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Potential Field-Based Decentralized Control Methods for Network Connectivity Maintenance

Permanent Link: http://ufdc.ufl.edu/UFE0043642/00001

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Title: Potential Field-Based Decentralized Control Methods for Network Connectivity Maintenance
Physical Description: 1 online resource (130 p.)
Language: english
Creator: Kan, Zhen
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: control -- network
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: In cooperative control for a multi-agent system, agents coordinate and communicate to achieve a collective goal (ocking, consensus, or pattern formation). As agents move to perform desired missions, ensuring the group remains close enough to maintain wireless communication (the group does not partition) is a great challenge in a decentralized control manner. The focus of this dissertation is to develop potential eld based decentralized controllers for a group of agents with limited sensing and communication capabilities to perform required mission objectives while preserving network connectivity. Articial potential eld based controllers are developed in Chapter 2 to maintain existing links connected in both low and high level graphs and ensure that a group of agents switch from one connected conguration to another without disconnecting the underlying network in process. In Chapter 3, based on the navigation function formalism, a decentralized control method is designed to enable a group of agents to achieve a desired global conguration from a given connected initial graph with desired neighborhood between agents, while maintaining global network connectivity and avoiding obstacles, using only local feedback and no radio communication between the agents for navigation. In Chapter 4, a novel strategy using a prex labeling and routing algorithm and a navigation function based control scheme is developed to achieve a desired formation for a group of identical agents from an arbitrarily connected initial graph. A decentralized continuous time-varying controller based on a modied dipolar navigation function is developed in Chapter 5 to reposition and reorient those mobile robots with nonholonomic constraints to a common setpoint with a desired orientation while maintaining network connectivity during the evolution, using only local sensing feedback from its one-hop neighbors. The work in Chapter 6 investigates and inuences emotions of people in a social network, where a distributed controller is designed to achieve emotion synchronization for a group of individuals in a social network (an agreement on the emotion states of all individuals). Chapter 7 concludes the dissertation by summarizing the work and discussing some remaining open problems that required further investigation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Zhen Kan.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Dixon, Warren E.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043642:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043642/00001

Material Information

Title: Potential Field-Based Decentralized Control Methods for Network Connectivity Maintenance
Physical Description: 1 online resource (130 p.)
Language: english
Creator: Kan, Zhen
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: control -- network
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In cooperative control for a multi-agent system, agents coordinate and communicate to achieve a collective goal (ocking, consensus, or pattern formation). As agents move to perform desired missions, ensuring the group remains close enough to maintain wireless communication (the group does not partition) is a great challenge in a decentralized control manner. The focus of this dissertation is to develop potential eld based decentralized controllers for a group of agents with limited sensing and communication capabilities to perform required mission objectives while preserving network connectivity. Articial potential eld based controllers are developed in Chapter 2 to maintain existing links connected in both low and high level graphs and ensure that a group of agents switch from one connected conguration to another without disconnecting the underlying network in process. In Chapter 3, based on the navigation function formalism, a decentralized control method is designed to enable a group of agents to achieve a desired global conguration from a given connected initial graph with desired neighborhood between agents, while maintaining global network connectivity and avoiding obstacles, using only local feedback and no radio communication between the agents for navigation. In Chapter 4, a novel strategy using a prex labeling and routing algorithm and a navigation function based control scheme is developed to achieve a desired formation for a group of identical agents from an arbitrarily connected initial graph. A decentralized continuous time-varying controller based on a modied dipolar navigation function is developed in Chapter 5 to reposition and reorient those mobile robots with nonholonomic constraints to a common setpoint with a desired orientation while maintaining network connectivity during the evolution, using only local sensing feedback from its one-hop neighbors. The work in Chapter 6 investigates and inuences emotions of people in a social network, where a distributed controller is designed to achieve emotion synchronization for a group of individuals in a social network (an agreement on the emotion states of all individuals). Chapter 7 concludes the dissertation by summarizing the work and discussing some remaining open problems that required further investigation.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Zhen Kan.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043642:00001


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POTENTIALFIELD-BASEDDECENTRALIZEDCONTROLMETHODSFOR NETWORKCONNECTIVITYMAINTENANCE By ZHENKAN ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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2011ZhenKan 2

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TomywifeYangChen,mymotherBaozhiHu,andmyfatherHepingKan,fortheir unwaveringsupportandconstantencouragement 3

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ACKNOWLEDGMENTS Iwouldliketoexpressmydeepestgratitudetomyadvisor,WarrenE.Dixon,forhis guidance,patienceandsupportinthecompletionofmyPh.D.Asanadvisor,heguided metostartmygraduatecareerontherightfoot,encouragedmetoexploremyownideas, andhelpedmerecoverwhenmystepsfaltered.Hisexperiencehelpedmegrowfastinthe fouryearsofPh.Dstudy.Iappreciateallhiscontributionsoftime,ideasandenthusiasm tomakemyPh.Dexperienceproductiveandstimulating.Inadditiontohelpmemature intheresearch,healsoprovidedinstrumentaladvices,likeanelderbrotheroraclose friend,onmyfuturecareerplan,jobhunting,andotheraspectsoflife.Ifeelsofortunate tohavehadDr.Dixonasmyadvisor,andIhopethatonedayIwouldbecomeasgoodas anadvisortomystudentsashehasbeentome. Iwouldliketoextendmygratitudetomycommitteemember,JohnShea.Iam deeplygratefultohimforthecarefullyreadingandinsightfulcommentsoncountless revisionsofmyconferenceandjournalpublications,andconstructivecriticismsatdierent stagesofmyresearch.IamalsothankfultoPrabirBarooah,andCarlD.Cranefor readingpreviousdraftsofthisdissertationandprovidingmanyvaluablecommentsthat improvedthepresentationandcontentsofthisdissertation. IwouldalsoliketothankallthepreviousandcurrentmembersoftheNCRlabfor theirvariousformsofsupportduringmygraduatestudy. Lastbutnottheleast,IwouldliketothankmywifeYang.Withouthersupport, encouragementandunwaveringlove,Iwouldneverbeabletonishmydissertation. Also,Iwouldliketothankmyparents,whohavebeenaconstantsourceoflove,concern, supportandstrengthalltheseyears. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFFIGURES....................................8 ABSTRACT........................................10 CHAPTER 1INTRODUCTION..................................12 1.1Motivation....................................12 1.2ProblemStatement...............................13 1.3LiteratureReview................................14 1.4Contributions..................................22 1.5DissertationOutline..............................27 2VISIONBASEDCONNECTIVITYMAINTENANCEOFANETWORKWITH SWITCHINGTOPOLOGY.............................29 2.1ProblemFormulation..............................29 2.1.1CommunicationGraph.........................30 2.1.2VisibilityGraph.............................31 2.1.3ConnectivityMaintenance.......................31 2.2ControlStrategy................................32 2.3ControlDesign.................................35 2.3.1PotentialField..............................35 2.3.2ControllerforSteadyState.......................37 2.3.3ControllerforSwitchingState.....................37 2.4ConnectivityAnalysis.............................38 2.5Simulation....................................40 2.6Summary....................................40 3NETWORKCONNECTIVITYPRESERVINGFORMATIONSTABILIZATIONANDOBSTACLEAVOIDANCEVIAADECENTRALIZEDCONTROLLER43 3.1ProblemFormulation..............................43 3.2ControlDesign.................................45 3.3ConnectivityandConvergenceAnalysis....................48 3.3.1ConnectivityAnalysis..........................50 3.3.2ConvergenceAnalysis..........................52 3.4Simulation....................................59 3.5Summary....................................62 5

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4NETWORKCONNECTIVITYPRESERVINGFORMATIONRECONFIGURATIONFORIDENTICALAGENTSFROMANARBITRARYCONNECTED INITIALGRAPH..................................63 4.1ProblemFormulation..............................64 4.2FormationReorganizationStrategy......................66 4.3NetworkTopologyLabelingAlgorithms....................67 4.3.1BasicAlgorithm.............................67 4.3.2RelabelingAlgorithm..........................70 4.3.2.1BranchRelabelingBRAlgorithm.............72 4.3.2.2NeighborRelabelingNRAlgorithm............73 4.4ControlDesign.................................74 4.4.1InformationFlow............................75 4.4.2NavigationFunction-BasedControlScheme..............77 4.4.3ConnectivityandConvergenceAnalysis................80 4.4.4ConvergenceAnalysis..........................81 4.5Simulation....................................83 4.6Summary....................................84 5ENSURINGNETWORKCONNECTIVITYFORNONHOLONOMICROBOTS DURINGDECENTRALIZEDRENDEZVOUS..................87 5.1ProblemFormulation..............................87 5.2ControlDesign.................................89 5.2.1DipolarNavigationFunction......................89 5.2.2ControlDevelopment..........................93 5.3ConnectivityandConvergenceAnalysis....................95 5.3.1ConnectivityAnalysis..........................95 5.3.2ConvergenceAnalysis..........................95 5.4Simulation....................................98 5.5Summary....................................99 6INFLUENCINGEMOTIONALBEHAVIORINSOCIALNETWORK.....102 6.1Preliminaries..................................103 6.1.1FractionalCalculus...........................103 6.1.2GraphTheory..............................105 6.2ProblemFormulation..............................106 6.3ControlDesign.................................108 6.4ConvergenceAnalysisandSocialBondMaintenance.............110 6.4.1SocialBondMaintenance........................111 6.4.2ConvergenceAnalysis..........................111 6.5Discussion....................................114 6.6Summary....................................115 6

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7CONCLUSIONANDFUTUREWORK......................117 7.1Conclusion....................................117 7.2Futurework...................................119 REFERENCES.......................................122 BIOGRAPHICALSKETCH................................130 7

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LISTOFFIGURES Figure page 2-1Modelofvisibilitygraph................................32 2-2Schematictopologyofunderlyingnetwork......................33 2-3Evolutionofnodesduringtimeintervalof t 2 ; 120 ...............41 2-4Evolutionofnodesduringtimeintervalof t 2 ; 220 ..............41 3-1Anexampleofthearticialpotentialeldgeneratedforadisk-shapedworkspace withdestinationattheoriginandanobstaclelocatedat [1 ; 1] T ..........48 3-2Aconnectedinitialgraphwithdesiredneighborhoodintheworkspacewithstatic obstacles........................................60 3-3Theachievednalconguration...........................61 3-4Theinter-nodedistanceduringtheevolution....................61 3-5TheplotoftheFiedlereigenvalueoftheLaplacianmatrixduringtheevolution. ThecircleindicatestheFiedlereigenvalueofthegraphateachtimeinstance..62 4-1Thesmalldiskareawithradius 1 denotesthecollisionregionandtheouter ringareadenotestheescaperegionsfornode i ...................65 4-2Theexampleofaninitiallyconnectedanddesiredgraphtopology,wherethe nodesdenotetheagent,andthelinesdenotetheavailablecommunicationlinks.70 4-3TheplotofagraphbeforeperformingBRalgorithmandafterperformingBR algorithm........................................73 4-4TheplotofagraphbeforeandafterperformingNRalgorithm...........74 4-5Theplotofdesiredformationwithandwithoutprexlabels............84 4-6Theplotofinitialgraphwithandwithoutprexlabels...............85 4-7Thetrajectoriesofallnodestoachievethedesiredformation,with"*"denotingtheirinitialpositionsandcirclesdenotingtheirnalpositions.........85 5-1Theplotofanexampledipolarnavigationfunction.................91 5-2Thetrajectoryforeachmobilerobotwiththearrowdenotingitscurrentorientation.........................................99 5-3Plotoflinearvelocityandangularvelocityforeachmobilerobot.........100 5-4Plotofpositionandorientationerrorforeachmobilerobot............100 8

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5-5Theevolutionofinter-robotdistance.........................101 6-1TheundirectedgraphmodelofZachary'skarateclubnetworkin[96].......108 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy POTENTIALFIELD-BASEDDECENTRALIZEDCONTROLMETHODSFOR NETWORKCONNECTIVITYMAINTENANCE By ZhenKan December2011 Chair:WarrenE.Dixon Major:MechanicalEngineering Incooperativecontrolforamulti-agentsystem,agentscoordinateandcommunicate toachieveacollectivegoale.g.,ocking,consensus,orpatternformation.Asagents movetoperformdesiredmissions,ensuringthegroupremainscloseenoughtomaintain wirelesscommunicationi.e.,thegroupdoesnotpartitionisagreatchallengeina decentralizedcontrolmanner.Thefocusofthisdissertationistodeveloppotential eldbaseddecentralizedcontrollersforagroupofagentswithlimitedsensingand communicationcapabilitiestoperformrequiredmissionobjectiveswhilepreserving networkconnectivity. ArticialpotentialeldbasedcontrollersaredevelopedinChapter2tomaintain existinglinksconnectedinbothlowandhighlevelgraphsandensurethatagroupof agentsswitchfromoneconnectedcongurationtoanotherwithoutdisconnectingthe underlyingnetworkinprocess.InChapter3,basedonthenavigationfunctionformalism, adecentralizedcontrolmethodisdesignedtoenableagroupofagentstoachieveadesired globalcongurationfromagivenconnectedinitialgraphwithdesiredneighborhoodbetweenagents,whilemaintainingglobalnetworkconnectivityandavoidingobstacles,using onlylocalfeedbackandnoradiocommunicationbetweentheagentsfornavigation.In Chapter4,anovelstrategyusingaprexlabelingandroutingalgorithmandanavigation functionbasedcontrolschemeisdevelopedtoachieveadesiredformationforagroupof identicalagentsfromanarbitrarilyconnectedinitialgraph.Adecentralizedcontinuous 10

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time-varyingcontrollerbasedonamodieddipolarnavigationfunctionisdevelopedin Chapter5torepositionandreorientthosemobilerobotswithnonholonomicconstraints toacommonsetpointwithadesiredorientationwhilemaintainingnetworkconnectivity duringtheevolution,usingonlylocalsensingfeedbackfromitsone-hopneighbors.The workinChapter6investigatesandinuencesemotionsofpeopleinasocialnetwork, whereadistributedcontrollerisdesignedtoachieveemotionsynchronizationforagroup ofindividualsinasocialnetworki.e.,anagreementontheemotionstatesofallindividuals.Chapter7concludesthedissertationbysummarizingtheworkanddiscussingsome remainingopenproblemsthatrequiredfurtherinvestigation. 11

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CHAPTER1 INTRODUCTION 1.1Motivation Multi-agentsystemsundercooperativecontrolprovideversatileplatformsthathave thepotentialtobeusedinvariouscommercialandmilitaryapplications.Forinstance,a listofsomeofthemainapplicationsforcooperativecontrolofmulti-vehiclesystemsis providedin[66],whichincludes: MilitarySystems:FormationFlight,CooperativeClassicationandSurveillance, CooperativeAttackandRendezvous,andMixedInitiativeSystems; MobileSensorNetworks:EnvironmentalSamplingandDistributedAperture Observing; TransportationSystems:IntelligentHighwaysandAirTracControl. Thesetypesoftasksusuallyrequireorcanbenetfromcollaborativemotionofagroup ofagents,andthustheagentsmustbeabletoexchangeinformationoversomeform ofcommunicationsnetwork.Formostapplications,communicationswillbeovera wirelessnetwork,inwhichthecommunicationlinksbetweenagentsaredependentonthe propagationofelectromagneticsignalsbetweentheagents,andtheelectromagneticpower densitydecreaseswithdistance.However,whenperformingdesiredtasks,theunderlying networkconnectivitycanbeimpactedduetothemotionofagents.Ifthenetworkis partitioned,theagentscannolongercoordinatetheirmovements,andthemissionmay fail.Hence,controlalgorithmsmustbedesignedinacooperativemannertopreserve networkconnectivitywhenperformingdesiredtasks. Someapplicationscanadoptacentralizedcontrolapproachwhereonealgorithm determinesandcommunicatesthenextrequiredmovementforeachagent.Forsomeapplications,thecentralizedapproachisnotpracticalduetothepotentialforcompromised communicationwithordemise/corruptionofthecentralcontroller.Decentralizedcontrol isanalternativeapproachinwhicheachagentmakesanindependentdecisionbasedon 12

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eitherglobalinformationcommunicatedthroughthenetworkorlocalinformationfrom one-hopneighbors.Methodsthatuseglobalinformationrequireeachagenttodetermine therelativetrajectoryofallotheragentsatalltimebypropagatinginformationthrough thenetwork,resultingindelaysinthetrajectoryinformationandconsumptionofnetwork bandwidth,eectsthatlimitthenetworksize.Methodsthatuselocalinformationneed onlyrelativetrajectoriesofneighboringagents;however,dicultiesarisingfromperformingrequiredmissionobjectivesfortheglobalnetworkusinglocalfeedbackcancausethe networktopartition.Whenthenetworkpartitions,communicationbetweengroupsof agentscanbepermanentlyseveredleadingtomissionfailure. Giventhewideapplicationofmulti-agentsystemsandthedesiretomaintainnetwork connectivityinadecentralizedmanner,thisdissertationismotivatedbythefollowing questions: 1.Howcanadecentralizedcontrolstrategybedesignedtoensureglobalnetwork connectivityusingonlylocalavailableinformation? 2.Howcantherequiredcollectivemissionobjectivebeachievedinacooperativeway whilepreservingnetworkconnectivity? 3.Isitpossibletoextendthemodelsandmethodsdevelopedformulti-agentsystems inengineeringbeleveragedtoyieldinsighttoinuencesocialgroups? 1.2ProblemStatement Accomplishingdesiredcollectivemissionobjectivesforanetworkedmulti-agent systemhighlydependsonthecoordinationoftheiractionsandthepeer-to-peer,wireless communicationamongagents.Inthisdissertation,limitedcommunicationandsensing capabilitiesforeachagentareconsidered,thatistwoagentscancommunicateand exchangeinformationiftheyarewithinaspeciedmaximumcommunicationrangeand cannotcommunicateiftheyareoutsideofthatrange.Hence,ensuringthattheoverall networkremainsconnectedrequiresthespeciedagentsstaywithinpredeterminedsensing andcommunicationranges,andthecooperativeobjectivesmustbeaccomplishedbyusing 13

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localinformationobtainedfromthelimitedsensingandcommunicationabilitiesofeach agent.Theworkpresentedinthisdissertationexaminesdecentralizedcontrolmethodsfor networkedmulti-agentsystemstoachieveglobalcollectiveobjectives,suchasformation controlandconsensus,whilepreservingconnectivityoftheglobalnetworkusinglocal informationfromimmediateneighbors. 1.3LiteratureReview Thissectionprovidesareviewofrelevantliteratureforeachchapter. Maintenanceofnetworkconnectivityforamulti-agentsystem: Motivated bythepracticalneedtokeepagentsinasinglegroup,recentresultssuchas[16,21,24, 33,40,69,99101]arefocusedonthenetworkconnectivitymaintenanceproblembased ontheconstructionofanarticialpotentialeld.Articialpotentialeldsarehavebeen widelyusedinpathplanningformulti-agentsystems,whereanattractivepotentialis usedtomodelthecontrolobjectiveandarepulsivepotentialisusedtopreventcollisions amongtheagentsandobstacles[80,81].In[99]and[69],apotentialeldbasedcentralized controlapproachisdevelopedtoensuretheconnectivityofagroupofagentsusingthe graphLaplacianmatrix.However,globalinformationoftheunderlyinggraphisrequired tocomputethegraphLaplacian.In[100],connectivitymaintenanceisperformedinthe discretespaceofgraphstoverifylinkdeletionswithrespecttoconnectivity,andmotion controlisperformedinthecontinuouscongurationspaceusingapotentialeld.In[101], apotentialeld-basedneighborcontrollawisdesignedtoachievevelocityalignmentand networkconnectivityamongdierenttopologies.In[24]and[16],arepulsivepotentialis usedforacollisionavoidanceobjective,andanattractivepotentialeldisusedtodrive agentstogether.Distributedcontrollawsareinvestigatedtoensureedgemaintenance in[40]byallowingunboundedpotentialforcewheneverpairsofagentsareabouttobreak theexistinglinks. Toensurenetworkconnectivityduringthemission,atwolevelcontrolstrategyis developedforamulti-agentsysteminChapter2,whereallagentsarecategorizedas 14

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clusterheadsorregularnodes.Ahighlevelgraphiscomposedofallclusterheadsandthe motionoftheclusterheadsiscontrolledtomaintainexistingconnectionsamongthem. Alowlevelgraphcomposedofallregularnodesiscontrolledtomaintainconnectivity withrespectitsspecicclusterhead.Articialpotentialeldbasedcontrollersarethen developedinChapter2tomaintaintheexistinglinksconnectedinbothlowandhigh levelgraphsallthetimeandtoensurethatagroupofagentsswitchfromoneconnected congurationtoanotherwithoutdisconnectingthenetwork. Formationcontrolwithnetworkconnectivityforamulti-agentsystem: Typicalapproachesinformationcontrolincludeleader-follower[52,94],behavioralbased[3,51],virtualstructures[5,53]andgraph-theory-based[29,39,56,69,71,73,102] methods,tonameafew.However,noconstraintsontheavailabilityofotheragents' statesandinformationabouttheenvironmentareconsidered:networkconnectivityisnot takenintoaccountinsuchresults.Whenconsideringnetworkconnectivity,overviewsof techniquesforformationcontrolaregivenin[11,66,97].Theearliestworksonformation controlwithnetworkconnectivityarediscussedin[11]withafocusontheimpactofa givennetworkconnectivityonthestabilityandcontrollabilityofformationsofrobots withoutconsideringthecontrolrequiredtoensurenetworkconnectivityduringthe mission.Althoughsomeresultsdescribedin[66,97]arefocusedonmaintainingnetwork connectivityduringformationcontrol,anopenproblemremainsindevelopingdesigna decentralizedcontrolapproachforagroupofagentsseekingadesiredformationinan uncertainenvironmentwhilepreservingnetworkconnectivity. Oneofthemostwidelyusedapproachesinformationcontrolistousearticial potentialeldstoguidethemovementoftheagents.Acommonproblemwitharticial potentialeld-basedcontrolalgorithmsistheexistenceoflocalminimawhenattractive andrepulsiveforcearecombined[19].InChapter2,networkconnectivityisensuredby usinganarticialpotentialeld-basedcontroller;however,theagentshavethepotential tobetrappedbylocalminima.Aspecictypeofarticialpotential,calledanavigation 15

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function,achievesauniqueminimumc.f.,[49,57,76]andhasbeenwidelyusedinmotion controlformulti-agentsystemsseee.g.,[8,20,22,25,100].Thenavigationfunction developedin[49]isareal-valuedfunctionthatisdesignedsothatthenegatedgradient elddoesnothavealocalminima.Thenegatedgradientofthenavigationfunctionis attractedtowardsthegoalandrepulsedbyobstaclesforalmostallinitialstates.As such,closed-loopnavigationfunctionapproachesguaranteeconvergencetoadesired destination.Thenavigationfunctionframeworkisextendedtomulti-agentsystems forobstacleavoidanceinresultssuchas[9,18,19,25,58];however,agentswithinthese resultsactedindependentlyandwerenotrequiredtoachieveanetworkobjective.In contrast,resultsin[15,90,91]usepotentialelds/navigationfunctionstoachieveobstacle avoidancewhiletheagentsarealsorequiredtoachieveacooperativenetworkobjective e.g.,formationcontrolorconsensus;however,theseresultsassumetheagentscanalways communicatei.e.,thegraphnodesareassumedtoremainconnected.Theassumption ofaconnectedgraphisrestrictiveforamobilenetwork,wherecommunicationdepends onthedistancebetweenagents,whichcanalsobeafunctionoftheenvironmentand availabletransmittingpower.In[21],apotentialeldisdesignedforagroupofmobile agentstoperformdesiredtaskswhilemaintainingnetworkconnectivity.Itisunclearhow thepotentialeldmethodin[21]canbeextendedtoincludestaticobstacles,sincethe resultingclosed-loopdynamicscannotbeexpressedasaMetzlermatrixwithzerosums asrequiredintheanalysisin[21].Moreover,theworkin[21]onlyprovesthatallstates convergetoacommonvaluethatcanbeinuencedbytheinitialstates[65]. Motivatedtoavoidlocalminimawhenusingarticialpotentialeld-basedapproach, anavigationfunctionbaseddecentralizedcontrollerisdevelopedinChapter3toensure networkconnectivityandstabilizeagroupofagentsinarequiredformationfroma connectedinitialgraphagentsareconsideredasnodesonagraphwithadesired neighborhood,whileavoidingcollisionswithotheragentsandexternalobstacles. 16

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Formationcontrolwithnetworkconnectivityfromanarbitraryinitial graph: TheresultinChapter3requiresthattheinitialgraphisconnectedinadesired waysothatnoinitialcommunicationlinkisallowedtobebrokenduringthemotion.Similarconstraintsontheinitialgraphconnectionsarealsopresentedinworkssuchas[21] and[20].However,assumptionsontheinitialgraphcanbelimiting,sincesomeapplicationsmayrequireagentstoachievedesiredformationsfromanarbitraryinitialgraphor dynamicallychangetheachievedformationstoadapttotheuncertainenvironment.For example,certainformationshaveproventobeparticularlyadvantageousforeciencyof datagathering,dataprocessing,andforecastingcf.,[68,103].Sincetheinitialtopology orthetopologyfromtheprevioustaskmaynotbeconducivetothecurrenttask,achievingadesiredformationortransformingfromonetopologytoanotherforagroupofagents withlimitedknowledgeisachallengingtask.Hence,maintainingtheoverallnetwork connectivityisparamount,andstabilizingamulti-agentsystematadesiredformation fromanarbitraryinitialtopologyusinglocalfeedbackcanbechallenging. InChapter4,eachagentpossessesonlylimitedknowledgei.e.,limitedsensing capabilitiesorknowledgeabouttheenvironmentandlimitedcommunicationcapabilities withnearbyagentstoperformtaskssuchasformationcontrolinacooperativemanner, whereagentsarerequiredtocoordinatetheirmotionwithrespecttootheragents. Limitedsensingcapabilitiesbyagentsinanetworkhasbeenexaminedinresultssuchas [15,26,27,38,91].In[26],assumingthateachagentisawareofitsowndestination,agroup ofmobileagentswithlimitedsensingrangeiscontrolledtoachievedesiredformations basedonapotential-eldbasedapproachinanon-cooperativeway,whereagentsarenot requiredtocoordinatetheirmotionwithrespecttootheragentstoachievethedesired formation.Theresultin[26]isextendedtoperformformationtrackinginacooperative wayin[27],wherelimitedsensingisusedforcollisionavoidanceonly.Acentralizedleaderfollowerapproachisdevelopedtoperformformationtrackingin[38],andacentralized navigationfunction-basedcontrolstrategyisdevelopedin[91]tosteeragroupofmobile 17

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agentswithlimitedsensingcapabilitiestoachieveadesiredformation.Motivatedbythe needtomaintainnetworkconnectivity,anarticialpotentialeld-baseddecentralized methodisusedtopreventthenetworkfrompartitioningandstabilizeagroupofagents withlimitedcommunicationandsensingcapabilitiesinadesiredformationin[21,40,43], wherethenetworkconnectivityisguaranteedbymaintainingtheinitiallyestablished neighborhoodallthetimeduringtheoperation.However,acommonassumptioninthe resultsof[21,40,43]isthattheinitialtopologyisrequiredtobeasupergraphofthe desiredtopologyensuringtheagentsareoriginallyinafeasibleinterconnectedstate.Such resultsmaynotbeapplicabletotheapplicationswhichrequireamulti-agentsystem tostartfromanarbitraryconnectedinitialgraphordynamicallychangetheachieved formationstoadapttotheuncertainenvironment,sincethereorganizationoftheinitial topologytoadesiredonemayrequirethebreakageofsomeprespeciedneighborhoodand resultsinthepartitionoftheunderlyingnetworkconnectivity. Contrarytotheworkof[15,21,26,27,38,43,91],formationcontrolforagroupof agentswithlimitedsensingandcommunicationcapabilitiesareconsideredinChapter4, inwhichtheagentsareidenticalandcantakeanypositioninthenaltopology.Basedon theconceptsofprexlabelingandprexroutingin[31,32,79],anovelnetworktopology labelingalgorithmdevelopedinourpreviouswork[67]ismodiedtodynamicallyspecify theneighborhoodofeachagentintheinitialgraphaccordingtothedesiredformation, anddeterminetherequiredmovementforallnodestoachievethedesiredformation.By modelingnetworkconnectivityasanarticialobstacle,anavigationfunctionbasedcontrol schemeisdevelopedinChapter4toguaranteenetworkconnectivitybymaintainingthe neighborhoodamongagentsdeterminedbytheprexlabelingalgorithm,andensurethe convergenceofallagentstothedesiredcongurationwithcollisionavoidanceamong agentsusinglocalinformationi.e.,localsensingandcommunication.Aninformation owisthenproposedfromtheworkof[86]and[87]tospecifytherequiredmovementfor extraagentstotheirdestinationnodes.Theinformationow-basedapproachgenerally 18

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providesapathwithmorefreedomforthemotionofextranodeswithoutpartitioningthe networkconnectivityandallowscommunicationlinkstobeformedorbrokeninasmooth mannerwithoutintroducingdiscontinuity.ConvergenceisprovenusingRantzer'sDual LyapunovTheorem[72]. Rendezvousofwheeledmobilerobotswithnetworkconnectivity: Results suchas[16,21,24,33,40,69,99101]aredevelopedtomaintainthenetworkconnectivity intheapplicationofformationcontrol,ocking,consensusandothertasksineither centralizedordecentralizedmanner.However,onecommonfeatureinmostofthe aforementionedworkisthatonlylinearmodelsofmotionaretakenintoaccount,i.e.,the rstorderintegrator.Althoughcontroldesignforthestabilizationofasinglerobotwith nonholonomicconstraintshasbeenextensivelystudiedinthepastdecades[50,55],such controllersmaynotbeapplicableforanetworkedmulti-robotsystemwithacooperative objective,e.g.,maintainingnetworkconnectivity.Motivatedtonavigateasystemwith nonholonomicconstraintstoadestinationwithadesiredorientation,adipolarnavigation functionwasproposedandadiscontinuoustime-invariantcontrollerwasdevelopedto navigateasinglerobotin[92].Theworkin[92]wasthenextendedtoamulti-robot systemwithbothholonomicandnonholonomicconstraintsin[58]andextendedto navigateanonholonomicsysteminthreedimensionsin[77].However,onlyatimeinvariantdiscontinuouscontrollerwasdevelopedin[58,77,92].In[16],whenconsidering maintenanceofthenetworkconnectivity,basedontheworkof[92],adiscontinuous controllerwasusedtosteeramulti-robotsystemwithnonholonomicconstraintsto rendezvousatacommonposition.However,eachrobotcanonlyachievethedestination witharbitraryorientationandhastoreorientatthedestination.Moreover,themultirobotsystemcanonlyconvergetoadestinationwhichdependsontheinitialdeployment in[16].Basedonourpreviousworkin[41],adecentralizedcontinuoustime-varying controller,usingonlylocalsensingfeedbackfromitsone-hopneighbors,isdesignedin Chapter5tostabilizeagroupofwheeledmobilerobotswithnonholonomicconstraints 19

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ataspeciedcommonsetpointwithadesiredorientation,whilemaintainingnetwork connectivityduringnetworkregulation. Consensusofhumanemotioninasocialnetwork: Socialinteractionsinuence ourthoughtsandactionsthroughsocialnetworkswhichprovideameansformore rapid,convenient,andwidespreadcommunication.Forinstance,ashmobsarebeing organizedthroughsocialmediaforeventsrangingfromentertainingpublicspontaneity tovandalism,violence,andcrime[1,14,28].Recentriotsandprotests[6,12,37,78]and ultimatelyrevolution[36,61],havebeenfacilitatedthroughsocialmediatechnologies suchasFacebook,Twitter,YouTube,andBlackBerryMessagingBBM.Inattempts toprevent,mitigate,orprosecutethesourcesofsuchsocialunrest,governmentsand lawenforcementagenciesareplacingagreateremphasisonexaminingandultimately controllingthestructureofsocialnetworks.ScotlandYardislookingtosocialmedia websitesaspartofinvestigationsintowidespreadlootingandriotinginLondon[6,37],and policeinSanFranciscodisabledaccesstosocialnetworksbycuttingocellphoneservice asameanstopreventriotsduetoapoliceshooting[12].OneU.S.Intelligencestrategy inAfghanistanistofocusonansweringrudimentaryquestionsaboutAfghanistan'ssocial andculturalfabricthroughtoolssuchasNexus7totapintotheexabytesofinformation forleveragingpopularsupportandmarginalizingtheinsurgency[82].Yetother'sargue thatNexus7lacksmodelsoralgorithms. Modelsandalgorithmshavebeenextensivelydevelopedforvariousengineered networksandmulti-agentsystems[66].Consensusisaparticularclassofnetworkcontrol problemthathasbeenextensivelystudiedwherethegoalisfortheindividualnodesto reachanagreementonthestatesofallagents[70,74,75].However,aninterestingquestion thathasreceivedlittleattentionishowcansuchmodelsandmethodsbeappliedtoward understandingandcontrollingasocialnetwork.Howcanoneproduceconsensusamonga socialnetworke.g.,tomanipulatesocialgroupstoadesiredend?Motivatedtowardsthis 20

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end,thefocusofChapter6istoinuencetheemotionsofasociallyconnectedgroupof individualstoaconsensusstate. Variousdynamicmodelshavebeendevelopedforpsychologicalphenomena,including eortstomodeltheemotionalresponseofdierentindividuals[34,88,89].Adynamic modelofloveisreportedintheworkof[88],whichdescribesthetime-variationofthe emotionsdisplayedbyindividualsinvolvedinaromanticrelationship.In[89],happiness ismodeledbyasetofdierentialequations,andthetimeevolutionofone'shappinessin responsetoexternalinputsisexamined.Amathematicalmodeloffearisalsodescribedin theworkof[34]. Fractional-orderdierentialequationsareageneralizationofinteger-orderdierential equationswhichexhibitanon-localpropertywherethenextstateofasystemnotonly dependsuponitscurrentstatebutalsouponitshistoricalstatesstartingfromtheinitial time[64].Thispropertyhasmotivatedresearcherstoexploretheuseoffractional-order systemsasamodelforvariousphenomenainnaturalandengineeredsystems,andin relationtothecurrentcontext,havealsobeenexploredasapotentiallymoreappropriate modelofpsychologicalbehavior.Forexample,theinteger-orderdynamicmodelsoflove andhappinessdevelopedin[88]and[89]wererevisitedin[2]and[85],wherethemodels weregeneralizedtofractional-orderdynamics,sinceaperson'semotionalresponseis inuencedbypastexperiencesandmemories.However,theresultsin[2,85,88,89]focus onanindividual'semotionmodel,withoutconsideringtheinteractionamongindividuals inthecontextofasocialnetworkwhererapidandwidespreadinuencesfromthesocial peerscanprevail. Insteadofstudyinganindividualmodelofaperson'semotionalresponse,Chapter6 developsanapproachtoinuencetheinteractionofaperson'semotionswithinasocial network.Motivatedbythenon-localpropertyoffractional-ordersystems,theemotional responseofindividualsinthenetworkaremodeledbyfractional-orderdynamicswhose statesdependoninuencesfromsocialbonds.Withinthisformulation,thesocialgroup 21

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ismodeledasanetworkedfractionalsystem.Therstapparentresultthatinvestigated thecoordinationofnetworkedfractionalsystemsis[7],inwhichlineartimeinvariant systemsareconsideredandwheretheinteractionbetweenagentsismodeledasalinkwith aconstantweight.Inthischapter,thesocialbondbetweentwopersonsisconsideredas aweightfortheassociatededgeinthegraphmeasuringtheclosenessoftherelationship betweentheindividuals.Incomparisonto[7],themainchallengeinthisworkisthat socialbondsaretimevaryingparameterswhichdependsontheemotionalstatesof individuals.PreviousstabilityanalysistoolssuchasexaminingtheEigenvaluesoflinear systemsforfractional-ordersystemscf.[7,10,85]arenotapplicabletothetime-varying systeminChapter6.Toachievetheseobjectivesofmaintainingexistingsocialbonds amongindividualsi.e.,socialcontrolsorinuencesshouldnotbesoaggressivethatthey isolateorbreakbondsbetweenpeopleinthesocialgroupandemotionsynchronization inthesocialnetworki.e.,anagreementontheemotionstatesofallindividuals,a decentralizedpotentialfunctionisdevelopedinChapter6,andasymptoticconvergenceof eachemotionstatetothecommonequilibriuminthesocialnetworkisthenanalyzedviaa MetzlerMatrix[65]andaMittag-Leerstability[54]approach. 1.4Contributions ThecontributionsofChapters2-6arediscussedasfollows: 1. VisionBasedConnectivityMaintenanceofaNetworkwithSwitching Topology: ThemaincontributioninChapter2isthedevelopmentofatwolevel controlframeworkforconnectivitymaintenanceandcooperationofmulti-agent systems.Eachagentisequippedwithanomnidirectionalcameraandwireless communicationcapabilities,whichindicatesthateachagentisabletoseethe otheragentswithinitseldofviewandcancommunicatewithotheragentswithin itscommunicationzonetoexchangeinformation.Motivatedtoreducetheuseof interagentradiocommunicationforthemaintenanceofnetworkconnectivity,atwo levelgraphisdeveloped,whereallagentsarecategorizedaseitherclusterheadsor 22

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regularnodes.Ahighlevelgraphiscomposedofallclusterheadsandthemotion oftheclusterheadsiscontrolledtomaintainexistingconnectionsamongthem.A lowlevelgraphcomposedofallregularnodesiscontrolledtomaintainconnectivity withrespectitsspecicclusterhead.Imagefeedbackisusedastheprimarymethod tomaintainconnectivityamongagentswhilewirelesscommunicationisonlyused tobroadcastinformationwhenaspecictopologychangeoccurs.Onebenetof usingimagefeedbackasaprimarytoolisthatradiocommunicationmaynotbe applicableinsomedynamic,hostile,ortacticalenvironments,andevenwhenradio communicationispossiblethenetworkbandwidthmayberequiredfordistributing otherdata.Anothercontributionofthisworkisthedevelopmentofthearticial potentialeldbasedcontrollerstomaintaintheexistinglinksconnectedinboth lowandhighlevelgraphallthetime,andtoensurethatagroupofagentsswitches fromoneconnectedcongurationtoanotherwithoutdisconnectingthenetworkin process. 2. NetworkConnectivityPreservingFormationStabilizationandObstacle AvoidanceviaADecentralizedController: decentralizedcontrolmethod isdevelopedinChapter3toenableagroupofagentstoachieveadesiredglobal congurationwhilemaintainingglobalnetworkconnectivityandavoidingobstacles, usingonlylocalfeedbackandnoradiocommunicationbetweentheagentsfornavigation.Eachagentisequippedwitharangesensore.g.,cameratoprovidelocal feedbackoftherelativetrajectoryofotheragentswithinalimitedsensingregion, andatransceivertobroadcastinformationtoimmediateneighbors.Bymodelingthe interactionamongtheagentsasagraph,andgivenaconnectedinitialgraphwith desiredneighborhoodbetweenagents,thedevelopedmethodachievesconvergence toadesiredcongurationandmaintenanceofnetworkconnectivityusingadecentralizednavigationfunctionapproachwhichusesonlylocalfeedbackinformation. Byusingalocalrangesensorandnotrequiringknowledgeofthecompletenetwork 23

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structureasinmethodsthatuseagraphLaplacian,anadvantageousfeatureofthe developeddecentralizedcontrolleristhatnointer-agentcommunicationisrequired i.e.,communicationfreeglobaldecentralizedgroupbehavior.Thatis,connectivity ismaintainedsothatradiocommunicationisavailablewhenrequiredforvarious task/missionscenarios,butcommunicationisnotrequiredtonavigate,enabling stealthmodesofoperation.Collisionavoidanceandnetworkconnectivityareembeddedasconstraintsinthenavigationfunction.Byprovingthatthedistributed controlschemeisavalidnavigationfunction,themulti-agentsystemisguaranteed toconvergetoandstabilizethedesiredconguration. 3. NetworkConnectivityPreservingFormationRecongurationforIdentical AgentsFromAnArbitraryConnectedInitialGraph: Achievingadesired formationforagroupofidenticalagentswithlimitedsensingandcommunication capabilitiesfromanarbitrarilyconnectedinitialgraphisconsideredinChapter 4.Thelocalinteractionamongagentsismodeledbyadynamicgraphandthe goalistoachieveadesiredformationwhichischaracterizedbyagiveninter-agent distancefromanarbitraryconnectedinitialgraphwhilemaintainingnetwork connectivityinadecentralizedmanner.Contrarytothelimitationinmostexisting workinformationcontrolcf.[11,66,97]andtheirreferenceswheretheabsoluteor relativeposesoftheagentsareprespecied,andtheinitialtopologyrequirestobea supergraphofthedesiredtopology,anovelformationcontroltechniqueisdeveloped inChapter4,inwhichtherobotsareidenticalandcantakeanypositioninthenal topology.Thatis,wedonotwishtospecifywhichnodesintheinitialtopologywill takewhichpositionsinthenaltopology;rather,weonlycarethatthereisanagent ineachpositionspeciedinthenaltopology.Assumingthatthenaltopologyis atree,aprexlabelingandroutingalgorithmfrom[67]ismodiedtospecifythe neighborhoodofeachagentaccordingtothedesiredformationallowingtheagents tointerchangetheirroles,anddeterminetherequiredmovementforallnodesto 24

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achievethedesiredformation.Bymodelingthenetworkconnectivityasanarticial obstacle,anavigationfunctionbasedcontrolschemeisdevelopedinthischapter toguaranteethenetworkconnectivitybymaintainingtheneighborhoodamong agentsdeterminedbytheprexlabelingalgorithm,andensuretheconvergenceof allagentstothedesiredcongurationwithcollisionavoidanceamongagentsusing localinformationi.e.,localsensingandcommunication.Aninformationowis thenproposedfromtheworkof[86]and[87]tospecifytherequiredmovement forextraagentstotheirdestinationnodes.Theinformationow-basedapproach generallyprovidesapathwithmorefreedomforthemotionofextranodeswithout partitioningthenetworkconnectivityandallowscommunicationlinkstobeformed orbrokeninasmoothmannerwithoutintroducingdiscontinuity.Convergenceis provenusingRantzer'sDualLyapunovTheorem[72]. 4. EnsuringNetworkConnectivityforNonholonomicRobotsDuringDecentralizedRendezvous: Assumingarangesensore.g.,cameraprovideslocal feedbackoftherelativetrajectoryofotherrobotswithinalimitedsensingregion andatransceiverisusedtobroadcastinformationtoimmediateneighborsoneach robot,theobjectiveinChapter5istorepositionandreorientagroupofwheeled robotswithnonholonomicconstraintstoacommonsetpointwithadesiredorientationwhilemaintainingnetworkconnectivityduringtheevolution.Adistinguishing featureofthisworkisthatitalsoconsidersacooperativeobjectiveofmaintaining thenetworkconnectivityduringnetworkregulationforagroupofmobilerobots. Anotherfeatureofthedevelopeddecentralizedcontrolleristhat,usinglocalsensing information,nointer-agentcommunicationisrequiredi.e.,communication-free globaldecentralizedgroupbehavior.Thatis,networkconnectivityismaintained sothattheradiocommunicationisavailablewhenrequiredforvarioustasks,but communicationisnotrequiredforregulation.Usingadipolarnavigationfunction framework,themulti-robotsystemisguaranteedtomaintainconnectivityandbe 25

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stabilizedatacommondestinationwithadesiredorientationwithoutbeingtrapped bylocalminima.Moreover,theresultcanbeextendedtootherapplicationsby replacingtheobjectivefunctioninthenavigationfunctiontoaccommodatedierent tasks,suchasformationcontrol,ocking,andotherapplications. 5. InuencingEmotionalBehaviorinSocialNetwork: InsteadofstudyingnetworkedcontrolproblemsinengineeringasinChapter2-5,Chapter6investigatesan approachtoinuencetheinteractionofaperson'semotionswithinasocialnetwork. Usinggraphtheory,asocialnetworkismodeledasanundirectedgraph,wherean individualinthesocialnetworkisrepresentedasavertexinthegraph,andthe socialrelationshipbetweentwoindividualsisrepresentedasanedgeconnecting twovertices.Thesocialbondbetweentwopersonsisconsideredasaweightforthe associatededgeinthegraphmeasuringtheclosenessoftherelationshipbetween theindividuals.Motivatedbythenon-localpropertyoffractional-ordersystems, wherethenextstateofasystemnotonlydependsuponitscurrentstatebutalso uponitshistoricalstatesstartingfromtheinitialtime,theemotionalresponseof individualsinasocialnetworkismodeledbyfractional-orderdynamicswhosestates dependoninuencesfromsocialbonds.Withinthisformulation,thesocialgroupis modeledasanetworkedfractionalsystem.Contrarytotherstapparentresultthat investigatedthecoordinationofnetworkedfractionalsystemsin[7],inwhichlinear timeinvariantsystemsareconsideredandwheretheinteractionbetweenagentsis modeledasalinkwithaconstantweight,themainfeatureinthischapteristhat socialbondsaretimevaryingparameterswhichdependsontheemotionalstatesof individuals.PreviousstabilityanalysistoolssuchasexaminingtheEigenvaluesof linearsystemsforfractional-ordersystemscf.[7,10,85]arenolongerapplicableto thetime-varyingsysteminthischapter.Thischapteralsoconsidersasocialbond thresholdontheabilityoftwopeopletoinuenceeachother'semotions.Toensure interactionamongindividuals,oneobjectiveistomaintainexistingsocialbonds 26

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amongindividualsabovetheprespeciedthresholdallthetimei.e.,socialcontrols orinuencesshouldnotbesoaggressivethattheyisolateorbreakbondsbetween peopleinthesocialgroup.Anotherobjectiveistodesignadistributedcontroller foreachindividual,usinglocalemotionalstatesfromsocialneighbors,toachieve emotionsynchronizationinthesocialnetworki.e.,anagreementontheemotion statesofallindividuals.Toachievetheseobjectives,adecentralizedpotentialfunctionisdevelopedtoencodethecontrolobjectiveofemotionsynchronization,where maintenanceofsocialbondsismodeledasaconstraintembeddedinthepotential function.Asymptoticconvergenceofeachemotionstatetothecommonequilibrium inthesocialnetworkisthenanalyzedviaaMetzlerMatrix[65]andaMittag-Leer stability[54]approach. 1.5DissertationOutline Chapter1servesasanintroduction,wherethemotivation,problemstatement, literaturereviewandthecontributionsofthedissertationarediscussed. Chapter2describesatwolevelcontrolframeworkforconnectivitymaintenance andcooperationofmulti-agentsystems.Articialpotentialeldbasedcontrollersare developedtomaintainexistinglinksconnectedinbothlowandhighlevelgraphsallthe time,andalsoensurethatagroupofagentsswitchesfromoneconnectedcongurationto anotherwithoutdisconnectingthenetworkinprocess. Chapter3providesadecentralizedcontrolmethodbasedonthenavigationfunction formalismtoenableagroupofagentstoachieveadesiredglobalcongurationfroma connectedinitialgraphwithdesiredneighborhoodbetweenagents,whilemaintaining globalnetworkconnectivityandavoidingobstacles,usingonlylocalfeedbackandnoradio communicationbetweentheagentsfornavigation.Theperformanceofthedecentralized controlmethodisillustratedthroughsimulations. Chapter4illustratesanovelformationcontrolstrategyforagroupofidenticalagents withlimitedsensingandcommunicationcapabilitiestoachieveadesiredformationfrom 27

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anarbitrarilyconnectedinitialcondition.Aprexlabelingandroutingalgorithmis modiedtospecifytheneighborhoodofeachagentaccordingtothedesiredformation allowingtheagentstointerchangetheirroles,anddeterminetherequiredmovementfor allnodestoachievethedesiredformation.Anavigationfunctionbasedcontrolscheme isdevelopedtoguaranteethenetworkconnectivitybymaintainingtheneighborhood amongagentsdeterminedbytheprexlabelingalgorithm,andensuretheconvergenceof allagentstothedesiredcongurationwithcollisionavoidanceamongagentsusinglocal informationi.e.,localsensingandcommunication.Simulationresultsareprovidedto illustratethedevelopedstrategy. Chapter5developsadipolarnavigationfunctionandcorrespondingtime-varyingcontinuouscontrollertorepositionandreorientagroupofwheeledrobotswithnonholonomic constraints,whilemaintainingthenetworkconnectivityduringthemission,byusingonly localsensingfeedbackinformationfromneighbors.Simulationresultsdemonstratethe performanceofthedevelopedapproach. Chapter6extendstheapproachesdevelopedinpreviouschapterstoprovideameans toinuencethehumanemotionforagroupofindividualinasocialnetwork.Thesocial interactionsamongindividualsinasocialnetworkaremodeledasanundirectedgraph witheachvertexrepresentinganindividualandeachedgerepresentingasocialbond betweenindividuals.Bymodelingtheemotionalresponseofindividualsinthenetwork asfractional-orderdynamicswhosestatesdependoninuencesfromsocialbonds,a decentralizedcontrolmethodisdevelopedtomanipulatethesocialgrouptoacommon emotionalstatewhilemaintainingexistingsocialbondsi.e.,withoutisolatingpeersinthe group.Asymptoticconvergencetoacommonequilibriumpointi.e.,emotionalstateof thenetworkedfractional-ordersystemisprovedbyusingMittag-Leerstability. Chapter7concludesthedissertationbysummarizingtheworkanddiscussingsome remainingopenproblemsthatrequirefurtherinvestigation. 28

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CHAPTER2 VISIONBASEDCONNECTIVITYMAINTENANCEOFANETWORKWITH SWITCHINGTOPOLOGY Inmostapplicationsofamulti-agentsystem,agentsneedtocoordinateandcommunicatetotakeappropriatedecisionstofulllapre-speciedgoal.Inthischapter,each robotisassumedtobeequippedwithanomnidirectionalcamerathatcanmeasurethe relativepositionoftheotheragentsinitssensingarea,andsomeformoftransceiverthat canbeusedtobroadcastinformationtolocalnodes.Twomovingagentscancommunicate witheachotheriftheyremainwithinaspecicdistance.Asagentsmovetoperformsome missionobjective,itisparamounttoensurethatthegroupofagentsremainconnected i.e.,thegroupdoesnotpartition.Motivatedtoreduceinteragentradiocommunication, anetworkconnectivitymaintenanceobjectiveisconsideredinthischapterthatrelies primarilyonimagefeedback.Atwolevelcontrolstrategyisdevelopedin[47],whereall agentsintheteamarecategorizedasclusterheadsorregularnodes.Ahighlevelgraphis composedofallclusterheadsandthemotionoftheclusterheadsiscontrolledtomaintain existingconnectionsamongthem.Alowlevelgraphcomposedofallregularnodesis controlledtomaintainconnectivitywithrespectitsspecicclusterhead.Connectivity ofthenetworkismaintainedusingimagefeedbackonlyunlessaclusterheadchangeis required.Iftheclusterheadchangesandthenetworkneedstoreorganizethetopology, onlythenisthewirelesscommunicationusedtoalertthenodesofthetopologychange. Articialpotentialeldbasedcontrollersarethendevelopedtomaintaintheexistinglinks connectedinbothlowandhighlevelgraphsallthetimeandtoensurethatagroupof agentsswitchesfromoneconnectedcongurationtoanotherwithoutdisconnectingthe networkinprocess. 2.1ProblemFormulation Consideranetworkcomposedof N agents,whereagent i movesaccordingtothe followingkinematics: x i t = u i t ;i 2V = f 1 ;:::;N g 29

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where x i t 2 R 2 denotesthepositionofagent i inatwodimensionalDplaneattime t ,and x t 2 R 2 N denotesthestackpositionvectorsofallagents.In2, u i t 2 R 2 denotesthevelocityofagent i i.e.thecontrolinput.Theinteractionofthegroupis modeledasa dynamicgraph ,inthesensethatitevolvesintimewithitsconnectivity governedbythekinematicsoftheagents2.Thistimevaryingpropertygivesriseto thenotionofadynamicgraph, G t = V ; E t ,inwhichthesetoflinks E t istime varyingandeachcomponentin V standsfortheindexofanagent.Giventheassumption thateachagentisequippedwithanomnidirectionalcameraandwirelesscommunication capabilities,twodierentgraphmodelsneedtobespecied:acommunicationgraph andavisibilitygraph.Eachgraphiscomposedofdierenttypesofnodes:clusterheads andregularnodes,andtheinteractionbetweenthenodesineachgraphismodeledina dierentway. 2.1.1CommunicationGraph Inter-agentcommunicationismodeledintermsofatime-varyingcommunication graph G c = V ; E c t withtheindexsetofnodes V andsetofedges E c t = f i;j 2VVjk x ij k R c g : In2,eachnodeislocatedataposition x i k x ij k2 R + isdenedas k x ij k = k x i )]TJ/F27 11.9552 Tf 11.955 0 Td [(x j k ; and R c denotesthemaximumcommunicationradius.Thecommunicationgraph G c isanundirectedgraphinthesensethatnodes i caninuencenode j andviceversa. Anundirectedcommunicationlinkbetweennodes i and j isdenotedby i;j when k x ij k R c .Theindexsetofneighborsofnode i isdenotedby N c i = f j : j 6 = i j j 2V ; i;j 2E c g : 30

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2.1.2VisibilityGraph Eachagentiscapableofsensingadiskareawiththemaximumradius R v R c ; so thatanytwoagentsareabletocommunicatewitheachotheraslongastheycanseeeach other.Thevisibilitygraphismodeledasaundirectedtime-varyinggraph G v = V ;" v t withtheindexsetofnodes V andsetofedges v = f i;j 2VVjk x ij k R v g : Forthevisibilitygraph,theedge i;j isundirectedindicatingthat,ifnode i cansee node j; node j canalsoseenode i .Theindexsetofneighborsofnode i isdenoted by N v i = f j : j 6 = i j j 2V ; i;j 2E v t g : Thesubsequentdevelopmentisbasedon theassumptionthatthedistancebetweentwonodescanbeestimatedfromtheimage feedbacke.g.,usingmethodsasin[30]. 2.1.3ConnectivityMaintenance Since R v R c ; asucientgoaltoensure G c remainsconnectedistoensurethe visibilitygraph G v remainsconnected.Forsimplicity,thefollowingdevelopmentis basedontheassumptionthat R v = R c = R withoutlossofgenerality.Tounderstand connectivityforeachgraph,considerFig2-1.Forthecommunicationgraph,ifnode i inFig2-1isconnectedtonode j andnode j isconnectedtonode k ,thennode i isalso connectedtonode k throughedge i;j and j;k : Node i andnode k mayexchange informationin G c ,toachieveadesiredcooperativemotion.IfFig2-1isconsideredasa visibilitygraph,thenalthoughnode j canbeseenbynode i andnode k canbeseenby node j; node i isnotcapableofsharinginformationwithnode j .Thecommunication graphisconsideredconnectedifeverynodein G c isreachablefromeveryothernodebya seriesofedges. Thegoalinthischapteristodevelopadecentralizedimage-feedbackcontrolleri.e., velocityinputforeachagentsothat G c remainsconnecteddespiteclusterheadshiftsi.e., whenaclusterheadroleshiftsfromonenodetoaregularnode.Theadvantageisthatthe 31

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Figure2-1.Modelofvisibilitygraph. networkmaintenanceisachievedwithoutradiocommunication,exceptwhenaclusterhead shiftneedstooccur.Whenthetopologychangesduetoaclusterheadshift,thenewrole ofnodeisbroadcastacrossthewirelessnetwork. 2.2ControlStrategy Motivatedbytheideaofacommunicationbackbone[4,95],atwolevelnetwork structureisproposed.Thebasicideaistogroupallnodesinto m subsets.Eachsubset containsaoneandonlyonespecialnodedenedas clusterhead, where V CH = f 1 ; ;m g denotestheindexsetofclusterheads,andthesetofclusterheadsformsahighlevel networkgraph,representedas G high t .Specically,thehighlevelnetworksubgraph iscomposedofclusterheadsonly,whichisasmallsubsetofthegroup,providinga hierarchicalorganizationoftheoriginalnetwork.Thehighlevelnetworksubgraphis denedas G high t = V CH ; E high ,where E high = f i;j 2V CH V CH jk x ij k R g : Alltheremainingnodesineachsubgrapharedenedasa regularnodes, where V RN = f m +1 ; ;N g denotestheindexsetofregularnodes.The m subsetsformthe lowlevelnetwork,representedas G low i t : Specically,thelowlevelnetworksubgraph isdenedas G low t = G low 1 ; ; G low m .Each G low i t formsaconnectedsubgraphof G v t andonlyoneparticularnodeisselectedasaclusterheadineach G low i t .Note that m i =1 G low i t =0 ; whichmeans G low i t ismutuallyexclusivetoeachother,and 32

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Figure2-2.Schematictopologyofunderlyingnetwork. [ m i =1 G low i t = G v t .Sinceonlylocalinformationcanbeobtainedbyvisionsensors,we requirethattheselectedclusterheadcanbeseenbyallregularnodesineach G low i t ; andeachregularnodein G low i t movesundertheconstraintthatitmuststayconnected toitsclusterheadforalltime.Hence,eachlowlevelnetworksubgraph G low i t hasa xedtopology.ThistwographstructureisdepictedinFig.2-2,whereCHstandsfor clusterheadandRNstandsforregularnode.AsindicatedinFig.2-2,{CH1,CH2,CH3, CH4}formsthehighlevelnetworksubgraph G high t ; while{{CH1,RN1,RN2},{CH2, RN3},{CH3,RN5,RN4},{CH4}}formsthelowlevelnetworkgraph G low t : Thekeytomaintainthenetworkconnectivityistomaintainconnectivitywithineach subseti.e.,ensureeach G low i t isindividuallyconnectedandmaintainconnectivityofthe G high t graph.Thegraphs G low i t and G high t areinitiallyspecied,buteventscanoccur thatrequireaclusterheadtochangeroleswitharegularnodein G low i t Informationdrivenmethodssuchasthosedescribedin[105]and[104]canbeusedtodynamically selectclusterheadsfordierenttasks.Thedevelopmentinthischaptersimplyassumes thatsomeprocessdeterminestheneedforaclusterheadandregularnodetochangeroles. Fromasystemstheoryperspective,theunderlyingnetworkgraphdynamicsare consideredtohaveatransientandsteady-stateresponse.Asteady-statetopologyiswhen 33

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therolesoftheclusterheadsandregularnodesremainconstant.Thecontrolobjective duringsteady-stateistoensurethatallregularnodesmovetomaintainconnectivity withtherespectiveclusterheadwithin G low i t andensurethatallclusterheadsmaintain connectivitywithin G high t .Insteady-state,nonewedgesareformedwhenonenode entersothernode'ssensingradius.Inotherwords,theunderlyinggraphhasaxed topologyinthesensethattheedgesof G low i t and G high t donotchange,buttherelative positionofthenodeswithinthegraphcandynamicallychange.Connectivityduring steady-stateismaintainedbyimagefeedbackalone. Atransienttopologyiswhentheoverallgraphswitchesfromoneconnectedcongurationtoanotherwithoutdisconnecting.Thenetworkcanbecometransientwhentherole ofanodeischangedandnewedgesarecreatedundersomerule.Toguaranteetheconnectivityduringatransientstage,wirelesscommunicationhastobeusedtobroadcastthe newroleofnodestotheneighbors.Oncethenewrolesofthenodeshasbeenbroadcast, thenallnodesresumesteady-statewherethenodesuseonlyimagefeedback. Thetopologywillbecometransientduetochangesinmissionobjectivesortopology disturbances.Forexample,theroleofRN1inFig.2-2mayneedtochangetobecome anewclusterhead.Whenthetopologyundergoesareconguration,atwostepstrategy isinvestigated.First,RN1broadcastsitsrole-changethroughimmediateneighborsto everynodeinthegroup.Radiocommunicationisterminatedwhenallnodeshavebeen updated.Then,underimagefeedback,thenodesstarttoformanewconnected G high and G low i .Sincenoradiocommunicationisallowed,eachnodeonlyhaslocalinformation withinitssensingregion.CH3needstomovetowardCH2rst,and,wheneveranedge betweenCH1andCH3iscreated,itmovestoCH1togetcloseenoughtoRN1.Likewise, newedgesarecreatedforCH4andCH2oncetheycanbeseenbyRN1.Althoughnew edgesarecreatedamongclusterheads,thereisnoedgecreatedforregularnodes,evenif someothernodesenteritssensingregion.Inotherwords,allregularnodesmovewithits respectiveclusterhead.Asaresult,eachsubgraph G low i canberepresentedasonesingle 34

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node,representedasaclusterhead.Aslongastheclusterheadsareconnected,thewhole graphisconnected.Onebenetofthisstructureisthatnetworkwithlargenumberof nodescanbescaleddown. 2.3ControlDesign 2.3.1PotentialField Thegoalinthissectionistodesigndistributedcontrollaws u i t forallnodesto guaranteetheconnectivityof G v t .Theset N v i t istimevaryinganddependentonthe relativepositionsofthenodes.Nodeswithindistanceslessthan R areinteractingwith eachotherthroughapotentialforceandapotentialfunctionisusedforconnectivity maintenance,aswellascollisionavoidance. Anattractivepotentialeldisdenedas ij : R 2 R + ,whichisanonnegative functionofthedistancebetweennodes i and j; i.e. ij = ij k x ij k : Thepurposeofthe attractiveforceistoguaranteethatnode j willneverleavethesensingzoneofnode i ,if node j isinitiallylocatedatadistancelessthan R fromnode i .Theattractivepotential eldistoregulatedistancesbetweenagentswithintherange ;R : Somepropertiesare requiredtomake ij aqualiedpotentialfunction: 1 ij k x ij k !1 as k x ij k! R 2 ij k x ij k is C 1 for k x ij k2 ;R and @' ij @ k x ij k > 0 ; if k x ij k2 ;R 3 ij k x ij k =0 when k x ij k >R: Arepulsivepotentialeldisdenedas ij : R 2 R + ,whichisadierentiableexceptat point k x ij k =0 ,nonnegativefunctionofthedistancebetweennodes i and j; i.e. ij = ij k x ij k : Thepurposeoftherepulsiveforceistoguaranteecollisionsavoidancebetween node i andnode j astheygetclosetoeachother.Somepropertiesarerequiredtomake ij aqualiedpotentialfunction: 1 ij k x ij k !1 as k x ij k! 0 2 ij k x ij k =0 ; and @ ij @x i =0 ; if k x ij k R 3 @ ij @ k x ij k < 0 ; if k x ij k2 ;R ; and @ ij @ k x ij k =0 ; if k x ij k R: 35

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Property2guaranteesthat X j 2N v i t @ ij @x i = X j 6 = i @ ij @x i : Theimportanceofthispropertyisthat,unlikeanattractiveforce,thetimevarying set N v i t doesnotintroduceanydiscontinuitytothesystemwhenonenodeenters thesensingzoneofanothernode.Inspiredby[90],[98],[17],afunction ij k x ij k is introducedtosmooththediscontinuitywhenanewedgeisformed.Tocapturethe newlyformededge,aset N i t isdenedas N i t = f j 2V ;j 6 = i jk x ij k R )]TJ/F27 11.9552 Tf 12.68 0 Td [( g ; where 0 < R: Thesetofedgesisupdatedas: E v t = E v t )]TJ/F15 11.9552 Tf 7.085 -4.339 Td [( [E v t ; where E v t = f i;j j i;j = 2E v t )]TJ/F15 11.9552 Tf 7.084 -4.338 Td [( ^ j 2N i t g : Thefunction ij isdenedwithfollowing properties: 1 ij = ij if k x ij k R )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 ij = const and @' ij @x i =0 if k x ij k R: 3 ij is C 1 everywhereandthepartialderivative @' ij @ k x ij k > 0 for R )]TJ/F27 11.9552 Tf 11.955 0 Td [(< k x ij k
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2.3.2ControllerforSteadyState Ineachsubgraph G low i ; aregularnodeisattractedbyitsclusterheadonlyandrepelled byalltheadjacentnodes.Thetotalpotentialofregularnode i;i 2 V RN ; is: U r i = ik + X j 2N v i t ij ; where k;k 2 V CH ; denotestheindexofthecorrespondingclusterheadin G low i : Thecontrol lawforaregularnodeisdesignedas u r i t = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(@U r i @x i : Themotionofaclusterheadisnotaectedbyregularnodes,andaclusterheadonly moveswiththeconstrainttoensureconnectivityandcollisionavoidancein G high : The compositepotentialofclusterhead i;i 2 V CH ; isgivenby: U c i = X i;j 2E high ij + X i;j 2E high ij + U T i ; where U T i denotesataskpotentialtomodelarequiredperformance,whichimposesan attractivepotentialonnode i: Thecontrollawfortheclusterheadsisdesignedas u c i t = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(@U c i @x i : 2.3.3ControllerforSwitchingState Collisionavoidanceandnetworkconnectivitymustbemaintainedevenwhenthe topologyundergoesatransition.AsdescribedinSection2.2,themotionofregularnodes isdictatedbythemotionoftheparentclusterhead.Thetotalpotentialandcontrollaw forregularnode i isthesameas2and2insteadystateconditions.However, thepotentialfortheclusterheadnodeschange.Specically,thecompositepotentialof 37

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clusterhead i 2 V CH isgivenby U c i = X i;j 2 E t ij + X i;j = 2 E t ij + X j 6 = i ij ; wheretheset E t high t denotesatimevaryingsetofedgesdevelopedbasedon theswitchingstrategyinSection2.2.Thegoalofnewset E t istoguideclusterheads toformanewsteadystate.Notethattherearetwomaindierencesbetween2and 2.First,thereisno U T i in2.Theterm U T i isdesignedtoperformtasksinsteady state.Thegoaloftheswitchingstateistoreshapethetopologytoanewsteadytopology. Thereisnoneedtokeep U T i duringtheswitchingprocess.Secondly,thefunction ij is usedtotakecareofthediscontinuitythatiscausedbynewedgeformation.Basedonthe developedcompositepotential,thecontrollawforclusterheadsisdesignedas u c i t = )]TJ/F27 11.9552 Tf 10.494 8.088 Td [(@U c i @x i : Aninitialconnectedunderlyinggraphisrequiredtoguaranteetheconnectivityforallthe futuretime. 2.4ConnectivityAnalysis Proposition2.1. Forsteadystate,ifthenetworkgraph G v t isconnectedat t = t 0 ,then connectivityandcollisionavoidanceisguaranteedwiththecontrollerproposedin2and 2for t>t 0 : Proof. Thetopologyof G v t isstaticinsteadystateinthesensethatnewedgesarenot formed.Ineachsubgraph G low i ; regularnodesmovewithrespecttoitsclusterhead,andin subgraph G high ; clusterheadsmovewiththeconstraintthattheconnectivityisensured.A Lyapunovcandidatefunctionalisdesignedas V = X i 2V RN U r i + X i 2V CH U c i : Basedon2and2,asanagentgetsclosetoacollisionorasthegraphgetscloser topartitioning,then V x t approachesinnity.Takingtimederivativeof2and 38

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substitutingfor2and2,yields V = X i 2V RN @U r i @x i x i + X i 2V CH @U c i @x i x i = )]TJ/F33 11.9552 Tf 15.765 11.358 Td [(X i 2V RN @U r i @x i 2 )]TJ/F33 11.9552 Tf 16.51 11.357 Td [(X i 2V CH @U c i @x i 2 0 : Theexpressionsin2and2implythat V x t V x t 0 : Sincethesystemis initiallycollisionfreeandconnectedat t 0 ; then V x t 0 < 1 ; andthegraphisensuredto remaincollisionfreeandconnectedforall t t 0 providedthegraphtopologyremainsina steadystatecondition. Proposition2.2. Duringtheswitchingprocess,connectivityandcollisionavoidanceof thenetworkgraph G t isguaranteedbythecontrollerproposedin2and2. Proof. Proposition2.1indicatesthatconnectivityisguaranteedineach G low i : Toshowthe graph G v t isconnectedduringaclusterheadswitch,weonlyneedtoshowthatonceany twoclusterheadscomeintoadistancelessthanorequalto R )]TJ/F27 11.9552 Tf 12.347 0 Td [( forthersttime,they willremainconnectedtoeachother,i.e.thedistancebetweenthemisstrictlylessthan R forallfuturetime.Toexaminethisscenario,aLyapunovcandidatefunctionalisdesigned as: V = X i 2V CH X i;j 2 E t ij + X i;j = 2 E t ij + X i 6 = j ij : Anattractivepotentialfunction ij isadiscontinuousfunctionatthepoint k x ij k = R whiletherepulsivefunction ij isadierentiablefunction.Wheneveranode j formsa newedgewithnode i; thefunction ij isswitchedto ij inasucientlysmoothmanner, sothat V iscontinuouslydierentiable.Takingthetimederivativeof V andsubstituting 2and2yields V 0 ; andhence, V x t V x t 0 < 1 : Itisknownthat V !1 when k x ij k! R foratleastonepairofnodes.Hence,allpairsofnodesthatdid 39

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notinitiallyformanedgemovesothatnewedgesareformedsothatthecommunication graphremainsconnected. 2.5Simulation Preliminarysimulationresultsillustratetheperformanceoftheproposedcontrol strategy.Agroupof 7 nodeswithkinematicsgivenin2aredistributedintheplane withaninitiallyconnectedunderlyinggraph.Assumethateachnodehasasensingzone of R =1 m.Whentwoagentsareadjacent,alineisdrawnbetweenthemtoshowthe connectivity. Agroupof7nodesevolvedunderthecontrollawproposedinSection2.3.InFig.2-3 andFig.2-4,therectangularnodesrepresentclusterheadsandcirclesrepresentregular nodes.Thedashedlinesidentifythelinkamongclusterheads,whilesolidlinesidentity thelinkbetweenaregularnodeandaclusterhead.At t =0 ; aninitialconnectedgraph isgenerated.Duringtimeinterval t 2 ; 120 ; thegroupofnodesmovesinsteadystate. Thetopologyismaintainedduringnodemotion,asshowninFig.2-3.Oneclusterhead issimulatedwithataskfunctiontomovealongadesignedtrajectory, P y =2sin : 2 P x where P y and P x denotesthestack x and y coordinatevectorrespectively.InFig.2-3,all nodesmoveinadesiredmanner. Tosimulatetheperformanceofaswitchingstate,thetopologychangesat t =121 inthesensethatoneregularnodeswitchesitsroletoanewclusterhead,whileone clusterheadchangesitsroletoaregularnode.Thenewclusterheadistaskedwiththe objectivetomovealongthedesiredtrajectory, P y = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2sin : 2 P x ,fromitscurrent position.Fig.2-4illustrateshowthesenodesmovetoreshapethetopologytoformanew steadystatetopologywithoutdisconnectingthegroup. 2.6Summary Atwostagecontrolframeworkisproposedforconnectivitymaintenanceandcooperationofamulti-agentsystemusingimagefeedback.Theideaistogroupallnodesintotwo subgraphs,ahighlevelnetworksubgraphandseverallowlevelnetworksubgraphs.The 40

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Figure2-3.Evolutionofnodesduringtimeintervalof t 2 ; 120 Figure2-4.Evolutionofnodesduringtimeintervalof t 2 ; 220 41

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keytomaintainthenetworkconnectivityistoensuretheconnectivityofhighlevelsubgraphandtheconnectivityofeachlowlevelsubgraph.Apotential-eld-basedcontrolleris usedtoguaranteetheconnectivity,aswellascollisionavoidance.Futureeortswillfocus onmoresimulationswithmorenodesandmoreswitchingtoexaminetheinterplayofthe nodes,includingcaseswheremultiplenodesshiftfromclusterheadsatthesametime. 42

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CHAPTER3 NETWORKCONNECTIVITYPRESERVINGFORMATIONSTABILIZATIONAND OBSTACLEAVOIDANCEVIAADECENTRALIZEDCONTROLLER Inthischapter,anavigationfunctionframeworkisusedtodevelopadecentralized controllerseealso[44]thatguaranteesamulti-agentsystemtoachieveadesiredcongurationwhilepreservingthenetworkconnectivityduringthemotion.Byusingalocal rangesensorandnotrequiringknowledgeofthecompletenetworkstructureasinmethodsthatuseagraphLaplacian,anadvantageousfeatureofthedevelopeddecentralized controlleristhatnointer-agentcommunicationisrequiredi.e.,communicationfreeglobal decentralizedgroupbehavior.Thegoalistomaintainconnectivitysothatradiocommunicationisavailablewhenrequiredforvarioustask/missionscenarios,butcommunication isnotrequiredtonavigate,enablingstealthmodesofoperation.Collisionavoidanceand networkconnectivityareembeddedasconstraintsinthenavigationfunction.Byproving thatthedistributedcontrolschemeisavalidnavigationfunction,themulti-agentsystem isguaranteedtoconvergetoandstabilizeatthedesiredconguration. 3.1ProblemFormulation Consideranetworkcomposedof N agentsintheworkspace F ,whereagent i moves accordingtothefollowingkinematics: q i = u i ;i =1 ; ;N where q i 2 R 2 denotesthepositionofagent i inatwodimensionalDplane,and u i 2 R 2 denotesthevelocityofagent i i.e.,thecontrolinput.Theworkspace F isassumedto becircularandboundedwithradius R; and @ F denotestheboundaryof F : Eachagent in F isrepresentedbyapoint-masswithalimitedcommunicationandsensingcapability encodedbyadiskarea.Itisassumedthateachagentisequippedwitharangesensor andwirelesscommunicationcapabilities.Twomovingagentscancommunicatewitheach otheriftheyarewithinadistance R c ,whiletheagentcansensestationaryobstacles orotheragentswithinadistance R s .Forsimplicityandwithoutlossofgenerality,the 43

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followingdevelopmentisbasedontheassumptionthatthesensingareacoincideswith thecommunicationarea,i.e., R c = R s .Further,itisassumedthatalltheagentshave equalactuationcapabilities.Asetofxedpoints, p 1 ; ;p M ; aredenedtorepresent M stationaryobstaclesintheworkspace F ,andtheindexsetofobstaclesisdenotedas M = f 1 ; ;M g .Theassumptionofpoint-obstaclesisnotrestrictive,sincealargeclass ofshapescanbemappedtosinglepointsthroughaseriesoftransformations[93],andthis point-worldtopologyisadegeneratecaseofthesphere-worldtopology[76]. Theinteractionofthesystemismodeledasa dynamicgraph ,inthesensethatthe systemevolvesintimegovernedbytheagentkinematicsin3.Thedynamicgraph isdenotedas G t = V ; E t ,where V = f 1 ; ;N g denotesthesetofnodes,and E t = f i;j 2VVj d ij R c g denotesthesetoftimevaryingedges,wherenode i and node j arelocatedataposition q i and q j ; and d ij 2 R + istherelativedistancedenedas d ij = k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k .Ingraph G t ,eachnode i representsanagent,andtheedge i;j denotes alinkbetweenagent i and j whentheystaywithinadistance R c .Nodes i and j arealso called one-hopneighbors ofeachother.Thesetofone-hopneighborsofnode i i.e.,allthe agentswithinthesensingzoneofagent i isgivenby N i = f j;j 6 = i j j 2V ; i;j 2Eg : Oneobjectiveinthisworkistohavethemulti-agentsystemconvergetoadesired conguration,determinedbyaformationmatrix c ij 2 R 2 representingthedesired relativepositionofnode i withanadjacentnode j 2N f i ,where N f i N i denotesthe setofnodesrequiredtoformaprespeciedrelativepositionwithnode i inthedesired conguration.Theneighborhood N i isatimevaryingsetsincenodesmayenterorleave thecommunicationregionofnode i atanytimeinstant,while N f i isastaticsetwhichis speciedbythedesiredconguration.Thedesiredpositionofnode i ,denotedby q di ,is denedas q di = n q i jk q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j )]TJ/F27 11.9552 Tf 11.955 0 Td [(c ij k 2 =0 ;j 2N f i o : Anedge i;j isonlyestablishedbetweennodes i and j if j 2N f i 44

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A collisionregion 1 isdenedforeachagent i asasmalldiskwithradius 1
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where 2 R + isatuningparameter, i : R 2 R + isthegoalfunction,and i : R 2 [0 ; 1] isaconstraintfunctionfornode i .Thegoalfunction i in3encodesthecontrol objectiveofnode i ,speciedintermsofthedesiredrelativepositionwithrespecttothe adjacentnodes n j 2N f i o ,anddrivesthesystemtoadesiredconguration 2 .Thegoal functionisdesignedas i q i ;q j = X j 2N f i k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j )]TJ/F27 11.9552 Tf 11.955 0 Td [(c ij k 2 : Theconstraintfunction i in3isdesignedas i = B i 0 Y j 2N f i b ij Y k 2N i [M i B ik ; toensurecollisionavoidanceandnetworkconnectivitybyonlyaccountingfornodes andobstacleslocatedwithinitssensingareaduringeachtimeinstant.Specically,the constraintfunctionin3isdesignedtovanishwhenevernode i intersectswithoneof theconstraintsintheenvironment,i.e.,ifnode i touchesaxedobstacle,theworkspace boundary,othernodes,ordepartsawayfromitsadjacentnodes n j 2N f i o toadistance of R c .In3, b ij b q i ;q j : R 2 [0 ; 1] ensuresconnectivityofthenetworkgraph i.e.,guaranteesthatnodes n j 2N f i o willneverleavethecommunicationzoneofnode i if node j isinitiallyconnectedtonode i andisdesignedas b ij = 8 > > > > < > > > > : 1 d ij R c )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F26 7.9701 Tf 12.475 4.708 Td [(1 2 2 d ij +2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(R c 2 + 2 2 d ij +2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(R c R c )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2
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Alsoin3, B ik B q i ;q k : R 2 [0 ; 1] ,forpoint k 2N i [M i ; where M i indicatesthe setofobstacleswithinthesensingareaofnode i ateachtimeinstant,ensuresthatnode i isrepulsedfromothernodesorobstaclestopreventacollision,andisdesignedas B ik = 8 > < > : )]TJ/F26 7.9701 Tf 12.475 4.707 Td [(1 2 1 d 2 ik + 2 1 d ik d ik < 1 1 d ik 1 : Similarly,thefunction B i 0 in3isusedtomodelthepotentialcollisionofnode i with theworkspaceboundary,wherethepositivescalar B i 0 2 R isdesignedsimilarto B ik byreplacing d ik with d i 0 ,where d i 0 2 R + istherelativedistanceofthenode i tothe workspaceboundarydenedas d i 0 = R )-222(k q i k Assumption3.2guaranteesthat i and i willnotbezerosimultaneously.The navigationfunctioncandidateachievesitsminimumof 0 when i =0 andachievesits maximumof 1 when i =0 .For i tobeanavigationfunction,ithastosatisfythe followingconditions[76]: 1smoothon F atleasta C 1 function[18]; 2admissibleon F ,uniformlymaximalon @ F andconstraintboundary; 3polaron F ; q di isauniqueminimum; 4aMorsefunction,criticalpoints 3 ofthenavigationfunctionarenondegenerate. If i isaMorsefunctionand q di isauniqueminimumof i i.e., q di ispolaron F ,then almostallinitialpositionsexceptforasetofpointsofmeasurezeroasymptotically approachthedesiredposition q di [76].Inaddition,thenegativegradientofthenavigation functionisboundedifitisanadmissibleMorsefunctionwithasingleminimumatthe desireddestination[76].Anexampleofthegeneratedarticialpotentialeldisshownin 3 Apoint p intheworkspace F isacriticalpointif r q i i j p =0 47

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Figure3-1.Anexampleofthearticialpotentialeldgeneratedforadisk-shaped workspacewithdestinationattheoriginandanobstaclelocatedat [1 ; 1] T Fig.3-1,inwhichthedestinationisassignedaminimumpotentialvalueandtheobstacle isassignedamaximumpotentialvalue. Basedonthedenitionofthenavigationfunctioncandidate,thedecentralized controllerforeachnodeisdesignedas u i = )]TJ/F27 11.9552 Tf 9.298 0 Td [(K r q i i ; where K isapositivegain,and r q i i isthegradientof i withrespectto q i .Hence,the controllerin3isboundedandyieldsthedesiredperformancebysteeringnode i along thedirectionofthenegativegradientof i if3isanavigationfunction. 3.3ConnectivityandConvergenceAnalysis Thefreecongurationworkspace F i F isacompactconnectedanalyticmanifold fornode i F i f q j i q > 0 g ,and q denotesthestackedpositionvectorofnode i .The boundaryof F i isdenedas @ F i )]TJ/F26 7.9701 Tf 6.587 0 Td [(1 i : Thenarrowsetaroundapotentialcollisionfor node i isdenedas B B i;k f q j 0 0 ;k 2N i [M i g ; 48

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andanarrowsetaroundapotentialconnectivityconstraintisdenedas B b i;j f q j 0 0 ;j 2N f i g : Theset B 0 = f q j 0 0 g isusedtodenoteanarrowsetaroundapotential collisionofnode i withworkspaceboundary.Inspiredbytheseminalworkin[76], F i is partitionedintovesubsets F 0 F 1 F 2 F 3 ,and F di as F i = F di [F 0 [F 1 [F 2 [F 3 ; wherethesetofdesiredcongurationsfornode i isdenedas F di f q j i q =0 g .The sets F 0 F 1 F 2 and F 3 describetheregionsneartheworkspaceboundary, nearthepotentialcollisionconstraint,neartheconnectivityconstraintandawayfromall constraintsfornode i ,respectively,andaredenedas F 0 B 0 )-222(F di ; F 1 [ i + # i k =1 B B i;k )-222(F di ; F 2 [ i j =1 B b i;j )-222(F di ; and F 3 F i )-222(fF di [F 0 [F 1 [F 2 g ; where i ;# i ; i 2 R + denotethenumberofnodesintheset N i M i and N f i ,respectively : Basedonthepartitionof F i ,Proposition3.1to3.8aresubsequentlyintroducedto ensurethatthedesignedfunctionin3isanavigationfunction.Proposition3.1shows thatnetworkconnectivityisensurediftheinitialgraphisconnected.Toestablishthe convergencepropertiesofthepotentialeld,Proposition3.2showsthesystemconverges tothesetofcriticalpointsunderthecontrollerin3,andProposition3.3to3.8 49

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togetherensurethat q di istheuniqueminimum,andtheothercriticalpointsaresaddle points,byprovingthat3hasthepropertiesofanavigationfunction.Proposition3.3 shows q di isaminimumin i .Proposition3.4showsthereisnominimaontheworkspace boundary.Proposition3.5ensuresallthecriticalpointscanbepushedawayfrom F 3 bychoosing bigenough.Proposition3.6showsthat i isaMorsefunction,while Proposition3.7and3.8indicatethatthecriticalpointsin F 0 F 1 and F 2 arenot minima.Proposition3.5to3.8togetherguaranteethat q di istheuniqueminimuminthe workspace.ThefollowingassumptionsareusedtoproveProposition3.6to3.8. Assumption3.3. Noobstaclesorotheragentsareassumedtostaywithinthecollision regionofnode i ,whennode i isveryclosetobreakingthecommunicationlinkwithanode j 2N f i i.e.,node i andnode j belongtotheregion B b i;j Assumption3.4. Theregion B B i;k for k 2N i [M i isdisjoint.Thisassumption impliesanegligibleprobabilityofmorethanonesimultaneouscollisionwithnode i 3.3.1ConnectivityAnalysis Proposition3.1. Ifthegraph G isconnectedinitiallyand j 2N f i ,then3ensures nodes i and j willremainconnectedforalltime. Proof. Proof: Considernode i locatedatapoint q 0 2F thatcauses Q j 2N f i b ij =0 ,which willbetruewheneitheronlyonenode j isabouttodisconnectfromnode i orwhenmore thanonenodeisabouttodisconnectwithnode i simultaneously.Thesetwopossibilities areconsideredinthefollowingtwocases. Case1:Thereisonlyonenode j 2N f i forwhich b ij q 0 ;q j =0 and b il q 0 ;q l 6 =0 8 l 2N f i l 6 = j .Thegradientof i withrespectto q i is r q i i = i r q i i )]TJ/F27 11.9552 Tf 11.955 0 Td [( i r q i i i + i 1 +1 : Since b ij =0 ,theconstraintfunction i =0 from3.Thus,thegradient r q i i evaluatedat q 0 canbeexpressedas r q i i j q 0 = )]TJ/F28 7.9701 Tf 10.494 6.604 Td [( i r q i i +1 i q 0 : Basedonthefactthat i can 50

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beexpressedastheproduct i = b ij b ij ,where b ij q 0 ;q j = B i 0 Y l 2N f i ;l 6 = j b il Y k 2N i [M i B ik ; and r q i b ij iscomputedas r q i b ij = 8 > < > : 0 d ij R c )]TJ/F26 7.9701 Tf 10.494 6.084 Td [(2 d ij + 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(R c q i )]TJ/F28 7.9701 Tf 6.586 0 Td [(q j 2 2 d ij R c )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 d ij R c ; thegradientof i evaluatedat q 0 canbeobtainedas r q i i j q 0 = )]TJ/F26 7.9701 Tf 10.494 6.084 Td [(2 b ij R c q i )]TJ/F27 11.9552 Tf 11.97 0 Td [(q j : Since K i ; i b ij and areallpositiveterms,3,and r q i i j q 0 canbeusedtodeterminethatthe controlleri.e.,thenegativegradientof r q i i isalongthedirectionof q j )]TJ/F27 11.9552 Tf 12.791 0 Td [(q i ; which impliesnode i isforcedtomovetowardnode j toensureconnectivity.Thatis,basedon thedesignof b ij in3anditsgradientin3,wheneveranodeenterstheescape regionofnode i; anattractiveforceisimposedonnode i toensureconnectivity. Case2 4 :Considertwonodes j;l 2N f i ,where b ij =0 and b il =0 i.e., k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k = R c and k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q l k = R c simultaneously.Inthiscase, i =0 and r q i i isazerovector,3 canbeusedtodeterminethat q 0 isacriticalpointi.e., r q i i j q 0 =0 ,andthenavigation functionachievesitsmaximumvalueatthecriticalpointi.e., i j q 0 =1 .Since i is maximizedat q 0 ,noopensetofinitialconditionscanbeattractedto q 0 underthecontrol lawdesignedin3. FromthedevelopmentinCase1andCase2,thecontrollawin3ensuresthatall nodes j 2N f i remainconnectedwithnode i foralltime. 4 Case2canbeextendedtomorethantwonodeswithoutlossofgenerality. 51

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3.3.2ConvergenceAnalysis Proposition3.2. Thesystemin3convergestothelargestinvariantseti.e.,theset ofcriticalpoints S = n q jr q i i j q =0 o underthecontrollerin3,providedthatthe tuningparameterin3satises > ; where =max q j c 1 j c 3 ; j c 2 j c 3 : Proof. ConsideraLyapunovcandidate V q = P N i =1 i ,where q isthestackedstatesof allnodes,i.e., q =[ q 1 ; ;q N ] T : Thetimederivativeof V iscomputedas V = r V T _q = )]TJ/F27 11.9552 Tf 9.298 0 Td [(K X N i =1 X N j =1 r q i i T r q i j ; whichcanbefurtherseparatedas V = )]TJ/F27 11.9552 Tf 9.299 0 Td [(K X i : r q i i =0 kr q i i k 2 + X j 6 = i r q i i T r q i j )]TJ/F27 11.9552 Tf 9.299 0 Td [(K X i : r q i i 6 =0 kr q i i k 2 + X j 6 = i r q i i T r q i j : Whenallnodesarelocatedatthecriticalpointsin3, V =0 .Toshowthatthesetof criticalpointsarethelargestinvariantset,itrequiresthat V isstrictlynegative,whenever thereexistsatleastonenode i suchthat r q i i 6 =0 : Since r q i i 6 =0 foratleastonenode, 3canberewrittenas V = )]TJ/F27 11.9552 Tf 9.299 0 Td [(K X i : r q i i 6 =0 kr q i i k 2 + X j 6 = i r q i i T r q i j : Toensurethat V< 0 in3,itissucienttorequirethat P j 6 = i r q i i T r q i j > 0 ; whichcanbeexpandedbyusing3as )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [( i r q i i )]TJ/F28 7.9701 Tf 13.151 5.256 Td [( i r q i i T i + i 1 +1 X j 6 = i )]TJ/F27 11.9552 Tf 5.48 -9.684 Td [( j r q i j )]TJ/F28 7.9701 Tf 13.151 6.084 Td [( j r q i j j + j 1 +1 > 0 : Since i ; i areallpositivefrom33and3,and i ; i cannotbezerosimultaneously fromAssumption3.2,theinequalityin3isvalidprovided i r q i i )]TJ/F27 11.9552 Tf 13.151 8.088 Td [( i r q i i T X j 6 = i j r q i j )]TJ/F27 11.9552 Tf 13.151 8.087 Td [( j r q i j > 0 ; 52

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whichcanbesimpliedas 1 2 c 1 + 1 c 2 + c 3 > 0 ; where c 1 = i r q i i T X j 6 = i j r q i j ; c 2 = )]TJ/F27 11.9552 Tf 9.298 0 Td [( i r q i i T X j 6 = i j r q i j )]TJ/F27 11.9552 Tf 11.955 0 Td [( i r q i i T X j 6 = i j r q i j ; and c 3 = i r q i i T X j 6 = i j r q i j : In3,since i and j arepositive,andnode i satises r q i i 6 =0 ;c 3 ispositivefrom 3.Usingthefactthat c 1 )-166(j c 1 j ;c 2 )-166(j c 2 j ,3canbewrittenas )]TJ/F15 11.9552 Tf 13.694 8.088 Td [(1 2 j c 1 j)]TJ/F15 11.9552 Tf 19.963 8.087 Td [(1 j c 2 j > )]TJ/F27 11.9552 Tf 9.299 0 Td [(c 3 ; whichsucestoshowthat > max q j c 1 j c 3 and > max n j c 2 j c 3 o : Therefore,if is chosensuchthat > ; where =max q j c 1 j c 3 ; j c 2 j c 3 ; thesystemconvergestothesetof criticalpoints. Proposition3.3. Thenavigationfunctionisminimizedatthedesiredpoint q di Proof. Thenavigationfunction i isminimizedatacriticalpointiftheHessianof i evaluatedatthatpointispositivedenite.Thegradientexpressionin3isusedto determineif q di isacriticalpoint.Fromthedenitionof q di and3,thegoalfunction evaluatedatthedesiredpointis i j q di =0 .Also,thegradientofthegoalfunction evaluatedatthedesiredpoint q di is r q i i j q di = X j 2N f i 2 q di )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j )]TJ/F27 11.9552 Tf 11.955 0 Td [(c ij =0 : 53

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Since i j q di =0 and r q i i j q di =0 ,3canbeusedtoconcludethat r q i i j q di =0 .Thus, thedesiredpoint q di intheworkspace F isacriticalpointof i .TheHessianof i is r 2 q i i = 1 i + i 1 +2 n i + i h r q i i r q i i T )-222(r q i i r q i i T + i r 2 q i i )]TJ/F27 11.9552 Tf 9.299 0 Td [( i r 2 q i i )]TJ/F27 11.9552 Tf 13.15 8.088 Td [( +1 [ i r q i i )]TJ/F27 11.9552 Tf 11.955 0 Td [( i r q i i ] )]TJ/F26 7.9701 Tf 6.587 0 Td [(1 i r q i i + r q i i T : Usingthefactsthat i j q di =0 and r q i i j q di =0 andtheHessianof i is r 2 q i i =2 i I 2 ; where I 2 istheidentitymatrixin R 2 2 ; theHessianof i evaluatedat q di isgivenby r 2 q i i q di =2 )]TJ/F18 5.9776 Tf 8.36 3.259 Td [(1 i I 2 i : Theconstraintfunction i > 0 atthedesiredcongurationbyAssumption3.2,and i isa positivenumber.Hence,theHessianof i evaluatedatthatpointispositivedenite. Proposition3.4. Nominimaof i areontheboundaryofthefreeworkspace F i Proof. Considerapoint q 0 2 @ F i .Fromthedenitionof @ F i theconstraintfunction i q 0 =0 .Thegoalfunction i iszeroonlyatthedesiredcongurationpoint,andfrom Assumption3.2,thedesiredcongurationcannotbeontheboundaryof F i .Thus,the goalfunction i evaluatedat q 0 isnotzero.Using3andthefactsthat i j q 0 =0 and i j q 0 6 =0 i j q 0 ismaximizedatanyarbitrarilychosenpoint q 0 ontheboundaryof F i Proposition3.5. Forevery "> 0 ,thereexistsanumber \050 suchthatif > \050 no criticalpointsof i arein F 3 Proof. From3,anycriticalpointmustsatisfy i r q i i = i r q i i : If ischosenas > sup i kr q i i k i kr q i i k ; 54

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where sup istakenover F 3 ,thenfrom3, i willhavenocriticalpointsin F 3 Since =inf b ij =inf B ik in F 3 ,anupperbound sup i k r q i i k i k r q i i k isgivenas sup i kr q i i k kr q i i k i \050 where \050 sup i kr q i i k X i j =1 ;j 6 = i sup kr q i b ij k + X i + # i k =0 ;k 6 = i sup kr q i B ik k : In3, kr q i b ij k kr q i B ik k and i k r q i i k areboundedtermsin F 3 from3,3 andthefactthat r q i B ik = 8 > < > : )]TJ/F26 7.9701 Tf 12.476 4.707 Td [(2 2 1 d ik + 2 1 q i )]TJ/F28 7.9701 Tf 6.586 0 Td [(q k d ik d ik < 1 0 d ik 1 : Proposition3.6. Thereexists 0 > 0 suchthatif "<" 0 ,then i isaMorsefunction. Proof. Thedevelopmentin[49]and[91]provesthatfor i tobeaMorsefunction,itis sucienttoshowthat ^ u T r 2 q i i q ci ^ u ispositiveforsomeparticularvector ^ u bychoosing asmall ,where q ci isacriticalpoint.Toshowthat ^ u T r 2 q i i q ci ^ u ispositivefortheunit vector ^ u q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k ; 3isusedandtheHessian r 2 q i i evaluatedat q ci is ^ u T r 2 q i i q ci ^ u i + i )]TJ/F18 5.9776 Tf 8.361 3.258 Td [(1 )]TJ/F26 7.9701 Tf 6.587 0 Td [(1 =^ u T i r 2 q i i + )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1 )]TJ/F26 7.9701 Tf 13.753 4.707 Td [(1 i i r q i i r q i i T )]TJ/F27 11.9552 Tf 11.955 0 Td [( i r 2 q i i ^ u Tofacilitatethesubsequentanalysis,thesetofcriticalpointsin F i isdividedintosets ofcriticalpointsinregions F 0 F 1 ,and F 2 .Foracasewhereacriticalpoint 55

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q ci 2F 2 ,usingthefactthat r q i i and r 2 q i i canbeexpressedas r q i i = b ij r q i b ij + b ij r q i b ij ; r 2 q i i = b ij r 2 q i b ij + )]TJ/F30 11.9552 Tf 5.479 -9.684 Td [(r q i b ij r T q i b ij + r q i b ij r T q i b ij + b ij r 2 q i b ij ; where b ij isdenedin3,andthefactthatthersttermontherighthandsideof 3isalwayspositivefrom3,thesubsequentexpressioncanbeobtainedas i + i 1 +1 ^ u T r 2 q i i q ci ^ u> i ; where = 1 b ij )]TJ/F27 11.9552 Tf 5.48 -9.683 Td [(a 1 b 2 ij + a 2 b ij + a 3 ; with a 1 = )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 r q i b ij 2 b ij )]TJ/F15 11.9552 Tf 12.685 0 Td [(^ u T r 2 q i b ij ^ u; a 2 = 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 r q i b ij T r q i b ij )]TJ/F15 11.9552 Tf 11.517 3.154 Td [( b ij ^ u T r 2 q i b ij ^ u )]TJ/F15 11.9552 Tf 12.685 0 Td [(^ u T )]TJ/F30 11.9552 Tf 5.48 -9.683 Td [(r q i b ij r T q i b ij + r q i b ij r T q i b ij ^ u; and a 3 = )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 b ij kr q i b ij k 2 : Since b ij > 0 ,anecessaryconditiontoshowthat > 0 istoprovethat a 1 b 2 ij + a 2 b ij + a 3 > 0 ; where a 3 > 0 if > 1 .Toprovetheinequalityin3,thefollowingtwocasesare analyzed. Case1:For a 1 < 0 ,theinequalityin3isvalidif b ij < )]TJ/F27 11.9552 Tf 9.298 0 Td [(a 2 )]TJ/F33 11.9552 Tf 11.955 10.394 Td [(p a 2 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(4 a 1 a 3 2 a 1 : 56

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Case2:For a 1 0 canberewrittenas a 2 + a 3 b ij ; whichispositiveif b ij < a 3 j a 2 j : Therefore, > 0 ,andfrom3, ^ u T r 2 q i i q ci ^ u> 0 forallcasesif b ij ischosenas b ij <" 0 0 min )]TJ/F27 11.9552 Tf 9.298 0 Td [(a 2 )]TJ/F33 11.9552 Tf 11.955 10.394 Td [(p a 2 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(4 a 1 a 3 2 a 1 ; a 3 j a 2 j : ByusingthesameprocessofevaluatingtheHessian r 2 q i i atcriticalpointsbelonging to F 0 and F 1 ,upperbounds 00 0 and 000 0 for canbeobtainedfor q ci 2F 1 and q ci 2F 0 respectively.Bychoosing "<" 0 =min 0 0 ;" 00 0 ;" 000 0 ; thefunction is guaranteedtobepositivewhichimpliesallthecriticalpointsarenon-degeneratecritical pointsof i Proposition3.7. Thereexists 1 > 0 ,suchthat i hasnolocalminimumin F 2 ,as longas "<" 1 Proof. Consideracriticalpoint q ci 2F 2 .Since i isaMorsefunction,thenif r 2 q i i q ci hasatleastonenegativeeigenvalue, i willhavenominimumin F 2 .Toshow r 2 q i i q ci hasatleastonenegativeeigenvalue,aunitvector ^ v r q i i kr q i i k ? isdenedasatestdirectiontodemonstratethat ^ v T r 2 q i i q ci ^ v< 0 ; where ? denotesavectorthatisperpendiculartosomevector .Using3and 3, i + i 1 +1 q ci ^ v T r 2 q i i q ci ^ v = )]TJ/F27 11.9552 Tf 9.299 0 Td [( i + b ij ; where =^ v T )]TJ/F30 11.9552 Tf 5.479 -9.683 Td [(r q i b ij r T q i b ij + r q i b ij r T q i b ij )]TJ/F15 11.9552 Tf 11.517 3.154 Td [( b ij r 2 q i b ij ^ v; 57

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=^ v T )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [( b ij r 2 q i i )]TJ/F27 11.9552 Tf 11.956 0 Td [( i r 2 q i b ij ^ v; and r 2 q i b ij = 8 > < > : 0 d ij R c )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 or d ij R c 2 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(R c q i )]TJ/F28 7.9701 Tf 6.586 0 Td [(q j q i )]TJ/F28 7.9701 Tf 6.586 0 Td [(q j T 2 2 d 3 ij )]TJ/F26 7.9701 Tf 13.151 6.084 Td [(2 d ij + 2 )]TJ/F28 7.9701 Tf 6.586 0 Td [(R c d ij 2 2 I 2 R c )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 0 .Toensure ^ v T r 2 q i i q ci ^ v< 0 ;" mustbe selectedas "<" 1 where 1 =inf F 2 j i j j j Proposition3.8. Thereexists 2 > 0 ; suchthat i hasnolocalminimumin F 1 and F 0 ; aslongas "<" 2 : Proof. Consideracriticalpoint q ci 2F 1 .SimilartotheproofforProposition3.7,the currentproofisbasedonthefactthatif ^ w T r 2 q i i q ci ^ w< 0 forsomeparticularvector ^ w q i )]TJ/F28 7.9701 Tf 6.586 0 Td [(q k k q i )]TJ/F28 7.9701 Tf 6.586 0 Td [(q k k ? ,then i willhavenominimumin F 1 .Tofacilitatethesubsequent analysis,similartothedenitionof b ij in3, i canbeexpressedastheproduct i = B ik B ik and B ik isdenedas B ik q i ;q k = B i 0 Y j 2N f i b ij Y l 2N i [M i ;l 6 = k B il : Using3,3and3, i + i 1 +1 q ci ^ w T r 2 q i i q ci ^ w = i B ik + i B ik ; where = r T q i B ik r q i i kr q i i k 2 i )]TJ/F15 11.9552 Tf 13.151 8.087 Td [(2 1 )]TJ/F27 11.9552 Tf 11.955 0 Td [(d ik d ik 2 1 ; =^ w T r T q i B ik r q i i kr q i i k r 2 q i i + )]TJ/F26 7.9701 Tf 13.753 4.707 Td [(1 ij r q i B ik r T q i B ik )-222(r 2 q i B ik ^ w; and r 2 q i B ik = 8 > < > : )]TJ/F26 7.9701 Tf 12.475 4.707 Td [(2 2 1 + 2 d ik 1 I 2 )]TJ/F26 7.9701 Tf 13.15 5.698 Td [(2 q i )]TJ/F28 7.9701 Tf 6.587 0 Td [(q k q i )]TJ/F28 7.9701 Tf 6.586 0 Td [(q k T 1 d 3 ik d ik < 1 0 d ik 1 : : 58

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Since d ik < 1 ,and r T q i B ik r q i i k r q i i k canbeupperboundedbyapositiveconstantin F 1 thenif d ik issmallenough, isguaranteedtobenegative.Hence,thereexistapositive scalar 21 = B ik d ik ; whichissmallenoughtoensure < 0 .Toensure ^ w T r 2 q i i q ci ^ w< 0 ; mustbeselectedas "< min f 21 ; inf F 1 B ik j j g : Let ^ x beanunitvectordenedas ^ x q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q 0 k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q 0 k ? : Thesameprocedurethatwasusedtoshow ^ w T r 2 q i i q ci ^ w< 0 in F 1 canbefollowedtoobtainanotherupperboundfor "; whichensures ^ x T r 2 q i i q ci ^ x< 0 in F 0 : Bychoosing 2 astheminimumoftheupperboundfor developedfor F 1 and F 0 i isensuredtohavenominimumin F 1 and F 0 aslongas "<" 2 BasedonPropositions3.2to3.8,if ischosensuchthat min f 0 ;" 1 ;" 2 g thenthe minimaacriticalpointisnotin F 0 F 1 F 2 F 3 ortheboundaryof F i .Thus, theminimahastobein F di if > max f 1 ; \050 ; g .Hence,nodesstartingfromany initialpositionsexceptfortheunstableequilibriawillconvergetothedesiredformation speciedbytheformationmatrix c ij 3.4Simulation Simulationresultsillustratetheperformanceoftheproposedcontrolstrategy.As shownintheFig.3-2,aconnectedinitialgraphof40nodeswithkinematicsin3are deployedwithdesiredneighborhoodinaworkspaceof R =30 mwithstaticobstacles. 59

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Figure3-2.Aconnectedinitialgraphwithdesiredneighborhoodintheworkspacewith staticobstacles. Eachnodeisassumedtohavealimitedcommunicationandsensingzoneof R c =3 m. Thesquaresanddotsdenotethemovingnodesandthestaticobstaclesrespectively, whilethesolidlineconnectingtwonodesrepresentsacommunicationlink,indicating thatthetwonodesarelocatedwithineachother'scommunicationandsensingzone. Thedesiredcongurationischaracterizedbyacirclewiththeinter-nodedistanceof 2 Thesystemissimulatedfor 600 s withthestepsizeof 0 : 1 .Thetuningparameter in 3issetas =1 : 5 ,and 1 = 2 =0 : 4 min3and3.ResultsinFig.3-3 indicatethatthesystemnallyconvergestothedesiredconguration.Fig.3-4shows theinter-nodedistancebetweennodesconvergestothedesiredvalue,andindicatesthat thecommunicationlinksaremaintainedduringtheevolutioni.e.,thedistancebetween connectednodesislessthan R c =3 m .Toshowtheconnectivityofthenetworkduring theevolution,theFiedlereigenvalueofthegraphLaplacianmatrixisplottedinFig.3-5. SincetheFiedlereigenvalueisalwayspositive,thegraphisconnected[35]. 60

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Figure3-3.Theachievednalconguration. Figure3-4.Theinter-nodedistanceduringtheevolution 61

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Figure3-5.TheplotoftheFiedlereigenvalueoftheLaplacianmatrixduringthe evolution.ThecircleindicatestheFiedlereigenvalueofthegraphateachtime instance. 3.5Summary Givenaninitialgraphwithadesiredneighborhood,anavigationfunctionbased decentralizedcontrollerisdevelopedtoensurethesystemasymptoticallyconvergestothe desiredcongurationwhilemaintainingnetworkconnectivityandavoidingcollisionswith otheragentsandobstacles.Adistinguishingfeatureofthedevelopedapproachisthat thedistributedagentsachieveacoordinatedglobalcongurationwithoutrequiringradio communication.Futureeortsarefocusedonenablingradio-silentnavigationfroman arbitrarilyconnecteddistributednetwork.Moreover,furthereortsarerequiredtoeliminateAssumption3.3sothatotherobstaclesoragentscanbewithinthecollisionregion ofnode i whennode i isabouttobreakthecommunicationlink.LikewiseAssumption3.4 becomeslesspracticalasapointgrowstoasphereinthepresenceofuncertainty,andas theworkspacebecomesmorecrowded.Futureworkisrequiredtoaddressthepervasive problemofobstacleavoidanceinaclutteredworkspacewithuncertainty. 62

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CHAPTER4 NETWORKCONNECTIVITYPRESERVINGFORMATIONRECONFIGURATION FORIDENTICALAGENTSFROMANARBITRARYCONNECTEDINITIALGRAPH AnavigationfunctionbaseddecentralizedmethodisdevelopedinChapter3to stabilizeagroupofagentsinadesiredformationwhilemaintainingnetworkconnectivity. However,anassumptioninChapter3isthattheinitialtopologyisrequiredtobe asupergraphofthedesiredtopologyensuringtheagentsareoriginallyinafeasible interconnectedstate.Suchinitialgraphassumptionmaylimititsapplicationswhich requireagentstoachievedesiredformationsfromanarbitraryinitialgraphordynamically changetheachievedformationstoadapttoanuncertainenvironment.Thefocusofthis chapteristocontrolagroupofidenticalagentstoachieveadesiredformationfroman arbitrarilyconnectedinitialcondition,whilepreservingnetworkconnectivityduringthe motion.Anovelnetworktopologylabelingalgorithmdevelopedinthework[67]isapplied inthischaptertodynamicallyspecifytheneighborhoodofeachagentintheinitialgraph accordingtothedesiredformation,anddeterminetherequiredmovementforallnodes toachievethedesiredformation.Onedistinguishingfeatureofthisapproachisthatwe donotwishtospecifywhichnodesintheinitialtopologywilltakewhichpositionsin thenaltopology;rather,weonlycarethatthereisanagentineachpositionspecied inthenaltopology.Bymodelingthenetworkconnectivityasanarticialobstacle,a navigationfunctionbasedcontrolschemedevelopedinourpreviousworkof[42]isapplied toguaranteethenetworkconnectivitybymaintainingtheneighborhoodamongagents determinedbytheprexlabelingalgorithm,andensuretheconvergenceofallagentsto thedesiredcongurationwithcollisionavoidanceamongagentsusinglocalinformation i.e.,localsensingandcommunication.Aninformationowisalsoproposedtospecify therequiredmovementforextraagentstotheirdestinationnodes.Theinformation ow-basedapproachgenerallyprovidesapathwithmorefreedomforthemotionofextra nodeswithoutpartitioningthenetworkconnectivityandallowscommunicationlinks 63

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tobeformedorbrokeninasmoothmannerwithoutintroducingdiscontinuity.Finally, convergenceisprovenusingRantzer'sDualLyapunovTheorem[72]. 4.1ProblemFormulation Consideranetworkcomposedof N agentsinworkspace F ,whereagent i moves accordingto q i = u i ;i =1 ; ;N where q i =[ x i y i ] T 2 R 2 denotesthepositionofagent i inatwodimensionalDplane, and u i 2 R 2 denotesthevelocityofagent i i.e.,thecontrolinput.Theworkspace F is assumedtobecircularandboundedwithradius R .Eachagentin F isassumedtobea point-masswithlimitedcommunicationandsensingcapabilityencodedbyalocaldisk area.Forsimplicityandwithoutlossofgenerality,thefollowingdevelopmentisbasedon theassumptionthatthesensingzoneisthesameasthecommunicationzone,bothwith radius R c .Twomovingagentscancommunicatewitheachotheriftherelativedistanceis lessthantheradius R c .Alltheagentsareassumedtobeidenticalwithequalactuation capabilities.Eachagentisassumedtohavereal-timeknowledgeofitsownstates. Sinceeachagentcanonlysenseandcommunicatewithotheragentslocated withinthedistance R c ,theinteractionamongagentsismodeledasanundirected graph G = V ; E t ,with V denotingtheindexsetofallnodesandthesetofedges E = f i;j 2VVj d ij R c g ; wherenode i and j representagentslocatedatposition q i and q j ; and d ij 2 R + isthedistancebetweenthem,denedas d ij = k q i )]TJ/F27 11.9552 Tf 11.956 0 Td [(q j k : Theedge i;j denotesabidirectionalcommunicationlinkbetweennode i and j; indicatingthatnode i and j haveaccesstothestatesofeachother : Theneighborhoodof node i; N i ; i.e.,alltheagentswithinthecommunicationzoneofagent i ,isgivenby N i = f j;j 6 = i j j 2V ; i;j 2Eg ; whichisatime-varyingset,sinceothernodesmayenter orleavethecommunicationregionofnode i atanytime.Thedesiredcongurationis characterizedbythespeciedrelativepositionbetweennodes i and j 2N f i ; where N f i denotesthesetofneighborsfornode i in G f ,andthedesiredposition q di fornode i is 64

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Figure4-1.Thesmalldiskareawithradius 1 denotesthecollisionregionandtheouter ringareadenotestheescaperegionsfornode i denedas q di = n q i jk q i )]TJ/F27 11.9552 Tf 11.956 0 Td [(q j )]TJ/F27 11.9552 Tf 11.955 0 Td [(c ij k 2 =0 ;j 2N f i o ; where c ij 2 R 2 representsthedesiredrelativepositionbetweennode i and j Acollisionregionisdenedforeachagent i asasmalldiskareawithradius 1
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Assumption4.1. Theinitialgraphassociatedwith G i isconnectedandtheinitial positionsdonotcoincidewithsomeunstableequilibriai.e.,saddlepoints. Assumption4.2. G f isatree,andallnodesin G i areassumedtohaveaknowledgeof G f beforeapplyingthenetworktopologylabelingalgorithms.Thedesiredformationassociated with G f isspeciedinadvanceandvalid,whichimpliesthatthedesiredcongurationis connectedandwillnotleadtoacollisionorthedesiredcongurationwillnotleadtoa partitionedgraph,i.e., 1 < k c ij k
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topologybymovingsomeagents.Thenovelideaisthattheagentscanbetreatedas packetsthatarethenroutedthroughthenetwork.Hence,theagentscanbemovedtoward itsdestinationasroutingapacketthroughthenetwork.Adistinguishingadvantage ofusingprexlabelingandroutingisthattopologyrecongurationcanbeachieved fromanarbitraryinitialtopology,sincethelabels,destinationsandneighborhoodfor agentsaredynamicallydetermined,insteadofbeingspeciedinadvancelikeresultssuch as[15,21,26,27,38,43,91].However,theprexlabelingandroutingapproachinSection 4.3requiresglobalcommunicationforeachagenttodetermineitsownlabelandbroadcast itslabeltootheragentsthroughtheoverallnetwork.Sincetheagentsarenotphysically movedinthisstage,theprexlabelingprocedurecanbedoneinano-linefashionfor thegiveninitialgraph.Oncetherststageiscomplete,thepotentialeldbasedmotion controlalgorithmdevelopedinSection4.4isthenappliedtosteertheagentstothe desiredformationwhilepreservingnetworkconnectivity.Notethatonlylocalsensingand communicationisusedfortheformationstabilizationinthesecondstage. 4.3NetworkTopologyLabelingAlgorithms Aprexlabelingandroutingalgorithmisdevelopedinthissectiontolabelthe agentswithprex,specifytheneighborhoodforeachnodein G i .Sincesomenodesi.e., extranodesarerequiredtoberepositionedtoachieve G f ,theproposedalgorithmalso determinesthedestinationandpathforeachextranodetoachieve G f whilepreserving networkconnectivity. 4.3.1BasicAlgorithm Atrie G t isgeneratedfrom G i byrandomlyselectinganodein G i tobetheroot andlettingtheselectedrootassignaprexlabeltoeachnodein G i inbreadth-rst fashion[13].Forinstance,inasimpleformofprexlabeling,onenodeiselectedtobethe root.Theneighborsoftherootarelabeledaschildrenoftherootinaprextree.Then thosenodeslabeltheirchildren,andtheprocesscontinuesuntileachnodehasaunique 67

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label 1 .Theprexlabelassignedtoeachnodeservesasitsnetworkaddress.Toillustrate theprexlabelingalgorithm,anexampleisprovidedFig.4-2,wheretherootisassigned thelabel 0 .Thechildrenoftherootarethenassignedthelabels f 01 ; 02 ; 03 g and f 01 ; 02 g inFig.4-2aandFig.4-2brespectively. Byidentifyingnodesin G t thatdonotmatchthedesiredtopology G f ,thenodethat ismissingchildrenisdenotedasa requestingnode ,andthenodewhoselabeldoesnot existin G f iscalledan extranode .Eachrequestingnodesendstherootamessage,M.Req, thatincludesalistofitsmissingchildrenaddresses.Acopyofthismessagewillalsobe storedatallnodeswhoarelocatedalongthepathtotheroot.Atthesametime,each extranodealsosendsamessage,M.Label,includingitsownlabeltotheroot.Oncethe rootobtainsallthemessagesfromthesenodes,therootwillaskeachextranodetomove one-by-onestartingfromanextranodewiththelongestlabeli.e.,theextranodethatis deepestinthetrie.Theextranodewiththelongestprexlabelismovedrsttopreserve thenetworkconnectivityduringtheprocess.Ifanextranodeisaparentofotherextra nodes,theotherextranodewillbeorphanedandthusmayloosenetworkconnectivityif theparentnodemovesrst. Inprexrouting,theextranoderstndsoutthedestinationnode'sprexlabel,and thenusesmaximumprexmatchinglogic[31,32,79]toroutethemselvestothedesired destinationthroughthenetwork.ThisextranodewillcheckforcachedM.Reqmessages inthenodeswhicharelocatedalongthewaytotherootforthemissingnodeaddress.If thereexistsmorethanonemissingnodeaddress,theextranodewilldecidetomoveto theclosestmissingnodebycomparingtheprexlabelofthemissingnodeaddressand thecurrentnodeaddress.Wheneverthisextranodereachesitsparentrequestingnode 1 Becauseoftheloopsinthegraph,thelabelingisnotunique.Thatis,foragiven graph,theremayexistseveraldierentsetsoflabeling.However,ineachindividualsetof labeling,eachagenthasauniqueprexlabel. 68

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in G f ,therequestingnodewillsendamessage,M.RCM,totherootconrmingthatit hasreceivedanextranode,andtheextranodewillberelabeledtotaketheroleofthe formerlymissingnodeinthedesiredtopology.Thismessagealsotellsallthenodesalong thepathtotheroottodeletethismissingnodeaddressfromtheircache.Whentheroot obtainsM.RCM,itwillinitiatetheprocessagainbyaskingtheextranodesatthenext levelinthetrietomove.Thisprocesswillcontinueuntilthedesirednetworktopologyis achieved. Fig.4-2isprovidedtoillustratetheprexroutingalgorithm.InFig.4-2aand b,therootisassignedthelabel 0 .Thechildrenoftherootarethenassignedthelabels f 01 ; 02 ; 03 g and f 01 ; 02 g inFig.4-2aandbrespectively.Othernodessuchas f 021 ; 022 ; 031 g and f 011 ; 012 ; 021 ; 022 g arelabeledbyfollowingasimilarprocedure.Thenode 03and031inFig.4-2aareextranodes,sincetheydonotexistintheFig.4-2bby comparingtheprexlabelsintheinitialandthedesiredtopologyinFig.4-2.Toachieve thedesiredtopologyshowninFig.4-2b,node031isrequiredtomoversttowardthe node01throughthepath{031,03,0,01},sincenode031hasthelongestlabelamongall theextranodes.Onceanewedgebetweennode01and031iscreated,031isrelabeledby 011,andfollowingasimilarprocess,node03thenstartstomovetowardnode01tollthe restvacancyof012inFig.4-2b. Thetimerequiredtotransitiontothenaltopologycanbereducedbyallowing someextranodestomovesimultaneously.InDestinationAssignmentAlgorithm,the rootassignseachextranodeadestinationbysendingthemamessageM.Dest,which includestherequestingnodeaddress.PseudocodeforDestinationAssignmentAlgorithm isprovidedinAlgorithm4.1.Inatreetopology,leafnodesarethosenodeswhodonot havechildren.Thus,leafnodescanberepositionedwithoutaectingnetworkconnectivity. Ifanextranodeisaleafnode,itcanmovewheneveritreceivesamessageM.Dest,but othernodeshavetowaitfortheirchildrentomoveupthetreebeforetheparentcan move.Anextranodethatreachesthedestinationrstwillbeimmediatelyrelabeledand 69

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Figure4-2.Theexampleofaninitiallyconnectedanddesiredgraphtopology,wherethe nodesdenotetheagent,andthelinesdenotetheavailablecommunication links. forwardedbytherequestingnodetofulllthedesirednetworktrie.Ifmorethanone extranodereachthedestinationrequestingnodeatthesametime,therequestingnode willcheckthehopnumberofeachextranodebycomparingtheextranodelabelwith itsprex.Thentherequestingnodewillassignthosenodestoitssubtreetominimize thenumberofhopseachnodemusttravel.Thatis,nodesthathavealreadytraveledthe largestnumberofhopswillbeassignedaschildrenoftherequestingnode,whereasextra nodesthathavetraveledfewhopswillbeassignedastheleafnodesofthedeepestpartof thesubtree. 4.3.2RelabelingAlgorithm Theperformanceofreorganizingthenetworktopologyishighlydependentonhow theprexlabelsareassignedin G i .Toimprovetheperformanceofreorganization,Branch RelabelingBRandNeighborRelabelingNRmethodsaredevelopedtorelabelthe nodesintrie G t .Beforeapplyingthesealgorithms, G i isrequiredtobeprex-labeledvia theBasicAlgorithmintheprevioussubsection,toformatrie G t .PseudocodeforBranch RelabelingandNeighborRelabelingareprovidedinAlgorithm4.2andAlgorithm4.3. 70

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Algorithm4.1 DestinationAssignment 1: procedure Input: extranodes extranodelabels 2: groups=GroupByPrexLengthextranodes; Startingfromthelongestlengthsgroup 3: for G 2 groups 3 G:length == max groups:length do 4: searchBackDepth=1; 5: for node 2 G do 6: node.DestinationAssigned=false; 7: endfor 8: while 9 node 2 G 3 node:DestinationAssigned == false do 9: for node 2 G 3 node:DestinationAssigned == false do Eachtimethroughthisloop,searchstartingatancestorthatis searchBackDepthtowardtheroot. 10: parent=node; 11: for i =1 to searchBackDepth do 12: parent=GetNodeParentparent; 13: endfor 14: requestingNode=SearchSubtreeForClosestMissingNodeparent; 15: if requestingNode 6 = ; then 16: node.DestinationAddress=requestingNode; 17: node.DestinationAssigned=true; 18: endif 19: endfor 20: searchBackDepth ++ ; 21: endwhile 22: groups=groups-G; 23: endfor 24: endprocedure 71

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4.3.2.1BranchRelabelingBRAlgorithm IntheBRalgorithm,thetotalamountofnodemovementrequiredtoachievethe desirednetworktopologycanbereducedbyswappingthenodesprexlabelsintwoor morebranches.Eachleafnodesendsamessageincludingitslabeltotheroot.Oncethe roothastheknowledgeofallthenodeprexlabelsinthenetwork,startingfromthe root,nodescanthenconsiderswappingtheprexlabelassociatedwithtwobranchesof descendants.Inthiswork,branchrelabelingisonlyappliedattheroot,sincetherootis mostlikelytohavethelargestimpactontheamountofmovementrequiredtoachievethe desiredtopology.AnexampleBRalgorithmisillustratedinFig.4-3,wheretheshaded nodesdenotetheextranodes,thesolidlinesindicatetherequirededgesinthedesired topologyinFig.4-2b,andthedashedlineindicatetheavailablecommunicationlinks butnotrequiredtomaintainforthedesiredtopology.BeforeapplyingtheBRalgorithm, bothnode03and031arerequiredtomoveallthewaytonode01toachievethedesired topology.AfterapplyingtheBRalgorithm,lessmovementisrequiredsinceonlynode03 isrequiredtomove. Algorithm4.2 BranchRelabelingAlgorithm 1: procedure Input: Node allnodeincludinglabelinthegraph 2: RootChild=GetRootChildNode; 3: RootChildSwap=AllSwapRootNeighbor; obtainallswappinglabelpatternofallrootneighbor 4: for p 2 RootChildSwap do 5: Nodetemp=BRLabelNode,p; Relabelallnodesineachbranchaccordingtop 6: HopNump=GetHopNumNodetemp; obtainthetotalhopnumberrequiredforp 7: endfor 8: p=MinHopHopNum; obtainpachievingminimumtotalhoprequired 9: Node=BRLabelNode,p; Relabeleachnodeaccordingtopachievingminimumtotalhoprequired 10: endprocedure 72

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Figure4-3.TheplotofagraphbeforeperformingBRalgorithmandafterperformingBR algorithm. 4.3.2.2NeighborRelabelingNRAlgorithm TheNRmethodisdesignedtorelabeltheextranodesbyexploitingadditional connectionsthatexistin G i toreducethenumberofnodesrequiredtomove.Allthe nodesrstcheckiftheyareextranodesandresetthelabelsofextranodesasnull. Oncetheroleofextranodesisidentied,allnodesthataremissingchildrenin G f will checkifsomeoftheextranodesaretheirneighborsin G i .Ifso,thenodethatismissing childrenwillrelabeltheneighboringextranodebysendingitamessage,M.Relabel.The processisrepeateduntilnomorerelabelingispossible.Then,allremainingextranodes willbelabeledbytheirneighborstoform G t .AnexampleNRalgorithmisillustrated inFig.4-4,wheretheshadednodesdenotetheextranodes,thesolidlinesindicatethe requirededgesinthedesiredtopologyinFig.4-2b,andthedashedlineindicatethe availablecommunicationlinksbutnotrequiredtomaintainforthedesiredtopology. GivenaninitialtopologywithprexlabelingasshowninFig.4-4a,twoextranodesare identied.AfterapplyingtheNRalgorithm,onlyoneextranodeisidentiedinFig.4-4 b. 73

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Figure4-4.TheplotofagraphbeforeandafterperformingNRalgorithm. Fordensenetworksinwhichthenumberofedgesishigh,NRmayoeradvantages overBRsinceNRcanexploitthemanynetworkconnections.Inmostcases,NRandBR canbecombinedtooerthegreatestbenetacrossinitialnetworktopologies. Algorithm4.3 NeighborRelabelingAlgorithm 1: procedure Input: Node allnodeincludinglabelinthegraph 2: RootChild=GetRootChildNode; 3: RootChildSwap=AllSwapRootNeighbor; obtainallswappinglabelpatternofallrootneighbor 4: for p 2 RootChildSwap do 5: Nodetemp=BRLabelNode,p; Relabelallnodesineachbranchaccordingtop 6: HopNump=GetHopNumNodetemp; obtainthetotalhopnumberrequiredforp 7: endfor 8: p=MinHopHopNum; obtainpachievingminimumtotalhoprequired 9: Node=BRLabelNode,p; Relabeleachnodeaccordingtopachievingminimumtotalhoprequired 10: endprocedure 4.4ControlDesign Section4.3describeshowtodeterminetheneighborhoodforeachnodein G i accordingtothedesired G f .Thissectionwilldiscussabouthowtophysicallymoveallnodesto 74

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achievethedesiredformationwithtopology G f ,whilemaintainingnetworkconnectivity. Informationowisintroducedtoindicatetherequiredcommunicationforextranodes.A potentialeldbaseddecentralizedcontrolstrategyisthendevelopedtoguidetheextra nodestotheirdestinations,andensureallnodesconvergetothedesiredformationwith collisionavoidanceamongnodeswhilemaintainingnetworkconnectivity. 4.4.1InformationFlow Assumethatthe G i and G f arealreadyprexlabeled.Toachievethedesired c ij in4for 8 i and j 2N f i ,aneighborhoodbetweennode i and j isrequired.Since theneighborhoodin G i isspeciedaccordingto G f ,mostnodesin G i startwithdesired neighborsi.e.,iftwonodesareneighborsintheinitialtopology,theyarealsoneighbors inthedesiredtopology.However,extranodesin G i arenotconnecteddirectlytotheir desiredneighbors,andextranodesarerequiredtomovetothedestinationnodeswhich aredeterminedbyAlgorithm4.1tobuildthedesiredneighborhood.Hence,itisnecessary fortheextranodetoalwayshaveaccesstothestatesofitsdestinationnode,andan informationow I ij isintroducedtoindicatetherequiredinformationexchangebetween extranode i anditsdestination j 2N f i Since G i isaconnectedgraphbyAssumption1,aninformationow I ij canbe realizedbyaseriesofnodesformingaconnectedpathconnectingtheextranode i andits destinationnode j 2N f i in G i .Ifthelengthof I ij istwowhichindicatesthatnode i and j areconnectedbyamutualneighbor,theconnectivityof I ij canbeensuredbymaintaining theconnectivityofnode i and j withthemutualneighborrespectively.Ifthelengthof I ij isgreaterthantwo,thisindicatesthatnode i and j areconnectedthroughmorethanone intermediatenodes.Theconnectivitybetweennodenode i and j isnotguaranteedbyjust maintainingtheconnectionwithitsimmediateneighbors,sincetheintermediatenodes haveapotentialtobreaktheexistingedgebetweenthemselves,resultinginthepartition ofnode i and j .Therefore,thefollowingdevelopmentisbasedontheassumptionthat thelengthof I ij isatmosttwo,whichisnotrestrictiveinthesensethatan I ij with 75

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pathlengthgreaterthantwocanbepartitionedintoseveralconnectedpartialpathi.e., I ik 1 ;I k 1 k 2 ; ;I k n j with k 1 ; ;k n denotingtheintermediatenodesof I ij withthelength ofeachsectionatmosttwo.Theextranode i canmoveinastep-by-stepfashionbyrst approachingnode k 1 ; thennode k 2 ; ;k n ; untilachievingitsdestinationnode j: Aninformationow I ij canberealizedbyseveraldierentpaths,wheretheinterest isnotonlymaintainingtheinformationow I ij ,butalsondingashortpathtoconnect i and j .Themutualnodeiscalledthe relaynode ,sinceitisusedtopassinformation betweennode i and j .Toindicatethefreedomofmotionthateachagentcantakewithout disconnectingthecommunicationlink,inspiredbytheworkof[86]and[87],alocally measurableedgerobustnessterm mn isdenedas mn = 1 2 R c )]TJ/F27 11.9552 Tf 11.955 0 Td [(d mn foranytwoimmediatenodes m and n inthegraph G i.e., m;n 2E .Theedge robustness mn isusedtomeasuretherobustnessoftheedge m;n ,sincenode m and n willremainconnectedwitheachother,unlessbothofthemaredisplacedbyadistance of mn .Therefore,alarger mn indicatesmorefreedomofmotion.Duetonodemotion, somenodesmayenterthecommunicationzoneofbothnode i and j atsometimeinstant foraninformationow I ij ,resultinginmultipleoptionsfortherelaynode.Using4, thelengthofthetwo-edgepath l ij isrepresentedas l ij = d ir + d rj =2 R c )]TJ/F15 11.9552 Tf 12.379 0 Td [(2 ir + rj ; where ir and rj aretherobustnessofeachcommunicationlink i;r and r;j computed from4respectively.Findingtheshortestpathfor I ij i.e.,minimizing l ij isequalto maximizingtheadditionof ir and rj ,since R c isaconstant.Pathrobustnessisdened as I ij = ir + rj ; andthegoalistominimizethetimedelayincommunicationby choosingtheshortestpath,andtherebymaximizingthepathrobustness.Basedonthe previousdiscussion,arelaynodeisdeterminedby r =argmax r 2N i N j I ij ; 76

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wherethemaximumtakenovertheintersectionofcommunicationneighbors, N i N j ; aimstondanodeprovidingtheshortestpathconnectingnode i and j .Motivationfor choosingtheadditionofedgerobustnessasthepathrobustness,insteadofchoosingthe minimumoftheedgerobustnesse.g.,[86]and[87]asthepathrobustness,istoavoid introducingadiscontinuityinthecontrolalgorithm. Theinformationow-basedapproachcanbeillustratedbyanexample,wherethe initialanddesiredtopologyareshowninFig.4-4bandFig.4-2brespectively.Since thelabelingalgorithmdevelopedinSection4.3ensuresthatallnodes,exceptforextra nodes,startwithinthedesiredneighborhoodi.e.,iftwonodesareneighborsintheinitial topology,theyarealsoneighborsinthedesiredtopologyintheinitialtopology,thenodes otherthanextranodesonlyneedtomaintainconnectivityofthedesiredneighborhood, whichisillustratedbythelabelednodesandsolidlineinFig.4-4bandFig.4-2b. FortheextranodewhichismarkedinFig.4-4bastheshadednode,theinformation ow I ij isusedtoindicatetherequiredcommunicationbetweentheextranodeandnode 01.Sincetheinformationexchangecanberealizedbyseveralpathsinthecommunication graphe.g.,throughthepath{0,01}or{02,01}orotherlongerpath,ashortpathcan beidentiedbychoosingarelaynodethatmaximizesthepathrobustnessin4.A navigationfunction-basedcontrolleristhendevelopedinthefollowingsectiontoensure theconnectivityoftherequiredcommunicationlinks,andensuresthesystemachievesthe desiredformation. 4.4.2NavigationFunction-BasedControlScheme Consideradecentralizednavigationfunctioncandidate i : F! [0 ; 1] fornode i as i = i i + i 1 = ; where 2 R + isatuningparameter, i : R 2 R + isthegoalfunction,and i : R 2 [0 ; 1] isaconstraintfunction. 77

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Thegoalfunction i in4drivesthesystemtoadesiredconguration,speciedin termsofthedesiredrelativeposewithrespecttotheinformationneighbor j 2N f i .The goalfunction isdesignedas i = X j 2N I i k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j )]TJ/F27 11.9552 Tf 11.956 0 Td [(c ij k 2 : ThegradientandHessianmatrixof i aregivenas r q i i = X j 2N I i 2 q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j )]TJ/F27 11.9552 Tf 11.955 0 Td [(c ij and r 2 q i i =2 I 2 i ; where I 2 istheidentitymatrixin R 2 2 ,and i 2 R + denotethenumberofinformation neighborsintheset N f i : SincetheHessianmatrixof i 4isalwayspositivedenite, thegoalfunction4 hasuniqueminimum,andtheminimumisreachedonlywhen r q i i =0 ; whichimpliesthat q i and q j achievesthedesiredrelativeposefrom4. Theconstraintfunction i in4isdesignedfornode i as i = B i 0 Y j 2N I i b r ij Y k 2N C i [M B ik ; toensureconnectivityofeveryinformationow I ij ; andcollisionavoidancewiththe workspaceboundary,adjacentnodesandmovingobstaclesateachtimeinstant.In4, b r ij b q i ;q r : R 2 [0 ; 1] ensuresconnectivityofaninformationow I ij i.e.,guarantees thattherelaynode r willalwaysbeconnectedtonode i andisdesignedas b r ij = 8 > > > > > > > < > > > > > > > : 1 d ir R c )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F26 7.9701 Tf 12.475 4.707 Td [(1 2 2 d ir +2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(R c 2 + 2 2 d ir +2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(R c R c )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2
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Node i isawareof rj and N j in4throughcommunicationwithnode j: Thus,the node r canbedeterminedlocallyfrom4.Alsoin4, B ik B q i ;q k : R 2 [0 ; 1] forpoint k 2N i ; ensuresthatnode i isrepulsedfromallnodeslocatedwithinitssensing zonetopreventacollision,andisdesignedas B ik = 8 > < > : )]TJ/F26 7.9701 Tf 12.475 4.707 Td [(1 2 1 d 2 ik + 2 1 d ik d ik < 1 1 d ik 1 : Similarly,thefunction B i 0 in4isusedtomodelthepotentialcollisionofnode i with theworkspaceboundary,wherethepositivescalar B i 0 2 R isdesignedsimilarto B ik with thereplacementof d ik by d i 0 ,where d i 0 2 R + istherelativedistanceofnode i tothe workspaceboundarydenedas d i 0 = R )-222(k q i k Basedonthedenitionofthenavigationfunctioncandidate,adecentralizedcontrollerforeachnodeisdesignedas u i = )]TJ/F27 11.9552 Tf 9.299 0 Td [(K i r q i i ; where K i isapositivegain,and r q i i isthegradientof i withrespectto q i ,givenas r q i i = i r q i i )]TJ/F27 11.9552 Tf 11.956 0 Td [( i r q i i i + i 1 +1 : In4and4, b r ij and B ik arebothdesignedtobecontinuousanddierentiablefunctionsin ;R c ; with b r ij achievingtheminimumwhenthecommunicationlink i;r isabouttobebrokene.g., d ir = R c and B ik achievestheminimumwhennodes i and k areabouttocollide.Theconstraintfunctiononlytakeseectwhenevernode i hasthepotentialtobreakanexistingcommunicationlinkorcollidewithothernodes. Thegradientof b r ij and B ik arethezerovectorinthefreemotionregion,i.e.,theinterval of 1 ;R c )]TJ/F27 11.9552 Tf 12.577 0 Td [( 2 ,asshowninFig.4-1,whichindicatesthatnode i isonlydrivenbyits goalfunction4 toformthedesiredrelativeposewithnode j 2N f i from4and 4.Ifnode i dynamicallybuildsnewcommunicationlinksorbreaksexistinglinksto 79

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theagentswithinthefreemotionregion,thecontrollerisstillcontinuousfrom4, since r q i i =0 and i =1 inthefreemotionregion.Incontrastwiththediscontinuity introducedintheswitchingtopologyincurrentliteratureseee.g.,[90],thishighlighted featureenablesasmoothtransitionbetweennode i andotherconnectednodes. 4.4.3ConnectivityandConvergenceAnalysis Thepreviousdevelopmentindicatesthat G isconnectediftheinformationow I ij is maintainedin G .Thefollowingproofindicatesthatthecontrollerin4canguarantee connectivityofinformationow I ij in G : Proposition4.1. Foranyinformationow I ij withnode r astherelaynode,thecontrol law4willguaranteethat I ij ismaintainedallthetime,thatisnode i and j are connectedinacommunicationpathin G Proof. Aninformationow I ij isrealizedinthecommunicationgraph G byapath fromnode i tonode j throughamutualnode r: Fromthedenitionofarelaynode, r 2N i N j ,whichmeansnode r islocatedinthecommunicationzoneofboth i and j: Toshowthattheedge i;r ismaintainedunderthecontrollaw4,considernode i locatedatapoint q 0 2F thatcauses b r ij =0 ,whichindicatesthatnode i isabout todisconnectwithnode r .Since b r ij =0 i =0 from4,thenavigationfunction achievesitsmaximumvaluefrom4.Since i ismaximizedat q 0 ; noopensetofinitial conditionscanbeattractedto q 0 underthenegatedgradientcontrollawdesignedin 4.Therefore,thecommunicationlinkbetweennode i and r ismaintainedbythe controller4.Followingthesameprocedure,theedge r;j canbemaintainedby asimilarcontrolappliedtonode j .Duetothemotionofthenodes,someothernode k mayprovideashorterpathconnectingnode i and j thannode r fromsometime instant.Whenthisoccurs,itisreasonabletocreateanewpathfromnode i tonode j throughnode k tomaintaintheinformationow I ij .Therelaynode k canbedetermined accordingto4anddoesnotintroduceadiscontinuity.Node k canbeswitchedfrom 80

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node r inthefreemotionregionofbothnode i and j: Followingtheanalysisabove,the connectivityofthenewpathcanalsobeguaranteed. 4.4.4ConvergenceAnalysis Ourpreviousworkin[43]provesthattheproposed i in4isaqualiednavigationfunction,whichguaranteesconvergenceofthesystemtothedesiredconguration. From[43],thecontrollaw4ensuresthatalmostallinitialconditionsareeither broughttoasaddlepointortotheuniqueminimum q di onacompactconnectedmanifold withboundary,aslongasthetuningparameter in4satisesthat > max f 1 ; \050 g ; where \050 isdevelopedin[43].ThefollowingdevelopmentusesRantzer'sDualLyapunov Theorem[72]toshowthattheundesiredcriticalpointsi.e.,saddlepointsareallmeasure zero,andthesystemcanonlyconvergetotheuniqueminimum q di : Forthebounded workspaceinthiswork,avariationofRantzer'sDualLyapunovTheoremisstatedas[19]: Theorem4.1. Suppose x =0 2 S where S isanopen,positivelyinvariant,bounded subsetof R n isastableequilibriumpointfor x t = f x t ; where f 2 C 1 S; R n ; f =0 : Furthermore,supposethereexistsafunction 2 C 1 S )-284(f 0 g ; R suchthat x f x = k x k isintegrableon f x 2 S : k x k 1 g and [ r f ] > 0 foralmostall x 2 S: Then,foralmostallinitialstates x 2 S; thetrajectory x t existsfor t 2 [0 ; 1 and tendstozeroas t !1 2 Theorem4.1requires x =0 2 S tobeastableequilibriumpoint.From4and 4,thegoalfunctionevaluatedatthedesiredpointis i j q di =0 ,and r q i i j q di =0 from 4 ; whichcanbeusedtoconcludethat r q i i j q di =0 from4.Thus,thedesired point q di intheworkspace F isacriticalpointof i .Usingthefactsthat i j q di =0 and 2 Forafunction f : R n R n ,thenotationofdivergenceisdenedas r f = @f 1 @x 1 + + @f n @x n 81

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r q i i j q di =0 andtheHessianof i is r 2 q i i =2 i I 2 from4 ; theHessianof i evaluated at q di isgivenby r 2 q i i q di =2 )]TJ/F18 5.9776 Tf 8.36 3.258 Td [(1 i I 2 i .Theconstraintfunction i > 0 atthedesired congurationbyAssumption2,and i isapositivenumber.Hence,theHessianof i evaluatedat q di ispositivedenite,andthenavigationfunction i isminimizedat q di Proposition4.2. Theclosed-loopkinematicsofsystem4withthecontroller4 aregivenby _q = f q ; where q denotesthestackedstatesofeachnodeas q = q T 1 q T N T and f q = f T 1 f T N with f T i = )]TJ/F27 11.9552 Tf 9.299 0 Td [(K i r q i i for 8 i 2N .Considerthesystem _q = f q for 8 i 2N ,andadensityfunctionas = )]TJ/F27 11.9552 Tf 9.299 0 Td [('; where = P N i =1 i inTheorem1.If thereexistsan 0 > 0 suchthat4issatised,theundesiredcriticalpointsaresetsof measurezerofromTheorem1,provided > max f 1 ; \050 ;" 0 g atanysaddlepoints ; where isaparameterinthenavigationfunction4. Proof. Thefunction isdenedforallpointsintheworkspaceotherthanthedesired equilibrium q di ; andeach i is C 2 andtakesavaluein [0 ; 1] : Thusboththefunction anditsgradientareboundedfunctionsintheworkspace,whichindicatesthatthe integrabilityconditioninTheorem1isfullled.Fromthedivergencecriterion, r f = r T f + r f ; andfromthedenitionofacriticalpoint, r q i i =0 : Hence, f T i = )]TJ/F27 11.9552 Tf 9.298 0 Td [(K i r q i i =0 for 8 i 2N ,whichindicatesthat f =0 ,and r f canbesimplied as r f = X N i =1 K i @ 2 i @x 2 i + @ 2 i @y 2 i : Since arepositiveatundesiredcriticalpointsfrom4,and K i isapositivegain,a sucientconditionfor4tobestrictlypositiveis @ 2 i @x 2 i + @ 2 i @y 2 i > 0 : Using4, @ 2 i @x 2 i and @ 2 i @y 2 i arecomputedas @ 2 i @x 2 i = @ i @x i @ i @x i + i @ 2 i @x 2 i )]TJ/F26 7.9701 Tf 13.753 4.707 Td [(1 @ i @x i @ i @x i )]TJ/F28 7.9701 Tf 13.151 5.255 Td [( i @ 2 i @x 2 i i + i 1 +1 @ 2 i @y 2 i = @ i @y i @ i @y i + @ 2 i @y 2 i )]TJ/F26 7.9701 Tf 13.753 4.707 Td [(1 @ i @y i @ i @y i )]TJ/F28 7.9701 Tf 13.151 5.256 Td [( i @ 2 i @y 2 i i + i 1 +1 : 82

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Observingthat @ 2 i @x 2 i and @ 2 i @y 2 i hassimilarstructure,itsucestoshowthat @ 2 i @x 2 i > 0 for 8 i 2N ,sincethesameresultscanbederivedfor @ 2 i @y 2 i .Since i and i arepositivefrom 4and4,andcannotbezerosimultaneouslyfromAssumption2,thepositivity of4canbeprovenbyshowingthatthenumeratoroftherightsideof4is positive.Usingthefactthat @ i @x i = i @ i @x i atacriticalpoint,thefollowingexpressioncan beobtainedfrom4: C 1 2 + C 2 + C 3 > 0 : where, C 1 = i i @ i @x i 2 ;C 2 = i i i @ 2 i @x 2 i )]TJ/F33 11.9552 Tf 11.955 13.271 Td [( @ i @x i 2 and C 3 = )]TJ/F28 7.9701 Tf 10.494 5.256 Td [( i @ 2 i @x 2 i .Notethat i =0 indicates i achievesitsmaximumfrom4.However,sincethesetofinitialconditions isopen,andnoopensetofinitialconditionscanbeattractedtothemaximaof i along thenegativegradientmotion )]TJ/F27 11.9552 Tf 9.299 0 Td [(K i r q i i [21],then i 6 =0 : Inaddition, i isevaluatedat theundesiredcriticalpointsi.e.,exceptthe q di ,so i 6 =0 and @ i @x i 6 =0 from4and 4.Tosatisfytheconditionin4,twocasesareconsideredfor C 1 2 + C 2 + C 3 =0 : Case1:Nosolutionof existsfor4 : Since i i @ i @x i 2 > 0 ; whichmeans can bearbitraryvalue.Notethat isapositivegainin4.Hence,aslongas > 0 ; the conditionin4isvalidinCase1. Case2:Twosolutions, S 1 and S 2 ; existfor in4:Inthiscase,thecondition in4issatisedaslongas > max f S 1 ;S 2 ; 0 g .CombiningCase1andCase2, indicatesthatif > max f 1 ; \050 ;" 0 g ; where 0 isdenedas 0 =max f S 1 ;S 2 ; 0 g ; allsaddlepointsaremeasurezero,andthesystemwillonlyconvergetothedesired conguration. 4.5Simulation Inthissection,theproposedstrategyissimulatedforagroupof10nodestoachieve adesiredletterAformationshowninFig.4-5afromanarbitrarilyconnectedinitial graph.InFig.4-5a,thecirclesrepresentthenodesandthesolidlinesindicatethe 83

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Figure4-5.Theplotofdesiredformationwithandwithoutprexlabels. desiredcommunicationlinksbetweenconnectednodes,andtheprexlabeleddesired graphisshowninFig.4-5bafterapplyingthelabelingalgorithmsinSection4.3. AnarbitraryconnectedinitialgraphisgeneratedinFig.4-7a.Notethattheinitial topologyinFig.4-7aisnotasupergraphofthedesiredtopologyinFig.4-5a.The existingresultsin[21,40,43]arenotapplicabletothecurrentexample.Whenapplyingthe networktopologylabelingalgorithmdescribedinSection4.3,theinitialtopologycanbe labeledasinFig.4-7b.ComparingFig.4-5bandFig.4-7b,allnodeshavedesired neighborsexcepttwonodes:node011ismissingachild,andnode011111doesnotexist inFig.4-5b.Node011111isidentiedasanextranodeandrequiredtomoveupthe trietowarditsdestinationnode011tollthevacancyofnode0112.Applyingthecontrol strategydevelopedinSection4.4,thedesiredformationisachievedasshowninFig.4-7. 4.6Summary Cooperativeformationcontrolforagroupofidenticalagentswithlimitedsensingand communicationcapabilitiesisachievedinthiswork.Throughtheuseofaprexlabeling strategy,theinitialtopologyassumptioni.e.,requiredtobeasupergraphthedesired topologypresentedinmostpreviousworkisrelaxed,whichenablesidenticalagentsto 84

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Figure4-6.Theplotofinitialgraphwithandwithoutprexlabels. Figure4-7.Thetrajectoriesofallnodestoachievethedesiredformation,with"*" denotingtheirinitialpositionsandcirclesdenotingtheirnalpositions. 85

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achieveadesiredformationfromanyconnectedinitialgraph,anddynamicallychange theirformationtoadapttoanuncertainenvironment.Adecentralizedcontrolmethod usinglocalinformationisdevelopedtoguaranteetheagentscooperativelyconvergeto thedesiredformationwithoutdisconnectingtheunderlyingnetworkgraphwithcollision avoidanceamongagents.Futureworkwillfocusonextendingtheproposedapproachto theapplicationofnon-homogeneousagents,andoptimizingtheinter-agentcommunication duringtheoperation. 86

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CHAPTER5 ENSURINGNETWORKCONNECTIVITYFORNONHOLONOMICROBOTS DURINGDECENTRALIZEDRENDEZVOUS ArticialpotentialeldbasedcontrollersaredevelopedinChapter2-4toperform desiredtaskswhilepreservingthenetworkconnectivityduringthemission.However, onlylinearmodelsofmotionaretakenintoaccountinChapter2-4,i.e.,therstorder integrator.Thefocusofthischapteristodesigndecentralizedcontrollersforagroup ofwheeledmobilerobotswithnonholonomicconstraints.Basedonourpreviouswork in[41],adecentralizedcontinuoustime-varyingcontroller,usingonlylocalsensing feedbackfromitsone-hopneighbors,isdesignedin[45]tostabilizeagroupofwheeled mobilerobotswithnonholonomicconstraintsataspeciedcommonsetpointwitha desiredorientation.Onefeatureofthecontrollerdevelopedinthischapteristhatit considersacooperativeobjectiveofmaintainingthenetworkconnectivityduringnetwork regulation.Anotherfeatureofthedecentralizedcontrolleristhat,usinglocalsensing information,nointer-agentcommunicationisrequiredi.e.,communication-freeglobal decentralizedgroupbehavior.Thatis,networkconnectivityismaintainedsothatthe radiocommunicationisavailablewhenrequiredforvarioustasks,butcommunicationis notrequiredfornavigation.Usingthenavigationfunctionframework,themulti-robot systemisguaranteedtomaintainconnectivityandbestabilizedatacommondestination withadesiredorientationwithoutbeingtrappedbylocalminima.Moreover,thisresult canbeextendedtootherapplicationsbyreplacingtheobjectivefunctioninthenavigation functiontoaccommodatedierenttasks,suchasformationcontrol,ocking,andother applications. 5.1ProblemFormulation Consideranetworkedmulti-robotsystemcomposedof N WheeledMobileRobots WMRsoperatinginaworkspace F ,where F isaboundeddiskareawithradius R w ; and @ F denotestheboundaryof F : Intheworkspace F ,eachrobotmovesaccordingto 87

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thefollowingnonholonomickinematics: q i = 2 6 6 6 6 4 cos i 0 sin i 0 01 3 7 7 7 7 5 2 6 4 v i t i t 3 7 5 ;i =1 ; ;N where q i t p T i t i t T 2 R 3 denotesthestatesofrobot i; with p i x i t y i t T 2 R 2 denotingthepositionofrobot i ,and i 2 )]TJ/F27 11.9552 Tf 9.298 0 Td [(; ] denotingits orientationwithrespecttotheglobalcoordinateframeintheworkspace F .In5, v i t ;! i t 2 R arethecontrolinputs,representingthelinearandangularvelocityof robot i; respectively. Assumethateachrobothasalimitedcommunicationandsensingcapabilityencoded byadiskareawithradius R c and R s respectively,and R c R s ,whichensuresthattwo robotsareabletocommunicatewitheachotheraslongastheycansenseeachother.For simplicityandwithoutlossofgenerality,itisassumedthatthesensingareacoincides withthecommunicationareai.e., R c = R s = R inthefollowingdevelopment.Two movingrobotscancommunicatewithandsenseeachotheraslongastheystaywithina distanceof R: Further,alltherobotsareassumedtohaveequalactuationcapabilities. TheinteractionamongtheWMRsismodeledasanundirectedgraph G t = V ; E t where V = f 1 ; ;N g denotesthesetofnodes,and E t = f i;j 2VVj d ij R g denotesasetoftime-varyingedges.Ingraph G t ,eachnode i 2V representsarobot locatedataposition p i ,andanundirectededge i;j 2E existsbetweennode i and j in G t iftheirrelativedistance d ij k p i )]TJ/F27 11.9552 Tf 11.955 0 Td [(p j k2 R + islessthan R ,whichindicatesthat node i and j areabletoaccessthestatesi.e.,positionandorientationofeachother throughlocalsensingandinformationexchange.Theneighborsetofnode i isdenoted as N i = f j j i;j 2Eg ,whichincludesthenodesthatcanbesensedandreachedfor communication.Sincethegraph G t isundirected, i 2N j j 2N i for 8 i;j 2V i 6 = j .Duetothelimitedsensingandcommunicationcapabilities,node i onlyknowsthe 88

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statesofthosenodeswithinitssensingrageandcanonlycommunicatewithnodeswithin itscommunicationrange.Oncenode j movesoutofthesensingandcommunication zoneofnode i; node i willnolongershareinformationwithnode j directly,whichmay leadtomissionfailure.Hence,tocompletedesiredtasks,maintainingconnectivityofthe underlyinggraphisnecessary. Themainobjectivesinthischapteraretoderiveasetofdistributedcontrollersusing onlylocalinformationi.e.,thestatesoftheotherrobotswithinitssensingareatolead thegroupofWMRstorendezvousatacommondestination p withadesiredorientation i ; i.e., q i = p T i T 8 i intheworkspace F ; whileguaranteeingtheunderlying graph G t remainsconnectedduringthesystemevolution,providedthegiveninitial graphisconnected.Toachievethesegoals,thefollowingassumptionsarerequiredinthe subsequentdevelopment. Assumption5.1. Theinitialgraph G isconnected,andthoseinitialconditionsdonot coincidewithunstableequilibriai.e.,saddlepoints. Assumption5.2. Thedestination p anddesiredorientation i isknownforeachrobot andachievable,whichindicatesthatthedestinationwillnotmeetanyconstraints,i.e., coincidewiththeworkspaceboundary,orleadtothepartitionoftheunderlyinggraph. 5.2ControlDesign 5.2.1DipolarNavigationFunction Articialpotentialeld-basedmethodsthatuseattractiveandrepulsivepotentials havebeenwidelyusedtocontrolmulti-robotsystems.Duetotheexistenceoflocal minimawhenattractiveandrepulsiveforcearecombined,robotscanbetrappedbylocal minimaandarenotguaranteedtoreachtheglobalminimumofthepotentialeld.A navigationfunctionisaparticularcategoryofpotentialfunctionswherethepotentialeld doesnothavelocalminimaandthenegativegradientvectoreldofthepotentialeld guaranteesconvergencetoadesireddestination,alongwithpossiblecollisionavoidance. Formally,anavigationfunctionisdenedas: 89

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Denition5.1. [49][76],Let F E n beacompactconnectedanalyticmanifoldwith boundary.Amap : F! [0 ; 1] isaNavigationFunction,ifitis:1smoothon F atleast a C 2 function;2admissibleon F ,uniformlymaximalon @ F andconstraintboundary; 3polaron F ; q d isauniqueminimum;and4aMorsefunction,criticalpointsofthe navigationfunctionarenon-degenerate. Specically,property2establishesthatthegeneratedtrajectoriesarecollision-free, sincetheresultingvectoreldistransversetotheboundaryof F ,whileproperty3 indicatesthat,usingapolarfunctiononacompactconnectedmanifoldwithboundary, allinitialconditionsareeitherbroughttoasaddlepointortotheuniqueminimum q d Thelastpropertyensuresthattheinitialconditionsthatbringthesystemtosaddlepoints aresetsofmeasurezero[49].Giventhisproperty,allinitialconditionsawayfromsetsof measurezeroarebroughttotheuniqueminimum. Thenavigationfunctionintroducedin[49]and[76]ensuresglobalconvergenceof theclosed-loopsystem;however,theapproachisnotsuitablefornonholonomicsystems, sincethefeedbacklawgeneratedfromthegradientofthenavigationfunctioncanleadto undesiredbehavior.Toovercometheundesiredbehaviors,theoriginalnavigationfunction wasextendedtoaDipolarNavigationFunctionin[92]and[93],wheretheowlines createdinthepotentialeldresembleadipole,sothattheowlinesarealltangentto thedesiredorientationattheoriginandthevehiclecanachievethedesiredorientation. OneexampleofthedipolarnavigationisshowninFig.5-1,wherethepotentialeldhas auniqueminimumatthedestinationi.e., p =[0 ; 0] T and =0 ,andachievesthe maximaattheworkspaceboundaryof R w =5 .Notethatthesurface x =0 dividesthe workspaceintotwoparts,andforcesalltheowlinestoapproachthedestinationparallel tothey-axis. Tomaintainnetworkconnectivityandnavigatetherobotstoacommondestination withadesiredorientationusinglocalinformation,thedipolarnavigationfunctionin[92] 90

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Figure5-1.Theplotofanexampledipolarnavigationfunction. Anexampleofadipolarnavigationfunctionwithworkspaceof R w =5 ,andthe destinationlocatedattheoriginwithadesiredorientation =0 and[93]ismodiedforeachnode i as i : F! [0 ; 1] ; i = i i + H i i 1 = ; where 2 R + isatuningparameter.Thegoalfunction i t : R 2 R + in5encodes thecontrolobjectiveofachievingthedesireddestinationfornode i ,speciedbythe distancefrom p i t 2 R 2 tothecommondestination p 2 R 2 ; whichisdesignedas i = k p i t )]TJ/F27 11.9552 Tf 11.955 0 Td [(p k 2 : Thefactor H i t 2 R + in5createstherepulsivepotentialofanarticialobstacle toalignthetrajectoriesatthedestinationwiththedesiredorientation.Therepulsive potentialfactorisdesignedas H i = nh + Y N i =1 i ; 91

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where nh isasmallpositiveconstant,and i t 2 R + isdesignedas i = p i )]TJ/F27 11.9552 Tf 11.955 0 Td [(p T n di 2 ; where n di 2 R 2 isdesignedas n di = cos i sin i T : Toensureconnectivityoftheexistinglinksbetweentwonodesandrestrictthemotion ofeachnodeinthespeciedworkspace,theconstraintfunction i : R 2 N [0 ; 1] in5 isdesignedas i = B i 0 Y j 2N i b ij : Specically,theconstraintfunctionin5isdesignedtovanishwhenevernode i meets oneoftheconstraintsintheworkspace,i.e.,ifnode i touchestheworkspaceboundary,or departsfromitsneighbornodes j 2N i toadistanceof R .Asmalldiskareawithradius 1 < > : )]TJ/F26 7.9701 Tf 12.476 4.707 Td [(1 2 1 d 2 i 0 + 2 1 d i 0 ;d i 0 < 1 1 ;d i 0 1 ; where d i 0 R w )-249(k p i k2 R istherelativedistanceofnode i totheworkspaceboundary. Toensureconnectivityoftheunderlyinggraph,anescaperegionforeachnodeisdened astheouterringofthesensingandcommunicationareawithradius r;R )]TJ/F27 11.9552 Tf 12.328 0 Td [( 2
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isinitiallyconnectedtonode i andisdesignedas b ij = 8 > > > > > > > < > > > > > > > : 1 d ij R )]TJ/F27 11.9552 Tf 11.955 0 Td [( 2 )]TJ/F26 7.9701 Tf 12.475 4.707 Td [(1 2 2 d ij +2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(R 2 + 2 2 d ij +2 2 )]TJ/F27 11.9552 Tf 11.955 0 Td [(R R )]TJ/F27 11.9552 Tf 11.956 0 Td [( 2
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di remainscontinuousalonganyapproachingdirectiontothegoalposition.Basedonthe denitionof di in5 r i i = )-167(kr i i k cos di sin di T ; where r i i = @' i @x i @' i @y i T denotesthepartialderivativeof i withrespectto p i and kr i i k denotestheEuclideannormof r i i .Thedierencebetweenthecurrent orientationandthedesiredorientationforrobot i ateachtimeinstantisdenedas ~ i t = i t )]TJ/F27 11.9552 Tf 11.956 0 Td [( di t ; where di t isgeneratedfromthedecentralizednavigationfunctionin5and5. Since i in5isaqualiednavigationfunction,thepropertiesofanavigationfunction guaranteesthat q di t q i as t !1 .Hence,toachievethenavigationcontrolobjective, i t musttrackthedesiredorientation di t Basedontheopen-loopsystemin5andthesubsequentstabilityanalysis,the controllerforeachroboti.e.,thelinearandangularvelocityofrobot i isdesignedas v i = k v kr i i k cos ~ i ; i = )]TJ/F27 11.9552 Tf 9.299 0 Td [(k w ~ i + di ; where k v k w 2 R + .Theterms r i i and di in5and5aredeterminedfrom 5and5as r i i = H i i r i i )]TJ/F27 11.9552 Tf 11.955 0 Td [( i r i H i i i + H i i 1 +1 ; where r i i and r i H i i areboundedintheworkspace F from5and5,and di = k v cos ~ i 2 6 4 sin di )]TJ/F15 11.9552 Tf 11.291 0 Td [(cos di 3 7 5 T r 2 i i 2 6 4 cos i sin i 3 7 5 ; 94

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where r 2 i i denotestheHessianmatrixof i withrespectto p i .Substituting5into 5,theclosed-loopsystemforrobot i canbeobtainedas p i = 2 6 4 x i y i 3 7 5 = k v kr i i k cos ~ i 2 6 4 cos i sin i 3 7 5 : Afterusingthefactthat cos i sin i r i i = )-167(kr i i k cos ~ i ; from5,theclosed-looperrorsystemsin5canbeexpressedas p i = )]TJ/F27 11.9552 Tf 9.298 0 Td [(k v r i i : 5.3ConnectivityandConvergenceAnalysis 5.3.1ConnectivityAnalysis Theorem5.1. Givenaninitiallyconnectedgraph G composedofnodeswithkinematics givenby5,thecontrollerin5and5ensurethegraphremainsconnected. Proof. Considernode i locatedatapoint p 0 2F thatcauses Q j 2N i b ij =0 ,whichwill betruewheneitheronlyonenode j isabouttodisconnectfromnode i orwhenmultiple nodesareabouttodisconnectwithnode i simultaneously.From5, i =0 ,which indicatesthatthenavigationfunctiondesignedin5achievesitsmaximumvalue.From thenegativegradientof i in5,noopensetofinitialconditionscanbeattractedto themaximaofthenavigationfunction[76].Therefore,theexistingedgebetweennode i andnode j 2N i willbemaintainedforalltime. 5.3.2ConvergenceAnalysis Theorem5.2. Givenaninitiallyconnectedgraph G composedofnodeswithkinematics givenby5,thecontrollerin5and5ensuretherobotsconvergestoa commonpointwithadesiredorientation,inthesensethat k p i t )]TJ/F27 11.9552 Tf 11.955 0 Td [(p k! 0 and ~ i t 95

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0 as t !18 i 2N ,providedthatthetuningparameter in5issucientlarge, > Proof. ConsideraLyapunovfunctioncandidate V P t = X N i =1 i ; where P t isthestackedpositionstatesofallnodes, P t = p T 1 t p T N t T ,and i istheassociatednavigationfunctionfornode i designedin5.Thetimederivative of V is V = X N i =1 r i 1 T p i + + X N i =1 r i N T p i = X N i =1 X N j =1 p T i r i j ; whichcanbefurtherseparatedas V = X i : r i i =0 p T i r i i + X j 6 = i p T i r i j + X i : r i i 6 =0 p T i r i i + X j 6 = i p T i r i j ; where r i j ; r i i 2 R 2 denotethepartialderivativeof j and i withrespectto p i respectively. Toshowtheobjectiveof k p i )]TJ/F27 11.9552 Tf 11.955 0 Td [(p k! 0 8 i 2N ,thesetofcriticalpoints, S = f p i jr i i =0 for 8 i 2Ng mustbeshowntobethelargestinvariantsetofthestackedclosed-loopsystemof5. Whenallnodesarelocatedatthecriticalpointsi.e.,thepositionofnode i satisfying r i i =0 in5, V =0 ; since p i =0 from5.Fornode i notlocatedatthecritical pointsi.e., r i i 6 =0 ,5canberewrittenas V = X i : r i i 6 =0 p T i r i i + X j 6 = i p T i r i j : 96

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Toshowthatthesetofcriticalpointsisthelargestinvariantset, V mustbestrictly negativewheneverthereexistsatleastonenode i suchthat r i i 6 =0 ,forwhichitis sucienttoshowthat p T i r i i + X j 6 = i p T i r i j < 0 : Substituting5and5into5,yields k v r i i T r i i + k v X j 6 = i r i i T r i j > 0 ; whichcanbesimpliedas 1 2 c 1 + 1 c 2 + c 3 > 0 ; where c 1 = k v i r T i H i i X j 6 = i j r i H j j ; c 2 = )]TJ/F27 11.9552 Tf 9.299 0 Td [(k v H i i r T i i X j 6 = i j r i H j j ; c 3 = k v r i i T r i i ; byusingthefactthat r i j =0 from5and H i i i arepositivefrom5,5 and5.Asucientconditionfortheinequalityin52tobesatisedis )]TJ/F15 11.9552 Tf 13.694 8.088 Td [(1 2 j c 1 j)]TJ/F15 11.9552 Tf 19.963 8.088 Td [(1 j c 2 j > )]TJ/F27 11.9552 Tf 9.299 0 Td [(c 3 : Hence,if > ; where =max 8 < : s j c 1 j c 3 ; j c 2 j c 3 9 = ; ; thesystemconvergestothesetofcriticalpoints.ApplyingLaSalle'sinvarianceprinciple, thetrajectoriesofthesystemconvergetothelargestinvariantsetcontainedintheset S = fkr i i k =0 ; 8 i 2Vg : Thesetin5isformedwheneverthepotentialfunctionseitherreachthedestination orasaddlepoint.Since i in5isanavigationfunction,thesaddlepointsof i are 97

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isolatedin[41].Thus,thesetofinitialconditionsthatleadtosaddlepointsaresetsof measurezero[63].Thelargestinvariantsetconstrainedisthesetofdestination[23]. Hence, kr i i k =0 indicatesthat k p i )]TJ/F27 11.9552 Tf 11.955 0 Td [(p k! 0 for 8 i: Toshowthat ~ i 0 ,wetakethetimederivativeof ~ i t in5anduse5to developtheopen-looporientationtrackingerrorsystemas ~ i = i )]TJ/F15 11.9552 Tf 14.276 3.155 Td [(_ di .Using5,the closed-looporientationtrackingerroris ~ i = )]TJ/F27 11.9552 Tf 9.298 0 Td [(k w ~ i ; whichhastheexponentiallydecayingsolution ~ i t = ~ i e )]TJ/F28 7.9701 Tf 6.586 0 Td [(k w t Basedon5and5,itisclearthat @' @x i ; @' @y i 2L 1 on F ;hence,5canbe usedtoconcludethat v i t 2L 1 .Provided di t 2L 1 in5on F ,5canbe usedtoshowthat i t 2L 1 5.4Simulation Apreliminarynumericalsimulationisperformedinthissectiontodemonstrate theperformanceofthecontrollerdevelopedin5and5inascenariowherea groupoffourmobilerobotswiththekinematicsin5arenavigatedtothecommon destination h p T ; i T = 000 T .Thefourmobilerobotsaredeployedina workspaceof R w =5 m withaninitiallyconnectedconditionof 2 6 6 6 6 6 6 6 4 q T 1 q T 2 q T 3 q T 4 3 7 7 7 7 7 7 7 5 = 2 6 6 6 6 6 6 6 4 )]TJ/F15 11.9552 Tf 9.298 0 Td [(21 : 5 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 131 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 250 : 7 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 7279 )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 : 5 )]TJ/F15 11.9552 Tf 9.298 0 Td [(0 : 71 : 8850 )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 : 25 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 : 50 : 9425 3 7 7 7 7 7 7 7 5 Thelimitedcommunicationandsensingzoneforeachrobotisassumedas R =2 m and 1 = 2 =0 : 5 m: Thetuningparameter in5isselectedas =1 : 5 ,andthe controlgains k v and k w areadjustedto k v =1 : 1 and k w =0 : 9 .Thecontrollawin5 and5yieldsthesimulationresultsshowninFig.5-2toFig.5-5.Fig.5-2shows 98

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Figure5-2.Thetrajectoryforeachmobilerobotwiththearrowdenotingitscurrent orientation. thetrajectoryevolutionforeachrobot,wheretherobotsarerepresentedbydots,and theassociatedarrowsindicatethecurrentorientation.Thelinearandangularvelocity controlinputsforeachrobotareshowninFig.5-3.InFig.5-4,theplotofpositionand orientationerrorforeachmobilerobotindicatesthateachrobotachievesthecommon destinationwiththedesiredorientation.Theevolutionofinter-robotdistanceisshownin Fig.5-5,whichimpliesthattheconnectivityoftheunderlyinggraphismaintained,since theinter-robotdistanceislessthantheradius R =2 m duringthemotion. 5.5Summary Basedonthedipolarnavigationfunctionformalism,adecentralizedtime-varying continuouscontrollerisdevelopedtoachievenetworkcooperativegoals,thatarenavigatingmobilerobotstoacommondestinationwithadesiredorientationandensuringthe networkconnectivityforalltime,byusingonlylocalsensinginformationfromone-hop neighbors.Adistinguishingfeatureofthedevelopeddecentralizedapproachisthatno inter-agentcommunicationisrequiredtocompletethenetworkrendezvousobjective, 99

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Figure5-3.Plotoflinearvelocityandangularvelocityforeachmobilerobot. Figure5-4.Plotofpositionandorientationerrorforeachmobilerobot. 100

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Figure5-5.Theevolutionofinter-robotdistance. whichresultsinradiosilenceduringthenetworkregulation.Futureeortsarefocusedon enablingcollisionavoidancewithobstaclesinadynamicenvironmentusinglocalsensing information. 101

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CHAPTER6 INFLUENCINGEMOTIONALBEHAVIORINSOCIALNETWORK Chapter2-5focusondesigningcontrolalgorithmsfornetworkedmulti-agentsystems inengineering,suchasocking,formationcontrolandrendezvousproblems.Thischapter aimstostudyhowcanthemodelsandmethodsdevelopedinengineeringbeapplied towardunderstandingandcontrollingasocialnetwork.Howcanoneproduceconsensus amongasocialnetworke.g.,tomanipulatesocialgroupstoadesiredend?Motivated towardsthisend,controllersdevelopedinthischapteraretoinuencetheemotionsofa sociallyconnectedgroupofindividualstoaconsensusstate.Usinggraphtheory,asocial networkismodeledasanundirectedgraph,whereanindividualinthesocialnetwork isrepresentedasavertexinthegraph,andthesocialbondbetweentwoindividuals isrepresentedasanedgeconnectingtwovertices.Duetothenon-localpropertyof fractional-ordersystems,theemotionalresponseofindividualsinthenetworkaremodeled byfractional-orderdynamicswhosestatesdependoninuencesfromsocialbonds. Withinthisformulation,thesocialgroupismodeledasanetworkedfractionalsystem. Thischapteralsoconsidersasocialbondthresholdontheabilityoftwopeopleto inuenceeachother'semotions.Toensureinteractionamongindividuals,oneobjective istomaintainexistingsocialbondsamongindividualsabovetheprespeciedthreshold allthetimei.e.,socialcontrolsorinuencesshouldnotbesoaggressivethatthey isolateorbreakbondsbetweenpeopleinthesocialgroup.Anotherobjectiveisto designadistributedcontrollerforeachindividual,usinglocalemotionalstatesfrom socialneighbors,toachieveemotionsynchronizationinthesocialnetworki.e.,an agreementontheemotionstatesofallindividuals.Toachievetheseobjectives,a decentralizedpotentialfunctionisdevelopedin[46]toencodethecontrolobjectiveof emotionsynchronization,wheremaintenanceofsocialbondsismodeledasaconstraint embeddedinthepotentialfunction.Asymptoticconvergenceofeachemotionstatetothe 102

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commonequilibriuminthesocialnetworkisthenanalyzedviaaMetzlerMatrix[65]and aMittag-Leerstability[54]approach. 6.1Preliminaries Fractionalcalculusandgraphtheorynotionsareintroducedinthissection,which serveasabasisforthesubsequentdevelopmentandanalysisinthischapter. 6.1.1FractionalCalculus Theuniformformulaofafractionalintegralwithorder 2 ; 1 isdenedas t 0 D )]TJ/F28 7.9701 Tf 6.587 0 Td [( t f t = 1 )-167( t t 0 f t )]TJ/F27 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [( d; where t 0 D )]TJ/F28 7.9701 Tf 6.587 0 Td [( t f t denotesthefractionalintegraloforder on [ t 0 ;t ] f t isanintegrable function,and )-167( denotestheGammafunction[64].CaputoandRiemann-Liouville R-Lfractionalderivativesarethetwomostwidelyusedfractionaloperators[64].Foran arbitraryrealnumber p 2 R ,theR-LandCaputofractionalderivativesaredenedas t 0 D p t f t = d [ p ]+1 dt [ p ]+1 )]TJ/F28 7.9701 Tf 5.479 -11.477 Td [(t 0 D )]TJ/F28 7.9701 Tf 6.587 0 Td [( t f t ; and C t 0 D p t f t = t 0 D )]TJ/F28 7.9701 Tf 6.586 0 Td [( t d [ p ]+1 dt [ p ]+1 f t ; respectively,where =[ p ]+1 )]TJ/F27 11.9552 Tf 12.28 0 Td [(p 2 ; 1 ; [ p ] representstheintegerpartof p ,and t 0 D p t and C t 0 D p t areR-LandCaputofractionalderivativeswithorder p on [ t 0 ;t ] ,respectively 1 SincetheR-Lfractionaloperatorrequiresafractional-orderinitialcondition,whichcanbe diculttointerpret[59],thesubsequentdevelopmentisbasedontheCaputofractional operator. 1 If p isaninteger,theCaputoderivativein6with =1 isequivalenttotheintegerorderderivative. 103

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Similartotheexponentialfunctionusedinsolutionsofinteger-orderdierential equations,theMittag-LeerM-Lfunctiongivenby E ; z = 1 X k =0 z k )-167( k + ; where ;> 0 and z 2 C ,isfrequentlyusedinsolutionsoffractional-ordersystems. Particularly,when = =1 ;E ; z in6isanexponentialfunction, E 1 ; 1 z = e x : Considerthefractionalordernon-autonomoussystem C t 0 D t x t = f t;x withinitialcondition x t 0 ; where 2 ; 1 ,and f t;x ispiecewisecontinuousin t and locallyLipschitzin x: Stabilityofthesolutionsto6aredenedbytheM-Lfunction asfollows[54]. Denition6.1. Mittag-LeerStabilityThesolutionof6issaidtobeMittagLeerstableif k x t kf m [ x t 0 ] E ; 1 )]TJ/F27 11.9552 Tf 9.299 0 Td [( t )]TJ/F27 11.9552 Tf 11.955 0 Td [(t 0 g b ; where t 0 istheinitialtime, 2 ; 1 ;b> 0 ;> 0 ;m =0 ;m x 0 ;m x islocally Lipschitz,and E ; 1 isdenedin6with =1 : Lyapunov'sdirectmethodisextendedtofractional-ordersystemsinthefollowing LemmatodetermineMittag-Leerstabilityforthesolutionsof6in[54]. Lemma6.1. [54]Let x =0 beanequilibriumpointforthesystem6,and D R n beadomaincontainingtheorigin.Let V t;x : ; 1 ] D R beacontinuously dierentiablefunctionandlocallyLipschitzwithrespectto x suchthat k 1 k x k a V t;x k 2 k x k ab ; C 0 D t V t;x )]TJ/F27 11.9552 Tf 28.56 0 Td [(k 3 k x k ab ; where x 2 D 2 ; 1 ;k 1 ;k 2 ;k 3 ;a and b arearbitrarypositiveconstants.Then x =0 is Mittag-Leerstable. 104

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6.1.2GraphTheory Graphtheoryseecf.[62]iswidelyusedtorepresentanetworkedsystem.Let G = V ; E denoteanundirectedgraph,where V = f v 1 ; ;v N g and EVV denotethe setofnodesandthesetofedges,respectively.Eachedge v i ;v j 2E representstheneighborhoodofnode i andnode j ,whichindicatesthatnode i andnode j areabletoaccess eachother'sstates.Theneighborsetofnode i isdenotedas N i = f v j j v i ;v j 2Eg : A pathbetween v 1 and v k isasequenceofdistinctnodesstartingwith v 1 andendingwith v k suchthatconsecutivenodesareadjacentingraph G .Graph G isconnectedifin G any nodecanbereachedfromanyothernodebyfollowingaseriesofedges.Theadjacency matrixisdenedas A [ a ij ] 2 R N N with a ij > 0 if v i ;v j 2E ,and a ij =0 otherwise. Amatrix L forthegraph G isdenedas L A )]TJ/F27 11.9552 Tf 12.309 0 Td [(D 2 R N N ,where D [ d ij ] 2 R N N isadiagonalmatrixwith d ii = P N j =1 a ij : The N N matrixwithpositiveorzeroodiagonalelementsandzerorowsumsisreferredasaMetzlermatrix[60].Zeroisatrivial eigenvalueofaMetzlermatrix,andalltheothereigenvaluesarepositive,ifandonlyifthe correspondingundirectedgraph G isconnected.Theeigenvectorassociatedwiththezero eigenvalueis 1 ,where 1 =[1 ; ; 1] T 2 R N Tofacilitatethefollowingdevelopment,acorollarytoTheorem1of[65]isintroduced asfollows. Corollary6.1. Theequilibriumpoint x 1 2 R N ofthesystem x t = L t x t isexponentiallystablei.e.,theelementsof x t 2 R N achieveexponentialconsensus, providedthatthetime-varyingmatrix L t 2 R N N in6isaMetzlermatrixwith piecewisecontinuousandboundedelements,andthetime-varyinggraphcorrespondingto L t isconnectedforall t 0 105

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6.2ProblemFormulation Considerasocialnetworkcomposedof N individuals.Usinggraphtheory,the interactionamongindividualsismodeledasanundirectedgraph G = V ; E .Forinstance, thekarateclubnetworkin[96]ismodeledasanundirectedgraphasshowninFig. 6-1,wherethevertexinthegraph G isrepresentedbyanindividual,thesolidarrow connectingtwoindividualsdenotestheedgein G ,representinganestablishedsocialbond i.e.,friendshipandindicatingthattheindividualsareabletoaccesseachother'sstates i.e.,senseandunderstandthesocialstateofapeer. Inasocialnetwork,thestateofanindividualcanbethesocialstatus,socialconnections,emotionalstatus,oretc.Inthefollowingdevelopment,thesocialstatedenotessome humanemotion,suchashappiness,love,angerorfear.Theemotionstate q i t 2 R isa realnumberindicatingthecurrentstateofanindividual i thatcanbedetectedfromother membersi.e.,socialneighborssuchasclosefriendsorfamilyinthesocialnetwork.For instance,agreatervalueof q i t impliesahappierstateofindividual i: Anintegerderivativeofafunctionisonlyrelatedtoitsnearbypoints,whilea fractional-orderderivativeinvolvesallthepreviouspoints.Sincehumanemotionsare alwaysinuencedbymemoriesandpastexperiences, q i t ismodeledasthesolutiontoa fractional-orderdynamicas C 0 D t q i t = u i t ;i =1 ; ;N; where u i 2 R denotesaninuencei.e.,controlinputovertheemotionalstate,and C 0 D t q i t isthe th derivativeof q i t with 2 ; 1] .Themodelin6isaheuristic approximationtoanemotionalresponse.Themodelindicatesthataperson'semotional stateisadirectrelationshiptoexternalinuenceintegratedoverthehistoryofaperson's previousemotionalstates.On-goingeortsbythescienticcommunityarefocusedonthe developmentofclinicallyderivedmodels;yet,todate,thereisnowidelyacceptedmodel ofaperson'semotionalresponsetoeventsinasocialnetwork. 106

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Socialbondsinanetworkcanbeestablishedthroughanumberofrelationships betweenindividualse.g.,studentandteacher,employerandemployee,patientand doctor,twostrangersthatshareacommoninterestandcanberepresentedasan undirectededgeingraph G .Eachbondhasaweightingfactordenotedas S ij 0 thatmeasurestheamountofinuencethatissharedbetweenindividuals i and j .The greaterthevalue S ij ; theclosertherelationshipbetweenindividuals i and j; and S ij =0 iftwoindividualshavenoinuenceovereachother.Throughananalysisofasocial graphovertime,onecoulddetermineaweightingforthelevelofinuencebetween individuals.However,thesubsequentdevelopmentonlyrequiresthatanindividualnode hasanunderstandingoftherelativeinuencebetweenitselfanditslocalsocialneighbors. Moreover,itisassumedthat,thereexistathreshold 2 R + ,andindividuals i and j are abletoinuenceeachother'semotionalstatesonlywhenthesocialbond S ij .Inother words,anedge ij ingraph G doesnotexistifthesocialbond S ij betweenindividuals i and j islessthanthethreshold .Theneighborsofindividual i ingraph G isdened as N i = f v j j S ij g ,whichdeterminesasetofindividualswhohaveaninuential relationshipwithindividual i: Inthesubsequentdevelopment,thesocialbondisdenedas S ij = f )]TJ/F30 11.9552 Tf 5.479 -9.684 Td [(k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k 2 ; where f isadierentiablefunction,mappingtheemotionstatesofindividuals i and j toarealnon-negativevalue.Somepropertiesfor S ij include:1 f )]TJ/F30 11.9552 Tf 5.48 -9.683 Td [(k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k 2 decreases as k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k increasesthefurtheraparttheemotionalstateoftwoindividualstheless inuencetheyhaveovereachother,whichindicatesthat @f @ k q i )]TJ/F28 7.9701 Tf 6.586 0 Td [(q j k < 0 ;2 f )]TJ/F30 11.9552 Tf 5.479 -9.684 Td [(k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k 2 achievestheminimumof 0 whenindividual i hasnoinuence/relationshipwithindividual j ;3thesecondpartialderivative @ 2 f @q i isbounded.Thesepropertiesarebasedonthe generalobservationthattheemotionalstatesofindividual i and j tendtoconsensusina closerelationship.Forexample,theemotionalstateofonespouse,parentorchildtends tomirrortheemotionalstateofanotherspouse,child,orparentrespectively.Hence,itis 107

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Figure6-1.TheundirectedgraphmodelofZachary'skarateclubnetworkin[96]. reasonabletoassumethat S ij isafunctionofthedierencebetween q i and q j ; designed as k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k 2 inthiswork.Whilesomediscreteeventscancauseadiscontinuousshiftin someone'ssocialbondse.g.,acheatingspouse,winningthelottery,unexpectedsickness ordeaththatwouldleadtoanunboundedsecondpartialderivative,mostsocialbonds tendtobecontinuousovertime. Basedontheproblemsetting,thesocialnetworkofhumanemotionsisnowformulatedasanetworkedfractional-ordersystemdescribedby6.Theemotionsynchronizationobjectiveinasocialnetworkistoregulatetheemotionalstatesofindividuals toadesiredstatei.e., q i t q forall i with q 2 R denotinganequilibriumpoint. Moreover,individualsgenerallyprefertoshareanemotionalresponseratherthanreact inanemotionalwaythatrendersthemanoutcast.Hence,theemotionsynchronization problemalsoincludesagoalthatgivenaninitiallyconnectedgraph G ,thesocialbonds betweenindividualsaremaintainedi.e.,maintainthesocialbonds S ij allthetimeso thatpeersremainpeers.Sincesocialbondsexistinitially anytwoindividualsareableto reacheachotherthroughapathofedgesassociatedwithasocialbondsatisfying S ij 6.3ControlDesign Articialpotentialeldbasedmethods,composedofattractiveandrepulsivepotentials,havebeenwidelyusedinthecontrolofmulti-agentsystems,wherethecontrol 108

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objectiveisencodedastheminimumpotentialvaluebytheattractivepotentialand constraintsareencodedasthemaximumpotentialvaluebytherepulsivepotential cf.[48,49].Drivenbythenegativegradientoftheproposedpotentialeld,thesystemwillasymptoticallyachievetheminimumofthepotentialeld.Inthischapter,the potentialeldapproachisappliedtosocialcontrol. Toachieveemotionsynchronization,adecentralizedpotentialfunctionisdevelopedas i : R N [0 ; 1] forindividual i of N as i = i )]TJ/F27 11.9552 Tf 5.479 -9.684 Td [( k i + i 1 =k ; where k 2 R + isatuningparameter, i : R 2 R + isthegoalfunction,and i : R N R + isaconstraintfunction.Thegoalfunctionin6isdesignedas i = X j 2N i 1 2 k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k 2 ; whichisminimizedwhenevertheemotionalstateofindividual i agreeswiththeemotions ofneighbors j j 2N i .Toensureexistingsocialbondsaremaintainedi.e., S ij ,the constraintfunctionin6isdesignedas i = Y j 2N i 1 2 b ij ; where b ij = S ij )]TJ/F27 11.9552 Tf 12.885 0 Td [( ,and S ij isdenedin6.Foranexistingsocialbondbetween individuals i and j; thepotentialfunction i in6willapproachitsmaximumwhenevertheconstraintfunction i decreasesto 0 i.e.,thesocialbond S ij decreasestothe thresholdof Basedonthedenitionofthepotentialfunctionin6,theemotionalinuenceis designedas u i = )]TJ/F27 11.9552 Tf 9.299 0 Td [(K i r q i i ; 109

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where K i isapositivegain.In6, r q i i denotesthegradientof i withrespectto q i as r q i i = k i r q i i )]TJ/F27 11.9552 Tf 11.955 0 Td [( i r q i i k k i + i 1 k +1 : From6and6, r q i i and r q i i in6canbedeterminedas r q i i = X j 2N i q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j ; and r q i i = X j 2N i b ij 1 2 r q i b ij = X j 2N i @f @ )]TJ/F30 11.9552 Tf 5.48 -9.684 Td [(k q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j k 2 b ij q i )]TJ/F27 11.9552 Tf 11.956 0 Td [(q j ; respectively,where b ij Q l 2N i ;l 6 = j b il : Substituting6and6into6, r q i i is rewrittenas r q i i = )]TJ/F33 11.9552 Tf 11.291 11.358 Td [(X j 2N i m ij q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j ; where m ij = k i )]TJ/F15 11.9552 Tf 11.517 3.155 Td [( b ij i @f @ k q i )]TJ/F28 7.9701 Tf 6.587 0 Td [(q j k 2 k k i + i 1 k +1 isnon-negative,basedontherstpropertyof S ij ,andthedenitionof i i k ,and b ij 6.4ConvergenceAnalysisandSocialBondMaintenance Toshowthatindividualsinthefractional-ordernetworkconvergetoacommondesiredemotionalstate,thefollowinganalysisissegregatedintothreeproofs.Intherst proof,theconnectivityofthesocialgroupisproventobeensuredbytheinuencefunctionin6.Inthesecondproof,aninteger-ordersimplicationofthedynamicsystem in6isconsideredandexponentialconvergenceisproven.Exponentialconvergenceof theinteger-ordersystemisusedtoestablishtheexistenceofaLyapunovfunctionandits derivativebyinvokingaconverseLyapunovtheorem.TheCaputofractionalderivativeof 110

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thedevelopedLyapunovfunctionisthendeterminedandusedwithinaMittag-Leerstabilityanalysisthatprovestheclosed-loopfractional-ordersystemasymptoticallyconverges totheequilibriumsetofconsensusstates. 6.4.1SocialBondMaintenance Assumingasocialnetworkisinitiallyconnected,thesocialgroupwillremainconnectedifeveryexistingedgeinthenetworkgraphismaintainedallthetimei.e., S ij ThefollowingLemmaisdevelopedtoshowthatconnectivityoftheunderlyinggraphis maintainedundertheinuencefunctionin6i.e.,socialpeersdonotbecomeisolated anddisconnectedfromthesocialgroup. Lemma6.2. Theinuencefunctionin6guaranteesconnectivityof G allthetime. Proof. Consideranemotionalstate q 0 forindividual i ,wherethebondbetweenindividual i andneighbor j 2N i satises b ij q 0 ;q j =0 ; whichindicatesthatthesocialbondis tooweaktoaecttheemotionofindividual i ,andtheassociatededgeisabouttobreak. From6, i =0 when b ij =0 ,andthenavigationfunction i achievesitsmaximum valuefrom6.Since i ismaximizedat q 0 ; noopensetofinitialconditionscanbe attractedto q 0 underthenegatedgradientcontrollawdesignedin6.Therefore, thesocialbondbetweenindividual i and j ismaintainedgreaterthan by6,and theassociatededgeisalsomaintained.Followingsimilararguments,everyedgein G is maintained,andconnectivityoftheunderlyinggraphisguaranteed. 6.4.2ConvergenceAnalysis Fortheparticularcaseof =1 ; thefractional-orderdynamicsin6simplies totheinteger-ordersystem q i t = u i t .Thefollowingtheoremestablishesexponential convergencetothecommonequilibriumfortheinteger-ordersystem. Theorem6.1. Theequilibriumpoint q 2 R oftheinitiallyconnectedgraphofnodeswith integer-orderdynamics q i t = u i t isexponentiallystableforall i ,giventheinuence function u i t developedin6. 111

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Proof. For =1 ; substituting6and6into6yieldsthefollowingclosedloopemotiondynamicsofindividual i : q i t = )]TJ/F33 11.9552 Tf 11.291 11.357 Td [(X j 2N i K i m ij q i )]TJ/F27 11.9552 Tf 11.955 0 Td [(q j : Using6andsimilarto[21],thedynamicsofallindividualsinthesocialnetworkcan berewritteninacompactformas _q t = t q t ; where q = q 1 ; ;q N T denotesthestackedvectorof q i ; andtheelementsof t 2 R N N aredenedas ik t = 8 > > > > < > > > > : )]TJ/F33 11.9552 Tf 11.291 8.967 Td [(P j 2N i K i m ij i = k K i m ij j 2N i ;i 6 = k 0 ;j= 2N i ;i 6 = k: From6, t ismatrixwithzerorowsums.Usingthefactthat m ij isnon-negative from6,and K i isapositiveconstantgainin6,theo-diagonalelementsof t arepositiveorzero,anditsrowsumsarezero.Hence, t isaMetzlermatrix. Giventhat t isaMetzlermatrixandthesocialnetworkisalwaysconnectedwiththe controllerdevelopedin6seeLemma6.2,Corollary6.1canbeappliedto6to concludethattheelementsof q t exponentiallyachieveconsensus. Theorem6.2. Theequilibriumpoint q 2 R oftheinitiallyconnectedgraphofnodeswith thefractional-orderdynamicsin6with 2 ; 1 isasymptoticallystableforall i giventheinuencefunction u i t developedin6. Proof. Let x i t q i t )]TJ/F27 11.9552 Tf 13.199 0 Td [(q 2 R and x t q t )]TJ/F27 11.9552 Tf 12.586 0 Td [(q 1 2 R n .Thefractional-order dynamicsin6with 2 ; 1 forallindividualscanbeobtainedfrom6as C 0 D t x t = t x t + q 1 g t; x ; 112

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where t isthesameasinTheorem6.1,sinceeachelementin t isafunctionof q i t )]TJ/F27 11.9552 Tf 12.901 0 Td [(q j t from6and q i t )]TJ/F27 11.9552 Tf 12.901 0 Td [(q j t = x i t )]TJ/F27 11.9552 Tf 12.901 0 Td [(x j t .Sincethestabilityofa fractional-ordersystemisdenedbyDenition6.1,andMittag-Leerstabilityimplies asymptoticstabilityasdiscussedin[54],thefollowingdevelopmentisfocusedonproving that6isMittag-Leerstable. Since i and i arenotzerosimultaneously,and i i andtheirpartialderivatives areboundedfrom6and6, t in6isbounded.Assumingthat t is boundedbyaconstant l 2 R + ,theLipschitzconditionfor g t; x in6is k g t; x k k x k l: Theorem6.1statesthattheequilibriumpoint q isexponentiallystablefortheintegerordersystemof6.TheconverseLyapunovtheorem,Theorem4.9in[84],indicates thatthereexistsafunction 2 V t; x : ; 1 ] R N R andstrictlypositiveconstants k 1 ; k 2 ; and k 3 suchthat k 1 k x k V t; x k 2 k x k ; V )]TJ/F27 11.9552 Tf 21.918 0 Td [(k 3 k x k : 2 Asdiscussedin[65],onevalidselectionfortheLyapunovfunctionis V x = max x 1 ; ;x n )]TJ/F15 11.9552 Tf 11.955 0 Td [(min x 1 ; ;x n : 113

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Let =1 )]TJ/F27 11.9552 Tf 11.483 0 Td [( 2 ; 1 .FollowingasimilarprocedureintheproofofTheorem8in[54] andusing6and6,theCaputofractionalderivativeof V iscomputedas C 0 D t V t; x = C 0 D 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [( t V t; x = C 0 D )]TJ/F28 7.9701 Tf 6.587 0 Td [( t V )]TJ/F27 11.9552 Tf 28.56 0 Td [(k 3 )]TJ/F28 7.9701 Tf 5.479 -4.747 Td [(C 0 D )]TJ/F28 7.9701 Tf 6.587 0 Td [( t k x k )]TJ/F27 11.9552 Tf 28.56 0 Td [(k 3 C 0 D )]TJ/F28 7.9701 Tf 6.587 0 Td [( t k g t; x k l )]TJ/F27 11.9552 Tf 29.755 8.088 Td [(k 3 l C 0 D )]TJ/F28 7.9701 Tf 6.586 0 Td [( t g t; x )]TJ/F27 11.9552 Tf 29.755 8.087 Td [(k 3 l k x k : Mittag-Leerstabilityofsystem6with 2 ; 1 canbeobtainedas x t V ;x k 1 E 1 )]TJ/F28 7.9701 Tf 6.586 0 Td [( )]TJ/F27 11.9552 Tf 12.369 8.087 Td [(k 3 k 2 l t 1 )]TJ/F28 7.9701 Tf 6.587 0 Td [( ; byapplyingLemma6.1to6and6,where a = b =1 .Theresultin6 impliestheequilibriumpoint q 1 2 R n oftheclosed-loopfractional-ordersystemin6 isasymptoticallystable. 6.5Discussion Thepreviousdevelopmentisbasedontheassumptionthat q isacommonequilibriumpointforalltheindividualsinasocialnetwork.Insomesituations,acommon equilibriumpointforanemotionalstatee.g.,groupangercouldbederivedfromadiscreteevente.g.,apoliceshooting[6,37]orlongtermeventse.g.,yearsofoppression fromadictator[36,61].Insuchsituations,thecontrollerin6providesinstructions foranindividualtocombineemotionaldierenceswithsocialpeers,whileconsidering thestrengthoftherespectivesocialbonds,sothatastheindividual'semotionalstate convergesto q ,socialbondsi.e.,theneedforpeerstoshareanemotionalstatebetween peerswillalsoinuencethemtoconvergetothesameemotionalstate.Ifapersoninstantlyconvergesto q ,theemotionaldierencebetweensocialpeersmaydecreasetothe pointwhere S ij < ,resultinginaseparationfromthesocialgroupandanendofthe 114

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individual'sinuenceoverthegroupi.e.,thechangeinemotionalstateisgreatenough thatbondsbetweensocialpeersarebrokenandthesocialpeersignoretheindividual's state.Thecontrollerin6accountsfortheweightedinteractionsandinuenceover peersbasedontheassumptionthatpeerswillintegrateanemotionalstateinanon-local fractional-ordersense. Ofcourse,individualsinasocialnetworkoftendonothaveacommonequilibrium point.Forexample,agroupoffriendsmaywishtoengageinanactivitythatdiersfrom thedesireofanindividual.Inthesescenarios,apersonmustresolvetheconictbetween theindividualequilibriumpointandthesocialbondconstraintthat S ij .Thatis, eitherpeerpressurewilldeviatethepersonfromthedesiredsocialstate,strengthening correspondingsocialbonds,orsocialbondswiththegroupwilldecrease/break.This observationindicatesthatlongtermpeerswithstrongsocialbondslikelyshareacommon equilibriumpoint.Follow-oneortstothecurrentchapterarebeingdevelopedtoincorporatethedynamicsoftheequilibriumpoint/socialbondarbitrationalongwithinuence strategiestoenablesocialpeerstodeviateapersonfromanequilibrium,orchangethe equilibrium. 6.6Summary Inthischapter,emotionsynchronizationforagroupofindividualsinasocialnetwork isstudied.Bymodelinghumanemotionasafractional-ordersystem,adecentralizedpotentialeld-basedfunctionisdevelopedtoensurethattheemotionstatesofallindividuals asymptoticallyconvergetoacommonequilibriumwhilemaintainingsocialbonds.Social bondsplayanimportroleinaperson'semotionalstate.Forinstance,apersontendsto putgreatertrustinaclosefriendthansomerandomperson,andthus,canbemoreeasily inuencedbytheclosefriend.However,thecurrentdevelopmentonlyexaminesthesocial bondasathresholdconstrainttoensurecontinuedinteractionbetweenfriends,without consideringthepotentialdynamicsofhowaperson'semotionscanbeaectedbydierent socialbondsinthenetwork.Hence,futureworkisbeingconsideredthatexploresthe 115

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relationbetweenaperson'semotionandtheassociateddierentlevelsofsocialbonds. Moreover,furthereortsarealsotargetinginuencestrategiestoenablesocialpeerstodeviateapersonfromanequilibrium,orchangetheequilibriumi.e.,peerpressurestrategies appliedtoreluctantpeers. 116

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CHAPTER7 CONCLUSIONANDFUTUREWORK Thischapterconcludesthedissertationbydiscussingthemaincontributionsdevelopedineachchapterandtheopenproblemsforfutureresearch. 7.1Conclusion Variousapplicationscanbenetfromcoordinationandcollaborationamongagroup ofagents,suchassensing,searchingandrescue.Toecientlyexchangeinformation andmakeappropriatedecisionsforamulti-agentsystem,agentsaretypicallyrequired tocollaborateoverawirelesscommunicationnetwork.Thefocusofthisdissertation istodevelopacontrolstrategyforagroupofagentswithlimitedcommunicationand sensingcapabilitiestoachievecollectivetasks.Sinceeachagentknowsthepositionsof onlythoseagentswithinitssensingrangeandcanonlycommunicatewithnodeswithin itscommunicationrange,thetasksmustbeaccomplishedwhileensuringthatspecied nodesstaywithineachother'ssensingandcommunicationrangesandthattheoverall communicationnetworkstaysconnected.Articialpotentialeldbasedcontrollersare developedtopreserveoverallnetworkconnectivityandenablecooperativetaskssuchas formationcontrolandtherendezvousproblemsinChapter2-Chapter5.Theresultsin Chapter2-Chapter5mainlyfocusoncontroldesignsformulti-agentsystemstoperform cooperativecontrolobjectivesinengineering.InChapter6,themodelsandmethods areextendedtowardunderstandingandexertinginuencewithinasocialnetwork.The speciccontributionsofeachchapteraresummarizedasfollows. InChapter2,atwostagecontrolframeworkisdevelopedtoachievethecooperative controlobjectiveofmaintainingglobalnetworkconnectivityduringthemissionby groupingallnodesintotwosubgraphs,ahighlevelnetworksubgraphandseverallowlevel networksubgraphs.Thekeytomaintainnetworkconnectivityistoensureconnectivity withinthehighlevelsubgraphandeachlowlevelsubgraph.Apotential-eld-based controlleristhendesignedtoensurethatthehighlevelsubgraphandeachlowlevel 117

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subgrapharealwaysconnectediftheyareconnectedinitially,andachievecollision avoidanceamongagentswhenperformingdesiredtasks. InChapter3,givenaninitialgraphwithadesiredneighborhood,anavigation functionbaseddecentralizedcontrollerisdevelopedtoensurethesystemasymptotically convergestothedesiredcongurationwhilemaintainingnetworkconnectivityand avoidingcollisionswithotheragentsandobstacles.Contrarytootherpotentialeld basedapproacheswhichcanbetrappedbylocalminima,usingthenavigationfunction frameworkinthischapter,thesystemisguaranteedtoachieveitsglobalminimum,which correspondstothedesiredconguration.Anotherfeatureofthedevelopedapproachis thatthedesiredglobalcongurationcanbeachievedbyagroupofagentsusingonlylocal sensingfeedbackwithoutrequiringradiocommunicationamongagents,whichenablesa stealthmodeofoperation. InChapter4,theassumptionofaninitialgraphwithadesiredneighborhoodin Chapter3iseliminated.Anovelstrategy,usingaprexlabelingandroutingalgorithm from[67]andanavigationfunctionbasedcontrolschemefrom[42],isdevelopedto achieveadesiredformationforagroupofidenticalagentsfromanarbitrarilyconnected initialgraph.Sincetheagentsconsideredinthischapterareidentical,theagentscantake anypositioninthenaltopology.Contrarytoanassumptioninmostexistingworkin formationcontrolcf.[11,66,97]andtheirreferenceswheretheabsoluteorrelativeposes oftheagentsareprespecied,andtheinitialtopologyisrequiredtobeasupergraphof thedesiredtopology,theapproachdevelopedinChapter4enablesformationcontrolfrom anarbitraryinitialcondition,onlyassumingthatthenaltopologyofthedesiredphysical congurationisatree.Basedonthenavigationfunctionframework,thedesiredformation isguaranteedtobeachievedwithcollisionavoidanceamongagentsduringthemotion, andnetworkconnectivityisensuredbymodelingtheunderlyinggraphconnectivityas anarticialconstraintinthenavigationfunction.Moreover,theconceptofinformation owisappliedtondapathwithmorefreedomforthemotionofextranodeswithout 118

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partitioningnetworkandallowssomecommunicationlinkstobeformedorbrokenina smoothmannerwithoutintroducingadiscontinuity. InChapter5,adecentralizedcontinuoustime-varyingcontroller,usingonlylocal sensingfeedbackfromitsone-hopneighbors,isdevelopedtorepositionandreorienta groupofwheeledrobotswithnonholonomicconstraintstoacommonsetpointwitha desiredorientationwhilemaintainingnetworkconnectivityduringtheevolution.Using thenavigationfunctionframework,themulti-agentsystemisguaranteedtomaintain connectivityandbestabilizedatacommondestinationwithadesiredorientationwithout beingtrappedbylocalminima.Moreover,theresultcanbeextendedtootherapplications byreplacingtheobjectivefunctioninthenavigationfunctiontoaccommodatefordierent tasks,suchasformationcontrol,ocking,andotherapplications. InChapter6,adistributedcontrollerforeachindividualinasocialnetworkis designed,usinglocalemotionalstatesfromsocialneighbors,toachieveemotionsynchronizationforagroupofindividualsinasocialnetworki.e.,anagreementontheemotion statesofallindividuals.Motivatedbythenon-localpropertyoffractional-ordersystems, theemotionalresponseofindividualsinthenetworkaremodeledbyfractional-orderdynamicswhosestatesdependoninuencesfromsocialbonds.Usinggraphtheory,asocial networkismodeledasanundirectedgraph,whereanindividualinthesocialnetwork isrepresentedasavertexinthegraph,andthesocialbondbetweentwoindividualsis representedasanedgeconnectingtwovertices.Encodingthecontrolobjectiveofemotion synchronizationandmodelingthemaintenanceofsocialbondsasaconstraint,apotentialfunctionisdevelopedtoensureasymptoticconvergenceofeachemotionstatetothe commonequilibriuminthesocialnetwork. 7.2Futurework TheworkinChapter2toChapter5illustratesthatpotentialeldbasedcontrol methodscanbesuccessfullyappliedformulti-agentsystemstoperformcooperative controltaskswhilemaintainingnetworkconnectivityduringthemission.Theworkin 119

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Chapter6thenappliesthecontroltechniquesdevelopedinengineeringtoinvestigate andinuenceemotionsofpeopleinasocialnetwork,whichopensnewavenuesforfuture researchinthedomainofsocialengineering.Sincethestudyforasocialnetworkisa promisingareaandresearchinthisareaisstillatanascentstage,severalinteresting openproblemsstillexist.Inthissection,openproblemsrelatedtomulti-agentsystemsin engineeringandthecontroldesigninsocialnetworksinthisdissertationarediscussed. Futureworkformulti-agentsysteminengineering: 1.InChapter3,itisassumedthatnootherobstaclesoragentscanbewithinthe collisionregionofnode i whennode i isabouttobreakthecommunicationlink. Inaddition,theprobabilityofmorethanonesimultaneouscollisionwithnode i is assumednegligible,whichmaybecomelesspracticalasapointgrowstoaspherein thepresenceofuncertainty,andastheworkspacebecomesmorecrowded. 2.InChapter4,thetopologyofthenalformationisrequiredtobeatree.Ifthis assumptioncanbeeliminated,theformationcontrolproblemcanbeformulatedas thedesiretoachieveanydesiredformationfromanyarbitraryinitialgraph. 3.InChapter5,oneobjectiveisforagroupofagentswithnonholonomicconstraints tomeetatthesamedestination.Aninterestingextensionistoconsideragentswith morecomplicatedconstraintsanddynamics,suchasunmannedairorunderwater vehicle. Futureworkincontroldevelopmentforasocialnetwork: 1.ThedynamicmodelgiveninChapter6isaheuristicapproximationtoanemotional response.Themodelindicatesthataperson'semotionalstateisadirectrelationship toexternalinuenceintegratedoverthehistoryofaperson'spreviousemotional states.On-goingeortsbythescienticcommunityarefocusedonthedevelopment ofclinicallyderivedmodels.Futureworkisrequiredtoincludemorepreciseclinical emotionalmodelstodescribeaperson'semotionalresponsetoeventsinasocial network,insteadofusingtheheuristicapproximationinChapter6. 120

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2.InChapter6,asocialnetworkismodeledasaundirectedgraph,whereanindividual inthesocialnetworkisrepresentedasavertexinthegraph,andthesocialinteractionbetweentwoindividualsisrepresentedasanundirectededgeofacertainweight connectingtwovertices.Iftwonodesareconnected,theymayinuenceeachother throughtheirsocialinteractioni.e.,theundirectededge.However,theinuence betweentwonodesmayormaynotbesymmetric.Inotherwords,thegraphcanbe undirected,directed,ormixed.Themagnitudeanddirectionorlackofdescribing theedgesbetweennodesmaybedenedthroughtheuseofaninuencefunction. Forinstance,peopleusuallyinteractwithdierentnumbersofindividualsandwith someindividualsmorethanothers.Hence,peoplecanbemoreinuencedbypeers withdierentsocialbonds.Modelingasocialnetworkasanundirected,directedor evenmixedgraph,andstudyingtheinteractionamongnodeswithineachtypeof graphshouldbepursuedasfuturework. 3.SocialbondaredenedinChapter6asaweightededgemeasuringtheamount ofinuencethatissharedbetweenindividuals.Itisassumedthat,thereexista threshold 2 R + ,andtwoindividualsareabletoinuenceeachother'semotional statesonlywhentheirsocialbondisgreaterthan d .Aninterestingproblemishow individualsinuenceeachother.Dothesocialbondsneedtobeaboveaninuence threshold?Doestherehavetobeadirectconnection? 4.Chapter6assumesthat q isacommonequilibriumpointforalltheindividuals inasocialnetwork.However,individualsinasocialnetworkcanhavedierent equilibriumpoints.Forexample,agroupoffriendsmaywishtoengageinan activitythatdiersfromthedesireofanindividual.Follow-oneortstothe currentworkarebeingdevelopedtoincorporatethedynamicsoftheequilibrium point/socialbondarbitrationalongwithinuencestrategiestoenablesocialpeers todeviateapersonfromanequilibrium,orforapersontochangetheequilibriumof thegroup. 121

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REFERENCES [1]ClevelandrapperMachineGunKellyarrestedatFlashMobevent, FoxNews August212011. [2]W.AhmadandR.El-Khazali,Fractional-orderdynamicalmodelsoflove, Chaos, SolitonsandFractals ,vol.33,no.4,pp.1367,2007. [3]T.BalchandR.Arkin,Behavior-basedformationcontrolformultirobotteams, IEEETrans.Robot.Autom. ,vol.14,no.6,pp.926,Dec1998. [4]L.BaoandJ.J.Garcia-Luna-Aceves,Topologymanagementinadhocnetworks, in Proc.ACMInt.Symp.Mob.AdHocNetw.Comput. NewYork,NY,USA:ACM, 2003,pp.129. [5]R.Beard,J.Lawton,andF.Hadaegh,Acoordinationarchitectureforspacecraft formationcontrol, IEEETrans.ControlSyst.Technol. ,vol.9,no.6,pp.777, 2001. [6]P.Bright,HowtheLondonriotsshowedustwosidesofsocialnetworking, Ars Technica ,August102011. [7]Y.Cao,Y.Li,W.Ren,andY.Chen,Distributedcoordinationofnetworked fractional-ordersystems, IEEETransactionsonSystems,Man,andCybernetics, PartB:Cybernetics, ,vol.40,no.2,pp.362,2010. [8]J.Chen,W.E.Dixon,D.M.Dawson,andM.McIntyre,Homography-basedvisual servotrackingcontrolofawheeledmobilerobot, IEEETrans.Robot. ,vol.22,pp. 406,2006.[Online].Available:http://ncr.mae.u.edu/papers/tr06.pdf [9]J.Chen,D.Dawson,M.Salah,andT.Burg,MultipleUAVnavigationwithnite sensingzone,in Proc.Am.ControlConf. ,June2006,pp.4933. [10]Y.Chen,H.Ahn,andI.Podlubny,Robuststabilitycheckoffractionalorderlinear timeinvariantsystemswithintervaluncertainties, SignalProcessing ,vol.86,no.10, pp.2611,2006. [11]Y.ChenandZ.Wang,Formationcontrol:areviewandanewconsideration,in Proc.IEEE/RSJInt.Conf.Intell.Robot.Syst. IEEE,2005,pp.3181. [12]S.Cherry,Peacefulproteststriggercellularshutdowns, IEEESpectrum ,August19 2011. [13]T.Cormen, Introductiontoalgorithms .TheMITpress,2001. [14]C.Dade,FlashMobsaren'tjustforfunanymore, NationalPublicRadio ,May26 2011. 122

PAGE 123

[15]M.DeGennaroandA.Jadbabaie,Formationcontrolforacooperativemulti-agent systemusingdecentralizednavigationfunctions,in Proc.Am.ControlConf. ,June 2006,pp.1346. [16]D.V.DimarogonasandK.J.Kyriakopoulos,Ontherendezvousproblemformultiple nonholonomicagents, IEEETrans.Autom.Control ,vol.52,no.5,pp.916,May 2007. [17]D.DimarogonasandK.Kyriakopoulos,Connectivitypreservingdistributedswarm aggregationformultiplekinematicagents,in Proc.IEEEConf.Decis.Control ,Dec. 2007,pp.2913. [18]D.V.Dimarogonas,S.G.Loizou,K.J.Kyriakopoulos,andM.M.Zavlanos,A feedbackstabilizationandcollisionavoidanceschemeformultipleindependent non-pointagents Automatica ,vol.42,no.2,pp.229243,2006. [19]D.DimarogonasandK.Johansson,Analysisofrobotnavigationschemesusing RantzerdualLyapunovtheorem,in Proc.Am.ControlConf. ,June2008,pp.201 [20],Decentralizedconnectivitymaintenanceinmobilenetworkswithbounded inputs,in Proc.IEEEInt.Conf.Robot.Autom. ,May2008,pp.1507. [21],Boundedcontrolofnetworkconnectivityinmulti-agentsystems, Control TheoryApplications,IET ,vol.4,no.8,pp.1330,Aug.2010. [22]D.DimarogonasandK.Kyriakopoulos,Decentralizedstabilizationandcollision avoidanceofmultipleairvehicleswithlimitedsensingcapabilities,in Proc.Am. ControlConf. ,June2005,pp.46674672vol.7. [23],Afeedbackcontrolschemeformultipleindependentdynamicnon-point agents, Int.J.Control ,vol.79,no.12,pp.1613,2006. [24],Connectednesspreservingdistributedswarmaggregationformultiplekinematic robots, IEEETrans.Robot ,vol.24,no.5,pp.1213,2008. [25]D.Dimarogonas,M.Zavlanos,S.Loizou,andK.Kyriakopoulos,Decentralized motioncontrolofmultipleholonomicagentsunderinputconstraints,in Proc.IEEE Conf.Decis.Control ,vol.4,Dec.2003,pp.33903395. [26]K.Do,Boundedcontrollersforformationstabilizationofmobileagentswithlimited sensingranges, IEEETrans.Autom.Control ,vol.52,no.3,pp.569,march 2007. [27],Formationtrackingcontrolofunicycle-typemobilerobotswithlimitedsensing ranges, IEEETrans.ControlSyst.Technol. ,vol.16,no.3,pp.527,may2008. [28]D.Downs,TheevolutionofFlashMobsfromprankstocrimeandrevolution, The SanFranciscoExaminer ,August282011. 123

PAGE 124

[29]J.FaxandR.Murray,Informationowandcooperativecontrolofvehicle formations, IEEETrans.Autom.Control ,vol.49,no.9,pp.1465,Sept.2004. [30]N.R.Gans,G.Hu,andW.E.Dixon,Simultaneousstabilityofimageandpose errorinvisualservocontrol,in Proc.IEEEInt.Symp.Intell.Control ,SanAntonio, Texas,September2008,pp.438. [31]J.J.Garcia-Luna-AcevesandD.Sampath,Scalableintegratedroutingusingprex labelsanddistributedhashtablesforMANETs,in IEEEInternationalConf.on MobileAdhocandSensorSystems ,2009,pp.188. [32]J.Garcia-Luna-AcevesandD.Sampath,EcientmulticastroutinginMANETsusing prexlabels,in IEEEInternatonalConf.onComputerCommun.andNetworks 2009,pp.1. [33]A.GhaarkhahandY.Mosto,Communication-awaretargettrackingusing navigationfunctions-centralizedcase,in Int.Conf.RobotCommun.Co-ord. ,March 31-April22009,pp.1. [34]K.Ghosh,Fear:Amathematicalmodel, MathematicalModelingandApplied Computing ,vol.1,no.1,pp.27,2010. [35]C.GodsilandG.Royle, AlgebraicGraphTheory ,ser.GraduateTextsinMathematics. Springer,2001. [36]S.Gustin,Socialmediasparked,acceleratedEgypt'srevolutionaryre, Wired February112011. [37]J.Halliday,Londonriots:howBlackBerryMessengerplayedakeyrole, The Guardian ,August82011. [38]J.Huang,S.Farritor,A.Qadi,andS.Goddard,Localizationandfollow-the-leader controlofaheterogeneousgroupofmobilerobots, IEEE/ASMETrans.Mechatron. vol.11,no.2,pp.205,april2006. [39]A.Jadbabaie,J.Lin,andA.Morse,Coordinationofgroupsofmobileautonomous agentsusingnearestneighborrules, IEEETrans.Autom.Control ,vol.48,no.6,pp. 988,June2003. [40]M.JiandM.Egerstedt,Distributedcoordinationcontrolofmultiagentsystemswhile preservingconnectedness, IEEETrans.Robot. ,vol.23,no.4,pp.693,2007. [41]Z.Kan,A.Dani,J.Shea,andW.E.Dixon,Ensuringnetworkconnectivityfor nonholonomicrobotsduringrendezvous,in Proc.IEEEConf.Decis.Control Orlando,FL,T.A. [42],Informationowbasedconnectivitymaintenanceofamulti-agentsystem duringformationcontrol,in Proc.IEEEConf.Decis.Control ,Orlando,FL,T.A. 124

PAGE 125

[43]Z.Kan,A.Dani,J.M.Shea,andW.E.Dixon,Ensuringnetworkconnectivity duringformationcontrolusingadecentralizednavigationfunction,in Proc.IEEE Mil.Commun.Conf. ,SanJose,CA,2010,pp.954. [44],Networkconnectivitypreservingformationstabilizationandobstacleavoidance viaadecentralizedcontroller, IEEETrans.Automat.Control ,T.A. [45]Z.Kan,J.Klotz,T.Cheng,andW.E.Dixon,Ensuringnetworkconnectivityfor nonholonomicrobotsduringdecentralizedrendezvous,in Proc.Am.ControlConf. Montreal,Canada,June27-292012,submitted. [46]Z.Kan,J.M.Shea,andW.E.Dixon,Inuencingemotionalbehaviorinsocial network,in Proc.Am.ControlConf. ,Montreal,Canada,June27-292012,submitted. [47]Z.Kan,S.Subramanian,J.Shea,andW.E.Dixon,Visionbasedconnectivity maintenanceofanetworkwithswitchingtopology,in IEEEMulti-Conf.Syst.and Contr. ,Yokohama,Japan,September2010,pp.1493. [48]O.Khatib,Real-timeobstacleavoidanceformanipulatorsandmobilerobots, Int. J.Robot.Res. ,vol.5,no.1,pp.90,1986. [49]D.E.KoditschekandE.Rimon,Robotnavigationfunctionsonmanifoldswith boundary, Adv.Appl.Math. ,vol.11,pp.412,Dec1990. [50]I.KolmanovskyandN.McClamroch,Developmentsinnonholonomiccontrol problems, IEEEControlSyst.Mag. ,vol.15,no.6,pp.20,1995. [51]J.Lawton,R.Beard,andB.Young,Adecentralizedapproachtoformation maneuvers, IEEETrans.Robot.Autom. ,vol.19,no.6,pp.933,2003. [52]N.LeonardandE.Fiorelli,Virtualleaders,articialpotentialsandcoordinated controlofgroups,in IIEEEConf.Decis.Control ,vol.3.IEEE,2001,pp.2968. [53]M.LewisandK.Tan,Highprecisionformationcontrolofmobilerobotsusingvirtual structures, AutonomousRobots ,vol.4,no.4,pp.387,1997. [54]Y.Li,Y.Chen,andI.Podlubny,Mittag-Leerstabilityoffractionalordernonlinear dynamicsystems, Automatica ,vol.45,no.8,pp.1965,2009. [55]Z.LiandJ.Canny, Nonholonomicmotionplanning .KluwerAcademicPub,1993, vol.192. [56]Z.Lin,M.Broucke,andB.Francis,Localcontrolstrategiesforgroupsofmobile autonomousagents, IEEETrans.Autom.Control ,vol.49,no.4,pp.622,2004. [57]S.LoizouandK.Kyriakopoulos,Closedloopnavigationformultipleholonomic vehicles,in Proc.IEEE/RSJInt.Conf.Intell.Robot.Syst. ,vol.3,2002,pp.2861 2866. 125

PAGE 126

[58],Navigationofmultiplekinematicallyconstrainedrobots, IEEETrans.Robot vol.24,no.1,pp.221,2008. [59]A.Loverro,Fractionalcalculus:history,denitionsandapplicationsfortheengineer, DepartmentofAerospaceandMechanicalEngineering,UniversityofNotreDame 2004. [60]D.Luenberger, Introductiontodynamicsystems:theory,models,andapplications JohnWiley&Sons,1979. [61]S.Mahmood,ThearchitectsoftheEgyptianrevolution, TheNation ,February14 2011. [62]R.Merris,Laplacianmatricesofgraphs:Asurvey, Lin.Algebra.Appl. ,vol.197-198, pp.143,1994. [63]J.Milnor, Morsetheory .PrincetonUnivPr,1963. [64]C.Monje,Y.Chen,B.Vinagre,D.Xue,andV.Feliu, Fractional-orderSystemsand Controls:FundamentalsandApplications .Springer,2010. [65]L.Moreau,Stabilityofcontinuous-timedistributedconsensusalgorithms,in Proc. IEEEConf.Decis.Control ,2004,pp.3998. [66]R.Murray,Recentresearchincooperativecontrolofmultivehiclesystems, Journal ofDynamicSystems,Measurement,andControl ,vol.129,pp.571,2007. [67]L.Navaravong,J.M.Shea,andW.Dixon,Physical-andnetwork-topologycontrol forsystemsofmobilerobots,in Mil.Commun.Conf. ,T.A. [68]P.Ogren,E.Fiorelli,andN.Leonard,Cooperativecontrolofmobilesensornetworks: Adaptivegradientclimbinginadistributedenvironment, IEEETrans.Autom. Control ,vol.49,no.8,pp.1292,2004. [69]R.Olfati-SaberandR.Murray,Consensusproblemsinnetworksofagentswith switchingtopologyandtime-delays, IEEETrans.Autom.Control ,vol.49,no.9,pp. 1520,Sept.2004. [70]R.Olfati-Saber,J.A.Fax,andR.M.Murray,Consensusandcooperationin networkedmulti-agentsystems, Proc.IEEE ,vol.95,no.1,pp.215233,January 2007. [71]D.Pais,M.Cao,andN.Leonard,Formationshapeandorientationcontrolusing projectedcollineartensegritystructures,in Proc.Am.ControlConf. ,june2009,pp. 610. [72]A.Rantzer,Adualtolyapunov'sstabilitytheorem, Systems&ControlLetters vol.42,no.3,pp.161168,2001.[Online].Available:http://www.sciencedirect. com/science/article/B6V4X-42BSJB9-1/2/afc4d7b165898322d9410de642da3f56 126

PAGE 127

[73]W.RenandR.Beard,Consensusseekinginmultiagentsystemsunderdynamically changinginteractiontopologies, IEEETrans.Autom.Control ,vol.50,no.5,pp. 655,2005. [74]W.Ren,R.Beard,andE.Atkins,Asurveyofconsensusproblemsinmulti-agent coordination,in Proc.Am.ControlConf. IEEE,2005,pp.1859. [75],Informationconsensusinmultivehiclecooperativecontrol, IEEEControlSyst. Mag ,vol.27,no.2,pp.71,2007. [76]E.RimonandD.Koditschek,Exactrobotnavigationusingarticialpotential functions, IEEETrans.Robot.Autom. ,vol.8,no.5,pp.501,Oct1992. [77]G.Roussos,D.Dimarogonas,andK.Kyriakopoulos,Dnavigationandcollision avoidanceforanon-holonomicvehicle,in Proc.Am.ControlConf. ,2008,pp.3512 [78]M.Saba,WallStreetprotestersinspiredbyArabSpringmovement, CNN September172011. [79]D.SampathandJ.J.Garcia-Luna-Aceves,Prose:scalableroutinginMANETsusing prexlabelsanddistributedhashing,in IEEECommun.SocietyConf.onSensor, MeshandAdHocCommun.andNetworks ,2009,pp.1. [80]R.Sepulchre,D.A.Paley,andN.E.Leonard,Stabilizationofplanarcollective motion:All-to-allcommunication, IEEETrans.Autom.Control ,vol.52,no.5,pp. 811,May2007. [81]R.Sepulchre,D.Paley,andN.Leonard,Stabilizationofplanarcollectivemotion withlimitedcommunication, IEEETrans.Autom.Control ,vol.53,no.3,pp.706 ,Apr2008. [82]N.Shachtman,Exclusive:InsideDarpa'ssecretAfghanspymachine, Wired ,July 212011. [83]R.Sinha,J.Zobel,andD.Ring,Cache-ecientstringsortingusingcopying, JournalofExperimentalAlgorithmicsJEA ,vol.11,pp.1,2007. [84]J.SlotineandW.Li, AppliedNonlinearControl .PrenticeHall,1991. [85]L.Song,S.Xu,andJ.Yang,Dynamicalmodelsofhappinesswithfractionalorder, CommunicationsinNonlinearScienceandNumericalSimulation ,vol.15,no.3,pp. 616,2010. [86]D.SpanosandR.Murray,Robustconnectivityofnetworkedvehicles,in Proc.IEEE Conf.Decis.Control ,vol.3,Dec.2004,pp.2893. [87],Motionplanningwithwirelessnetworkconstraints,in Proc.Am.Control Conf. ,Jun.2005,pp.8792. 127

PAGE 128

[88]J.Sprott,Dynamicalmodelsoflove, Nonlineardynamics,psychology,andlife sciences ,vol.8,no.3,pp.303,2004. [89],Dynamicalmodelsofhappiness, NonlinearDynamics,Psychology,andLife Sciences ,vol.9,no.1,2005. [90]H.G.Tanner,A.Jadbabaie,andG.J.Pappas,Flockinginxedandswitching networks, IEEETrans.Autom.Control ,vol.52,no.5,pp.863,May2007. [91]H.TannerandA.Kumar,Towardsdecentralizationofmulti-robotnavigation functions,in Proc.IEEEInt.Conf.Robot.Autom. ,April2005,pp.41324137. [92]H.TannerandK.Kyriakopoulos,Nonholonomicmotionplanningformobile manipulators,in IEEEInternationalConferenceonRoboticsandAutomation ,vol.2, 2000,pp.1233vol.2. [93]H.Tanner,S.Loizou,andK.Kyriakopoulos,Nonholonomicnavigationandcontrol ofcooperatingmobilemanipulators, IEEETrans.Robot.Autom. ,vol.19,no.1,pp. 53,Feb2003. [94]H.Tanner,G.Pappas,andV.Kumar,Leader-to-formationstability, IEEETrans. Robot.Autom. ,vol.20,no.3,pp.443,2004. [95]Z.YaoandK.Gupta,Backbone-basedconnectivitycontrolformobilenetworks,in Proc.IEEEInt.Conf.Robot.Autom. ,May2009,pp.1133. [96]W.Zachary,Aninformationowmodelforconictandssioninsmallgroups, J. Anthropol.Res. ,pp.452,1977. [97]M.Zavlanos,M.Egerstedt,andG.Pappas,Graphtheoreticconnectivitycontrolof mobilerobotnetworks,2011. [98]M.Zavlanos,A.Jadbabaie,andG.Pappas,Flockingwhilepreservingnetwork connectivity,in Proc.IEEEConf.Decis.Control ,Dec.2007,pp.2919. [99]M.ZavlanosandG.Pappas,Potentialeldsformaintainingconnectivityofmobile networks, IEEETrans.Robot. ,vol.23,no.4,pp.812,Aug.2007. [100],Distributedconnectivitycontrolofmobilenetworks, IEEETrans.Robot. vol.24,no.6,pp.1416,Dec.2008. [101]M.Zavlanos,H.Tanner,A.Jadbabaie,andG.Pappas,Hybridcontrolfor connectivitypreservingocking, IEEETrans.Autom.Control ,vol.54,no.12,pp. 2869,2009. [102]F.Zhang,Geometriccooperativecontrolofparticleformations, IEEETrans. Autom.Control ,vol.55,no.3,pp.800,2010. [103]F.ZhangandN.Leonard,Cooperativeltersandcontrolforcooperativeexploration, IEEETrans.Autom.Control ,vol.55,no.3,pp.650,2010. 128

PAGE 129

[104]F.Zhao,J.Liu,J.Liu,L.Guibas,andJ.Reich,Collaborativesignalandinformation processing:aninformation-directedapproach, Proc.IEEE ,vol.91,no.8,pp. 1199,Aug.2003. [105]F.Zhao,J.Shin,andJ.Reich,Information-drivendynamicsensorcollaboration, IEEESignal.Proc.Mag. ,vol.19,no.2,pp.61,Mar2002. 129

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BIOGRAPHICALSKETCH ZhenKanwasborninMarch,1983inHefei,China.HereceivedhisBachelorof EngineeringdegreeinMechanicalEngineeringin2005,andaMasterofSciencedegreein MechatronicEngineeringin2007fromHefeiUniversityofTechnologyHFUT,China.He thenjoinedtheNonlinearControlsandRoboticsNCRresearchgroupattheUniversity ofFloridaUFtopursuehisdoctoralresearchundertheadvisementofDr.WarrenDixon andcompletedhisPh.D.inDecember2011.FromJanuary2012onwardhewillbea post-doctoralfellowwiththeUniversityofFlorida. 130