Lyapunov-based Nonlinear Estimation Methods with Applications to Machine Vision

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Lyapunov-based Nonlinear Estimation Methods with Applications to Machine Vision
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english
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Dani,Ashwin P
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University of Florida
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Doctorate ( Ph.D.)
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University of Florida
Degree Disciplines:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
Dixon, Warren E
Committee Members:
Crane, Carl D
Barooah, Prabir
Khargonekar, Pramod

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Subjects / Keywords:
control -- estimation -- motion -- nonlinear -- slam -- structure -- vision
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
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Mechanical Engineering thesis, Ph.D.
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Abstract:
Recent advances in image-based information estimation has enabled the use of vision sensor in many robotics and surveillance applications. The work in this dissertation, is focused on developing online techniques for image-based structure and motion (SaM) estimation. Since traditional batch methods are not useful for the online vision-based control tasks, observer-based approaches to the problem have been developed. Starting from the Kalman-filter for SaM problem by L. Matthies, many contributions to the observer approach for the SaM problem exist in literature. Various models are introduced in literature for SaM estimation but two models are prevalent, viz; a kinematic relative motion affine model with implicit outputs and a transformed nonlinear state model with the linear output equation. The existing SaM observers are designed for the case of a stationary object, requires full camera velocity information and cannot be used for certain camera motions. In this dissertation, new solutions to the SaM are presented using the transformed nonlinear state model which can be used for larger set of camera motions, does not require full camera velocity information, and are reduced-order. Solutions for the stationary as well as moving objects viewed by a moving camera are presented. In Chapter 3, a reduced order observer is developed to estimate the structure of a static object using a moving camera, where full camera velocity and linear acceleration are known. Chapter 4 focuses on the development of a reduced order observer for the SaM estimation of a stationary object when only a single camera linear velocity is known. In Chapter 5, an observer design is presented for a specific class of nonlinear systems where the output dynamics are affine in the unmeasurable state and the dynamics of the unmeasurable state are nonlinear. The method is applied to simultaneously estimate the structure and motion of a moving object seen by a moving camera. Another strategy to the observer design in the presence of an unmeasurable disturbance is an unknown input observer (UIO). Chapter 6 provides a solution to an UIO design for a general class of nonlinear systems and it?s application to structure estimation of a moving object is shown in Chapter 7.
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In the series University of Florida Digital Collections.
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Statement of Responsibility:
by Ashwin P Dani.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Dixon, Warren E.

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LYAPUNOV-BASEDNONLINEARESTIMATIONMETHODSWITHAPPLICATIONS TOMACHINEVISION By ASHWINP.DANI ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c 2011AshwinP.Dani 2

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Tomyparents,ChhayaandKishor,wife,Uttara,andbrother,Ketanfortheirunwavering support 3

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ACKNOWLEDGMENTS Iwouldliketoexpresssinceregratitudetomyadvisor,WarrenE.Dixon,whose experienceandmotivationwereinstrumentalforthisdissertation.Asanadvisor,he providedguidanceinmyresearch,andencouragementindevelopingmyownideas. ThedynamicandfreeworkculturehehasdevelopedintheNCRlabhasencouraged andfacilitatedmylearning.Asamentor,hehelpedmeunderstandtheintricaciesof workinginaprofessionalenvironmentbyexposingmetovariousprofessionalactivities suchasjournal/conferencereviews,grantproposalwriting,teachingclassroomlectures, conductingrobotcampsasapartofsocietaloutreachactivity,etc.Ifeelfortunatetohave hadtheopportunitytoworkwithhimandwouldliketothankhimforallthetechnicalas wellasnon-technicalknowledgehehasgivenme. IwouldliketothankPrabirBarooahforhissuggestionsandtechnicaldiscussions duringvariousoccasionsovermygraduateschoolcareer.Duringhisclassandother discussions,Inotonlyreceivedatrainingabouttechnicalmatterbutalsoreceiveda trainingonquestioningeachandeverystep.Iamslowlyassimilatingthishabitandthis hasplayedacrucialroleforthetechnicaldevelopmentinthisdissertation. IwouldalsoliketoexpressmythankfulnesstoCarlD.CraneIIIfortimeandthe helpheprovidedmeduringtheclassroommeetings,variousprojectdiscussionmeetings, andoralqualiers.Iwouldliketothankhimforthesupportandencouragementhe providedmeduringourvisittoAirForcebaseinVirginia,foraprojectdemonstration. IwouldliketoextendmygratitudetoPramodKhargonekarforhistechnicalsuggestions,duringoralqualiersandduringone-on-onemeetings,toimprovethequality ofthedissertation.Duringhis`AdvancedTopicsinSystemsandControls'classIhave learntmanydierentaspectsofsystemsandcontrolswhichIwastotallyignorantof.His encouragingandphilosophicallecturesintheclassroomchangedmyperspectiveforthe systemsandcontrolseldandfordoingresearchingeneral. 4

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Iwouldalsoliketothankmycoworkers,andfriendsfortheirsupportandencouragement.VarioustechnicaldiscussionswithmycoworkersfromtheNCRlabhavehelpedme learnandunderstandthesubjectindetail.Also,Iappreciateallthehelptheyprovided meforconductingvariousexperiments. Lastbutnottheleast,Iwouldliketothankmyparentsfortheirloveandsupport, mywife,forallthesupportandencouragementsheprovided,mybrotherforhisconstant supporttomyhighereducation. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................9 LISTOFFIGURES....................................10 ABSTRACT........................................12 CHAPTER 1INTRODUCTION..................................14 1.1Motivation....................................14 1.2ProblemStatementandOverview.......................14 1.3LiteratureReview................................15 1.4Contributions..................................23 1.5DissertationOutline..............................26 2CAMERAMOTIONMODEL............................28 2.1EuclideanandImageSpaceRelationships..................28 2.2Camera-ObjectRelativeMotionModel....................31 2.3Assumptions...................................32 3GLOBALLYEXPONENTIALLYSTABLEOBSERVERFORVISION-BASED RANGEESTIMATION...............................34 3.1Vision-basedRangeEstimation........................35 3.1.1RangeObserver.............................35 3.1.2StabilityAnalysis............................37 3.2StabilityAnalysisinthePresenceofDisturbances..............41 3.3Discussion....................................43 3.4SimulationsandExperiments.........................45 3.5Summary....................................52 4STRUCTUREANDMOTIONUSINGASINGLEKNOWNCAMERALINEARVELOCITY...................................56 4.1StructureandMotionEstimation.......................57 4.1.1EstimationwithaKnownLinearVelocity...............57 4.1.1.1StepI:Angularvelocityestimation.............59 4.1.1.2StepII:Structureestimation.................60 4.1.2StabilityAnalysis............................61 4.2Simulation....................................63 4.3Summary....................................65 6

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5ALYAPUNOV-BASEDOBSERVERFORACLASSOFNONLINEARSYSTEMSWITHAPPLICATIONTOIMAGE-BASEDSTRUCTUREANDMOTIONESTIMATION.................................68 5.1NonlinearObserver...............................68 5.1.1SystemDynamics............................68 5.1.2StateEstimator.............................69 5.1.3StabilityAnalysis............................71 5.2ApplicationtoStructureandMotionProblem................77 5.2.1StructureandMotionfromMotionSaMfMObjective.......77 5.2.2StateDynamicsFormulation......................78 5.2.3StructureandMotionObserver....................80 5.2.4ConditionsontheMovingObjectTrajectory.............81 5.3Simulation....................................83 5.4Summary....................................86 6LYAPUNOV-BASEDUNKNOWNINPUTOBSERVERFORACLASSOF NONLINEARSYSTEMS..............................89 6.1NonlinearUnknownInputObserver......................89 6.1.1NonlinearDynamics...........................89 6.1.2UIODesign...............................90 6.1.3SucientCondition...........................92 6.1.4ConditionsforChoosingMatrixA...................96 6.1.5LMIFormulation............................100 6.2Summary....................................101 7APPLICATIONOFTHEUNKNOWNINPUTOBSERVERTOTHESTRUCTUREESTIMATION................................102 7.1StructureandMotionEstimation.......................102 7.1.1StructureandMotionfromMotionSaMfMObjective.......102 7.1.2NonlinearUnknownInputObserver..................103 7.1.3StabilityAnalysis............................105 7.1.4ConditionsforStability.........................106 7.1.5ConditionsonObjectTrajectories...................106 7.2Simulation....................................107 7.3Summary....................................111 8CONCLUSIONANDFUTUREWORK......................112 8.1Conclusion....................................112 8.2FutureWork...................................114 APPENDIX APROOFOFPOSITIVENESSOF P ........................116 7

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BOBSERVABILITYCONDITIONS.........................118 B.1ObservabilityConditioninChapter4.....................118 B.2ObservabilityConditioninChapter5.....................118 REFERENCES.......................................120 BIOGRAPHICALSKETCH................................129 8

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LISTOFTABLES Table page 3-1Comparisonofthepresentedobserverwithobserversin[1]and[2]........44 3-2ComparisonoftheRMSdepthestimationerrors..................49 9

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LISTOFFIGURES Figure page 2-1Coordinateframerelationshipsofamovingcameraandanobject.........28 2-2Movingcameralookingatthestaticscene......................30 3-1Comparisonbetweenthetrueandestimateddepthinthepresenceofmeasurementnoise.......................................49 3-2Comparisonofthedepthestimation.........................50 3-3Evolutionofthesignal h 2 1 t + h 2 2 t againsttime..................50 3-4Depthestimationusingtheobserverin[1]......................51 3-5Estimationof ^ y 3 t startingfromlargeinitialcondition t 0 =300 usingthe proposedobserver...................................51 3-6Depthestimationwithlargeinitialcondition t 0 =300 usingtheproposed observer.........................................52 3-7Stateestimationusingtheobserverinforlargeinitialconditions.........53 3-8AUVexperimentalsetup...............................53 3-9AnimageframedisplayingthetrackedbuoytargetbytheAUV.........54 3-10Comparisonoftheestimatedandgroundtruthrangeofthebuoywithrespect totheunderwatervehicle...............................54 3-11States y 1 t and y 2 t computedusingimagepixels.................55 4-1State y 3 u 1 u 2 T : ................................65 4-2State y 3 u 1 u 2 T : ................................66 4-3State y 3 u 1 u 2 T : ................................66 5-1Thevelocityofthemovingobjectmeasuredintheinertialreferenceframe....85 5-2Stateestimationerrorsusingtheproposedobserver.................85 5-3Stateestimationerrorsusingtheobserverin[3]...................86 5-4Comparisonoftheactualandestimated3Drelativepositionoftheobjectand thecamera.......................................87 5-5Thevelocityofthemovingobjectmeasuredintheinertialreferenceframe....87 10

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5-6Stateestimationerrorsfortimevaryingvelocity, v p t ; usingtheproposedobserver..........................................88 5-7Comparisonoftheactualandestimated3Drelativepositionoftheobjectand thecamera.......................................88 7-1ComparisonoftheactualandestimatedX,YandZpositionsofamovingobjectwithrespecttoamovingcamera.........................108 7-2Errorintherangeestimationofthemovingobject.................108 7-3ComparisonoftheactualandestimatedX,YandZpositionsofamovingobjectwithrespecttoamovingcamera.........................110 7-4Errorintherangeestimationofthemovingobjectinthepresenceofnoise....110 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy LYAPUNOV-BASEDNONLINEARESTIMATIONMETHODSWITHAPPLICATIONS TOMACHINEVISION By AshwinP.Dani August2011 Chair:WarrenE.Dixon Major:MechanicalEngineering Recentadvancesinimage-basedinformationestimationhasenabledtheuseofvision sensorinmanyroboticsandsurveillanceapplications.Theworkinthisdissertation,is focusedondevelopingonlinetechniquesforimage-basedstructureandmotionSaM estimation.Sincetraditionalbatchmethodsarenotusefulfortheonlinevision-based controltasks,observer-basedapproachestotheproblemhavebeendeveloped.Starting fromtheKalman-lterforSaMproblembyL.Matthies,manycontributionstothe observerapproachfortheSaMproblemexistinliterature.Variousmodelsareintroduced inliteratureforSaMestimationbuttwomodelsareprevalent,viz;akinematicrelative motionanemodelwithimplicitoutputsandatransformednonlinearstatemodelwith thelinearoutputequation.TheexistingSaMobserversaredesignedforthecaseofa stationaryobject,requiresfullcameravelocityinformationandcannotbeusedforcertain cameramotions.Inthisdissertation,newsolutionstotheSaMarepresentedusingthe transformednonlinearstatemodelwhichcanbeusedforlargersetofcameramotions, doesnotrequirefullcameravelocityinformation,andarereduced-order.Solutionsforthe stationaryaswellasmovingobjectsviewedbyamovingcameraarepresented. InChapter3,areducedorderobserverisdevelopedtoestimatethestructureofa staticobjectusingamovingcamera,wherefullcameravelocityandlinearacceleration areknown.Chapter4focusesonthedevelopmentofareducedorderobserverforthe SaMestimationofastationaryobjectwhenonlyasinglecameralinearvelocityisknown. 12

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InChapter5,anobserverdesignispresentedforaspecicclassofnonlinearsystems wheretheoutputdynamicsareaneintheunmeasurablestateandthedynamicsofthe unmeasurablestatearenonlinear.Themethodisappliedtosimultaneouslyestimatethe structureandmotionofamovingobjectseenbyamovingcamera.Anotherstrategyto theobserverdesigninthepresenceofanunmeasurabledisturbanceisanunknowninput observerUIO.Chapter6providesasolutiontoanUIOdesignforageneralclassof nonlinearsystemsandit'sapplicationtostructureestimationofamovingobjectisshown inChapter7. 13

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CHAPTER1 INTRODUCTION 1.1Motivation Cameraimagesprovideadensedatasetinanencryptedform.Estimationofthedata encodedintheimagesisapervasiveproblemacrossmultipledisciplines.Fromthelarge amountofinformationencodedintheimages,structureandmotionSaMoftheobjects inthesceneisofparticularinterest.SaMestimationisimportantforroboticapplications suchasvision-basedurbannavigationofanautonomousagent,manipulationofunknown andmovingtargets,orhuman-machineinteractionapplications.Formilitaryapplications suchas,automatedmissilenavigation,guidance,andcontrol,GPS-deniedautonomous ight,autonomousvisualsurveillanceandvideo-basedgeo-location;formedicalrobotics applications,image-basedestimationofSaMiscritical.Giventhecrosscuttingneedfor SaMestimation,thisdissertationismotivatedbythefollowingquestions. 1.Howcanthestructureofstationaryobjectsbeaccuratelyestimatedinreal-timefor astaticscenegivenallthecameramotioninformation? 2.Howcanthestructureofastationaryobjectbeestimatedgivenminimalinformation aboutthecameramotion? 3.Howcanthestructureandmotionindynamicscenesbeestimatedwhentheobjects aremovingwithunknownmotion? 1.2ProblemStatementandOverview ObserversforSaMestimationofatargetobjectintheeld-of-viewofamoving cameraaredeveloped.ForthegeneralSaMproblem,thecameraandthetargetare allowedtomovefreely.TheobjectiveoftheclassicstructurefrommotionSfMproblem istoestimatetheEuclideancoordinatesoffeaturepointsattachedtoanobjecti.e.,3D structureprovidedtherelativemotionbetweenthecameraandtheobjectisknown. TheconverseoftheSfMproblemisthemotionfromstructureMfSproblemwhere therelativemotionbetweenthecameraandtheobjectisestimatedbasedonknown 14

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geometryofthefeaturepointsattachedtoanobject.Anextendedproblemisstructure andmotionSaMwheretheobjectiveistoestimatetheEuclideangeometryofthe featurepointsaswellastherelativemotionbetweenthecameraandfeaturepoints. TheSaMproblemisfundamentalandsomeexamplesindicatethatSaMestimationis onlypossibleuptoascalewhenapinholecameramodelisused[4].Theworkpresented inthisdissertationexaminesamovingcameracapturingimagesofeitherastaticor movingobject.Forthesescenarios,areducedorderobserverisdevelopedusingaspecic state-spacemodeltoestimatestructureofastationaryobjecteithergivenallthecamera velocitiesseeChapter3orgivenpartialknowledgeofthecameramotionseeChapter 4.NonlinearobserverisdevelopedinChapter5forageneralclassofsystems,andthis observerisappliedtoestimatethestructureandmotionofamovingobjectgivenall cameravelocities.Anunknowninputobserverforageneralclassofsystemsisdeveloped inChapter6andit'sapplicationtothestructureestimationofamovingobjectgivenall cameravelocitiesispresentedinChapter7.TheobserverdesignsinChapters5and6are basedonanonlinearmodelofthemovingobjectandmovingcamerarelativemotion.A pinholecameraprojectionmodelisassumedandthemeasurementoffeaturepointsin eachcameraframecanbeobtainedusingexistingfeaturepointidentication,andtracking algorithmssee[57]. 1.3LiteratureReview Structureestimationofastationaryobjectgivenallsixcameravelocities: SolutionstotheSfMproblemcanbebroadlyclassiedasoinemethodsbatchmethods andonlinemethodsiterativemethods.Referencesandcritiquesofbatchmethodscan befoundin[813]andthereferencestherein.Onlinemethodstypicallyformulatethe SfMproblemasacontinuousdierentialequation,wheretheimagedynamicsarederived fromacontinuousimagesequencesee[3,1423]andthereferencestherein.Online methodsoftenrelyontheuseofanExtendedKalmanlterEKF[14,2426].Kalman lterbasedapproachesalsolackaconvergenceguaranteeandcoulddivergeinpractical 15

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scenarios.Also, apriori knowledgeaboutthenoiseisrequiredforsuchsolutions.In comparisontoKalmanlter-basedapproaches,someresearchershavedevelopednonlinear observersforSfMwithanalyticalproofsofstability.Forexample,ahigh-gainobserver calledtheidentier-basedobserverIBOispresentedforrangeestimationin[23]under theassumptionofknowncameramotion.In[3],adiscontinuoussliding-modeobserver isdevelopedwhichguaranteesexponentialconvergenceofthestatestoanarbitrarily smallneighborhood,i.e.,uniformlyultimatelyboundedUUBresult.Acontinuous observerwhichguaranteesasymptoticrangeestimationispresentedin[19]underthe assumptionofknowncameramotion.Asemi-globallyasymptoticallystablereduced-order observerispresentedin[27]toestimatetherange,givenknowncameramotion.In[21], anasymptoticallyconvergingnonlinearobserverisdevelopedbasedonLyapunov'sindirect method.AnapplicationofIBOispresentedin[28]toestimatetherangeoffeaturesinthe staticscene. Thedynamicsoftheunknownstatearenonlinearandtheunknownstateappears linearlyinthedynamicsoftheknownstates.Inthepreviouswork,thenonlinearities aredominatedusingslidingmodetechniquesasshownin[3,19]whichachievesUUB orasymptoticstability.Recently,in[2]anonlinearobserverisdevelopedandusing converseLyapunovtheoremexponentialconvergenceoftheestimationerrorisshown, buttheresultislocalinnature,meaningtheinitialconditionhastobewithinsome boundofthe`true'depth.ThenonlinearitiesaretreatedasaperturbationandLyapunovbasedstabilityanalysisispresentedtoshowthelocalstabilityoftheerrorsystem.Itis pointedoutin[2]thattheobserverisguaranteedtoconvergewithinitialconditionsinan arbitrarilylargecompactsetifthelinearvelocityinZ-direction,andangularvelocities inXandYdirectionsaresmall.Theauthorsin[2]proposethatthesevelocitiescan bescaleddownsuitablyinavisualservoingcontrollerscheme.Thus,theobservercan beinitializedwithinanarbitrarilylargecompactsetonlywhenthecameravelocities aresmall,whichrestrictsthedomainofapplicationsofthisobserver.Inanotherrecent 16

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work[1],animmersionandinvarianceI&Ibasedobserverisdesignedwhichcomputes theoutputinjectionfunctionsbysolvingapartialdierentialequationandachievesglobal exponentialstability.Theobserverrequirescameraaccelerationmeasurementsalong withcameravelocityandimagefeaturemeasurements.In[1],theauthorsstatethatthe observerhastosatisfytheExtendedOutputJacobianEOJobservabilityrankcondition, whichisstricterthanthepersistencyofexcitationcondition.Thus,theobserverin[1] cannotaddressallthecameramotions,whichcanbeaddressedbytheobserverin[2].The gainconditionoftheobserverin[1]isafunctionofimagefeatures,cameravelocitiesand cameraacceleration.Motivatedbythedesiretoachievegloballyexponentiallyconvergence observerfortherangeestimationwithlessstringentobservabilityconditionsanew nonlinearobserverisdevelopedinChapter3. Structureestimationofastationaryobjectwithoneknowncameralinear velocity: VariousbatchanditerativemethodshavebeendevelopedtosolvetheSaM problemuptoascale,suchas[11,29].However,incomparisontoSfMandMfSresults, sparseliteratureisavailablewheretheSaMproblemisformulatedintermsofcontinuous imagedynamicswithassociatedanalyticalstabilityanalysis.In[30],analgorithmis presentedtoestimatethestructureandmotionparametersuptoascalingfactor.In[31], aperspectiverealizationtheoryfortheestimationoftheshapeandmotionofamoving planarobjectobservedusingastaticcamerauptoascaleisdiscussed.Recently,a nonlinearobserverisdevelopedin[21]toasymptoticallyidentifythestructuregiventhe cameramotioni.e.,theSfMproblemortoasymptoticallyidentifythestructureand theunknowntime-varyingangularvelocitiesgivenallthreelinearvelocities.In[14,32] structureandlinearvelocitiesareestimatedgivenpartialstructureinformationsuchas lengthbetweentwopointsonanobject,whichmaybedicultinpracticeforrandom objects.Inanotherrecentresultin[28],theIBOapproachin[23]isusedtoestimate thestructureandtheconstantangularvelocityofthecameragivenallthreelinear velocities,whichmaynotbepossibleinpracticalscenariossuchasacameraattached 17

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toaunmannedvehiclewhereside-slipvelocitiesmaynotbeavailable.Theproblemof estimatingstructure,timevaryingangularvelocities,andtimevaryinglinearvelocities ofthecamerawithoutknowledgeofpartialstructureinformationremainsanunsolved problem. ThetechnicalchallengepresentedbytheSaMproblemisthattheimagedynamicsare scaledbyanunknownfactor,andtheunknownstructureismultipliedbyunknownmotion parameters.Thechallengeistoestimateastateintheopenloopdynamicsthatappears nonlinearlyinsideamatrixthatismultipliedbyavectorofunknownlinearandangular velocitytermssee212.Byassumingthatthevelocitiesareknown,orsomemodel knowledgeexists,previousonlineeortshavebeenabletoavoidtheproblemofseparately estimatingmultiplicativeuncertainties.Thecontributionofthisworkisastrategyto segregatethemultiplicativeuncertainties,andthentodevelopareducedordernonlinear observertoaddresstheSaMproblemwherethestructurei.e.,theproperlyscaledrelative Euclideancoordinatesoffeaturepoints,thetime-varyingangularvelocities,andtwo unknowntime-varyinglinearvelocitiesareestimatedi.e.,onerelativelinearvelocityis assumedtobeknownalongwithacorrespondingacceleration. Structureandmotionestimationofamovingobjectusingamoving camera: SolutionstotheSfMproblemwhentheobjectisstationary,canbeusedforselflocalizationandmapbuildingofanenvironmentusingamovingcamera.Sincetheobject isstationary,amovingcameracancapturesnapshotsoftheobjectfromtwodierent locationsandtriangulationcanbeusedtoestimatethestructure.IfobjectismovingSfM techniquescannotbeusedtorecoverthestructure.Inthisdissertation,thestructureand motionestimationofamovingobjectusingamovingcameraisreferredtoasSaMfM. Thepioneeringworkin[33]providesasolutiontotheSaMfMproblemwhereatleast veviewsarerequiredifthemotionoftheobjectisconstrainedtoastraightlineandat leastnineviewsarerequirediftheobjectismovingwithconictrajectories.In[34],the structureandmotionoftheobjectsmovingwithlinearorconictrajectoriesarerecovered 18

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fromtangentprojections,providedatleastnineviewsareavailable.In[35],anobjectis assumedtobemovingwithconstantvelocitieswhilebeingobservedbyanapproximate orthographicprojectioncameramodel.In[36],astereocameraisusedtoprovidea solutiontotheSaMfMwithatleastfourviews.In[37],abatchalgorithmispresentedfor objectmotionsrepresentedbymoregeneralcurves.In[38],afactorization-basedbatch algorithmisproposedwhereobjectsareassumedtobemovingwithconstantspeedina straightline,observedbyaweakperspectivecamera.Analgebraicgeometryapproach ispresentedin[39]toestimatethemotionofobjectsuptoascalegivenaminimum numberofpointcorrespondences.In[40],authorsproposeabatchalgorithmtoestimate thestructureandmotionofobjectsmovingonagroundplaneobservedbyamoving airbornecamera.Themethodreliesonastaticsceneforestimatingtheprojectivedepth, approximatedbythedepthoffeaturepointsonastaticbackgroundassumingthatone ofthefeaturepointsofthemovingobjectliesonthestaticbackground.In[41],abatch algorithmisdevelopedbyapproximatingthetrajectoriesofamovingobjectusingalinear combinationofdiscretecosinetransformDCTbasisvectors. Traditionally,SaMfMproblemistackledusingbatchalgorithmswhichusesalgebraic relationshipsbetween3Dcoordinatesofpointsinthecameracoordinateframeand corresponding2Dprojectionsontheimageframecollectedovernimagestoestimate thestructure.Batchalgorithmsarenotusefulinreal-timecontrolalgorithmssuchas formationcontrolofunmannedgroundvehicles,autonomousmissileguidance,navigation, andcontrolwherestateestimationisrequiredateverycameraimagecapture.Insteadof algebraicrelationshipsandgeometricconstraintsusedbybatchalgorithms,arigidbody kinematicmotionmodelcanbeusedtodesignnonlinearobservers/estimatorsusingthe datafromimagesuptothecurrenttimestep.Nonlinearobserveralgorithmgenerate theestimatesateverytimeinstantandhencecanbeimplementedonlinealongwiththe controlalgorithm. 19

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In[42],an H 1 approachtotheSaMfMproblemispresentedwherethemovingobject velocitiesareconsideredasexternaldisturbances,andonlythestructureestimation problemissolved.Theestimatorin[42]convergestoaballaroundtheorigini.e., anUUBresult.Inthisdissertation,twoapproachestotheproblemofstructureand motionestimationforamovingobjectarepresented.Intherstapproach,anextended statespaceiscreatedbycombiningthemovingobjectvelocitywiththeunknownstate. Theextendedstatespacedynamicsareexpressedasanonlinearsystemwhereoutput measurablestatedynamicsisaneintheunmeasurablestateandthedynamicsof theunmeasurablestateisnonlinear.Anonlinearobserverisdesignedforthisclassof nonlinearsystems.Inthesecondapproach,thelinearvelocityofthemovingobjectis viewedasanunknownexogenousinputandanunknowninputobserverapproachis developed.Inthefollowing,variousobserverdesigntechniquesfornonlinearsystems presentinliteraturearediscussedandtheproposedapproachiscontrastedagainstthe existingliterature. Oneoftheearliestobserverdesigntechniquesfornonlinearsystemsisbasedon astrategyoflinearizingaplantuptoanonlinearfunctionofoutputsbyachangeof coordinates.Themethodrequirestheoutputequationtobelinearinthestate.A Luenbergerobservercanbedevelopedforthetransformedsystem,buttheconditions forsimultaneoustransformationofthestatedynamicsandoutputequationarevery stringentandarebasedonasolutiontoapartialdierentialequationPDE[43,44]. In[45],amethodtoapproximatelysolveastatetransformationPDEisdeveloped.The observerdesignin[45]doesnotrequiretheoutputequationtobelinearinthestate, hence,itimposeslessstringentconditionsonsystemtransformation.Theresultsin[45] areextendedin[46]byincreasingthedomainoffeasiblecoordinatetransformations. Extensionoftheresultin[45]canalsobefoundin[47].Anotherapproachtononlinear observerdesignutilizesastrategywheregainsareselectedlargeenoughtodominatethe unmeasurablestatedependentnonlinearity[4850]. 20

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Lyapunov-basedobserverdesignsolutionsforLipschitznonlinearsystemscan befoundin[5154].In[52],anecessaryandsucientobserverexistenceconditionis developedforLipschitznonlinearsystems.In[53],theconditionsin[52]arerelatedtoa H 1 problemwhichsatisfyalltheregularityassumptions.However,forthedesignmethods in[52]and[53]theLipschitzconstantofthenon-linearityhastobesmall.Thislimitation isovercomein[54]bydevelopingarobustobserverbasedontheloop-transferrecovery LTRobserverdesigntechnique[55].Alloftheseobserverdesignsrequirethesystem dynamicstocontainalineartime-invariantterm,i.e., Ax alongwithanonlinearterm. In[3,23,27],observersforaclassofnonlinearsystemsaredeveloped,wherethe dynamicsofthemeasurablepartofthestateisaneintheunmeasurablepartofthe state,andthedynamicsoftheunmeasurablepartofthestateisnonlinear.Systemsof thisclassdonotcontainalineartime-invariantpartinthesystemdynamics.In[23], theobserverdesignisbasedontheuseofhighgainandparameteridenticationtheory from[56].However,theobserverin[23]canonlybedesignedforsystemswherethe dimensionoftheunmeasurablepartofthestateislessthanthemeasurablepartofthe state.Anobserverisdevelopedin[3]whichisnotrestrictedbythedimensionsofthe measurableandtheunmeasurablepartsofthestate.Thedesignin[3]isbasedona slidingmodestrategyandyieldsastateestimationwhichisuniformlyultimatelybounded UUBaroundtheoriginofthesystem.Recently,areduced-orderobserverdesignfora generalclassofnonlinearsystemswherethedynamicsofmeasurablepartofthestate isnotrequiredtobeaneintheunmeasurablepartispresentedin[57].Theobserver in[57]canalsobeusedtodesignanobserverfortheclassofsystemsin[3,23,27]butthe observerdesignreliesonndinganappropriateinvariantmanifoldwhichcanberendered attractive.FindingsuchinvariantmanifoldinvolvessolvingaPDEwhichmaybea tedioustask.Motivatedbythedesiretodesignanasymptoticallyconvergingobserverfor theclassofnonlinearsystemspresentedin[3]andovercomethelimitationsin[3,23,27],a newobserverispresentedinChapter5. 21

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Unknowninputobserverandit'sapplicationtostructureandmotion estimationofamovingobject: Intherelativerigidbodymotiondynamicsfor structureandmotionestimation,themovingobject'slinearvelocitycanbeviewedas anexogenoustime-varyingdisturbance.AnunknowninputobserverUIOapproachis usedtoestimatethestateofthedynamicalsystemwherethemovingobject'svelocityis consideredasanunknowninput.Thisapproachdoesnotrequirecreatinganextended statespaceasrequiredbytheobserverdesignapproachintheChapter5.Tomotivatethe UIOdesigninthisdissertation,theexistingUIOdesignapproachesarediscussedinthe followingsection. OneoftheearliestUIOresultsintroducedtheconceptofsystemobservabilitywith respecttounknowninputs[58].Presently,severalUIOalgorithmsexistinliteraturefor estimatingthestatewhenatime-varyingdisturbance,consideredasanexogenousinput,is presentinthesystemcf.,[5971].UIOsolutionsforlineartime-invariantLTIsystems arebroadlystudiedcf.,[5968,7275]. LinearUIOalgorithmsareextendedtononlinearsystemsin[6971,7681].In[69], anUIOisdesignedforSISOnonlinearsystems.In[70],anonlinearUIOispresented basedon H 1 optimization.TheobserveriscalledadynamicUIOwhichprovidesanextra degreeofdesignfreedombutincreasestheorderofthesystem.In[78],anonlinearUIO ispresentedforaclassofnonlinearsystemsbasedonanLMIapproachbutnonecessary andsucientobserverexistenceconditionsaredeveloped.In[79],ahighgainobserverfor aclassofnonlinearsystemsispresentedforstateandunknowninputestimationbutis achievedonlyuptoasmallboundwhichcanbereducedbyincreasingtheobservergains, i.e.,anuniformlyultimatelyboundedUUBresult.AhigherorderslidingmodeUIO ispresentedfornonlinearsystemsin[80]whichrequirestheoriginalnonlinearsystemto satisfygeometricconditionsfortransformingthesystemintotheBrunovskycanonical form.Basedonadetectabilitynotion,asucientconditionfortheexistenceofanUIOis derivedin[81]forstate-anesystemsuptoanoutputinjection.Basedonthegeometric 22

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approachof[59],necessaryandsucientconditionsarederivedin[71]fortheexistence ofanUIOforstateanesystemsuptoanonlinearunknowninputdynamics;hence, theUIOcanbeusedforalargerclassofunknowninputs.UIOsareusedextensivelyin faultdetectionandisolationforvariousclassesofsystemssuchaslinearsystems[82], controlanesystems[83],bilinearsystems[84],andnonlinearsystems[76,85].In[76],an unknowninputobserverforaclassofnonlinearsystemsispresentedforfaultdiagnosis. Theobserverdesignreliesonacoordinatetransformationwhichdecouplesthenonlinear systemintoasystemindependentofunknowninputsandasystemwiththestates thatcanbeexpressedaslinearcombinationsoftheoutputsandthestatesoftherst subsystem.TheobservergainisobtainedbysolvingaparametricLyapunovequation whichcanbechallengingtocompute[78].ToovercomethelimitationsofthecurrentUIO designs,asolutionforageneralclassofnonlinearsystemsisdesiredandispresentedin Chapter6. 1.4Contributions Thisdissertationfocusesondevelopingnonlinearobserversforvision-basedrange andmotionestimation.Theobserversaredevelopedtoovercomethetechnicalchallenges suchasglobalestimationinthepresenceoflocallyLipschitznonlinearityandnonlinear multiplicativeuncertainty.Thecontributionsoftheproposedresultsareasfollows. Globallyexponentiallyconvergentobserverforvision-basedrangeestimation: Thischapterpresentsanewreducedorderobserverforvisionbasedrange estimation.Themaincontributionsoftheobserveraresummarizedasfollows. 1.Globalexponentialconvergenceisachievedunderagainconditionsandtheobservabilityconditionrequiredbytherangeobserverspresentintheliterature. 2.Theobserverisproventobeexponentiallyconvergentevenunderarelaxedobservabilityconditionwhichallowscameramotiontobezeroalongallthreedirectionfor sucientlysmalldurationoftime. 23

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3.Theobserverisshowntobenitegain L p stablewithrespecttoanexogenous disturbanceinput.Thus,theobservererrorsremainboundedevenifthestationary objectassumptionisviolatedwheretheobjectmotionisconsideredasanexogenous input. Incomparisonwiththeobserverin[1],thegainconditioninthischapterisonlyafunction ofupperboundsoncameravelocitiesandimagesize. Structureandmotionwithasingleknownlinearvelocity: Inthischapteran observerforrangestructureandmotionestimationispresented.Thetechnicalchallenge presentedbytheSaMproblemisthattheimagedynamicsarescaledbyanunknown scalefactor,andtheunknownstructureismultipliedbyunknownmotionparameters.As describedinChapter4,thechallengeistoestimateastateintheopenloopdynamics thatappearsnonlinearlyinsideamatrixthatismultipliedbyavectorofunknownlinear andangularvelocitytermsseeEq.2.Byassumingthatthevelocitiesareknown,or somemodelknowledgeexists,previousonlineeortshavebeenabletoavoidtheproblem ofseparatelyestimatingmultiplicativeuncertainties.Thecontributionofthisworkis astrategytosegregatethemultiplicativeuncertainties,andthentodevelopareduced ordernonlinearobservertoaddresstheSaMproblemwherethestructurei.e.,the properlyscaledrelativeEuclideancoordinatesoffeaturepoints,thetime-varyingangular velocities,andtwounknowntime-varyinglinearvelocitiesareestimatedi.e.,onerelative linearvelocityisassumedtobeknownalongwithacorrespondingacceleration.The resultexploitsanuncertainlocallyLipschitzmodeloftheunknownlinearvelocitiesofthe camera.Thestrategicuseofastandardhomographydecompositionisusedtoestimate theangularvelocities,providedtheintrinsiccameracalibrationparametersareknown andfeaturepointscanbebetweenimages.ApersistencyofexcitationPEconditionis formulated,whichprovidesanobservabilityconditionthatcanbephysicallyinterpreted astheknowncameralinearvelocityshouldnotbezerooveranysmallintervaloftime, andthecamerashouldnotbemovingalongtheprojectedrayofapointbeingduring 24

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anysmallintervaloftime.ALyapunov-basedanalysisisprovidedthatindicatesthe SaMobservererrorsareasymptoticallyregulatedprovidedthePEconditionissatised. Bydevelopingareducedorderobservertosegregateandestimatethemultiplicative uncertainties,newapplicationscanbeaddressedincluding:rangeandvelocityestimation usingacameraxedtoamovingvehiclewhereonlytheforwardvelocity/accelerationof thevehicleisknown,rangeandvelocityestimationusinganunmannedairvehicleUAV usingonlyaforwardvelocity/accelerationsensors,etc. Observerdesignforaclassofnonlinearsystemswithanapplicationto structureandmotion: Inthischapter,theproblemofstructureandmotionestimation ofamovingobjectiscasteintoaspecicclassofanonlinearsystemsandanobserver designispresented.Forthesystemsofthisclass,outputdynamicsmeasurablepartofthe stateisaneintheunmeasurablepartofthestate,anddynamicsoftheunmeasurable partofthestateisnonlinear.Thereisnorestrictiononthedimensionsofthemeasurable partandtheunmeasurablepartofthestate.Theobserverdesigndoesnotrequirethe transformationoftheplantdynamicsintotheobservercanonicalformorsolvingaPDE, hence,assumptionofuniformobservabilityofthesystemdynamicsisnotrequired. Additionally,thesystemdynamicsdoesnotrequirealineartimeinvariantterm.Thestate estimationerrorasymptoticallyconvergestozerointhepresenceof L 2 [0 ; 1 time-varying disturbances.Thedesignisbasedonidentifyingthelinearunmeasurablepartofthestate fromthedynamicsofthemeasurablepartofthestateusingarobustidentiercalledthe robustintegralofthesignumoftheerrorRISE[86,87].Theidentieristhenusedto stabilizetheestimationerrordynamicsoftheunmeasurablestate.Theproposedobserver providesareal-timesolutiontostructureandmotionandextendstheexistingresults toobjectsmovingindependentlywithunknowntime-varyingvelocityconvergingtoa constant.Theproposedmethodhasnorequirementsoftheminimumnumberofpoint correspondencesortheminimumnumberofviews. 25

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Lyapunov-basedunknowninputobserverdesignforaclassofnonlinear systems: Inthischapter,anonlinearUIOisdevelopedforageneralclassofmulti-input multi-outputMIMOnonlinearsystems.Basedontheexistenceofasolutiontothe Riccatiequation,necessaryandsucientexistenceconditionsarederived.Theconditions provideguidelinesforchoosingtheobservergainmatrixbasedontheLipschitzconstantof thenonlinearitypresentinthedynamics.Analgorithmforchoosingthegainmatrixbased ontheEigenvalueplacementissuggestedin[52].Thegainmatrixisobtainedbysolving anLMIfeasibilityproblem.ContributionsofChapter6includethedesignofanUIOfor ageneralclassofnonlinearsystemsandanextensionoftheobserverexistenceconditions derivedin[52]forsystemswithknowninputstoageneralclassofnonlinearsystemswith unknowninputs. Applicationoftheunknowninputobserverforthestructureandmotion: Thecontributionofthisworkistoprovideacausalalgorithmforestimatingthestructure ofamovingobjectusingamovingcamerawithrelaxedassumptionsontheobject's motion.Theobjectisassumedtobemovingonagroundplanewitharbitraryvelocities observedbyadownwardlookingcamerawitharbitrarylinearmotionin3Dspace.No assumptionsaremadeontheminimumnumberofpointsorminimumnumberofviews requiredtoestimatethestructure.Featurepointdataandcameravelocitydatafromeach imageframeisrequired.Estimatingthestructureofamovingobjectisre-castintoan unknowninputobserverdesignproblem. 1.5DissertationOutline Chapter1servesasanintroduction.Themotivation,problemstatementandthe contributionsofthedissertationarediscussedinthischapter.Chapter2describesthe relativemotionmodelofacamera-objectsystemandstatesthecommonassumptions ofthemodelusedinrestofthechapters.Chapter3illustratesanonlinearobserverfor rangeestimationwhenallsixcameravelocitiesareknown.Theobserverpresentedin thischapterachievesglobalexponentialconvergenceresultforanonlinearsystemwitha 26

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locallyLipschitznonlinearity.Simulationresultscomparetheperformanceoftheobserver withtherecentlypublishedobserversfortherangeestimationproblem.Theobserver performanceisvalidatedusingexperimentsconductedonautonomousunderwatervehicle AUV.Chapter4providesasolutiontoanextendedstructureandmotionproblemwhere outofthesixcameravelocitiesthreeangularandthreelinearonlyoneofthelinear velocitiesisknown.Areducedorderobserverisdevelopedwhichasymptoticallyestimates theunknowndepthandtwounknownlinearcameravelocitiesgivenapersistencyof excitationPEconditionissatised.Angularvelocitiesareestimatedusinghomography matrixdecompositionbetweenconsecutivecameraframes.Simulationresultsareprovided toillustratetheperformanceoftheobserver.Chapter5developsanewobserverfora classofnonlinearsystemswhichcanbeusedtosolvethestructureandmotionestimation problemwhenanobjectismoving.Physicalconstraintsontheobject'smotionare discussed.Performanceoftheobserverisillustratedviacomparisonwiththeexisting observersinsimulation.Chapter6providesanewsolutiontoageneralclassofunknown inputobserversUIO.Necessaryandsucientexistenceconditionsaredevelopedfor theobserver.AlinearmatrixinequalityLMIisdevelopedtocomputetheobserver gainmatrix.Chapter7presentsanapplicationoftheUIOtotheSaMfMproblem. SpecicscenariosarediscussedwhentheSaMfMstatedynamicscanbetransformed intothestructureofUIOdevelopedinChapter6.Chapter8concludesthedissertation bysummarizingtheworkinthisdissertationandgivingdirectionstoafewfutureopen problemsintheobserverdesignandstructureandmotionestimation. 27

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CHAPTER2 CAMERAMOTIONMODEL Thepurposeofthischapteristoprovidebackgroundinformationpertainingto coordinateframesattachedtothecameraandobject,theirrelationship,geometric projectionsoftheobjectonacameraimageandcamera-objectrelativemotionmodel. 2.1EuclideanandImageSpaceRelationships Figure2-1.Coordinateframerelationshipsofamovingcameraandanobject. Amovingcameraobservingasceneinducesamotionintheimageplane.Point correspondencesinsuccessiveimagetoimagecanbecomputedusingexistingfeature trackingtechniques[57].Consideramovingcamerathatviewsfourormoreplanar 1 and non-collinearfeaturepointsdenotedby j = f 1 ; 2 ;::::;n g8 n 4 lyingxedinavisible plane r ,attachedtoanobjectinfrontofthecameraasshowninFig.2-1.Let F r bea 1 Fourplanarpointsareneededtocomputethehomography.Thehomographycanalso becomputedwith8non-coplanarandnon-collinearfeaturepointsusingthevirtualparallax"algorithm[88]. 28

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staticcoordinateframeattachedtotheobject.Astaticreferenceorthogonalcoordinate frame F c isattachedtothecameraatthelocationcorrespondingtoaninitialpointin time t 0 wheretheobjectisinthecameraeldofviewFOV.Aftertheinitialtime,an orthogonalcoordinateframe F c attachedtothecameraundergoessomerotation R t 2 SO andtranslation x f t 2 R 3 awayfrom F c TheEuclideancoordinates m t 2 R 3 ofapoint 2 observedbyacameraexpressedin thecameraframe F c andtherespectivenormalizedEuclideancoordinates m t 2 R 3 are denedas m t = x 1 t ;x 2 t ;x 3 t T ; m t = x 1 t x 3 t ; x 2 t x 3 t ; 1 T : TheconstantEuclideancoordinatesandthenormalizedcoordinatesofthefeaturepoints expressedinthecameraframe F c aredenotedby m 2 R 3 ,and m 2 R 3 respectivelyand aregivenbyEq.2super-scriptedbya` '.Tofacilitatethesubsequentdevelopment,the statevector y t =[ y 1 t y 2 t y 3 t ] T 2Y R 3 isconstructedfromEq.2as y = x 1 x 3 ; x 2 x 3 ; 1 x 3 T : Thecorrespondingfeaturepoints m and m t viewedbythecamerafromtwo dierentlocationsandtwodierentinstancesintimearerelatedbyadepthratio t x 3 x 3 t 2 R andahomographymatrix H t 2 R 3 3 as m = Hm where H x 3 x 3 R + x f d n T : Thehomographymatrixcanbeestimatedusingfour coplanarpointsoreightnon-coplanarpoints.Thehomographymatrix H t canbe 2 Subsequenttechnicaldevelopmentisshownforasinglepoint.Inpractice,resultscan beextendedtomultiplepointsinasimilarmanner. 29

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Figure2-2.Movingcameralookingatthestaticscene. decomposedusingvariousmethodsseee.g.,[89,90]toobtaintherotation R t ,depth ratio t ,thescaledtranslation x f t d where x f t istheabsolutetranslationbetween F c and F c d and n representstheperpendiculardistanceandnormalvectorbetween F c andtheobjectplane.Thedecompositionofthehomographyleadstotwosolutions,oneof whichisphysicallyrelevant.Anin-depthdiscussionaboutthehomographyestimationand decompositionandhowtoobtainthephysicallyrelevantsolutioncanbefoundin[4,89]. Usingprojectivegeometry,thenormalizedEuclideancoordinates m and m t canbe relatedtothepixelcoordinatesintheimagespaceas p = A c m;p = A c m where p t =[ u v 1] T isavectoroftheimage-spacefeaturepointcoordinates u t v t 2 R denedontheclosedandboundedset I R 3 ,and A c 2 R 3 3 isaconstant, 30

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known,invertibleintrinsiccameracalibrationmatrix[4]givenby A c = 2 6 6 6 6 4 m )]TJ/F23 11.9552 Tf 9.299 0 Td [( m cot u 0 0 m sin v 0 001 3 7 7 7 7 5 : InEq.2 u 0 v 0 2 R denotethepixelcoordinatesoftheprincipalpoint, m 2 R representsthefocallengthinpixelsand 2 R istheskewanglebetweenthecameraaxes. Since A c isknown,Eq.2canbeusedtorecover m t ,whichcanbeusedtopartially reconstructthestate y t 2.2Camera-ObjectRelativeMotionModel AsseenfromFig.2-2,thestaticscenepoint q canbeexpressedinthecoordinate system F c as m = x f + Rx oq where x oq isavectorfromtheoriginofcoordinatesystem F c tothepoint q expressedin thecoordinatesystem F c .DierentiatingEq.2,therelativemotionof q asobservedin thecameracoordinatesystemcanbeexpressedbythefollowingkinematics[4,91] m =[ ] m + v r where m t isdenedinEq.2, [ ] 2 R 3 3 denotesaskewsymmetricmatrixformed fromtheangularvelocityvectorofthecamera t = 1 2 3 T 2W R 3 ; and v r t representstherelativevelocityofthecamerawithrespecttothemovingpoint, denedas v r = v c )]TJ/F15 11.9552 Tf 14.509 3.022 Td [( R v p : InEq.2, v c t = v cx v cy v cz T 2V c R 3 denotesthecameravelocityinthe inertialreferenceframe, R v p t v p t = v px v py v pz T 2V p R 3 denotes thevelocityofthemovingpoint q expressedincamerareferenceframe F c ,and v p t = 31

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v px v py v pz T 2 V p R 3 denotesthevelocityofthemovingpointintheinertial referenceframe F : SubsequentdevelopmentinChapters3and4isbasedonthefact that,forstationaryobject, v p t =0 ; hence, v r t = v c t : UsingEqs.2and2,thedynamicsofthepartiallymeasurablestate y t canbe expressedas y 1 = v cx )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 v cz y 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 y 2 1 ++ y 2 1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v px )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 v pz y 3 y 2 = v cy )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 v cz y 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [(+ y 2 2 1 + y 1 y 2 2 + y 1 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( v py )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 v pz y 3 y 3 = )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 2 3 v cz )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 y 3 1 + y 1 y 3 2 + y 2 3 v pz wherethestates y 1 t and y 2 t canbemeasuredastheoutputofthesystemthrough theinvertibletransformationgivenbyEq.2.Thefollowingsymbolsaredenedto streamlinethenotationsthroughoutthedissertation: h 1 t v cx t )]TJ/F23 11.9552 Tf 12.91 0 Td [(y 1 t v cz t ; h 2 t v cy t )]TJ/F23 11.9552 Tf 12.197 0 Td [(y 2 t v cz t ;p 1 t )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 t y 2 t 1 t ++ y 2 1 t 2 t )]TJ/F23 11.9552 Tf 12.197 0 Td [(y 2 t 3 t and p 2 t )]TJ/F15 11.9552 Tf 9.299 0 Td [(+ y 2 2 t 1 t + y 1 t y 2 t 2 t + y 1 t 3 t .Thedevelopmentin Chapters3and4usesthefactthatinthegenericcamera-objectrelativemotionstate spacedynamics, v px t = v py t = v pz t =0 : 2.3Assumptions Thefollowingphysicallyinspiredassumptionsareusedinthedevelopmentofthe Chapters3-7. Assumption2.1. TherelativeEuclideandistance x 3 t betweenthecameraandthe featurepointsobservedonthetargetisupperandlowerboundedbysomeknownpositive constantsi.e.,theobjectremainswithinsomenitedistanceawayfromthecamera. Assumption2.2. Thecameravelocitiesareassumedtobebounded,andthelinear velocitiesareassumedtobedierentiablewithboundedaccelerations. 32

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Remark 2.1 Thestates y 1 t and y 2 t representpixellocations.Fromthenitesizeof theimage, y 1 t and y 2 t areboundedbyknownconstantsas y 1 y 1 t y 1 ;y 2 y 2 t y 2 : TherelativeEuclideandistance x 3 t betweenthecameraandthefeaturepointislower boundedbythecamerafocallength m inmeters,andisnotassumedtobeupper bounded.Therefore,thestate y 3 t ,aninverseofthestate x 3 t ,canbeupperandlower boundedas[2] 0
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CHAPTER3 GLOBALLYEXPONENTIALLYSTABLEOBSERVERFORVISION-BASEDRANGE ESTIMATION TheobjectiveoftheclassicstructurefrommotionSfMproblemistoestimate theEuclideancoordinatesoftrackedfeaturepointsattachedtoanobjecti.e.,3D structureprovidedtherelativemotionbetweenthecameraandtheobjectisknown. InthemotionmodeldescribedinChapter3thedynamicsoftheunknownstateare nonlinearandtheunknownstateappearsinthedynamicsoftheknownstates.Inthe previouswork,thenonlinearitiesaremitigatedusingslidingmodetechniquesorthe nonlinearpartisconsideredasaperturbationtermandstabilityisprovenusinga converseLyapunovtheoremoranimmersionandinvariancetechniqueisusedtond outputinjectionbysolvingpartialdierentialequations.Ingeneral,theobservability conditionfortheexistingresultsstatesthatcameramustbetranslatingalongatleast oneofthethreedirectionsandshouldnotbetranslatingparalleltotherayprojected fromafeaturepointatanyinstantoftime.Inthischapter,agloballyexponentially stablereduced-orderobserverisdesigned.Thecontributionsofthisworkisthreefold. Theobserverisgloballyexponentiallystableundersucientobservabilityandgain conditions.Second,theobserverisproventobeexponentiallyconvergentevenundera relaxedobservabilityconditionwhichallowsthecameramotiontobezeroalongallthree directionsforsucientlysmalldurationoftime.Finally,theobserverisshowntobenite gain L p stablewithrespecttoanexogenousdisturbanceinput.Thus,theobservererrors remainboundedevenifthestationaryobjectassumptionisviolatedwheretheobject motionisconsideredasanexogenousinput.Incomparisonwiththeobserverin[1],the gainconditionisonlyafunctionofupperboundsoncameravelocitiesandimagesize. Simulationandexperimentalresultsareprovidedtoshowtheperformanceoftheproposed observer.Comparisonoftheperformanceofproposedobservertotheperformanceof observersin[2]and[1]isprovided. 34

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3.1Vision-basedRangeEstimation Theobjectiveoftherangeestimationproblemi.e.,SfMistoestimatetheEuclidean coordinatesoffeaturepointsinastaticsceneusingamovingcamerawithknowncamera velocities b t and t .Theprojectivetransformationontotheimageplanelosesdepth information,butitcanberecoveredfrom2Dpointcorrespondencesintheimages. OncetheEuclideandepthisrecoveredusingEq.s2and2,thecompleteEuclidean coordinatescanbecomputed. 3.1.1RangeObserver Inthissection,anewnonlinearobserverforrangeestimationispresented.The dynamicsoftherange,givenby m t ,arerepresentedusingtheperspectivedynamic systeminEq.2.Allsixvelocitiesandlinearaccelerationsofthecameraareavailable assensormeasurements Scenarioswheretherelativemotion t and b t areknown includeacameraattachedtotheend-eectorofarobotmanipulator,mobilerobot, autonomousunderwatervehicleAUV,ormicroairvehicleMAV.Linearandangular cameravelocity,andlinearcameraaccelerationcanbeacquiredusingawidearrayof sensorcongurationsutilizinganinertialmeasurementunitIMU,globalpositioning systemGPS,orothersensors. Thestate y 3 t containsdepthinformationwhichislostduetoaperspectivetransformation.Toobtaintherangeofafeaturepoint m t ,itisnecessarytoscalethemeasured states y 1 t and y 2 t usingthedepth.Thus,themainmotivationoftheobserveristoestimatethestate y 3 t .Lettheestimatesofthestate y 3 t bedenedas ^ y 3 t .Toquantify thedepthestimationmismatch,anestimateerror e t isdenedas e y 3 )]TJ/F15 11.9552 Tf 12.747 0 Td [(^ y 3 : 35

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Toensuretheestimate ^ y 3 t isbounded,alocallyLipschitzprojectionlaw[92]isdesigned toupdate ^ y 3 t as ^ y 3 t = proj ^ y 3 ; = 8 > > > > > > > > > > < > > > > > > > > > > : 8 > > > > < > > > > : if y 3 ^ y 3 t y 3 or ^ y 3 t > y 3 and t 0 or ^ y 3 t y 3 and t > 0 if ^ y 3 t 0 .Thesignal t canbe integratedtoeliminatethecomputationofopticalow,i.e., y 1 and y 2 ,andthesignal ^ y 3 canbegeneratedusing ^ y 3 = + : InsteadofEq.3,inEq.3theupdatelawforthefunction y 1 ;y 2 ; ^ y 3 ;!;v c ; v c is givenby =^ y 2 3 v cz + y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 ^ y 3 )]TJ/F23 11.9552 Tf 9.298 0 Td [(k 3 h 2 1 + h 2 2 ^ y 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 3 h 1 p 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 3 h 2 p 2 )]TJ/F23 11.9552 Tf 9.298 0 Td [(k 3 y 1 v cx )]TJ/F23 11.9552 Tf 11.955 0 Td [(k 3 y 2 v cy + k 3 v cz y 2 1 + y 2 2 2 36

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and y 1 ;y 2 ;v c isdenedas k 3 v cx y 1 + v cy y 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(v cz y 2 1 + y 2 2 2 where k 3 2 R + .Theinitialconditionoftheobserverisselectedas t 0 = 0 where 0 isanarbitraryconstant. Assumption3.1. Thesubsequentdevelopmentisbasedontheassumptionthat h 2 1 + h 2 2 "> 0 8 t 0 forapositiveconstant .Thisassumptionisanobservabilitycondition fortheobserverinEqs.3-3,andisthesameasobtainedpreviouslyintheliterature [1,19,23,93].Theconditionphysicallyimpliesthat v cx t v cy t v cz t arenotequalto zerosimultaneouslyandthemotionofthecamerashouldnotbealongtheprojectedrayof thepointbeingobserved. 3.1.2StabilityAnalysis Theorem3.1. TheobserverpresentedinEqs.3-3isagloballyexponentiallystable observerprovidedAssumptions2.2and3.1aresatisedalongwiththesucientcondition k 3 2 v cz m + v cz + y 2 1 + y 1 2 where v cz 1 and 2 areknownupperboundson v cz t 1 t and 2 t Proof. ForthreecasesofprojectionlawdescribedbyEq.3the e t errordynamicsare givenby Case1: y 3 ^ y 3 t y 3 or ^ y 3 t > y 3 and t 0 or ^ y 3 t y 3 and t > 0 37

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UsingEqs.2and3-3,theerrordynamicsof e t canbeexpressedas e = & )]TJETq1 0 0 1 303.063 687.091 cm[]0 d 0 J 0.478 w 0 0 m 6.137 0 l SQBT/F23 11.9552 Tf 303.063 680.27 Td [(y 3 )]TJ/F15 11.9552 Tf 12.747 0 Td [(^ y 3 : Case3: ^ y 3 t
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Case3:Takingthederivativeof V e andutilizingEq.3yields V = y 3 +^ y 3 v cz )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 1 + y 1 2 e 2 )]TJ/F23 11.9552 Tf 9.298 0 Td [(k 3 h 2 1 + h 2 2 e 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(e ^ y 3 )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 3 ; wherethelasttermontherighthandsideofEq.3isalwaysnegative,andhence,the inequalityinEq.34canbeachieved. ForallthreecasesofprojectiontheGronwall-Bellmanlemma[94]canbeappliedto Eq.3toyield V t V exp )]TJ/F23 11.9552 Tf 9.298 0 Td [(k 1 t : Hence,fromEq.6,thefollowingupperboundfor e t canbeobtained k e t k k e k exp )]TJ/F23 11.9552 Tf 9.299 0 Td [(k 1 t where 2 R + FromEq.3, e t 2L 1 1 Since e t 2L 1 ,andusingRemark2.1, y 3 t 2L 1 thus ^ y 3 t 2L 1 .Fromtheboundednessof y t v c t and t ,Eq.3canbeusedto provethat k 3 2L 1 .Basedonthefactthat e t y t t v c t k 3 2L 1 ,standard linearanalysismethodscanbeusedtoprovethat e t 2L 1 .Thus, y 3 t isexponentially estimatedandEqs.2-2canbeusedtorecovertheEuclideancoordinates m t ofthe featurepoint. IftheconditioninAssumption3.1isnotsatisedandthegain k 3 ischosenaccording toEq.3,theproposedobserverisstillexponentiallyconvergent,providedthePE conditionin[2]issatised. 1 Forafunction s t 2 R n 8 n 2 [1 ; 1 s t 2L 1 meansthefunction s t hasanite L 1 norm,i.e., k s t k L 1 = sup t 0 k s t k 2 < 1 where kk 2 denotesthe2-normin R n 39

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Theorem3.2. TheobserverpresentedinEqs.3-3isaexponentiallystableobserver provided k 3 ischosenaccordingtoEq.3,Assumption2.2issatised,andthefollowing PEconditionissatised t + T t )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(h 2 1 + h 2 2 d > 0 ; 8 t>t 0 where T; 2 R + Proof. ToexaminethestabilityoftheestimationerrordynamicsinEq.3underthe assumptionthatEq.3issatised,considerthenominalsystem e = )]TJ/F23 11.9552 Tf 9.298 0 Td [(k 3 h 2 1 + h 2 2 e: UsingTheorem2.5.1of[95]theerrorsysteminEq.3isgloballyexponentiallystable iftheconditioninEq.3issatised.SincethenominalsysteminEq.3isglobally exponentiallystableusing4.14of[96]basedontheConverseLyapunovTheorem,there existsafunction V :[0 ; 1 R R thatsatisestheinequalities c 1 jj e jj 2 V t;e c 2 jj e jj 2 ; @ V @t + @ V @e )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 11.9552 Tf 9.299 0 Td [(k 3 h 2 1 + h 2 2 e )]TJ/F23 11.9552 Tf 28.56 0 Td [(c 3 jj e jj 2 ; @ V @e c 4 jj e jj where c i 2 R + 8 i = f 1 ;::; 4 g .AfterusingEq.6withthepropertiesinEq.3and substitutingintheperturbedsystemEq.3,thefollowinginequalitiescanbeobtained V @ V @t + @ V @e )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F23 11.9552 Tf 9.299 0 Td [(k 3 h 2 1 + h 2 2 e ; + @ V @e y 3 +^ y 3 v cz )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 2 1 + y 1 2 e ; V )]TJ/F23 11.9552 Tf 28.56 0 Td [(c 3 jj e jj 2 + c 4 jj e jj 2 ; where = 2 v cz m + v cz + y 2 1 + y 1 2 ,and V t canbeupperboundedas V )]TJ/F15 11.9552 Tf 21.917 0 Td [( c 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(c 4 jj e jj 2 : 40

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Since k 3 isselectedaccordingtoEq.3withsucientlysmall c 3 satises c 3 >c 4 Hence,theoriginoftheperturbedsystemEq.3isexponentiallystable. Remark 3.1 Asstatedin[2],thePEconditionphysicallyimpliesthatallthelinear velocitiesshouldnotbeidenticallyzeroandthatthecamerashouldnotbetranslating alongtheprojectedrayofanyfeaturepointduringanysmallintervaloftime [ t;t + T ] .If allofthelinearvelocitiesarezeroatanyinstantoftime h 2 1 t + h 2 2 t =0 andthestability oftheobserverinEqs.3-3cannotbeshownusingTheorem3.1,Theorem3.2ensures stabilityofsysteminsuchcases. 3.2StabilityAnalysisinthePresenceofDisturbances Inthissection,thestabilityoftheobserverinEqs.3-3isanalyzedinthe presenceofanexogenousinputsuchasadisturbanceactingonthecameramotionora targetobjectbeginstomove.Thedisturbanceentersthesystemas m = 2 6 6 6 6 4 1000 )]TJ/F23 11.9552 Tf 9.299 0 Td [(x 3 x 2 010 x 3 0 )]TJ/F23 11.9552 Tf 9.298 0 Td [(x 1 001 )]TJ/F23 11.9552 Tf 9.298 0 Td [(x 2 x 1 0 3 7 7 7 7 5 2 6 4 v c + v c + 3 7 5 ; where v c t t representtheexogenousinputssuchthat v c t t 2L pe 2 with sup 0 t jj v c t jj r b and sup 0 t jj t jj r forsome r b ;r 2 R + .UsingEqs.2 and3,thedynamicsoftheunmeasurablestate y 3 t canbeexpressedas y 3 = )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 2 3 v cz )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 y 3 1 + y 1 y 3 2 + y 3 where y 3 = )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 2 3 v cz )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 y 3 1 + y 1 y 3 2 : 2 Thespace L pe = f u j u 2L p ; 8 2 [0 ; 1 ] g ; and u isatruncationof u denedby u t = u y ; 0 t 0 t> 41

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Theorem3.3. TheobserverpresentedinEqs.3-3isnite-gain L p 3 stablewhere p 2 [1 ; 1 ] withrespecttotheexogenousinput v T c T T and L p gainlessthanor equalto 1 k 1 Proof. UsingEqs.322and3-3theerrorsystemcanbewrittenas e = y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 e + y 3 +^ y 3 v cz e )]TJ/F23 11.9552 Tf 9.298 0 Td [(k 3 h 2 1 + h 2 2 e + y 3 k 3 : TheerrorsysteminEq.3canbeexpressedinthefollowingform e = f e;u ; r = h e where u t = k 3 y 3 t isanexogenousdisturbance/noiseinput, r t = e t .Let R bea domaincontaining e t =0 and u t =0 ,thefunction f : R R R islinearandglobally Lipschitzin u t h : R R iscontinuousin e t UsingTheorem3.1,theunforcedsystem e = f e; 0 isgloballyexponentiallystablewiththeLyapunovfunctioninEq.6whichsatisesthe followingbounds 0 : 5 k e k 2 V e 0 : 5 k e k 2 ; @V @t + @V @e f e; 0 )]TJ/F23 11.9552 Tf 28.559 0 Td [(k 1 k e k 2 ; @V @e k e k : 3 Amapping F : L m e !L n e isnite-gain L stableifthereexistnon-negativeconstants % and suchthat k Fu k L % k u k L + forall u 2L m e and 2 [0 ; 1 wheretheextended space L m e isdenedas L m e = f u j u 2L m ; 8 2 [0 ; 1 g : 42

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Sincethefunction f e;u isgloballyLipschitzin u t ,thefollowinginequalityissatised jj f e;u )]TJ/F23 11.9552 Tf 11.956 0 Td [(f e; 0 jjjj u jj : SinceEqs.3and3aresatised,usingTheorem5.1of[96]theerrorsysteminEq. 3isnitegain L p stablewhere p 2 [1 ; 1 ] with L p gainlessthanorequalto 1 k 1 foreach e 2 R ,i.e., jj e jj L p 1 k 1 jj u jj L p + jj e 0 jj ; where = 8 > > < > > : 1 ;ifn = 1 1 k 1 n 1 =n ;ifn 2 [1 ; 1 : Thevelocitiesoftheobjectdenotedby b O and O canbeassumedtobe L pe disturbancesactingonthesystemasshowninEq.3.Thus,Theorem3.3impliesthatevenif thestationaryobjectassumptionisviolated,theobservererrorsarebounded.The L p gain isthemeasureofaccuracyoftheestimatesandgivesanupperboundontheestimation errors.The L p gaincanbereducedbyincreasingthegain k 3 whichinturnreducesthe constant k 1 seeEqs.6and3. 3.3Discussion AcomparisonbetweentheproposedobserverwiththeI&Iobserver[1]andthe observerin[2]isprovidedinTable3-1andthenumberedlistbelow. 1.Thepresentedobserverachieves global exponentialestimationofthe3DEuclidean coordinatesoffeaturepoints,whichisasimilarresultachievedbytheobserver developedin[1].Theobserverpresentedin[2]onlyachieves local exponential convergenceoftheestimationerrors.Thus,theproposedobserverandtheobserver in[1]canhavearbitraryinitialconditionsasopposedtotheinitialconditions requiredbytheobserverpresentedin[2].Alimitationofthelocalnatureofthe resultin[2]isillustratedinthesubsequentsimulations. 43

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Table3-1.Comparisonofthepresentedobserverwithobserversin[1]and[2]. ProposedobserverObserverviaI&I[1]Observerin[2] Globalexponentialerror convergence Globalexponentialerror convergence Localexponentialerror convergence Observability: h 2 1 t + h 2 2 t "> 0 ; 8 t 0 Observability: h 2 1 t + h 2 2 t > 0 ; 8 t 0 Observability: @ t : 8 t> t; h 2 1 t + h 2 2 t =0 stableif 9 t : h 2 1 t + h 2 2 t =0 singularif h 2 1 t + h 2 2 t =0 foranytime t stableif 9 t : h 2 1 t + h 2 2 t =0 Requirescameravelocities andlinearaccelerations Requirescameravelocities andlinearaccelerations Requiresonlycamera velocities ReducedorderReducedorderFullorder Adaptedfrom[1]F.MorbidiandD.Prattichizzo,Rangeestimationfromamovingcamera:animmersionandinvarianceapproach,inProc.IEEEInt.Conf.Robot.Autom., Kobe,Japan,May2009,pp.2810.[2]A.D.Luca,G.Oriolo,andP.R.Giordano, Featuredepthobservationforimagebasedvisualservoing:Theoryandexperiments,Int. J.Robot.Res.,vol.27,no.10,pp.1093,2008. 2.Oneoftheadvantagesoftheobserverpresentedin[2]overtheobserverin[1],is theuseofalessrestrictiveobservabilityconditionwhichenablestheobservertobe usedforalargersetofcameramotions.Theobservabilityconditionoftheproposed observeristhesameasthatin[1],butiftheobservabilityconditionin[1]isnot satised,theI&Iobserverbecomessingular.Theadvantageoftheproposedobserver isthateveniftheobservabilityconditioninAssumption3.1isnotsatised,the observerisstilllocallyexponentiallystableandthuscanencompassalargersetof cameramotions.Thelimitationsofthesingularityissuewiththeobserverin[1]is illustratedinthesubsequentsimulationsection. 3.Theproposedobserverrequiresmeasurementsofthecameralinearaccelerationalong withcameravelocitiesandimagefeatures,whicharealsorequiredbytheobserver in[1].Thus,theproposedobserverandtheobserverin[1]aremoresensitiveto noisyinputmeasurementscomparedtotheobserverin[2].Improvedsteady-state performanceisillustratedbytheobserverin[2]inthepresenceofnoiseinthe subsequentsimulationsection. 44

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4.Thegainconditionin[1]isafunctionoftheimagesize,cameravelocitiesand acceleration.Onthecontrary,thegainconditionfortheproposedobserverisonlya functionofimagesizeandcameravelocities. 3.4SimulationsandExperiments Simulationsareconductedtoevaluatetheperformanceoftheobserver.Theperformanceoftheobserveriscomparedwiththeobserversin[2]and[1].Foreachsimulation thefocallengthofthecameraissetto m =30 andthegainsfortheestimatorsare adjustedtoachievethebestperformancei.e.,theleastestimationerror.Incontrastto thetrial-and-errorapproach,methodologicalapproachessuchas[97102]couldbeusedto adjusttheobservergains.Fortherstsimulation,theinitiallocationofthepointonthe targetwithrespecttotheinitialcameraframeisselectedas m t 0 = 1050 : 5 T m Thecameravelocitiesareselectedas v c = 0 : 30 : 4+0 : 1sin t 4 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 3 T m/s, = 0 )]TJ/F24 7.9701 Tf 12.153 4.707 Td [( 30 0 T rad/s. AdditivewhiteGuassiannoisewithasignal-to-noiseratioSNRof20dBisadded totheimagepixelmeasurements,andnoisewithzeromeanandavarianceof0.01is addedtothevelocitymeasurements.Thevelocitysignalisdierentiatedusingthe DerivativeblockinSimulinktoobtainalinearaccelerationsignal.Theestimatesare integratedwithastepsizeof 0 : 01 sec usingtheode4Matlabcommandwhichusesa Runge-KuttaR-Kintegrator.Theinitialconditionoftheobserverissetto t 0 =5 with k 3 =1 : 55 10 )]TJ/F22 7.9701 Tf 6.586 0 Td [(3 .Fortheobserverin[1],theinitialconditionischosentobe 4 t 0 = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 9 andtheobservergainissetto 2 : 5 10 )]TJ/F22 7.9701 Tf 6.587 0 Td [(5 : Theinitialconditionsandthe 4 Thesymbol t istakenfrom[1]anddenotesanauxiliarystate. 45

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observergainsfortheobserverin[2]areselected 5 as k 1 = k 2 =200 ;k 3 =0 : 1 and ^ y 1 t 0 =600 ; ^ y 2 t 0 =300 ; ^ y 3 t 0 =50 : Theinitialconditionsareselectedsothatthe initialvalueoftheestimateddepthisequalforallthreeobservers.Acomparisonofthe depthestimationperformanceoftheobserversisshowninFigure3-1.AsshowninTable 3-2,theroot-meansquareRMSofthedepthestimationerrorisalsocomparedforthe transientandthesteady-stateresponse.Thetransientperiodisselectedtobetherst 0 : 2 sec .TheproposedobserverhastheleasttransientRMSerror,andtheobserverin[2] hastheminimumsteady-stateRMSerror. AsecondsimulationisperformedbasedonDiscussionPoint2ofSection3.3.The cameravelocitiesforthissimulationareselectedas v c = 000 : 5cos t= 2 T m=s = 000 T rad=s whichviolatestheobservabilityconditioninAssumption3.1butsatisesthecondition inEq.3.Again,theimagepixeldataiscorruptedwiththeadditivewhiteGaussian noisewithanSNRof20dB.Noiseofzeromeanand0.01varianceisaddedtothecamera velocitymeasurements.UsingtheRunge-Kuttaintegratorwithatimestepof 0 : 03 sec; thestateestimatesarecomputed.Figure3-2showsthedepthestimationperformanceof theproposedobserverandtheobserverin[2]forthesameinitialconditions ^ y 3 t 0 : The proposedobserverexhibitsabettertransientperformancecomparedtotheobserverin[2]. Figure3-3showstheevolutionof h 2 1 t + h 2 2 t : At 1 sec;h 2 1 t + h 2 2 t =0 : 1 andat 3 sec; h 2 1 t + h 2 2 t =10 e )]TJ/F15 11.9552 Tf 12.368 0 Td [(4 .InFigure3-4,thereisapeakinthedepthestimateof[1]near t =1 sec: Theresponserecoversfromthepeakat t =1 sec butat t =3 sec theobserver 5 Thesymbols k 1 ;k 2 ;k 3 ; ^ y 1 t ; ^ y 2 t aretakenfrom[2].Theobserverin[2]isathird orderobserverand ^ y 1 t ; ^ y 2 t denotestheestimatesof y 1 t and y 2 t 46

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in[1]becomessingular.TheresultsinFigure3-4coincidewiththetheoreticalprediction discussedinPoint2ofSection3.3. Athirdsimulationisperformedusingcameravelocitiesof v c = 0 : 30 : 4+0 : 1sin t 4 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 T m=s; = 0 3 0 T rad=s: todemonstratethatforlargeinitialconditionstheproposedobserverconvergeswhile thelocalobserverin[2]isunstable.In[2],thedomainofinitialconditionsissmall forlarge v cz t ;! 1 t and 2 t : Theinitialrelativepositionofthetargetpointis m t 0 = 1055 T m .Theproposedobserverisinitializedto t 0 =300 andthe gainisselectedas k 3 =0 : 09 : Fortheobserverin[2],theinitialconditionsandgainsare setto ^ y 1 t 0 =60 ; ^ y 2 t 0 =30 ; ^ y 3 t 0 =144 and k 1 = k 2 =12 ;k 3 =10 : 2 .The observersareintegratedusingtheRunge-Kuttaintegratorwithatimestepof 0 : 01 sec ThestateestimationresultsareshowninFigures3-5-3-7.Sincethestates y 1 t and y 2 t aremeasurable,theinitialconditionsof ^ y 1 t and ^ y 2 t aresetequaltotheinitialvalues of y 1 t and y 2 t : Thegainsoftheobserverin[2]aretunedandtheinitialconditionis progressivelyincreaseduntiltheobservererrorconverges.Convergenceisobservedfor ^ y 3 t 0 143 butnotfor ^ y 3 t 0 144 : Fortheproposedobserver,theobservererror convergesevenforaninitialconditionaslargeas t 0 =300 : Inthissimulationavalue of t 0 =300 correspondsto ^ y 3 t 0 =583 : 5 fortheproposedobserver.Thesimulation demonstratesthattheobserverin[2]isunstablewhentheinitialconditionsarechosen outsidealocaldomain. Experimentsareconductedtoestimatetherangeofa9-inchMooringbuoyoatingin themiddleofawatercolumnasobservedbyacamerarigidlyattachedtoanautonomous underwatervehicleAUV.Figure3-8showstheAUVexperimentalplatform.TheAUVis equippedwithaMatrixVisionmvBlueFox-120acolorUSBcamera,aDopplervelocitylog DVL,apressuretransducer,acompassandaninertialmeasurementunitIMU.Two 47

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computersrunningMicrosoftWindowsServer2008areusedontheAUV.Onecomputeris dedicatedforrunningimageprocessingalgorithmsandtheothercomputerexecutessensor datafusion,lowlevelcomponentcommunicationandcontrol,andmissionplanning.An unscentedKalmanlterUKFisusedtofusetheIMU,DVLandpressuretransducer dataat 100 Hz toaccuratelyestimatetheposition,orientationandvelocityoftheAUV withrespecttoaninertialframebycorrectingtheIMUbias.Thispositiondataisusedto comparetheresultsoftheobserverwitharelativegroundtruthmeasurementoftheAUV byrotatingthelocalizedAUVpositionintothecameraxedframe.Thebuoyistracked inthevideoimageofdimension 640 480 usingastandardfeaturetrackingalgorithmas showninFigure3-9,andpixeldataofthecentroidofthebuoyisrecordedat 15 Hz: The cameraiscalibratedusingastandardcameracalibrationalgorithm[103]andisgivenby A c = 2 6 6 6 6 4 749 : 822310321 : 05569 0750 : 19507292 : 41939 001 3 7 7 7 7 5 : Thelinearandangularvelocity,andlinearaccelerationdataobtainedfromtheUKFis loggedatthecameraframerate.Usingthevelocity,linearaccelerationandpixeldata obtainedfromtheAUVsensors,therangeofthebuoyisestimatedwithrespecttothe camera.Theinitialconditionischosenas t 0 =0 : 08 andtheobservergainisselected tobe k 3 =2 10 e )]TJ/F15 11.9552 Tf 12.858 0 Td [(6 : TheobserverequationsareintegratedusingaRunge-Kutta integratorwithatimestepof 1 15 sec: Acomparisonoftheestimatedrangewiththeground truthmeasurementisshowninFigure3-10.InFigure3-11,thefeaturetrackingalgorithm failsforseveralframesneartime t =6 sec .Therangeestimationalgorithmshowsrobust performanceeveninthepresenceoffeaturetrackingerrorsasillustratedinFigure3-10. 48

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Table3-2.ComparisonoftheRMSdepthestimationerrors. ProposedobserverObserverin[2]Observerin[1] TransientRMSerror0.31280.34770.3547 Steady-stateRMSerror0.17170.01550.2150 Adaptedfrom[1]F.MorbidiandD.Prattichizzo,Rangeestimationfromamovingcamera:animmersionandinvarianceapproach,inProc.IEEEInt.Conf.Robot.Autom., Kobe,Japan,May2009,pp.2810.[2]A.D.Luca,G.Oriolo,andP.R.Giordano, Featuredepthobservationforimagebasedvisualservoing:Theoryandexperiments,Int. J.Robot.Res.,vol.27,no.10,pp.1093,2008. Figure3-1.Comparisonbetweenthetrueandestimateddepthinthepresenceof measurementnoiseAtheestimateddepthusingtheproposedobserver.B theestimateddepthusingtheobserverin[1].Ctheestimateddepthusing theobserverin[2].Adaptedfrom[1]F.MorbidiandD.Prattichizzo,"Range estimationfromamovingcamera:animmersionandinvarianceapproach,"in Proc.IEEEInt.Conf.Robot.Autom.,Kobe,Japan,May2009,pp. 2810.[2]A.D.Luca,G.Oriolo,andP.R.Giordano,"Featuredepth observationforimagebasedvisualservoing:Theoryandexperiments,"Int.J. Robot.Res.,vol.27,no.10,pp.1093,2008. 49

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Figure3-2.Comparisonofthedepthestimationusingtheproposedobserverandthe observerin[2]whencameramotiondoesnotsatisfyAssumption3.1. AdaptedfromA.D.Luca,G.Oriolo,andP.R.Giordano,"Featuredepth observationforimagebasedvisualservoing:Theoryandexperiments,"Int.J. Robot.Res.,vol.27,no.10,pp.1093,2008. Figure3-3.Evolutionofthesignal h 2 1 t + h 2 2 t againsttime. 50

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Figure3-4.Depthestimationusingtheobserverin[1].Adaptedfrom[1]F.Morbidiand D.Prattichizzo,"Rangeestimationfromamovingcamera:animmersionand invarianceapproach,"inProc.IEEEInt.Conf.Robot.Autom.,Kobe,Japan, May2009,pp.2810. Figure3-5.Estimationof ^ y 3 t startingfromlargeinitialcondition t 0 =300 usingthe proposedobserver. 51

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Figure3-6.Depthestimationwithlargeinitialcondition t 0 =300 usingtheproposed observer. 3.5Summary Anonlinearobserverispresentedfortherangeestimationoffeaturepointsusing amovingcamera.Theobserverisgloballyexponentiallystableprovidedanobservabilityconditionissatised.Theobserverisalsoshowntobeexponentiallystableundera relaxedobservabilitycondition.Theobserverrequiresvelocityandlinearaccelerationmeasurementsofthecamera.Theobserverisshowntoberobustagainstexternaldisturbances actingonthecameramotionandpixelnoise.SimulationresultsanddatafromanAUV experimentdemonstratetheperformanceoftheobserver. 52

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Figure3-7.Stateestimationusingtheobserverin[2]forlargeinitialconditions.Sincethe stateestimateisverylarge,simulationfailstointegrateat t =0 : 03 sec AdaptedfromA.D.Luca,G.Oriolo,andP.R.Giordano,"Featuredepth observationforimagebasedvisualservoing:Theoryandexperiments,"Int.J. Robot.Res.,vol.27,no.10,pp.1093,2008. Figure3-8.AUVexperimentalsetup. 53

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Figure3-9.AnimageframedisplayingthetrackedbuoytargetbytheAUV. Figure3-10.Comparisonoftheestimatedandgroundtruthrangeofthebuoywith respecttotheunderwatervehicle. 54

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Figure3-11.States y 1 t and y 2 t computedusingimagepixels. 55

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CHAPTER4 STRUCTUREANDMOTIONUSINGASINGLEKNOWNCAMERALINEAR VELOCITY Inthischapter,asolutiontotheSaMestimationispresentedwhenonlyoneof thelinearcameravelocitiesofthecameracanbemeasured.Theangularvelocityof thecameraisestimatedbydecomposingthehomographyestimatedfromtwoimages. Arobustderivativeestimatorisusedfortheangularvelocityestimation.Inthestate dynamics,theunknowntime-varyinglinearvelocitiesismultipliedbyanunknownstate posesaproblemofmultiplicativetime-varyinguncertainties.Inthisworkastrategyis presentedtosegregatethemultiplicativeuncertainties,andthentodevelopareduced ordernonlinearobservertoaddresstheSaMproblemwherethestructurei.e.,the properlyscaledrelativeEuclideancoordinatesoftrackedfeaturepoints,thetime-varying angularvelocities,andtwounknowntime-varyinglinearvelocitiesareestimated.The resultexploitsanuncertainlocallyLipschitzmodeloftheunknownlinearvelocitiesof thecamera.ApersistencyofexcitationPEconditionisformulated,whichprovidesan observabilityconditionthatcanbephysicallyinterpretedastheknowncameralinear velocityshouldnotbezerooveranysmallintervaloftime,andthecamerashould notbemovingalongtheprojectedrayofapointbeingtracked.ALyapunov-based analysisisprovidedthatindicatestheSaMobservererrorsaregloballyasymptotically regulatedprovidedthePEconditionissatised.Bydevelopingareducedorderobserver tosegregateandestimatethemultiplicativeuncertainties,newapplicationscanbe addressedincluding:rangeandvelocityestimationusingacameraxedtoamoving vehiclewhereonlytheforwardvelocity/accelerationofthevehicleisknown,range andvelocityestimationusinganunmannedairvehicleUAVusingonlyaforward velocity/accelerationsensors,etc. 56

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4.1StructureandMotionEstimation 4.1.1EstimationwithaKnownLinearVelocity Inthissection,anestimatorisdesignedfortheperspectivedynamicsysteminEq.2 12,wheretheangularvelocityisconsideredunknownandonlyoneofthelinearvelocities i.e., v cz 1 andrespectiveaccelerationi.e., v cz isavailable.Moreover,anuncertain dynamicmodelofthelinearvelocity v c t isassumedtobeavailableas[3,21] v ci t = q v ci ;t v ci ; 8 i = f x;y g where q v ci ;t 2 R isaknownlocallyLipschitzfunctionofunknownstates. Tofacilitatethedesignandanalysisofthesubsequentobserver,anewstate u t 2 U R 2 u 1 t = y 3 v cx ;u 2 t = y 3 v cy T ,isdenedwhere U isaclosedandbounded set.AfterutilizingEqs.2and4,thedynamicsfor u 1 t u 2 t canbeexpressedas u i = y 3 v cz u i + y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 u i + q v ci u i ; 8 i = f 1 ; 2 g : FromEqs.2and4thedynamicsoftheknownstates y 1 t y 2 t andtheunknown state t = y 3 u 1 u 2 T are 2 6 4 y 1 y 2 3 7 5 = 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 v cz 10 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 2 v cz 01 3 7 5 2 6 6 6 6 4 y 3 u 1 u 2 3 7 7 7 7 5 + 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 y 2 1 ++ y 2 1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 3 )]TJ/F15 11.9552 Tf 9.298 0 Td [(+ y 2 2 1 + y 1 y 2 2 + y 1 3 3 7 5 and 2 6 6 6 6 4 y 3 u 1 u 2 3 7 7 7 7 5 = g ;!;y 1 ;y 2 ;v cz = 2 6 6 6 6 4 y 2 3 v cz + y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 y 3 y 3 v cz u 1 + y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 u 1 + q b 1 u 1 y 3 v cz u 2 + y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 u 2 + q b 2 u 2 3 7 7 7 7 5 : 1 Anobservercanbedevelopedwithanyofthethreelinearvelocitiesknown.Inthis chapter, v cz t isassumedtobeknownw.l.o.g. 57

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Let J y;v cz = 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 v cz 10 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 2 v cz 01 3 7 5 ; and y;! = 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 y 2 1 ++ y 2 1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(+ y 2 2 1 + y 1 y 2 2 + y 1 3 3 7 5 : Since y 1 t and y 2 t aremeasurable,fromEqs.2and25theEuclideanstructure m t canbeestimatedoncethestate y 3 t isdetermined.Sincethedynamicsoftheoutputs y 1 t y 2 t areaneintheunknownstate t ,areducedorderobservercanbedeveloped basedonthisrelationshipfortheunknownstate t .Thesubsequentdevelopmentis basedonthestrategyofconstructingtheestimates ^ t ^ y 3 ^ u 1 ^ u 2 T 2 R 3 .To quantifytheSaMestimationobjective,anestimationerror ~ t = ~ 1 ~ 2 ~ 3 T 2 R 3 is denedas ~ t y 3 )]TJ/F15 11.9552 Tf 12.747 0 Td [(^ y 3 u 1 )]TJ/F15 11.9552 Tf 12.685 0 Td [(^ u 1 u 2 )]TJ/F15 11.9552 Tf 12.685 0 Td [(^ u 2 T : Assumption4.1. Thefunction q v ci ;t 8 i = f x;y g islocallyLipschitzwhere q v cx )]TJ/F23 11.9552 Tf -438.325 -23.908 Td [(q ^ v cx = 1 v cx )]TJ/F15 11.9552 Tf 12.828 0 Td [(^ v cx and q v cy )]TJ/F23 11.9552 Tf 12.384 0 Td [(q ^ v cy = 2 v cy )]TJ/F15 11.9552 Tf 12.827 0 Td [(^ v cy where 1 and 2 areLipschitz constants. Remark 4.1 ThelinearvelocitymodelinEq.41andintheresultsin[3,21]isrestrictedtomotionsthataresatisedbyAssumption4.1;yet,variousclassesoftrajectories satisfythisassumptione.g.,straightlinetrajectories,circles,someperiodictrajectories, etc.. Assumption4.2. Thefunction J y 1 ;y 2 ;v cz denedinEq.43satisesthepersistencyof excitationcondition,i.e., 9 > 0: t + t J T y 1 ;y 2 ;v cz J y 1 ;y 2 ;v cz d I 8 t 0 : Remark 4.2 Assumption4.2isviolatedi 9 t 1 j8 t>t 1 ;v cz t =0 or y 1 t = c 1 ;y 2 t = c 2 Thatis,Assumption4.2isvalidunlessthereexistsatime t 1 suchthatforall t>t 1 the cameratranslatesalongtheprojectedrayofanobservedfeaturepoint. 58

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Assumption4.3. Thelinearcameravelocities v c t areupperandlowerboundedby constants. Remark 4.3 ThefollowingboundscanbedevelopedusingAssumption2.1,Remark2.1 andthedenitionsof u 1 t and u 2 t u 1min u 1 u 1max ;u 2min u 2 u 2max : 4.1.1.1StepI:Angularvelocityestimation Solutionsareavailableinliteraturethatcanbeusedtodeterminetherelativeangular velocitybetweenthecameraandatarget[16].Toquantifytherotationmismatchbetween F c and F c ,arotationerrorvector e t 2 R 3 isdenedbytheangle-axisrepresentation as e u t t where u t 2 R 3 representsaunitrotationaxis,and t 2 R denotestherotationangle about u t thatisassumedtobeconnedtoregion )]TJ/F23 11.9552 Tf 9.299 0 Td [(< t < .Theangle t and axis u t canbecomputedusingtherotationmatrix R t obtainedbydecomposingthe Homographymatrix H t givenbytherelationinEq.2.TakingtimederivativeofEq. 4yields e = L ! where L t 2 R 3 3 denotesaninvertibleJacobianmatrix[16].Arobustintegralofthe signoftheerrorRISE-basedobserver ^ e t 2 R 3 isgeneratedin[16]as ^ e = K + I 3 3 ~ e t + t t 0 K + I 3 3 ~ e d + v v = sgn ~ e where K ! 2 R 3 3 arepositiveconstantdiagonalgainmatrices,and ~ e t 2 R 3 quantiestheobservererroras ~ e t e )]TJ/F15 11.9552 Tf 12.933 0 Td [(^ e : ALyapunov-basedstabilityanalysisis 59

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providedin[16]thatproves ^ e t )]TJ/F15 11.9552 Tf 13.692 0 Td [(_ e t 0 as t !1 andthatallclosed-loopsignalsarebounded.BasedonEqs.4and4,theangular velocitycanbedeterminedas ^ t = L )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 ^ e t as t !1 : Anangularvelocityestimationerror ~ t 2 R 3 ~ 1 t ; ~ 2 t ; ~ 3 t T isdenedas ~ i t = i t )]TJ/F15 11.9552 Tf 13.14 0 Td [(^ i t ; 8 i = f 1 ; 2 ; 3 g : Asshownin[16],theangularvelocityestimatorgiven byEq.4isasymptoticallystable;thus,theangularvelocityestimationerror k ~ t k! 0 as t !1 4.1.1.2StepII:Structureestimation Areducedorderobserverfor t isdesignedas 2 6 6 6 6 4 ^ y 3 ^ u 1 ^ u 2 3 7 7 7 7 5 = 2 6 6 6 6 4 y 3 u 1 u 2 3 7 7 7 7 5 +)]TJ/F28 11.9552 Tf 21.068 38.377 Td [(2 6 6 6 6 4 )]TJ/F24 7.9701 Tf 10.494 6.18 Td [(v cz y 2 1 + y 2 2 2 y 1 y 2 3 7 7 7 7 5 wherethestatevector y 3 u 1 u 2 T isupdatedusingthefollowingupdatelaw 2 6 6 6 6 4 y 3 u 1 u 2 3 7 7 7 7 5 = 2 6 6 6 6 4 ^ y 2 3 v cz + y 2 ^ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 ^ 2 ^ y 3 ^ y 3 v cz ^ u 1 + y 2 ^ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 ^ 2 ^ u 1 + q ^ v cx ^ u 1 ^ y 3 v cz ^ u 2 + y 2 ^ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 ^ 2 ^ u 2 + q ^ v cy ^ u 2 3 7 7 7 7 5 | {z } g ^ ; ^ !;y 1 ;y 2 ;b 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T 0 B B B B @ 2 6 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 v cz 10 )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 2 v cz 01 3 7 5 2 6 6 6 6 4 ^ y 3 ^ u 1 ^ u 2 3 7 7 7 7 5 + 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 y 2 ^ 1 ++ y 2 1 ^ 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 ^ 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(+ y 2 2 ^ 1 + y 1 y 2 ^ 2 + y 1 ^ 3 3 7 5 | {z } y; ^ 1 C C C C C C A +)]TJ/F28 11.9552 Tf 21.068 38.376 Td [(2 6 6 6 6 4 v cz y 2 1 + y 2 2 2 0 0 3 7 7 7 7 5 : 60

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InEq.4, )]TJ/F21 11.9552 Tf 12.498 0 Td [(2 R 3 3 ^ i t aregivenbyEq.4,and J y;v cz isdenedinEq.4 3.DierentiatingEq.4andusingEqs.4,4,4and4yieldsthefollowing closed-loopobservererrordynamics 2 6 6 6 6 4 ~ 1 ~ 2 ~ 3 3 7 7 7 7 5 = 2 6 6 6 6 4 y 3 +^ y 3 v cz ~ 1 + y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 ~ 1 y 3 v cz ~ 2 + v cz ^ u 1 ~ 1 + y 2 ^ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 ^ 2 ~ 2 + q v cx ~ 2 y 3 v cz ~ 3 + v cz ^ u 2 ~ 1 + y 2 ^ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 ^ 2 ~ 3 + q v cy ~ 3 + y 2 ~ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 ~ 2 ^ y 3 + y 2 ~ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 ~ 2 ^ u 1 + 1 v cx )]TJ/F15 11.9552 Tf 12.398 0 Td [(^ v cx ^ u 1 + y 2 ~ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 ~ 2 ^ u 2 + 2 v cy )]TJ/F15 11.9552 Tf 12.398 0 Td [(^ v cy ^ u 2 3 7 7 7 7 5 +)]TJ/F23 11.9552 Tf 16.419 0 Td [(J T 0 B B B B @ 2 6 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 b 3 10 )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 2 b 3 01 3 7 5 2 6 6 6 6 4 ^ y 3 ^ u 1 ^ u 2 3 7 7 7 7 5 )]TJ/F28 11.9552 Tf 11.955 27.617 Td [(2 6 4 y 1 y 2 3 7 5 + 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 y 2 ^ 1 ++ y 2 1 ^ 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 ^ 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(+ y 2 2 ^ 1 + y 1 y 2 ^ 2 + y 1 ^ 3 3 7 5 1 C A UsingtheoutputdynamicsfromEq.4,theerrordynamicscanberewrittenas ~ = g ;!;y 1 ;y 2 ;v cz )]TJ/F23 11.9552 Tf 11.955 0 Td [(g ^ ; ^ !;y 1 ;y 2 ;v cz )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T J ~ + y; ~ : UsingAssumption4.2,andlocallyLipschitzpropertyof g ,followingrelationshipcanbe developed g ;!;y 1 ;y 2 ;v cz )]TJ/F23 11.9552 Tf 11.955 0 Td [(g ^ ; ^ !;y 1 ;y 2 ;v cz ~ + k ~ k where 2 R + : TheresultsfromtheangularvelocityestimatorinSection4.1.1.1prove that ~ 1 ; ~ 2 ; ~ 3 0 therefore, 0 as t !1 4.1.2StabilityAnalysis Theorem4.1. IfAssumptions4.1-4.3aresatised,thereducedorderobserverinEqs. 4and4asymptoticallyestimates t inthesensethat ~ t 0 as t !1 61

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Proof. ThestabilityoftheerrorsysteminEq.4canbeprovedusingaconverse Lyapunovtheorem[96].Considerthenominalsystem ~ = f ~ = )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T J ~ : UsingTheorem2.5.1of[95]theerrorsysteminEq.4isgloballyexponentially stableifAssumption4.2issatised.Hence, ~ t satisestheinequality ~ t ~ t 0 1 e )]TJ/F24 7.9701 Tf 6.586 0 Td [( 2 t )]TJ/F24 7.9701 Tf 6.587 0 Td [(t 0 where 1 2 2 R + ,and 2 isdirectlyproportionalto )]TJ/F19 11.9552 Tf 11.215 0 Td [(andinverselyproportionalto [3,95] Consideraset D = f ~ t 2 R 3 ~ t < 1g .Usinga converseLyapunovtheoremthereexistsafunction V :[0 ; 1 D! R thatsatises c 1 ~ t 2 V t; ~ c 2 ~ t 2 ; @V @t + @V @ ~ )]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F23 11.9552 Tf 7.315 0 Td [(J T J ~ )]TJ/F23 11.9552 Tf 28.559 0 Td [(c 3 ~ t 2 ; @V @ ~ c 4 ~ t forsomepositiveconstants c 1 ;c 2 ;c 3 ;c 4 .Using V t; ~ asaLyapunovfunctioncandidate fortheperturbedsysteminEq.4,thederivativeof V t; ~ alongthetrajectoriesofEq. 4isgivenby V t; ~ = @V @t + @V @ ~ )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T J ~ + @V @ ~ g ;!;y 1 ;y 2 ;v cz )]TJ/F23 11.9552 Tf 11.955 0 Td [(g ^ ; ^ !;y 1 ;y 2 ;v cz + @V @ ~ )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T y; ~ : UsingtheboundsinEq.4thefollowinginequalityisdeveloped V t; ~ )]TJ/F15 11.9552 Tf 21.918 0 Td [( c 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(c 4 ~ 2 + c 4 d ~ where isintroducedinEq.4and d t = k )]TJ/F21 11.9552 Tf 7.315 0 Td [(k J T k y; ~ k + c 4 k ~ k where d t 0 as t !1 .UsingTheorem4.14of[96]theestimatesof c 3 and c 4 aregivenby c 3 = 1 2 ;c 4 = 2 1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(L 1 )]TJ/F23 11.9552 Tf 11.956 0 Td [(e )]TJ/F18 5.9776 Tf 7.782 5.036 Td [( 2 )]TJ/F25 5.9776 Tf 5.756 0 Td [(L ln 2 1 2 2 62

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where L 2 R + isanupperboundonthenormofJacobianmatrix @f ~ @ ~ ,where f ~ is denedinEq.4.Notethat lim 2 L c 4 = lim 2 L 2 1 0 B @ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(e )]TJ/F18 5.9776 Tf 7.782 7.014 Td [( 2 )]TJ/F25 5.9776 Tf 5.756 0 Td [(L ln 2 2 1 2 2 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(L 1 C A 6 = 1 : Since 2 isdirectlyproportionaltothegain )]TJ/F19 11.9552 Tf 7.314 0 Td [(,theinequality c 3 )]TJ/F23 11.9552 Tf 13.163 0 Td [(c 4 > 0 canbe achievedbychoosingthegain )]TJ/F19 11.9552 Tf 11.215 0 Td [(sucientlylarge.UsingEqs.4,4,andbasedon thedevelopmentinSection9.3of[96],thefollowingboundisobtained ~ t r c 2 c 1 e c 4 2 c 1 ~ t 0 e )]TJ/F24 7.9701 Tf 6.586 0 Td [( t )]TJ/F24 7.9701 Tf 6.587 0 Td [(t 0 + c 4 2 c 1 e c 4 2 c 1 t 0 e )]TJ/F24 7.9701 Tf 6.586 0 Td [( t )]TJ/F24 7.9701 Tf 6.586 0 Td [( d d whereaconstantconvergencerate > 0 canbeincreasedbyincreasing c 3 .FromEq. 4, ~ t 2L 1 ,thus ~ 1 t ~ 2 t ~ 3 t 2L 1 .Since ~ 1 t ~ 2 t ~ 3 t 2L 1 andthefactthat 1 t 2 t 3 t 2L 1 canbeusedtoconcludethat ^ 1 t ^ 2 t ^ 3 t 2L 1 .UsingtheresultfromSection4.1.1.1that k ~ t k! 0 as t !1 ,the function k y; ~ k! 0 as t !1 .Hence, d t 0 as t !1 and d t 2L 1 .Since d t 0 as t !1 and d t 2L 1 ,bytheLebesguedominatedconvergencetheorem[104] lim t !1 t 0 e )]TJ/F24 7.9701 Tf 6.586 0 Td [( t )]TJ/F24 7.9701 Tf 6.586 0 Td [( d d = t 0 e )]TJ/F24 7.9701 Tf 6.586 0 Td [( lim t !1 d )]TJ/F23 11.9552 Tf 12.644 0 Td [( d = lim t !1 d t =0 [seeTheorem 3.3.2.33of[105]].Lemma9.6.3of[96]cannowbeinvokedtoshowthat ~ t 0 as t !1 .Hence,thereducedorderestimatorinEqs.4and4identiesthestructure ofobservedfeaturepointsandunknowncameramotionasymptotically.Since y 3 t u 1 t and u 2 t canbeestimated,themotionparameters b 1 t and b 2 t canberecoveredbased onthedenitionof u t 4.2Simulation Inthisaddendum,anumericalsolutionispresentedtoillustratetheperformanceof theproposedestimationinestimatingdepthofafeaturepointandlinear,andangular 63

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velocitiesofthecamera.Thetimevaryingangularmotionofthecameraisselectedas = 0 : 01sin t 2 0 : 01sin t 2 0 T : ThelinearvelocitiesintheXandYdirectionsareupdatedusingtheequation v c t = )]TJ/F23 11.9552 Tf 9.299 0 Td [(v 2 cx t )]TJ/F23 11.9552 Tf 9.299 0 Td [(v 2 cy t T whichisoftheformdescribedbyinthechapter.ThelinearvelocityinZdirection measuredusingasensorischosenas v cz t =cos t : TheinitiallinearvelocityintheXandYdirectionsarechosenas v c t 0 = 11 T ; andtheinitialEuclideancoordinatesoftherstpointarechosenas m t 0 = 1010100 T : Thecameramotioninducesamotionofthefeaturepointsintheimageframe.The cameracalibrationmatrixisarbitrarilychosentobe A c = 2 6 6 6 6 4 8000300 0800200 001 3 7 7 7 7 5 : Pointsaretrackedintheimagewhilethecameraismoving.Imagecoordinatesoftherst pointandlinearvelocity v cz t ofthecameraarefedbacktotheestimator.Thestatesof theestimatorareinitializedto y 3 t 0 =0 : 1 ; u 1 t 0 =0 : 1 ; u 2 t 0 =0 : 1 : 64

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Figure4-1.State y 3 u 1 u 2 T : Theestimatorgainischosenas )-278(= 2 6 6 6 6 4 3 : 600 00 : 440 000 : 44 3 7 7 7 7 5 : Ameasurementnoisewithmeanzeroandvariance0.1isaddedusingMatlab's`randn' commandtotheimagepointvector p denedinandlinearvelocityinZdirection v cz t .Thestatevector t =[ y 3 t ;u 1 t ;u 2 t ] isshowninFig.4-1.ThestateestimatedbyanobserverinandisshowninFig.4-2.InFig.4-3,anasymptotic convergenceoftheestimationerrorisshowninthepresenceofnoisymeasurementinputs. 4.3Summary Areducedorderobserverisdevelopedfortheestimationofthestructurei.e.range tothetargetandEuclideancoordinatesofthefeaturepointsofastationarytargetwith respecttoamovingcamera,alongwithtwounknowntime-varyinglinearvelocitiesand theangularvelocity.TheangularvelocityisestimatedusingHomographyrelationships 65

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Figure4-2.State y 3 u 1 u 2 T : Figure4-3.State y 3 u 1 u 2 T : 66

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betweentwocameraviews.Theobserverrequirestheimagecoordinatesofthepoints, asinglelinearcameravelocity,andthecorrespondinglinearcameraaccelerationinany oneofthethreecameracoordinateaxes.UnderaphysicallymotivatedPEcondition, asymptoticconvergenceoftheobserverisguaranteed.However,futureeortscould potentiallyeliminatetheneedforanymodelofthevehicletrajectoryevenifuncertainas inthisresultandeliminatetheneedforanaccelerationmeasurement. 67

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CHAPTER5 ALYAPUNOV-BASEDOBSERVERFORACLASSOFNONLINEARSYSTEMS WITHAPPLICATIONTOIMAGE-BASEDSTRUCTUREANDMOTION ESTIMATION Inthischapter,anonlinesolutionispresentedtoanswerthequestion: Givenobservationsofpointcorrespondencesineveryimageofavideostreamwithknowncamera motion,isitpossibletorecovertheEuclideanstructureandmotioni.e.linearand angularvelocitiesofindependentlymovingobjectsobservedbythemovingcamera? A nonlinearobserverisdevelopedtoestimatethestructureandmotionoftheobjectviewed byamovingcamera.Theobserveralgorithmusescameravelocitiesandthefeaturepoint dataobtainedfromanimagesequence.Theproposedmethodhasseveraladvantagesover theexistingmethods.Therearenorequirementsofminimumnumberofpointcorrespondencesornumberofviews.Thenonlinearobserverprocessesthedataineveryimageas itarrives,andthus,canperformreal-timecomputationofthestructureandmotionofa movingobject.Astabilityanalysisoftheproposedobserverispresentedwhichguarantees convergenceoftheobserver,providedanobservabilityconditionbasedonthepersistency ofexcitationPEofthecameramotionissatised. 5.1NonlinearObserver 5.1.1SystemDynamics Consideraclassofnonlinearsystemsdescribedbythefollowingdynamics y = f y;u + J y;u x; x = g y;x;u + d t where y t 2Y R n 1 isthemeasuredstate, x t 2X R n 2 istheunmeasured state, u t 2U R m istheinput, d t 2D R n 2 isanexternaldisturbance, J : R n 1 R m R n 1 n 2 isaknownfunctionofinputsandoutputs,and f : R n 1 R m R n 1 and g : R n 1 R n 2 R m R n 2 areknownnonlinearfunctions.Thesets Y ; X ; U and D arecompactsets.Thesystemsatisesthefollowingassumptions. 68

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Assumption1: Thefunctions g y;x;u and g y;x;u areboundedforall y 2Y x 2X and u 2U ; andislocallyLipschitzwithrespectto x inthesensethat 9 2 R + : k g y;x 1 ;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(g y;x 2 ;u k k x 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(x 2 k ; where x 1 ;x 2 2 and isacompactsubsetof R n 2 Assumption2: Theunknowntime-varyingdisturbancesatisestheproperties: d t 2L 1 and d t 2L 2 L 1 [106,107]. Assumption3: Thefunctions J y;u ; J y;u ; J y;u areboundedforall y 2Y and u 2U : Assumption4: Thereexists 1 ;" 2 R + suchthattheinequality 1 t + t J T y ;u J y ;u d 1 I n 2 issatisedforall t 0 .Thisisthewell-knownpersistenceofexcitationP.E.condition [95]. 5.1.2StateEstimator AstateestimatorisdevelopedforthesysteminEq.5underAssumptions1-4.To quantifytheestimationobjective,errorsdenotedby e 1 y;t 2 R n 1 and e 2 x;t 2 R n 2 are denedas e 1 y )]TJ/F15 11.9552 Tf 12.747 0 Td [(^ y;e 2 x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x where ^ y t 2 R n 1 and ^ x t 2 R n 2 aretheestimatesof y t and x t : Tofacilitatethe stabilityanalysis,alterederror r e 1 ; e 1 2 R n 1 isdenedas r e 1 + e 1 1 Foranytwomatrices X and Y ,theexpression X > Y meansthematrix X)-289(Y is positivesemi-denitepositivedenite.ThesubscriptoftheIdentitymatrix I denesthe dimensionof I 69

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where 2 R + isatuningparameter.BasedonthestructureofEq.5,afull-order continuousnonlinearobserverisdesignedas ^ y = f y;u + + e 1 + v; ^ x = g y; ^ x;u +)]TJ/F23 11.9552 Tf 26.285 0 Td [(J T y;u v )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T y;u J y;u ^ x where )]TJ/F21 11.9552 Tf 10.635 0 Td [(2 R n 2 n 2 isagainmatrix, v e 1 2 R n 1 isthegeneralizedsolutionto v = e 1 + sgn e 1 ;v =0 where ; 2 R n 1 n 1 arediagonalgainmatrices,and sgn e 1 = sgn e 11 ::sgn e 1 n 1 T : Theclosed-looperrordynamicsfor e 1 y;t and e 2 x;t aredeterminedbydierentiatingEq.5andusingEqs.5and5as e 1 = Jx )]TJ/F15 11.9552 Tf 11.955 0 Td [( + e 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(v; e 2 = g y;x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(g y; ^ x;u )]TJ/F15 11.9552 Tf 11.956 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T v )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 1 +)]TJ/F23 11.9552 Tf 26.284 0 Td [(J T J ^ x + d: Theclosed-loopdynamicsfor r e 1 ; e 1 aredeterminedbydierentiatingEq.5andusing Eq.5as r = J x + Jx )]TJ/F23 11.9552 Tf 11.955 0 Td [( e 1 )]TJ/F15 11.9552 Tf 13.698 0 Td [(_ v: Let 1 y;u;x; x J y;u x t + J y;u x t : FromEq.5andAssumption1-3, x t ; x t 2L 1 ; thesefacts,alongwithAssumption3,indicatethat k 1 y;u;x; x k 1 ; k 1 y;u;x; x k 2 where 1 ; 2 2 R + areknownconstants.UtilizingEq.5,theexpressioninEq.5can bewrittenas r = 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r )]TJ/F23 11.9552 Tf 11.955 0 Td [(sgn e 1 : 70

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5.1.3StabilityAnalysis StabilityoftheobserverinEq.5isanalyzedbyrststudyingthestabilityof the e 1 t dynamics.Since 1 y;u;x; x satisestheboundsinEq.5, 1 y;u;x; x canbeconsideredasatime-varyingboundeddisturbance.Therobustterm v t isused tocompensateforthedisturbance 1 y;u;x; x andasymptoticallystabilize e 1 t and r t .HenceinEq.5,thesignal v t identiestheterm J y;u x t andcanbeusedto stabilizethe e 2 t dynamicsinEq.5.ThemainresultofthepaperfollowsinTheorem 5.1whichproves k e 2 t k! 0 as t !1 usingtoolsfromconverseLyapunovtheory.To facilitatetheproofforTheorem1,thefollowingLemmaisestablished. Lemma5.1. TheerrorsysteminEq.5isgloballyasymptoticallystableinthesense that k e 1 t k! 0 as t !1 providedAssumptions1-3andfollowingsucientconditionsaresatised > 1 + 1 2 ;> 1 2 ;> 1 2 : Proof. Let r;e;P 2 R 2 n 1 +1 bedenedas r T e T 1 p P T suchthat R 2 n 1 +1 contains r;e;P =0 .InEq.5,theauxiliaryfunction P e 1 ;r; 1 ;t 2 R isageneralizedsolutiontothedierentialequation P = )]TJ/F23 11.9552 Tf 9.299 0 Td [(L;P = n 1 X i =1 e 1 i )]TJ/F23 11.9552 Tf 11.955 0 Td [(e T 1 1 wherethefunction L e 1 ;r; 1 2 R isdenedas L r T 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(sgn e 1 : ProvidedthesucientconditionsinEq.511aresatised, P e 1 ;r; 1 ;t 0 asshownin theAppendix.Let V : R 2 n 1 +1 R beaLipschitz,regularpositivedenitefunction 71

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denedas V 1 2 r T r + 1 2 e T 1 e 1 + P whichsatisesthefollowinginequalities 1 2 k k 2 = U 2 V U 1 = k k 2 : Theclosed-loopsysteminEqs.5,5,and5canbedescribedby = F ;t ; where F ;t 2 R 2 n 1 +1 : Theright-handsideoftheclosed-loopsystem F ;t isdiscontinuousintheset f ;t j e 1 =0 g : Asshownin[108,109],auniquegeneralizedsolution canbeestablishedintheFilippov'ssensebystudyingadierentialinclusion 2 F ;t ; where r;e;P isabsolutelycontinuousi.e.,dierentiablealmosteverywherea.e. and F isLebesguemeasurableandlocallybounded.UnderFilippov'sframework, generalizedLyapunovstabilitytheorycanbeusedtoestablishstrongstabilityofthe closed-loopsysteminEqs.5,5,and5see[110112]forfurtherdetails.Since V isLipschitzandregular,and r;e;P isabsolutelycontinuous,Theorem2.2 of[112]canbeinvokedtoconcludethat V isabsolutelycontinuous, V existsa.e., and V 2 a:e: ~ V where ~ V = 2 @V T K r e 1 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.258 Td [(1 2 P T where @V isthegeneralizedgradientof V [111],and K [ ] isdenedin[110,112]. Since V isLipschitzandregular,Eq.5canbesimpliedas[110] ~ V = r V T K r e 1 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.258 Td [(1 2 P T = re 1 2 p P K r e 1 1 2 P )]TJ/F18 5.9776 Tf 7.782 3.258 Td [(1 2 P T : 72

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UtilizingEqs.53,5,and5,andusingProperty2,5,and7of K [ ] giveninTheorem 1of[110]yields ~ V r T 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r )]TJ/F23 11.9552 Tf 11.956 0 Td [(K [ sgn e 1 ]+ e T 1 r )]TJ/F23 11.9552 Tf 11.956 0 Td [(e T 1 e 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(r T 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(K [ sgn e 1 ] = )]TJ/F23 11.9552 Tf 9.298 0 Td [(r T r + e T 1 r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e T 1 e 1 where )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(r T )]TJ/F23 11.9552 Tf 11.955 0 Td [(r T i SGN e 1 i =0 and K [ sgn e 1 ]= SGN e 1 areused[110],where SGN isdenedsuchthat SGN e 1 i = )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 if e 1 i < 0 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 ; 1] if e 1 i =0 ,and 1 if e i 1 > 0 : UsingYoung'sinequalitytoshowthat k e 1 kk r k 1 2 k e 1 k 2 + 1 2 k r k 2 ; thefollowinginequality isobtained ~ V )]TJ/F15 11.9552 Tf 21.918 0 Td [( )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(1 2 k r k 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( )]TJ/F15 11.9552 Tf 13.15 8.088 Td [(1 2 k e 1 k 2 wherethesymbol meanseveryelementof ~ V islessthanorequaltotherighthand side[112].Choosing > 1 2 and > 1 2 ,thefollowingupperboundcanbeestablished ~ V )]TJ/F15 11.9552 Tf 23.602 3.155 Td [( k r k 2 )]TJ/F15 11.9552 Tf 13.115 0 Td [( k e 1 k 2 = )]TJ/F23 11.9552 Tf 9.298 0 Td [(U where )]TJ/F22 7.9701 Tf 13.15 4.707 Td [(1 2 , )]TJ/F22 7.9701 Tf 13.151 4.707 Td [(1 2 .TheresultinEq.5indicatesthat V )]TJ/F23 11.9552 Tf 21.918 0 Td [(U 8 V 2 a:e: ~ V : TheinequalitiesinEqs.5and5,indicatethat V 2L 1 ; thus r t ;e 1 t 2L 1 ; andfrom y t 2L 1 ; ^ y t 2L 1 : Since r t and e 1 t 2L 1 ; Eq.5canbeusedto showthat e 1 t 2L 1 ; andEqs.5,5,andAssumption1canbeusedtoshowthat v t ; r t 2L 1 : Since r t ;e 1 t 2L 1 ; Eq.5canbeusedtoprovethat r t is uniformlycontinuous.FromEq.5andthefactthat V 2L 1 ;r t ;e 1 t 2L 2 : Since e 1 t 2L 1 ;e 1 t 2L 1 L 2 ; r t 2L 1 ; and r t 2L 1 L 2 ; Barbalat'slemmacan beinvokedtoprovethat k r t k! 0 and k e 1 t k! 0 as t !1 : 73

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Theorem5.1. GiventhedynamicsinEq.5withinputs u t andoutput y t ; the observerinEq.5asymptoticallyestimatesthestate x t inthesensethat k e 2 t k! 0 as t !1 providedAssumptions1-4aresatised. Proof. Substituting v t fromthe e 1 t dynamicsintothe e 2 t dynamicsinEq.5yields e 2 = g )]TJ/F15 11.9552 Tf 12.371 0 Td [(^ g )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 11.9552 Tf 7.315 0 Td [(J T Je 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T )]TJ/F23 11.9552 Tf 9.299 0 Td [(r )]TJ/F23 11.9552 Tf 11.955 0 Td [(e 1 + d: Since g y;x;u islocallyLipschitz,theMeanValueTheoremMVTcanbeinvokedto yield g y;x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(g y; ^ x;u = y; ^ x;u e 2 t ; where y; ^ x;u isboundedforalltime t as =sup t k y; ^ x;u k : The e 2 t dynamicsinEq.5canbewrittenas e 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T Je 2 + e 2 +)]TJ/F23 11.9552 Tf 19.075 0 Td [(J T r +)]TJ/F23 11.9552 Tf 19.076 0 Td [(J T e 1 + d: Thenominalsystem e 2 = f e 2 = )]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T Je 2 isgloballyexponentiallystableifAssumption4issatisedusingTheorem2.5.1of[95]. Hence,trajectoriesofthenominalsysteminEq.5satisfytheinequality: k e 2 t k k k e 2 k e )]TJ/F24 7.9701 Tf 6.587 0 Td [( t )]TJ/F24 7.9701 Tf 6.587 0 Td [(t 0 ; 8 t t 0 0 74

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where k; 2 R arepositiveconstants.UsingtheconverseLyapunovtheoremthereexistsa Lyapunovfunction V 1 :[0 ; 1 R n 2 R thatsatises c 11 k e 2 k 2 V 1 t;e 2 c 21 k e 2 k 2 ; V 1 t;e 2 = @ V 1 @t + @ V 1 @e 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F15 11.9552 Tf 9.299 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T Je 2 )]TJ/F23 11.9552 Tf 21.918 0 Td [(c 31 k e 2 k 2 ; @ V 1 @e 2 c 41 k e 2 k ; where c 11 ;c 21 ;c 31 ;c 41 2 R + : Takingthetimederivativeof V 1 t;e 2 alongthetrajectoriesof thesystem e 2 = )]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F23 11.9552 Tf 7.315 0 Td [(J T Je 2 + e 2 thefollowingexpressionisobtained V 1 t;e 2 = @ V 1 @t + @ V 1 @e 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [()]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T Je 2 + e 2 )]TJ/F23 11.9552 Tf 38.522 0 Td [(c 31 k e 2 k 2 + c 41 k e 2 k 2 = )]TJ/F28 11.9552 Tf 19.261 9.683 Td [()]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(c 31 )]TJ/F23 11.9552 Tf 11.955 0 Td [(c 41 k e 2 k 2 : UsingTheorem4.14of[96],theestimatesfor c 31 and c 41 aregivenby c 31 = 1 2 ;c 41 = 2 1 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(L 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(e )]TJ/F18 5.9776 Tf 7.782 7.014 Td [( 2 )]TJ/F25 5.9776 Tf 5.756 0 Td [(L ln 2 2 1 2 2 # where 1 ; 2 2 R + and L 2 R + isanupperboundonthenormoftheJacobianmatrix @ f e 2 @e 2 ,where f e 2 isdenedinEq.5.Notethat lim 2 L c 41 = lim 2 L 2 1 0 B @ 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(e )]TJ/F18 5.9776 Tf 7.782 7.013 Td [( 2 )]TJ/F25 5.9776 Tf 5.756 0 Td [(L ln 2 2 1 2 2 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(L 1 C A 6 = 1 : Since 2 isdirectlyproportionaltothegain )]TJ/F19 11.9552 Tf 11.215 0 Td [([95],theinequality c 31 )]TJ/F23 11.9552 Tf 12.391 0 Td [(c 41 > 0 canbe achievedbychoosingthegain )]TJ/F19 11.9552 Tf 11.216 0 Td [(sucientlylarge.UsingtheupperboundsinEq.5 andtheinequalityinEq.5,theerrorsysteminEq.5isgloballyexponentially 75

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stable.Hence,thereexistsaLyapunovfunction V 2 :[0 ; 1 R n 2 R thatsatises c 12 k e 2 k 2 V 2 t;e 2 c 22 k e 2 k 2 ; V 2 t;e 2 = @ V 2 @t + @ V 2 @e 2 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [()]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F23 11.9552 Tf 7.315 0 Td [(J T Je 2 + e 2 )]TJ/F23 11.9552 Tf 21.918 0 Td [(c 32 k e 2 k 2 ; @ V 2 @e 2 c 42 k e 2 k where c 12 ;c 22 ;c 32 ;c 42 2 R + : Takingthetimederivativeof V 2 t;e 2 alongthetrajectoriesof theerrorsysteminEq.5,thefollowingexpressionisobtained: V 2 = @ V 2 @t + @ V 2 @e 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [()]TJ/F15 11.9552 Tf 9.298 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T Je 2 + e 2 +)]TJ/F23 11.9552 Tf 19.076 0 Td [(J T r +)]TJ/F23 11.9552 Tf 19.076 0 Td [(J T e 1 + d )]TJ/F23 11.9552 Tf 28.56 0 Td [(c 32 k e 2 k 2 + c 42 k e 2 k )]TJ/F23 11.9552 Tf 7.315 0 Td [(J T r + e 1 + c 42 k e 2 kk d k )]TJ/F23 11.9552 Tf 28.56 0 Td [(c 32 k e 2 k 2 + c 42 & 1 k e 2 kk r k + c 42 & 2 k e 2 kk e 1 k + c 42 k e 2 kk d k wherethefactthatthematrix J y;u isnormboundedisusedsothat )]TJ/F23 11.9552 Tf 7.314 0 Td [(J T & 1 and )]TJ/F23 11.9552 Tf 7.314 0 Td [(J T & 2 forconstants & 1 ;& 2 2 R + : Completingthesquares,thefollowinginequality canbeobtained V 2 )]TJ/F23 11.9552 Tf 21.917 0 Td [( 1 k e 2 k 2 + 2 where 1 c 32 )]TJ/F23 11.9552 Tf 10.506 0 Td [( 1 )]TJ/F23 11.9552 Tf 10.506 0 Td [( 2 > 0 ; 2 t c 2 42 & 2 1 k r t k 2 + & 2 2 k e 1 t k 2 4 1 + c 2 42 k d t k 2 4 2 and 1 ; 2 t ; 1 ; 2 2 R + : Usingtheboundson V 2 t;e 2 inEq.5,theinequalityinEq.69canbeexpressedas V 2 )]TJ/F23 11.9552 Tf 24.713 8.088 Td [( 1 c 22 V 2 + 2 where 2 max 2 : FromEqs.5and5, V 2 t;e 2 decreasesalongthetrajectories ofEq.5untilthesolutionreachesacompactset c n e 2 t jk e 2 t k q 2 1 o : Hence, allsolutionsoftheclosed-loopsystemEq.5convergetothecompactball c and allsignalsoftheclosed-loopsystemEq.5areuniformlyultimatelybounded.Since e 2 t 2L 1 ; usingEq.5, V 2 t;e 2 2L 1 : IntegratingEq.6,followinginequalitycan 76

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beobtained t t 0 1 k e 2 k 2 d V 2 t 0 + c 2 42 4 1 0 @ & 2 1 t t 0 k r k 2 d + & 2 2 t t 0 k e 1 k 2 d 1 A + c 2 42 4 2 t t 0 k d k 2 d UsingAssumption2andthefactthat r t ;e 1 t 2L 2 fromEq.5,itcanbe concludedthat t t 0 k e 2 k 2 d< 1 : Hence, e 2 t 2L 2 L 1 .SinceallsignalsontherighthandsideofEq.5arebounded, e 2 t 2L 1 : Thus,Barbalat'slemmacanbeinvokedtoprovethat k e 2 t k! 0 as t !1 5.2ApplicationtoStructureandMotionProblem Inthissection,thenonlinearobserverdevelopedinSection5.1isappliedtoa wellknownmachinevisionproblemcalled`structureandmotionfrommotion'.In contrasttothetraditional`structurefrommotion'problemwheretheobjectiveisto estimatestructureofastationaryobject,theSaMfMproblemsolvesthestructureand motionestimationoftheobjectmovingwithunknownvelocities.Inthefollowing, theestimationobjectiveisdescribedandaphysicalplantmodelisprovidedwhich canbetransformedintoastatespacemodeloftheformgiveninEq.5undersome assumptions.AnobserverisdevelopedbyfollowingtheguidelinespresentedinSection 5.1.2.Theperformanceoftheobserverisdemonstratedviaanumericalsimulation. 5.2.1StructureandMotionfromMotionSaMfMObjective TheobjectiveofSaMfMistorecoverthestructurei.e.Euclideancoordinatesand motioni.e.Euclideanlinearandangularvelocitiesofmovingobjectsobservedbya movingcamera,assumingallcameravelocitiesareknown.Theobjectcanbetrackedasa 77

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singlepointoracollectionofmultiplepoints,wheretherangei.e., 1 x 3 t andthemotion ofeachpointareestimated. 5.2.2StateDynamicsFormulation Thecamera-objectrelativemotionmodeldevelopedinChapter2isusedforthe subsequenttechnicaldevelopmentinthischapter.Someassumptionsaremadeonthe movingobject'svelocitiestofacilitatethedevelopment. ThestatesdenedinEq.23containunknownstructureinformationoftheobject.Tofacilitatetheobserverdesign,statesaredenedinthissectiontoincorporate unknownstructureandvelocityinformation.Specically,anauxiliarystatevector p t = p 1 t p 2 t p 3 t T 2 R 3 isdenedas p v px y 3 t v py y 3 t v pz y 3 t T whichincorporatestheunknownobjectvelocityinformation.Torecoverthe3Dstructure, thestate y 3 t shouldbeestimatedbecauseitcontainsrangeinformation.Since,thestates y 1 t ;y 2 t canbemeasuredfromtheimages,theestimatedstate y 3 t canbeusedto scale y 1 t and y 2 t ,andthus m t ,i.e.the3Dstructurecanberecovered.Torecover velocityinformation,thestate p t mustbeestimated.Once y 3 t and p t areestimated, velocityinformationcanberecoveredbyscalingtheestimated p t bytheestimated y 3 t UsingEqs.23and2,thedynamicsofthestatevector y t areexpressedas y 1 = 1 + v cx )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 1 v cz y 3 )]TJ/F23 11.9552 Tf 11.956 0 Td [(p 1 + y 1 p 3 ; y 2 = 2 + v cy )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 v cz y 3 )]TJ/F23 11.9552 Tf 11.955 0 Td [(p 2 + y 2 p 3 ; y 3 = )]TJ/F23 11.9552 Tf 9.298 0 Td [(v cz y 2 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 2 1 )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 1 2 y 3 + v pz y 2 3 where 1 t 2 R and 2 t 2 R aredenedas 1 t )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 y 2 1 + )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 2 1 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 2 3 ; 2 t )]TJ/F28 11.9552 Tf 11.291 9.684 Td [()]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(1+ y 2 2 1 + y 1 y 2 2 + y 1 3 : 78

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DierentiatingEq.5andusingEq.5,thedynamicsofthestate p t canbe representedbythefollowingsetofdierentialequations p 1 = y 3 v px )]TJ/F23 11.9552 Tf 11.955 0 Td [(v cz p 1 y 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 p 1 + p 3 p 1 ; p 2 = y 3 v py )]TJ/F23 11.9552 Tf 11.955 0 Td [(v cz p 2 y 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 p 2 + p 3 p 2 ; p 3 = y 3 v pz )]TJ/F23 11.9552 Tf 11.955 0 Td [(v cz p 3 y 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 p 3 + p 2 3 : Bydeningthevector z t 2 R 2 andthevector t 2 R 4 as z t y 1 y 2 T ; t y 3 p 1 p 2 p 3 T thestatedynamicsinEqs.5and5canbeexpressedas z = z;u + J z;u ; = g z;;u + d t where t = 1 t 2 t T u t = v c t t T ;d t = y 3 t v p t ; andthe functions J z;u 2 R 2 4 and g z;;u 2 R 4 aregivenby J = 2 6 4 v cx )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 v cz )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 y 1 v cy )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 v cz 0 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 y 2 3 7 5 ; and g = 2 6 6 6 6 6 6 6 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [(v cz y 2 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 y 3 + p 3 y 3 )]TJ/F23 11.9552 Tf 9.299 0 Td [(v cz p 1 y 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 2 1 )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 1 2 p 1 + p 3 p 1 )]TJ/F23 11.9552 Tf 9.299 0 Td [(v cz p 2 y 3 )]TJ/F15 11.9552 Tf 11.955 0 Td [( y 2 1 )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 1 2 p 2 + p 3 p 2 )]TJ/F23 11.9552 Tf 9.298 0 Td [(v cz p 3 y 3 )]TJ/F15 11.9552 Tf 11.956 0 Td [( y 2 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 2 p 3 + p 2 3 3 7 7 7 7 7 7 7 5 : Assumption5.1. Thevelocityofobjectexpressedinthecamerareferenceframesatises v p t 2L 2 L 1 and v p t 2L 1 : 79

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Assumption5.2. Thecameravelocities t ;v c t andthepointvelocity v p t belongsto class C 2 withboundedderivatives. Assumption5.3. Thereexists 2 ; 2 R + suchthattheinequality t + t J T z ;u J z ;u d 2 I 4 8 t 0 : Remark 5.1 Assumption5.5indicatesthatthelinearvelocityofthemovingobject v p t ; measuredinthecamerareferenceframe,canbetime-varyingbutconvergestoaconstant. Assumption5.5holdsforaspecialcaseof v p t = c; where c isaconstant. Remark 5.2 BasedonAssumptions5.5and5.6,Eqs.5,5and5thefollowing inequalitiescanbedeveloped k 1 t k = J z;u t + J z;u t 3 k 1 t k 4 where 3 ; 4 2 R + denoteknownboundingconstants. Remark 5.3 .ObservabilityCondition Eventhoughtherankof J T z;u J z;u can beatmost2,theintegrationof J T z;u J z;u canachievefullrank[3,95,113,114]. TheconditioninAssumption5.5-5.7failifthecameraistranslatingparalleltotheray projectedbythemovingobjectonthecamera,i.e., y 1 t ;y 2 t =0 orifthecameraisnot translatinginanydirection,i.e., v cx = v cy = v cz =0 8 t 0 : 5.2.3StructureandMotionObserver BasedontheworkpresentedinSection5.1.2,anobserverisdesignedtoestimate t whichcontainsunknowndepthandunknownvelocityinformationofthemovingobject. Let ^ z t 2 R 2 and ^ t 2 R 4 denotetheestimatesof z t and t : Basedonthestructure ofEq.5andtheobserverdesigninEq.5,afull-ordercontinuousnonlinearobserver isdesignedas ^ z = z;u + + e 1 + v; ^ = g z; ^ ;u +)]TJ/F23 11.9552 Tf 26.285 0 Td [(J T z;u v )]TJ/F23 11.9552 Tf 11.956 0 Td [(e 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [()]TJ/F23 11.9552 Tf 7.314 0 Td [(J T J ^ 80

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where )]TJ/F21 11.9552 Tf 11.038 0 Td [(2 R 4 4 isagainmatrix, ; 2 R + aretuningparameters,thesignal v t 2 R 2 is ageneralizedsolutionto v = e 1 + sgn e 1 ;v =0 where 2 R 2 2 ;sgn e 1 = sgn e 11 sgn e 12 T ; and e 1 t 2 R 2 istheoutput estimationerrordenedas e 1 y )]TJ/F15 11.9552 Tf 12.747 0 Td [(^ y: IfAssumptions5and6aresatised,Assumptions1-4aresatisedbythedynamicsystem inEq.5.BasedonthestabilityanalysispresentedinSection7.1.3, ^ t t as t !1 : 5.2.4ConditionsontheMovingObjectTrajectory Inthissection,physicalconstraintsonthetrajectoryofthemovingtargetdueto Assumption5arediscussed.AccordingtoAssumption2,thetime-varyingdisturbance, d t = y 3 t v p t 2L 2 L 1 and d t =_ y 3 v p + y 3 v p 2L 1 : UsingRemark1,Assumption 5,Assumption6andEq.5, y 3 v p t 2L 1 and y 3 v p + y 3 v p 2L 1 : For y 3 t v p t 2L 2 ; lim t !1 t 0 k y 3 _v p k 2 d < 1 : Using k y 3 t v p t k 2 k y 3 t k 2 k v p t k 2 ; thefollowingnorminequalitycanbedeveloped lim t !1 t 0 k y 3 v p k 2 d )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(sup t 0 k y 3 t k 2 0 @ lim t !1 t 0 k v p k 2 d 1 A : UsingRemark1, y 3 t 2L 1 hence, sup t 0 k y 3 t k 2 < 1 : FromAssumption5, v p t 2L 2 ; thus,Eq.6issatised. AccordingtoRemark2,oneofthecasesforwhichAssumption5issatisesiswhen theobjectvelocity, v p t ; isconstant,i.e., R v p = c 81

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where c 2 R 3 isavectorofconstants.TheexpressioninEq.5canbeinterpretedin twoways: Case1:Therotationmatrixbetweenthecurrentcameracoordinateframeandthe inertialcoordinateframe, R t ; andthevelocityofthemovingobjectintheinertial coordinateframe, v p t ; donotchangewithtime.Therotationmatrix R t isaconstant matrixiftheangularvelocitiesofthecameraarezero.Hence,ifthecameraexhibitsonly translationalmotionandtheobjectvelocityisconstantintheinertialcoordinateframe thentheexpressioninEq.5holds. Case2:ThetimederivativeoftheequalityinEq.5iszero,i.e., R t v p t + R t v p t =0 : Using R t =[ t ] R t ; thefollowingordinarydierentialequation ODEcanbedeveloped v p = )]TJ/F15 11.9552 Tf 11.851 3.022 Td [( R T [ ] R v p : Basedonthefactthat R T t [ t ] R t = R T t t ; thefollowingODEis obtained v p = )]TJ/F28 11.9552 Tf 11.291 9.684 Td [( R T v p : Since R T t t = t ; theODEforobjectvelocitycanbeexpressedas v p = )]TJ/F15 11.9552 Tf 11.291 0 Td [([ ] v p : Foraspecialcaseofconstantcameraangularvelocity, !; theODEinEq.5has ananalyticalsolutiongivenby v p t = e )]TJ/F22 7.9701 Tf 6.587 0 Td [([ ] t v p t 0 : UsingRodrigues'formulaforamatrixexponential,Eq.5canbetransformedinto v p t = I + [ ] k k sin k k + [ ] 2 k k 2 )]TJ/F23 11.9552 Tf 11.955 0 Td [(cos k k v p t 0 82

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where t )]TJ/F23 11.9552 Tf 9.298 0 Td [(!t: Fortimevaryingobjectlinearvelocity, v p t ; andtheconstantcamera angularvelocity, !; whichincombinationsatisestheequationEq.5,theobjectlinear velocity v p t satisestheAssumption5.Hence,theobserverinSection5.1canbeused fortheclassofobjectlinearvelocityandcameraangularvelocityforwhichEq.56is satised. Forthecaseoftime-varyingcameraangularvelocity,ananalyticalsolutiontoEq. 5isawideopenprobleminliterature.For v p t and t satisfyingtheODEin Eq.5,Assumption5issatisedandtheobserverinEq.5canbeusedtoachieve asymptoticrangeandmotionestimation. 5.3Simulation TheperformanceoftheobserverinEq.5istestedusinganumericalsimulationin Matlab.Theresultsarecomparedwiththeobserverin[3].Thecameracalibrationmatrix isselectedas A c = 2 6 6 6 6 4 7200320 0720240 001 3 7 7 7 7 5 : Thevelocitiesofthecameraareselectedas v c t = )]TJ/F15 11.9552 Tf 9.298 0 Td [(3 sin t= 5+22 cos t= 10+15 sin t= 2+0 : 5 T m=s; t = )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 01 sin t= 50 : 01 cos t= 20 T rad=s andthetime-varyingobjectvelocity v p t ; expressedintheinertialframe,isshownin Fig.5-1.Theequivalentobjectvelocityexpressedinthecamerareferenceframeis v p t = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 T m=s: Theinitialrelativerangebetweenthecameraandtheobjectis m t 0 = 4450 T m TheestimatesareintegratedusingafourthorderRunge-Kuttaintegratorwithatimestep of 0 : 01 sec: Fortheproposedobserver,theinitialconditionsandobservergainsareselected 83

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as 2 ^ z t 0 = 4 50 4 50 T ; ^ t 0 = 0 : 015 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 02 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 01 )]TJ/F15 11.9552 Tf 9.299 0 Td [(0 : 01 T ; =70 ; =80 ; = diag 0 : 050 : 05 T ; 3 and )-420(= diag 0 : 0910 : 0950 : 20 : 165 T : The initialconditionsfortheobserverin[3]areselectedexactlythesameasfortheproposed observerandtheobservergainsareselectedas 4 1 = 2 =5 ; 1 = 2 =0 : 01 ; and )-298(= diag 0 : 0910 : 0950 : 20 : 165 T : Theerrorsintheestimationof t usingthe proposedobserverandtheobserverin[3]areshowninFigs.5-2and5-3.FromFig.5-3,it isdeterminedthattheestimationerrorsconvergetoasmallballaroundtheoriginaUUB result.Theproposedobservershowsanimprovedtransientperformanceovertheobserver in[3].InFig.5-4,acomparisonoftheestimatedandactual3Drelativepositionofthe targetispresented. Inthesecondsimulation,thecameravelocitiesareselectedtobethesameastherst simulationandtheobjectvelocity, v p t ; isselectedas v p t = )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 e )]TJ/F22 7.9701 Tf 6.586 0 Td [(0 : 01 t )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 sin : 001 t )]TJ/F15 11.9552 Tf 9.298 0 Td [(1 T m=s: Theobjectvelocity, v p t ; isslowlytime-varyingandsatisesAssumption5.5.The initialconditionsarechosenthesameastherstsimulation.The )]TJ/F19 11.9552 Tf 11.215 0 Td [(gainisselectedas )-412(= diag 0 : 0900 : 230 : 20 : 215 T : Theobjectvelocityintheinertialframeis showninFig.5-5.Theobservershowsrobustperformanceeveninthepresenceoftimevaryingobjectvelocities, v p t asseeninthestateestimationerrorsinFig.5-6,andthe estimatedandactualpositioncomparisoninFig.5-7. 2 Since z t ismeasurablefromtheimage,theinitialconditionsof ^ z t areselected equalto z t 0 : 3 diag fg representsadiagonalmatrixconstructor. 4 Thesymbols 1 ; 2 ; 1 ; 2 areintroducedin[3]. 84

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Figure5-1.Thevelocityofthemovingobjectmeasuredintheinertialreferenceframe. Figure5-2.Stateestimationerrorsusingtheproposedobserver. 85

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Figure5-3.Stateestimationerrorsusingtheobserverin[3].AdaptedfromX.Chenand H.Kano,"Stateobserverforaclassofnonlinearsystemsanditsapplicationto machinevision,"IEEETrans.Autom.Control,vol.49,no.11,pp. 2085,2004. 5.4Summary Anobserverforaclassofnonlinearsystemsisdesigned.Thedesignisbasedon anidentierapproachwheretheunmeasurablepartofthestateisidentiedusinga robustidentierfromtheoutputdynamicsandtheidentierisusedtostabilizetheerror dynamicsoftheunmeasurablepartofthestate.Itisshownthattheobserverdesign improvesuponexistingsolutionstotheproblempresentedin[3]byprovingasymptotic estimationerrorconvergenceeveninthepresenceofexternaldisturbances.Anapplication oftheobservertothestructureandmotionprobleminmachinevisionispresented.The observerhassomeadvantagesoverbothexistingbatchsolutionsandonlinesolutions. Newinsightonmovingobjecttrajectoriesaredevelopedforonlinestructureandmotion estimationwhentheobjectismoving. 86

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Figure5-4.Comparisonoftheactualandestimated3Drelativepositionoftheobjectand thecamera. Figure5-5.Thevelocityofthemovingobjectmeasuredintheinertialreferenceframe. 87

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Figure5-6.Stateestimationerrorsfortimevaryingvelocity, v p t ; usingtheproposed observer. Figure5-7.Comparisonoftheactualandestimated3Drelativepositionoftheobjectand thecamerafortime-varyingobjectvelocity v p t 88

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CHAPTER6 LYAPUNOV-BASEDUNKNOWNINPUTOBSERVERFORACLASSOF NONLINEARSYSTEMS Inthischapter,anonlinearUIOisdevelopedforageneralclassofmulti-inputmultioutputMIMOnonlinearsystems.BasedontheexistenceofasolutiontotheRiccati equation,sucientexistenceconditionsarederived.Theconditionsprovideguidelinesfor choosingtheobservergainmatrix K basedontheLipschitzconstantofthenonlinearity presentinthedynamics.Analgorithmforchoosing K basedontheEigenvalueplacement issuggestedin[52].Inthischapter, K isobtainedbysolvinganLMIfeasibilityproblem. 6.1NonlinearUnknownInputObserver 6.1.1NonlinearDynamics ConsiderageneralclassofMIMOnonlinearsystemsexpressedas x = f x;u + g y;u + Dd y = Cx where x t 2 R n isthestateofthesystem, u t 2 R m istheknowncontrolinput, d t 2 R q isanunknowninput, y t 2 R p istheoutputofthesystem 1 C 2 R p n isfull rowrank, D 2 R n q isfullcolumnrank 2 g : R p R m R n isnonlinearin y t and u t ;f : R n R m R n isnonlinearin x t and u t ,andsatisestheLipschitzcondition jj f x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(f ^ x;u jj 1 jj x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x jj where 1 2 R + ; and ^ x t 2 R n isanestimateofthe unknownstate x t : 1 Itisassumedthat p q: ThisisastandardconditionpresentintheUIOliterature cf.[5970,7276,78]. 2 Thisconditionisnotrestrictivesinceif rank D = q 1
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ThesysteminEq.6canbewrittenas x = Ax + f x;u + g y;u + Dd y = Cx where A 2 R n n ; and f x;u = f x;u )]TJ/F23 11.9552 Tf 12.248 0 Td [(Ax .Theauxiliaryfunction f x;u satisesthe Lipschitzcondition[115,116] jj f x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(f ^ x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(A x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x jj 1 + 2 jj x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x jj where 2 2 R + : 6.1.2UIODesign TheUIOobjectiveistodesignanasymptoticallyconvergingstateobserverto estimate x t inthepresenceofanunknowninput d t .Toquantifythisobjectivean estimationerrorisdenedas e t ^ x t )]TJ/F23 11.9552 Tf 11.955 0 Td [(x t : BasedonEq.6andthesubsequentstabilityanalysis,theUIOforthesysteminEq.6 isdesignedas z = Nz + Ly + M f ^ x;u + Mg y;u ^ x = z )]TJ/F23 11.9552 Tf 11.955 0 Td [(Ey where z t 2 R n and N 2 R n n L 2 R n p M 2 R n n aredesignedas[66] 3 M = I n + EC N = MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC L = K I p + CE )]TJ/F23 11.9552 Tf 11.955 0 Td [(MAE 3 ThesubscriptoftheIdentitymatrix I denesthedimensionof I 90

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where E 2 R n p issubsequentlydesigned,and K 2 R n p isagainmatrixwhichsatises theinequality Q N T P + PN + 1 + 2 2 PMM T P + I n < 0 where P 2 R n n isapositivedenite,symmetricmatrix.UsingEq.6,theequality NM + LC )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA =0 n n issatised,where 0 i j denotesazeromatrixofthedimensions i j .If E isselectedso that E = F + YG where Y 2 R n p canbechosenarbitrarily,and F and G aregivenby F )]TJ/F23 11.9552 Tf 9.299 0 Td [(D CD y ;G )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(I p )]TJ/F15 11.9552 Tf 11.955 0 Td [( CD CD y then ECD = )]TJ/F23 11.9552 Tf 9.298 0 Td [(D andthefollowingequalityissatised: MD = I n + EC D =0 n q : Notethatthegeneralizedpseudoinverseofthematrix CD; denedas CD y = CD T CD )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 CD T existsprovided rank CD = q .SubstitutingEq.6intoEq.6,takingthetime derivativeoftheresult,andusingEqs.62and6yields e = Nz + Ly + M f ^ x;u )]TJ/F15 11.9552 Tf 11.955 0 Td [( I n + EC Ax )]TJ/F15 11.9552 Tf 11.955 0 Td [( I n + EC f x;u )]TJ/F15 11.9552 Tf 11.955 0 Td [( I n + EC Dd: UsingEqs.6and6,andtheconditionsinEqs.6and6theerrorsysteminEq. 6canbewrittenas e = Ne + M )]TJ/F15 11.9552 Tf 8.027 -6.529 Td [( f ^ x;u )]TJ/F15 11.9552 Tf 14.503 3.155 Td [( f x;u : 91

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6.1.3SucientCondition Lemma6.1providesaconditiononthegainmatrix K fortheinequalityinEq.6to hold.TheresultofLemma6.1isusedinTheorem6.1todevelopasucientconditionfor theexistenceoftheobserverpresentedinSection6.1.2. Lemma6.1. ThematrixinequalityinEq.6issatisedifthepair MA;C isobservable, K isselectedsothat MA )]TJ/F23 11.9552 Tf 12.766 0 Td [(KC isHurwitz 4 ,andthefollowingconditionis satised min 2 R + min MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n > p 3 1 + 2 where min denotestheminimumsingularvalueofamatrix,and 3 max )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(MM T : Proof. Theproofofthistheoremisinspiredbytheworkin[52]whichdevelopedsucient conditionsforanobserverdesignfornonlinearsystemswithknowninputs.Thedevelopmentoftheproofisbasedontheconditionsfortheexistenceofasolutiontothealgebraic RiccatiequationARE.ConsideraHamiltonianmatrix H ,denedas H = 2 6 4 AR )]TJ/F23 11.9552 Tf 9.299 0 Td [(Q )]TJETq1 0 0 1 330.034 360.245 cm[]0 d 0 J 0.478 w 0 0 m 8.775 0 l SQBT/F23 11.9552 Tf 330.034 350.402 Td [(A T 3 7 5 where A;Q and R arerealmatrices, Q and R aresymmetric.IftheHamiltonian,hasno imaginaryeigenvalues, R iseitherpositivesemi-deniteornegativesemi-deniteandthe pair )]TJETq1 0 0 1 101.536 280.547 cm[]0 d 0 J 0.478 w 0 0 m 8.775 0 l SQBT/F23 11.9552 Tf 101.536 270.704 Td [(A;R isstabilizable,thenthereexistsasymmetricsolutiontotheARE A T X + X A + XRX + Q =0 n n : ConsideraHamiltonianmatrix, H; denedas 4 Aspointedoutin[52],stabilityofthematrix MA )]TJ/F23 11.9552 Tf 14.621 0 Td [(KC isnotsucientfortheerror systemoftheform6tobestable. 92

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H = 2 6 4 MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC 1 + 2 2 MM T )]TJ/F23 11.9552 Tf 9.298 0 Td [("I n )]TJ/F23 11.9552 Tf 11.955 0 Td [(I n )]TJ/F15 11.9552 Tf 11.291 0 Td [( MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC T 3 7 5 associatedwiththeARE MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC T P + P MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC + 1 + 2 2 PMM T P + I n + "I n =0 n n where 2 R + isasmallconstant.Since MA;C isobservable,thematrix K canbe selectedsothat MA )]TJ/F23 11.9552 Tf 10.636 0 Td [(KC isHurwitz,andhencethepair )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC; 1 + 2 2 MM T is stabilizable.If H hasnoeigenvaluesontheimaginaryaxisthenthereexistsasymmetric positivedenitesolutiontotheAREinEq.6.Inthefollowing,itisproventhatifthe conditioninEq.6issatisedthentheeigenvaluesof H donotlieontheimaginary axis.Theproofisgivenbycontradiction. Considerthecharacteristicequationof H as det 2 6 4 I n )]TJ/F15 11.9552 Tf 11.955 0 Td [( MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F15 11.9552 Tf 11.291 0 Td [( 1 + 2 2 MM T "I n + I n I n + MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC T 3 7 5 =0 where 2 C denotesaneigenvalueof H: Usingthefactthatforanytworealmatrices A and B ,if det )]TJ/F15 11.9552 Tf 8.276 -6.662 Td [( B =1 ; then det )]TJ/F15 11.9552 Tf 8.566 -6.662 Td [( A = det )]TJ/F15 11.9552 Tf 8.566 -6.662 Td [( A det )]TJ/F15 11.9552 Tf 8.277 -6.662 Td [( B = det )]TJ/F15 11.9552 Tf 8.567 -6.662 Td [( A B ; thecharacteristic equationcanbewrittenas det 2 6 4 I n )]TJ/F15 11.9552 Tf 11.955 0 Td [( MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F15 11.9552 Tf 11.291 0 Td [( 1 + 2 2 MM T "I n + I n I n + MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC T 3 7 5 2 6 4 I n )]TJ/F22 7.9701 Tf 15.743 4.708 Td [(1 1+ h I n + MA )]TJ/F23 11.9552 Tf 11.956 0 Td [(KC T i 0 I n 3 7 5 =0 : Lettheeigenvaluesof H beontheimaginaryaxisi.e., = j!; thenthecharacteristic equationbecomes det h [ j!I n )]TJ/F15 11.9552 Tf 11.955 0 Td [( MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC ] h j!I n + MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC T i ++ 1 + 2 2 MM T i =0 : 93

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Using MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n = j!I n + MA )]TJ/F23 11.9552 Tf 11.956 0 Td [(KC T ; thecharacteristicequationis transformedintofollowingequality, det MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n )]TJ/F15 11.9552 Tf 11.955 0 Td [(+ 1 + 2 2 MM T =0 : Now,itisshownthatifEq.6istruethentheequalityinEq.6cannotbesatised whichprovesthat H cannothaveeigenvaluesontheimaginaryaxis. Sincethesingularvaluesofamatrixarecontinuousfunctionsoftheelements ofthematrix, min MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n isacontinuousfunctionof 2 R + and min MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n !1 as !1 : Hence,thereexistsanite 2 R + for which min MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n hasaminimum[52,104],i.e., 9 1 : min 2 R + min MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n = min MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j! 1 I n = min : UsingEq.6 min = min MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j! 1 I n > p 3 1 + 2 andthefollowing inequalityholds: MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n 2 min I n 8 5 where min 2 R + ; and denotesthecomplexconjugatetransposeofamatrix.By choosing suchthat 2 min I n > 1 + 2 2 + 3 I n 1 + 2 2 + )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(MM T ; the followinginequalitycanbeobtained: MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n )]TJ/F15 11.9552 Tf 11.956 0 Td [(+ 1 + 2 2 MM T > 0 n n : whichcontradictstheconclusionofEq.6that MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I n )]TJ/F15 11.9552 Tf -473.31 -23.908 Td [(+ 1 + 2 2 MM T isasingularmatrix.Hence,ifEq.6holds,then H cannot 5 Foranytwomatrices X and Y ,theexpression X > Y meansthematrix X)-289(Y is positivesemi-denitepositivedenite. 94

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haveeigenvaluesontheimaginaryaxis.Thus,asymmetricsolutiontoEq.6existsand Eq.6issatised. Theorem6.1. ThenonlinearUIOinEq.6isexponentiallystablesuchthat 9 ; 2 R + k e t k k e t 0 k exp )]TJ/F23 11.9552 Tf 9.299 0 Td [(t if rank CD = q andtheconditioninEq.6issatised. Proof. If rank CD = q thenthesolutiontoEq.6exists.Forprovingthesuciency oftheconditioninEq.6,consideraLyapunovcandidatefunction V e : R n R denedas V = e T Pe: TheLyapunovfunctionsatises min P k e k 2 V max P k e k 2 where min and max aretheminimumandmaximumeigenvaluesofthematrix P: Taking thetimederivativeofEq.6alongthetrajectoriesofEq.6yields V = e T )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(N T P + PN e +2 e T PM )]TJ/F15 11.9552 Tf 8.027 -6.529 Td [( f ^ x;u )]TJ/F15 11.9552 Tf 14.503 3.155 Td [( f x;u V e T )]TJ/F23 11.9552 Tf 5.479 -9.683 Td [(N T P + PN e +2 e T PM k f ^ x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(f x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(A x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x k V e T )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(N T P + PN e +2 e T PM 1 k e k +2 e T PM 2 k e k : Usingthenorminequality 2 i e T PM k e k 2 i e T PM 2 + k e k 2 ; 8 i = f 1 ; 2 g ; 95

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theupperboundon V isgivenby V e T )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(N T P + PN e + 1 + 2 2 e T PMM T Pe + e T e V e T )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(N T P + PN + 1 + 2 2 PMM T P + I n e V e T Qe: IftheconditioninEq.6issatised,thenLemma6.1canbeinvokedtoconclude Q< 0 andhence V< 0 : UsingEqs.617-6theupperboundsfor V e and e t canbe developedas V e V e exp )]TJ/F23 11.9552 Tf 9.299 0 Td [(t where = max Q min P and k e t k k e k exp )]TJ/F23 11.9552 Tf 9.299 0 Td [(t where = max P min P Remark 6.1 In[52],theconditioninEq.6isalsoclaimedtobeanecessaryconditionfortheerrordynamicsinEq.6tobestable.However,theclaimin[52]isnot entirelycorrectbecausethenecessityoftheconditioninEq.6isonlyshownfora fewparticularcaseswhenthefunction f x;t takesalinearform,viz.; f x;t = Lx or f x;t = L t x; where L;L t 2 R n n : Also,theotherargumentagainstthenecessity oftheconditioninEq.6isthatifalargebound 1 intheEq.6isselected,then theconditioninEq.6willfailforagiven K; buttheerrordynamicsinEq.6can stillbestablewiththegiven K .Hence,conditioninEq.6isclearlynotanecessary conditionforthestabilityoferrordynamicsinEq.6. 6.1.4ConditionsforChoosingMatrixA Inthissection,conditionsonchoosing A areprovidedbasedontheobserverexistence conditions.ThesucientexistenceconditionsfortheUIOinEq.6canbesummarized asfollows: 1. MA;C isobservable, 96

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2. 6 rank CD = rank D = q 3. MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC isHurwitz,and 4.Eq.6holds. Hence,matrix A shouldbechosensuchthat MA;C isobservable.Since rank CD = rank D = q; itisprovensubsequentlythattheobservabilityofthepair MA;C is equivalenttothefollowingrankcondition[66,72] rank 2 6 4 I n )]TJ/F23 11.9552 Tf 11.955 0 Td [(AD C 0 p q 3 7 5 = n + q; 8 2 C : Thus, A inEq.6shouldbeselectedsothatEq.6issatised.TheconditioninEq. 6facilitatestheselectionof A basedonthesystemmatricesandhencecircumvents thecomputationof M forcheckingtheobservabilityof MA;C .Anothercriteriaonthe selectionof A istominimizetheLipschitzconstantinEq.6. Inthefollowing,Theorem6.2provestherelationshipbetweenEq.6andthe observabilityofthepair MA;C andusestheresultprovedinLemma6.2. Lemma6.2. If rank CD = rank D = q D y denotesaleftinverseofmatrix D then 7 Ker )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(D y Ker M = ; where isanullset,andif rank M = n )]TJ/F23 11.9552 Tf 12.568 0 Td [(q then rank 2 6 4 M D y 3 7 5 = n: Proof. FromEq.6 MD =0 n q ,hence, Ker M = Im D : Also, Ker )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(D y = Ker )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(D T and Ker )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(D T = Im D ? [117].Hence, Ker )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(D y Ker M = Im D ? Im D = : Usingthefactthatthenullspacesof M and D y donothaveanycommon 6 Since rank CD = min f rank C ;rank D g andbyassumption rank C = p;p q therankof D mustbe q forrankof CD tobe q: 7 Ker denotesakernelofanullspaceofamatrixand Im denotesanimagespace ofamatrix. 97

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elements, 2 6 4 M D y 3 7 5 h = 2 6 4 Mh D y h 3 7 5 6 =0 n + q 8 h 2 R n ; where 0 n + q isazerovectorofdimension n + q ,and dim 0 B @ Ker 2 6 4 M D y 3 7 5 1 C A =0 : Using rank 2 6 4 M D y 3 7 5 + dim 0 B @ Ker 2 6 4 M D y 3 7 5 1 C A = n; [117] itcanbeconcludedthat rank 2 6 4 M D y 3 7 5 = n: Theorem6.2. [66]Assume rank CD = rank D = q ,and rank M = n )]TJ/F23 11.9552 Tf 12.013 0 Td [(q .Thepair MA;C isobservableiEq.620holds. Proof. Therankcondition rank CD = rank D = q isobtainedasanecessaryand sucientconditionfortheexistenceofanunknowninputobserverforlinearsystems in[60,62,66,72].UsingthePopov-Bellman-HautusPBHtestofobservability,thepair MA;C isobservablei rank 2 6 4 I n )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA C 3 7 5 = n 8 2C : Considerthefollowingmatrixrankproperties Property1: rank ~ A ~ B = rank ~ B if ~ A 2 R l r withrank r and ~ B 2 R r s ; Property2: rank ~ A ~ B = rank ~ A if ~ B 2 R l r withrank l and ~ A 2 R s l : UsingEqs.6,6andtheProperty1,followingequalityisobtained rank 2 6 4 M )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA C 3 7 5 = rank 2 6 4 I n E 0 p n I p 3 7 5 2 6 4 I n )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA C 3 7 5 = rank 2 6 4 I n )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA C 3 7 5 = n 8 2C : 98

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Givenmatrices S 2 R n + q + p n + p and T 2 R n + q n + q as S = 2 6 6 6 6 4 M 0 n p D y 0 q p 0 p p I p 3 7 7 7 7 5 ;T = 2 6 4 I n 0 n q )]TJ/F28 11.9552 Tf 11.291 9.683 Td [()]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(D y )]TJ/F23 11.9552 Tf 11.955 0 Td [(D y A I q 3 7 5 theresultinLemma6.2showsthat S isfullcolumnranki.e., rank S = n + p ,and T is fullrowrankmatrixi.e., rank T = n + q .UsingProperties1and2,thefollowingrank conditioncanbeobtained rank 2 6 4 I n )]TJ/F23 11.9552 Tf 11.955 0 Td [(AD C 0 p q 3 7 5 = rank 0 B @ S 2 6 4 I n )]TJ/F23 11.9552 Tf 11.955 0 Td [(AD C 0 p q 3 7 5 T 1 C A = rank 2 6 6 6 6 4 M )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA 0 n q 0 q n I q C 0 p q 3 7 7 7 7 5 = q + rank 2 6 4 M )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA C 3 7 5 8 2C ; whichimpliesthat rank 2 6 4 M )]TJ/F23 11.9552 Tf 11.956 0 Td [(MA C 3 7 5 = n: Hence,if rank M = n )]TJ/F23 11.9552 Tf 10.769 0 Td [(q; theobservability of MA;C andEq.6areequivalent. Remark 6.2 TheconditioninEq.6impliesthatthepair C;A;D hasnoinvariant zeros.ForaLTIsystemwithunknowninputs,Eq.6impliesobservability[72].In[72], itismentionedthatsystemobservabilityisnotsucientfortheexistenceofanUIOfor LTIsystems.Thecondition rank CD = rank D = q isalsorequiredfortheexistence oftheobserver.Thecondition rank CD = rank D = q andEq.6aredenedas thestrong*detectability 8 conditionandarenecessaryandsucientconditionsforthe 8 Thenotionof' strong*detectability' isintroducedin[72]todistinguishfromastrong detectabilityconditionwhichimpliesminimum-phaseconditionforlinearsystemswith unknowninputs. 99

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existenceofUIOforLTIsystems.Forageneralnonlinearsystem,necessaryandsucient UIOexistenceconditionsareunknownandisanopenproblemintheliterature. 6.1.5LMIFormulation Inthefollowingsection,theconditioninEq.6isreformulatedasanLMIfeasibility problem.Thematrices P K and Y shouldbeselectedsuchthatthesucientcondition fortheobservererrorstabilityinEq.6issatised.Substituting N and M fromEq.6 intoEq.6yields MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC T P + P MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC + I + 1 + 2 2 P I + EC I + EC T P< 0 : AfterusingEq.6,theinequalityinEq.6canbeexpressedas A T I + FC T P + P I + FC A + A T C T G T P T Y + P Y GCA )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T P T K )]TJ/F23 11.9552 Tf 11.955 0 Td [(P K C + I + 1 + 2 2 P + PFC + P Y GC P + PFC + P Y GC T < 0 where P Y = PY and P K = PK .Fortheobserversynthesis,thematrices Y K and P> 0 shouldbecomputedsuchthatthematrixinequalityinEq.6issatised.Since P> 0 P )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 exists,and Y and K canbecomputedusing Y = P )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 P Y ,and K = P )]TJ/F22 7.9701 Tf 6.586 0 Td [(1 P K UsingSchur'scomplement,theinequalityinEq.6canbetransformedintothematrix inequality 2 6 4 P 1 R R T )]TJ/F23 11.9552 Tf 9.299 0 Td [(I 3 7 5 < 0 where P 1 = A T I + FC T P + P I + FC A + A T C T G T P T Y + P Y GCA )]TJ/F23 11.9552 Tf 11.955 0 Td [(C T P T K )]TJ/F23 11.9552 Tf 11.955 0 Td [(P K C + I; R = P + PFC + P Y GC; = 1 + 2 : 100

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ThematrixinequalityinEq.6isanLMIinvariables P P Y ,and P K .TheLMI feasibilityproblemcanbesolvedusingstandardLMIalgorithms[118]andisaproblem ofnding P P Y and P K suchthat ismaximized.Maximizing isequivalentto maximizing 1 whichmeanstheobservercanbedesignedfornonlinearfunctionswitha largerLipschitzconstant.IftheLMIinEq.6isfeasible,thenasolutiontoEq.6 exists.Hence,theLMIfeasibilityproblemisasucientconditionforthestabilityofthe observer.Alternatively,asucientnumericalalgorithmispresentedin[52]tocompute K suchthatEq.63issatised.Thealgorithmisbasedontheeigenvalueplacement approachandisasucientconditionforEq.6tobesatised. 6.2Summary AnUIOforaclassofnonlinearsystemsisdeveloped.Sucientconditionsforthe existenceoftheUIOaredevelopedandcomputationoftheobservergainisachievedvia anLMIformulation.Thesucientconditionsdevelopedfortheexistenceofdeveloped UIOextendthenecessaryandsucientconditionsofUIOsforLTIsystemsandthe existenceconditionsofnonlinearobserverforLipschitznonlinearsystemswithknown inputs. 101

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CHAPTER7 APPLICATIONOFTHEUNKNOWNINPUTOBSERVERTOTHESTRUCTURE ESTIMATION Inthischapter,theunknowninputobserverdevelopedinChapter6isappliedto thestructureandmotionestimationproblemwithamovingobject.Thedeveloped causalalgorithmrequireslessrestrictiveassumptionsontheobject'smotionthanexisting approaches.Theobjectisassumedtobemovingonagroundplanewitharbitrary velocitiesobservedbyadownwardlookingcamerawitharbitrarylinearmotion.No assumptionsaremadeontheminimumnumberofpointsorminimumnumberofviews requiredtoestimatethestructure.Featurepointdataandcameravelocitydatafromeach imageframeisrequired.Simulationresultsarepresentedtoshowtheeectivenessofthe proposedapproach. 7.1StructureandMotionEstimation 7.1.1StructureandMotionfromMotionSaMfMObjective TheobjectiveofSaMfMistorecoverthestructurei.e.,Euclideancoordinateswith ascalingfactorandmotioni.e.,velocitiesofmovingobjectsobservedbyamoving camera,assumingthatallcameravelocitiesareknown.Inthischapter,anobserveris presentedwhichestimatesthestructureofthemovingobjectwithrespecttothemoving camera.Itisassumedthatoneormorefeaturepointsontheobjectaretrackedineach imageframeandcameravelocitiesarerecordedusingsensorssuchasanIMU.Thecamera isassumedtobeinternallycalibrated.Estimatingthestructureofanobjectisequivalent toestimatingthestate x t ineachimageframe.Basedonthedenitionofthestatein Eq.2,thestructureofthemovingobjectcanbeestimatedbyscaling ^ x 1 t and ^ x 2 t by ^ x 3 t 102

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7.1.2NonlinearUnknownInputObserver Considerthefollowingsystem x = f x;u + g y;u + Dd y = Cx where x t 2 R 3 isastateofthesystem, u t 2 R 6 isameasurablecontrolinput, d t 2 R isanunmeasurableinput, y t 2 R 2 isoutputofthesystem,thefunction f x;u is nonlinearin x t and u t andsatisestheLipschitzcondition jj f x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(f ^ x;u jj 1 jj x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x jj where 1 2 R + .ThesysteminEq.2canberepresentedintheformofEq. 7with f x;u = 2 6 6 6 6 4 v cx )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 1 v cz x 3 v cy )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 v cz x 3 )]TJ/F23 11.9552 Tf 9.299 0 Td [(x 2 3 v cz )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 x 3 1 + y 1 x 3 2 3 7 7 7 7 5 ; g y;u = 2 6 6 6 6 4 )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 y 2 1 ++ y 2 1 2 )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 2 3 )]TJ/F15 11.9552 Tf 9.299 0 Td [(+ y 2 2 1 + y 1 y 2 2 + y 1 3 0 3 7 7 7 7 5 ; and C = 2 6 4 100 010 3 7 5 isfullrowrank, D 2 R 3 1 isfullcolumnrank,and q =1 Thesystemin7canbewritteninthefollowingform x = Ax + f x;u + g y;u + Dd y = Cx where A 2 R 3 3 ; and f x;u = f x;u )]TJ/F23 11.9552 Tf 12.262 0 Td [(Ax .Thefunction f x;u satisestheLipschitz condition jj f x;u )]TJ/F23 11.9552 Tf 11.956 0 Td [(f ^ x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(A x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x jj 1 + 2 jj x )]TJ/F15 11.9552 Tf 12.68 0 Td [(^ x jj ; where 2 2 R + : An exponentiallyconvergingstateobserverisdesignedtoestimate x t inthepresenceofan unknowninput d t i.e.,themovingobject'svelocity. 103

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AnunknowninputreducedorderstateobserverforthesysteminEq.7isdesigned as z = Nz + Ly + M f ^ x;u + Mg y;u ^ x = z )]TJ/F23 11.9552 Tf 11.955 0 Td [(Ey where ^ x t 2 R 3 isanestimateoftheunknownstate x t z t 2 R 3 isanauxiliarysignal, thematrices N 2 R 3 3 L 2 R 3 2 E 2 R 3 2 M 2 R 3 3 aredesignedas[66] M = I 3 + EC N = MA )]TJ/F23 11.9552 Tf 11.956 0 Td [(KC L = K I + CE )]TJ/F23 11.9552 Tf 11.956 0 Td [(MAE where E issubsequentlydesignedand K 2 R 3 2 isagainmatrixwhichsatisesthe inequality Q N T P + PN + 1 + 2 2 PMM T P + I 3 < 0 where P 2 R n n isapositivedenite,symmetricmatrix.UsingEq.7theequality NM + LC )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA =0 3 3 issatised,andif E isselectedas E = F + YG where Y 2 R 3 2 canbechosenarbitrarily,and F and G aregivenby F )]TJ/F23 11.9552 Tf 9.299 0 Td [(D CD y ;G )]TJ/F23 11.9552 Tf 5.479 -9.684 Td [(I 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( CD CD y then ECD = )]TJ/F23 11.9552 Tf 9.299 0 Td [(D; andthefollowingequalityissatised: MD = I 3 + EC D =0 3 1 : 104

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Since rank CD =1 ; thegeneralizedpseudoinverseofthematrix CD existsandisgiven by CD y = CD T CD )]TJ/F22 7.9701 Tf 6.587 0 Td [(1 CD T : Toquantifytheestimationobjectiveanestimationerrorisdenedas e t ^ x t )]TJ/F23 11.9552 Tf 11.955 0 Td [(x t = z )]TJ/F23 11.9552 Tf 11.956 0 Td [(Ey )]TJ/F23 11.9552 Tf 11.955 0 Td [(x: TakingthetimederivativeoftheestimationerrorandusingEqs.7and7yields e =_ z )]TJ/F15 11.9552 Tf 11.955 0 Td [( I + EC x; e = Nz + Ly + M f ^ x;u )]TJ/F15 11.9552 Tf 11.955 0 Td [( I + EC Ax )]TJ/F15 11.9552 Tf 9.298 0 Td [( I + EC f x;u )]TJ/F15 11.9552 Tf 11.955 0 Td [( I + EC Dd: UsingEqs.7and7,theerrorsysteminEq.7canbewrittenas e = Ne + NM + LC )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA x + M )]TJ/F15 11.9552 Tf 8.027 -6.529 Td [( f ^ x;u )]TJ/F15 11.9552 Tf 14.503 3.155 Td [( f x;u )]TJ/F23 11.9552 Tf 11.955 0 Td [(MDd: UsingEq.7and NM + LC )]TJ/F23 11.9552 Tf 11.955 0 Td [(MA =0 ; theerrordynamicscanbewrittenas e = Ne + M )]TJ/F15 11.9552 Tf 8.027 -6.529 Td [( f ^ x;u )]TJ/F15 11.9552 Tf 14.503 3.155 Td [( f x;u : 7.1.3StabilityAnalysis Theorem7.1. ThenonlinearunknowninputobserverinEq.7isexponentiallystableif min 2 R + min MA )]TJ/F23 11.9552 Tf 11.955 0 Td [(KC )]TJ/F23 11.9552 Tf 11.955 0 Td [(j!I 3 > p 3 1 + 2 where min denotestheminimumsingularvalueofamatrix,and 3 max )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(MM T : Proof. TheprooffollowsdirectlyfromtheproofofTheorem6.1. 105

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7.1.4ConditionsforStability TheSaMfMobserverdevelopedinthischapterfollowsthesameexistenceconditions developedinChapter6.Inthissection,conditionsspecictotheSaMfMdynamicsare developed.TheinequalityinEq.7issatisedifthepair MA;C isobservable[66] andtheconditioninEq.711issatised.Ifthepair MA;C isobservablethenthe gainmatrix K canbeselectedsothat N = MA )]TJ/F23 11.9552 Tf 12.492 0 Td [(KC isHurwitz.Since rank CD = rank D =1 thecondition rank 2 6 4 sI n )]TJ/F23 11.9552 Tf 11.955 0 Td [(AD C 0 3 7 5 =4 ; 8 s 2 C impliesthatthepair MA;C isobservable[66]. 7.1.5ConditionsonObjectTrajectories ThedynamicsinEq.2cannotbetransformedintotheformofEq.7because oftheconstraintsonnumberofoutputs p andnumberofunknowninputs q: However,by makingsomeassumptionsonthemotionofacameraandtheviewedobject,thedynamics inEq.2canbetransformedintheformofEq.7.Inthissection,twospecic scenariosarediscussed. Example1:Thecameraisundergoingarbitrarypurelytranslationalmotion,i.e.,the angularvelocitiesofthecameraarezeroandtheobjectismovingalongastraightline withtime-varyingunknownvelocities.Forthiscase,choicesof R t and v p t inEq.2 become 1 R = I 3 ; and v p t = d 1 t 00 T ; or v p t = 0 d 2 t 0 T ; or v p t = 00 d 3 t T ; or v p t = d 4 t d 5 t 0 T ; etc.,where d i t 2 R 8 i = f 1 ;::; 5 g is theunknowntime-varyingobjectvelocity. 1 w.l.o.g.thecameracoordinateframewhentheimagecapturestartscanbesetasInertialcoordinateframe. 106

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Example2:Adownwardlookingcameraisundergoingarbitrarytranslationalmotion alongwiththeangularvelocityalongtheZ-directionanaxispointingdownwards. Theobjectismovingonagroundplanei.e.,X-Yplanewitharbitrarytime-varying velocities.Inthiscase,therotationmatrix R t isgivenby R = 2 6 6 6 6 4 cos t sin t 0 )]TJ/F23 11.9552 Tf 9.299 0 Td [(sin t cos t 0 001 3 7 7 7 7 5 = 2 6 4 R s 0 2 1 0 1 2 1 3 7 5 where t 2 [ )]TJ/F15 11.9552 Tf 9.299 0 Td [(2 ; 2 istherotationanglebetweenthecurrentcameracoordinate frameandinertialcoordinateframe,and R s t 2 R 2 2 representstheupperleft 2 2 blockofthe R t : Theobjectvelocityintheinertialframeisrepresentedas v p t = d 1 t d 2 t 0 T ; where d 1 t ; d 2 t 2 R : Thecameraangularvelocityissuchthat R v p = u where u t = d 3 t 00 T or u t = 0 d 4 t 0 T ; and d 3 t ; d 4 t 2 R areunknowntime-varyingquantities.TheequalityinEq.7canbeachievedif R s d 1 t d 2 t T = u; where u t = d 5 0 T or u t = 0 d 6 T ; where d 5 t ; d 6 t 2 R : Physically,theconditioninEq.7representscameramotionssuch thattheheadingdirectionofthecameraisparallelorperpendiculartotheobject's headingdirectionintheX-Yplane.Forexample,consideranobjectundergoingacircular motionintheX-Yplanewithunknowntime-varyingvelocitiesobservedbyacamera undergoinganarbitrarylinearmotionintheX-Y-Zplaneandcircularmotionalongthe downwardlookingZ-direction. 7.2Simulation Twosimulationsareperformedforamovingcameraobservinganobjectmovingina plane.Forrstsimulation,cameravelocitiesaregivenby v c t = 210 : 5 cos t= 2 T and t = 001 T .Theobjectvelocityisselectedsuchthat v p t = 0 : 500 T 107

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Figure7-1.ComparisonoftheactualandestimatedX,YandZpositionsofamoving objectwithrespecttoamovingcamera. Figure7-2.Errorintherangeestimationofthemovingobject. 108

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Thecameracalibrationmatrixisgivenby A c = 2 6 6 6 6 4 7200320 0720240 001 3 7 7 7 7 5 : Matrices A C and D are A = 2 6 6 6 6 4 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(12 101 000 3 7 7 7 7 5 ;C = 2 6 4 100 010 3 7 5 ;D = 2 6 6 6 6 4 1 0 0 3 7 7 7 7 5 : AnLMIisformedasshowninSection6.1.5ofChapter6.Thematrix Y andthegain matrix K arecomputedusingtheLMIfeasibilitycommand' feasp 'inMatlabandare givenby K = 2 6 6 6 6 4 0 : 82780 00 : 8278 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 53740 3 7 7 7 7 5 ;Y = 2 6 6 6 6 4 00 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 : 5374 3 7 7 7 7 5 : Fig.7-1showscomparisonoftheactualandestimatedX,YandZcoordinatesofthe objectinthecameracoordinateframe.Fig.7-2showstherangeestimationerrorbetween themovingobjectandthemovingcamera. Inthesecondsimulation,cameravelocitiesaremovingobjectsvelocitiesaregivenby v c t = 210 : 5 cos t= 2 T ;! t = 001 T ; and v p t = 0 : 5 sin t 00 T : Measurementnoisewith20dBSNRisaddedtotheimagepixelcoordinatesandthe cameravelocitiesusing awgn commandinSimulink.Alltheothersimulationparameters areselectedsameasthepreviouscase.Fig.7-3showsthecomparisonofactualand estimatedX,Y,andZcoordinatesoftheobjectinthecameracoordinateframeinthe presenceofmeasurementnoise.Fig.7-4showstheerrorinrangeestimation.From thetworesultsintwosimulations,itcanbeseenthattheobserverisinsensitivetothe disturbanceinput. 109

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Figure7-3.ComparisonoftheactualandestimatedX,YandZpositionsofamoving objectwithrespecttoamovingcamerainthepresenceofnoise. Figure7-4.Errorintherangeestimationofthemovingobjectinthepresenceofnoise. 110

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7.3Summary Inthischapter,anonlinearobserverisdevelopedtosolvetheSaMfMproblem.The proposedalgorithmestimatesthestructureofamovingobjectusingamovingcamera withlessrestrictiveassumptionsontheobjectmotion.Theobjectmotionisassumed tobealongastraightlineorinaplaneobservedbyamovingairbornecamera.The algorithmimprovesonourpreviousworkin[119]byrelaxingtheconstantobjectvelocity assumptiontoarbitraryobjectmotioninastraightlineorinaplane.Theobserver-based approachiscausalanddoesnotassumeaminimumnumberofviewsorfeaturepoints. Thestructureestimationisinsensitivetotheobjectmotioninthesensethatthestate estimationiscompletelydecoupledfromtheobjectmotionwhichactsasanexogenous disturbanceinputundercertainconditionsontheobjectmotion.Futureeortswill focusedondesigningareduced-orderobserverforstructureaswellasmotionestimation. 111

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CHAPTER8 CONCLUSIONANDFUTUREWORK 8.1Conclusion Newreal-timesolutionstoageneralstructureandmotionproblemincomputervision arepresented.Asopposedtothetraditionalbatchmethods,anobserver-basedapproach isdeveloped.Anonlinearstatemodelisusedtoderivenewreduced-orderstructure estimationalgorithmswhentheobjectisstationaryandthecameraismovingwithfullor partialcameravelocityinformation.Solutionsofthestructureestimationofthestationary objecthavebeenextendedtothemovingobjectmovingcamerascenario.Forthemoving objectmovingcamerascenario,themovingobject'svelocityisconsideredasanexternal unknowntime-varyingdisturbance.Twoapproachestotheobserverdesigninthepresence ofexternalunknowndisturbancearepresented.Intherstapproach,thedisturbanceis consideredtobeavanishingdisturbance.Conditionsontheobject'smotionaredeveloped whichsatisestheassumptionsrequiredbytheobserverdesign.Theseconditionsput certainconstraintsintheformofdierentialequationsonthemovingobject'smotion andtheangularvelocityofthecamera.Foraspecialcaseofconstantcameraangular velocities,ananalyticalrelationbetweentheobject'smotionandthecameraangular velocitiesisdevelopedwhereasymptoticestimatorconvergencecanbeachieved.Inthe secondapproach,atimevaryingobjectvelocityisconsideredasanunknowninputto thesystemandanunknowninputobserverforageneralclassofnonlinearsystemsis developed.TheUIOapproachdoesnotrequiretheexternaldisturbancestobevanishing butrequiresittosatisfyaminimumphaseconditionforthetransferfunctionfromthe disturbancetotheoutputofthesystem.Forthemovingcameramovingobjectdynamics theminimumphaseconditionprovidesnewconstraintsonthevelocityofthemoving objectandtheangularvelocityofthecamera.Theseconstraintscanaccommodatea moregeneralmovingcameramovingobjectscenario.ResultsinChapters5and6forthe movingcameramovingobjectscenarioadvancesthestate-of-the-artintermsofamount 112

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ofdatarequiredandbyenablingreal-timeestimation.Thespeciccontributionsofeach resultarementionedbelow. InChapter3,areducedordernonlinearobserverispresentedforthestructure estimationofpointsonastationaryobjectusingamovingcamera.Theobserveris globallyexponentiallystableprovidedanobservabilityconditionissatised.Theobserver isalsoshowntobeexponentiallystableunderarelaxedobservabilitycondition.Hence, theresultimprovesthedomainofapplicabilitytoalargersetofcameramotionsthan previousalgorithms.Theobserverisshowntoberobustagainstexternaldisturbances. Comparisonoftheobserverperformanceagainsttwoexistingobserversispresentedin theoryandinnumericalsimulations. Chapter4developsareducedorderobserverfortheestimationofthestructureofa stationarytargetwithrespecttoamovingcamera,alongwithtwounknowntime-varying linearandangularcameravelocities.Theobserverrequirestheimagecoordinatesofthe featurepointsontheobject,asinglelinearcameravelocity,andthecorrespondinglinear cameraaccelerationinanyoneofthethreecameracoordinateaxes.Underaphysically motivatedPEcondition,asymptoticconvergenceoftheobservererrorisguaranteed.The resultadvancesstateoftheartSaMalgorithmsintermsofafewernumberofmeasurable signalsrequiredbythealgorithm. InChapter5,anonlinearobserverformovingcameramovingobjectscenariois developed.Theobject'svelocityisconsideredasanexternaldisturbance.UnderaPE condition,Lyapunovanalysisisperformedtoshowthattheobservationerrorconvergesto zero.Theobserverisrobustinthesensethattheestimationerrorconvergeseveninthe presenceof L 2 [0 ; 1 time-varyingdisturbances. InChapter6,anUIOisdevelopedforageneralclassofnonlinearsystems.Necessary andsucientconditionsaredevelopedfortheUIOdesign.Theresultimprovesthestate oftheartUIOalgorithmsbydevelopinganUIOforamoregeneralclassofnonlinear systems.TheapplicationoftheUIOtothemovingcameramovingobjectscenariois 113

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presentedinChapter7.Conditionsontheobjectmotionaredevelopedwhichsatisesthe assumptionsoftheUIO.TheapplicationoftheUIOtothemovingcameramovingobject scenarioenablessolutionstotheproblemsinreal-timeandsignicantlyadvancesthestate oftheartstructureandmotionalgorithmsgivenmovingobjectsintermsofreal-time computation,andamountofdatarequiredbytheestimationalgorithm. 8.2FutureWork Theworkinthisdissertationopensnewdoorsforresearchinthedomainofnonlinear observerdesign,andstructureandmotion.Inthissection,openproblemsrelatedtothe workinthisdissertationarediscussed.Theopenproblemsaresegregatedintotwobroad technicaldisciplines,viz.;NonlinearObserverandStructureandMotion. 1.Foralltheobserverdesignsinthisdissertation,theoutputequationisconsidered tobecontinuous.Apracticallyinspiredproblemintheobserverdesignis:howto designnonlinearobserverswiththeintermittentand/ortime-delayedobservations. Recently,KalmanlterandextendedKalmanlterwithintermittentobservations havebeendevelopedin[120,121].Theparallelsoftheseresultstothegeneral nonlinearobserverdesignshouldbepursued. 2.InChapter5,averyspecialclassofnonlinearsystemsisconsidered.Oneinteresting questioniscanonegeneralizetheoutputdynamicsinEq.5tocontainnonlinear termsoftheunmeasurablestate x t ?Thisextensionwillincreasetheapplication domainoftheobserverdesignapproachtomultipleareas. 3.InChapter5,asymptoticerrorconvergenceisachievedbyassumingthattheexternaldisturbanceisasymptoticallyconvergingtozero.Undersuitableconditions, istheasymptoticerrorconvergenceachievablewithboundednon-vanishingdisturbance? 4.InChapter5,canthePEassumptionberelaxedtoshowtheerrorconvergence? Thereisrelatedworkin[122]whichintroducedrelaxed-PEnotionforparameter 114

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identicationandadaptivecontrolliterature.Similarideasmaybeadaptedforthe observerdesigntorelaxthePEassumptions. 5.InChapter6,theclassofnonlinearsystemsconsideredfortheUIOhasaconstant matrix D multipliedbyatime-varyingdisturbance.DesigningaUIOfortimevaryingorstatedependent D matrixforthegeneralclassofnonlinearsystemsisan openproblem. 6.InChapter6,designingnecessaryandsucientconditionsunderwhichUIOexists forageneralclassofnonlinearsystemsisanopenproblem.Recently,necessaryand sucientconditionsaredevelopedfortheexistenceofUIOforstateanesystems in[71].Theextensionofsuchresultsforamoregeneralclassofnonlinearsystems shouldbepursued. 7.InChapter7,theapproachforstructureandmotionestimationofthemoving objectrequiressomeconstraintsontheobjectandcameramotions.Existingbatch methodssolvestheproblembyputtingsomegeometricconstraintsontheobject trajectorybutrequiresmorethantwoimagestoperformtheestimation.Future eortsshouldtrytoincorporatesomegeometricconstraintsontheobjectmotion withthestatespacedierentialequationmodelsothatageneralizedsolutionto thestructureandmotionproblemcanbedevelopedwhichwillpotentiallyrelaxthe existingconditionsonthecameraandobjectmotion. 8.Inthisdissertation,thesurfaceoftheobjectisconsideredasaLambertiansurface, i.e.,thereisnoreectionofthesurroundingsceneandthefeaturepointsonthe objectcanbetrackedeasily.Thisobservationraisesaquestion:canonlinesolutions bedevelopedforthestructureandmotionestimationproblemwhentheobject's surfaceisspecular,i.e.,reectsthesurroundings.Recently,in[123,124]some geometricsolutionstotheproblemaredeveloped.Anonlinesolutiontothisproblem doesnotappeartoexist. 115

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APPENDIXA PROOFOFPOSITIVENESSOF P Integrating5,thefollowingexpressioncanbeobtained P t = P )]TJ/F24 7.9701 Tf 19.061 18.664 Td [(t 0 L d: A Using5and5theintegralof L t canbewrittenas t 0 L d = t 0 e T 1 1 d )]TJ/F23 11.9552 Tf 11.955 0 Td [( t 0 e T 1 sgn e 1 d + t 0 e T 1 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [(sgn e 1 d: A UsingintegrationbypartsfortherstintegralinAandusingtheproperty t 0 e T 1 sgn e 1 d = 2 X i =1 j e 1 t j)]TJ/F23 11.9552 Tf 17.932 0 Td [( 2 X i =1 j e 1 j theintegralof L t canbeexpressedas t 0 L d = e T 1 t 1 t )]TJ/F23 11.9552 Tf 11.955 0 Td [(e T 1 1 )]TJ/F23 11.9552 Tf 11.955 0 Td [( 2 X i =1 j e 1 i t j + 2 X i =1 j e 1 i j + t 0 e T 1 1 d )]TJ/F24 7.9701 Tf 19.06 18.663 Td [(t 0 )]TJ/F23 11.9552 Tf 5.48 -9.683 Td [(e T 1 1 + j e 1 j d: A UsingAandA, P t canbewrittenas P t = )]TJ/F23 11.9552 Tf 9.299 0 Td [(e T 1 t 1 t + e T 1 1 + n 1 X i =1 j e 1 i t j)]TJ/F23 11.9552 Tf 17.933 0 Td [( n 1 X i =1 j e 1 i j )]TJ/F23 11.9552 Tf 9.299 0 Td [( t 0 e T 1 1 d + t 0 e T 1 1 + n 1 X i =1 j e 1 i j d + n 1 X i =1 j e 1 i j)]TJ/F23 11.9552 Tf 17.933 0 Td [(e T 1 1 116

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P t = )]TJ/F23 11.9552 Tf 9.298 0 Td [(e T 1 t 1 t + n 1 X i =1 j e 1 i t j )]TJ/F23 11.9552 Tf 9.298 0 Td [( t 0 e T 1 1 d + t 0 )]TJ/F23 11.9552 Tf 5.48 -9.684 Td [(e T 1 1 + j e 1 j d P t n 1 X i =1 j e 1 i t j i )-222(j 1 i t j + t 0 n 1 X i =1 j e 1 i j i )]TJ/F23 11.9552 Tf 11.955 0 Td [( j 1 i j)-222(j 1 i j d A If isselectedaccordingto5,thenusing5andA P 0 : 117

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APPENDIXB OBSERVABILITYCONDITIONS B.1ObservabilityConditioninChapter4 FromChapter4, J y 1 ;y 2 ;b 3 2 R 2 3 isgivenby J = 2 6 4 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 b 3 10 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 2 b 3 01 3 7 5 and J T J = 2 6 6 6 6 4 y 2 1 + y 2 2 b 2 3 )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 b 3 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 2 b 3 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 b 3 10 )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 2 b 3 01 3 7 7 7 7 5 : Takingtheintegralof J T y 1 ;y 2 ;b 3 J y 1 ;y 2 ;b 3 yields t + t J T Jd = 2 6 6 6 6 4 t + t y 2 1 + y 2 2 b 2 3 d t + t )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 b 3 d t + t )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 2 b 3 d t + t )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 b 3 d 0 t + t )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 2 b 3 d 0 3 7 7 7 7 5 : B TheconditionsforwhichBbecomesrank2are: 1. b 3 t =0 8 t 2 [ t;t + ] 2. y 1 t = c 1 ;y 2 t = c 2 ; and b 3 t = c 3 8 t 2 [ t;t + ] where c i 8 i = f 1 ; 2 ; 3 g are constants. TherstconditionindicatesthatthecameramustbemovinginZ-directionduring anysmalldurationoftime [ t;t + ] : Thesecondconditionissatisediftheimagepointis notmovingforasmalldurationoftime.Theimagepointisconstantonlyifthecamera isnotmovingorifthecameraistravelingalongtherayprojectedbythefeaturepointon thecameraimage. B.2ObservabilityConditioninChapter5 FromChapter5, 118

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J = 2 6 4 v cx )]TJ/F23 11.9552 Tf 11.956 0 Td [(y 1 v cz )]TJ/F15 11.9552 Tf 9.299 0 Td [(10 y 1 v cy )]TJ/F23 11.9552 Tf 11.955 0 Td [(y 2 v cz 0 )]TJ/F15 11.9552 Tf 9.299 0 Td [(1 y 2 3 7 5 : Let h 1 t = v cx t )]TJ/F23 11.9552 Tf 12.893 0 Td [(y 1 t v cz t and h 2 t = v cy t )]TJ/F23 11.9552 Tf 12.894 0 Td [(y 2 t v cz t : Theintegralof J T v cx ;v cy ;v cz ;y 1 ;y 2 J v cx ;v cy ;v cz ;y 1 ;y 2 isgivenby t + t J T Jd = 2 6 6 6 6 6 6 6 4 t + t h 2 1 + h 2 2 d t + t )]TJ/F23 11.9552 Tf 9.298 0 Td [(h 1 d t + t )]TJ/F23 11.9552 Tf 9.299 0 Td [(h 1 d t + t )]TJ/F23 11.9552 Tf 9.299 0 Td [(h 2 d 0 t + t y 1 h 1 + y 2 h 2 d t + t )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 1 d B t + t )]TJ/F23 11.9552 Tf 9.298 0 Td [(h 2 d t + t y 1 h 1 + y 2 h 2 d 0 t + t )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 1 d t + t )]TJ/F23 11.9552 Tf 9.298 0 Td [(y 2 d t + t )]TJ/F23 11.9552 Tf 9.299 0 Td [(y 2 d t + t y 2 1 + y 2 2 d 3 7 7 7 7 7 7 7 5 TheintegralinBisnotfullrankif: 1. y 1 t = c 1 ; and y 2 t = c 2 8 t 2 [ t;t + ] where c 1 ; c 2 areconstants. 2. h 1 t = c 3 ; and h 2 t = c 4 8 t 2 [ t;t + ] where c 3 ; c 4 areconstants. Condition1canbesatisedonlyifcameraisstationaryorifthecameraismoving alongtheprojectedrayofthefeaturepointontheimageplane.Forboththecases, h 1 t = h 2 t =0 ; hence,therstconditionisasubsetofthesecondcondition.The secondconditionimpliesthefollowingratioinequalities, v cx )]TJ/F15 11.9552 Tf 12.198 0 Td [( c 3 v cz = X Z ; and v cy )]TJ/F15 11.9552 Tf 12.198 0 Td [( c 4 v cz = Y Z : B TheconditionsinBaresatisedifthecameraismovingalongtherayprojected bythefeaturepointontheimageplaneorifthecameravelocitiesarenotpersistently changing. 119

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BIOGRAPHICALSKETCH AshwinDaniwasborninNagpur,India.HereceivedhisBachelorofEngineering degreeinmechanicalengineeringfromCollegeofEngineeringPuneCOEP,India.After hisgraduationin2005,hewashiredbyInternationalBusinessMachinesIBMasa graduateengineerfrom2005to2006.HethenjoinedtheNonlinearControlsandRobotics NCRresearchgrouptopursuehisdoctoralresearch,undertheadvisementofWarrenE. Dixon.HehasworkedasaninternatMitsubishiElectricResearchLabs,CambridgeMA, duringJanuaryof2009toMay2009andMay2010toAugust2010onobserverdesignfor distributedparametersystems.HereceivedhisPh.D.fromtheUniversityofFloridain theSummerof2011.FromAugust2011onwardhewillbeapost-doctoralfellowwiththe UniversityofIllinoisatUrbanaChampaign. 129