowsthateachoftheIMMmodelsout-performsthesinglefiltermodels.ThisfindingindicatesthatthepropermodelofeachIMM,theCVwithlowprocessnoise,wasprimarilyselectedforthenonmaneu-veringtrajectory.ThesingleCVfiltermodelcouldnotachieve the same variance reduction on nonmaneu-veringtracksastheIMMfiltersbecausetheqvaluewasselected high to limit lags on target maneuvers. ThesingleCAfiltermodelhasthelargestpositionandrateerrors.Sincethemajorityofmosttrackperiodsinpracti-calapplicationsarenonmaneuvering,thatwouldmakethisfiltermodelundesirable.

The third row of Table 3 shows the peak RMSerrorsfortheweavingtarget.NoneofthefiltermodelsselectedfortheIMMmethodsismatchedtothechang-ingdynamicsoftheweavingtarget.Thus,theresultsforthistargetdonotshowaclearadvantageofonefiltermodeloveranyoftheothers.Figure2showstheRMSvelocityerrorsforallfiltermodelsasafunctionoftimefortheweavingtarget.Thisplotshowstherelativeper-formanceofeachfiltermodelduringthenonmaneuver-ingperiod(<60s),aswellasthemagnitudeoftheerrorsduringtheweave.AlthoughtheIMMdoesnotreduceerrors during the weaving period, the performance isnot degraded from any single model. Figure 3 shows

Table 3. Peak RMS errors for selected target periods.

Positionerrors(m)/Velocityerrors(m/s)Target Sensor CV CA CV-CV CV-CA CV-CA-TURNNomaneuver Scan 289/42 360/133 243/20 233/17 241/22Nomaneuver HP 50/23 59/49 40/10 34/7 36/9Weave HP 51/55 44/60 54/60 49/56 48/5710-gacceleration HP 64/112 37/64 37/73 43/76 44/80Diver Scan 246/206 180/136 175/154 172/132 167/129Diver HP 48/73 36/45 36/56 36/44 34/411-gturn HP 85/36 100/63 86/39 85/47 81/365.6-gturn Scan 968/323 445/179 432/188 444/174 391/1205.6-gturn HP 107/105 77/60 78/84 76/67 65/385.6-gturn(postmaneuver) HP 70/25 70/67 66/32 63/32 66/53

CVCACV-CVCV-CACV-CA-TURN

RM

S r

ate

erro

rs (

m/s

)

80

70

60

50

40

30

20

10

00 20 40 60 80 100 120 140 160

Time (s)

Figure 2. The filtered RMS velocity errors for the weaving target plotted as a function of time for each of the five filtering methods. Note that the weave maneuver begins at 60 s.

Figure 3. The average model probabilities for the CV-CA-TURN IMM on the weaving target. The model probabilities are averaged over 100 Monte Carlo realizations.

CV

CA

TURN

Ave

rage

mod

el p

roba

bilit

y

0.1

00 20 40 60 80 100 120 140 160

Time (s)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0


A. F. GENOVESE

theaveragemodelprobabilitiesfortheCV-CA-TURNIMMon theweaving target.TheCVmodel isdomi-nantduringthenonmaneuveringperiod,buttheIMMalgorithm is not able to find a single preferred modelduringthetargetweave.

ThefourthrowofTable3showsthepeakRMSerrorsforthe10-glinearspeedacceleration.Thistargetmaneu-vershouldbewellmatchedtotheCAfiltermodel.How-ever, the results indicate that the single CA is onlymarginallybetterthantheCV-CVIMMfiltermodel.Inaddition,itappearsthattheCV-CVfilterisbetterthanthe more complex CV-CA and CV-CA-TURN filtermodels.Thisisadeceptiveresultbecausethepeakofthefilter error is in the initialmaneuver transitionperiod.Figure4showstheRMSvelocityerrorsasafunctionoftime.Thisplotshowsthatalthoughtheerrorsatthestartofthemaneuveraresimilar,theCA,CV-CA,andCV-CA-TURNfiltersaremuchbetter in steadystate.TheplotalsoshowsthattheCAfilterisslowtorecovernon-maneuveringerrorlevelsafterthemaneuverends.Thus,the additional filter models in the more complex fil-tering methods allow for a quicker recovery from theinitiallag.

Thefifthand sixth rowsofTable3 showthepeakRMSerrorsforthehigh-altitudedivingtargetusingeachsensormodel.Theresultsshowamarginalimprovementin the filter errors as the model complexity increases.Figure5showstheRMSvelocityerrorsasafunctionoftime for thedivingtargetusingtheHPsensormodel.This plot shows the peak errors at the start of themaneuver with the CV-CA-TURN IMM as best insteadystate.ThisdemonstratesthattheTURNmodelimplemented for thiscomparison studyworkswell formaneuversinthreedimensions.

The seventh row of Table 3 shows the peak RMSerrors for the 1-g constant-speed turn maneuver. Allfilter methods yield similar errors, with the exceptionof the single CA model, which had the worst overallperformance. The lag errors produced from a smallmaneuverdonotrequirethedynamicfilteradaptabil-ityofacomplexIMM.Theerrorsdidincreasefromthenonmaneuveringsteady-statevalues,indicatingthattheIMMdiddetectthemaneuverandadapt.

TheeighthandninthrowsofTable3showthepeakRMSerrorsforthe5.6-gconstant-speedturnmaneuverusingeachsensormodel.Theerrorsforthesecasesshowa significant improvement as filter model complexityincreases. This is expected since the TURN model ismatchedtothetargetmaneuver.Thisisalsotruewheneithersensorwasusedtoprovidemeasurementstothefiltermodels.Figure6showstheRMSvelocityerrorsasafunctionoftimeforthe5.6-gturnusingtheHPsensormodel.ThisplotconfirmstheerrorreductionachievedinthemaneuversteadystateusingtheCV-CA-TURNfiltermodel.NotefromFigure6theimprovementfromtheCV-CVfiltertotheCV-CAfiltermodel.Thisplotshows the short-duration increase in errors followingthe target maneuver for the CA and CV-CA-TURNmodels.Thepeaksof theseerrorsare reflected in row10ofTable3.ShowninFigure7aretheaveragemodelprobabilitiesfortheCV-CA-TURNIMMonthe5.6-gconstant-speed turning target.Thesemodelprobabili-tiesreflectthecorrectselectionofeachfiltermodelforthe target duration that yielded the best overall errorperformance.

In general, the filter performance improved, withrespect to RMS errors, as the complexity of the filtermodelsincreased.TheCVfiltermodelcouldnotachievethesamevariancereductiononnonmaneuveringtracksastheIMMfiltersbecauseitsqvaluewasselectedhigh


RM

S r

ate

erro

rs (

m/s

)

120

60

40

20

00 20 40 60 80 100

Time (s)

80

100

10 30 50 70 90


RM

S r

ate

erro

rs (

m/s

)

80

60

40

20

00 50 100 150

Time (s)

100

120

Figure 4. The filtered RMS velocity errors for the 10-g linear-speed accelerating target plotted as a function of time for each of the five filtering methods. Note that the maneuver occurs from 60 to 70 s.

Figure 5. The filtered RMS velocity errors for the high-altitude diving target plotted as a function of time for each of the five filter-ing methods. Note that the dive starts at 110 s.



tolimitlagsontargetmaneuvers.However,thismodelalsoprovedtohavethelargesterrorsduringmostofthetargetmaneuvers.TheCAmodelperformedwellwiththetargetmaneuversbuthadtheworstperformanceonthenonmaneuveringtracks.

All of the IMM models were equally effective onnonmaneuvering tracks. The key to the IMM filtermodelperformanceduringtargetmaneuversisamatchof the filter state models to the target dynamics. TheCV-CV IMMdoesnot try tomodel targetmaneuversbut instead limits lags by increasing filter gains. Thusthis model had the largest errors of the IMM modelsduring target maneuvers. The CV-CA IMM out-per-formed the CV-CV IMM on most target maneuversbecauseoftheaccelerationestimatesinthefiltermodel.

CV

CA

TURN

Ave

rage

mod

el p

roba

bilit

y

0.1

00 20 40 60 80 100 120 140 160

Time (s)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 7. The average model probabilities for the CV-CA-TURN IMM on the 5.6-g constant-speed turning target. The model prob-abilities are averaged over 100 Monte Carlo realizations.

The CV-CA-TURN yielded the best overall perfor-manceontheturningtargetswhileprovidingacompa-rableperformanceonallothertargets.

PARAMETER SELECTION FOR THE IMM FILTER

DesignparametersfortheIMMfiltersareselectedtocontrolfilteroperatingcharacteristicssuchasgainandresponsetomaneuvers.Therequireddesignparametersfor the filtering methods defined in the performancecomparison are the IMM state switching matrix ()andthefiltermodelqvalues.

State Switching MatrixThestateswitchingmatrixisselectedaspartofthe

IMMalgorithmtogoverntheunderlyingmodeswitch-ing probabilities. This matrix defines the probabilitythat a target will make the transition from one filtermodel state toanother state.Anexampleofa typicalstateswitchingmatrix,foranIMMwithtwomodels,isgivenhere:

� =

=

p pp p

11 12

21 22

0 95 0 050 12 0 88

. .

. ..

The first filter model within an IMM is typicallyselected to handle the nonmaneuvering periods of atargettrajectory.Undermostconditions,thisisbestrep-resentedbyaconstant-velocityfiltermodelwithsmallprocess noise. As a general guideline, the first modelisselectedtohavethehighestprobability.Thesecondmodel,representingatargetmaneuver,isselectedtobelessprobablethanthefirstmodel.Thisisdonetorepre-sentthebehavioroftypicalairbornetracks.

Figure 8 shows the RMS velocity errors using theCV-CAIMMfilterwiththreedifferentstateswitching

Π = [0.95 0.05; 0.12 0.88]

RM

S r

ate

erro

rs (

m/s

)

120

60

40

20

00 20 40 60 80 100

Time (s)

80

100

120 140 160

Π = [0.80 0.20; 0.05 0.95]Π = [0.60 0.40; 0.60 0.40]

ΠΠΠ

Figure 8. The filtered RMS velocity errors using a CV-CA IMM filter model with three different switching matrices plotted as a function of time using the 5.6-g constant-speed turning target.


RM

S r

ate

erro

rs (

m/s

)

120

60

40

20

00 20 40 60 80 100

Time (s)

80

100

120 140 160

Figure 6. The filtered RMS velocity errors for the 5.6-g constant-speed turning target plotted as a function of time for each of the five filtering methods. Note that the turn maneuver occurs from 60 to 110 s.


A. F. GENOVESE

matrices.Theredcurverepresentsthestateswitchingmatrix selected for the filter performance comparisonstudy.AcomparisonoftheplotsofFigs.6and8showsthattheselectionofthefiltermodelshasalargereffectonstateerrors thansubtlevariations intheswitchingmatrix.TheblackandbluecurvesinFig.8showthatmoderate changes to the state switching matrix willeffectsmallchangestothefilterperformance.Thelarg-esteffectsareseenduringthenonmaneuveringperiodswhenthetotalerrorisdominatedbystatenoise.Ingen-eral, theperformanceof the IMMappears tobe rela-tivelyinsensitivetotheselectionofthestateswitchingmatrix.

Process Noise SelectionThefiltermodelsdefined for thefilterperformance

comparisonuseprocessnoiseasa selectedfilter inputtocontrol the steady-stategains.Theprocessnoise isdefinedbyasingleselectableparameterspecifiedastheq value. The relationship between the q value andsteady-stateKalmanfiltergainsisknown.Reference9provides closed form expressions for the CV processnoisemodel.Smallqvaluesyieldsmallgainsthatpro-videgoodmeasurementnoisereductionbutleadtolargelagsduringmaneuvers.Largegainsprovidelittlenoisereductionbutgiveabetterlagresponseduringmaneu-vers. Conceptually, a multiple model filter algorithmcoulduse twofilters,oneateachextremeof thegainspectrum, and the algorithm would select the properbalancebetweenthetwofilters.ThisisnottruefortheIMMbecauseselectionofprocessnoiseneedstocon-sidertheinteractionbetweenfiltermodels.

Mixing filter model states and covariances in theIMMalgorithmallowsforapromptreactiontochang-ingtargetmodes.However,thismixingwillalsoaffectthe individual filter model gains. An example of thiseffectisshownintheplotsofFigs.9and10.Figure9shows thebearingpositiongain as a functionof timeonanonmaneuveringtargetforasingleCVfilter.Fourcurvesareshown,representingdifferentlevelsofinputprocess noise. This plot shows reduction of the filtergainasqvaluesaredecreased.TheplotinFig.10showsthesamecurvestakenfromthefirstfilterofaCV-CVIMMwhere theprocessnoiseof the secondfilterwaskeptconstant.IncontrasttotheplotinFig.9,thegainsinthisplotareshowntoreachapracticalfloorastheprocess noise is decreased. This floor varies with theqvaluesofthesecondfilter,indicatinganinherentlimi-tationintherealizabledynamicrangeofgainsinatwo-modelIMM.

TheIMMmodelprobabilitycalculationsareaffectedbytheprocessnoiseselectionforeachfilter.Theselec-tion of process noise parameters for filters within theIMM requires a balance between the high and lowmodels to achieve the best model interaction. Whenthedifferencebetweenprocessnoiseinthetwomodels

q = 0.01

Bea

ring

gain

0.1

00 20 40 60 80 100 120 140 160

Time (s)

0.2

0.3

0.4

0.5

0.6

q = 0.1

q = 1

q = 10

Figure 9. Single CV Kalman filter bearing position gains as a function of time on a nonmaneuvering target. The four curves rep-resent different levels of input process noise.

q = 0.01Bea

ring

gain

0.1

00 20 40 60 80 100 120 140 160

Time (s)

0.2

0.3

0.4

0.5

0.6

q = 0.1q = 1

q = 10

Figure 10. The bearing position gains for the first filter model from a CV-CV IMM plotted as a function of time on a nonmaneuvering target. The four curves represent different levels of input process noise on the first filter model. The q value for the second filter model was kept fixed at 15,000 m2/s3.

istoolarge,theprobabilityofthemaneuvermodelwillbe low during target maneuvers. The effect of this isadegradedperformanceonmaneuveringtargetswhena filter with a higher process noise is used. Figure 11showsRMSvelocityerrors fromaCV-CVIMMfiltermodelwiththreesetsofqvalues.Theplotrepresentedbytheredcurvehastheworstlagerrorduringthetargetmaneuvereventhoughtheprocessnoiseforthemaneu-veringmodelisthelargest.Ofnote,thebluecurveinFig.11usesthesameparametersfromthefiltercompari-sonalsoplotted inFig.6.Similar to the state switch-ing matrix, selection of the filter models has a largerimpact on model performance than the selection offilterprocessnoise.



q = [1 3600]R

MS

rat

e er

rors

(m

/s)

120

60

40

20

00 20 40 60 80 100

Time (s)

80

100

120 140 160

q = [0.1 15000]q = [100 1600]

Figure 11. The filtered RMS velocity errors using a CV-CV IMM filter model with three different sets of input process noise plotted as a function of time for the 5.6-g constant-speed turning target.

CONCLUSIONSThe comparative results for five filter models on a

varietyofmaneuveringtargetsshowthatmethodsthatusetheIMMalgorithmprovidethebestoverallresultswithrespecttofilteredpositionandrateerrors.Theper-formance improvement of the IMM is dependent onhavingfiltermodelsthatarewellmatchedtothetargetbehavior.Thefiltermodelthatisbestmatchedtothetarget dynamics will provide the best state estimatesandwillhavethehighestprobability.Resultspresentedhere also show that behavior of the IMM algorithmis robust when none of the filter models matches thetargetdynamics.TheoverallIMMperformancewillat

all timesbe similar to thebest individualfiltermodelwithintheIMM.

The performance of the IMM algorithm has beenshown to be relatively insensitive to state switchingmatrix and filter process noise parameter selection.Resultsindicatethatvariationsinthemodelparameterswilleffectsmallchangesintheperformanceofthefilteralgorithm. Parameter selection needs to be consideredtooptimizetheperformanceofanIMMgiventhecom-ponentfiltermodels.Selectionof the componentfiltermodelsshouldbetheprimaryconsiderationfordesignofanIMMsincethebestoverallperformanceisachievedwhenafiltermodelismatchedtothetargetkinematics.

REFERENCES 1Bar-Shalom, Y., Chang, K. C., and Blom, H. A. P., “Tracking a

ManeuveringTargetUsing InputEstimationVersus the InteractingMultiple Model Algorithm,” IEEE Trans. Aerosp. Electronic Syst.EAS-25(2),296–300(1989).

2Blackman,S.S.,andPopoli,R.,Design and Analysis of Modern Track-ing Systems,ArtechHouse,Norwood,MA,pp.221–240(1999).

3Yeddanapudi,M.,Bar-Shalom,Y., andPattipati,R., “IMMEstima-tionforMultitarget-MultisensorAirTrafficSurveillance,”Proc. IEEE 85(1),80–94(1997).

4Brown,R.G.,andHwang,P.Y.C.,Introduction to Random Signals and Applied Kalman Filtering,Wiley,NewYork,pp.230–236(1992).

5Castella, F. R., “Multisensor, Multisite Tracking Filter,” IEE Proc. Radar, Sonar Navigation141(2),75–82(Apr1994).

6Castella,F.R.,Upgrading to a 9-State EKF for TBM Tracking,F2A-96-0-021,JHU/APL,Laurel,MD(9May1996).

7Blair,W.D.,Watson,G.A.,andAlouani,A.T.,Use of Kinematic Constraint for Tracking Constant Speed Maneuvering Targets,NAVSWCTR91-561,NavalSurfaceWarfareCenter,Dahlgren,VA(1991).

8Watson,G.A.,andBlair,W.D.,“IMMAlgorithmforTrackingTar-getsThatManeuverThroughCoordinatedTurns,” SPIE—Signal and Data Processing of Small Targets1698,236–247(1992).

9Ekstrand,B.,“AnalyticalSteadyStateSolutionforaKalmanTrack-ingFilter,”IEEE Trans. Aerosp. Electronic Syst.AES-19(6),815–819(1983).

THE AUTHOR

ANTHONYF.GENOVESE is amemberofAPL’sSeniorProfessionalStaff.HereceivedaB.S.inelectricalengineeringfromDrexelUniversityin1992andanM.S.in electrical engineering fromThe JohnsHopkinsUniversityWhitingSchoolofEngineeringin1995.SincejoiningAPLin1992,hisworkhasfocusedondevelop-mentandtestingoftrackeralgorithmsforavarietyofprojects,includingtheMk92RadarProcessor,theCooperativeEngagementCapability,andtheItalianNavyArchobolenoProgram.Mr.GenoveseiscurrentlyamemberoftheSensorSignaland Data Processing Group of the Air Defense Systems Department. His [email protected].

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