IEEE TRANSACTIONS ON AUTOMATIC CONTROL VOL - PDF document

IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.49,NO.8,AUGUST2004CooperativeControlofMobileSensorNetworks:AdaptiveGradientClimbinginaDistributedEnvironmentPetterÖgren,Member,IEEE,EdwardFiorelli,Member,IEEE,andNaomiEhrichLeonard,SeniorMember,IEEEWepresentastablecontrolstrategyforgroupsofvehiclestomoveandreconÞgurecooperativelyinresponsetoasensed,distributedenvironment.Eachvehicleinthegroupservesasamobilesensorandthevehiclenetworkasamobileand etal.:COOPERATIVECONTROLOFMOBILESENSORNETWORKSeachindividualinthegroupusescontrolforcesthatderivefrominter-vehiclepotentialssimilartothoseusedtomodelnaturalswarms[10],[20].Theseprovidegroupcohesionandhelppreventcollisions.Theframeworkisbasedonthatpresentedin[16].Thisframeworkleadstodistributedcontroldesignsinwhicheachvehiclerespondstoitslocalenvironment.Noorderingofvehiclesisnecessaryandthisprovidesrobustnesstovehiclefailuresorotherchangesinthenumberofoperatingvehicles.Toaccomplishthedecouplingoftheformationstabilizationproblemfromtheoverallperformanceofthenetworkmission,weintroducetothegroupavirtualbody.Thevirtualbodyisacollectionoflinked,movingreferencepoints.Thevehiclegroupmoves(andrecongures)withthevirtualbodybymeansofforcesthatderivefromarticialpotentialsbetweenthevehiclesandthereferencepointsonthevirtualbody.Thevirtualbodycantranslateandrotateinthree-dimensionalspace,expandandcontract.Thedynamicsofthevirtualbodyaredesignedintwosteps.Inonestep,extending[21],weregulatetheofthevirtualbodyusingafeedbackformationerrorfunctiontoen-surestabilityandconvergencepropertiesoftheformation.Intheotherstep,weprescribethedirectionofmotionofthevirtualbodytoaccomplishthedesiredmission,e.g.,adaptivegradientclimbinginadistributedenvironment.Thedevelopmentof[21]concernscoordinationalongprespeciedtrajectories.Theprescriptionofvirtualbodydynamicsrequiressomecentralizedcomputationandcommunication.Eachvehicleinthegroupcommunicatesitsstateandeldmeasurementstoacentralcomputerwheretheupdatedstateofthevirtualbodyiscomputed.Thecongurationofthevirtualbodyiscommunicatedbacktoeachvehicleforuseinitsownlocal(decentralized)controllaw.ThisscenariowasmostpracticalintheAOSN-IIexperimentbecausethegliderssurfacedregularlyandestablishedtwo-waycommunicationwiththeshorestation.Forgradientclimbingtasks,thegradientofthemeasuredeldisapproximatedatthevirtualbodyspositionusingthe(noisy)dataavailablefromallvehicles.Centralizedcomputa-tionisused.Wepresentaleast-squaresapproximationofthegradientandstudytheproblemoftheoptimalformationthatminimizesestimationerror.WealsodesignaKalmanlterandusemeasurementhistorytosmoothouttheestimate.Ourframeworkmakesitpossibletopreservesymmetrywhenthereislimitedcontrolauthorityinadynamicenvironment.Forexample,inthecaseofunderwaterglidersinastrongthegroupcanbeinstructedtomaintainauniformdistributionasneeded,butbefreetospin,andpossiblywiggle,withtheOfequalimportancearetheconsequencesofdelays,asyn-chronicity,andotherreliabilityissuesincommunications.In[17],forexample,stabilityofchain-likeswarmsisconsid-eredinthepresenceofsensingdelaysandasynchronism.Weassumeinthispaperthatthecommunicationissynchronizedandcontinuous(theimplementationintheMontereyBayex-perimentwasmodiedtoaddressthesekindsofrealities[9]).Wenotethatcentralizedcomputationmaybecomeburden-someforlargegroupsofvehicles,e.g.,whenaddressingthenonconvexoptimizationofformationsforminimizationofes-timationerror.cialpotentialswereintroducedtoroboticsforobstacleavoidanceandnavigation[13],[29].Inthemodelingofanimalaggregations,forcesthatderivefrompotentialsareusedtonelocalinteractionsbetweenindividuals(see[23]andthereferencestherein).Inrecentworkalongtheselines,theauthorsof[10]and[20]investigateswarmstabilityundervariouspotentialfunctionproles.Articialpotentialshavealsorecentlybeenexploitedtoderivecontrollawsforautonomous,multiagent,roboticsystemswhereconvergenceproofstodesiredgurationsareexplicitlyprovided(see,forexample,[19],[16],[24],and[31]).Translation,rotationandexpansionofagroupistreatedin[33]usingasimilarnotionofavirtualrigidbodycalledavirtualstructurewhichhasdynamicsdependentonaformationerrorfunction.However,theformationcontrollawsandthedynamicsofthevirtualstructuredifferfromthosepresentedhere,andanorderingofvehiclesisimposedin[33].Gradientclimbingwithavehiclenetworkisalsoatopicofgrowinginterestintheliterature(see,forexample,[2]and[10]).In[18],gradientclimbingisperformedinthecontextofdistributingvehiclenetworksaboutenvironmentalboundaries.In[6],theauthorsuseVoronoidiagramsandaprioritionaboutanenvironmenttodesigncontrollawsforavehiclenetworktooptimizesensorcoveragein,e.g.,surveillanceThepaperisorganizedasfollows.InSectionII,wereviewtheformationframeworkof[16]basedonarticialpotentialsandvirtualleaders.FormationmotionisintroducedinSectionIIIandthepartiallydecoupledproblemsofformationstabilizationandmissioncontrolaredescribed.Themaintheoremforforma-tionstabilizationispresentedinSectionIV.AdaptivegradientclimbingmissionsaretreatedinSectionV.Weprovidenalre-marksinSectionVI.Anearlierversionofpartsofthispaperappearedin[22].II.ARTIFICIALOTENTIALSIRTUALODIESANDYMMETRYInthissection,wedescribetheunderlyingframeworkfordis-tributedformationcontrolbasedonarticialpotentialsandavir-tualbody.Theframeworkfollowsthatpresentedin[16](withsomevariationinnotation).Eachvehicleinthegroupismod-eledasapointmasswithfullyactuateddynamics.Extensiontounderactuatedsystemsispossible.In[14],theauthorsusefeed-backlinearizationtotransformthedynamicsofanoff-axispointonanonholonomicrobotintofullyactuateddoubleintegratorequationsofmotion.Letthepositionofthe thvehicleinagroupof vehicles,withrespecttoaninertialframe,begivenbyavector , asshowninFig.1.Thecontrolforceonthe vehicleisgivenby .Thedynamicsare for Weintroduceawebof referencepointscalledleadersanddenethepositionofthe thvirtualleaderwithre-specttotheinertialframetobe ,for Assumethatthevirtualleadersarelinked,i.e.,letthemformvirtualbody.Thepositionvectorfromtheoriginoftheiner-tialframetothecenterofmassofthevirtualbodyisdenoted ,asshowninFig.1.In[16],thevirtualleadersmove IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.49,NO.8,AUGUST2004 Fig.1.Notationforframework.Solidcirclesarevehiclesandshadedcirclesarevirtualleaders.withconstantvelocity.Inthissectionwespecializetothecaseinwhichallvirtualleadersareatrest.MotionofthevirtualbodywillbeintroducedinSectionIII. and .Betweeneverypairofvehicles and wedeneanarticialpotential whichdependsonthedistancebetweenthe thand vehicles.Similarly,betweeneveryvehicle andeveryvirtual wedeneanarticialpotential whichdependsonthedistancebetweenthe thvehicleand thvirtualleader.Thecontrollaw isdenedasminusthegradientofthesumofthesepotentialsplusalineardampingterm (II.1)where isapositivenitematrix.Weconsidertheformofpotential thatyieldsaforcethatisrepellingwhenapairofvehiclesistooclose,i.e.,when ,attractingwhenthevehiclesaretoofar,i.e.,when andzerowhenthevehiclesareveryfarapart ,where and areconstantdesignparameters.Thepo- isdesignedsimilarlywithpossiblydifferentdesign and (amongothers),seeFig.2.Eachvehicleusesexactlythesamecontrollawandisinencedonlybynearneighborvehicles,i.e.,thosewithinaballof ,andnearbyvirtualleaders,i.e.,thosewithinaballof .Theglobalminimumofthesumofalltheartipotentialsconsistsofacongurationinwhichneighboringve-hiclesarespacedadistance fromoneanotherandadistance fromneighboringvirtualleaders.In[16],wediscussfurtherhowtodeneavirtualbodyforcertainvehicleformations.Forexample,thehexagonallatticeformationshowninFig.3isanequilibriumfor ,and Theglobalminimumwillexistforappropriatechoiceof and ,butitwillnotingeneralbeunique.FortheexampleofFig.3,thelatticeisattheglobalminimum;however,itisnotuniquesincethereisrotationalandtranslationalsymmetryoftheformationanddiscretesymmetries(suchaspermutationsofthevehicles).Translationalsymmetryofthegroupresultsbecausethepotentialsonlydependuponrelativedistance.The Fig.2.Representativecontrolforcesderivedfromarticialpotentials. Fig.3.Hexagonallatticeformationwithtenvehiclesandonevirtualleader. Fig.4.Equilibriumsolutionsforaformationintwodimensionswithtwovehicles.(a)Withonevirtualleaderthereis symmetryandafamilyofsolutions(twoareshown).(b)Withtwovirtualleadersthe symmetrycanbebrokenandtheorientationofthegroupxed.rotationalsymmetrycan,ifdesired,bebrokenwithadditionalvirtualleadersasshowninFig.4.Itissometimesofinteresttohavetheoptionofbreakingsym-metryornot.Breakingsymmetrybyintroducingadditionalvir-tualleaderscanbeusefulforenforcinganorientation,butitmaymeanincreasedinputenergyfortheindividualvehicles.Undercertaincircumstances,itmaynotbefeasibletoprovidesuchinputenergyandinsteadmorepracticaltosettleforagroupshapeandspacingwithoutaprescribedgrouporientation.Wedenethestateofthevehiclegroupas .In[16],localasymptoticstability correspondingtothevehiclesatrestattheglobalminimumofthesumofthearticialpotentialsisprovedwiththeLyapunovfunction III.FORMATIONOTIONRANSLATIONOTATIONXPANSIONInthissection,weintroducemotionoftheformationbypre-scribingmotionofthevirtualbody.Thismotioncanincludetranslation,rotation,expansionandcontractionofthevirtualbodyand,therefore,thevehicleformation.Byparameterizingthevirtualbodymotionbythescalarvariable ,weenableadecouplingoftheproblemofformationstabilizationfromthe etal.:COOPERATIVECONTROLOFMOBILESENSORNETWORKSproblemofformationmaneuveringandmissioncontrol.InSec-tionIV,weprescribethevirtualbodyspeed whichde-pendsonafeedbackofaformationerror(seealso[21]),andweproveconvergencepropertiesoftheformation.InSectionIV,weprescribethedirectionofthevirtualbodymotion,e.g., forgradientclimbinginadistributedenvironmentandproveconvergencepropertiesofthevirtualbody,andthusthevehiclenetwork,toamaximumorminimumoftheenvironmentalA.TranslationandRotationThestabilityproofof[16]isinvariantwithrespectto actiononthevirtualbody.Thissimplymeansthattheframe-workdescribedinSectionIIisindependentofthepositionandorientationofthevirtualbody.Giventhepositions ofthevir-tualleaders,theSE(3)actionproducesanothersetofvirtualleaderpositions where .Thisactioncanbeviewedasxingthepositionsofthevirtualleaderswithrespecttoavirtualbodyframeandthenmovingthevirtualbodytoanyarbitrarycongurationin Weexploitthissymmetrybyprescribingatrajectoryofthevirtualbodyin ,whichweparameterizeby suchthat with the3 3identitymatrix.Here, ,istheinitialpositionofthe thvirtualleaderwithrespecttoavirtualbodyframeorientedastheinertialframebutwithoriginatthevirtualbodycenterofmass.B.ExpansionandContractionWesimilarlyobservethattheframeworkofSectionIIisin-varianttoascalingofalldistancesbetweenthevirtualleadersandalldistanceparameters ,byafactor .Wedenethecongurationspaceofthevirtualbodytobe andexploitthisadditionalsymmetrybyintroducingaprescribedtrajectoryofthevirtualbodyin ,againparameterizedby ,whichnowincludesexpansion/contraction: suchthat (III.3)with and C.RetainedSymmetriesAsdiscussedinSectionII,itmaybedesirabletokeepcer-tainsymmetrieswhilecontrollingtheformation.Inthespecialcasewherethevirtualbodyisapointmass,rotationalsymmetrywouldbepreservedwhiletranslationalsymmetriesarebroken.Moregenerally,certainsymmetriescanbekeptbyallowingthevehiclestoinuencethevirtualleaderdynamics;see[22]forD.Sensor-DrivenTasksandMissionTrajectoriesAthirdmaneuvercontroloption,distinctfromprescribedtrajectoriesandfreevariables,istolettranslation,rotation,ex-pansion,andcontractionevolvewithfeedbackfromsensorsonthevehiclestocarryoutamissionsuchasgradientclimbing.Thisresultsinanaugmentedstatespaceforthesystemgiven .However,itisonlythedirectionsandnotthemagnitudeofthevirtualbodyvectoreldsthatwecanin isprescribedtoenforceformationstability.Toseethisdecouplingofthemissioncontrolproblemfromthefor-mationstabilizationproblem,notethatthetotalvectoreldsforthevirtualbodymotioncanbeexpressedas Theprescriptionof ,giveninSectionIV,controlsthespeedofthevirtualbodyinordertoguaranteeformationstabilityandconvergenceproperties.ForthemissioncontrolproblemSec-tionV,wedesignthedirections and IV.SPEEDOFRAVERSALANDORMATIONTABILIZATIONWenowexplorehowfastthesystemcanmovealongatrajec-torywhileremaininginsidesomeuserdenedsubsetofthere-gionofattraction.InTheorem4.1,weprovethatthevirtualbodytraversingthetrajectory from to withspeedprescribedby(IV.4)willguaranteetheformationtoconvergetothenaldestinationwhilealwaysremaininginsidetheregionofattraction,formulatedasanupperbound theLyapunovfunction .Here,wewillbeinterestedintheLyapunovfunction ,[21],thatextendstheLyapunov givenby(II.2)where and arereplaced and accordingto(III.3).A.ConvergenceandBoundednessTheorem4.1(ConvergenceandBoundedness): beaLyapunovfunctionforeveryxedchoiceof with .Let beadesiredupperboundonthevalueofthisLyapunovfunctionsuchthattheset isbounded.Let beanominaldesiredformationspeedand asmallpositivescalar.Let beacontinuousfunc-tionwithcompactsupportin and Iftheendpointisnotreached, ,let begivenby (IV.4)withinitialcondition .Attheendpointandbeyond, ,set .Then,thesystemisstableandasymptoticallyconvergesto .Furthermore,ifatinitial ,then forall Proof::Wedirectlyhave IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.49,NO.8,AUGUST2004 weget .If,ontheother ,weget (IV.5)Now,assumethat .Thisgives and .Thus Therefore,if then alongtrajecto-riesinbothcases.Thus, forall if AsymptoticStability:Lettheextendedstateofthesystem ,and Since and ,thelimit exists.Bytheboundednesspropertyof , isinvariantandbounded.Thus,on the -limitset exists,isinvariant .For ,wemusthave and,therefore, .Wewillnowshowthat isthelargestinvariantsetin and,therefore, . impliesthat isaLyapunovfunctionwithre-spectto (since xed).Thus,everytrajectorycandidate ,where .Thisimplies(bythechoiceof in(IV.4))that (duetothe term,un- wherethetrajectoryiscompletedandwelet halt).Therefore, istheonlyinvariantsetin .Thus, andthesystemisasymptotically Simulationofatwo-vehicleplanarrotationusingavirtualbodyconsistingoftwovirtualleaders[asshowninFig.4(b)]ispresentedin[22].Remark4.2:Atypicalchoiceof is if if Here, guaranteeingasymptoticstabilityandgiving at . becauseofthemin-operatorin(IV.4).Itssupportislimitedto ,thusnotaffecting property.Remark4.2:IftheLyapunovfunction islocallypositiveniteanddecresentand islocallypositivedenite,thenonecanndaclass function suchthat Inthiscase,strongerresultscanbeproved,asin[21].V.GLIMBINGINAISTRIBUTEDNVIRONMENTInthissection,wepresentourstrategyforenablingthevehiclegrouptoclimb(ordescend)thegradientofanoisy,distributedenvironment.WeassumethattheÞeldisunknownapriori,butcanbemeasuredbythevehiclesalongtheirpaths.Inourframework,thevirtualbodyisdirectedtoclimbthegradientestimatedfromallthe(noisy)measurements.Thevehiclesmovewiththevirtualbodytoclimbthegradient.Wecomputealeast-squaresapproximationofthegradientof usingnoisymeasurements fromasinglesensorpervehicle.Wealsostudytheoptimalformationproblemtomini-mizeerrorinthegradientestimate.Inthecaseofgradientde-scent,wherethegradientisestimatedtobe ,weprescribe sothatthevirtualbodymovesinthedirectionofsteepestde-scent.(Forgradientclimbing,weuse .)Usingthissetupandourleast-squaresestimatewithKalmanltering,weproveconvergencetoasetwherethemagnitudeofthegradientisclosetozero,thuscontainingallsmoothlocalminima.Givenacomputedoptimalintervehiclespacing,wecanalsoadapttheresolutionofourgrouptobestsortoutthesignalfromthenoise.Forinstance,onewouldexpecttowantatighterformation,forincreasedmeasurementresolution,wherethescalar variesgreatly.Givenadesiredintervehicle ,wecouldlet evolveaccordingto ,with ascalarconstantand theinitialinter-vehicleequilibriumdistance,seeFig.2.The canbetakeneitherfromtheclosed-formanalysisinLemma5.3orfromanumericalsolutionoftheoptimizationproblem(V.7)inLemma5.2.Wedescribeanalternateapproachtogradientestimationusingthegradientoftheaveragevalueoftheeldcontainedwithinaclosedregion.Asshownin[32],thisaveragecanbeexpressedasafunctionoftheeldvaluesalongonlytheboundaryoftheclosedregion.Wepresentacaseinwhichthisapproachisequivalenttotheleast-squaresapproach;thiselucidateswhenwecanviewtheleast-squaresestimateasanaveragingprocess.A.LeastSquaresandOptimalDistancesFixacoordinateframetotheformationatthecenterofmass ofthevirtualbodydepictedinFig.1,andletthepositionofthe thvehiclebethevector or ,inthisframe.Givenisasetofmeasurements where isasingle,possiblynoisymeasurement,taken etal.:COOPERATIVECONTROLOFMOBILESENSORNETWORKSbythe thvehicleatitscurrentposition .Weseektoesti- ,i.e.,thetruegradientandvalue ofthescalar .(Notethat iscompletelydifferentfromthe usedabovetorepresentthevehiclegroupstate.)Tondtheestimatewemakeanaf oftheeld,andthenuse .Wecalculate and usingaleast-squaresfor-mulaandcalltheestimate Lemma5.1(Least-SquaresEstimate):Thebest,inaleast-squaressense,approximation ofacontinuouslydifferentiablescalar fromasetofmeasurements atpositions givenby ...... Itisassumedthatthe saresuchthat hasfullrank.Fur-thermore,theerrorduetosecond-ordertermsandmeasurementnoisecanbewritten where ... ... .. ... ismeasurementnoiseand istheHessianoftheProof:ATaylorexpansionaroundtheorigintogetherwithanassumedmeasurementnoise ateachpointgivesthemea-suredquantity Ignoringthehigherordertermsandwritingtheequationsinma-trixformweget .Applyingtheleastsquaresestimate[30],minimizing ,weget yieldingtheestimationerror Remark5.1:Notethatifthemeasurements aretobeusefulforestimating ,thenthedistances besmallenoughtomakethelowerordertermsintheTaylorexpansiondominate.Toformulatetheoptimalformationproblem,wemovetoastochasticframeworkanddenethestochasticvariables ,and correspondingtothedeterministic (measurementerror)and (combinedmeasurementandhigherordertermserror).Let ,i.e., Gaussianwithzeromeanandvariance .Sincethesecondderivativeoftheeldingeneralisunknownandhardtoestimatewell(fromnoisymeasurements),wereplaceitwithastochasticscalarvariabletimesamatrix, ,where and isaveryroughestimateoftheHessian .Welet beafunctionof and Lemma5.2(OptimalFormationProblem):Viewtheesti-mateerrorasastochasticvariable ... . . ... Let ,where pendson asinLemma5.1.Theexpectedvalueofthesquareerrornormis (V.6)Anoptimalformationgeometryproblemcannowbeformulated (V.7)Proof:FromLemma5.1,wedirectlyhave Further, giving since and areuncorrelated. Remark5.2:Onecanarguethat .Thisgivesaslightlydifferentexpression,butthenumericalresultsaresim-ilar. canbearguedtoincorporateuncertainhigherorderterms.Inthiscase, isnotsomucharoughHessianestimateasanestimateofwhichdirectionshavelargehigherordertermsingeneral.Remark5.3:Ifthesymmetryisbrokenbythedemandsonsensing,i.e.,anirregularformationshapeisrequired,thenthecontrollawmustbreakthesymmetryaswell.Forexample,anorderingofvehiclescouldbeimposedandthedifferentvaluesoftheparameters ,etc.,communicatedtothedifferentvehicles.Theaforementionedoptimizationproblemisnonconvexandthereforenumericallynontrivial,i.e.,standardalgorithmsonlyachievelocalresults.Examinationofthoseresults,however,re-vealsaclearpatternofregularpolyhedra(deformedif aroundtheoriginwithsizedependenton and .Resultsforthetwo-dimensionalcasewiththreetoeightvehiclescanbefoundinTableI.Forlargernumbersofvehiclestheminimiza-tionalgorithmterminatesinlocalminimadifferentfromregularInthethree-dimensional,four-vehicle,casewegetanequilat-eraltetrahedron.Thescalinganddeformationeffectsof IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.49,NO.8,AUGUST2004TABLEIESULTSFORI;  areinvestigatedinclosedformforanonsymmetricspe-cialcaseinLemma5.3.Toillustratenonconvexityofproblem(V.7)welookatthelocalminimumcorrespondingtoanequilateraltriangle.Fixingthepositionsoftwoofthevehiclesandplottingtheobjectivefunctionasthepositionofthethirdoneisvaried,wegetthesurfaceofFig.5.Notehowtheerrorincreaseswhenallthreevehiclesarealigned.Forabetterunderstandingof dependence,wecon-sideraspecialcasewithrestrictedtwo-dimensionalgeometryandinvestigatetheclosed-formgradientapproximationerrorandoptimaldistancesindetail.Lemma5.3(OptimalFormation:Three-VehicleCase):Inthetwo-dimensionalcasewiththreevehiclesat and ,wegetthefamiliar,nite-differenceapproximationestimate Theestimationerrorvarianceisfurthermore (V.8)whichisminimizedbychoosing (V.9)Proof:For Wenowhave Lookingattheerrors,weget Evaluating accordingto(V.6)gives(V.8).Settingthepartialderivativesof(V.8)withrespect and tozeroweget(V.9). Remark5.4:Theexpressionsfortheoptimalchoiceof and impliesthefollowingreasonableruleofthumbfordifferentnumbersofvehicles:whenthenoise,i.e., ,increasesthenthedistancebetweenvehicles shouldincreaseandwhenthesecondderivative,i.e., ,increasesthen shoulddecrease.B.KalmanFilterToadditionallyimprovethequalityofthegradientestimates,weuseaKalmanlterandthustakethetimehistoryofmea-surementsintoaccount.Usingthesimplestpossiblemodelofthetimeevolutionofthetruequantities, ,wegetanobserverdrivingtheestimationtowardthemomentaryleastsquaresesti- Lemma5.4(KalmanFilterEstimate):Letthetimeevolutionofthetruegradientandscalar ,andthe ,begivenbythelinearsystem where and arewhitenoisevectors, isascalarand isgiveninLemma5.1. sothat and .Then,thetimeevolutionoftheoptimalestimate is (V.10)If,ontheotherhand,weusethesimplernoisemodel where istheidentitymatrixand .Then,(V.10)simpliesto (V.11)Proof:Foragenerallinearsystem, ,thesteady-stateKalmanlter[28]is Letting and weget(V.10).Plugging intothismakes whichisequivalentto(V.11). Remark5.5: theKalmanlterestimateisdriventowardthemomentaryLeastSquaresestimate, .Thespeedofthismotionisproportionalto (since );fasterif changesfast,i.e.,large ,andslowerifthereisalotofmea-surementnoise,i.e.,large . etal.:COOPERATIVECONTROLOFMOBILESENSORNETWORKS Fig.5.Nonconvexity.Thepositionsoftwovehicles, andp ,arexedintheoptimaltriangularformationat (0:9;0:p ,andthepositionofthethirdvehicle isvariedwhileevaluating .Notehowhaving closetothelinethrough andp givesalargeerrorduetothelossofrankof .Duetothissingularity,log(isplotted.C.ConvergenceToinvestigatehowclosetheformationdescendingthegra-dientwillgettothetruelocalminimuminthemeasuredrsttranslatethequanticationoftheestimationerrorbacktoadeterministicframeworkandtheninvestigatethesizeoftheestimationerror.nition5.1:Letthefunction beimplicitlydenedbytheequation where istheprobabilitydensityfunction[4]for Theintegrationisin and isthe partofthecovariancematrix ,i.e., Remark5.6:Notethat andthatitismonoton-icallyincreasingin .Italsoincreaseswithascalarresizingof . shouldbeinterpretedasacondencelevelinthegradientestimate.Forexample,ifonesets ,thenwith99.9% Assumption1(StochastictoDeterministic):Inordertomovebacktoaframework,welettheupper onthegradientestimationerrorbegivenbythe nedpreviouslyintermsofthecovariance andthecondencelevel .Thatis,weassumethat hasbeenchosenlargeenoughsothatitisreasonabletoignore %worstcaseswhenstudyingthelongtermevolutionofthedynamics.Weproceedundertheassumption Theorem5.1(Convergence):Lettheformationbegivenbyasetofvectors .Furthermore,usetheeldmeasurementsatthesepoints tocalculateanestimate ,fromLemma5.4.IfAssumption1holds, iscontinuouslydiffer-entiableandboundedbelowandthedirectionofmotionofthevirtualbodyissetto ,thenthevirtualbodyposition willconvergetoaset Here, denotesthematrixsquarerootand isanarbitrarysmallpositivescalar.Remark5.7: iscontinuouslydifferentiable,the containsalllocalminima. growswithincreasednoiselevel andincreasedcondencelevel .Larger alsoimpliesalargerset butasshownlateryieldsfasterconvergenceto ProofofTheorem:Weusethenotation ,where iscalculatedfrom .Since wecanuse asaLyapunovfunction.Thismakes IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.49,NO.8,AUGUST2004InTheorem4.1itwasshownthat convergestotheusernedendpoint .Therefore,ifwecan suchthat (V.12)outsidetheregion andchoose bigenough,thenthetrajec-torymustconvergeto .Toseethiswechoose ,where isalowerboundon and . convergingto nowimpliesthat convergesto ,(since ),orenters wheretheboundisnotvalid.Converging howeverimpliesconvergingtoaglobalminimum,whichbythepreviousremarkmustbein .Thus,weneedtoshowthatthebound(V.12)isindeedvalidoutside .Letting withthebound weget Outsideof ,weget implying andwecanchoose .Asarguedabovethismakesthevehiclesconvergeto . Remark5.8:Inthis,wehaveassumedthe stobeconstant.Thisis,however,notthecaseinmanyapplications.Dependingonthemagnitudeofthedeviationsonecaneitherletthembeac-countedforbythesensornoise, ,ormakethe matrixtimedependent.Thelastoptionrequiresanonsteady-stateKalmanltermakingtheequationsabitlonger.The usedinAssump-tion1mustfurthermorebereplacedbyanupperboundon D.Gradient-of-the-AveApproximationMotivatedbytheproblemofgradientclimbinginanoisyenvironment,weexamineinthissubsectionanalternativeapproachtogradientestimation:wecomputethegradientofanaverageofthescalarenvironmentaleldvalues(measurements)overaclosedregion.Thisgradientisformulatedasanintegraloveracontinuoussetofmeasurementsandisapproximatedusingthenitesetofmeasurementsprovidedbythevehiclegroup.Wefocusonshowingthatforaparticularchoiceofdiscretization,i.e.,numericalquadrature,andforcertaindistributionsofvehiclesoveracirclein andasphere ,thisgradient-of-the-averageestimateisequivalenttotheleast-squaresestimate.TheclassofvehicleformationsforwhichthisequivalenceholdsincludestheoptimalformationscomputedinSectionV-A.Theaverageofascalar ,insideadiscofradius givenby where isthediscofradius centeredat Foragradientclimbing(ordescent)problem,weseekthegradientof withrespectto .Inviewofourdiscex-ample,wecanview asspecifyingthebestdirectiontomovethecenterofthediscsoastomaximize(minimize)theaverageof over .AsshownbyUryasev[32],thisgradientcanbewrittenas where istheboundaryof Supposewearegivenonly noisymeasurements of atpoints ,i.e.,onefromeachvehicle.Wecanap-proximatetheaforementionedintegralusingnumericalquadra-ture.Considerthecaseinwhichthe vehiclesareuniformlydistributedovertheboundary.Usingthecompositetrapezoidal,weobtain where ,i.e.,themeasurementlocationrelativeto ,and .Changingcoordinatessuchthattheorigincoincideswith (V.13)Similarly,tocomputethegradientoftheaveragevalueof withinaballofradius in ,weobtain (V.14)forvehicledistributionsthatpermit equalareapartitionsofthespherewitheachvehiclelocatedatacentroidofapartition,andwhereallvehiclesdonotlieonthesamegreatcircle.Lemma5.5(LeastSquaresEquivalence): vehi-clesin .Supposefor thatthevehiclesareuniformlydistributedaroundacircleofradius .Supposeinthecase thatthevehiclesaredistributedoverasphereofradius suchthattheformationpartitionsthesphereintoequal-areasphericalpolygons,whereeachvehicleislocatedatacentroid,andallvehiclesdonotlieonthesamegreatcircle. ,thepositionvectorofthe thvehiclerelativetothecenterofthecircleorsphere.Eachvehicletakesanoisymeasurement .De where isthe thcoordinateof Assumethatthegroupgeometrysatis and where and for for and isthestandardinnerproducton .Then,theleastsquaresgradientestimateisequivalenttothegradient-of-the-averageestimateasgivenin(V.13)and(V.14).Proof:Aproofisonlypresentedforformationsin ;theresultin followsanalogously. etal.:COOPERATIVECONTROLOFMOBILESENSORNETWORKSIntermsof Itfollowsfromthehypothesesongroupgeometrythat .Furthermore ... .. Thus,theleastsquaresestimate isgivenby whichisequivalentto (V.13). Remark5.9:For vehiclesin ,theassumptionsonthegroupgeometryaresatisedforequallyspacedvehiclesonthecircle.Theseformationsare -sided,regularpolyhedrathatco-incidewiththeoptimalformationsfor Remark5.10:Forvehiclesin ,thegroupgeometryas-sumptionsarenotsoeasilysatised;indeed,thespecimaynotbeachievableforarbitrary .ExamplesofformationsmeetingtheassumptionsincludevehiclesplacedattheverticesofoneofthevePlatonicsolids,i.e.,tetrahedron ,oc- ,cube ,icosahedron ,and .Recallthatthetetrahedronwasfoundtobeanoptimalformation 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IEEETRANSACTIONSONAUTOMATICCONTROL,VOL.49,NO.8,AUGUST2004[29]E.RimonandD.E.Koditschek,Exactrobotnavigationusingartipotentialfunctions,IEEETrans.Robot.Automat.,vol.8,pp.501Apr.1992.[30]G.Strang,LinearAlgebraandItsApplications.Florence,KY:Brooks/Cole,1988.[31]H.Tanner,A.Jadbabaie,andG.J.Pappas,ockingofmobileagents,partI:Fixedtopology,Proc.42ndIEEEConf.DecisionCon-trol,2003,pp.2010[32]S.Uryasev,Derivativesofprobabilityfunctionsandsomeapplica-Ann.Oper.Res.,vol.56,pp.287311,1995.[33]B.J.Young,R.W.Beard,andJ.M.Kelsey,Acontrolschemeforimprovingmulti-vehicleformationmaneuvers,Proc.Amer.ControlConf.,2001,pp.704 Pettergren04)wasborninStockholm,Sweden,in1974.HereceivedtheM.S.degreeinengineeringphysicsandthePh.D.degreeinappliedmathematicsfromtheRoyalInstituteofTechnology(KTH),Stockholm,Sweden,in1998and2003,respectively.Forfourmonthsin2001,hevisitedtheMechanicalandAerospaceEngineeringDepartment,PrincetonUniversity,Princeton,NJ.Heiscurrentlyworkingasafull-timeResearcherattheSwedishDefenceResearchAgency(FOI).Hisresearchinterestsincludemultirobotsystems,formations,navigation,andobstacleavoidance. EdwardFiorelli04)receivedtheB.S.degreefromtheCooperUnionfortheAdvancementofScienceandArt,NewYork,in1997.HeiscurrentlyworkingtowardthePh.D.degreeintheDepart-mentofMechanicalandAerospaceEngineering,PrincetonUniversity,Princeton,NJ.Heiscurrentlyafth-yeargraduatestudentintheDynamicalControlSystemsLaboratory,underthedirectionofDr.NaomiLeonard.From19971999,heworkedasaMechanicalEn-gineeratMarconiAerospace,Greenlawn,NYwherehedesignedradarandtelecommunicationequipmentenclosuresformilitaryaerospaceapplications.Hisresearchinterestsincludemultiagentcontroldesignanditsapplicationtodistributedsensornetworks. NaomiEhrichLeonard02)re-ceivedtheB.S.E.degreeinmechanicalengineeringfromPrincetonUniversity,Princeton,NJ,in1985andtheM.S.andPh.D.degreesinelectricalengi-neeringfromtheUniversityofMaryland,CollegePark,in1991and1994,respectively.SheiscurrentlyaProfessorofMechanicalandAerospaceEngineeringatPrincetonUniversityandanAssociatedFacultyMemberofthePrograminAppliedandComputationalMathematicsatPrinceton.Herresearchfocusesonthedynamicsandcontrolofmechanicalsystemsusingnonlinearandgeometricmethods.Currentinterestsincludeunderwatervehicles,mobilesensornetworks,adaptivesamplingwithapplicationtoobservingandpredictingphysicalprocesses,andbiologicaldynamicsintheocean.From1985to1989,sheworkedasanEngineerintheelectricpowerindustryforMPRAssociates,Inc.,Alexandria,VA.