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IEEETRANSACTIONSONROBOTICS,VOL.25,NO.4,AUGUST2009851VisionBased,DistributedControlLawsforMotionCoordinationofNonholonomicRobotsNimaMoshtagh,Member,IEEE,NathanMichael,Member,IEEE,AliJadbabaie,SeniorMember,IEEE,andKostasDaniilidis,SeniorMember,IEEEAbstractInthispaper,westudytheproblemofdistributedmotioncoordinationamongagroupofnonholonomicgroundrobots.Wedevelopvisionbasedcontrollawsforparallelandbalancedcircularformationsusingaconsensusapproach.Theproposedcontrollawsaredistributedinthesensethattheyrequireinformationonlyfromneighboringrobots.Furthermore,thecontrollawsarecoordinatefreeanddonotrelyonmeasurementorcommunicationofheadinginformationamongneighborsbutinsteadrequiremeasurementsofbearing,opticalflow,andtimetocollision,allofwhichcanbemeasuredusingvisualsensors.Collisionavoidancecapabilitiesareaddedtotheteammembers,andtheeffectivenessofthecontrollawsaredemonstratedonagroupofmobilerobots.IndexTermsCooperativecontrol,distributedcoordination,visionbasedcontrol.I.INTRODUCTIONCOOPERATIVEcontrolofmultipleautonomousagentshasbecomeavibrantpartofroboticsandcontroltheoryresearch.Themainunderlyingthemeofthislineofresearchistoanalyzeand/orsynthesizespatiallydistributedcontrolarchitecturesthatcanbeusedformotioncoordinationoflargegroupsofautonomousvehicles.Someofthisresearchfocussesonflockingandformationcontrol9,14,16,22,31,andsynchronization2,39,whileothersfocusonrendezvous,distributedcoverage,anddeployment1,5.Akeyassumptionimpliedinallofthepreviousreferencesisthateachvehicleorrobothereaftercalledanagentcommunicatesitspositionand/orvelocityinformationtoitsneighbors.Inspiredbythesocialaggregationphenomenainbirdsandfish6,30,researchersinroboticsandcontroltheoryhaveManuscriptreceivedFebruary23,2008revisedJanuary31,2009.FirstpublishedJune10,2009currentversionpublishedJuly31,2009.ThispaperwasrecommendedforpublicationbyAssociateEditorZ.W.LuoandEditorJ.P.Laumonduponevaluationofthereviewerscomments.TheworkofA.JadbabaiewassupportedinpartbytheArmyResearchOffice–MultidisciplinaryUniversityResearchInitiativeARO/MURIunderGrantW911NF0510381,inpartbytheOfficeofNavalResearchONR/YoungInvestigatorProgram542371,inpartbyONRN000140610436,andinpartunderContractNSFECS0347285.TheworkofK.DaniilidiswassupportedinpartunderContractNSFIIS0083209,inpartunderContractNSFIIS0121293,inpartunderContractNSFEIA0324977,andinpartunderContractARO/MURIDAAD190210383.N.MoshtaghwaswiththeGeneralRobotics,Automation,Sensing,andPerceptionLaboratory,UniversityofPennsylvania,Philadelphia,PA19104USA.HeisnowwithScientificSystemsCompany,Inc.,Woburn,MA01801USAemailnmoshtaghssci.com.N.Michael,A.Jadbabaie,andK.DaniilidisarewiththeGeneralRobotics,Automation,Sensing,andPerceptionLaboratory,UniversityofPennsylvania,Philadelphia,PA19104USAemailnmichaelgrasp.upenn.edujadbabaigrasp.upenn.edukostasgrasp.upenn.edu.Colorversionsofoneormoreofthefiguresinthispaperareavailableonlineathttp//ieeexplore.ieee.org.DigitalObjectIdentifier10.1109/TRO.2009.2022439developedtools,methods,andalgorithmsfordistributedmotioncoordinationofmultivehiclesystems.Twomaincollectivemotionsthatareobservedinnatureareparallelmotionandcircularmotion21.Onecaninterpretstabilizingthecircularformationasanexampleofactivityconsensus,i.e.,individualsaremovingaroundtogether.Stabilizingtheparallelformationisanotherformofactivityconsensusinwhichindividualsmoveofftogether33.Circularformationsareobservedinfishschooling,whichisawellstudiedtopicinecologyandevolutionarybiology6.Inthispaper,wepresentasetofcontrollawsforcoordinatedmotions,suchasparallelandcircularformations,foragroupofplanaragentsusingpurelylocalinteractions.Thecontrollawsareintermsofshapevariables,suchastherelativedistancesandrelativeheadingsamongtheagents.However,theseparametersarenotreadilymeasurableusingsimpleandbasicsensingcapabilities.Thismotivatestherewritingofthederivedcontrollawsintermsofbiologicallymeasurableparameters.Eachagentisassumedtohaveonlymonocularvisionandisalsocapableofmeasuringbasicvisualquantities,suchasbearingangle,opticalflowbearingderivative,andtimetocollision.Rewritingthecontrolinputsintermsofquantitiesthatarelocallymeasurableisequivalenttoexpressingtheinputsinthelocalbodyframe.Suchachangeofcoordinatesystemfromaglobalframetoalocalframeprovidesuswithabetterintuitiononhowsimilarbehaviorsarecarriedoutinnature.Verificationofthetheorythroughmultirobotexperimentsdemonstratedtheeffectivenessofthevisionbasedcontrollawstoachievethedesiredformations.Ofcourse,inreality,anyformationcontrolrequirescollisionavoidance,andindeed,collisionavoidancecannotbedonewithoutrange.Inordertoimprovetheexperimentalresults,weprovidedinteragentcollisionavoidancepropertiestotheteammembers.Inthispaper,weshowthatthetwotasksofformationkeepingandcollisionavoidancecanbedonewithdecoupledadditivetermsinthecontrollaw,wherethetermsforkeepingparallelandcircularformationsdependonlyonvisualparameters.Thispaperisorganizedasfollows.InSectionII,wereviewanumberofimportantrelatedworks.SomebackgroundinformationongraphtheoryandothermathematicaltoolsusedinthispaperareprovidedinSectionIII.TheproblemstatementisgiveninSectionIV.InSectionsVandVI,wepresentthecontrollersthatstabilizeagroupofmobileagentsintoparallelandbalancedcircularformations,respectively.InSectionVII,wederivethevisionbasedcontrollersthatareintermsofthevisualmeasurementsoftheneighboringagents.InSectionVIII,collisionavoidancecapabilitiesareaddedtothecontrollaws,andtheireffectivenessistestedonrealrobots.15523098/26.00©2009IEEEAuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2002fromIEEEXplore.Restrictionsapply.852IEEETRANSACTIONSONROBOTICS,VOL.25,NO.4,AUGUST2009II.RELATEDWORKANDCONTRIBUTIONSTheprimarycontributionofthispaperisthepresentationofsimplecontrollawstoachieveparallelandcircularformationsthatrequireonlyvisualsensing,i.e.,theinputsareintermsofquantitiesthatdonotrequirecommunicationamongnearestneighbors.IncontrastwiththeworkofJusthandKrishnaprasad17,MoshtaghandJadbabaie27,Paleyetal.32,33,andSepulchreetal.35,whereitisassumedthateachagenthasaccesstothevaluesofitsneighborspositionsandvelocities,wedesigndistributedcontrollawsthatuseonlyvisualcluesfromnearestneighborstoachievemotioncoordination.Ourapproachonderivingthevisionbasedcontrollawscanbeclassifiedasanimagebasedvisualseroving41.Inimagebasedvisualservoing,featuresareextractedfromimages,andthenthecontrolinputiscomputedasafunctionoftheimagefeatures.In8,12,and38,authorsuseomnidirectionalcamerasastheonlysensorforrobots.In8and38,input–outputfeedbacklinearizationisusedtodesigncontrollawsforleaderfollowingandobstacleavoidance.However,theyassumethataspecificverticalposeofanomnidirectionalcameraallowsthecomputationofbothbearinganddistance.IntheworkofPrattichizzoetal.12,thedistancemeasurementisnotusedhowever,theleaderusesextendedKalmanfilteringtolocalizeitsfollowers,andcomputesthecontrolinputsandguidestheformationinacentralizedfashion.Inourpaper,thecontrolarchitectureisdistributed,andwedesigntheformationcontrollersbasedonthelocalinteractionamongtheagentssimilartothatof14and22.Furthermore,forourvisionbasedcontrollers,nodistancemeasurementisrequired.In25and34,circularformationsofamultivehiclesystemundercyclicpursuitisstudied.Theirproposedstrategyisdistributedandsimplebecauseeachagentneedstomeasuretherelativeinformationfromonlyoneotheragent.Itisalsoshownthattheformationequilibriaofthemultiagentsystemaregeneralizedpolygons.Incontrastto25,ourcontrollawisanonlinearfunctionofthebearingangles,andasaresult,oursystemconvergestoadifferentsetofstableequilibria.III.BACKGROUNDInthissection,webrieflyreviewanumberofimportantconceptsregardinggraphtheoryandregularpolygonsthatweusethroughoutthispaper.A.GraphTheoryAnundirectedgraphGconsistsofavertexsetVandanedgesetE,whereanedgeisanunorderedpairofdistinctverticesinG.Ifx,y∈Vandx,y∈E,thenxandyaresaidtobeadjacent,orneighbors,andwedenotethisbywritingx∼y.Thenumberofneighborsofeachvertexisitsdegree.Apathoflengthrfromvertexxtovertexyisasequenceofr1distinctverticesthatstartwithxandendwithysuchthatconsecutiveverticesareadjacent.IfthereisapathbetweenanytwoverticesofagraphG,thenGissaidtobeconnected.TheadjacencymatrixAGaijofanundirectedgraphGisasymmetricmatrixwithrowsandcolumnsindexedbytheverticesofG,suchthataij1ifvertexiandvertexjareneighbors,andaij0otherwise.Wealsoassumethataii0foralli.ThedegreematrixDGofagraphGisadiagonalmatrixwithrowsandcolumnsindexedbyV,inwhichthei,ientryisthedegreeofvertexi.ThesymmetricsingularmatrixdefinedasLGDG−AGiscalledtheLaplacianofG.TheLaplacianmatrixcapturesmanytopologicalpropertiesofthegraph.TheLaplacianLisapositivesemidefinitematrix,andthealgebraicmultiplicityofitszeroeigenvaluei.e.,thedimensionofitskernelisequaltothenumberofconnectedcomponentsinthegraph.Thendimensionaleigenvectorassociatedwiththezeroeigenvalueisthevectorofones,1n1,...,1T.Formoreinformationongraphtheory,see13.B.RegularPolygonsLetd1andnanddarecoprime,thentheedgesintersect,andthepolygonisastar.Ifnanddhaveacommonfactorl1,thenthepolygonconsistsofltraversalsofthesamepolygonwith{n/l}verticesandedges.Ifdn,thepolygon{n/n}correspondstoallpointsatthesamelocation.Ifdn/2withneven,thenthepolygonconsistsoftwoendpointsandalinebetweenthem,withpointshavinganevenindexononeendandpointshavinganoddindexontheother.Formoreinformationonregulargraphs,see7.IV.PROBLEMSTATEMENTConsideragroupofnunitspeedplanaragents.Eachagentiscapableofsensinginformationfromitsneighbors.Theneighborhoodsetofagenti,thatis,Ni,isthesetofagentsthatcanbeseenbyagenti.Theprecisemeaningofseeingwillbeclarifiedlater.Thesizeoftheneighborhooddependsonthecharacteristicsofthesensors.TheneighboringrelationshipbetweenagentscanbeconvenientlydescribedbyaconnectivitygraphGV,E,W.Definition1ConnectivitygraphTheconnectivitygraphGV,E,Wisagraphconsistingof1asetofverticesVindexedbythesetofmobileagents2asetofedgesE{i,j|i,j∈V,andi∼j}3asetofpositiveedgeweightsforeachedgei,j.TheneighborhoodofagentiisdefinedbyNi.{j|i∼j}⊆V\{i}.Letrirepresentthepositionofagenti,andletvibeitsvelocityvector.ThekinematicsofeachunitspeedagentisAuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2002fromIEEEXplore.Restrictionsapply.MOSHTAGHetal.VISIONBASED,DISTRIBUTEDCONTROLLAWSFORMOTIONCOORDINATIONOFNONHOLONOMICROBOTS853Fig.1.TrajectoryofeachagentisrepresentedbyaplanarFrenetframe.givenby˙rivi˙viωiv⊥i˙v⊥i−ωivi1wherev⊥iistheunitvectorperpendiculartothevelocityvectorviseeFig.1.Theorthogonalpair{vi,v⊥i}formsabodyframeforagenti.WerepresentthestackvectorofallthevelocitiesbyvvT1,...,vTnT∈R2n1.Thecontrolinputforeachagentistheangularvelocityωi.Sinceitisassumedthattheagentsmovewithconstantunitspeed,theforceappliedtoeachagentmustbeperpendiculartoitsvelocityvector,i.e.,theforceoneachagentisagyroscopicforce,anditdoesnotchangeitsspeedandhence,itskineticenergy.Thus,ωiservesasasteeringcontrol16foreachagent.Letusformallydefinetheformationsthatwearegoingtoconsider.Definition2ParallelformationTheconfigurationinwhichtheheadingsofallagentsarethesameandvelocityvectorsarealignediscalledtheparallelformation.Notethatinthisdefinition,wedonotconsiderthevalueoftheagreeduponvelocitybutjustthefactthattheagreementhasbeenreached.Attheequilibrium,therelativedistancesoftheagentsdeterminetheshapeoftheformation.Anotherinterestingfamilyofformationsisthebalancedcircularformation.Definition3BalancedcircularformationTheconfigurationwheretheagentsaremovingonthesamecirculartrajectoryandthegeometriccenteroftheagentsisfixediscalledthebalancedcircularformation.Theshapeofsuchaformationcanberepresentedbyanappropriateregularpolygon.Inthefollowingsections,westudyeachformationanddesignitscorrespondingdistributedcontrollaw.V.PARALLELFORMATIONSOurgoalinthissectionistodesignacontrollawforeachagentsothattheheadingsofthemobileagentsreachanagreement,i.e.,theirvelocityvectorsarealigned,thusresultinginaswarmlikepattern.ForanarbitraryconnectivitygraphG,considertheLaplacianmatrixL.We,therefore,defineameasureofmisalignmentasfollows27,35wv12summationdisplayi∼j|vi−vj|212〈v,¯Lv〉2wherethesummationisoverallthepairsi,j∈E,and¯LL⊗I2∈R2n2n,withI2beingthe22identitymatrix.Thetimederivativeofwvisgivenby˙wvnsummationdisplayi1〈˙vi,¯Lvi〉nsummationdisplayi1ωi〈v⊥i,¯Lvi〉where¯Lvi∈R2isthesubvectorof¯Lvassociatedwiththeithagent.Thus,thefollowinggradientcontrollawguaranteesthatthepotentialwvdecreasesmonotonicallyωiκ〈v⊥i,¯Lvi〉−κsummationdisplayj∈Ni〈v⊥i,vij〉3whereκ0.6Thefollowingtwotheorems28presenttheresultswhenbalancedcircularformationsareattainedforagroupofunitspeedagentswithfixedconnectivitygraphs.Theorem2isforthecasewhenGisacompletegraph,andTheorem3isfortheringgraph.Theorem2Considerasystemofnagentswithkinematics5.GivenacompleteconnectivitygraphGandapplyingcontrollaw6,thenagentsystemalmostgloballyasymptoticallyconvergestoabalancedcircularformation,whichisdefinedinDefinition3.ProofSee28fortheproof.squaresolidThereasonforalmostglobalstabilityofthesetofbalancedstatesisthatthereisameasurezerosetofstateswheretheequilibriumisunstable.Thissetischaracterizedbythoseconfigurationsthatmagentsareatantipodalpositionfromtheothern−magents,where1≤mn/2.Next,weconsiderthesituationthattheconnectivitygraphhasaringtopologyGring.Theorem3Considerasystemofnagentswithkinematics5.SupposetheconnectivitygraphhastheringtopologyGringandthateachagentappliesthebalancingcontrollaw6.Then,therelativeheadingswillconvergetothesameangleφo.Ifφo∈π/2,3π/2,thebalancedstateislocallyexponentiallystable.ProofSee28fortheproof.squaresolidAttheequilibrium,thefinalconfigurationforGringisaregularpolygon{n/d}inwhichtherelativeanglebetweentwoconnectednodesisφo2πd/n.FromTheorem3,ifthisanglesatisfiesφo∈π/2,3π/2,thenthebalancedstateisstable.Thus,thestableconfigurationcorrespondstoapolygonwithd∈n/4,3n/4.Forexample,forn5,thestableformationsarepolygons{5/3}and{5/4},whicharethesamepolygonsasobtainedwithreverseorderingofthenodes.Forn4,thestableformationis{4/2}.Actually,simulationssuggestthatthelargestregionofattractionfornevenbelongstoapolygon{n/d},withdn/2,andfornodd,itisastarpolygon{n/d},withdn±1/2.VII.VISIONBASEDCONTROLLAWSNotethatthecontrolinputs4and6forparallelandcircularformationsdependontheshapevariables,i.e.,relativeheadingsandpositions,whicharenotdirectlymeasurableusingvisualsensors,suchasasinglecameraonarobot,becauseestimationoftherelativepositionandmotionrequiresbinocularvision.Thismotivatesustorewriteinputs4and6intermsofparametersthatareentirelymeasurableusingasimplevisualsensor.Next,wedefinethevisualparametersthatwewillusetoderivethevisionbasedcontrollaws.BearingangleLetrixiyiTbethelocationofagentiinafixedworldframe,andletvi˙xi˙yiTbeitsvelocityvector.Theheadingororientationofagentiisthengivenbyθiatan2˙yi,˙xi.7AuthorizedlicenseduselimitedtoNanchangUniversity.DownloadedonJanuary12,2010at2002fromIEEEXplore.Restrictionsapply.
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