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Trajectory Optimization Using Pseudospectral Methods for a Multiple Autonomous Underwater Vehicle Target Tracking Problem

Permanent Link: http://ufdc.ufl.edu/UFE0044152/00001

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Title: Trajectory Optimization Using Pseudospectral Methods for a Multiple Autonomous Underwater Vehicle Target Tracking Problem
Physical Description: 1 online resource (50 p.)
Language: english
Creator: Franklin, Adam J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: auv -- control -- optimal -- pseudospectral -- trajectory
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

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Abstract: In this work, a method is presented for solving a multiple AUV target tracking problem using pseudospectral methods. The multiple AUV target tracking problem involves generating a trajectory that minimizes the sum of the distances from each AUV to the target over the entire trajectory. The trajectory generated for each vehicle must not violate the distance constraint between any other vehicle or the target. The multiple AUV target tracking problem is solved as an optimal control problem using the open-source optimal control software GPOPS and a nonlinear programming problem solver SNOPT. The results of the multiple AUVs target tracking problem for two different target scenarios are analyzed and validated.
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 Adam J Franklin.
Thesis: Thesis (M.S.)--University of Florida, 2012.
Local: Adviser: Rao, Anil.
Local: Co-adviser: Dixon, Warren E.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044152:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044152/00001

Material Information

Title: Trajectory Optimization Using Pseudospectral Methods for a Multiple Autonomous Underwater Vehicle Target Tracking Problem
Physical Description: 1 online resource (50 p.)
Language: english
Creator: Franklin, Adam J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: auv -- control -- optimal -- pseudospectral -- trajectory
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this work, a method is presented for solving a multiple AUV target tracking problem using pseudospectral methods. The multiple AUV target tracking problem involves generating a trajectory that minimizes the sum of the distances from each AUV to the target over the entire trajectory. The trajectory generated for each vehicle must not violate the distance constraint between any other vehicle or the target. The multiple AUV target tracking problem is solved as an optimal control problem using the open-source optimal control software GPOPS and a nonlinear programming problem solver SNOPT. The results of the multiple AUVs target tracking problem for two different target scenarios are analyzed and validated.
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 Adam J Franklin.
Thesis: Thesis (M.S.)--University of Florida, 2012.
Local: Adviser: Rao, Anil.
Local: Co-adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044152:00001


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TRAJECTORYOPTIMIZATIONUSINGPSEUDOSPECTRALMETHODSFORAMULTIPLEAUTONOMOUSUNDERWATERVEHICLETARGETTRACKINGPROBLEMByADAMJ.FRANKLINATHESISPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFMASTEROFSCIENCEUNIVERSITYOFFLORIDA2012

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c2012AdamJ.Franklin 2

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TomywifeKristieandmyparentsMarkandKathy 3

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ACKNOWLEDGMENTS IwouldliketoexpressmyappreciationtomysupervisorycommitteechairandadvisorDr.AnilV.Raoforhissupport,encouragementandtechnicalguidancethroughoutmyresearch.Also,Iwouldliketothankmycommitteeco-chairDr.WarrenE.Dixonforhisteachingandsupportduringmycourseofstudy.IwouldliketoalsothankMichaelA.Pattersonforhelpingmetobetterunderstandmyresearch.Mostimportantly,IwouldliketoexpressmydeepestappreciationtomywifeKristieandmyparentsMarkandKathy.Theirloveandsupportmadethisthesispossible. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 9 CHAPTER 1INTRODUCTION ................................... 10 1.1Motivation .................................... 10 1.2LiteratureReview ................................ 10 1.2.1PathPlanning .............................. 10 1.2.2MultipleAUVsCoordination ...................... 11 1.2.3TargetDetectionandLocalization ................... 13 1.3AbouttheThesis ................................ 14 1.4ThesisOutline ................................. 14 2MULTIPLEAUVTARGETTRACKINGPROBLEM ................ 15 2.1StructureofanOptimalControlProblem ................... 15 2.1.1FirstOrderOptimalityConditions ................... 17 2.1.2TransversalityConditions ........................ 17 2.2MethodsforSolvingOptimalControlProblems ............... 18 2.2.1IndirectMethods ............................ 18 2.2.2DirectMethods ............................. 19 2.3ProblemFormulation .............................. 19 2.3.1EquationsofMotion .......................... 20 2.3.2BoundaryConditions .......................... 22 2.3.3PathConstraints ............................ 24 2.3.3.1Stateinequalityconstraints ................. 24 2.3.3.2Targetcollisionavoidanceconstraints ........... 25 2.3.3.3Friendlycollisionavoidanceconstraints .......... 25 2.3.3.4Controlconstraints ...................... 25 2.3.3.5Costfunctional ........................ 26 3PSEUDOSPECTRALMETHODSANDGPOPS .................. 27 3.1LG,LGR,andLRLCollocationPoints .................... 27 3.2FormulationofPseudospectralMethodUsingLGRPoints ......... 28 3.3hp-AdaptivePseudospectralMethod ..................... 29 3.4ProblemFormulationinGPOS ........................ 30 3.4.1ObjectiveandInputParameters .................... 30 5

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3.4.2CostFunctionFormulation ....................... 31 3.4.3PathConstraintFormulation ...................... 31 4RESULTSANDDISCUSSION ........................... 34 4.1UnderwaterTarget ............................... 34 4.1.1AnalysisofResults ........................... 34 4.1.2ValidationofResults .......................... 34 4.2SurfaceTarget ................................. 36 4.2.1AnalysisofResults ........................... 36 4.2.2ValidationofResults .......................... 36 5CONCLUSION .................................... 47 REFERENCES ....................................... 48 BIOGRAPHICALSKETCH ................................ 50 6

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LISTOFTABLES Table page 2-1Notation ........................................ 15 2-2Equationsofmotionparameters .......................... 21 3-1User-speciedinitialandnalstatesofthefourAUVs .............. 30 4-1Lagrangecostvalidationforanunderwatertarget ................. 35 4-2Lagrangecostvalidationforasurfacetarget .................... 36 7

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LISTOFFIGURES Figure page 3-1SchematicshowingLGL,LGRandLGcollocationpoints ............. 33 4-1Three-dimensionaltrajectoryofAUVstrackingtheunderwatertarget ...... 38 4-2Top-downviewoftrajectoryfortrackingtheunderwatertraget .......... 38 4-3Forwardpositionversustimefortrackingtheunderwatertraget ......... 39 4-4Lateralpositionversustimefortrackingtheunderwatertraget .......... 39 4-5Depthversustimefortrackingtheunderwatertraget ............... 40 4-6Thelagrangecostversustimefortheunderwatertarget ............. 40 4-7DistancefromAUVstotargetversustimefortheunderwatertarget ....... 41 4-8AveragedistancefromoneAUVtotheothersversustimefortheunderwatertarget ......................................... 42 4-9Three-dimensionaltrajectoryofAUVstrackingthesurfacetarget ........ 42 4-10Top-downviewoftrajectoryfortrackingthesurfacetraget ............ 43 4-11Forwardpositionversustimefortrackingthesurfacetraget ........... 43 4-12Lateralpositionversustimefortrackingthesurfacetraget ............ 44 4-13Depthversustimefortrackingthesurfacetraget ................. 44 4-14Thelagrangecostversustimeforthesurfacetarget ............... 45 4-15DistancefromAUVstotargetversustimeforthesurfacetarget ......... 45 4-16AveragedistancefromoneAUVtotheothersversustimeforthesurfacetarget 46 8

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AbstractofThesisPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofMasterofScienceTRAJECTORYOPTIMIZATIONUSINGPSEUDOSPECTRALMETHODSFORAMULTIPLEAUTONOMOUSUNDERWATERVEHICLETARGETTRACKINGPROBLEMByAdamJ.FranklinMay2012Chair:AnilV.RaoCochair:WarrenE.DixonMajor:MechanicalEngineeringInthiswork,amethodispresentedforsolvingamultipleAUVtargettrackingproblemusingpseudospectralmethods.ThemultipleAUVtargettrackingprobleminvolvesgeneratingatrajectorythatminimizesthesumofthedistancesfromeachAUVtothetargetovertheentiretrajectory.Thetrajectorygeneratedforeachvehiclemustnotviolatethedistanceconstraintbetweenanyothervehicleorthetarget.ThemultipleAUVtargettrackingproblemissolvedasanoptimalcontrolproblemusingtheopen-sourceoptimalcontrolsoftwareGPOPSandanonlinearprogrammingproblemsolverSNOPT.TheresultsofthemultipleAUVstargettrackingproblemfortwodifferenttargetscenariosareanalyzedandvalidated. 9

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CHAPTER1INTRODUCTION 1.1MotivationTheDepartmentofTransportationisresponsibleforthesafetyofmorethan300seaandriverportswithmorethan3,700terminals[ 1 ].TheUnitedStatesNavyhas285deployableshipswhich,whenunderwayinhostilewaters,arepossibletargetsforterroristattacks[ 2 ].TheprotectionoftheseassetsisatoppriorityoftheDepartmentsofTransportationandDefense,consumingmoney,equipmentandmanpower.Currently,theresponsibilityofmonitoringthesonarsweepsofportsandwaterssurroundingshipsforpossiblethreatsfallstotheindividualonwatch.Thisdullandrepetativetaskisbettersuitedforautonomousunderwatervehicles(AUVs). 1.2LiteratureReview 1.2.1PathPlanningAUVsarebeingusedtocompletetaskssuchasoceansampling,mapping,oceanoorsurveying,oceanographicdatacollection,minesweepingandportandshipsecurity.Inordertoaccomplishthesetasks,AUVsmustbeabletotrackadesiredpath.Thetwomostcommondivisionsofpathplanningaretrajectorytrackingandpathfollowing.[ 3 4 ]Trajectorytrackingreferstodrivingavehicletotrackatime-parameterizedreferencecurve.Ghommam,J.andothers[ 3 ]proposedatrajectorytrackingcontrolthatcouldbeappliedtoagroupofunderactuatedAUVs.Lyapunov-basedcontroltechniqueshadbeenusedfortrajectorytrackingofasingleAUV,butGhommam,J.andothers[ 3 ]wereabletodecentralizethecontroltoagroupofAUVsbyusinggraphtheory.Throughdecentralization,thecontrolstructuretakesintoaccountthedynamicsofallvehiclesaswellastheconstraintsoftheinter-vehiclecommunicationsnetwork.TheirworkwasvalidatedbysimulatingagroupofthreeunderwaterAUVswithgoodtrackingresults.Thegoalofpathfollowingistoforceavehicletoconvergetoandfollowadesiredspatialpath,withoutregardtotime.Wang,Y.andothers[ 4 ]developeda 10

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backstepping-basedcontrollawforaccuratepathfollowingforanunderactuatedAUV.TheyachievedthiscontrollawbydevelopingapathtrackingcontrolfromthekinematicsandthenextendingthecontroltotheAUVdynamicsthroughbacksteppingtechniques.Wang,Y.andothers[ 4 ]provedtheconvergenceofthevehicle'strajectorytothedesiredpathviaLyapunovstabilityanalysis.However,theircontrollerreliedheavilyoncomplexdynamicsthatmaynotalwaysbeknown. 1.2.2MultipleAUVsCoordinationInmanyapplications,agiventaskistoocomplicatedtobeaccomplishedbyasingleAUV.Therefore,multipleAUVs,workingtogether,arerequiredtocompletethetask.Furthermore,asystemofmultipleAUVsismorerobustthanasinglevehicle,becauseateamofvehiclesprovidesredundancy.However,onefundamentalproblemofasystemofAUVsisvehiclecoordinationwhilecompletinganassignedtask.Vehiclecoordinationstrategiesusingeitherabehavioral,avirtualstructure,queuesandarticialpotentialtrenchesoraleader-followerapproachhavebeenproposedinliterature[ 5 10 ].Behavioralcoordinationisadistributedautonomouscontrolapproachforasystemofmultiplevehicles.Articialforcelawsaredenedbetweenvehiclesinagroup.Theselawsareinverse-powerforcelaws,incorporatingbothattractionandrepulsion.Theforcelawsarewell-denedandtheyreecttheinteractionsamongvehicles.Anindividualvehicle'smotionisdictatedbytheforcesimposedbyothervehiclesinthesystem.Furthermore,thisapproachisdistributedbecauseeachvehicledeterminesitsmotionbyobservingtheforcelawsbetweenitselfandtheothervehicles.Reif,J.andothers[ 5 ]studiedthebehavioralcooperationapproachteamsofmultiplerobots.Theyobtainedcomputersimulatedresultswherethesystemofmultiplerobotsconvergedtothedesiredformation.Invirtualstructurecoordination,thecontrolisderivedinthreesteps.First,thedesireddynamicsofthevirtualstructureisdened.Second,themotionofthevirtualstructureistranslatedintothedesiredmotionforeachvehicle.Finally,trackingcontrols 11

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foreachvehiclearederived.Beard,R.andothers[ 6 ]developedavirtualstructureapproachforthecoordinationofmultiplespacecrafts.Thestrengthofthevirtualstructureapproachisthatitisfairlyeasytoprescribeacoordinatedbehaviorfortheformationofspacecrafts.Furthermore,thefeedbacktothevirtualstructureisnaturallydened,foreachspacecraftreactsaccordingtothepositionsofothersintheformation.However,thedisadvantageofthisapproachisthatitscurrentdevelopmentlendsitselftoacentralizedimplementation.Ge.S,andothers[ 8 ]developedanapproachforrepresentingateamofmultiplevehiclesintermsofqueuesandvertices,ratherthannodes,aswellastheintroductionofanewconceptofarticialpotentialtrenches,foreffectivelycontrollingtheformation.Thisapproachimprovesthescalabilityandexibilityofavehicleteamwhenthenumberofvehiclesintheformationchangesandallowstheformationtoadapttoobstacles.Furthermore,theteamisprotectedagainstfailuresofindividualvehicles,fortheformationscalesandadaptsautomatically.Throughsimulation,Ge.S,andothers[ 8 ]showthecommunicationbandwidthbetweenvehiclesplacesanupperboundonthemaximumnumberofvehiclesintheformation.Intheleader-followerapproachtomultiplevehiclecoordination,onevehicleisdesignatedastheleader.Theleadermovesalongapredenedtrajectorywhiletheothervehicles,thefollowers,maintainadesireddistanceandorientationwithrespecttotheleader[ 9 ].IntheworkbyEmrani,S.andothers[ 10 ],theleader-followerformationcoordinationofmultipleAUVsisexpandeduponbyincorporatinguncertaintiesinthehydrodynamics.Todealwiththeseuncertainties,anadaptivecontrollawbasedontheinversedynamicsoftheplantwasdeveloped.TheygoontoprovideaLyaponov-basedclosed-loopstabilityanalysisfortheproposedcontroller.Simulationresultshavedemonstratedtheeffectivenessoftheproposedapproachforleader-followerformationcontroloftheAUVs. 12

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1.2.3TargetDetectionandLocalizationInapplicationswhereAUVsareminesweepingorprovidingsecurityforportsorships,theabilitytodetectandlocalizetargetsiscrucial.Inrecentyears,amajorityoftheresearchtoimprovetheseabilitieshasbeeninimprovingeitherthesensorsanddataanalysistechniquesortheinformationsharingbetweenmultipleAUVs[ 11 12 ].Wiegert,R.andothers[ 11 ]workedtodevelopanewmagneticsensorsystemforanAUVinordertoimproveminedetectioninlitoralwaters.Thepreviousmethod,calledthescalartriangulationandranging(STAR)method,usedrotationallyinvariantscalarquantitiesinordertodetectmagnetictargets.Toimproveuponthismethod,Weigert,R.andothers[ 11 ]usedmulti-tensordatafrommagneticgradiometers,whichexploitedtherotationallyinvariantandrobustsymmetrypropertiesofthegradientcontractionscalareld.SimulationresultsshowedanimprovementinmagnetictargetdetectionovertheoriginalSTARmethod.Rauch,C.andothers[ 12 ]workedtodevelopamodulethatcouldbeaddedtoanAUV,allowingthevehicletoholdsixtotendeployablesonartransponders.ThesemarkersweretobejettisonednearatargetandthedeployingAUVwouldsurveythemarkerstodeterminetheirrelativepositiontothetarget.SubsequentAUVswerethengiventhebestknowncoordinatesofthetargetandtherelativeoffsetstothemarkers,allowingforguidancetothetarget.Rauch,Candothers[ 12 ]alsoworkedtoimprovetheacousticsignalprocessingrequiredtosurveythemarkersandthencommunicatetheirrelativelocationstootherAUVs.Otherworkintargetlocalizationhasbeendonenotbyimprovingsensors,butbyimprovinghowinformationissharedbetweenvehicles.Belbachir,A.andothers[ 13 ]workedtodeneacontrolstrategywhichadaptseachAUV'smotionaccordingtoitsorothervehicles'sensoryinformation.Theirgoalwastominimizetherangebetweenvehiclesduringinformationexchangesinordertoreduceexchangetime.Belbachir,A. 13

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andothers[ 13 ]whereabletovalidatedtheircontrolstrategythroughexperiment,usingtwoAUVsandonesurfacevehicletoactasarelaybetweenthetwoAUVs. 1.3AbouttheThesisInthisresearch,multipleAUVspatrolaregionofinterest(ROI)forunidentiedcontacts.Upondetectionaunidentiedcontact,allvehiclesaretoclosewithandtrackthetargetuntilitleavestheROI.Anhp-adaptivepseudospectralmethodthatusesRadaucollocationpointsisimplementedtogeneratethetrajectoriesthatminimizesthesumofthedistancesfromeachAUVtothetarget.Thesetrajectoriessatisfythedynamicsofthevehicles.ThisworkcouldbeadaptedsothatinsteadoftrackingthetargetunitlitleavestheROI,theAUVscouldblock,observeorevendisablethetarget.TheresearchcanbeusedtoassistnotonlytheDepartmentofTransportationinprotectingdomesticportsandwaterways,butalsotheDepartmentofDefenseinprotectingshipsinhostilewaters.Withfutureworkinthisarea,nearreal-timeimplementationofoptimalcontrolispossible. 1.4ThesisOutlineThisthesisisorganizedasfollows.Chapter 1 isanintroductiononthemotivationfortheresearchoftargettrackingusingmultipleAUVsandareviewofthepreviousworkrelatedtotheresearch.InChapter 2 ,themultipleAUVstargettrackingproblemisformulatedalongwithageneraldescriptionofoptimalcontrolproblemswithitsoptimalityandtransversalityconditions.Chapter 3 givesabriefdescriptionoftheRadaupsedospectralandhp-adaptivemethodsanddescribestheformulationoftheoptimalcontrolproblem.InChapter 4 theresultsofthisresearcharepresentedanddiscussed. 14

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CHAPTER2MULTIPLEAUVTARGETTRACKINGPROBLEMThischapterprovidesabriefdescriptionofandthemethodsforsolvinganoptimalcontrolproblem.Furthermore,themultipleAUVtargettrackingproblemisdescribed.Finally,thischapterdescribestheequationsofmotionforthevehiclesandthecostfunctionformulation. 2.1StructureofanOptimalControlProblemAnoptimalcontrolproblemiscomprisedofthedynamicequationsofmotion,theobjectivefunctional,boundaryconditionsandpathconstraints.Belowisatableofthenotationusedindescribingtheformulationofanoptimalcontrolproblem. Table2-1. Notation SymbolDescription t0InitialTimetfFinalTimex(t0)StateValueattheInitialTime,t0x(tf)StateValueattheFinalTime,tfJCostFunctionalJaAugmentedCostFunctionalQMayerCostLLagrangeCostBoundaryConditiongPathConstraintHHamiltonian#LagrangeMultiplierforBoundaryConditionCostateoftheDifferentialEquation Thedynamicequationsofmotionneedtobecontinuousanddifferentiable.Thedynamicequationsofmotionareconvertedintostatespacerepresentationinordertohavetheindividualstateequationsandtheirrelationtothecontrolinput.Therefore,thenumberofequationsisequaltothenumberofstatesintheproblem.ThedynamicequationsofmotionarerepresentedasEquation( 2 )below, _x=f(x,u,t)(2) 15

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wherex(t)2Rnisthestate,u(t)2Rpisthecontrol,andtisthetime.Theobjectivefunctional,alsocalledthecostfunctional,istheparametertobeoptimizedwhilesolvingtheoptimalcontrolproblem.Thecostfunctionalisafunctionalofthestatevariables,controlvariables,and/ortime.Specically,thecostfunctionalcanbetheendpointstates,theendpointtimeorthestateand/orcontrolovertheentireperiodoftime.ThepartofthecostfunctionalconsistingoftheendpointsistheMayercostandthepartconsistingofstateorcontrolvariablesfortheentireperiodoftimeistheLagrangecost. J=Q(x(t0),t0,x(tf),tf)+Ztft0L(x(t),u(t),t)dt(2)Intheabovecostfunctional,Q(x(t0),t0,x(tf),tf)istheMayercostandL(x(t),u(t),t)istheLagrangecost.Theboundaryconditionsaretheinitialandterminalvaluesofthestatesandtime,eitherknownatthebeginningoftheproblemortobeachieved.Theboundaryconditionsbecometheinitialandterminalconstraintsoftheoptimalcontrolproblem. (x(t0),t0,x(tf),tf)=0(2)Thepathconstraintsarethelinearornonlinearconstraintstobesatisedbythetrajectoryofthesystem.Theycanbeeitherequalityorinequalityconstraints.Thenonlinearinequalityconstraintsarerepresentedas g(x(t),u(t),t)0(2)Thenonlinearequalityconstraintsarerepresentedas g(x(t),u(t),t)=0(2) 16

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2.1.1FirstOrderOptimalityConditionsTheaugmentedcostfunctionalisformulatedas Ja=Q)]TJ /F9 11.955 Tf 11.96 0 Td[(#T+Ztft0L+T(f)]TJ /F8 11.955 Tf 13.54 0 Td[(_x)dt(2)TheHamiltonianisrepresentedas H=L+Tf=H(x,,u)(2)TheaugmentedcostfunctionalintermsoftheHamiltonianisgivenby Ja=Q)]TJ /F9 11.955 Tf 11.96 0 Td[(#T+Ztft0H)]TJ /F9 11.955 Tf 11.95 0 Td[(T_xdt(2)Theoptimalityconditionforcostatedynamicsisgivenas _x=@H @T(2)Theoptimalcontrol,u,isfoundusingtheconditiongivenas @H @uT=0(2) 2.1.2TransversalityConditionsTherstvariationoftheaugmentedcostfunctionalJaisusedtondthetransversalityconditions.Theseconditionsareusedtosolvethedifferentialequations,whicharederivedfromtherstorderoptimalityconditions.Therearedifferenttransversalityequationsforthedifferentboundaryconditionsonthestateandtimeofthesystem.Thetransversalityequationsaregivenbelow: Fornoboundaryconditionsonthestateandtimeofthesystem,#=0,theboundaryconditionis(x(t0),t0,x(tf),tf)=0 Foraxedinitialstate,x0=0,thecostateattheinitialtimeisgivenby(t0)=)]TJ /F11 11.955 Tf 11.29 16.85 Td[(@Q @x(t0)T+@ @x(t0)T# 17

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Foraxednalstate,xf=0,thecostateatthenaltimeisgivenby(t0)=@Q @x(tf)T)]TJ /F11 11.955 Tf 11.95 16.86 Td[(@ @x(tf)T# Foraxedinitialtime,t0=0,theHamiltonianattheinitialtimeisgivenbyH(t0)=@Q @t0)]TJ /F9 11.955 Tf 11.95 0 Td[(#T@ @t0 Foraxednaltime,tf=0,theHamiltonianatthenaltimeisgivenbyH(tf)=)]TJ /F9 11.955 Tf 10.49 8.09 Td[(@Q @tf+#T@ @tf 2.2MethodsforSolvingOptimalControlProblemsInordertosolveanoptimalcontrolproblem,thedifferentialequationsneedtobesolved,subjecttoconstraints,whileoptimizingaperformanceindex.Thetwomainclassicationsofmethodstosolvesuchoptimalcontrolproblemswithpathconstraintsandboundaryconditionsareindirectmethodsanddirectmethods. 2.2.1IndirectMethodsIndirectmethodsstemfromthecalculusofvariations.Indirectmethodsrequiresolvingatwo-pointboundaryvalueproblem.Here,theoptimalityconditionsforsolvingtheoptimaltrajectoryarederived.Therstorderoptimalityconditionsareformedbytakingtherstvariationofthecostfunctional,usingcalculusofvariations.Thisproducestherst-orderdifferentialequationsforthestatesandthecostates.ThetransversalityconditionsoftenproducenonlinearequationswhichcannotbesolvedwiththeRicattiequation.Thus,thesenonlinearequationsformthenonlinearconstraintsoftheproblem.Indirectmethodsconverttheoptimalcontrolproblemintoapurelydifferential-algebraicsystemofstates,costatesanddynamics.Hence,theoptimalcontrolproblembecomesarootndingproblem,whereasetofdifferentialequationshastobesolvedandrootshavetobefoundtosatisfytheconstraints. 18

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2.2.2DirectMethodsWithdirectmethods,allofthefunctionsintheoptimalcontrolproblemareapproximatedthentranscribedintoanonlinearoptimizationproblem.Bymakingthisapproximation,thecontinuoustimeinnite-dimensionalproblemisconvetedintoanonlinearprogrammingproblem,whichisanite-dimensionalproblem.Thequadratureapproximationoftheintegralisusedinthecontinuouscostfuntionasgivenbelow J=Q+Ztft0Ldt=QuadratureApproximationofIntegral,JJa.(2)Thepathconstraintsareevaluatedonlyatspecicpointsofthequadrature.Thepseudospectralmethodisadirectmethodofsolvinganoptimalcontrolproblem,whichisexplainedindetailinChapter 3 2.3ProblemFormulationConsiderahomogenousgroupofAAUVspatrollingaregionofinterest(ROI).TheROIforthisproblemisarectangularprism.EachoftheAAUVsisresponsibleforitsownsubsectionoftheROI,whichisalsoarectangularprism.ThegoalisforalloftheAAUVstotrackatargetfromthetimeitenterstheROItothetimeitleavestheROI,whileobeyingtargetandotherAUVcollisionavoidanceconstraintsandoptimizingaperformanceindex.TheperformanceindexusedinthisproblemisminimizingthesumofthedistancefromeachAUVtothetargetovertheentiretrajectory. 19

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2.3.1EquationsofMotionThetranslationalkinematicequationsforAAUVsmaneuveringunderthesurfaceofaatbodyofwaterare _xi(t)=uicos( i)cos(i)+v1(cos( i)sin(i)sin(i))]TJ /F8 11.955 Tf 11.96 0 Td[(cos(i)sin( i))+wi(sin(i)sin( i)+cos(i)cos( i)sin(i)),_yi(t)=uicos(i)sin( i)+vi(cos(i)cos( i)+sin(i)sin( i)sin(i))+wi()]TJ /F8 11.955 Tf 11.29 0 Td[(cos( i)sin(i)+cos(i)sin( i)sin(i)),_zi(t)=)]TJ /F5 11.955 Tf 9.3 0 Td[(uisin(i)+vicos(i)sin(i)+wicos(i)cos(i),(2)wherei=1,...,A,xi(t)andyi(t)arethehorizontalCartesiancomponentsofposition,zi(t)aretheinvertedverticalCartesiancomponentofposition(depth)andi,iand iaretheroll,pitchandyawanglesoftheAUVs.TherotationalkinematicequationsforAAUVsare i(t)=qisin(i)sec(i)+ricos(i)sec(i),_i(t)=qicos(i))]TJ /F5 11.955 Tf 11.95 0 Td[(risin(i),_i(t)=pi+qisin(i)tan(i)+ricos(i)tan(i),(2)wherei=1,...,A,i,iand iaretheroll,pitchandyawangles,ui,viandwiarethevelocities,andpi,qiandriaretheangularvelocitiesoftheAUVs.Thedifferentialequationsdescribingthesix-degreeoffreedommotionofeachAUVisgivenbelow.ThefollowingequationsofmotionwereadaptedfromtheworkofArslan, 20

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M.andothers[ 14 ] _ui(t)=1 m(Fix+Xuuu1juij)+viri)]TJ /F5 11.955 Tf 11.96 0 Td[(wiqi,_vi(t)=1 m(k2Miy+Yvvvijvij)+wipi)]TJ /F5 11.955 Tf 11.95 0 Td[(uiri,_wi(t)=1 m(k2Miz+Zwwwijwij)+uiqi)]TJ /F5 11.955 Tf 11.95 0 Td[(vipi,_pi(t)=1 Ix(Mix+Kpppijpij))]TJ /F8 11.955 Tf 11.95 0 Td[((Iz)]TJ /F5 11.955 Tf 11.96 0 Td[(Iy)qiri,_qi(t)=1 Iy(Miy+Mqqqijqij))]TJ /F8 11.955 Tf 11.95 0 Td[((Ix)]TJ /F5 11.955 Tf 11.96 0 Td[(Iz)piri,_ri(t)=1 Iz(Miz+Nrrrijrij))]TJ /F8 11.955 Tf 11.96 0 Td[((Iy)]TJ /F5 11.955 Tf 11.96 0 Td[(Ix)piqi,(2)wherei=1,...,A.Inthismodel,thecontrolinputstotheAUVsarethethrustFixandthemomentsMix,MiyandMiz.Below,Table 2-2 liststheparametersusedinEquation( 2 ). Table2-2. Equationsofmotionparameters SymbolValueDescription m30Vehiclemass(kg)Ix,Iy,Iz0.177,3.45,3.45Momentsofinertia(kgm2)Xuu-3.9Axialdragcoef.(kg=m)Yvv-131Cross-owdragcoef.(kg=m)Zww-131Cross-owdragcoef.(kg=m)Kpp-0.13Rollingresistancecoef.(kgm2=rad2)Mqq-94Rollingresistancecoef.(kgm2=rad2)Nrr-94Rollingresistancecoef.(kgm2=rad2)k10.025Ratioofthethrusttotherollingmoment(m)k22Reciprocalofdist.fromnshafttoCB(m)]TJ /F6 7.97 Tf 6.59 0 Td[(1) 21

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2.3.2BoundaryConditionsTheinitialconditionsofeachAUV'spositionandorientationaregivenas xi(t0)=xi0,yi(t0)=yi0,zi(t0)=zi0,i(t0)=i0,i(t0)=i0, i(t0)=i0,(2)wherei=1,...,Aandt0istheinitialtime.ThesearetheinitialpositionsandorientationsofthepatrollingvehiclesatthetimethetargetenterstheROI.Next,theinitialconditionsonthevelocitiesandangularvelocitiesofeachvehiclearegivenas ui(t0)=ui0,vi(t0)=vi0,wi(t0)=wi0,pi(t0)=pi0,qi(t0)=qi0,ri(t0)=ri0,(2)wherei=1,...,Aandt0istheinitialtime.ThesearetheinitialvelocitiesandangularvelocitiesofthepatrollingAUVsatthetimethetargetenterstheROI.Next,theterminal 22

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conditionsonthepositionsandtheorientationforeachvehiclearegivenas xi(tf)=xif,yi(tf)=yif,zi(tf)=zif,i(tf)=if,i(tf)=if, i(tf)=if,(2)wherei=1,...,AandtfisthetimewhenthetargetexiststheROI.Also,theterminalconditionsonthevelocitiesandtheangularvelocitiesforeachvehiclearegivenbelow ui(tf)=uif,vi(tf)=vif,wi(tf)=wif,pi(tf)=pif,qi(tf)=qif,ri(tf)=rif,(2)wherei=1,...,AandtfisthetimewhenthetargetexiststheROI.Atthestartoftheproblem,onlytheterminalvelocitiesuif,vifandwifareknown,andtheyaresettozero.ThisisdonetoensurethevehiclesdonotleavetheROIafterthetargetegressestheregion.Alloftheremainingterminalconditionsareleftfreeandwillbedeterminedbysolvingthetrackingproblem. 23

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2.3.3PathConstraints 2.3.3.1StateinequalityconstraintsInthisproblem,itisassumedtheAUVsdonotleavetheROI.Therefore,thestateinequalityconstraintsare xminxi(t)xmax,yminyi(t)ymax,0zi(t)zmax,(2)wherei=1,...,Aandxmin,xmax,ymin,ymaxandzmaxaretheallowablelimitsonthethree-dimensionalROI.ItshouldbenotedthatzminiszerobecauseitisassumedthattheAUVswillnotbreakthesurfaceofthewater.Furthermore,thevehicles'orientations,velocitiesandangularvelocitiesmuststaywithintheoperationallimitsgivenbelow mini(t)max,mini(t)max, min i(t) max,uminui(t)umax,vminvi(t)vmax,wminwi(t)wmax,pminpi(t)pmax,qminqi(t)qmax,rminri(t)rmax,(2)wherei=1,...,Aandmin,max,min,max, minand maxarethelimitsontheorientationofthevehicles,umin,umax,vmin,vmax,wminandwmaxarethevelocitylimitsofthevehiclesandpmin,pmax,qmin,qmax,rminandrmaxaretheangularvelocitylimitsofthevehicles. 24

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2.3.3.2TargetcollisionavoidanceconstraintsItisarequirementthattheAUVsdonotcollidewiththetarget.Thetargetcollisionavoidanceconstraintsareformulatedasfollows.Let(xi,yi,zi)betheCartesiancoordinatesoftheAAUVsand(xT,yT,zT)betheCartesiancoordinatesofthetarget.ThenumberoftargetcollisionavoidanceconstraintswillbeA,whereAisthenumberofAUVsoperatingintheROI.Thetargetcollisionavoidanceconstraintsaregivenas (xi(t))]TJ /F5 11.955 Tf 11.96 0 Td[(xT(t))2+(yi(t))]TJ /F5 11.955 Tf 11.96 0 Td[(yT(t))2+(zi(t))]TJ /F5 11.955 Tf 11.95 0 Td[(zT(t))2d2T,(2)wherei=1,...,AanddTistheminimumsafeoperatingdistancebetweenanAUVandthetarget. 2.3.3.3FriendlycollisionavoidanceconstraintsAlso,itisnecessarythattheAUVsdonotcollidewitheachother.Thefriendlycollisionavoidanceconstraintsareformulatedasfollows.Let(xa,ya,za)and(xb,yb,zb)betheCartesiancoordinatesofAUVsaandb,respectively,wherea,b=1,...,A,anda6=b.ThenumberoffriendlycollisionavoidanceconstraintswillbeA(A)]TJ /F8 11.955 Tf 12.5 0 Td[(1)=2,whereAisthenumberofvehiclesoperatingintheROI.Thefriendlycollisionavoidanceconstraintsaregivenas (xa(t))]TJ /F5 11.955 Tf 11.95 0 Td[(xb(t))2+(ya(t))]TJ /F5 11.955 Tf 11.96 0 Td[(yb(t))2+(za(t))]TJ /F5 11.955 Tf 11.95 0 Td[(zb(t))2d2A,(2)wherei=1,...,AanddAistheminimumsafeoperatingdistancebetweentwoAUVs. 2.3.3.4ControlconstraintsThefollowingcontrolconstraintsareappliedtoeachAUV. FminFix(t)Fmax,MxminMix(t)Mxmax,MyminMiy(t)Mymax,MzminMiy(t)Mzmax,(2) 25

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wherei=1,...,AandFminandFmaxcorrespondtothethrustlimitsontheAUVsandMxmin,Mxmax,Mymin,Mymax,MzminandMzmaxarethelimitsontheappliedmomentscausedbytheAUVs'nsaboutthevehicles'bodyaxes. 2.3.3.5CostfunctionalInthisproblem,thecostfunctionaltobeminimizedisthesumofthedistancefromeachAUVtothetarget.Thedistanceofeachvehicletothetargetisgivenby di)]TJ /F7 7.97 Tf 6.59 0 Td[(t=p (xi)]TJ /F5 11.955 Tf 11.96 0 Td[(xt)2+(yi)]TJ /F5 11.955 Tf 11.95 0 Td[(yt)2+(zi)]TJ /F5 11.955 Tf 11.96 0 Td[(zt)2,(2)wherei=1,...,A.Thus,thecostfunctionalis J=Ztft0AXi=1di)]TJ /F7 7.97 Tf 6.58 0 Td[(t,(2) 26

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CHAPTER3PSEUDOSPECTRALMETHODSANDGPOPSPseudospectralmethodsareaclassofdirecttrajectoryoptimizationmethodswhichusedirectcollocationinordertotranscribetheoptimalcontrolproblemtoanonlinearprogrammingproblem(NLP).Garg,D.andothers[ 15 ]showthatoptimalcontrolproblemscanbesolvedusingcollocationateitherLegendre-Gauss,Legendre-Gauss-RadauorLegendre-Gauss-Lobattopoints.Psuedospectralmethodsuseglobalpolynomialstoparameterizethestateandthecontrolandthencollocatethedifferential-algebraicequationsusingnodesfromaGaussianquadrature.Thismethodproducesaccuratesolutionsforproblemswithsmoothsolutions.Forproblemswithsolutionswhicharenotsmooth,thetimeintervalfrom[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,1]isbrokenintoseveralintervals,sothatdifferentglobalpolynomialapproximationsareusedovereachinterval. 3.1LG,LGR,andLRLCollocationPointsThethreemostcommonsetsofcollocationpointsareLegendre-Gauss(LG),Legendre-Gauss-Radau(LGR),andLegendre-Gauss-Labatto(LGL)points.ThesepointsareobtainedfromtherootsoftheLegendrepolynomialand/orlinearcombinationsofaLegendrepolynomialanditsderivatives.Allthreesetsofcollocationpointsaredenedonthedomain[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,1].ThedifferencebetweenthesetsisthattheLGsetdoesnotincludeeitherendpoint,theLGRsetincludesoneendpointandtheLGLsetincludesbothoftheendpoints.Also,theLGRsetisasymmetricrelativetotheoriginandcanbedenedusingtheendpointaseithertheinitialpointortheendpoint.Figure 3-1 illustratesthedifferencebetweenLGL,LGRandLGcollocationpoints[ 16 ].LetNbethenumberofcollocationpointsandPN()betheNth-degreeLegendrepolynomial,thentheLG,LGR,andLGLcollocationpointsareobtainedfromtherootsofthepolynomial.LGpointsaretherootsobtainedfromPN(),LGRpointsaretheroots 27

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obtainedfromPN)]TJ /F6 7.97 Tf 6.59 0 Td[(1()+PN(),andLGLpointsaretherootsobtainedfromPN)]TJ /F6 7.97 Tf 6.59 0 Td[(1()togetherwithpoints)]TJ /F8 11.955 Tf 9.29 0 Td[(1and1.[ 15 ] 3.2FormulationofPseudospectralMethodUsingLGRPointsGarg,D.andothers[ 16 ]presenttheRadauPseudospectralmethodfordirecttrajectoryoptimizationandcostateestimationofnite-horizonandinntie-horizonoptimalcontrolproblemsusingglobalcollocationatLegendre-Gauss-Radau(LGR)points.TheauthorshaveshowntheuseofLGRcollocationaidsinndingaccurateprimalanddualsolutionsforbothniteandintnite-horizonoptimalcontrolproblems.Therefore,theRadauPseudospectralmethodisusedinthiswork.Tosimplifytheproblem,consideranunconstrainedoptimalcontrolproblemwithaterminalcost,onthetimeinterval[)]TJ /F8 11.955 Tf 9.29 0 Td[(1,1].Thetimeintervalcanbetransformedfrom[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,1]tothetimeinterval[t0,tf]viatheafnetransformation t=tf)]TJ /F5 11.955 Tf 11.95 0 Td[(t0 2+tf+t0 2(3)Thegoalistodeterminethestatex()2Rnandthecontrolu()2Rmthatminimizethecostfunctional (x)=x(1),(3)subjecttotheconstraintsbelow dx d=f(x(),u()),x()]TJ /F8 11.955 Tf 9.3 0 Td[(1)=x0,(3)wheref:RnRm!Rnandx0istheknowninitialcondition.ConsiderNLGRcollocationpoints,1,2,3,...,Nintheinterval[)]TJ /F8 11.955 Tf 9.3 0 Td[(1,1],where1=)]TJ /F8 11.955 Tf 9.3 0 Td[(1andn+1.Anadditionalnon-collocatedpointN+1=1,whichisusedtoapproximatedthestatevariable,isintroduced[ 16 ].Eachcomponentofthestatexis 28

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approximatedbyaLagrangepolynomial,Li,asgiveninEquation( 3 )below, Li()=N+1Yj=1j6=i)]TJ /F9 11.955 Tf 11.95 0 Td[(j i)]TJ /F9 11.955 Tf 11.95 0 Td[(j,i=1,...,N+1(3)Thejthcomponentofthestateisapproximatedastheseries xj()=N+1Xj=1xijLi(),(3)DifferentiatingandevaluatingtheseriesinEquation( 3 )atthecollocationpointsk,wherek=1,2,...,N,gives _xj()N+1Xi=1xij_Li(k)=N+1Xi=1Dkixij,(3)whereDki=_Li(k).TheN(N+1)matrixDiscalledthedifferentiationmatrix.CollocatingthesystemdynamicsateachoftheNcollocationpoints,thediscreteoptimizationproblembecomes DX=F(X,U).(3)FromEquation( 3 ),thenite-dimensionalNLPisformulatedasbelow, Minimize(XN)subjecttoDX=F(X,U), whereX0=x0.Thesystemdynamicsarethenrewrittenas, D1:NX1:N=F(X,U))]TJ /F5 11.955 Tf 11.95 0 Td[(D0x0,(3)whereD1:NistheNNdifferentiationmatrix,whichisinvertible. 3.3hp-AdaptivePseudospectralMethodAnhp-adaptivepseudospectralmethodwithcollocationatRadaupointsischoseninthiswork.ThismethodisproposedbyDarby,C.L.andothers[ 17 ],whichiterativelyandsimultaneouslydeterminesthenumberofsegmentbreaks,thewidthofeach 29

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segmentandthepolynomialdegreerequiredineachsegmentforapproximation,untiltheuserspeciedaccuracyisachieved.Thismethodleadstohigheraccuracysolutionswithlesscomputationaleffortandmemorythanisrequiredinglobalpseudospectralmethods. 3.4ProblemFormulationinGPOS 3.4.1ObjectiveandInputParametersTheobjectiveistohavethefourAUVstrackatarget,whileminimizingthedistanceofeachvehicletothetargetandmaintainingasafeoperatingdistance.ThenalpositionsandorientationsoftheAUVsareleftfree,fortheproblemendswhenthetargethaslefttheROI.TheROIisdenedtobe[0100;-50-50;0100]inmeters.TheROIrangesfrom0to100metersalongthex-axis(horizontallength),-50to50metersalongthey-axis(horizontalwidth)and0to100metersalongthez-axis(verticaldepth).Table 3-1 ,showstheinitialandterminalvaluesofthestatesandtime. Table3-1. User-speciedinitialandnalstatesofthefourAUVs InitialConditionsTerminalConditions t0=0stf=distancetraveledbytarget/targetspeedsxi(t0)=75,75,25,25mxi(tf)freeyi(t0)=)]TJ /F8 11.955 Tf 9.3 0 Td[(25,25,25,)]TJ /F8 11.955 Tf 9.29 0 Td[(25myi(tf)freezi(t0)=5mzi(tf)freei(t0)=0degi(tf)freei(t0)=0degi(tf)free i(t0)=)]TJ /F8 11.955 Tf 9.3 0 Td[(45,45,135,)]TJ /F8 11.955 Tf 9.29 0 Td[(135deg i(tf)freeui(t0)=0m=sui(tf)=0m=svi(t0)=0m=svi(tf)=0m=swi(t0)=0m=swi(tf)=0m=spi(t0)=0deg=spi(tf)freeqi(t0)=0deg=sqi(tf)freeri(t0)=0deg=sri(tf)free Inthisproblem,itisassumedthatthetarget'strajectoryisknown.BecausetheproblemendswhenthetragetleavestheROI,tfcanbecalculatedbydividingthedistancetraveledbythetargetbythespeedofthetarget.Also,thenalvelocitiesare 30

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settozerotoensurethattheAUVsdonotleavetheROIafterthetargetleavestheregion. 3.4.2CostFunctionFormulationThecostfunctionalfortheproblemisthesumofthedistancesfromeachofthefourAUVstothetarget,whichisformulatedas J=Ztft0AXi=1p (xi)]TJ /F5 11.955 Tf 11.95 0 Td[(xt)2+(yi)]TJ /F5 11.955 Tf 11.96 0 Td[(yt)2+(zi)]TJ /F5 11.955 Tf 11.95 0 Td[(zt)2,(3)wherei=1,...,Aandxi,yiandziaretheCartesiancoordinatesoftheithvehicleandxt,ytandztaretheCartesiancoordinatesofthetarget. 3.4.3PathConstraintFormulationForthisproblem,targetandfriendlycollisionavoidanceconstraintsareneeded.BecausethecostfunctionisdrivingthedistancebetweentheAUVsandthetargettozero,fourtargetcollisionavoidanceconstraintsarerequired.Letdi)]TJ /F7 7.97 Tf 6.58 0 Td[(TbetheminimumoperatingdistancebetweentheithAUVandthetarget,wherei=1,...,4,thenthetarget-avoidancepathconstraintsare PathConsttarget=[d1)]TJ /F7 7.97 Tf 6.59 0 Td[(T;d2)]TJ /F7 7.97 Tf 6.59 0 Td[(T;d3)]TJ /F7 7.97 Tf 6.59 0 Td[(T;d4)]TJ /F7 7.97 Tf 6.59 0 Td[(T][1;1;1;1].(3)OncethefourAUVsbegintrackingatarget,thevehiclesareoperatingrelativelyclosetogether.InordertoavoidcollisionamongAUVs,friendlycollisionavoidanceconstraintsareneeded.Therearefourvehicles,eachavoidingtheotherthreevehicles,whichproduces12friendlycollisionavoidancepathconstraints.However,duetothereciprocityofthedistancebetweentwovehicles,onlysixfriendlycollisionavoidancepathconstraintsareneeded.Letdi)]TJ /F7 7.97 Tf 6.58 0 Td[(jbetheminimumoperatingdistancebetweentheithandjthAUV,wherei,j=1,...,4andi6=j,thenthefriendlycollisionavoidancepath 31

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constraintsare PathConstfriendly1=[d1)]TJ /F6 7.97 Tf 6.59 0 Td[(2;d1)]TJ /F6 7.97 Tf 6.59 0 Td[(3;d1)]TJ /F6 7.97 Tf 6.59 0 Td[(4][1;1;1]PathConstfriendly2=[d2)]TJ /F6 7.97 Tf 6.59 0 Td[(3;d2)]TJ /F6 7.97 Tf 6.59 0 Td[(4][1;1]PathConstfriendly3=[d3)]TJ /F6 7.97 Tf 6.58 0 Td[(4][1](3)InChapter 4 ,theresultsofthemultipleAUVtargettrackingproblemfortwodifferenttargetscenariosareplottedanddiscussed.Thetrajectoriesofeachofthevehiclesareplotted.Furthermore,theresultsareanalyzedthenvalidated. 32

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Figure3-1. SchematicshowingLGL,LGRandLGcollocationpoints 33

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CHAPTER4RESULTSANDDISCUSSIONInthischapter,theresultsofthemultipleAUVtargettrackingproblemfortwotargetscenariosarepresented.Intherstscenario,thetargetbeingtrackedisanunderwatervehicletravelingatadepthof30meterswithaspeedof7knots.Thetargetinthesecondscenarioisasurfacevehiclewithaspeedof20knots.TheseproblemsaresolvedbyrstconvertingtheoptimalcontrolproblemsintoNLPsusingGPOPS.TheNLPsarethensolvedusingtheSparseNonlinearOPTimimzer(SNOPT)[ 18 ].Thesolutionsobtainedfortheoptimalcontrolproblemsarethenplotted.Furthermore,theresultsareanalyzedandvalidated. 4.1UnderwaterTarget 4.1.1AnalysisofResultsThetrajectoriesofthefourAUVsandthetargetareplottedinthethreedimensionalplotinFigure 4-1 .TheplotshowsthetrajectoriesofthevehiclesfromthetimetheunderwatertargetenterstheROItothetimethetargetegressesfromtheROI.Figure 4-10 providesatop-downviewofthetrajectories.Also,Figures 4-3 4-4 and 4-5 displaythex,yandz-axispositionsversustimeplotsofeachAUV. 4.1.2ValidationofResultsTheLagrangecostofminimizingthesumofthedistancesfromeachAUVtothetargetissatisedandshowninFigure 4-6 .Furthermore,thecostfunctionalisveriedfromTable 4-1 ,whichdisplaystheLagrangecostandtheaveragedistanceoftheAUVstothetargetovertheentiretrajectory.ThetargetandfriendlycollisionavoidanceconstraintsdenedinChapter 3 arevalidatedfromtheresultsplottedinFigures 4-7 and 4-8 .TheseplotsvalidatethateachAUVavoidsthetargetandtheotherAUVsbytheminimumsafeoperatingdistance,whichissetat1meter. 34

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Table4-1. Lagrangecostvalidationforanunderwatertarget Time(s)LagrangeCost(m)AverageAUVDistancetoTarget(m) 0.0000252.433863.10840.0503251.899262.97480.1684250.642562.66060.3533248.712162.17800.6042246.123961.53100.9197242.984960.74621.2984239.239659.80991.7384235.019858.75492.2376230.202557.55062.7935225.143056.28583.4034219.699654.92494.0644213.691453.42284.7732205.751551.43795.5264195.358048.83956.3203182.113345.52837.1511166.276941.56928.0146147.494536.87368.9068126.450231.61259.8232104.611526.152910.759382.184920.546211.710760.152015.038012.672740.651910.163013.640625.64926.412314.609613.91653.479115.57515.90081.475216.53244.08071.020217.47675.52761.381918.40365.04881.262219.30844.26881.067220.18684.00001.000021.03444.00001.000021.84734.00001.000022.62134.00001.000023.35284.00001.000024.03814.00001.000024.67394.00001.000025.25724.00001.000025.78514.00001.000026.25494.00001.000026.66454.00001.000027.01194.00001.000027.29524.00001.000027.51324.00001.0000 35

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Table 4-1 .Continued Time(s)LagrangeCost(m)AverageAUVDistancetoTarget(m) 27.66494.00341.000927.74944.03861.009727.76924.04311.0108 4.2SurfaceTarget 4.2.1AnalysisofResultsThetrajectoriesofthefourAUVs,aswellasthetarget,areplottedinthethreedimensionalplotinFigure 4-9 .ThisplotshowsthetrajectoriesofthevehiclesfromthetimethesurfacetargetenterstheROItothetimethetargetegressestheROI.Figure 4-10 providesatop-downviewofthetrajectories.Also,Figures 4-11 4-12 and 4-13 displaythex,yandz-axispositionsversustimeplotsofeachAUV. 4.2.2ValidationofResultsTheLagrangecostofminimizingthesumofthedistancesfromeachAUVtothetargetissatisedandshowninFigure 4-14 .Furthermore,thecostfunctionalisveriedfromTable 4-2 ,whichdisplaystheLagrangeCostandtheaveragedistanceoftheAUVstothetargetovertheentiretrajectory. Table4-2. Lagrangecostvalidationforasurfacetarget Time(s)LagrangeCost(m)AverageAUVDistancetoTarget(m) 0.0000300.320775.08020.0553299.562574.89060.1852297.789374.44730.3885295.032073.75800.6643291.322272.83061.0113286.628871.65721.4277280.892370.22311.9115273.942568.48562.4603265.752766.43823.0716255.054763.76373.7422241.568260.39204.4690224.419756.10495.2484204.193951.04856.0766182.293745.57346.9496158.587739.6469 36

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Table 4-2 .Continued Time(s)LagrangeCost(m)AverageAUVDistancetoTarget(m) 7.8630133.231133.30788.8126110.511827.62809.793589.589122.397310.801169.378617.344711.830551.347012.836712.876634.77478.693713.934420.71365.178414.998613.96513.491316.064110.72072.680217.12578.84652.211618.178310.37022.592519.216610.35032.587620.23588.90062.225221.23076.80021.700022.19655.22791.307023.12864.33611.084024.02234.00001.000024.87344.00001.000025.67774.00001.000026.43124.00001.000027.13044.00001.000027.77174.00001.000028.35214.00001.000028.86884.00001.000029.31924.00001.000029.70114.00001.000030.01274.00001.000030.25244.00321.000830.41914.06941.017430.51204.23991.060030.53384.25721.0643 ThetargetandfriendlycollisionavoidanceconstraintsdenedinChapter 3 arevalidatedfromtheresultsplottedinFigures 4-15 and 4-16 .TheseplotsvalidatethateachAUVavoidsthetargetandtheotherAUVsbytheminimumnsafeoperatingdistance. 37

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Figure4-1. Three-dimensionaltrajectoryofAUVstrackingtheunderwatertarget Figure4-2. Top-downviewoftrajectoryfortrackingtheunderwatertraget 38

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Figure4-3. Forwardpositionversustimefortrackingtheunderwatertraget Figure4-4. Lateralpositionversustimefortrackingtheunderwatertraget 39

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Figure4-5. Depthversustimefortrackingtheunderwatertraget Figure4-6. Thelagrangecostversustimefortheunderwatertarget 40

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Figure4-7. DistancefromAUVstotargetversustimefortheunderwatertarget 41

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Figure4-8. AveragedistancefromoneAUVtotheothersversustimefortheunderwatertarget Figure4-9. Three-dimensionaltrajectoryofAUVstrackingthesurfacetarget 42

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Figure4-10. Top-downviewoftrajectoryfortrackingthesurfacetraget Figure4-11. Forwardpositionversustimefortrackingthesurfacetraget 43

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Figure4-12. Lateralpositionversustimefortrackingthesurfacetraget Figure4-13. Depthversustimefortrackingthesurfacetraget 44

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Figure4-14. Thelagrangecostversustimeforthesurfacetarget Figure4-15. DistancefromAUVstotargetversustimeforthesurfacetarget 45

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Figure4-16. AveragedistancefromoneAUVtotheothersversustimeforthesurfacetarget 46

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CHAPTER5CONCLUSIONThisworkpresentedanapproachtosolveamultipleAUVtargettrackingproblemusingpseudospectralmethods.Anexampleproblemthatgeneratesthetrajectorywhichminimizesthesumofthedistancesfromeachvehicletothetargethasbeensolved.ThistrajectorysatisesthetargetandfriendlyavoidanceconstraintsplacedontheAUVs.TheproblemhasbeenformulatedinChapter2.TheoptimalcontrolproblemhasbeenformulatedinChapter3.FromtheresultsplottedinChapter4,itwasfoundthatthesolutionsatisedalltheconstraintsintheproblemandgaveanoptimalsolution.ThisresearchworkcanbeusedtoaidintheprotectionofU.S.portsandwaterways,aswellasshipsinforeignwaters.Futureworkinoptimizingthenumberofcollocationpointsusedtosolvetheoptimalcontrolproblemwilldecreasethecomputationtimerequiredtothesetrajectories.Furthermore,improvementsinCPUprocessorsaswellasfasterNLPsolverswillsignicantlyimprovethecomputationtime.Thiswillhelpinnearreal-timeimplementationofthisresearch. 47

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REFERENCES [1] Frittelli,J.,PortandMaritimeSecurity:BackgroundandIssuesforCongress,CRSReportforCongress,May2005. [2] StatusoftheNavy, http://www.navy.mil/navydata/navy_legacy_hr.asp?id=146 ,Feb2012. [3] Ghommam,J.,Calco,O.,andRozenfeld,A.,CoordinatedPathFollowingforMultipleUnderactuatedAUVs,OCEANS,April2008,pp.1. [4] Wang,Y.,Yan,W.,Gao,B.,andCui,R.,Backstepping-basedPathFollowingControlofanUnderactuatedAutonomousUnderwaterVehicle,InternationalConferenceonInformationandAutomation,June2009,pp.466. [5] Reif,J.andWang,H.,SocialPotentialFields:ADistributedBehavioralControlforAutonomousRobots,RoboticsandAutonomousSystems,Vol.27,No.4,1999,pp.171. [6] Beard,R.,Lawton,J.,andHadaegh,F.,ACoordinationArchitectureforSpacecraftFormationControl,IEEETransactionsonControlSystemsTechnology,Vol.9,No.6,Nov2001,pp.777. [7] Fua,C.,Ge,S.,Do,K.,andLim,K.,Multi-robotFormationsBasedontheQueue-FormationSchemeWithLimitedCommunication,IEEETransactionsonRobotics,Vol.23,No.6,2007,pp.1160. [8] Ge,S.andFua,C.,QueuesandArticialPotentialTrenchesforMultirobotFormations,IEEETransactionsonRobotics,Vol.21,No.4,Aug2005,pp.646. [9] Cui,R.,Ge,S.,How,B.,andChoo,Y.,Leader-FollowerFormationControlofUnderactuatedAUVswithLeaderPositionMeasurement,IEEEInternationalConferenceonRoboticsandAutomation,2009,pp.979. [10] Emrani,S.,Dirafzoon,A.,Talebi,H.,Nikravesh,S.,andMenhaj,M.,AnAdaptiveLeader-FollowerFormationControllerforMultipleAUVsinSpatialMotions,36thAnnualConferenceonIEEEIndustrialElectronicsSociety,Nov2010,pp.59. [11] Wiegert,R.andOeschger,J.,GeneralizedMagneticGradientContractionBasedMethodforDetection,LocalizationandDiscriminationofUnderwaterMinesandUnexplodedOrdnance,OCEANS,Vol.2,Sept2005,pp.1325. [12] Rauch,C.,Austin,T.,Grosenbaugh,M.,Jaffre,F.,Stokey,R.,andMacDonald,J.,GeneralizedMagneticGradientContractionBasedMethodforDetection,LocalizationandDiscriminationofUnderwaterMinesandUnexplodedOrdnance,OCEANS,Sept2008,pp.1. 48

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[13] Belbachir,A.andPerrier,M.,Cooperative-AdaptiveAlgorithmsforTargetsLocalizationinUnderwaterEnvironment,AutonomousUnderwaterVehicles(AUV),2010IEEE/OES,Sept2010,pp.1. [14] Arslan,M.,Fukushima,N.,andHagiwara,I.,NonlinearOptimalControlofanAUVanditsActuatorFailureCompensation,InternationalConferenceonControl,Automation,RoboticsandVision,Dec2008,pp.668. [15] Garg,D.,Patterson,M.,Hager,W.,Rao,A.,Benson,D.,andHuntington,G.,AUniedFrameworkfortheNumericalSolutionofOptimalControlProblemsUsingPseudospectralMethods,Automatica,Vol.46,No.11,Nov2010,pp.1843. [16] Garg,D.,Patterson,M.,andFrancolin,C.,DirectTrajectoryOptimization;CostateEstimationsofFinite-horizon;Innite-horizonOptimalControlProblemsUsingaRadauPseudospectralMethod,ComputationalOptimizationandApplications,Oct2009,pp.1. [17] Darby,C.,Hager,W.,andRao,A.,Anhp-adaptivePseudospectralMethodforSolvingOptimalControlProblems,OptimalControlApplicationsandMethods,Aug2010,pp.1. [18] StanfordBusinessSoftwareInc.,SNOPT7.2, http://www.sbsi-sol-optimize.com/asp/sol_product_snopt.htm ,2011. 49

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BIOGRAPHICALSKETCH AdamFranklinwasborninColumbus,Mississippiin1987.HeearnedhisBachelorofSciencedegreeinsystemsengineeringandhiscommissionintheUnitedStatesMarineCorpsfromtheNavalAcademyinMayof2010.HealsograduatedwithaMasterofSciencedegreeinmechanicalengineeringfromtheDepartmentofMechanicalandAerospaceEngineeringattheUniversityofFloridainMay2012.Hecompletedhisresearchworkandmaster'sthesisunderthesupervisionofDr.AnilV.Raoandwasco-advisedbyDr.WarrenE.Dixon.Henowcontinueshisofcertraining,withasecondarymilitaryoccupationspecialtyofordnancesystemsengineer. 50