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ImprovedParticleFilteringforPseudo-UniformBeliefDistributionsinRobotLocalisationDavidBuddenandMikhailProkopenkoInformationandCommunicationsTechnologiesCentre,AdaptiveSystemsCommonwealthScienticandIndustrialResearchOrganisation(CSIRO)POBox76,Epping,NSW2121,Australia{david.
budden,mikhail.
prokopenko}@csiro.
auAbstract.
Self-localisation,ortheprocessofanautonomousagentde-terminingitsownpositionandorientationwithinsomelocalenviron-ment,isacriticaltaskinmodernrobotics.
Althoughthistaskmaybeformallydenedasasimpletransformationbetweenlocalandglobalco-ordinatesystems,theprocessofaccuratelyandecientlydeterminingthistransformationisacomplextask.
Thisisparticularlythecaseinanenvironmentwherelocalisationmustbeinferredentirelyfromnoisyvisualdata,suchastheRoboCuprobotsoccercompetitions.
Althoughmanyeectiveprobabilisticltersexistforsolvingthistaskinitsgen-eralform,pseudo-uniformbeliefdistributions(suchasthosearisingfromcourse-grainobservations)exhibitpropertiesallowingforfurtherperfor-manceimprovement.
ThispaperexplorestheRoboCup2DSimulationLeagueasonesuchscenario,approximatingthearticiallyconstrainednoisemodelsasuniformtoderiveanimprovedparticlelterforself-localisation.
Thedevelopedsystemisdemonstratedtoyieldfrom38.
2to201.
3%reductioninlocalisationerror,whichisfurthershownascor-respondingwitha6.
4%improvementingoaldierenceacrossapproxi-mately750games.
Keywords:Robotics,localisation,particlelter,robotsoccer.
1IntroductionTheRoboCup2DSimulationLeagueincorporatesanumberofcriticalchal-lengesintheareasofarticialintelligence,machinelearninganddistributedcomputing[6,9].
Theseinclude,butarenotlimitedto:–Distributedclient/servermodel,introducingthechallengesoffragmented,localisedandimpreciseinformation(bothintermsofnoiseandlatency)abouttheenvironment[8].
–Asynchronousperception-actionactivity,andalimitedwindowofopportu-nitytoperformadesiredaction[5].
–Nocentralisedcontrollersorcentralworldmodel,resultinginalackofglobalvisionorlocalisationinformation[10,11].
S.
Behnkeetal.
(Eds.
):RoboCup2013,LNAI8371,pp.
385–395,2014.
cSpringer-VerlagBerlinHeidelberg2014386D.
BuddenandM.
ProkopenkoThispaperfocusesonthenalpoint;specically,howtoinferlocalisationin-formationfromnoisyobservationsoftheenvironment.
Initssimplestform,thetaskoflocalisation(particularlyself-localisationofanagent)canbeviewedasaproblemofcoordinatetransformation[13];specically,determiningthetransformationbetweentheagent'slocalcoordinatesystemandtheenvironment'sglobalcoordinatesystem.
Knowledgeofthistransformationallowstheagenttoconsiderglobalfeatures(suchasuniquemarkersandgoal-posts)withreferencetoitsowncoordinateframe,facilitatingnavigationandexecutionofmorecomplexbehaviours.
Knowingthepositionandorientationofanagentisbothsucientandneces-saryfordeterminingthiscoordinatetransformation.
AstheRoboCup2DSim-ulationLeaguedoesnotprovideeachagentwithglobalvisioninformation,thepositionandorientationmustbeinferredfromobservinguniqueandstation-arymarkerswithintheenvironment.
Inatraditionalroboticsscenario,theagentemploysphysicalsensors(suchascamerasorrange-nders)andcom-putervisiontechniques[3,4]toinfertherelativepositionandorientationofsuchmarkers.
Theseobservationstypicallycorrespondwithawell-denedGaussianerror,allowingfortheeectiveapplicationsofKalmanandparticlelteringtechniques[12,13].
Althoughtheneedforimageprocessingisremoved,theRoboCup2DSim-ulationLeagueintroducesanumberofuniquechallengestothetaskofself-localisation.
Particularly,thearticialobservationalnoiseintroducedbytheserverisnon-Gaussian(andapproximatelyuniformwithinwell-denedradialregions).
Theremainderofthispaperpresentsaformaldenitionoftheapprox-imatepositionalerrorregiongivenanarbitrarynumberofobservations,anddescribesamodiedparticlelterthatemploysknowledgeofthenoisemodeltoincreaseself-localisationaccuracy.
Alinearapproximationstepisintroducedtoreducethealgorithm'stimecomplexity,withthenalperformanceresultscomparedtothewell-knownagent2DimplementationdevelopedbyAkiyamaet.
al.
[1].
2ParticleFilterTheparticlelterisanonparametricimplementationoftheBayeslter,wheretheposteriordistributionbel(xt)isapproximatedbyasetofrandomstatesam-ples(particles)drawnfromthisposterior[13].
Concretely,thesetXtofMpar-ticlesaredenotedXt=x[1]t,x[2]tx[M]t,whereeachparticlex[m]t(1≤m≤M)isaninstantiationofthestateattimet.
SuchparticlesareproportionaltotheBayeslterposteriordistributionx[m]tbel(xt)=p(xt|z1:t,u1:t),wherez1:tisthesetofallpreviousobservationsandu1:tthesetofallpreviousactions.
Conceptually,theparticlelterconsistsofthefollowingsteps[13]ImprovedParticleFilteringforPseudo-UniformBeliefDistributions387Algorithm1.
particleFilter(Xt1,ut,zt)Xt=Xt=form=1.
.
.
MdoSamplex[m]tp(xt|ut,x[m]t1)w[m]t=p(zt|x[m]t)Xt=Xt+x[m]t,w[m]tendforform=1.
.
.
Mdodrawiwithprobability∝w[i]taddx[i]ttoXtendforreturnXt1.
Actionupdate:SampleMparticlesXt=x[1]t,x[2]tx[M]tatrandomfromthestatetransitiondistributionp(xt|z1:t,u1:t).
Theseparticlesformanonparametricapproximationofbel(xt),wherebel(xt)=p(xt|z1:t1,u1:t)isthepredictionposteriorbeforeconsideringobservationzt.
2.
Calculateweights:Calculateanimportancefactor(weight)w[m]t=p(zt|x[m]t)foreachparticlex[m]tgivenobservationzt.
Addeachweightedparticlex[m]t,w[m]ttothetemporaryparticlesetXt.
3.
Resample:DrawMparticles(withreplacement)fromXtandaddtotheparticlesetXt.
Theprobabilityofdrawingtheparticlex[m]tisproportionaltothecorrespondingweightw[m]t,andoftenresultsintheinclusionofduplicateparticles(andconsequentexclusionofparticleswithlowimportanceweight).
SimilarlytotheBayeslter,theparticlelterrecursivelycalculatestheposte-riorbel(xt)Xtfromthepriorbel(xt1)Xt1byconsideringtheobservationztandactionutattimet.
AbasicimplementationparticleFilter(Xt1,ut,zt)ofsuchaparticlelterisdenedinAlgorithm1.
3ProblemModelAsdescribedinSec1,alackofcentralisedcontrollersoracentralworldmodelwithintheRoboCup2Dsimulationleagueresultsinalackofglobalvisionorlocalisationinformation.
Agentsarethereforerequiredtoself-localisebyobserv-ingthepositionsofanumberofuniquelyidentiablemarkers,asillustratedinFig.
1.
Thesoccerserverthenintroducesarticialnoisetoeachobservation.
Thissectionprovidesaconcretedenitionofthiserrormodel,aswellasderivingtheapproximatedself-localisationerrorthatresultsfromNmarkerobservations.
3.
1Non-gaussianObservationNoiseAsdescribedinSec.
1,anagentrinaRoboCup2DSimulationLeaguesce-narioself-localisesbyobservinganumberofxedmarkersm[n],withtheglobal388D.
BuddenandM.
ProkopenkoFig.
1.
Visualisationofanagentrmakingobservationsr[n]mofmarkersm[n].
Thesoccerserver[6]introducesnoise(asdenedin(3.
1))toeachobservation,resultingintheself-localisationerrormodeldenedinSec.
3.
2.
positionofeachmarkerknownapriori.
Ifrwereabletomakenoise-freemea-surementsoftheenvironment,asingleobservationofanysuchmarkerwouldresultinzerolocalisationerror.
Instead,thesoccerserver[6]introducesarticialnoisetoeachobservation,suchthatthenoisemagnitudeisproportionaltotheactualdistancebetweenagentandmarker.
Inatraditionallocalisationproblem,suchnoisecouldbeaccuratelymod-eledbyaGaussiandistributionN(rm,σ)centeredabouttheobservedmarkerlocationrm,allowingfortheimplementationofwell-knownKalmanorparticlelteringtechniques[12,13].
ThesoccerserverinsteadimplementsamorecomplexnoisemodelNs(d,Δ1,Δ2),involvingthequantisationofexponentialterms[6].
Concretely,Giventheactualdistancedofanagentfromanobservedmarker,thenoisydistancedcommunicatedbytheservertotheagentisdenedasNs:d→d=QeQ(log(d),Δ1),Δ2,whereΔ1andΔ2representthequantisationstepsizes(setto0.
01and0.
1respectively),andthequantisationfunctionQisdenedasQ(x,y)=yxy.
Althoughthenonparametricnatureofaparticlelterissuitableforsuchadistribution(byapproximatingtheposteriordistributionbel(xt)asasetofparticlesXt),furtherapproximatingthiscomplexdistributionasuniformwithinaradialregion(correspondingwiththeagent'sradialeldofview1)allowsforasimpliedlterimplementation.
Theremainderofthissectionderivesaformaldenitionoftheresultantself-localisationnoisedistribution.
1Theeldofviewofanagentrischosentobeeitherπ,2π/3orπ/3radians,dependingontheobserveddistanceoftheballfromr.
ImprovedParticleFilteringforPseudo-UniformBeliefDistributions3893.
2Self-localisationErrorModelOneObservedMarker.
Theuncertaintyinthepositionofamarkerobservedatrm=(xm,ym)resultingfromobservationrbyanagentratrr=(xr,yr)maybeapproximatedasanannulussectorDofuniformprobability,asillus-tratedinFig.
2a.
DmaybeconcretelydenedasD=p∈R2:||prr||||rrrm||≤Δr,|φp|≤Δθ,whereΔθandΔrparameterisethemaximumangularanddistanceerrors(asdemonstratedinFig.
2),andφp=arccos(rmrr)·(prr)||rmrr||||prr||.
Theuniformdistri-butionDonlyhasphysicalmeaningwhenconsideringcoordinatesrelativetotherobot(astheexactpositionofthemarkerisknownapriori).
Forthepurposeofself-localisation,weareinterestedinsteadinthe(closed)regionE:thesetofallpossiblelocationsqoftheagentrresultingfromobservationr.
DrmrθrΔrΔθrDEpqqp(a)(b)Fig.
2.
VisualisationoftheregionDdeningallpossiblelocationsofamarkerm,giventhatanagentratrrobservesmatrm(a).
Asthepositionofmisknownapriori,thisobservationresultsintheregionEofallpossiblelocationsoftheagent(b).
Fig.
2billustratestheconstructionoftheregionE.
Asthepositionofthemarkerisknowntoberm,itfollowsthataobservationofthemarkerresidingatp=(px,py)∈Dcorrespondswithanerrorofprm.
Astheagentknowstheabsolutepositionofthemarkerapriori,itfollowsthatpositionalself-beliefmaybedenedasq=(qx,qy)=rr(prm).
ItmaybedemonstratedthattheequivalentmappingT:p→q=(rr+rm)p(1)resultsintheimageE=q∈R2:q=T(rr)R(π)T(rm)p,390D.
BuddenandM.
Prokopenkowhereq=(qx,qy,1)andp=(px,py,1)arethehomogeneousrepresenta-tionsofp∈Dandq∈Erespectively,T(v)3,3isthehomogenoustranslationmatrixby2-dimensionalvectorv,andR(θ)3,3isthehomogenousrotationma-trixbyangleθ2.
Concretely,q=q1=T(rr)R(π)T(rm)p=I2rr01I2001I2rm01p1=rr+rmp1∴q=(rr+rm)p,whereI2isthe2-dimensionalidentitymatrix.
ThisresultisequivalenttothemappingTdenedin(1),conrmingthatEistheimageofDunderT.
TheregionE,representingallpossiblelocationsofanagentrobservingasinglemarkeratrm(asillustratedinFig.
2),maynallyberepresentedasE=q∈R2:||qrm||||rrrm||≤Δr,|φq|≤Δθ.
(2)ThisrepresentstheintuitivedenitionoftheregionE;theregionDrotated180degreesaboutthemidpointofrrandrm.
MultipleObservedMarkers.
InaRoboCup2Dsimulationleaguescenario,itisfarmorecommonplaceforanagenttoobservenotone,butNmarkersm[n](n∈[1,N]),eachataseparatepositionknownapriori.
ThisresultsinNregionsofuniformobservationbeliefD[1]D[N],whichmaptoNcor-respondingregionsofpositionalself-beliefE[1]E[N]asperSec.
3.
2.
Astheprobabilityp(q/∈E[n])=0(n∈[1,N]),itfollowsthatthenalregionofself-beliefEresultingfromNsuchobservationsmaybeexpressedasE=Nn=1E[n],E=(3)whereE[n]istheregionrepresentingallpossiblelocationsofanagentrgivenanobservationofmarkerm[n],denedasper(1).
4ImplementationThissectiondetailstheimplementationoftheGliders2013localisationsystem,basedontheparticlelterframeworkdetailedinSec.
2andproblemmodelofSec.
3.
2Homogeneousrepresentationofan(n1)-dimensionaltransformationasann-dimensionalmatrixallowsforanyanetransformationtobedecomposedintotheproductofelementaryshear,scaling,rotationandtranslationmatrices[7].
ImprovedParticleFilteringforPseudo-UniformBeliefDistributions391D(1)D(2)rrrm(1)rm(2)r(1)r(2)EFig.
3.
VisualisationoftheregionEdeningallpossiblelocationsofanagentratrr,giventhatrobservesmarkersm[1]andm[2]atr[1]mandr[2]mrespectively4.
1ActionUpdateAsdescribedinSec.
2,theparticlelteractionupdatestepinvolvessamplingMparticlesXt=x[1]t,x[2]tx[M]tatrandomfromthestatetransitiondis-tributionp(xt|z1:t,u1:t).
IntheGliders2013implementation,thisnonparametricapproximationofbel(xt)isdenedasXt=x[m]t∈R2:x[m]t=x[m]t1+dt1m∈[1,M],wheredt1isthedisplacementresultingfromtheagent'sactionattimet1,andtheinitialsetofparticlesX0arechosenasMrandompositionssatisfyingasingleobservationofthenearestmarkerm[n].
4.
2CalculateWeightsAsdescribedinSec.
2,theparticleltercalculateweightsstepinvolvescalculat-inganimportancefactor(weight)w[m]t=p(zt|x[m]t)foreachparticlex[m]tgivenobservationzt.
AstheproblemmodeldescribedinSec.
3involvesonlyuniformprobabilitydistributions,theseweightsaresimplydenedasw[m]t=1ifx[m]t∈Et0otherwise,(4)whereEtistheregionEdenedin(3),forobservationsofmarkersm[n](1≤m≤M)attimet.
4.
3ResampleAsdescribedinSec.
2,theparticlelterresamplestepinvolvesdrawingMparticlesfromXttoaddtotheparticlesetXt,withtheprobabilityofdrawing392D.
BuddenandM.
Prokopenkotheparticlex[m]tproportionaltothecorrespondingweightw[m]t.
Astheparticleweightsdenedin(4)takeonlybinaryvalues,itfollowsthatanyparticlex[m]t/∈Etwillneverbedrawn.
Similarly,anyparticlex[m]t∈Etwillbedrawnwithequalprobability.
Followingthisobservation,theGliders2013implementationoftheposteriorXtbel(xt)isdenedasXt=Xt∩Et∪Yt,(5)whereYtisasetof|XtXt|pointsdrawnatrandomfromtheuniformdistri-butionEt.
ThismaintainsasetofMparticlesbetweeniterations,whereeachparticleisavalidpositionalself-beliefgivenallobservationsattimet3.
5ResultsTheperformanceofthepresentedGliders2013localisationsystemwasevaluatedagainsttwometrics:theeectonself-localisationerror,explicitlymeasurablebycalculatingthedierencebetweenserverandself-beliefplayerpositions;andtheeectonoverallteamperformance,implicitlymeasuredthroughthegoalstatisticsofastatisticallysignicantnumberofgames.
Fig.
4demonstratestheresultsforself-localisationerror.
Specically,averageself-localisationerror(forallplayers)isplottedasafunctionofballdistance,forthepreviousagent2D[1]localisationsystem(blue)andintroducedGliders2013particleltersystem(red).
Meanerroracross10games(60,000cycles,orap-proximately750,000measurements)ispresentedasasolidline,withthecorre-spondingstandarddeviationpresentedasadashedline.
Asthex-axisrepresentsthemaximumdistancefromconsideredplayerstotheball,valuesapproachingtherightofthegraphindicatethetotalmean,whereasvaluestowardtheleftindicatetheperformanceofplayersincriticalpositions(i.
e.
veryclosetotheball).
ItcanbeseenfromFig.
4thattheGliders2013localisationsystemyieldsameanself-localisationerrorreductionof1.
7cm;anoverallimprovementof38.
2%.
Moresignicantistheimprovementforplayersincloseproximitytotheball,wherelocalisationperformanceisespeciallycritical.
Concretely,self-localisationerrordecreasesby6.
1cmforplayerswithin1meteroftheball;animprovementof201.
3%.
Althoughtheseresultsappearsignicantasapercentage,areductionoflocal-isationerrorbycentimetersmayseeminsignicantinthecontextofafull-sizedsoccereld.
Itisthereforecriticalthattheseresultsbedemonstratedtocor-respondwithastatisticallysignicantimprovementingameperformance.
Toestablishtheextentoftheoverallteamimprovementbroughtuponbythein-creasedlocalisationaccuracy,multipleiterativeexperimentswerecarriedoutmatchingGliders2012upagainstteamsnotbasedonagent2D[1].
Amongthe3AlthoughthisdenitionoftheposteriorXtdoesnotmapexactlytotheformalparticlelterframeworkpresentedinSec.
2,itisbettersuitedtothescenarioofuniformdistributionsofpositionalself-belief.
ImprovedParticleFilteringforPseudo-UniformBeliefDistributions393010203040506070809010000.
010.
020.
030.
040.
050.
060.
070.
080.
090.
1Maximumdistancefromplayerstoball(m)Meanlocalisationerror(m)010203040506070809010000.
010.
020.
030.
040.
050.
060.
070.
080.
090.
1Standarddeviation(m)0.
0170.
061Fig.
4.
Self-localisationerrorasafunctionofballdistance,forthepreviousagent2D[1]localisationsystem(blue),andintroducedGliders2013particleltersystem(red).
Meanerrorispresentedasasolidline,withstandarddeviationpresentedasadashedline.
Anoveralllocalisationimprovementof38.
2%(or1.
7cm)isevident,andadra-maticimprovementof201.
3%(6.
1cm)forplayerswithin1meteroftheball(whenaccurateperformanceismostcritical).
Resultspresentedarefor10games(60,000cycles,or750,000measurements)betweenGliders2013andWrightEagle.
latterclass,WrightEagle[2](therunner-upofRoboCup2012)wasselected.
Ap-proximately750gameswereconductedagainstthisbenchmarkteamintotal,bothforthebaselineGliders2012team,andGliders2012incorporatingtheim-provedGliders2013localisationsystem(suchthatthissystemistheonlyvariablebetweenteams).
TheseresultsarepresentedinTable1.
Table1.
GamestatisticsbetweentheGliders2012baselineteamandWrightEagle[2](therunner-upofRoboCup2012),forboththeold(agent2D[1])andnew(Gliders2013)localisationsystems.
Astheseresultshavebeengeneratedoverapproximately750gamesandthelocalisationsystemistheonlyvariablebetweenGliders2012teams,itcanbecondentlyinferredthatthe6.
4%improvementingoaldierenceresultsdirectlyfromthe6.
1cmreductioninself-localisationerror(forcritically-positionedplayers;1.
7cmoverall).
OldLocalisationNewLocalisationImprovementGoalsScored0.
360.
4114%GoalsConceded2.
192.
132.
8%GoalDierence-1.
83-1.
726.
4%394D.
BuddenandM.
Prokopenko6ConclusionSelf-localisationisacomplexyetcriticaltask,bothwithintheRoboCup2DSim-ulationLeague,andthemulti-billiondollarroboticsindustryatlarge.
Itallowsanagenttoinferitsownpositionwithinsomelocalenvironment,thusfacilitatingnavigationaroundorinteractionwithobstacles,featuresandotheragents.
Thispaperhasdescribedamodiedparticlelteralgorithmfornon-Gaussianbeliefdistributions,anddemonstratedhowapproximatingthesedistributionsasuni-form(givenconstrainednoisemodels,suchasthoseresultingfromcourse-grainobservationsorarticialenvironments)facilitatesreductioninself-localisationerror.
Theproposedsystemyieldedanimprovementof38.
2%foragentswithinaRoboCup2DSimulationLeagueenvironment,whichincreasedto201.
3%asagentsapproachperformance-criticalregionsoftheeld.
Itwasfurtherdemon-stratedthatthisimprovementresultsina14%increaseingoalsscoredand2.
8%reductioningoalsconcededfortheGliders2012baselineteam,evaluatedacrossapproximately750gamesagainsttheWrightEagleteam[2](RoboCup2012runners-up).
Futureresearchwillfocusonapplyingsimilarmethodologiestoballandplayerlocalisationsystems(includingtheincorporationandoptimisationofteamcom-municationstrategies),inanattempttoapproachtheperformanceleveragedbyenablingfullgamestateinformationforallplayeragents.
Acknowledgements.
TheauthorswouldliketothanktheUniversityofNew-castle'sNUbotsRoboCupteaminassistingwiththepreparationofthismanuscript.
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