Localization
B.J.A.Kr¨ose,N.Vlassis,andR.Bunschoten
RealWorldComputingPartnership,NovelFunctionsLaboratorySNN,DepartmentofComputer
Science,UniversityofAmsterdam,
Kruislaan403,NL-1098SJAmsterdam,TheNetherlands
{krose,vlassis,bunschot}@science.uva.nl
Abstract.Mobilerobotsneedaninternalrepresentationoftheirenvironmenttodousefulthings.Usuallysucharepresentationissomesortofgeometricmodel.Forourrobot,whichisequippedwithapanoramicvisionsystem,wechooseanappearancemodelinwhichthesensoricdata(inourcasethepanoramicimages)havetobemodeledasafunctionoftherobotposition.Becauseimagesareveryhigh-dimensionalvectors,afeatureextractionisneededbeforethemodelingstep.VeryoftenalineardimensionreductionisusedwheretheprojectionmatrixisobtainedfromaPrincipalComponentAnalysis(PCA).PCAisoptimalforthereconstructionofthedata,butnotnecessarilythebestlinearprojectionforthelocalizationtask.Wederivedamethodwhichextractslinearfeaturesoptimalwithrespecttoariskmeasurereflectingthelocalizationperformance.WetestedthemethodonarealnavigationproblemandcompareditwithanapproachwherePCAfeatureswereused.
1Introduction
Aninternalmodeloftheenvironmentisneededtonavigateamobilerobotoptimallyfromacurrentstatetowardadesiredstate.Suchmodelscanbetopologicalmaps,basedonlabeledrepresentationsforobjectsandtheirspatialrelations,orgeometricmodelssuchaspolygonsoroccupancygridsinthetaskspaceoftherobot.
Awidevarietyofprobabilisticmethodshavebeendevelopedtoobtainarobustestimateofthelocationoftherobotgivenitssensoryinputsandtheenvironmentmodel.Thesemethodsgenerallyincorporatesomeobservationmodelwhichgivestheprobabilityofthesensormeasurementgiventhelocationoftherobotandtheparameterizedenvironmentmodel.Sometimesthisparametervectordescribesexplicitpropertiesoftheenvironment(suchaspositionsoflandmarks[8]oroccupancyvalues[4])butcanalsodescribeanimplicitrelationbetweenasensorpatternandalocation(suchasneuralnetworks[6],radialbasisfunctions[10]orlook-uptables[2]).
Ourrobotisequippedwithapanoramicvisionsystem.Weadopttheimplicitmodelapproach:wearenotgoingtoestimatetheparametersofsomesortofCADmodelbutwemodeltherelationbetweentheimagesandtherobotlocationdirectly(appearancemodeling).
Insection2wedescribehowthismodelisusedinaMarkovlocalizationprocedure.Thenwediscusstheproblemofmodelinginahighdimensionalimagespaceanddescribethestandardap-proachforlinearfeatureextractionbyPrincipalComponentAnalysis(PCA).Inordertoevaluatethemethodweneedacriterion,whichisdiscussedinsection5.Thecriterioncanalsobeusedtofindanalternativelinearprojection:thesupervisedprojection.Experimentsonrealrobotdataarepresentedinsections6and7wherewecomparethetwolinearprojectionmethods.
2Probabilisticappearance-basedrobotlocalization
Letxbeastochasticvector(e.g.,2-Dor3-D)denotingtherobotpositionintheworkspace.Similarto[1]weemployaformofMarkovlocalizationforourmobilerobot.Thismeansthatateachpointintimewehaveabeliefwheretherobotisindicatedbyaprobabilitydensityp(x).
Markovlocalizationrequirestwoprobabilisticmodelstomaintainagoodpositionestimate:amotionmodelandanobservationmodel.
Themotionmodeldescribestheeffectamotioncommandhasonthelocationoftherobotandcanberepresentedbyaconditionalprobabilitydensity
p(xt|u,xt−1)
(1)
whichdeterminesthedistributionofxt(thepositionoftherobotafterthemotioncommandu)iftheinitialrobotpositionisxt−1.
Theobservationmodeldescribestherelationbetweentheobservation,thelocationoftherobot,andtheparametersoftheenvironment.Inoursituationtherobottakesanomnidirectionalimagezatpositionx.Weconsiderthisasarealizationofastochasticvariablez.Theobservationmodelisnowgivenbytheconditionaldistribution
p(z|x;θ),
(2)
inwhichtheparametervectorθdescribesthedistributionandreflectstheunderlyingenvironmentmodel.
UsingtheBayes’rulewecangetanestimateofthepositionoftherobotafterobservingz:
p(x|z;θ)=p(z|x;θ)p(x)
.
p(z|x;θ)p(x)dx
(3)
Herep(x)givestheprobabilitythattherobotisatxbeforeobservingz.Notethatp(x)canbederivedusingtheoldinformationandthemotionmodelp(xt|u,xt−1)repeatedly.Ifbothmodelsareknownwecancombinethemanddecreasethemotionuncertaintybyobservingtheenvironmentagain.
Inthispaperwewillfocusontheobservationmodel(2).Inordertoestimatethismodelweneedadatasetconsistingofpositionsxandcorrespondingobservationsz1.Wearenowfacedwiththeproblemofmodelingdatainahigh-dimensionalspace,particularlysincethedimensionalityofz(inourcasetheomnidirectionalimages)ishigh.Thereforethedimensionalityofthesensordatahastobereduced.Herewerestrictourselvestolinearprojections,inwhichtheimagecanbedescribedasasetoflinearfeatures.WewillstartwithalinearprojectionobtainedfromaPrincipalComponentAnalyis(PCA),asisusuallydoneinappearancemodeling[5]
3PrincipalComponentAnalysis
LetusassumethatwehaveasetofNimages{zn},n=1,...,N.Theimagesarecollectedatrespective2-dimensionalrobotpositions{xn}.Eachimageconsistsofdpixelsandisconsideredasad-dimensionaldatavector.InaPrincipalComponentAnalysis(PCA)theeigenvectorsofthecovariancematrixofanimagesetarecomputedandusedasanorthogonalbasisforrepresentingindividualimages.Although,ingeneral,forperfectreconstructionalleigenvectorsarerequired,onlyafewaresufficientforvisualrecognition.Theseeigenvectorsconstitutetheq,(q thattheeigenvectorwiththelargesteigenvaluerepresentsthedirectioninwhichthevariationinthesetofimagesismaximal.WenowstacktheNimagevectorstoformtherowsofanN×d 1 Inthispaperweassumewehaveasetofpositionsandcorrespondingobservations:ourmethodissupervised.Itisalsopossibletodoasimultaneouslocalizationandmapbuilding(SLAM).Inthiscasetheonlyavailabledataisastreamofdata{z(1),u(1),z(2),u(2),...,z(T),u(T)}inwhichuisthemotioncommandtotherobot.Usingamodelabouttheuncertaintyofthemotionoftherobotitispossibletoestimatetheparametersfromthesedata[8]. imagematrixZ.Thenumericallymostaccuratewaytocomputetheeigenvectorsfromtheimagesetisbytakingthesingularvaluedecomposition[7]Z=ULVToftheimagematrixZ,whereVisad×qorthonormalmatrixwithcolumnscorrespondingtotheqeigenvectorsviwithlargesteigenvaluesλiofthecovariancematrixofZ[3]. Theseeigenvectorsviarenowthelinearfeatures.Notethattheeigenvectorsarevectorsinthed-dimensionalspace,andcanbedepictedasimages:theeigenimages.TheelementsoftheN×qmatrixY=ZVaretheprojectionsoftheoriginald-dimensionalpointstothenewq-dimensionaleigenspaceandaretheq-dimensionalfeaturevalues. 4Observationmodel Thelinearprojectiongivesusafeaturevectory,whichwewilluseforlocalization.TheMarkovlocalizationprocedure,aspresentedinSection2,isusedonthefeaturevectory: p(x|y;θ)=p(y|x;θ)p(x) , p(y|x;θ)p(x)dx (4) wherethedenominatoristhemarginaldensityoverallpossiblex.Wenowhavetofindamethodtoestimatetheobservationmodelp(y|x;θ)fromadataset{xn,yn},n=1,...,N. WeusedakerneldensityestimationorParzenestimator.InaParzenapproachthedensityfunctionisapproximatedbyasumofkernelfunctionsovertheNdatapointsfromthetrainingset.Notethatinastrictsensethisisnota‘parametric’techniqueinwhichtheparametersofsomepre-selectedmodelareestimatedfromthetrainingdata.Instead,thetrainingpointsthemselvesaswellasthechosenkernelwidthmaybeconsideredastheparametervectorθ.Wewritep(y|x;θ)as p(y,x;θ) (5)p(y|x;θ)= p(x;θ)andrepresenteachofthesedistributionasasumofkernelfunctions: N1 p(x,y;θ)=gy(y−yn)gx(x−xn) Nn=1 N1 gx(x−xn).p(x;θ)= Nn=1 (6)(7) where 1||y||2 gy(y)=exp− 2h2(2π)q/2hqand 1||x||2 exp−gx(x)= 2πh22h2(8) aretheq-andtwo-dimensionalGaussiankernel,respectively.Forsimplicityinourexperimentswe usedthesamewidthhforthegxandgykernels. 5Featurerepresentation Asismadeclearintheprevioussections,theperformanceofthelocalizationmethoddependsonthelinearprojection,thenumberofkernelsintheParzenmodel,andthekernelwidths.Firstwediscusstwomethodswithwhichthemodelcanbeevaluated.Thenwewilldescribehowalinearprojectioncanbefoundusingtheevaluation. 5.1Expectedlocalizationerror Amodelevaluationcriterioncanbedefinedbytheaverageerrorbetweenthetrueandtheesti-matedposition.Suchariskfunctionforrobotlocalizationhasbeenproposedin[9].Supposethedifferencebetweenthetruepositionx∗oftherobotandthetheestimatedpositionbyxisdenotedbythelossfunctionL(x,x∗).Iftherobotobservesy∗,theexpectedlocalizationerrorε(x∗,y∗)is(usingBayes’rule)computedas ∗∗ L(x,x∗)p(x|y∗)dxε(x,y)=x∗ ∗p(y|x)p(x)=dx.(9)L(x,x)∗)p(yxToobtainthetotalriskfortheparticularmodel,theabovequantitymustbeaveragedoverall possibleobservationsy∗obtainedfromx∗andallpossiblex∗togive ε(x∗,y∗)p(y∗,x∗)dy∗dx∗(10)RL= x∗ y∗ Theempiricalriskiscomputedwhenestimatingthisfunctionfromthedata: N1ˆL=Rε(xn,yn)Nn=1 NN 1l=1L(xl,xn)p(yn|xl)=.NNn=1p(y|x)nll=1 (11) Thisriskpenalizespositionestimatesthatappearfarfromthetruepositionoftherobot.A problemwiththisapproachisthatifatafewpositionsthereareverylargeerrors(forexampletwodistantlocationshavesimilarvisualfeaturesandmaybeconfused),theaverageerrorwillbeveryhigh.5.2 Measureofmultimodality Analternativewayofevaluatingthelinearprojectionfromztoyistoconsidertheaveragedegreeofmodalityofp(x|y)[11].Theproposedmeasureisbasedonthesimpleobservationthat,foragivenobservationznwhichisprojectedtoyn,thedensityp(x|y=yn)willalwaysexhibitamodeonx=xn.Thus,anapproximatemeasureofmultimodalityistheKullback-Leiblerdistancebetweenp(x|y=yn)andaunimodaldensitysharplypeakedatx=xn,givingtheapproximateestimate−logp(xn|y=yn)plusaconstant.Averagingoverallpointsynwehavetominimizetherisk N1ˆK=−logp(xn|y=yn)(12)R Nn=1withp(x|y=yn)computedwithkernelsmoothingasin(5).Thisriskcanberegardedasthenegativeaveragelog-likelihoodofthedatagivenamodeldefinedbythekernelwidthsandthespecificprojectionmatrix.ThecomputationalcostsofthisisO(N2),incontrastwiththeO(N3) ˆL.forR5.3 Supervisedprojection Weusetheriskmeasuretofindalinearprojectiony=WTzalternativetothePCAprojection. ˆKasafunctionoftheprojectionmatrixWallowstheminimiza-ThesmoothformoftheriskR tionoftheformerwithnonlinearoptimization.Forconstrainedoptimizationwemustcompute ˆKandthegradientoftheconstraintfunctionWTW−IqwithrespecttoW,thegradientofR andthenplugtheseestimatesinaconstrainednonlinearoptimizationroutinetooptimizewith ˆK.WefollowedanalternativeapproachwhichavoidstheuseofconstrainednonlinearrespecttoR optimization.TheideaistoparameterizetheprojectionmatrixWbyaproductofGivens(Ja-cobi)rotationmatricesandthenoptimizewithrespecttotheangleparametersinvolvedineachmatrix(see[12]fordetails).SuchasupervisedprojectionmethodmustgivebetterresultsthananunsupervisedonelikePCA(seeexperimentsinthispaperand[11]).Inthenextsectionswewillexperimentallytestbothmethods. 6ExperimentsusingPCAfeatures FirstwewanttoknowhowgoodthelocalizationiswhenusingPCAfeatures.Inparticularweinvestigatehowmanyfeaturesareneeded.6.1 Datasetsandpreprocessing Wetestedourmethodsonrealimagedataobtainedfromarobotmovinginanofficeenvironment,ofwhichanoverviewisshowninfigure1.WemadeuseoftheMEMORABLErobotdatabase.ThisdatabaseisprovidedbyTsukubaResearchCenter,Japan,fortheRealWorldComputingPart-nershipandcontainsadatasetofabout8000robotpositionsandassociatedmeasurementsfromsonars,infraredsensorsandomni-directionalcameraimages.Themeasurementsinthedatabasewereobtainedbypositioningtherobot(aNomad200)onthegrid-pointsofavirtualgridwithdistancesbetweenthegrid-pointsof10cm.OneofthepropertiesoftheNomad200robotisthatitmovesaroundinitsenvironmentwhilethesensorheadmaintainsaconstantorientation.Becauseofthis,thestateoftherobotischaracterizedbythepositionxonly. Fig.1.Theenvironmentfromwhichtheimagesweretaken. Theomni-directionalimagingsystemconsistsofaverticallyorientedstandardcolorcameraandahyperbolicmirrormountedinfrontofthelens.Thisresultsinimagesasdepictedinfigure2.Usingthepropertiesofthemirrorwetransformedtheomni-directionalimagesto360degreespanoramicimages.Toreducethedimensionalitywesmoothedandsubsampledtheimagestoaresolutionof×256pixels(figure3).Asetof2000imageswasrandomlyselectedfromthetotalsettoderivetheeigenimagesandassociatedeigenvalues.Wefoundthatfor80%reconstructionerrorweneededabout90eigenvectors.However,wearenotinterestedinthereconstructionofimages,butintheuseofthelow-dimensionalrepresentationforrobotlocalization. Fig.2.Typicalimagefromacamerawithahyperbolicmirror Fig.3.Panoramaimagederivedwiththeomnidirectionalvisionsystem. 6.2Observationmodel Insection4wedescribedthekernelestimatorasawaytorepresenttheobservationmodelp(y|x;θ).Insuchamethodusuallyalltrainingpointsareusedforthemodeling.Inourdatabasewehave8000pointswecanuse.Ifweusethiswholedatasetthismeansthatintheoperationalstageweshouldcalculatethedistancetoall8000pointsforthelocalization,which,eventhoughthedimensionsofxandyarelow,iscomputationallytooslow.Wearethereforeinterestedintakingonlyapartofthesepointsinthekerneldensityestimationmodel.Inthefollowingsectionsasetofabout300imageswasselectedasatrainingset.Theseimagesweretakenfromrobotpositionsonthegrid-pointsofavirtualgridwithdistancesbetweenthegrid-pointsof50cm. AnotherissueinthekernelmethodisasensiblechoiceforthewidthoftheGaussiankernel.Theoptimalsizeofthekerneldependsontherealdistribution(whichwedonotknow),thenumberofkernelsandthedimensionalityoftheproblem.Whenmodelingp(x,y)foraone-dimensionalfeaturevectorywithourtrainingsetwefoundthath≈0.1maximizedthelikelihoodofanindependenttestset.Thetestsetconsistedof100images,randomlyselectedfromtheimagesinthedatabasenotdesignatedastrainingimages.Whenusingmorefeaturesforlocalization(ahigherdimensionalfeaturevectory)theoptimalsizeofthekernelwasfoundtobehigher.Weusedthesevaluesinourobservationmodel.6.3 Localization InaMarkovlocalizationprocedure,aninitialpositionestimateisupdatedbythefeaturesofanewobservationusinganobservationmodel.Theinitialpositionestimateiscomputedusingthemotionmodel,andgivesaninformedpriorintheBayesrule.Sinceweareonlyinterestedintheperformanceoftheobservationmodel,weassumeaflatpriordistributionontheadmissiblepositionsx. 45403530p(x|y)p(x|y)10−1−2−3−1−0.500.511.5252015105022454035302520151050210−1−2−3−1−0.500.511.52a)x2−1.5x1b)x2−1.5x1Fig.4.Animageatposition(1.74,-0.96)istaken.Thefiguredepictstheprobabilitydistributionoverthelearnedlocations.a)Thefirsteigenvector(withthehighesteigenvalue)isusedasfeature.b)Thefirst5eigenvectorsareused. InthecurrentexperimentswestudiedhowmanyoftheprincipalcomponentsareneededforgoodlocalizationInfigure4weseethedistributionp(x|y)foranimagewhichwastakenatposition(1.74,-0.96),fortwodifferentnumberoffeatures:fiveeigenimagesoroneeigenimage.Weobservethatthedistributionwhenusingasinglefeaturehasmultiplepeaks,indicatingmultiplehypothesesfortheposition.Thisissolvedifmorefeaturesaretakenintoaccount.Inbothsituationsthemaximumaposteriorivalueisclosetotherealrobotposition.Thisillustratesthatthemodelgivesagoodpredictionoftherobot’srealposition.Insomecasesweobservedamaximalvalueofthedistributionatanerroneouslocationiftoofewfeatureswereused.Soweneedsufficientfeaturesforcorrectlocalization.Theeffectofthenumberoffeaturesisdepictedinfigure5whereweplottedlocalizationriskfordifferentnumberoffeatures.Weseethatforthisdataset(300positions)theperformancelevelsoutafterabout10-15features. loss−based risk2.521.5risk10.50024681012feature vector dimensionality14161820Fig.5.Performancefordifferentnumberfeatures 7ComparingPCAwithasupervisedprojection WealsocomparedthelocalizationoftherobotwhenusingPCAfeaturesandwhenusingsupervisedprojectionfeatures.Hereweuseddatacollectedinourownlaboratory.TheNomadrobotfollowsapredefinedtrajectoryinourmobilerobotlabandtheadjoininghallasshowninFig.6.Thedatasetcontains104omnidirectionalimages(320×240pixels)capturedevery25centimetersalongtherobotpath.Eachimageistransformedtoapanoramicimage(×256)andthese104panoramicimagestogetherwiththerobotpositionsalongthetrajectoryconstitutethetrainingsetofouralgorithm.AtypicalpanoramicimageshotatthepositionAofthetrajectoryisshowninFig.7. 11000011B 1001A 1001C Fig.6.Therobottrajectoryinourbuilding. Fig.7.ApanoramicsnapshotfrompositionAintherobottrajectory. Inordertoapplyoursupervisedprojectionmethod,wefirstspheredthepanoramicimage data.Spheringisanormalizationtozero-meanandidentitycovariancematrixofthedata.ThisisalwayspossiblethroughPCA.Wekeptthefirst10dimensionsexplainingabout60%ofthetotalvariance.Therobotpositionswerenormalizedtozeromeanandunitvariance.Thenweappliedthesupervisedprojectionmethod(see[12]fordetailsofoptimization)projectingthesphereddatapointsfrom10-Dto2-D. InFig.8weplottheresultingtwo-dimensionalprojectionsusing(a)PCA,and(b)oursuper-visedprojectionmethod.WeclearlyseetheadvantageoftheproposedmethodoverPCA.Theriskissmaller,whilefromtheshapeoftheprojectedmanifoldweseethattakingintoaccounttherobotpositionduringprojectioncansignificantlyimprovetheresultingfeatures:therearefewerself-intersectionsoftheprojectedmanifoldinourmethodthaninPCAwhich,inturn,meansbet-terrobotpositionestimationontheaverage(smallerrisk).In[11]wealsoshowthatlocalizationismoreaccuratewhenusingthesupervisedprojectionmethod. 2.52 21.51.5110.5C 0.50B 0B −0.5−0.5−1−1−1.5−1.5−2C −2−2−1.5−1−0.500.511.52−2.5−2.5−2−1.5−1−0.500.511.52(a)PCAprojection:RK=−0.69(b)supervisedprojection:RK=−0.82Fig.8.Projectionofthepanoramicimagedatafrom10-D.(a)Projectiononthefirsttwoprincipalcomponents.(b)SupervisedprojectionoptimizingtheriskRK.ThepartwiththedashedlinescorrespondstoprojectionsofthepanoramicimagescapturedbytherobotbetweenpositionsBandCofitstrajectory. Finally,thetwofeaturevectors(directionsintheimagespaceonwhichtheoriginalimagesareprojected)thatcorrespondtotheabovetwosolutionsareshowninFig.9.InthePCAcasethesearethefamiliarfirsttwoeigenimagesofthepanoramicdatawhich,asisnormallyobservedintypicaldatasets,exhibitlowspatialfrequencies.WeseethattheproposedsupervisedprojectionmethodyieldsverydifferentfeaturevectorsthanPCA,namely,imageswithhigherspatialfrequenciesanddistinctlocalcharacteristics. 1steigenvector1stsupervisedfeaturevector 2ndeigenvector2ndsupervisedfeaturevector Fig.9.ThefirsttwofeaturevectorsusingPCA(left)andoursupervisedprojectionmethod(right). 8Discussionandconclusions Weshowedthatappearance-basedmethodsgivegoodresultsonlocalizingamobilerobot.IntheexperimentswiththePCAfeatures,theaverageexpectedlocalizationerrorfromourtestsetisabout40cmifaround15featuresareusedandtheenvironmentisrepresentedwith300trainingsamples.Notethatwestudiedtheworst-casescenario:therobothasnopriorinformationaboutitsposition(the‘kidnappedrobot’problem),andcombinedwithamotionmodelthelocalizationaccuracyshouldbebetter.Asecondobservationisthattheenvironmentcanberepresentedbyonlyasmallnumberofparameters.Forthe30015-dimensionalfeaturevectorsthestoragecapacityisalmostnegligibleandthelook-upcanbedoneveryfast. Theexperimentswiththesupervisedprojectionshowedthatthismethodresultedinalowerrisk,andthereforeabetterexpectedlocalization.In[11]wedescribeanexperimentwhereweusedthefullMarkovproceduretolocalizetherobot.ThesupervisedprojectionmethodgavesignificantlybetterresultsthanthePCAfeatures. Bothexperimentswerecarriedoutwithextensivedatasets,withwhichwewereabletogetgoodestimatesontheaccuracyofthemethod.However,thedatawereobtainedinastaticenvironment,withconstantlightingconditions.Ourcurrentresearchinthislinefocusesoninvestigatingwhichfeaturesaremostimportantifchangesintheilluminationwilltakeplace. 9Acknowledgment WewouldliketothankthepeopleintheRealWorldComputingPartnershipconsortiumandtheTsukubaResearchCenterinJapanforprovidinguswiththeMEMORABLErobotdatabase. 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