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NEEDLEINSERTIONFORROBOTICSURGERY By CLINELAPLASSOTTE ATHESISPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF MASTEROFSCIENCE UNIVERSITYOFFLORIDA 2012

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c 2012ClineLaplassotte 2

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Tomyfamilyandmyfriendsfortheirconstantencouragementandlove 3

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ACKNOWLEDGMENTS Firstofall,IwouldliketothankmyadvisorDr.Dixonforgivingmetheopportunityto workinhislaboratory,NonlinearControlsandRobotics,forthetimehespentwithmeandfor hisvaluableadvicesthathelpedmethroughoutmyproject.Ishallalsothankmyresearchgroup inwhichIwasparticularlywellreceivedandforhavingbroughtupalivelyatmosphereinthe dailywork. Furthermore,IamdeeplygratefultoDr.Collet,Pr.Tran-Son-TayandtheAtlantisprogram forhavingallowedmetotakeparttothisgreatexperiencewhichisstudyinginalargeand well-knownuniversityastheUniversityofFloridaforoneyear.AndIwouldliketothankDr. Bayleforhishelpandthefeedbackhegavemeonmywork. IwouldalsoliketoextendmygratitudetomycommitteememberDr.CarlCraneforthe timeandhelphehasprovided. Finally,Iwouldliketothankmyfamilyandmyfriendsfortheirencouragementandallthe peopleIsharemyworkwithattheUniversityofFlorida. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS....................................4 LISTOFFIGURES.......................................7 ABSTRACT...........................................8 CHAPTER 1INTRODUCTION....................................9 1.1MotivationandProblemStatement.........................9 1.2LiteratureReview..................................9 1.3OutlineandContributions.............................11 2NEEDLEINSERTIONFORCEDESIGN........................13 2.1SoftTissueDeformation..............................13 2.2NeedleInsertionForceModeling..........................13 2.2.1StiffnessForce...............................15 2.2.2FrictionForce................................16 2.2.3CuttingForce................................16 3ROBOTICNEEDLEINSERTIONINTOVISCOELASTICTISSUE..........18 3.1DynamicModel...................................18 3.2ControlDevelopment................................19 3.2.1ControlObjective..............................19 3.2.2Closed-LoopErrorSystem.........................20 3.3StabilityAnalysis..................................24 3.4SimulationResults.................................27 4TELEOPERATEDROBOTFORNEEDLEINSERTIONINTOVISCOELASTIC TISSUE..........................................32 4.1DynamicModel...................................32 4.2ControlDevelopment................................33 4.2.1ControlObjectiveandModelTransformation...............33 4.2.2Closed-LoopErrorSystem.........................35 4.3StabilityAnalysis..................................38 4.4SimulationResults.................................42 5CONCLUSION......................................49 5.1SummaryofResults................................49 5.2RecommendationsforFutureWork........................49 5

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REFERENCES.........................................51 BIOGRAPHICALSKETCH..................................57 6

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LISTOFFIGURES Figure page 2-1Needleinsertionsteps...................................14 3-1Multilayerneuralnetworkforjumpfunctionapproximation...............21 3-2Positionsforthesimulation................................28 3-3Positionoftheneedletip x t ...............................29 3-4Positiontrackingerror e t .................................30 3-5Needleforce f needle asafunctionoftime.........................30 3-6Needleforce f needle asafunctionoftheneedletipposition x t .............31 4-1Trajectoryformasterandslaverobotsfor F 1 = 15sin 1 : 1 t ...............44 4-2Positionerrorbetweenmasterandslaverobotfor F 1 = 15sin 1 : 1 t ...........44 4-3Desiredtrajectory x d 2 andpositionof q 1 + q 2 for F 1 = 15sin 1 : 1 t ............45 4-4Errorbetweenthedesiredtrajectory x d 2 and q 1 + q 2 for F 1 = 15sin 1 : 1 t ........45 4-5Trajectoryformasterandslaverobotsfor F 1 = 8.....................46 4-6Positionerrorbetweenmasterandslaverobotfor F 1 = 8.................46 4-7Desiredtrajectory x d 2 andpositionof q 1 + q 2 for F 1 = 8.................47 4-8Errorbetweenthedesiredtrajectory x d 2 and q 1 + q 2 for F 1 = 8..............47 4-9Needleforce f needle asafunctionoftimefor F 1 = 8...................48 4-10Needleforce f needle asafunctionoftheneedletippositionfor F 1 = 8..........48 7

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AbstractofThesisPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofMasterofScience NEEDLEINSERTIONFORROBOTICSURGERY By ClineLaplassotte August2012 Chair:WarrenE.Dixon Major:MechanicalEngineering Manymodernclinicalpracticesinvolvepercutaneousneedleinsertion.Thisthesisfocuses onmodelingandautomationaspectsrelatedtoroboticneedleinsertion.Medicalroboticsmay offermethodsforimprovingsuchpractices.Therstcontributionisthedevelopmentofa controllertoensurethataneedletiptracksatrajectorybeginninginanon-contactposition andendingwithinviscoelastictissue.Throughemploymentofaslidingmodecontroller andaneuralnetworkNN,thecontrollerguaranteessemi-globalasymptotictrackingofthe desiredtrajectory.Thesecondcontributionisthedevelopmentofacontrollertoensurethata needletipmountedonaslaverobottracksthetrajectorygivenbythesurgeonmanipulatingthe masterrobot,inthepresenceofuncertaintiesintheuserandenvironmentforces.Thecontrol developmentleadstosemi-globalasymptotictrackingofthedesiredtrajectoryusingasliding modecontrollerandaNN.Lyapunov-basedstabilityanalysisandsimulationsareprovidedto demonstratetheperformanceofthecontroldesignsthroughoutthethesis. 8

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CHAPTER1 INTRODUCTION 1.1MotivationandProblemStatement Medicalroboticshasgainedpopularityoverthelastdecade.Indeed,surgeonsallaroundthe worldusemanipulatorstoperformsurgicalprocedures.Thedevelopmentoftheseprocedures aremotivatedbyandhaveimprovedduetotherapidadvancementofminimallyinvasive procedures[14].Automatedandteleoperatedsystemshavethepotentialtoimprovethesafety andeffectivenessofsurgeriesbyenhancingvisualization,decreasingbleedingandtransfusion rates,andspeedingrecovery[5]. Manyclinicalpracticesinvolvepercutaneousneedleinsertions.Minimallyinvasive percutaneousproceduresincludebiopsies[6]andbrachytherapy[7]butneedleinsertionis alsousedforproceduressuchasbloodsampling[8],neurosurgery[9],andothers.Inthese procedures,oneorseveralneedlespenetrateintothepatient'sbodytoreachtheplannedtarget. Whileautomatedorteleoperatedneedleinsertionsystemscanleadtovariousadvantages, severalissuesmustbeconsideredincluding:thelackofvisibilityofthetarget,thedifcultaccess tothetarget,andrestrictedmaneuverabilitywiththetool.Forinstance,thetargetmaybeclose toasensitiveorganmandatingtheneedforextracautionandhighprecision.Targetingerrorcan occurduetoimaginglimitations,targetuncertaintiesduetophysiologicalorpatientmotion, humanerrorsduetofatigueorhandtremor,tissuedeformationandneedledeection[10].The efciencyofsuchamedicaltreatmentisveryoftenlinkedtotheaccuracyoftheneedleinsertion andtothecontroloftheinsertionforce.Thedesiredaccuracydependsontheapplicationand usuallyrangesfrommillimetertomicro-millimeter.Givensuchaccuracydemands,roboticand teleoperatedsystemshavebecomeincreasinglypopulartoolstoassistmedicalpersonnel. 1.2LiteratureReview Themodelingofneedleinsertionforceintosofttissuecanfacilitateaccuratesurgical simulationsandrobotictechnologiesappliedtopercutaneoustherapy.Thedevelopmentofsuch modelshasbeenthetopicofmanystudies[1117].Knowledgeofforcesduringneedleinsertion 9

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canhelptoidentifyandmodeldifferenttissuetypes.Humanbiologicaltissuesareknown toexhibitnonlinearpropertiesandconsistofinhomogeneousstructures.TheHunt-Crossley model[18]hasbeenconrmedasbeingsuitablefordescribingthepropertiesofviscoelastic tissues[19],especiallywhensmalldeformationsareinvolved[20].HuntandCrossleyshowed thatitispossibletoobtainabehaviorthatisinbetteragreementwiththephysicalintuitionifthe dampingcoefcientismadedependentonthebody'srelativepenetration.Nevertheless,some studiespresumealineartissuemodel,especiallyforcomputationalperformance[21].Oneofthe keyissuesisthattheinsertionforcevariesfromonepatienttoanother.Forthesametissue,the insertionforcecanbedifferentdependingontheage,thegender,orthebodymassofthepatient. Evenforonepatient,theinsertionforceneededforonetissuecanvary,forexample,ifthetissue isdiseased.Moreover,acquiringdatafrombiologicaltissuesanddevelopingappropriatemodels forapplicationinsimulationorrobot-assistedsurgeryisdifcultduetotissuedeformation, inhomogeneity,nonlinearity,andopacity[2224].Asaresult,itisnecessarytodesigntheneedle insertionforcesothatitaccountsfortheuncertaintyintissuecomposition. Inmedicalrobotics,ateleoperatedsystemconsistsofaslaverobotwhichtracksthemotion ofamasterrobotcommandedbyasurgeon,oftenwiththeassistanceofmedicalimaging.Many clinicalapplicationsbenetfromteleoperatedsystems.Exampleproceduresrangesfromteleechography[25,26]tominimallyinvasivesurgery[1,4,5,27,28].Teleoperatedsystemshavethe abilitytoreducethemorbidityofclinicalproceduresbyimprovingthesterileeld,decreasing bleeding,andreducingrecoverytime.However,sincetheclinicianisremovedfromdirect contactwiththepatient,researcheffortshavefocusedonmethodstoprovideimprovedforce reection,compensateforrobotic/tissueuncertainties,andimprovethestabilityandpassivityof thesystem. Thegoalofteleoperationsystemsistoachievepassivityandtransparencywhilemaintaining stability.Passivityisrelatedtoenergydissipation,apassivesystemconsumesenergyanddoes notproduceenergy.Toachieveidealtransparency,theslaverobothastoexactlyreproducethe positiontrajectoryofthemastermanipulator,andthemasterrobothastoaccuratelydisplaythe 10

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environmentforcetothehuman.Manybilateralcontrolarchitectureshavebeendevelopedto reachthesetwoaims[2933].Linearcircuittheory[34]andlinearrobustcontroltheory[35,36] havebeenstudiedinthepast.Someworkshavealsobeendonefornonlinearsystemsusing adaptivecontrol[3740],howeverthesedesignsneedexactmodelknowledge.Someprevious workshighlightedthestabilityandsafeoperationoftheteleoperatorusingthepassivityconcept asin[36,4143].Themethodproposedin[44]makestheteleoperatedsystempassiveusing ctitiousenergystorage.Researchesthataimtoachieveidealtransparencyusuallyrequire knowledgeabouttheenvironmentinputsasin[35],orestimatetheimpedanceoftheslaverobot asin[45].In[46],anadaptivecontrollerisdesignedforteleoperatedsystemswithparametric uncertaintiesinthemasterandslaverobotsdynamics.Timedelaymayalsobeanissue.In[47], abilateralteleoperatorprovidesrobuststabilityagainstconstantdelaybutdoesnotguarantee positiontracking. 1.3OutlineandContributions Chapter 1 servesasanintroduction,thatprovidesmotivation,problemstatement,literature review,andcontributionsofthethesis. Chapter 2 providesabackgrounddiscussiononsofttissuedeformation.Thischapter presentsalsoanovelneedleinsertionforcemodelingforviscoelastictissue.Theforcemodeling isdesignedasthesumofastiffnessforce,africtionforce,andacuttingforce[48].Thesethree forcesarecarefullychosentobeasclosetotherealityaspossible.Thestiffnessforceisdesigned usingtheHunt-Crossleymodel.Thefrictionforceismodeledasin[49].Thecuttingforceis modeledasaconstant. Chapter 3 detailsthedesignofanautomatedcontrollerthatensuressemi-globalasymptotic trackingofatrajectoryforwhichtheneedletipmovesfromanon-contactpositionintoviscoelastictissue.Thestudyisbasedonpreviousworks[5052],wheretheobjectivewastodesign acontrollerforarobotinteractingwithanuncertainHunt-Crossleyviscoelasticenvironment andundergoinganon-contacttocontacttransitionbuttherobotdidnotgointotheviscoelastic environment. 11

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Chapter 4 describesthedevelopmentofateleoperatedcontrollertoensurethataneedle tipmountedonaslaverobottracksthetrajectorygivenbythesurgeonmanipulatingthemaster robot,andgoingfromanon-contactpositionintoviscoelastictissue.Thestudyisbasedona previouswork[53],wheretheobjectivewastodesigntwocontrollersforateleoperatorsystem thattargetscoordinationofthemasterandslavemanipulatorsandpassivityoftheoverallsystem. Asin[53],thereisnoneedtoknowtheuserandenvironmentforcesinthispaper.However,the controldevelopmentusedin[53]isnotapplicableinthecaseofadiscontinuousneedleinsertion force.Then,thecontrollerisdesignedusingaslidingmodetermandneuralnetworkmethod. Chapter 5 givessomeconcludingcommentsandrecommendationsforfuturework. 12

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CHAPTER2 NEEDLEINSERTIONFORCEDESIGN 2.1SoftTissueDeformation Realisticmodelingofsofttissuedeformationduringneedleinsertioncanbeusedand improvedfortrainingandplanningtoreduceerrorsbetweendesiredandactualplacementof theneedletip.Thismodelingiscomplexbecauseoftheinhomogeneous,nonlinear,anisotropic, elasticandviscouspropertiesofsofttissue.Todetermineandunderstandtheseproperties,it isessentialtodosomemeasurementsonsofttissue[54].Manyultrasonicmethodshavebeen developedformeasuringbiomechanicalpropertiesofsofttissues[55,56]. Skinandsofttissueexhibitparticularproperties[57,58].Thecharacteristicsubstances ofthiskindoftissuearethecollagen,elastinandgroundsubstance[59].Atsmallstrains, elastinconfersstiffnesstothetissueandstoresmostofthestrainenergy.Thecollagenbers arecomparativelyinextensibleandareusuallyloose.Softtissueshavethepotentialtoundergo bigdeformationsandstillcomebacktotheinitialcongurationwhenunloaded.Thenonlinear stress-strainrelationshipresultsinforcenotbeinglinearlyproportionaltodisplacement[60].For computationalefciency,however,manyresearchersassumeasimplelineartissuemodel. 2.2NeedleInsertionForceModeling Theforcemodelingusedinthisstudyisinspiredby[48],whereanexperimentalprocedure foracquiringdatafromexvivotissueisgivenandtheneedleinsertionforceisdesignedasthe sumofastiffnessforce,africtionforce,andacuttingforce.Inthisstudy,thestiffnessforceis designedusingthenonlinearviscoelasticHunt-Crossleymodel.Thefrictionforceismodeled asin[49].Thismodeloffersanaccuraterepresentationofnonlinearfrictioneffects.Thecutting forcerequiredtoslicethroughtissueismodeledasaconstantdependingontheneedlesizeand onthetissueproperties[48]. Aneedleinsertionprocedurecanbedividedintothreestages.Therststageisafree-space motionandoccursbeforetheneedletouchesthetissue.Thesecondstageistheneedle-tissue viscoelasticinteractionandoccurswhen x t 2 R ,thepositionoftherobotend-effectoratthe 13

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Figure2-1.Needleinsertionsteps needletip,rangesbetween x t t 2 R ,thepositionoftheviscoelastictissue,and x m t 2 R ,the positionofthemaximallydeformedtissuesurfacebeforepuncture.Thelaststageistheinsertion throughthetissue,whichoccurswhen x t isgreaterthan x m t .Thedynamicsofthetissue dependsonforcesfromsurroundingtissueandorgans,physiologicalmovements,etc.,which resultintheevolutionof x t t overtime.Figure2-1illustrateseachstage. Theforce f needle x ; x isdiscontinuousbecauseofthetransitionbetweenneedle-tissue contactandinsertionthroughthetissue.Theneedleinsertionforcecanbemodeledas[48] 14

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f needle L 1 f stiffness + L 2 f friction + L 2 f cutting ; where L 1 x ; x t ; x m and L 2 x ; x m 2 R arefunctionswhichswitchatcontactandperforation, respectively,denedas L 1 8 > > < > > : 1 x t x x m 0 otherwise ; L 2 8 > > < > > : 1 x m < x 0 otherwise : 2.2.1StiffnessForce Thestiffnessforcecorrespondstoaviscoelasticinteractionbetweenthetissueandthe needletip[61].Thisinteractionoccursbeforethepuncture.Theneedlecompressesthesoft tissueuntilthepunctureofthesurface.In2,thestiffnessforce f stiffness x ; x 2 R is describedbytheHunt-Crossleymodelas[18] f stiffness ld n + m dd n ; where l 2 R istheunknowncontactstiffnessoftheviscoelasticmass, m 2 R istheunknown dampingcoefcient, n 2 R istheunknownHertziancompliancecoefcient,and d t 2 R isthe localdeformationofthetissue,denedas d x )]TJ/F54 11.9552 Tf 10.95 0 Td [(x t : Theviscoelasticforce f stiffness x ; x dependsonthelocaldeformationofthetissue,whilethe positionofthetissueisthesumofthedeformationandthepositionofthetissueunderthe pressureofphysiologicalmotionorneedletip. 15

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2.2.2FrictionForce Thefrictionforceoccursinsidethetissueafterthepunctureandalongtheneedleshaft. Frictionisanaturalphenomenonthatcanbefoundinmanymechanicalapplicationshowever itsmodelingisnotentirelyunderstood.In2,thefrictionforce f friction x 2 R ismodeled accordingto[49]as f friction g 1 tanh g 2 x )]TJ/F23 11.9552 Tf 10.949 0 Td [(tanh g 3 x + g 4 tanh g 5 x + g 6 x ; where g i 2 R ,for i = 1 ; 2 ;::: 6,areunknownpositiveconstants.Themodelin2exhibitsthe followingproperties: 1.itissymmetricabouttheorigin, 2.ithasastaticcoefcientoffriction,givenby g 1 + g 4 3.itincludestheStribeckeffect,givenbytanh g 2 x )]TJ/F23 11.9552 Tf 10.949 0 Td [(tanh g 3 x 4.ithasaviscousdissipationterm,givenby g 6 x 5.ithasaCoulombicfrictioncoefcientintheabsenceofviscousdissipation,givenby g 4 tanh g 5 x See[49]and[62]forfurtherdetails. 2.2.3CuttingForce Alsoin2,thecuttingforce f cutting 2 R representstheforcerequiredfortheneedleto penetrateintothetissue.Thisforceonlydependsontheneedlesizeandonthetissueproperties andisdenedas f cutting c ; where c 2 R isaunknownpositiveconstant. Remark 2.1 Inmanyneedleinsertionapplications,thedifferentparametersofthestiffnessforce, frictionforceandcuttingforce,denedpreviouslyin2,2and2,havetobeknown. Theidenticationoftheneedleinsertionforcecanbeperformedbeforetheoperationusing ex 16

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vivo testsasin[48].Foramedicalintervention, exvivo testscannotbedoneonapatientbut theseparameterscanbedeterminedduringtheintervention.In[11],anapproachforestimating needleforceisgivenbutitisnoteasilyapplicableformedicalproceduresbecauseoftheneed toputmarkersonthesurface.[22]describesanonlineestimationtodetermineHunt-Crossley parameters.Forthecontrolanalysisdevelopedinthefollowingchapters,theseparametersare assumedtobeuncertain. 17

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CHAPTER3 ROBOTICNEEDLEINSERTIONINTOVISCOELASTICTISSUE Thischapterdescribesthedevelopmentofanautomatedcontrollertoensurethataneedle tiptracksadesiredtrajectorybeginninginanon-contactpositionandendingwithinviscoelastic tissue. 3.1DynamicModel Thedynamicmodelforaone-degree-of-freedomtranslationrobotinteractingwitha viscoelasticenvironmentis M x x + h x + f needle x ; x = F : In3, x t ; x t ; x t 2 R denotetheplanarCartesianposition,velocity,andacceleration oftherobotend-effectorattheneedletip,respectively, M x 2 R denotestheuncertaininertia, h x 2 R denotesuncertainconservativeforces, f needle x ; x 2 R ,introducedinChapter2, denotestheinteractionforcebetweentherobotattheneedletipandtheviscoelastictissueduring theneedleinsertionprocedure,and F t 2 R denotestheforcecontrolinput. Remark. Thisstudyhasbeendevelopedforaone-degree-of-freedomtranslationrobotbutcould beextendedtotheresolutionofaredundancymanipulatorsproblem[63]. Thefollowingpropertyandassumptionsareappliedinthecontroldevelopment. Property1. Thefollowingrelationshipsarevalidforall x 2 R [64]: x tanh x tanh x 2 ; j tanh x j 1 : Assumption3.1. Therobot,tissue,andmaximaltissuesurfacepositions, x t x t t ,and x m t ,introducedinChapter2,andthecorrespondingvelocities, x t and x t t ,aremeasurable. Further,itisassumedthattherobottrajectory x t isboundedduetothegeometryoftherobot. 18

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Remark 3.1 Thepositionofthemaximallydeformedtissuesurfacebeforepuncture x m t canbe measuredusingthetechniquedescribedin[65]. Assumption3.2. Thelocaldeformationoftheviscoelasticmaterialduringcontact d x ; x t denedin2,isassumedtobebounded;hence, d n canbeupperboundedas d n d n ; where d n 2 R isaknownpositiveboundingconstant. Assumption3.3. Thedampingconstant m ,in2,isassumedtobeupperboundedas m m ; where m 2 R isaknownpositiveboundingconstant. 3.2ControlDevelopment 3.2.1ControlObjective Thecontrolobjectiveistoensurethattheone-degree-of-freedomtranslationrobottracks adesiredposition,denotedby x d t 2 R ,whichbeginsinfreespaceandendswithinthe viscoelastictissue.Thecontrollerisdesignedsuchthattheforcerequiredtoachievethis objectiveisboundedbyanarbitrarysmallvalue,whichisdesiredforproceduralsafety.A positiontrackingerrorandalteredtrackingerroraredesignedtoquantifythecontrolobjective as e x d )]TJ/F54 11.9552 Tf 10.949 0 Td [(x ; r e + a e ; where e t 2 R representsthepositionerrorattheneedletip, r t 2 R isalteredtrackingerror thatfacilitatesthesubsequentcontroldevelopment,and a 2 R isapositiveconstantcontrolgain. 19

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3.2.2Closed-LoopErrorSystem Premultiplyingthelteredtrackingerror r t in3bytherobotinertiamatrix M x takingthetimederivativeoftheresultingexpression,andusing3and3yieldsthe followingopen-looproboterrorsystem: M r = M e + M x d + h + f needle )]TJ/F54 11.9552 Tf 10.95 0 Td [(F + M a e + M a e )]TJ/F23 11.9552 Tf 15.372 2.379 Td [( Mr : Usingthedenitionoftheneedleforce f needle denedin2,2,2,and2,the expressionin3becomes M r = f + M e + L 2 g 1 tanh g 2 x )]TJ/F23 11.9552 Tf 10.949 0 Td [(tanh g 3 x + L 2 g 4 tanh g 5 x + L 2 g 6 x + L 2 c )]TJ/F54 11.9552 Tf 10.949 0 Td [(F + M a e )]TJ/F23 11.9552 Tf 15.373 2.379 Td [( Mr ; where f t 2 R isanauxiliarynonlinearanddiscontinuousfunctiondenedas f M x d + M a e + h + L 1 ld n + m dd n : Basedontheuniversalfunctionapproximationpropertyandresultsfrom[66]forapproximation ofjumpfunctions,thediscontinuousfunction f t in3canbeapproximatedbyathree-layer input,hidden,andoutputneuralnetworkNNas f = W T 1 s )]TJ/F54 11.9552 Tf 4.878 -9.69 Td [(V T 1 y + W T 2 j )]TJ/F54 11.9552 Tf 4.877 -9.69 Td [(V T 2 y + e y ; wheretheNNinput y t isdenedas y t = 1 x t xer dd d T 2 R 7 W 1 ; W 2 2 R N + 1 and V 1 ; V 2 2 R 7 N areidealNNweights, N 2 R isthenumberofhiddenlayerneuronsof theNN, s )]TJ/F54 11.9552 Tf 4.877 -9.689 Td [(V T 1 y = s 2 R N + 1 isasigmoidactivationfunction, j )]TJ/F54 11.9552 Tf 4.878 -9.689 Td [(V T 2 y = j 2 R N + 1 isasigmoid jumpapproximationfunction,and e y 2 R isthefunctionalreconstructionerroroftheNN. Theweights V 2 areknown,givenbythedesigneranddependingonthelocationofthejumps. Figure3-1showstheaugmentedmultilayerneuralnetworkforjumpfunctionapproximation. 20

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Figure3-1.Multilayerneuralnetworkforjumpfunctionapproximation. Thesubsequentstabilityanalysisindicatesthat,providedsomesufcientgainconditionsare satised,if y 0 isinacompactset,then y t remainsinacompactset 8 t Property2. BoundednessoftheIdealWeightsTheidealweightsareassumedtoexistandto beboundedbyknownpositivevaluessothat k V i k 2 F = tr V T i V i V iB ; k W i k 2 F = tr W T i W i W iB ; where i = 1 ; 2, V iB and W iB arepositiveconstants, k k F istheFrobeniusnormofamatrix,and tr isthetraceofamatrix. Theestimatefor f t ,denotedas f t 2 R ,isdenedas f W T 1 s )]TJ/F23 11.9552 Tf 7.597 -7.311 Td [( V T 1 y + W T 2 j )]TJ/F54 11.9552 Tf 4.878 -9.69 Td [(V T 2 y ; where W 1 t ; W 2 t 2 R N + 1 and V 1 t 2 R 7 N aretheestimatesoftheidealweightsandare generatedbyintegratingtheadaptiveupdatelaws 21

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W 1 = proj )]TJ/F55 11.9552 Tf 5.476 -9.69 Td [(G w 1 s r )]TJ/F55 11.9552 Tf 10.949 0 Td [(G w 1 s 0 V T 1 yr ; V 1 = proj )]TJ/F55 11.9552 Tf 5.476 -9.69 Td [(G v 1 yr W T 1 s 0 ; W 2 = proj G w 2 j r ; where G w 1 ; G w 2 2 R N + 1 N + 1 and G v 1 2 R 7 7 areconstant,positivedenite,diagonal, gainmatrices, s 0 2 R N + 1 N denotesthepartialderivativeof s = s )]TJ/F23 11.9552 Tf 7.598 -7.311 Td [( V T 1 y withrespectto itsargument,and proj denotesasmoothprojectionoperator[67,68].Basedonthefact that W 1 t and W 2 t areboundedbytheprojectionoperator,and s and j arebounded activationfunctions,then f t canbeupperboundedas f k ; where k 2 R isaknownpositiveconstant. Basedon3andthesubsequentstabilityanalysis,therobotcontrolforceinputis designedas F = f + k p tanh w e + b sgn r ; where k p ; w ; b 2 R arepositiveconstantcontrolgains.Thesmoothsaturationfunction tanh in3isusedtosaturatethetermsinthecontrollertolimitthecontrolforceduringcontact andpenetration.Using3,3,andtheNNprojectionboundsin[64],thecontrolforcein 3canbeboundedas j F j k + k p + b : Using3,3,and3,theexpressionin3canberewrittenas 22

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M r = W T 1 s + W T 2 j + e y )]TJ/F23 11.9552 Tf 14.375 2.379 Td [( W T 1 s )]TJ/F23 11.9552 Tf 14.375 2.379 Td [( W T 2 j )]TJ/F54 11.9552 Tf 10.949 0 Td [(k p tanh w e )]TJ/F52 11.9552 Tf 10.949 0 Td [(b sgn r + M e + L 2 g 1 tanh g 2 x )]TJ/F23 11.9552 Tf 10.949 0 Td [(tanh g 3 x + L 2 g 4 tanh g 5 x + L 2 g 6 x + L 2 c + M a e )]TJ/F23 11.9552 Tf 15.373 2.379 Td [( Mr : UsingtheTaylorseriesexpansion[66],theterm s = s )]TJ/F23 11.9552 Tf 13.508 0.52 Td [( s canbewrittenas s = s 0 V 1 T y + O V 1 T y 2 ; where W 1 t 2 R N + 1 and V 1 t 2 R 7 N areestimateerrorsoftheidealweightsandaredened as W 1 = W 1 )]TJ/F23 11.9552 Tf 14.374 2.379 Td [( W 1 ; V 1 = V 1 )]TJ/F23 11.9552 Tf 13.072 2.379 Td [( V 1 ; W 2 = W 2 )]TJ/F23 11.9552 Tf 14.375 2.379 Td [( W 2 : Aftersomealgebraicmanipulations,theexpressionin3canbeexpressedas M r = W T 1 s + W T 1 s 0 V T 1 y )]TJ/F23 11.9552 Tf 14.375 2.379 Td [( W T 1 s 0 V T 1 y + W T 2 j + D )]TJ/F54 11.9552 Tf 10.95 0 Td [(k p tanh w e )]TJ/F52 11.9552 Tf 10.949 0 Td [(b sgn r )]TJ/F23 11.9552 Tf 12.145 8.094 Td [(1 2 Mr )]TJ/F54 11.9552 Tf 10.949 0 Td [(kr )]TJ/F54 11.9552 Tf 10.949 0 Td [(e ; where k 2 R isapositiveconstant,thestatevector z 2 R 2 isdenedas z e ; r e t r t T and D z 2 R isdenedas D = W T 1 s 0 V T 1 y + W T 1 O )]TJ/F23 11.9552 Tf 7.597 -7.311 Td [( V T 1 y 2 + e y + L 2 g 1 tanh g 2 x )]TJ/F23 11.9552 Tf 10.949 0 Td [(tanh g 3 x + L 2 g 6 x + M e )]TJ/F23 11.9552 Tf 12.145 8.093 Td [(1 2 Mr + L 2 g 4 tanh g 5 x + L 2 c + M a e + kr + e : Using3,3,and[69],anupperboundfor D z canbedeterminedas j D j z + r k z k k z k ; 23

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where r 2 R isapositive,globallyinvertiblefunction,and z 2 R isaknownpositiveconstant. 3.3StabilityAnalysis Theorem3.1. Thecontrollergivenin3ensuressemi-globaltrackinginthesensethat e t 0 ast ; providedcontrolgainsareselectedsufcientlylargeseethesubsequentstabilityanalysis. Proof. Let D R 3 beadomaincontaining v t = 0,where v t 2 R 3 isdenedas v t z T t p Q t T ; andtheauxiliaryfunction Q t 2 R isdenedas Q t 1 2 tr )]TJ/F23 11.9552 Tf 7.598 -7.31 Td [( V T 1 G )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 v 1 V 1 + 1 2 tr )]TJ/F23 11.9552 Tf 8.901 -7.31 Td [( W T 1 G )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 w 1 W 1 + 1 2 tr )]TJ/F23 11.9552 Tf 8.901 -7.31 Td [( W T 2 G )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 w 2 W 2 : Since G 1 v G w 1 and G w 2 areconstant,symmetric,andpositivedenitematrices,itisstraightforwardthat Q t 0. Let V v ; t : D [ 0 ; R beaLipschitzcontinuousregularpositivedenitefunction denedas V 1 2 Mr 2 + 1 2 e 2 + k p w ln cosh w e + Q ; whichsatisesthefollowinginequalities: U 1 v V v ; t U 2 v ; wherethecontinuouspositivedenitefunctions U 1 v ; U 2 v 2 R aredenedas U 1 v h 1 k v k 2 ; U 2 v h 2 k v k 2 ; where h 1 ; h 2 2 R areknownpositiveconstants. 24

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Thedifferentialequationsoftheclosedloopdynamicsgivenin3arecontinuousexcept insets f v j x = x t g and f v j r = 0 g .UsingFilippov'sdifferentialinclusion[70],theexistenceof solutionscanbeestablishedfor v = f v ,where f v 2 R 3 denotestheright-handsideofthe closed-looperrorsignals.UnderFilippov'sframework,ageneralizedLyapunovstabilitytheory canbeusedtoestablishstrongstabilityoftheclosed-looperrorsystem.Thegeneralizedtime derivativeof3existsalmosteverywherea.e.,and V v 2 a : e : V v where V = x 2 V v K r e 1 2 Q )]TJ/F23 6.9738 Tf 8.162 3.532 Td [(1 2 Q T ; where V isthegeneralizedgradientof V v [71], K [ ] isdenedas[72,73] K [ f ] d > 0 m = 0 cof B x ; d )]TJ/F55 11.9552 Tf 10.949 0 Td [( ; where m = 0 denotestheintersectionofallsets ofLebesguemeasurezero, co denotesconvex closure,and B x ; d = u 2 R 3 jk u )]TJ/F54 11.9552 Tf 10.949 0 Td [(v k < d .Since V v isaLipschitzcontinuousregular function V = V T K r e 1 2 Q )]TJ/F23 6.9738 Tf 8.163 3.533 Td [(1 2 Q T Mre k p w tanh w e 2 Q 1 2 T K r e 1 2 Q )]TJ/F23 6.9738 Tf 8.162 3.533 Td [(1 2 Q T : Using3,3,and3,theexpressionin3becomes V r D )]TJ/F52 11.9552 Tf 10.949 0 Td [(b j r j )]TJ/F54 11.9552 Tf 10.95 0 Td [(kr 2 )]TJ/F52 11.9552 Tf 10.949 0 Td [(a e 2 )]TJ/F54 11.9552 Tf 10.651 0 Td [(tanh w e k p a e : Using3and3,theexpressionin3canbeupperboundedas V a : e : r k z k j r jk z k )]TJ/F56 11.9552 Tf 10.949 0 Td [( b )]TJ/F52 11.9552 Tf 10.949 0 Td [(z j r j )]TJ/F54 11.9552 Tf 10.949 0 Td [(kr 2 )]TJ/F52 11.9552 Tf 10.95 0 Td [(a e 2 )]TJ/F54 11.9552 Tf 12.145 8.284 Td [(k p a w j tanh w e j 2 : Letthecontrolgain k in3bedenedas k k 1 + k 2 ; 25

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where k 1 ; k 2 2 R areknownpositiveconstants.Using3andthegaincondition b > z ; theexpressionin3canbeupperboundedas V a : e : )]TJ/F66 11.9552 Tf 24.567 9.69 Td [()]TJ/F54 11.9552 Tf 5.475 -9.69 Td [(k 1 r 2 )]TJ/F52 11.9552 Tf 10.95 0 Td [(r k z k r k z k )]TJ/F54 11.9552 Tf 10.949 0 Td [(k 2 r 2 )]TJ/F52 11.9552 Tf 10.949 0 Td [(a e 2 : Completingthesquaresontheterminparenthesesin3yields V a : e : r k z k 2 4 k 1 k z k 2 )]TJ/F54 11.9552 Tf 10.949 0 Td [(k 2 r 2 )]TJ/F52 11.9552 Tf 10.949 0 Td [(a e 2 : Theexpressionin3canbefurtherupperboundedas V a : e : )]TJ/F52 11.9552 Tf 23.239 0 Td [(l k z k 2 + r k z k 2 4 k 1 k z k 2 ; where l = min f k 2 ; a g isaknownpositiveconstant.Finally,giventhegaincondition l > r k z k 2 4 k 1 ; theexpressionin3becomes V a : e : )]TJ/F54 11.9552 Tf 22.641 0 Td [(U v ; where U v = J k z k 2 ,forsomepositiveconstant J 2 R isacontinuouspositivesemi-denite functionsuchthat D n v 2 R 3 j k v k r )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 2 p l k 1 o : Theexpressionsin3and3canbeusedtoshowthat V v ; t 2 L ;hence, e t ; r t and Q t 2 L in D .Giventhat e t ; r t 2 L in D ,itcanbeproventhat e t 2 L in D from3.Since e t ; r t 2 L in D ,theassumptionthat x d t ; x d t existandarebounded 26

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canbeusedtoconcludethat x t ; x t 2 L in D .Similarly,itcanbeshownthat r t 2 L in D .Since e t ; r t 2 L in D ,thedenitionsfor U v and z t canbeusedtoprovethat U v isuniformlycontinuousin D Let S D denotesasetdenedasfollows: S v t D U 2 v t < h 1 r )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 2 p l k 1 2 : [74]cannowbeinvokedtostatethat k z t k 2 0 ast 8 v 0 2 S : Basedonthedenitionof v t in3,3canbeusedtoshowthat j e t j 0 ast 8 v 0 2 S : 3.4SimulationResults Thedevelopedcontrollerissimulatedforasystemwhosedynamicmodelisgivenby m x + b x + f needle x ; x = F ; where F t and f needle x ; x areintroducedin3, m = 0 : 152 kg b = 1 : 426 N s m )]TJ/F23 8.9664 Tf 6.967 0 Td [(1 ,which correspondtotheneedleinsertionrobotdescribedin[75].Thedifferentpositionandparameter valuesarechosentoagreewithadirectinsertionintotheliver.Theinitialneedletippositionis supposedtobeat x = 0for t = 0.Forsakeofsimplicity,itisassumedthatthetissueposition x t doesnotdependontimeanditsvalueisxedto x t = 20 mm .Thepositionofthemaximally deformedtissuesurfacebeforepunctureischosenas x m = 36 mm ,whichmeansthattheneedle progresses16mmwhileincontactwiththeliverbeforethepunctureoccurs.Thedesiredposition ischosenas x d = 60 mm ,whichcorrespondto40 mm intotheliver.Figure3-2showsthechoice ofthedifferentpositionsforthatsimulation.Theparametersfor f stiffness x ; x f friction x ,and 27

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Figure3-2.Positionsforthesimulation f cutting ,introducedinChapter2,arechosenas l = 0 : 2 N m )]TJ/F23 8.9664 Tf 6.967 0 Td [(1 ; m = 5 : 5 N s m )]TJ/F23 8.9664 Tf 6.967 0 Td [(2 ; n = 1 : 5 ; g 1 = g 4 = 0 : 1 N ; g 2 = g 3 = g 5 = 0 : 2 s m )]TJ/F23 8.9664 Tf 6.967 0 Td [(1 ; g 6 = 0 : 5 N s m )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 ; c = 0 : 94 N : Theparametersin3areselectedusingresultsfromexperimentsonliver[48,65].The controllergainsintroducedin3andthecontrolgain a introducedin3areselectedas k p = 5 ; w = 1 ; b = 2 ; a = 10 : ThenumberofhiddenlayerneuronsfortheNNischosenas N = 15,andtheNNweight updationgainsareselectedas G w 1 = G w 2 = 5 I 16 ; G v 1 = 5 I 5 ; where I p 2 R p p denotestheidentitymatrix. Figure3-3showsthepositionoftheneedletip x t ,whichasymptoticallyapproaches thedesiredposition x d = 60 mm .Then,theerrorgoestozeroastimegoestoinnityasshown inFigure3-4.Duringtherststage,between0and20 mm orbetween0and54 ms ,theforce betweentheneedleandthetissueisequaltozerobecausetheneedledoesnottouchthetissue yetasitcanbeseenonFigures3-5and3-6.Then,between20 mm andthemaximallydeformed 28

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Figure3-3.Positionoftheneedletip x t tissueat36 mm ,theneedleforceincreases,astheneedlecontactsthetissue;theneedleforce isthenequaltotheHunt-Crossleyforce.Themaximumforce : 6 N isfollowedbyasudden dropinforceastheneedlepuncturesthetissueandnowonlyneedstoovercomethefrictionand cuttingforces,whicharesmallerthanthetissuestiffnessforce.Thelaststageistheinsertion throughthetissuetoreachthetarget. 29

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Figure3-4.Positiontrackingerror e t Figure3-5.Needleforce f needle asafunctionoftime. 30

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Figure3-6.Needleforce f needle asafunctionoftheneedletipposition x t 31

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CHAPTER4 TELEOPERATEDROBOTFORNEEDLEINSERTIONINTOVISCOELASTICTISSUE Thischapterdescribesthedevelopmentofacontrollertoensurethataneedletipmounted onaslaverobottracksthetrajectorygivenbythesurgeonmanipulatingthemasterrobot.The trajectorymovesfromanon-contactpositionintoviscoelastictissue. 4.1DynamicModel Thedynamicmodelforaone-degree-of-freedomtranslationmasterandaone-degree-offreedomtranslationslaverobotisdescribedby g T 1 + F 1 = g M 1 q 1 q 1 + h 1 q 1 ; T 2 )]TJ/F54 11.9552 Tf 10.95 0 Td [(F 2 = M 2 q 2 q 2 + h 2 q 2 : In4and4, g 2 R denotesapositiveadjustablepowerscalingterm, q i t ; q i t ; q i t 2 R denotetherobotend-effectorposition,velocity,andacceleration,respectively, 8 i = 1 ; 2where i = 1denotesthemastermanipulatorand i = 2denotestheslavemanipulator, M i q i 2 R denotestheinertia, h i q i 2 R denotesconservativeforces, T i t 2 R denotestheforcecontrol input, F 1 t 2 R denotestheuserinputforce,and F 2 t 2 R denotestheforceinputfrom theenvironment,i.e.,theinteractionforcebetweentherobotandthetissueduringtheneedle insertion.Theforce F 2 t isdiscontinuousbecauseofthetransitionbetweenneedle-tissue contactandinsertionthroughthetissue. Assumption4.1. Theposition q i t andthevelocity q i t aremeasurable. Assumption4.2. Theuserforce F 1 t andtheenvironmentforce F 2 t arebounded. Assumption4.3. Thedynamicmodelsofthetworobotsareknown. 32

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4.2ControlDevelopment 4.2.1ControlObjectiveandModelTransformation Therstcontrolobjectiveistoensurethattheslaverobottracksthemasterrobotposition, whichgoesfromafree-spacepositionintoaviscoelastictissue,inthefollowingsense: q 2 t q 1 t ast : Energeticpassivityisimportanttoensuretherobotinteractswiththetissueinastableandsafe manner.Then,theotherobjectiveistoensurethatthesystemremainspassivewithrespecttothe scaleduserandenvironmentalpowerinthesensethat[76] t t 0 g q 1 t F 1 t )]TJ/F23 11.9552 Tf 12.844 0 Td [( q 2 F 2 t d t )]TJ/F54 11.9552 Tf 21.235 0 Td [(c ; where c 2 R isapositiveconstantwhichdependsontheinitialcondition,and g wasintroduced in4.Theequationin4meansthattheenergyproducesbytheslaverobotcannotbe biggerthanthesumoftheenergyfromthemasterrobotandtheinitialenergyinthesystem.An auxiliarycontrolobjectiveisemployedtoensurethepassivityobjectiveandforcereection,in thesensethat[41] q 1 t + q 2 t x d 2 t ast ; where x d t = x d 1 t x d 2 t T 2 R 2 isadesiredboundedtrajectory. Tofacilitatethesubsequentdevelopment,agloballyinvertibletransformationisdenedthat encodesboththecoordinationandthepassivityobjectives,i.e., x Sq + 2 6 4 x d 1 0 3 7 5 ; where x t x 1 t x 2 t T 2 R 2 q t q 1 t q 2 t T 2 R 2 ,and S 2 R 2 2 isdened asfollows: S 2 6 4 1 )]TJ/F23 11.9552 Tf 9.289 0 Td [(1 11 3 7 5 ; S )]TJ/F23 8.9664 Tf 6.967 0 Td [(1 1 2 2 6 4 11 )]TJ/F23 11.9552 Tf 9.289 0 Td [(11 3 7 5 : 33

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Basedon4,thedynamicmodelgivenin4and4canbeexpressedas M x x )]TJ/F23 11.9552 Tf 15.373 2.379 Td [( M x 2 6 4 x d 1 0 3 7 5 + h x = T t + F t ; where M x S )]TJ/F54 8.9664 Tf 6.967 0 Td [(T 2 6 4 g M 1 0 0 M 2 3 7 5 S )]TJ/F23 8.9664 Tf 6.967 0 Td [(1 2 R 2 2 ; h x S )]TJ/F54 8.9664 Tf 6.967 0 Td [(T 2 6 4 g h 1 h 2 3 7 5 2 R 2 ; T t S )]TJ/F54 8.9664 Tf 6.967 0 Td [(T 2 6 4 g T 1 T 2 3 7 5 2 R 2 ; F t 2 6 4 F 1 F 2 3 7 5 = S )]TJ/F54 8.9664 Tf 6.967 0 Td [(T 2 6 4 g F 1 )]TJ/F54 11.9552 Tf 9.289 0 Td [(F 2 3 7 5 2 R 2 : Property3. Thesubsequentdevelopmentisbasedonthepropertythat M x ,denedin4,is apositivedeniteandsymmetricmatrixinthesensethat m 1 k x k 2 x T M x x m 2 k x k 2 ; where x 2 R 2 ,and m 1 ; m 2 2 R arepositiveconstants. Apositiontrackingerror e 1 t 2 R 2 andalteredtrackingerror e 2 t 2 R 2 aredesignedto quantifythecontrolobjectiveas e 1 x )]TJ/F54 11.9552 Tf 10.95 0 Td [(x d ; e 2 e 1 + a 1 e 1 ; where a 1 2 R isapositiveconstantcontrolgain,and x d t 2 R 2 isintroducedin4.Based onthedenitionof x t in4and e 1 t in4,itisclearthatif k e 1 k 0as t then 34

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q 2 t q 1 t and q 1 t + q 2 t x d 2 t as t .Toensurethatthesystemremainspassive asdenedin4,thedesiredtrajectory x d t isgeneratedbythefollowingexpression M x d + B T x d + K T x d + 1 2 M x d = F ; where B T ; K T 2 R representpositiveconstants, M 2 R 2 2 isintroducedin4,and F 2 R 2 isa subsequentlydesignedforceestimator. 4.2.2Closed-LoopErrorSystem Premultiplyingthesecondtimederivativeofthetrackingerror e 1 t in4bytherobot inertiamatrix M x ,andusingthesystemdynamics4andthedesiredtrajectorydynamics 4,theopen-looproboterrorsystemcanbewrittenas M e 1 = T + F + M 2 6 4 x d 1 0 3 7 5 )]TJ/F23 11.9552 Tf 11.948 2.851 Td [( h )]TJ/F23 11.9552 Tf 14.01 2.379 Td [( F + B T x d + K T x d + 1 2 M x d : Basedontheassumptionofexactmodelknowledgeoftherobotdynamicsandthesubsequent stabilityanalysis,therobotcontrolinput T t isdesignedas T )]TJ/F23 11.9552 Tf 13.712 2.379 Td [( M 2 6 4 x d 1 0 3 7 5 + h )]TJ/F54 11.9552 Tf 10.95 0 Td [(B T x d )]TJ/F54 11.9552 Tf 10.95 0 Td [(K T x d )]TJ/F23 11.9552 Tf 12.145 8.094 Td [(1 2 M x d )]TJ/F23 11.9552 Tf 15.373 2.379 Td [( M a 1 e 1 )]TJ/F52 11.9552 Tf 10.949 0 Td [(b sgn e 2 )]TJ/F23 11.9552 Tf 12.145 8.093 Td [(1 2 e T 2 Me 2 ; where b 2 R isapositiveconstantcontrolgain.Using4,4canbewrittenas M e 1 = F )]TJ/F23 11.9552 Tf 14.01 2.379 Td [( F )]TJ/F23 11.9552 Tf 15.373 2.379 Td [( M a 1 e 1 )]TJ/F52 11.9552 Tf 10.949 0 Td [(b sgn e 2 )]TJ/F23 11.9552 Tf 12.145 8.093 Td [(1 2 e T 2 Me 2 : Using4andthetimederivativeofthelteredtrackingerror e 2 t ,4becomes M e 2 = F )]TJ/F23 11.9552 Tf 14.01 2.379 Td [( F )]TJ/F52 11.9552 Tf 10.95 0 Td [(b sgn e 2 )]TJ/F23 11.9552 Tf 12.145 8.094 Td [(1 2 e T 2 Me 2 : 35

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Basedontheuniversalfunctionapproximationpropertyandresultsfrom[66]forapproximation ofjumpfunctions,thediscontinuousforce F t ,denedin4,canbeapproximatedbya three-layerinput,hidden,andoutputneuralnetworkNNas F = W T 1 s )]TJ/F54 11.9552 Tf 4.878 -9.689 Td [(V T 1 y + W T 2 j )]TJ/F54 11.9552 Tf 4.878 -9.689 Td [(V T 2 y + e y ; wheretheNNinput y t isdenedas y t = 1 x T e T 1 e T 2 T 2 R 7 W 1 ; W 2 2 R N + 1 2 and V 1 ; V 2 2 R 7 N areidealNNweights, N 2 R isthenumberofhiddenlayerneuronsofthe NN, s )]TJ/F54 11.9552 Tf 4.878 -9.69 Td [(V T 1 y = s 2 R N + 1 isasigmoidactivationfunction, j )]TJ/F54 11.9552 Tf 4.878 -9.69 Td [(V T 2 y = j 2 R N + 1 isasigmoid jumpapproximationfunction,and e y 2 R 2 isthefunctionalreconstructionerroroftheNN. Theweights V 2 areknown,givenbythedesigneranddependonthelocationofthejumps.The subsequentstabilityanalysisindicatesthat,providedsomesufcientgainconditionsaresatised, if y 0 isinacompactset,then y t remainsinacompactset 8 t Property4. BoundednessoftheIdealWeightsTheidealweightsareassumedtoexistandto beboundedbyknownpositivevaluessothat k V i k 2 F = tr V T i V i V iB ; k W i k 2 F = tr W T i W i W iB ; where V iB and W iB arepositiveconstantsfor i = 1 ; 2, k k F istheFrobeniusnormofamatrix,and tr isthetraceofamatrix. Theestimatefor F t ,denotedas F t 2 R 2 ,isdenedas F W T 1 s )]TJ/F23 11.9552 Tf 7.597 -7.311 Td [( V T 1 y + W T 2 j )]TJ/F54 11.9552 Tf 4.878 -9.69 Td [(V T 2 y ; where W 1 t ; W 2 t 2 R N + 1 2 and V 1 t 2 R 7 N aretheestimatesoftheidealweightsandare generatedbyintegratingtheadaptiveupdatelaws 36

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W 1 proj )]TJ/F55 11.9552 Tf 5.476 -9.69 Td [(G w 1 s e T 2 )]TJ/F55 11.9552 Tf 10.949 0 Td [(G w 1 s 0 V T 1 ye T 2 ; V 1 proj )]TJ/F55 11.9552 Tf 5.476 -9.69 Td [(G v 1 ye T 2 W T 1 s 0 ; W 2 proj )]TJ/F55 11.9552 Tf 5.476 -9.689 Td [(G w 2 j e T 2 ; where G w 1 ; G w 2 2 R N + 1 N + 1 and G v 1 2 R 7 7 areconstant,positivedenite,diagonal, gainmatrices; s 0 2 R N + 1 N denotesthepartialderivativeof s = s )]TJ/F23 11.9552 Tf 7.597 -7.311 Td [( V T 1 y withrespectto itsargument,and proj denotesasmoothprojectionoperator[67,68].Basedonthefact that W 1 t and W 2 t areboundedbytheprojectionoperator,and s and j arebounded activationfunctions,then F t canbeupperboundedas F k ; where k 2 R isaknownpositiveconstant.Using4,4,and4,theexpressionin 4canberewrittenas M e 2 = W T 1 s + W T 2 j + e y )]TJ/F23 11.9552 Tf 14.375 2.379 Td [( W T 1 s )]TJ/F23 11.9552 Tf 14.375 2.379 Td [( W T 2 j )]TJ/F52 11.9552 Tf 10.949 0 Td [(b sgn e 2 )]TJ/F23 11.9552 Tf 12.145 8.093 Td [(1 2 e T 2 Me 2 : Theestimateerrorsoftheidealweights W 1 t 2 R N + 1 2 V 1 t 2 R 7 N ,and W 2 2 R N + 1 2 are denedas W 1 = W 1 )]TJ/F23 11.9552 Tf 14.374 2.379 Td [( W 1 ; V 1 = V 1 )]TJ/F23 11.9552 Tf 13.071 2.379 Td [( V 1 ; W 2 = W 2 )]TJ/F23 11.9552 Tf 14.375 2.379 Td [( W 2 : UsingtheTaylorseriesexpansion[66],theestimateerroroftheactivation s 2 R N + 1 ,denedas s = s )]TJ/F23 11.9552 Tf 13.507 0.52 Td [( s ,canbewrittenas s = s 0 V 1 T y + O V 1 T y 2 : Using4andtheexpressionin4,theclosed-looperrorsystemcanbewrittenas 37

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M e 2 = W T 1 s + W T 1 s 0 V T 1 y )]TJ/F23 11.9552 Tf 14.375 2.379 Td [( W T 1 s 0 V T 1 y + W 2 T j + D )]TJ/F52 11.9552 Tf 10.95 0 Td [(b sgn e 2 )]TJ/F23 11.9552 Tf 12.145 8.094 Td [(1 2 e T 2 Me 2 )]TJ/F54 11.9552 Tf 10.95 0 Td [(ke 2 )]TJ/F54 11.9552 Tf 10.95 0 Td [(e 1 ; aftersomealgebraicmanipulations.Thestatevector z 2 R 4 isdenedas z e 1 ; e 2 e 1 t e 2 t T In4, k 2 R isapositiveconstant,and D z 2 R 2 isdenedas D W T 1 s 0 V T 1 y + W T 1 O V 1 T y 2 + e y + ke 2 + e 1 : Using4,4,and[69],anupperboundfor D z canbedeterminedas k D k z + r k z k k z k ; where r 2 R isapositive,globallyinvertible,nondecreasingfunction,and z 2 R isaknown positiveconstant. 4.3StabilityAnalysis Theorem4.1. Thecontrollergivenin4ensuressemi-globalasymptotictrackinginthe sensethat q 2 t q 1 t ast ; providedcontrolgainsareselectedsufcientlylargeseethesubsequentstabilityanalysis. Proof. Let D R 5 beadomaincontaining v t = 0,where v t 2 R 5 isdenedas v t z T t p Q t T ; andtheauxiliaryfunction Q t 2 R isdenedas Q t 1 2 tr )]TJ/F23 11.9552 Tf 7.598 -7.311 Td [( V T 1 G )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 v 1 V 1 + 1 2 tr )]TJ/F23 11.9552 Tf 8.901 -7.311 Td [( W T 1 G )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 w 1 W 1 + 1 2 tr )]TJ/F23 11.9552 Tf 8.901 -7.311 Td [( W T 2 G )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 w 2 W 2 : 38

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Since G 1 v G w 1 ,and G w 2 areconstant,symmetric,andpositivedenitematrices,itisstraightforwardthat Q t 0. Let V v ; t : D [ 0 ; R beaLypschitzcontinuous,regular,positivedenitefunction denedas V 1 2 e T 2 Me 2 + 1 2 e T 1 e 1 + Q ; whichsatisesthefollowinginequalities: U 1 v V v ; t U 2 v ; wherethecontinuouspositivedenitefunctions U 1 v ; U 2 v 2 R aredenedas U 1 v h 1 k v k 2 ; U 2 v h 2 k v k 2 ; where h 1 ; h 2 2 R areknownpositiveconstants. Thedifferentialequationsoftheclosedloopdynamicsgivenin4arecontinuousexcept insets f v j x = x t g and f v j e 2 = 0 g .UsingFilippov'sdifferentialinclusion[70],theexistenceof solutionscanbeestablishedfor v = f v ,where f v 2 R 5 denotestheright-handsideofthe closed-looperrorsignals.UnderFilippov'sframework,ageneralizedLyapunovstabilitytheory canbeusedtoestablishstrongstabilityoftheclosed-looperrorsystem.Thegeneralizedtime derivativeof4existsalmosteverywherea.e.,and V v 2 a : e : V v where V = x 2 V K e 2 e 1 1 2 Q )]TJ/F23 6.9738 Tf 8.163 3.533 Td [(1 2 Q T ; where V isthegeneralizedgradientof V v [71], K [ ] isdenedin[72]and[73]as K [ f ] d > 0 m = 0 cof B x ; d )]TJ/F55 11.9552 Tf 10.949 0 Td [( ; where m = 0 denotestheintersectionofallsets ofLebesguemeasurezero, co denotesconvex closure,and B x ; d = u 2 R 3 jk u )]TJ/F54 11.9552 Tf 10.949 0 Td [(v k < d .Since V v isaLipschitzcontinuousregular 39

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function, V = V T K e 2 e 1 1 2 Q )]TJ/F23 6.9738 Tf 8.163 3.532 Td [(1 2 Q T ; Me 2 e 1 2 Q 1 2 T K e 2 e 1 1 2 Q )]TJ/F23 6.9738 Tf 8.162 3.533 Td [(1 2 Q T : Using4,4,and4,theexpressionin4becomes V e T 2 D )]TJ/F52 11.9552 Tf 10.949 0 Td [(b k e 2 k )]TJ/F54 11.9552 Tf 10.95 0 Td [(ke T 2 e 2 )]TJ/F52 11.9552 Tf 10.95 0 Td [(a 1 e T 1 e 1 : Using4,theexpressionin4canbeupperboundedas V a : e : r k z k k e 2 kk z k )]TJ/F56 11.9552 Tf 10.949 0 Td [( b )]TJ/F52 11.9552 Tf 10.95 0 Td [(z k e 2 k )]TJ/F54 11.9552 Tf 10.95 0 Td [(k k e 2 k 2 )]TJ/F52 11.9552 Tf 10.95 0 Td [(a 1 k e 1 k 2 : Letthecontrolgain k in4bedenedas k k 1 + k 2 ; where k 1 ; k 2 2 R areknownpositiveconstants.Using4andthegaincondition b > z ; theexpressionin4canbeupperboundedas V a : e : )]TJ/F66 11.9552 Tf 24.567 13.277 Td [( k 1 k e 2 k 2 )]TJ/F52 11.9552 Tf 10.949 0 Td [(r k z k k e 2 kk z k )]TJ/F54 11.9552 Tf 10.95 0 Td [(k 2 k e 2 k 2 )]TJ/F52 11.9552 Tf 10.95 0 Td [(a 1 k e 1 k 2 : Completingthesquaresontheterminparenthesesin4yields V a : e : r k z k 2 4 k 1 k z k 2 )]TJ/F54 11.9552 Tf 10.949 0 Td [(k 2 k e 2 k 2 )]TJ/F52 11.9552 Tf 10.95 0 Td [(a 1 k e 1 k 2 : Theexpressionin4canbefurtherupperboundedas 40

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V a : e : )]TJ/F52 11.9552 Tf 23.239 0 Td [(l k z k 2 + r k z k 2 4 k 1 k z k 2 ; where l = min f k 2 ; a 1 g isaknownpositiveconstant.Finally,giventhegaincondition l > r k z k 2 4 k 1 ; theexpressionin4becomes V a : e : )]TJ/F54 11.9552 Tf 22.641 0 Td [(U v ; where U v = m k z k 2 ,forsomepositiveconstant m 2 R isacontinuouspositivesemi-denite functioninthedomain D n v 2 R 5 j k v k r )]TJ/F23 8.9664 Tf 6.966 0 Td [(1 2 p l k 1 o : Theexpressionsin4and4canbeusedtoshowthat V v ; t 2 L ;hence, e 1 t ; e 2 t and Q t 2 L in D .Giventhat e 1 t ; e 2 t 2 L in D ,itcanbeproventhat e 1 t 2 L in D from4.Since e 1 t ; e 2 t 2 L in D ,theassumptionthat x d t ; x d t existandare boundedcanbeusedtoconcludethat x t ; x t 2 L in D and q t ; q t 2 L in D using 4.Similarly,itcanbeshownthat e 2 t 2 L in D .Since e 1 t ; e 2 t 2 L in D ,the denitionsfor U v and z t canbeusedtoprovethat U v isuniformlycontinuousin D Let S D denoteasetdenedasfollows: S v t D U 2 v t < h 1 r )]TJ/F23 8.9664 Tf 6.967 0 Td [(1 2 p l k 1 2 : [74]cannowbeinvokedtostatethat k z t k 2 0 ast 8 v 0 2 S : Basedonthedenitionof v t in4,4canbeusedtoshowthat k e 1 t k 0 ast 8 v 0 2 S : 41

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Using4and4,thecontroldevelopmentensuresthat q 2 t q 1 t ast 8 v 0 2 S : Theorem4.2. Thecontrollergivenin4ensuresthattheteleoperatedsystemispassivewith respecttothescaleduserandenvironmentalpower. Proof. See[53]. 4.4SimulationResults Inthissection,simulationresultsaregivenfortwodifferentuserinputforcestodemonstrate theperformanceofthecontrollergivenin4.Themasterandslavesystemdynamicsare simulatedusingthefollowingmodel m q 1 + b q 1 = T 1 )]TJ/F54 11.9552 Tf 10.949 0 Td [(F 1 ; m q 2 + b q 2 = T 2 )]TJ/F54 11.9552 Tf 12.743 0 Td [(f needle q 2 ; q 2 ; where T 1 t and T 2 t areintroducedin4and4, m = 0 : 152 kg b = 1 : 426 N s m )]TJ/F23 8.9664 Tf 6.967 0 Td [(1 whichcorrespondtotheneedleinsertionrobotdescribedin[75].Theneedleinsertionforce f needle q 2 ; q 2 issimulatedusingthedesigndescribedinChapter2,wheretheneedleinsertion forceisthesumofastiffnessforce,africtionforce,andacuttingforce.AsinChapter3,the differentpositionsandparametervaluesarechosentoagreewithadirectinsertionintotheliver. Theinitialpositionsare q 1 = 0 mm and q 2 = )]TJ/F23 11.9552 Tf 9.289 0 Td [(20 mm for t = 0.Thetissuepositionisxedto 200 mm fromtheorigin.Thepositionofthemaximallydeformedtissuesurfacebeforepuncture is216 mm ,whichmeansthattheneedleprogresses16mmwhileincontactwiththeliverbefore thepunctureoccurs.Theparametersforthedesiredtrajectoryintroducedin4arechosenas B T = 15 ; K T = 2 : 42

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ThenumberofhiddenlayerneuronsfortheNNischosenas N = 15,andtheNNweight updationgainsareselectedas G w 1 = G w 2 = 30 I 16 ; G v 1 = 30 I 5 ; where I p 2 R p p denotestheidentitymatrix.Fortherstsimulation,theuserinputforce F 1 t ,whichcorrespondstotheforceprovidedbythesurgeononthemasterrobot,isgivenbya sinusoidalforceas F 1 = 15sin 1 : 1 t : Thissimulationdoesnothaveapracticalmeaningbecauseitwouldmeanthatthesurgeon insertstheneedleintoapatient,removesitandinsertsitagain.However,thisusertrajectorywas simulatedtodemonstratetheperformanceofthecontrollerundersomearbitrarymotion.Figure 4-1showsthemasterposition q 1 t andtheslaveposition q 2 t .AsshowninFigure4-2,the errorbetweenthesetwopositiongoestozeroastimegoestoinnity.Thepassivityobjective, introducedin4,ismetwhenthetrajectoryof q 1 t + q 2 t followsthedesiredtrajectory x d 2 whichcanbeseeninFigure4-3and4-4. Forthesecondsimulation,theuserforce F 1 t issimulatedas F 1 = 8 : Figures4-5and4-6showthepositiontrackingbetweenthemasterrobotposition q 1 t andthe slaverobotposition q 2 t .ThepassivityobjectivecanbeseeninFigures4-7and4-8.InFigure 4-9and4-10,theneedleforce f needle isgivenasafunctionoftimeandpositionoftheneedletip, respectively.Itcanbeseenthatduringtherststagetheforcebetweentheneedleandthetissue isequaltozerobecausetheneedledoesnottouchthetissueyet.Then,theneedleforceincreases toreachamaximumforcewhichisfollowedbyasuddendropinforceastheneedlepunctures thetissueandnowonlyneedstoovercomethefrictionandcuttingforces. 43

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Figure4-1.Trajectoryformasterandslaverobotsfor F 1 = 15sin 1 : 1 t Figure4-2.Positionerrorbetweenmasterandslaverobotfor F 1 = 15sin 1 : 1 t 44

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Figure4-3.Desiredtrajectory x d 2 andpositionof q 1 + q 2 for F 1 = 15sin 1 : 1 t Figure4-4.Errorbetweenthedesiredtrajectory x d 2 and q 1 + q 2 for F 1 = 15sin 1 : 1 t 45

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Figure4-5.Trajectoryformasterandslaverobotsfor F 1 = 8. Figure4-6.Positionerrorbetweenmasterandslaverobotfor F 1 = 8. 46

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Figure4-7.Desiredtrajectory x d 2 andpositionof q 1 + q 2 for F 1 = 8. Figure4-8.Errorbetweenthedesiredtrajectory x d 2 and q 1 + q 2 for F 1 = 8. 47

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Figure4-9.Needleforce f needle asafunctionoftimefor F 1 = 8. Figure4-10.Needleforce f needle asafunctionoftheneedletippositionfor F 1 = 8. 48

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CHAPTER5 CONCLUSION 5.1SummaryofResults InChapter2,adiscussiononsofttissuedeformationisprovided.Theneedleinsertionforce modelingforviscoelastictissueispresentedasthesumofaHunt-Crossleystiffnessforce,a frictionforce,andaconstantcuttingforce. InChapter3,aone-degree-of-freedomtranslationrobotcontrollerisdesignedtoasymptoticallytrackadesiredtrajectorygoingfromanon-contactpositionintoaviscoelastictissue. Theneedleforceisdesignedconsideringthattheviscoelastictissuemodelisthesumofa stiffnessforce,africtionforce,andacuttingforce.Aslidingmodecontrollercombinedwitha multi-layerNNisusedtoensureasymptotictracking.ALyapunov-basedstabilityanalysisis providedtoprovethesemi-globalasymptotictracking.Theefcacyoftheproposedcontrolleris demonstratedthroughsimulations. InChapter4,acontrollerisdesignedtopermitaneedleinsertionslaverobottoasymptoticallytrackthepositionofthemasterrobotgoingfromanon-contactpositionintoatissue.A globallyinvertibletransformationisdenedtoshowstabilityandpassivity.ALyapunov-based stabilityanalysisisprovidedtoprovethesemi-globalasymptotictracking.Simulationresults demonstratethatthepositiontrackingandthepassivityobjectivearemet. Medicalroboticsresearchhasbecomeanimportanttooltoassistthedevelopmentof advancedmedicineandhighprecisionsurgery.Differentmethodshavebeenstudiedtoinserta needleconsideringtheconstraintsimposedbythephysiologicalpropertiesofapatient,butalso togivehapticfeedback,toreducehumanerrorsduetofatigueorhandtremor,andtodevelop medicalsimulatorstotrainmedicalstudentsandsurgeonsforsurgicalprocedures.Robotic needleinsertioncanleadtosaferandmoreaccurateneedleinsertions. 5.2RecommendationsforFutureWork Inthisstudy,undesiredbendingoftheneedleduringinsertionisnottakeintoconsideration. Abeveltipandtissuedeformationscancausetheneedletobendduringinsertionwhenusinga 49

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exibleneedle.Inclinicalpractices,adeviationoftheneedlefromthedesiredpathoftenreduces theeffectivenessoftheprocedures.Experimentally,sensorsorimagingdevicescanbeusedto acquiredataandthencontroltheneedlepositionusingrotationandtranslation. ToimprovethecontrollerdevelopedinChapter3and4,anaccuratedesignfortheposition oftheviscoelastictissuecouldbeused.Thedynamicsofthetissuedependsonforcesfrom surroundingtissueandorgans,andphysiologicalmovementsasheartbeatingorbreathing.In thesechapters,itisonlysupposedthatthepositionofthetissuedependsontimebutaspecic dynamicsisnotused.Adetailedstudyofphysiologicalmovementscouldgiveinformationabout thetissuemovementandthenitcouldbeappliedforthecontrollerdevelopmenttogetamore accurateresult.Thegoalistobeinperfectconformitywiththephysiologicalmovementsbutan easierapproachcouldbetoemployamass-springdynamic. InChapter4,nospecialcareisgivenfortimedelay.Timedelayaffectstheperformance ofdynamicsystem.Somemechanismsandcontrolstrategiescanbeappliedtothesesystemsto compensateforthem.Foramedicalapplication,thereisnorealneedtocompensatefortime delayinpracticeifmasterandslaverobotsareclosetoeachotherandaredirectlyconnected. Foralongdistancesurgeryitisfundamentaltodevelopacontrollerwhichguaranteesstability independentofthedelay. 50

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BIOGRAPHICALSKETCH ClineLaplassottewasborninWissembourg,Francein1989.In2009,sheenteredat TlcomPhysiqueStrasbourg,France.Theyearafter,shepursuedherstudywiththeMaster ImagerieRobotiqueetIngnieriepourleVivantImaging,RoboticsandEngineeringfor SurgeryattheUniversityofStrasbourg.Herinterestslieintheeldofnonlinearcontrol, biomedicalengineeringandrobotics. ThankstotheAtlantisprogram,ClinepursuedadualMasterofSciencedegreebetween theUniversityofStrasbourgandtheNonlinearControlsandRoboticsgroupintheDepartmentof MechanicalandAerospaceEngineeringattheUniversityofFlorida,underthesupervisionofDr. WarrenE.Dixon.Duringherlaboratorytime,shehadtheopportunitytoworkonroboticneedle insertion. 57