ROLEOFFLEXIBILITYINROBOTICFISHBySanazBazazBehbahaniADISSERTATIONSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofElectricalEngineeringŠDoctorofPhilosophy2016ABSTRACTROLEOFFLEXIBILITYINROBOTICFISHBySanazBazazBehbahaniUnderwatercreatures,especiallyhavereceivedattentionoverthepastseveraldecadesbecauseoftheirfascinatingswimmingabilitiesandbehaviors,whichhaveinspireden-gineerstodeveloprobotsthatpropelandmaneuverlikerealThisdissertationisfocusedontheroleofxibilityinroboticperformance,includingthedesign,dynamicmodeling,andexperimentalvalidationofxiblepectoralxiblepassivejointsforpectoralandwithactivelycontrolledstiffness.First,theswimmingperformanceandmechanicalefyofxiblepectoralconnectedtoactuatorshaftsviarigidlinks,arestudied,whereitisfoundthatxibledemonstrateadvan-tagesoverrigidinspeedandefyatrelativelylowtfrequencies,whiletherigidoutperformthexibleathigherfrequencies.Thepresentedmodeloffersapromisingtoolforthedesignofxibilityandswimminggait,toachievespeedandefyobjectivesfortheroboticThetraditionalrigidjointforpectoralrequiresdifferentspeedsforpowerandrecoverystrokesinordertoproducenetthrustandconsequentlyresultsincontrolcomplexityandlowspeedperformance.Toaddressthisissue,anovelxiblepassivejointispresentedwheretheisrestrictedtorowingmotionduringbothpowerandrecoverystrokes.Thisjointallowsthepectoraltosweepbackpassivelyduringtherecoverystrokewhileitfollowstheprescribedmotionoftheactuatorduringthepowerstroke,whichresultsinnetthrustevenundersymmetricactuationforpowerandrecoverystrokes.Thedynamicmodelofaroboticequippedwithsuchjointsisdevelopedandvalidatedthroughextensiveexperiments.Motivatedbytheneedfordesignoptimization,themodelisfurtherutilizedtoinvestigatetheofthejointlengthandstiffnessontherobotlocomotionperformanceandefy.Analternativexiblejointforpectoralisalsoproposed,whichenablesthepectoraltooperateprimarilyintherowingmode,whileundergoingpassivefeatheringduringtherecoverystroketoreducehydrodynamicdragontheAdynamicmodel,vexperimentally,isdevelopedtoexaminethetrade-offbetweenswimmingspeedandmechanicalefyinthedesign.Finally,weinvestigatexiblewithactivelytunablestiffness,enabledbyelectrorheolog-ical(ER)Thetunablestiffnesscanbeusedinoptimizingtheroboticspeedormaneu-verabilityindifferentoperatingregimes.FinswithtunablestiffnessareprototypedwithERenclosedbetweenlayersofliquidurethanerubberx10).Freeoscillationandbase-excitedoscillationbehaviorsofthearemeasuredunderwaterwhendifferentelectricareappliedfortheERwhicharesubsequentlyusedtodevelopadynamicmodelforthestiffness-tunableCopyrightbySANAZBAZAZBEHBAHANI2016ToMohsen,forhisadvice,patience,andunconditionallove;becausehealwaysunderstood.TomybelovedMomandDad,whohavebeenaconstantsourceofteachingandsupport.Tothelovingmemoryofmygrandpa,Sartip.vACKNOWLEDGMENTSItisapleasuretothankthemanypeoplewhomadethisthesispossible.Iamdeeplyindebtedtomyadvisor,Prof.XiaoboTan,forgivingmetheopportunitytopursuemydreams,withoutthesupportandadviceofwhomIwouldneverhavecompletedthisproject.IamreallyhappyandluckythatIhadthechancetoworkunderhissupervision.Ilearnedalotfromhisknowledge,enthusiasm,inspiration,andpatientguidancethroughoutmytimeatMichiganStateUniversity.IwouldliketothankProf.RanjanMukherjee,Prof.PhilipK.McKinley,andProf.LixinDong,forkindlyservingonmythesiscommittee,andbeinginspiringinmanyways.Theygenerouslygavetheirtimetooffermeinsightfulcommentsandadvicestowardimprovingthiswork.ThereisnowaytoexpresshowmuchitmeanttometohavebeenamemberoftheMichi-ganStateUniversityandSmartMicrosystemsLaboratory.Thesebrilliantfriendsandcolleaguesinspiredmeoverthemanyyears:Dr.JianxunWang,Dr.FeitianZhang,Dr.HongLei,Dr.JunZhang,Dr.AnthonyJ.Clark,OsamaEn-Nasr,JasonGreenberg,MontassarSharif,MariaCastano,ThassyoPinto,PratapBhanuSolanki,MohammedAl-Rubaiai,CodyThon,andalltheothercur-rentandformerSMLstudentsandvisitorsthatIknow.SpecialthanksmustgotoMr.JohnThon,forhisassistanceinprototypingtheroboticplatforms.Hetrulywentaboveandbeyondtomakesuremyroboticwouldrunsmoothly.IwouldliketoexpressmygratitudetothetechnicalandadministrativestaffoftheElectricalandComputerEngineeringDepartmentfortheirassistanceduringmystudyatMichiganStateUni-versity,inparticular,BrianWright,GreggMulder,RoxannePeacock,MeaganKroll,andLaurieRashid.IwouldliketoacknowledgethesupportofmyresearchbyNationalScienceFounda-tion(GrantsDBI0939454,CNS1059373,CCF1331852,IIS0916720,IIS1319602,IIP1343413,viECCS1029683,ECCS1446793),whichIhopewillfuelthefuturelearningofthosewalkinginmyfootsteps.Icannotforgetmyfriendswhowentthroughhardtimestogether,cheeredmeup,andcelebratedeachaccomplishment:Dr.JiankunLiu,Dr.MingwuGao,Dr.ElahehEsfahanian,AlirezaAmeli,MalihehGholami,DenaIzadi,AliLahouti,ShimaLahouti,andmanyothers.Ideeplythankmyparents,SamadBazazBehbahaniandAfsanehAzarifortheirunconditionalsupport,timelyencouragement,andendlesspatience,andthankmysister,Roxana,andmybrother,Rouzbeh,towhomIwillalwaysbeindebtfortheirloveandencouragementdespitethelongdistancebetweenus.Wordsalonecannotfullyexpressmyappreciationtowhattheyhavedoneforme.Lastly,andmostimportantly,Iwishtothankmyhusbandandmyhero,Dr.MohsenMosleh-pour,foralwaysbeingbesidemeandgivingmemotivationtomoveforward.Whobelievedinme,oftenfarmorethanIbelievedinmyself.Hehasbeenmybestfriendandgreatcompanion,wholoved,supported,encouraged,entertained,andhelpedmegetthroughthisagonizingperiodinthemostpositiveway.YouarethebesttreasureIcaneverdreamof.viiPREFACEThisdissertationissubmittedforthedegreeofDoctorofPhilosophyattheMichiganStateUni-versity.TheresearchdescribedhereinwasconductedundersupervisionofProf.XiaoboTanintheElectricalandComputerEngineeringDepartment,MichiganStateUniversity,betweenAugust2011andDecember2016.Thisworkistothebestofmyknowledgeoriginal,exceptwhereacknowledgmentandrefer-encesaremadetopreviouswork.Thisworkhasbeenpresentedinthefollowingpublications:Chapter2S.B.Behbahani,X.Tan,fiRoleofPectoralFinFlexibilityinRoboticFishPerformance,flJournalofNonlinearScience,Underreview,2016.Chapter3S.B.Behbahani,X.Tan,fiDesignandModelingofFlexiblePassiveRowingJointsforRoboticFishPectoralFins,flIEEETransactionsonRobotics,Vol.32,No.5,pp.1119-1132,2016.Chapter4S.B.Behbahani,X.Tan,fiBio-inspiredFlexibleJointswithPassiveFeatheringforRoboticFishPectoralFins,flBioinspiration&Biomimetics,Vol.11,No.3,pp.036009,2016.Chapter5S.B.Behbahani,X.Tan,fiDesignanddynamicmodelingofvariablestiffnessforroboticusingelectrorheologicalflUnderpreparation,2016.viiiTABLEOFCONTENTSLISTOFTABLES.......................................xiiChapter1INTRODUCTION...............................11.1Motivation.......................................21.2LiteratureReview...................................21.3ResearchObjectives..................................71.4OverviewofContributions..............................81.5DissertationLayout..................................101.6Publications......................................111.6.1JournalArticles................................111.6.2ConferenceProceedings...........................12Chapter2ROLEOFPECTORALFINFLEXIBILITYINROBOTICFISHPERFOR-MANCE....................................132.1Introduction......................................132.2DynamicsoftheRoboticFishBody.........................152.2.1RigidBodyDynamics............................162.3DynamicsofFlexibleRowingPectoralFins.....................182.3.1ForceCalculationsforElementi.......................212.3.2MomentCalculationsforElementi.....................222.4KinematicsofFlexibleRowingPectoralFins....................242.5MechanicalEfy.................................252.6MaterialsandMethods................................262.6.1RoboticFishPrototype............................262.6.2PectoralFinFabrication...........................272.6.3ExperimentalSetup..............................292.6.4Parameter............................302.7ResultsandAnalysis.................................312.7.1DynamicModelValidation..........................312.7.2ImpactofFinStiffness............................342.7.3ImpactofthePowertoRecoveryStrokeSpeedRatio............352.7.4ImpactofFlexibilityonMechanicalEfy...............382.8Conclusion......................................41Chapter3DESIGNANDMODELINGOFFLEXIBLEPASSIVEROWINGJOINTFORROBOTICFISHPECTORALFINS......................423.1Introduction......................................423.2JointDesign......................................453.3DynamicModeling..................................473.3.1RigidBodyDynamics............................48ix3.3.2HydrodynamicForcesfromPectoralFinswithFlexibleRowingJoints...513.3.2.1BladeElementTheory.......................523.3.2.2ModelingoftheFlexibleJoint...................533.4ExperimentalModelValidation............................563.4.1RoboticFishPrototypeandExperimentalSetup...............563.4.2Parameter............................583.4.3ComparisonbetweenFlexibleandRigidJoints...............603.4.4DynamicModelValidation..........................613.4.4.1Dynamiccharacteristicsofpectoral..............613.4.4.2Anchoredexperiments.......................623.4.4.3Free-swimmingexperiments....................643.5EffectofFlexibleJointLengthandStiffness.....................653.6MechanicalEfy.................................693.7Conclusion......................................75Chapter4BIO-INSPIREDFLEXIBLEJOINTSWITHPASSIVEFEATHERINGFORROBOTICFISHPECTORALFINS......................764.1Introduction......................................764.2DesignofFlexibleFeatheringJoint..........................794.3DynamicModelofFin-ActuatedRoboticFishIncorporatingtheFlexibleFeather-ingJoint........................................814.3.1HydrodynamicForcesontheFin.......................814.3.2SolvingtheFeatheringDynamics......................864.3.3HydrodynamicForcesandMomentsontheRoboticFish..........884.3.4Rigid-BodyDynamicsofaPectoralFin-actuatedRoboticFishUndergoingPlanarMotion.................................894.4ExperimentalResults.................................904.4.1RoboticFishPrototypeandExperimentalSetup...............904.4.2Parameter............................924.4.3ComparisonbetweenFlexibleFeatheringandRigidJoints.........944.4.4DynamicModelValidation..........................964.5EffectofFlexibleJointDepthandStiffness.....................994.6MechanicalEfy.................................1014.7Conclusion......................................106Chapter5DESIGNANDDYNAMICMODELINGOFELECTRORHEOLOGICALFLUID-BASEDVARIABLE-STIFFNESSFINFORROBOTICFISH.........1075.1FabricationProcedure.................................1095.1.1Materials...................................1095.1.2ManufacturingProcedure...........................1105.2DynamicModelfortheVariable-StiffnessFin....................1125.2.1EvaluationofHydrodynamicForceUsingLighthill'sLarge-AmplitudeElongated-BodyTheory.................................1125.2.2DynamicModelingoftheVariable-StiffnessCaudalFinFilledwithERFluid1145.2.2.1ERFluid..............................115x5.2.2.2Multi-layerBeam.........................1165.2.2.3DiscretizationusingFiniteElementMethod(FEM)........1205.3ExperimentalResults.................................1235.3.1ofShearModulusValues...................1235.3.2ofHydrodynamicCoef.................1275.3.3BaseActuationExperimentsforModelValidation..............1295.4Conclusion......................................132Chapter6CONCLUSIONSANDFUTUREWORK....................1346.1ConcludingRemarks.................................1346.2FutureWork......................................135References...........................................137xiLISTOFTABLESTable2.1:ofpectoralwithdifferentxibilities..............28Table3.1:offourdifferentxiblerowingjoints...............47Table3.2:ModelParameters............................59Table3.3:ComparisonbetweenthetwomethodsofcomputingWb..............70Table3.4:Comparisonofnon-dimensionalizedparameters..................74Table4.1:ModelParameters............................93Table4.2:MechanicalefyversustheStrouhalnumber.................105xiiLISTOFFIGURESFigure1.1:Examplesofreportedroboticprototypeswithpairedpectoral.....4Figure1.2:Axiblepectoralwheretheindividualtendonspermitactuationofeachray[61].....................................5Figure1.3:Elasticpectoralactuatorsforbiomimeticunderwatervehicles[62]:(a)Ac-tivepneumaticactuatorpectoraland(b)passivexible........5Figure1.4:Roboticwithxiblepectoral[27,63]..................6Figure1.5:Examplesofreporteddesignsfortunablestiffness.............8Figure2.1:Topviewoftheroboticactuatedbyxiblepectoralandrestrictedtoplanarmotion...................................16Figure2.2:Topviewofithelementofthexibleanditsparametersandvariables...19Figure2.3:Illustrationofforcesontheithelementofthexible............20Figure2.4:Illustrationofthepectoralmotionduringpowerandrecoverystrokes:(a)Thesnapshotsatdifferenttimeinstancesofonecycle,whereTprep-resentsthetotaltimeforeachcycle;(b)orientationandangularvelocityofthebaseofthepectoralwithrespecttothemainaxisoftheroboticwherez=5,gA=50deg,andTp=1sareused.....................25Figure2.5:3D-printedroboticbodyandothercomponentsoftherobot.........27Figure2.6:Solidworksdesignofacompositepectoral.................28Figure2.7:Anactual3D-printedpectoralRight:originalleft:printedtreatedwithUltra-EverDryomniphobicmaterial.....................29Figure2.8:Experimentalsetup:(a)Actual,(b)outputoftheMotivesoftware........29Figure2.9:Timehistoryofxiblepectoral(a)Evolutionoftherigidelementanglesg1Œg3,foronemovementcycle,(b)variationoftheanglesofattackforallelements,wherez=5,gA=50deg,andTp=3s,(c)zoom-inviewoftheangleofattackforonemovementcycle......................32xiiiFigure2.10:CaseofF2withz=2:Comparisonbetweendynamicmodelsimulationandexperimentalmeasurementoftheforwardswimmingvelocity,fordifferentfrequencies................................33Figure2.11:CaseofF2withz=2:Comparisonbetweendynamicmodelsimulationandexperimentalmeasurementoftheturningperiod,fordifferentfrequen-cies.........................................33Figure2.12:CaseofF2withz=2:Comparisonbetweendynamicmodelsimulationandexperimentalmeasurementoftheturningradius,fordifferentfrequen-cies.........................................33Figure2.13:Experimentalandsimulationresultsontheforwardswimmingvelocityver-susfrequencyfordifferentxibilitiesofthepectoralwherethepower/recoverystrokeratioz=2forallcases..................34Figure2.14:Experimentalcomparisonoftheforwardswimmingvelocitiesatdifferentpower/recoverystrokespeedratiosz=2andz=4.(a)Velocityversusatfrequency,and(b)Velocityversuspowerstroketime.....................36Figure2.15:Simulationcomparisonoftheforwardswimmingvelocitiesatdifferentpower/recoverystrokespeedratiosz=1.5,2,3,4,and5.(a)Velocityversusfre-quencyforcaseofF1,(b)velocityversuspowerstroketimeforcaseofF1,(c)velocityversusfrequencyforcaseofF2,(d)velocityversuspowerstroketimeforcaseofF2,(e)velocityversusfrequencyforcaseofF3,(f)velocityversuspowerstroketimeforcaseofF3.............37Figure2.16:Calculatedmechanicalefyversusfrequencyandspringcon-stantofthexible.............................38Figure2.17:Comparisonofnon-dimensionalizedparametersoftheroboticactuatedbypairedpectoralwithdifferentxibilitiesversusdifferentfre-quenciesforthecaseofz=2:(a)CalculatedefyandStrouhalnumber,(b)dimensionlessvelocityandReynoldsnumber,and(c)forwardswimmingvelocityincm/sandefy..........................40Figure3.1:Typesofpectoralmotion(Adaptedfrom[81]).Therotationaxesfortherowing,feathering,andmotionsarevertical,transverse,andlongitu-dinal,respectively.................................42Figure3.2:Illustrationofthemotionofthepectoralwithxiblerowingjoints(topview):(a)Powerstroke,(b)recoverystroke.Thexiblejointsaremarkedwithreddots....................................45xivFigure3.3:TheproposedxiblerowingjointdesignedinSolidWorkssoftware:(a)Dur-ingthepowerstroke,themechanicalstopperpreventsthefromsweepingforwardpassively(whichwouldreducethrust),(b)duringtherecoverystroke,thebendsbackpassivelyunderthehydrodynamicforces,whichreducesthedragontheandthusontherobot,and(c)3D-printedrowingpassivejointassembledontherobotic.......................46Figure3.4:Topviewoftheroboticactuatedbypectoralinaplanarmotion.....48Figure3.5:Illustrationofarigid,rectangularpectoralanditsparametersandvariables.51Figure3.6:Dynamicofthepectoralwithxiblerowingjointduring:(a)Powerstroke,and(b)recoverystroke.A1representsthexiblejoint....53Figure3.7:Roboticprototype:(a)DesignedSolidWorksmodel;(b)3D-printedroboticbodyalongwithmounted........................57Figure3.8:Experimentalsetup:(a)Schematic;(b)actual...................58Figure3.9:Experimentalresultsoftheforwardswimmingvelocityversus(a)powerstroketime,and(b)effectiveactuationfrequency,forthecasesofrigidjointandx-iblerowingjoint..................................61Figure3.10:Withfrequencyof1Hz:(a)Variationofthepectoralandservoarmangleforonemovementcycle,(b)Variationofthepectoralangleofattackforonemovementcycle,(c)Variationofthetotalhydrodynamicforceexertedtoroboticbodyinx-direction(Fhx)versussimulationtime,(d)Variationofroboticvelocityinx-direction(VCx)versussimulationtime........63Figure3.11:Comparisonbetweenmodelprediction(whitedashedline)andexperimentalmeasurement(bluesolidline)ofthemaximumrowingangleduringtherecov-erystroke,withfrequenciesof(a)0.75Hz,(b)1Hz,(c)1.25Hz,(d)1.5Hz,(e)1.75Hz,and(f)2Hz.Theblackverticallineindicatestheroboticheadingdirection,thegreendottedlineshowstheservoarmdirectionandtherightpectoralisshown...........................64Figure3.12:Comparisonbetweenexperimentalmeasurementofthetime-dependentrecov-erystrokeanglewithmodelpredictions.Thepectoralbeatsat1Hz.Thebluesolidlineandwhitedashedlineimplytheexperimentalmeasurementandmodelprediction,respectively,andthegreendottedlineshowstheservoarmdirection......................................65Figure3.13:Caseofrowingjoint(JR1):Comparisonbetweenthemodel-predictedandmeasuredforwardswimmingspeed,fordifferentfrequencies......66xvFigure3.14:Caseofrowingjoint(JR1):Comparisonbetweenthemodel-predictedandmeasuredturningperiod,fordifferentfrequencies............66Figure3.15:Caseofrowingjoint(JR1):Comparisonbetweenthemodel-predictedandmeasuredturningradius,fordifferentfrequencies............67Figure3.16:Modelpredictionandexperimentalmeasurementoftheforwardswimmingvelocityoftheroboticwiththeuseofthreexiblejoints(allmadeofFLX980)withdifferentlengths..........................68Figure3.17:Modelpredictionandexperimentalresultsoftheforwardswimmingvelocityoftheroboticwiththeuseoftwoxiblejointswithdifferentstiffnessvalues.......................................68Figure3.18:Calculatedmechanicalefyandforwardvelocityofdifferentxiblerowingjointsatdifferentfrequencies...................71Figure3.19:Calculatedmechanicalefyversusfrequencyandspringcon-stantofthexiblejoint..............................71Figure3.20:Comparisonofnon-dimensionalizedparametersforjointsJR1,JR2,andJR3atdifferentfrequencies:(a)CalculatedmechanicalefyandStrouhalnumber,(b)DimensionlessvelocityandReynoldsnumber.......72Figure4.1:schematicsofthedrag-basedlabriformswimmingmode,usingxiblefeath-eringjoints(topview):(a)Powerstroke,(b)recoverystroke.Thexiblefeatheringjointisshownwitharedcircle.....................77Figure4.2:Theproposedxiblefeatheringjoint:(a)Duringthepowerstroke,theme-chanicalstopperpreventsthefromfeathering,(b)duringtherecoverystroke,therotatesandtendstoalignwiththehorizontalsurface,toreducethedragforceontheand(c)3D-printedfeatheringpassivejointassembledontherobotic....................................80Figure4.3:Topviewoftheroboticactuatedbypectoralinplanarmotion......82Figure4.4:Illustrationofarigid,rectangularpectoralanditsparameters,duringthepowerstroke....................................83Figure4.5:Illustrationofarigid,rectangularpectoralanditsparameters,duringtherecoverystroke...................................85Figure4.6:3D-printedroboticprototypealongwithmounted...........91Figure4.7:Experimentalsetupforfree-swimmingrobotic...............92xviFigure4.8:Experimentalresultsoftheforwardswimmingvelocityintermsof(a)thepowerstroketime,and(b)theeffectiveactuationfrequency,forthecasesofrigidjointandxiblefeatheringjointfiJF1fl...................95Figure4.9:Comparisonbetweenmodelprediction(whitedashedline)andexperimentalmeasurement(bluesolidline)ofthemaximumfeatheringangleduringtherecoverystroke,withbeatfrequenciesof(a)0.75Hz,(b)1Hz,(c)1.25Hz,(d)1.5Hz,(e)1.75Hz,and(f)2Hz.Theyellowsolidlineindicatesthe‹kdirection......................................97Figure4.10:Caseoffeatheringjoint(JF1):Comparisonbetweenthemodel-predictedandmeasuredforwardswimmingspeed,fordifferentbeatfrequencies.Theforwardswimmingvelocityisreportedincm/secscaleonthelefty-axisandinBL/secontherighty-axis............................97Figure4.11:Caseoffeatheringjoint(JF1):Comparisonbetweenthemodel-predictedandmeasuredturningperiod,fordifferentbeatfrequencies............98Figure4.12:Caseoffeatheringjoint(JF1):Comparisonbetweenthemodel-predictedandmeasuredturningradius,fordifferentbeatfrequencies............98Figure4.13:Modelpredictionandexperimentalmeasurementoftheforwardswimmingvelocityoftheroboticusingthreedifferentxiblefeatheringjoints,madeofFLX980,withdifferentdepths.Theforwardswimmingvelocityisreportedincm/secscaleonthelefty-axisandinBL/secontherighty-axis........100Figure4.14:Modelpredictionandexperimentalresultsoftheforwardswimmingvelocityoftheroboticwiththeuseoftwoexiblefeatheringjointswithdifferentstiffnessvalues.Theforwardswimmingvelocityisreportedincm/secscaleonthelefty-axisandinBL/secontherighty-axis................101Figure4.15:Calculatedmechanicalefy(blue)andforwardswimmingvelocity(red)fordifferentxiblefeatheringjointsatdifferentbeatfrequencies......103Figure4.16:Mechanicalefy:(a)Calculatedmechanicalefyversusfrequencyandspringconstantofthexiblefeatheringjoint,(b)Springcon-stantofthefeatheringjointwithmaximumefyversusfrequency,(c)Maximumefyversusspringconstant..................104Figure5.1:3D-printedmoldsusedtoprototypethevariablestiffness..........110Figure5.2:Prototypingthevariable-stiffness(a)Thehalvesofthewithelectrodesincorporated;(b)degassingthepartsinavacuumoven;(c)product:hol-lowwithwiresattachedtotheelectrodes...................111xviiFigure5.3:Prototypedvariable-stiffnesswithER..............111Figure5.4:Planarviewoftheroboticandthedetailedillustrationofthexibletailcoordinatesystem.................................113Figure5.5:SchematicforactuationoftheERcaudal(a)Topview,(b)sideview........................................115Figure5.6:ERreactiontotheelectric......................115Figure5.7:SchematicsoftheERxible.............117Figure5.8:SchematicofthexibleinFEM:(a)oftheelementandnodenumbers,(b)elementi,itstwonodes,anddegreesoffreedom.......121Figure5.9:Experimentalsetupforobservingpassive,dampedvibrationoftheERxibleunderdifferentelectric(a)Schematic,(b)actual....124Figure5.10:parametersunderdifferentelectricintheair.(a)Dampingratio,(b)naturalfrequency.............................125Figure5.11:complexshearmodulusvaluesandthecurveunderdifferentelectric(a)Storagemodulus(G0),(b)lossmodulus(G00).........126Figure5.12:Asampleofexperimentallymeasureddampedoscillationsofthebeamtip,alongwithsimulationresultscorrespondingtothematchedparameters,foreachappliedelectric(a)Ef=0Vm,(b)Ef=0:25106Vm,(c)Ef=0:5106Vm,(d)Ef=0:75106Vm,(e)Ef=1106Vm,(f)Ef=1:25106Vm,(g)Ef=1:5106Vm...............................127Figure5.13:Experimentalsetupformeasuringdampedoscillationsofthexibleinwater.128Figure5.14:parametersunderdifferentelectricinwater.(a)Dampingra-tio,(b)naturalfrequency..............................128Figure5.15:Passive,dampedvibrationoftheERlledxiblebeaminwater:(a)Ef=0Vm,(b)Ef=0:25106Vm,(c)Ef=0:5106Vm,(d)Ef=0:75106Vm,(e)Ef=1106Vm,(f)Ef=1:25106Vm,(g)Ef=1:5106Vm.......130Figure5.16:Experimentalsetupformeasuringtheshapeunderbaseactuation(topview).131xviiiFigure5.17:Base-actuatedstiffness-tunablewithamplitudeof22:5andfre-quencyof2Hz:(a)-(b)Comparisonbetweenexperimentalmeasurementofthetime-dependentbeamshapeswithmodelpredictions,wheretheelectricappliedtothebeamisEf=0Vm,(c)-(d)comparisonbetweenexperimen-talmeasurementofthetime-dependentbeamshapeswithmodelpredictions,wheretheelectricappliedtothebeamisEf=1:5106Vm,wherethesolidblacklinecorrespondstotheservoarmdirectionandtheyellowdashedlinerepresentstheshapebythedynamicmodel,and(e)comparisonbe-tweenthesimulatedtipdisplacementsunderthetwodifferentelectric..132Figure5.18:Base-actuatedstiffness-tunablewithamplitudeof10andfre-quencyof4Hz:(a)-(c)Comparisonbetweenexperimentalmeasurementofthetime-dependentbeamshapeswithmodelpredictions,wheretheelectricappliedtothebeamisEf=0Vm,(d)-(f)comparisonbetweenexperimen-talmeasurementofthetime-dependentbeamshapeswithmodelpredictions,wheretheelectricappliedtothebeamisEf=1:5106Vm,wherethesolidblacklinecorrespondstotheservoarmdirectionandtheyellowdashedlinerepresentstheshapebythedynamicmodel................133xixChapter1INTRODUCTIONHowswim,theirremarkablebehaviors,andtheiruniqueswimmingpatternshavelongin-triguedbiologistsandengineerslookingforcluestounderstandandimitatereal[1Œ5].Scien-tistsandengineershavebeenmotivatedbytocreateaclassofautonomousunderwatervehiclescalledroboticDifferentswimmingmodesandcharacteristicsareappliedtothedesignofthesebiomimeticroboticwiththevirtuesofbeingmaneuverable,efandlifelike[6Œ9].Therearecomplexinteractionsbetweenabody,andsurroundingwhichmakestheroboticdesign,development,andcontrolchallenging.Improvedefy,maneuverability,andstealtharesomeofthepotentialadvantagesofroboticovertraditionalpropeller-drivenun-derwatervehicles[10Œ12].Amongtheirmanyapplications,roboticcanprovideanunderwaterplatformforaquaticenvironmentalmonitoring[13,14],aswellasservingasatoolforstudyingthebehaviorsoflivethroughrobot-animalinteractions[15].Inthischapter,themotivationbehindthisstudyisdescribedinSection1.1.Areviewofthestateoftheartinroboticresearch,especiallyresearchinvolvingxibleisgiveninSection1.2.TheobjectivesofthisstudyarediscussedinSection1.3.ThecontributionsofthisworkandthestructureofthisdissertationareexplainedinSection1.4and1.5,respectively.Theauthor'spublicationsduringthecourseofPh.D.studiesarelistedinSection1.6.11.1MotivationBiologistsandengineersstudyswimmingpatternstounderstandlocomotionenabledbybodyandmovements[16,17].Numerousdesignsforrobotichavebeenreportedintheliter-ature[18],withdifferentactuationmechanismsandlevelsofsystemcomplexity[19Œ26].Roboticcangeneratethrustusingatail(caudal)[27Œ29],pairedpectoral[30Œ32],orblendingbothmovements[33].Basedontheobservationsfromnature,differentkindsofactuatorshavebeensuggestedtoincreaseefyandmaneuverabilityofaroboticAtypicalstrategyforactuatingaroboticistousemotors,whichentailssingle/multiplejointsandrigidlinksforeachtailorpectoral[20,27,34Œ36].Analternativeapproachtoactuationistheuseofsmartmaterials[23,37Œ40].Variousstudieshaveindicatedthattheshapeandelasticityoftheplayrolesinthehydrodynamicperformanceofa[11,41].Fromthebiologicalpointofview,achangesitsxuralstiffnesstoperformdifferenttasks,suchasincreasingswimmingspeed[42,43].Thegoalofthisworkisnotjustthedevelopmentofaerobotsinceitishardtopursuethisgoalevenwiththemostadvancedtechnologies,duetothecomplexityofrealanditscom-plicatedmotion.Instead,westudytheeffectofxibilityoftheandjointsontheperformanceofrobotic1.2LiteratureReviewThereisliteratureavailableondesignandmodelingofroboticBesidesmimickinglivelocomotion,biomimeticroboticallowsresearcherstostudyvariousdesignsandkinematicsthatwouldresultinhigherlevelsofperformanceinspeed,maneuverability,stabilityandefy.2Forthemajorityofthisstudy,wefocusontheperiodicmotiongeneratedbypairedpec-toralforroboticAlthoughthismodeofswimmingisproventobelessefathigherspeeds,itprovidesremarkablemaneuveringandefswimmingatlowerswimmingspeeds.Thereisalargebodyofworkconductedonpectoralmorphology,kinematics,andhy-drodynamics,bothanalytically[44Œ46]andcomputationallyusingcomputationaldynamics(CFD)methods[47Œ49]toobtaintheforcesandtorquesproducedbytheseWebb[50,51]andBlake[52,53]modeledthepectoralpropulsionandmotionofaliveh,providingin-sightintothecomputationsofhydrodynamicforcesinvolvedinpectoralactuation.Thereareseveralstudiesonroboticwithpectoralmostofwhich,however,haveconsideredonlyrigidpectoralwithoneormoredegreesoffreedom,eachcontrolledbyaseparatemo-tor[32,34,35,44,54Œ60].SomeoftheseroboticareshowninFig.1.1.SeveralgroupshaveanalyzedthecaseofxiblepectoralinroboticLauderetal.[30,61]exploredthehydrodynamicsassociatedwithpectoralusingaself-propelledpectoralwithbi-laminarrays,bothexperimentallyandwithComputationalFluidDynamics(CFD)simulation.ThispectoralisshowninFig.1.2.Katoetal.[62]designedtwodifferenttypesofelasticpectoralAnactivepneumaticactuatorpectoralshowninFig.1.3(a),andapassivexibleshowninFig.1.3(b).TheyconductedexperimentstoestimatethepropulsiveforcesandusedFiniteElementMethod(FEM)toanalyzethebehaviorofthesepectoralDengetal.[27,63]modeledaroboticwithtwopectoralandacaudalandusedittodesigncontrollersforroboticHowever,thexibilityofthepectoralintheirexperimentalprototypewasnotincludedintheirmodel.ThedesignoftheirroboticshisshowninFig.1.4.WithCFDsimulation,Shoeleetal.[64]numericallyexaminedtheinteractionandforcegenerationbypectoralofaduringlabriformswimmingmode.Palmisanoetal.[32]studiedapectoralwithsoftraysinamotionandusedCFDanalysisforoptimal3(a)BassII[36](b)Boxybot[34](c)Labriformswimmingrobotic[35](d)UniversityofWashington'srobotic[44](e)Ghostswimmer[60]Figure1.1:Examplesofreportedroboticprototypeswithpairedpectoraldesign.Insummary,thepairedpectoralswimmingthatprovidesdragandlift,canbemimickedundercomplicatedmotorcontrolschemes[65,66].Whilexibilityisanimportantcharacteristicofpectoralforliveandisexpectedtothehydrodynamicsofroboticsignif-icantly,relevantstudieshavemostlyfocusedonexperimentalorCFDexplorationofthismatter4Figure1.2:Axiblepectoralwheretheindividualtendonspermitactuationofeachray[61].(a)(b)Figure1.3:Elasticpectoralactuatorsforbiomimeticunderwatervehicles[62]:(a)Activepneu-maticactuatorpectoraland(b)passivexibleandthesystematicanalysisoftheroleofxibilityinpectoralanditsimpactonroboticperformancehasbeenlimited.Inadditiontoxiblethestiffnessoftheshouldideallychangewiththeoperatingconditiontoimprovetheperformanceoftherobotic[42,43].Namely,theshouldbestifferwhentheoscillatingfrequencyincreases,tomaximizetheamountofthrust.Morerecently,limitedstudieshavebeenconductedtoaddressthisissue.Ziegleretal.[67]developedafree-swimming5Figure1.4:Roboticwithxiblepectoral[27,63].roboticwhichwasactuatedbyatailcapableofvaryingitselasticity.Thetailwasactuatedbyaprimarymotor,andtwosmallservoswereusedtovarytheelasticityoftheThisconsistedoftwomainfoils.Additionalfoilsofvariousthicknessandmaterialcouldbeinsertedintothemainfoil,utilizingthesmallservos.Thisresultsinchangeofthestiffnessofthexibletail.TheschematicoftheirproposeddesignisshowninFig.1.5(a).Parketal.[68]developedawithavariable-stiffnessmechanism,whichwasrealizedbycompressingacompliantmaterialtoincreasethestiffness.Theirdesignwasbasedontheendoskeletonstructure,whichusedasimplemechanism.Whenthecompliantmaterialwascompressed,thestiffnessincreased.Theirdesignconsistedofsixrigidplates,whichwereusedasthebackboneoftheroboticTheplateshadintervalsbetweenthem.Twotendonswereusedfordrivingthetailandtwoothertendonswereusedtochangeitsstiffness.Basically,whenthetendonswerepulled,thevariable-stiffnessstructurewascompressed,whichwouldresultinanincreaseinaxialstiffness,andwhenthetendonswerereleased,theaxialstiffnesswoulddecrease.Thisstructureisshown6indetailinFig.1.5(b).Nakabayashietal.[69]developedaroboticwithavariable-stiffnessmechanismusingavariable-effective-lengthspring.Thebendingstiffnessofthetailncouldvarydynamicallyinthiswork,whichworkedbyalteringthelengthofarigidplateusingamotor.Thiswouldresultinchangingtheeffective-lengthofthespring,andthuschangeofstiffness.However,thisdesignneedsenoughspacetoactuallyadjustthelengthofthespring.ThestructurefortheproposedvariablestiffnessmechanismisshowninFig.1.5(c).Inalltheworksmentionedabove,thestiffnesswascontrolledbyservos,andconsequently,therobotswererelativelybulkyduetotheneedforadditionalservostoperformthejob.1.3ResearchObjectivesThisdissertationaddressesfourissuesofincreasingcomplexity.(1)Howthexibilityofpectoralaffectstheswimmingperformanceandmechanicalefoftherobot?Ananalyticalstudyoftheeffectofxibilityoftheontheperformanceofroboticisavitalcontributiontotheroboticsociety.(2)Howcanweachievemoreefswimmingbyintroducingxiblepassivejoints,andreducingthecomplexityoftherobot?Analternativeapproachthatreducestheeffectiveareaofthepectoralintherecoverystrokeundersymmetricalactuationcanimprovetheperformanceoftherobotic(3)Howcanweutilizethexiblepassivejointideaandachieveamorecomplexmotion,mimickingdrag-basedlabriformswimmingofliveRealrarelyexclusivelyusemotionofpectoralinstead,theyuseacombinationofdifferentmotionstomoveforward.(4)Canweachieveadesignthatactivelychangesthestiffnessoftheroboticbasedontheoperationconditions,andyetisquickandcompact?Itisimportanttomodifythestiffnessoftheroboticbasedontheperformancestates.Thischangeneedstobeappliedinstantlytothebaseontheoperationregimetheroboticisin,tobemosteffective.7(a)Mechanismforstiffnesschangeintail[67](b)Designofdrivingpartwithvariable-stiffnessstructureandtendons[68](c)Structureofwithvariableeffectivelengthspring[69]Figure1.5:Examplesofreporteddesignsfortunablestiffness1.4OverviewofContributionsThegoalofthisresearchistoinvestigatethefundamentalsofepropulsionusingxiblecomponentsforperformanceenhancementinaquaticrobotics.Tothisend,thecontributionsof8thisresearcharefurthersummarizedasfollows:First,weexploretheimpactofthexibilityofroboticpectoralontherobotlocomo-tionperformanceandmechanicalefy.Adynamicmodelfortheroboticispresented,wherethexibleismodeledasmultiplerigidelementsconnectedviatorsionalspringsanddampers.BladeelementtheoryisusedtocapturethehydrodynamicforceonthexibleThemodelisvalidatedwithexperimentalresultsobtainedonaroboticprototype,equippedwith3D-printedofdifferentxibility.Themodelisthenusedtoanalyzetheimpactsofxibilityandpower/recoverystrokespeedratioontherobotswimmingspeedandmechanicalefy.Second,weintroducethedesignofanovelxiblepassivejointthatconnectstheservomotorarmtothepectoraltoovercomethedifthatarisewithatraditionalrigidjoint,whichhasslowerrecoverystrokespeedandfasterpowerstrokespeedinordertogenerateanetthrust.Adynamicmodelisdevelopedforthejointandforaroboticequippedwithsuchjoints.Thedesignandthemodelareevaluatedwithextensiveexperimentalresults.Motivatedbytheneedfordesignoptimization,themodelisfurtherutilizedtoinvestigatetheofthejointlengthandstiffnessontherobotlocomotionperformanceandefy.Third,anovelxiblejointisproposedforroboticpectoralwhichenablesaswimmingbehavioremulatingthemotionsofmanyaquaticanimals.Inparticular,thepectoraloperatesprimarilyintherowingmode,whileundergoingpassivefeatheringduringtherecoverystroketoreducehydrodynamicdragontheThelatterenableseffectivelocomotionevenwithsymmetricbaseactuationduringpowerandrecoverystrokes.Adynamicmodelisdevelopedtofacilitatetheunderstandinganddesignofthejoint,wherebladeelementtheoryisadoptedtocalculatethehydrodynamicforcesonthepectoralns,andthejointismodeledasapairedtorsionspringanddamper.Experimentalresultsonaroboticprototypearepresentedtoillustratetheeffectiveness9ofthejointmechanism,validatetheproposedmodel,andindicatetheutilityoftheproposedmodelfortheoptimaldesignofjointdepthandstiffnessinachievingthetrade-offbetweenswimmingspeedandmechanicalefy.Finally,westudythedesign,prototyping,anddynamicmodelingofatunable-stiffnessforroboticusinganelectrorheological(ER)whichenablesadaptingthexuralstiffnessofthecompliantAmulti-layercompositewithanERcoreisprototypedandutilizedtoinvestigatetheofelectriconitsperformance.Lighthill'slargeamplitudeelongated-bodytheoryisadoptedtoevaluatethehydrodynamicforcesontheandHamilton'sprincipleisusedtoderivethedynamicequationsofmotionofthexiblecompositecaudalThedynamicequationsarethendiscretizedusingtheelementmethod,toobtainanapproximatenumericalsolution.Experimentsareconductedontheprototypedxiblebeamtoidentifysomeparameters,validatetheproposeddynamicmodel,andassesstheefyoftheproposedstiffness-tuningapproach.1.5DissertationLayoutThisdissertationconsistsof6chaptersandisorganizedasfollows:InChapter2,acompletedy-namicmodelofaroboticactuatedbyapairofxiblepectoralisdiscussed.AdetailedexplanationofthevariousforcesactingontheroboticandthefullNewton-Eulerequationsarederived,wherebladeelementtheoryisusedtoevaluatethehydrodynamicforcesontheoscillatingxiblepectoralTheswimmingperformanceandmechanicalefyofxiblepectoralconnectedtoactuatorshaftsviarigidlinks,arestudied,whereitisfoundthatxibledemonstrateadvantagesoverrigidinspeedandefyatrelativelylowfrequen-cies,whiletherigidoutperformthexibleathigherfrequencies.Thepresentedmodel10offersapromisingtoolforthedesignofxibilityandswimminggait,toachievespeedandefyobjectivesfortheroboticChapter3providesdetailedinformationaboutthenovelxiblerowingjointmechanism.Thejointdesign,dynamicmodelingandexperimentalresultsarepresentedinthischapter.Theofthejointlengthandstiffnessontherobotlocomotionperformanceandefyareinvestigated.InChapter4,wefocusonanotherxiblepassivejoint,calledxiblefeatheringjoint.Thisjointenablestherobotictoperformthedrag-basedlabriformswimmingwithoutaddingadditionalservomotorstotheThejointdesign,dynamicmodeling,andexperimentalresultsofaroboticutilizingthisjointarepresentedinthischap-ter.Theeffectofjointdepthandstiffnessinachievingthetrade-offbetweenswimmingspeedandmechanicalefyisstudiedaswell.Chapter5focusesonthedesign,prototyping,anddynamicmodelingofthetunable-stiffnessforroboticThetunablestiffnesscanbeusedinoptimizingtheroboticspeedormaneuverabilityindifferentoperatingregimes.Finswithtunablestiffnessareprototyped,andfreeoscillationandbase-excitedoscillationbehaviorsofthearemeasuredunderwaterwhendifferentelectricdsareappliedfortheERwhicharesubsequentlyusedtodevelopadynamicmodelforthestiffness-tunableFinally,thelastchaptersummarizesthemostimportantaspectsandresultsachievedinthiswork.Possiblefuturedevelopmentsareoutlinedaswell.1.6Publications1.6.1JournalArticles1.S.B.Behbahani,X.Tan,fiDesignandModelingofFlexiblePassiveRowingJointsforRoboticFishPectoralFins,flIEEETransactionsonRobotics,Vol.32,No.5,pp.1119-1132,2016.112.S.B.Behbahani,X.Tan,fiBio-inspiredFlexibleJointswithPassiveFeatheringforRoboticFishPectoralFins,flBioinspiration&Biomimetics,Vol.11,No.3,pp.036009,2016,[Featuredarticle].3.S.B.Behbahani,X.Tan,fiRoleofPectoralFinFlexibilityinRoboticFishPerformance,flJournalofNonlinearScience,Underreview,2016.4.S.B.Behbahani,X.Tan,fiDesignanddynamicmodelingofvariablestiffnessforroboticusingelectrorheologicalflUnderpreparation,2016.1.6.2ConferenceProceedings1.S.B.Behbahani,X.Tan,fiDynamicModelingofRoboticFishCaudalFinwithElectrorhe-ologicalFluid-EnabledTunablestiffness,flProceedingsoftheASMEDynamicSystemsandControlConference(DSCC),Columbus,OH,USA,Oct.2015,pp.V003T49A006(9pages).2.S.B.Behbahani,X.Tan,fiDesignandDynamicmodelingofaFlexibleFeatheringJointforRoboticFishPectoralFins,fl[Invited],ProceedingsoftheASMEDynamicSystemsandControlConference(DSCC),SanAntonio,TX,USA,Oct.2014,pp.V001T05A005(9pages),[FinalistforBestStudentPaperAward].3.S.B.Behbahani,X.Tan,fiAFlexiblePassiveJointforRoboticFishPectoralFins:Design,DynamicModeling,andExperimentalResults,flProceedingsoftheIEEE/RSJInternationalConferenceonIntelligentRobotsandSystems(IROS),Chicago,IL,USA,Sept.2014,pp.2832-2838.4.S.B.Behbahani,J.Wang,X.Tan,fiAdynamicmodelforroboticwithxiblepec-toralflProceedingsoftheIEEE/ASMEInternationalConferenceonAdvancedIntelligentMechatronics(AIM),Wollongong,Australia,Jul.2013,pp.1552-1557.12Chapter2ROLEOFPECTORALFINFLEXIBILITYINROBOTICFISHPERFORMANCE2.1IntroductionModelingtherobotichdynamicsisoftencriticalforthedesign,control,andunderstandingofroboticbehavior,andithasreceivedextensiveattentionintheliterature[20,28,39,45,70Œ72].Themostchallengingstepindynamicallymodelingroboticiscapturingtheinteractionsbetweentheandtheandcalculatingtheresultingforceandmomentexertedonthebody.ComputationalFluidDynamics(CFD)modeling[47Œ49,73]iscapableofdescribingsuchinteractionswithhighandofferingphysicalinsight,butitscomputationalcostoftenmakesitinfeasiblefordesigningandcontrolpurposes.Severalalternativesareavailable.Forexample,quasi-steadyliftanddragmodelsfromairfoiltheorycanbeappliedtothebodyandsurfacesofunderwaterrobots[27,44,74Œ76].Onecouldalsoassumeperfect(irrotationalpotentialw)andexploitthesymmetrytoobtainamodelfortheinteractions[45,77].Effectsofvorticitycanbeaccommodatedbyassuming,forexample,vorticesperiodicallyshedfromthetail[46,78].InthischapterwearefocusedonpairedpectorallocomotionofaroboticAlthoughthecaudalistheprimaryappendageusedforpropulsioninroboticsh[20,23,28,34],pectoral13playavitalroleinmaneuveringandstabilityoflivewhileprovidingorassistingpropulsion[79,80].Kinematicsandhydrodynamicsoflivepectoralhavebeenstudiedbeseveralresearchers[50Œ53].Anumberofinvestigationshavebeenconductedonrigidpectoralwithoneormoredegreesoffreedom,tostudytheireffectonroboticswimming[27,35,36,44,55,81,82].Recently,studieshavebeenpursuedonxiblepectoral[21,30,32,61Œ64,83],andpectoralwithxiblejoints[84Œ87],tomimiclivebehaviormoreclosely.Whilethexibilityofpectoralnsforliveandroboticisappreciatedandhasbeenexploredwithbothexperimental[30,61]andCFD[49]methods,systematicanalysisoftheroleofpectoralxibilityinroboticperformancehasbeenlimited.Thegoalofthischapteristodevelopasystematic,computation-efframeworkforana-lyzinghowthexibilityofpectoralaffectstheswimmingperformanceandmechanicalefciencyoftherobot.Whilepectoralmotionscangenerallybeintothreemodesbasedontheaxisofrotation,rowing,feathering,andwefocusontherowingmotionsinceitcanbeutilizedforanumberofin-planelocomotionandmaneuveringtasks,suchasforwardswim-ming,sidewayswimming,andturning.Adynamicmodelfortheroboticisproposed.WeuseNewton-Eulerequationstomodeltherigidbodydynamicsoftherobot.Thexibleareapproximatedasmultiplerigidelements,connectedviatorsionalspringsanddampers.Suchanapproachhasbeenproveneffectiveandcomputationallyefincapturinglargedeformationinxiblecaudal[29].Bladeelementtheoryisadoptedtoevaluatethehydrodynamicforcesontheelements.Theproposeddynamicmodelisverthroughexperimentsonafree-swimmingroboticprototypeequippedwithpectoralTheare3D-printedwithdifferentstiffnesslevels,rangingfromveryxibletorigid.Thedynamicmodelisusedtoanalyzetheimpactofxibilityonswimmingspeedbehavioratdifferentfrequencies.Itisfoundthat,whiletherobotwithrigidachievesalinearly14increasingspeedwiththefrequency,thereisanoptimalfrequencyinthecaseofaxibleatwhichthespeedismaximized.Furthermore,withmoderatexibilityoutperformtherigidonthespeedperformanceforalargerangeofoperatingfrequencies.TheimpactofspeedratiozbetweenthepowerstrokeandtherecoverystrokeontheswimmingbehaviorisalsostudiedfordifferentFinally,theofxibilityandzonthemechanicalefyoftherobotareexamined.Thestudyrevealsinterestingtrade-offsbetweenthedifferentobjectives(speedandefy)andsupportstheuseoftheproposedmodelasapromisingtoolforxibilityandgaitdesign.Theremainderofthischapterisorganizedasfollows.Section2.2reviewsthedynamicsoftheroboticbody.Section2.3presentsthedynamicmodelforthexiblepectoralThekinematicsofthexiblepectoraladoptedinthisworkispresentedinSection2.4,whilethemethodformechanicalefcycalculationisdiscussedinSection2.5.InSection2.6wepresenttheroboticprototype,thepectoralfabricationmethod,theexperimentalsetup,andthemethodsformodelparameterSection2.7isdevotedtoresults.Finally,concludingremarksandfutureworkdirectionsarediscussedinSection2.8.2.2DynamicsoftheRoboticFishBodyTheroboticunderstudycontainstwosubsystems,thexiblepectoralforwhichthedy-namicsarecoveredinSection2.3,andtherigidbody.Theroboticprototypeusedinthisstudydoeshaveacaudalbuttheconsiderationofcaudalhydrodynamicsisoutsidethescopeofthiswork.Tostudythemotionoftherobot,weincludetherigidbodydynamicsbasedonKirch-hoff'sequationsofmotioninaninviscid[28,88],withtheaddedmasseffectincorporated.15Figure2.1:Topviewoftheroboticactuatedbyxiblepectoralandrestrictedtoplanarmotion.2.2.1RigidBodyDynamicsFig.2.1illustratesaroboticrestrictedtotheplanarmotion,where[X;Y;Z]Tdenotestheglobalcoordinates,and[x;y;z]Twithunitvectors[‹i;‹j;‹k]indicatestheedcoordinates,attachedtothecenterofmassoftheroboticbody.Eachpectoralismodeledasmultiple,connected,rigidelements,and‹miand‹nidenotetheunitvectorsparalleltoandperpendiculartotheithel-ementofthepectoralrespectively,wheresuperscriptsrandlindicatetherightandtheleftrespectively.Theroboticbodyandthexiblepectoralareconsideredtobeneutrallybuoyant,withcenterofgravityandgeometrycoinciding.Theequationsoftherobotic16bodyintheedcoordinatesarerepresentedas[29,63,87](mbXVCx)VCx=(mbYVCy)VCywCz+Fx;(2.1)(mbYVCY)VCy=(mbXVCx)VCxwCz+Fy;(2.2)(IzNwCz)wCz=Mz;(2.3)wherembisthemassoftheroboticbody,Izistherobotinertiaaboutthez-axis,XVCx,YVCy,andNwCzarethehydrodynamicderivativesthatrepresenttheeffectoftheaddedmass/inertiaontherigidbody[88].VCx,VCy,andwCzarethesurge,sway,andyawvelocities,respectively.ThevariablesFx,FyandMzdenotetheexternalhydrodynamicforcesandmomentexertedonthebody,whicharedescribedasFx=FhxFDcosb+FLsinb;(2.4)Fy=FhyFDsinbFLcosb;(2.5)Mz=Mhz+MD;(2.6)whereFhx,FhyandMhzarethehydrodynamicforcesandmomenttransmittedtothebodybythepectoralthedetailsofwhichareprovidedinSection2.3.AsdepictedinFig.2.1,FD,FL,andMDarethebodydrag,lift,andmoment,respectively.Theseforcesandmomentareexpressedas[28,29,44]FD=12rjVCj2SACD;(2.7)FL=12rjVCj2SACLb;(2.8)MD=CMw2Czsgn(wCz);(2.9)17wherejVCjisthelinearvelocitymagnitudeoftheroboticshbody,jVCj=qV2Cx+V2Cy,risthemassdensityofwater,SAisthewettedareaofthebody,CD,CLandCMarethedimensionlessdrag,lift,anddampingmomentcoefrespectively,bistheangleofattackofthebody,andsgn(:)isthesignumfunction.Finally,thekinematicsoftheroboticaredescribedas[29]X=VCxcosyVCysiny;(2.10)Y=VCycosy+VCxsiny;(2.11)y=wCz;(2.12)whereydenotestheanglebetweenthex-axisandtheX-axis.2.3DynamicsofFlexibleRowingPectoralFinsThissectionfocusesoncalculatingthehydrodynamicforcesonthexiblepectoralanddeter-miningitsdynamicsusingbladeelementtheory[5],bydividingthespanintomultiplerigidelements.Therowingmotionofthepectoralhasbeenasafidrag-basedflswimmingmechanism,wherethedragelementofdynamicsgeneratesthethrust[89,90].ThepectoralisconsideredtoberectangularwithspanlengthofSandchordlengthofC.Weconsideracoordinatesystemwithunitvectors‹miand‹niforthepectoralseeFig.2.1.Therelationshipbetweentheseunitvectorsandtheroboticedcoordinatesisgivenby‹mi=cosgi‹i+singi‹j;(2.13)‹ni=singi‹i+cosgi‹j;(2.14)18Figure2.2:Topviewofithelementofthexibleanditsparametersandvariables.wheregiistheanglebetween‹miand‹i;seeFig.2.1forillustration.Weusetheleftpectoraltoillustratethecalculations.However,itisstraightforwardtoextendthecalculationstotherightpectoralFig.2.2providesavisualizationoftheparametersandvariablesofaxibleelement.WedividethexibleintoNrigidelementswithanequallengthofl=SN,whereeachsegmentisconnectedtoitsneighborsviaapairoftorsionalspringanddamper.Theconstantsofthespringanddampercanbederivedfromthepropertiesofthexiblematerial.ThespringconstantKSisevaluatedas[91]KS=ECh312l;(2.15)wherehisthebeamthickness,andEistheYoung'smodulusofthexiblematerialusedforthepectoralThedampercoefKDcanbeevaluatedasKD=kKS,wherekisapropor-tionalconstant.Fig.2.3illustratestheforcesontheithelementofthexibleTheangleg1isdictatedbythepectoralactuator,andweneedtotheanglesg2togN,toknowthetra-jectoryofthexibleateachinstantoftime,which,subsequently,willallowonetoevaluatethehydrodynamicforcegeneratedbytheInthefollowingcalculationsweassumeananchoredroboticbody.Thissimplifyingassumptionisoftenadoptedintheliteratureforsimilarprob-19Figure2.3:Illustrationofforcesontheithelementofthexiblelems[2,19,29],andtheresultingerroristypicallynegligibleconsideringthemuchgreaterpectoralvelocitycomparedtotheroboticbodyspeed.Thepositionofeachpointsontheithelement(seetheofsinFig.2.2)attimetontheithelementcanbedescribedasri(s;t)=i1åk=1l‹mk+s‹mi:(2.16)Thecorrespondingvelocityatthepointsisvi(s;t)=ni1åk=1lgkcos(gigk)+sgio‹ni+ni1åk=1lgksin(gigk)o‹mi;(2.17)wheregkindicatesthetimederivativeofgk.Respectively,thecorrespondingaccelerationatthe20pointsisevaluatedasai(s;t)=ni1åk=1l¨gkcos(gigk)+lg2ksin(gigk)+s¨gio‹ni+ni1åk=1l¨gksin(gigk)lg2kcos(gigk)sg2io‹mi;(2.18)where¨gkindicatesthesecondtimederivativeofgk.2.3.1ForceCalculationsforElementiThehydrodynamicforceoneachelementofthexibleisevaluatedbasedonthebladeelementtheory[5]asfollowsdFhi(s;t)=12Cnai(s;t)rCjvi(s;t)j2ds‹evi;(2.19)where‹eviisaunitvectorinthedirectionofthevelocityoftheithelement.Cnistheforcecoefcient,whichdependsontheangleofattackateachpoint,ai(s;t),andhasaformofCn(ai)=lsinai;(2.20)wheretheparameterlisevaluatedempiricallythroughexperiments.NotethatEq.(2.19)capturesbothnormalandspan-wisecomponentsofthehydrodynamicforce.Theangleofattackateachpoint,ai(s;t),isviatanai(s;t)=<vi(s;t);‹ni><vi(s;t);‹mi>;(2.21)where<;>denotestheinnerproduct.ThetotalhydrodynamicforceactingontheithelementisbyintegratingEq.(2.19)21alongthelengthoftheelementFhi(t)=Zl0dFhi(s;t):(2.22)Theinteractionbetweentwoconsecutiveelementsiscapturedviaforcebalanceoneachele-ment:Fhi+FAi1FAi=miai;(2.23)whereFAiistheforceappliedbyelementionelementi+1,FAi1istheforceappliedbyelementi1onelementi,miistheeffectivemassoftheithelement(whichcontainsthemassandtheaddedmassofelementi,wheretheaddedmassiscalculatedbasedonarigidplatemovinginthewater),andaidenotestheaccelerationofthemidpointoftheithelement,whichcanbeevaluatedusingEq.(2.18)withs=l2.SeeFig.2.3forillustrationoftheforces.Notethatforthelastelement,FAN=0;therefore,Eq.(2.23)canbesolvediterativelyforFAi,where0iN1.2.3.2MomentCalculationsforElementiThetotalhydrodynamicmomentontheithelementiscalculatedasMhi=Zl0s‹midFhi(s;t);(2.24)wheredFhi(s;t)isasexpressedinEq.(2.19).ThetotalmomentrelativetopointAi1forelementiisevaluatedasMi=Mhi+lFAi+Mi+1+M(S+D)i=Ii¨gi;(2.25)22wheredenotesthecrossproduct,Iirepresentstheeffectiveinertiaoftheithelement,andM(S+D)i,themomentinducedbythetorsionalspringanddamperatAi1,isevaluatedasM(S+D)i=[KS(gigi1)+KD(gigi1)]‹k;(2.26)whereKSandKDarethespringanddampercoefusedtomodelthexiblepectoralNotethatfortheelementN,MN+1=0,which,throughtherecursionintheequalityin[25],allowstheexplicitexpressionofMi,2iN,intermsofothervariables.Thesecondequalityin(2.25)thusprovides(N-1)nonlinearsecond-orderequationsforgi,where2iN,whichfullydescribethedynamicsoftheN-elementpectoralEqs.(2.23)and(2.25)areusedtothetotalforce/momentexertedontheroboticbody:Fhx=<FA0;‹i>;(2.27)Fhy=<FA0;‹j>;(2.28)Mhz=cp<FA0;‹i>M1;(2.29)wherecpisthedistancebetweenroboticcenterofmassandthebaseofthexibleFA0isthetotalforceappliedbythexiblepectoraltothecenterofmassoftheroboticbody,andM1isthemomentappliedbythexiblepectoraltothecenterofmassoftheroboticbody.Theseforcesandmomentsarethenpluggedinto(2.4)-(2.6)tosolvetherigidbodydynamicsoftherobotic232.4KinematicsofFlexibleRowingPectoralFinsThepectoralinthisstudysweepbackandforthwithinthefrontalplane.Therowingmotionhasdistinctpowerandrecoverystrokes.Duringthepowerstroke,thepectoralmovesbackwardtoproducethrustthroughinduceddragonthepectoralsurface.Ontheotherhand,duringtherecoverystroke,themovestowardthefrontofthewithminimalloading,togetreadyfornextcycle.Thepectoralisactuatedatdifferentspeedsduringthepowerandrecoverystrokes(fasterpowerstroke)toproduceanetthrust.Inparticular,wetheratioz=PR=PowerstrokespeedRecoverystrokespeed:(2.30)Foreachcycle,thepectoralbaserotatesaccordingtog1(t)=8>>><>>>:p2gAcoshpz+1Tpti;0tTpz+1p2+gAcoshpz+1zTptTpz+1i;Tpz+1<tTp(2.31)wheregAistheamplitudeoftheactuationandTPdenotestheperiodofonecycle.Fig.2.4illustratesthepectoralkinematics.ThevisualizationofthepectoralmotionduringonecycleisshowninFig.2.4(a).Notethatforz=1,thepectoralsym-metricallyduringthepowerstrokeandtherecoverystroke,andforz>1,thepectoralslowsdownandspendsmoretimeintherecoveryphase.Fig.2.4(b)illustratestheorientationangleg1andtheangularvelocityg1ofthebaseofthepectoralwithrespecttothex-axisoftheroboticduringonecycle.Forturningtheroboticwejustactuateoneofthepectoralandkeeptheotherstill.24(a)(b)Figure2.4:Illustrationofthepectoralmotionduringpowerandrecoverystrokes:(a)Thesnapshotsatdifferenttimeinstancesofonecycle,whereTprepresentsthetotaltimeforeachcycle;(b)orientationandangularvelocityofthebaseofthepectoralwithrespecttothemainaxisoftheroboticwherez=5,gA=50deg,andTp=1sareused.2.5MechanicalAsidefromtheswimmingperformance(forexample,swimmingspeedorturningradius),anothercriticalfactorabouttheroboticisitsmechanicalefy.WewillanalyzetheefyoftherobotunderdifferentpectoralpropertiesusingthemodelpresentedinSections2.3and2.4.Inparticular,itisofinteresttoinvestigatetheeffectofxibilityofthepectoralonthemechanicalefy.Duringsteady-stateswimming,themechanicalefyoftherobotisevaluatedas[5,92,93]h=WbWT;(2.32)whereWbistheusefulworkneededtomovetheroboticduringeachcycle,andWTisthetotalworkdonebythepectoralduringthesameperiod.Inthisstudy,wedonotconsider25otherenergylosses,suchastheamountofelectricalpowerusedbyotherelectronics,orfrictionallossesinmotorsandgears.TheusefulworkWbisdeterminedbyWb=Zt0+Tpt0FThrust(t)VCx(t)dt;(2.33)wheret0denotesthebeginningofthecycle,andFThrust=Fhxisthetotalhydrodynamicforceproducedbythepectoralinthexdirection,exertedontheroboticbody.ThetotalworkdonebythepairedpectoralWT,isobtainedviaWT=2Zt0+Tpt0maxf0;Nåi=1Zl0<dFhi(s;t);vi(s;t)>gdt:(2.34)Atsomeinstantsoftime,themechanicalpowerofthepectoralcouldbenegative.However,theservoscannotreclaimthisenergyfromthewater.Therefore,wetreattheinstantaneouspowerforthesecasestobezero,whichexplainsthemax(0;)operationin(2.34).2.6MaterialsandMethods2.6.1RoboticFishPrototypeAroboticprototype,similartothatin[86],wasusedtotestandvalidatetheproposeddynamicmodel,andtosupporttheperformanceanalysis.Thisroboticincludedarigidbody,twopectoralandacaudalTherigidbodywasdesignedinSoildWorkssoftwareandprintedintheVeroWhitePlusmaterialfromaPolyJetmulti-material3Dprinter(Objet350Connex3DSystemfromStratasys),andwascoatedwithacrylicpainttominimizewaterabsorptionofthe3D-printedmaterial.Thebodyhadalengthof15cm,heightof8cm,andwidthof4.6cm.Three26Figure2.5:3D-printedroboticbodyandothercomponentsoftherobot.waterproofservomotors(Traxxas2065WaterproofSub-MicroServofromTraxxas)wasusedtomovethepectoralandcaudalindividually.Therobotwasbattery-operated,usingaLi-ionrechargeablebattery(7.4V,1400mAhfromPowerizer),andacustomizedpowerconverterPCBwasdesignedtoregulatethevoltageto5V(throughLM2673)and3.3V(throughLP38690)fortheservomotorsandthemicrocontroller,respectively.Therobotmotionwascontrolledbyamicrocontroller(ArduinoProMini,3.3V).Apictureofthe3D-printedbody,alongwithalltheothercomponentsfortherobot,isshowninFig.2.5.2.6.2PectoralFinFabricationThepectoralusedinthisstudyweredesignedtohaveacompositestructureandwere3D-printed.Thecouldbeeasilydetachedfromthemounts,toenabletestingoftherobotwithdifferentWefollowedtheapproachproposedin[94]forcreatingthecompositetheeffectivestiffnessofwhichcanbeeasilyadjustedbychangingthethicknessesofdifferentlayers.Thecompositeweremadeoftwodifferentmaterials,arigidplasticmaterial(VeroWhitePlus)andaxiblerubber-likematerial(TangoBlackPlus).ThedetailsofthedesignareillustratedinFig.2.6.ThexibilityofthewascontrolledbyadjustingthethicknessoftheVeroWhitePlus27Figure2.6:SolidworksdesignofacompositepectoralTable2.1:ofpectoralwithdifferentxibilities.Finnametinner(mm)KS(Nm)KD(Nms)F11.2NANAF20.31:131032:59104F301:831044:37105innerlayerwhilekeepingthetotalthicknessoftheconstant(1.2mm).Pectoralofsize45mm37.5mm,withthreedifferentstiffnessvalueswerefabricated.TheofthesearesummarizedinTable2.1.ThearenamedF1ŒF3forlaterreferenceinthispaper.The3D-printedpectoralweretreatedwithathincoatofUltra-EverDryomniphobicma-terial(UltraTechInternationalInc.)topreventchangesofpropertiesthatmayoccurwhentheycomeincontactwithwater.Fig.2.7showsanactual3D-printedpectoralbeforeandaftertheUltra-EverDrytreatment.TheapplicationofUltra-EverDryresultsinawhitecastonthetreatedpart.28Figure2.7:Anactual3D-printedpectoralRight:originalleft:printedtreatedwithUltra-EverDryomniphobicmaterial.(a)(b)Figure2.8:Experimentalsetup:(a)Actual,(b)outputoftheMotivesoftware.2.6.3ExperimentalSetupExperimentswereconductedina6ftlong,2ftwide,and2ftdeeptank.AnOptitrackmotioncapturesystem,containingfourFlex13cameras,eachmountedonheavy-dutyuniversalstandviaManfrottosuperclampand3Djuniorcamerahead,wereusedtotracktheroboticswimming.AcomputerequippedwithMotive1.7.5software,capableofsupportingreal-timeandofwork-ws,wasusedtoextractthedesireddata.ThedetailsofthissetupareshowninFig.2.8.Twodifferentlocomotionmodes,forwardswimmingandturning,wereadoptedfortheroboticintheexperiments.Foreachexperiment,theroboticswamforapproximately30secondstoreachitssteady-statemotion,andthenthesteady-statedatawerecapturedandextracted.Inthe29forwardswimmingcase,werecordedthetimeittookfortherobottoswimadistanceof50cm.Theexperimentforeachsettingwasrepeatedtentimes,tominimizetheimpactofrandomfactorsontheexperimentalresults.2.6.4ParameterAlltheparametersusedinthesimulationswereeithermeasureddirectlyorexperimen-tally.Detailsoftheprocedureforparametersoftheroboticbodycanbefoundin[86,87].Themassoftheroboticwasmb=0:502kgandtheinertiawasevaluatedtobeIz=7:26104kg/m2.Theaddedmassesandaddedinertiawerecalculatedbasedonapro-latespheroidapproximationfortheroboticbody[28,88],whichwereXVCx=0:1619kg,YVCy=0:3057kg,andNwCx=5:52105kg/m2.ThewettedsurfaceareaoftheroboticbodywasSA=0:0325m2.Thedrag,lift,andmomentcoefwereempiri-cally,usingtherigidpectoralandz=2.Inparticular,theseparametersweretunedtomatchtheforwardvelocity,turningradius,andturningperiodobtainedinsimulationwiththeexper-imentalmeasurement,whentwodifferentpowerstrokespeedsareused,completingthepowerstrokein0.5sand0.3s,respectively.TheresultingcoefwereCD=0:42,CL=4:86,andCM=7:6104kg/m2.TheparameterlfromEq.(2.20),whichrepresentsthehydrodynamicforcecoefofthepectoralwasempiricallytobe0:6,bymatchingtheforwardswimmingvelocityoftheroboticutilizingrigidpectoralforthefrequenciesof1Hzand1:667Hz,andpower/recoverystrokeratioz=2.Thisparameterwasthenusedforalltheotherfrequencies,pectoralxibilities,andzratios.Thespringanddampercoefofthexibleweretunedtomatchtheforwardswim-mingvelocityandtheturningperiodoftheroboticcollectedfromexperimentsforz=2,andfrequenciesof1Hzand1:667Hz.Eachisdiscretizedintothreeelements.Thepa-30rametersforeacharesummarizedinTable2.1.Thesenumberswerethenusedforalltheotherfrequenciesandzratiosthroughoutthesimulations.2.7ResultsandAnalysis2.7.1DynamicModelValidationFirst,wehaveincludedplotsofsimulationresultstoillustratehowthexiblepectoralmoves.Thetimehistoryoftheanglesofdifferentelementsofthexiblepectoralandtherespectiveanglesofattack,isshowninFig.2.9.Inthissimulation,thexibleF2wasused.Notetheclearphaselagsbetweentheconsecutiveelements'anglesgi.Theanglesofattackforallelementsareconstant(90or-90degrees)formostofthetime,buttheyexperienceabruptchangesduringthetransitionbetweenpowerandrecoverystrokes.Next,wepresentresultsthatvalidateourproposeddynamicmodel,wherethecaseofpectoralF2withz=2isused.AdditionalresultssupportingthemodelcanbefoundinSection2.7.2.Fig.2.10showsthecomparisonbetweenmodelpredictionsandexperimentalmeasurementsoftheforwardswimmingvelocity,reportedinbothcm/sandBL/s(bodylengthpersecond)versusthefrequency.Thefrequencyisas1Tp,whereTpisthedurationofeachcycle.Intheexperiments,forz=2,thespeedlimitoftheservomotorstranslatedtoamaximumactuationfrequencyof2Hz,sowehaveextendedthesimulationresultstofrequencyof3HzinordertocapturetheperformancetrendoftheroboticFromFig.2.10,thespeedoftheroboticdropsafterthefrequencyreachesanoptimalvalue,beyondwhichitgetsharderforthexiblepectoraltofollowtheprescribedservomotion.Figs.2.11and2.12showacomparisonbetweensimulationandexperimentalresultsontheturningperiodandtheturningradiusversusfrequency,respectively.InFig.2.11,theturningperiod31(a)(b)(c)Figure2.9:Timehistoryofxiblepectoral(a)Evolutionoftherigidelementanglesg1Œg3,foronemovementcycle,(b)variationoftheanglesofattackforallelements,wherez=5,gA=50deg,andTp=3s,(c)zoom-inviewoftheangleofattackforonemovementcycle.32Figure2.10:CaseofF2withz=2:Comparisonbetweendynamicmodelsimulationandexperi-mentalmeasurementoftheforwardswimmingvelocity,fordifferentfrequencies.Figure2.11:CaseofF2withz=2:Comparisonbetweendynamicmodelsimulationandexperi-mentalmeasurementoftheturningperiod,fordifferentfrequencies.Figure2.12:CaseofF2withz=2:Comparisonbetweendynamicmodelsimulationandexperi-mentalmeasurementoftheturningradius,fordifferentfrequencies.dropswiththebeatrate,asexpected,uptoaparticularoptimalfrequency,beyondwhichitstartstoincrease.TheoptimalfrequencyinFig.2.11coincideswiththatinFig.2.10,whichis33Figure2.13:Experimentalandsimulationresultsontheforwardswimmingvelocityversusbeatfrequencyfordifferentxibilitiesofthepectoralwherethepower/recoverystrokeratioz=2forallcases.notsurprising.ThesimulationresultsinFig.2.12suggestthattheturningradiusoftheroboticisnearlyindependentoffrequencies.Theexperimentaldatasupportthesimulationtosomeextent.Themeanoftheturningradiusremainsalmostthesame(between16and18cm)fordifferentfrequencies.Duetothedisturbancesfromtheinteractionbetweentheandthetankwall,therobotictypicallywouldnotstayrepeatedlyonthesameorbitineachturningexperiment,whichinevitablyintroducednoticeableerrorintheturningradiusmeasurement.Overall,itcanbeconcludedfromFigs.2.10-2.12thattheproposeddynamicmodelcanwellcapturethemotionoftheroboticactuatedbyxiblepectoralAdditionalresultsinSection2.7.2willfurthervalidatethemodelthroughgoodmatchbetweenexperimentalandsimulationresults.2.7.2ImpactofFinStiffnessHerewecomparetheperformanceofpectoralwithdifferentxibilitiestogaininsightintotheofxibility.Weutilizedapatternasedin(2.31),wherez=2.Thesimulationandexperimentalresults,showninFig.2.13,arereportedbothincm/sandBL/s.Again,wehaveextendedthesimulationresultstofrequencyof3Hzinordertocapture34theperformancetrendoftheroboticFromFig.2.13,itisinterestingtonotethatwithrigid(F1),theswimmingvelocityincreasesnearlylinearlywiththefrequency,whileforthexible(F2andF3),thereisaclearoptimalfrequencyatwhichthevelocityismaximum.Finswithmoderatexibilityoutperformtherigidfortheentirefrequencyrangeachievablebytheroboticprototype.Theoptimalfrequencyofoperationforthexiblecaseisbelievedtobecorrelatedtotheresonantfrequencyoftheninwater,anditdropsasthexibilityoftheincreases.Whentheistooxible(F3),theroboticsspeedperformancebecomespoor.2.7.3ImpactofthePowertoRecoveryStrokeSpeedRatioThissubsectionisdevotedtostudyingtheeffectofthepowertorecoverystrokespeedratio(z)ontheperformanceoftheroboticForaedpowerstrokespeed,ahigherzresultsinasloweroverallfrequency,whichcouldimplyaslowerswimmingspeedfortherobot;ontheotherhand,ahigherzmeansslowerrecoverystrokespeed,whichmeansweakerfibrakingflforceandpotentiallyhelpsincreasetheoverallswimmingspeed.Fig.2.14showstheexperimentalresultsontheforwardswimmingvelocity,comparingthecasesofz=2and4.HerethexibilityspansF1ŒF3.Fig.2.14(a)showsthevelocityversusfrequency.Overall,itcanbeseenthatthehigherzprovidesbetterperformanceforeachxibilityateachfrequency.Notethattherightmostpointineachcurvecorrespondstothemaximumspeedthattheservocanhandle.Fig.2.14(b)presentsthecomparisonontheswimmingvelocityversusthepowerstroketime(duration).Notethatwhiletheperformanceoftheroboticwiththelargerratio,z=4,outperformsthecaseofz=2mostofthetime,forthemostxibleF3,thecaseofz=2outperformsthatofz=4whenthepowerstroketimeissmall(orequivalently,athighpowerstrokespeeds).Therefore,inthefollowingweusemodel-basedsimulationtofurtherstudytheimpactofthepowertorecoverystrokespeedratio.35(a)(b)Figure2.14:Experimentalcomparisonoftheforwardswimmingvelocitiesatdifferentpower/recoverystrokespeedratiosz=2andz=4.(a)Velocityversusfrequency,and(b)Velocityversuspowerstroketime.Fig.2.15providesacomparisonwithinawiderrangeofpower/recoverystrokespeedratiosz=1.5,2,3,4,and5,forpectoralF1ŒF3,andforanextendedfrequency.Inallsimulationsthemaximumservospeedised,whichcorrespondstoafrequencyof3Hzwhenz=2.Therefore,therightmostpointoneachcurverepresentsthehighestfrequencyachievablewiththecorrespondingzundertheservoconstraint.FromFig.2.15(a),onecanconcludethatfortherigidpectoralF1,ahigherzprovidesafasterswimmingspeedateachfrequency,whichmatchestheresultsfromFig.2.14.Ontheotherhand,the36(a)(b)(c)(d)(e)(f)Figure2.15:Simulationcomparisonoftheforwardswimmingvelocitiesatdifferentpower/recoverystrokespeedratiosz=1.5,2,3,4,and5.(a)VelocityversusfrequencyforcaseofF1,(b)velocityversuspowerstroketimeforcaseofF1,(c)velocityversusfrequencyforcaseofF2,(d)velocityversuspowerstroketimeforcaseofF2,(e)velocityversusfrequencyforcaseofF3,(f)velocityversuspowerstroketimeforcaseofF3.highestachievableswimmingspeedundertheconstraintoftheservospeed,doesnothappenatthehighestz.Instead,thehighestswimmingspeedisobtainedifamoderatez(2or3)isused.FromFigs.2.15(c)and(e),itcanbeobservedthat,forxibletheswimmingspeedisnolongeramonotoneincreasingfunctionofzateachfrequency.Thereisacriticalvaluez,beyond37Figure2.16:Calculatedmechanicalefyversusfrequencyandspringconstantofthexiblewhichahigherzresultsinaslowerrobotspeedforagivenfrequency.Inaddition,bycomparingFigs.2.15(c)and(e),themorexibletheis,thelowerz;inparticular,z=4forF2,andz=3forF3.Foreachxiblethemaximumswimmingspeedachievablewithservoconstraintstakesplaceattheoptimalfrequencywhenthepower/recoverystrokespeedratioisz.Similarly,Figs.2.15(b),(d),and(f)presentthecomparisonontheswimmingvelocityversusthepowerstroketime.SincewetheamplitudegAofthebeat,eachpowerstroketimecorrespondstoaparticularpowerstrokespeed.Fromtheitisclearthat,foreachforanygivenpowerstrokespeed,thereisanintermediatevaluezthatachievesthebestswimmingspeed.Inotherwords,foragivenpowerstrokespeed,thereisanoptimalrecoverystrokespeed,nottoohigh,nottoolow.2.7.4ImpactofFlexibilityonMechanicalFinallywefocusonthemechanicalefyoftheroboticFig.2.16showstheefycurveversusfrequencyandspringconstantvaluesofthexibleHerethecalculationsarebasedonpower/recoverystrokespeedratioz=2.Weconsideredeightdifferentwithdifferentspringconstants,wherethedampertospringconstantratio,k,waskeptat0.24.This38valuematchestheratiosobtainedfromthetestedpectoralF2andF3inthiswork(0.23and0.24,respectively,fromtheparametersinTable2.1).Thethatthereisanoptimalxibilityforeachfrequency,thatresultsthehighestmechanicalefyfortheroboticAdditionalinsightcanbedrawnbasedonFig.2.17(a),wherethecomputedmechanicalefyoftherobotisshownasafunctionofthefrequency,forallthreeF1ŒF3.Itcanbeseenthatthemechanicalefyoftherigid(F1)slightlyincreaseswiththefrequencies.Ontheotherhand,xibletendtobemoreefatlowerfrequencies.Infact,theefywithF2ishigherthanthatwithF1whenforfrequencieslowerthan2.7Hz;andevenF3outperformsF1onefyuntilthefrequencyreachesabout1.2Hz.Tobringadditionalinsightintotheroboticperformanceanalysis,onecanfurtherstudysomenon-dimensionalizeparametersoftheroboticTheseparametersincludetheReynoldsnumber,Re=jVCjLn,andtheStrouhalnumberSt=2fSsingAjVCj,wherejVCjistheswimmingspeedoftherobot,Listheroboticlength,nisthekinematicviscosityofwater,Sisthepectoralspanlength,gAistheamplitude[27,35],andfisthefrequency.Anothernon-dimensionalizeparameterofinterestisthedimensionlessvelocity,whichisasVDL=jVCjSwA,wherewA=p(z+1)Tp.ThesedimensionlessparametersarereportedinFig.2.17forpectoralwithdifferentxibilitiesandz=2,versusdifferentfrequencies.Fig.2.17(a)showsadistinctinversecorrelationbetweentheefyandtheStrouhalnumber.Foreachxibility,whentheStrouhalnumberisatitslowest,theefyishighest.Fig.2.17(b)showsthattheroboticdemonstratesthehighestdimensionlessvelocitywhentheReynoldsnumberisatthelowerend,exceptfortherigid(F1),wherethedimensionlessvelocityremainsalmostconstant.Fig.2.17(c)showsthattheoptimalfrequencyfortheforwardswimmingvelocityoftheroboticforeachdoesnotcoincidewiththeoptimalfrequencyfortheefy.Therefore,themodelpresentedinthispapercanbeusedasatooltoaddressanoptimal,multi-objectivedesignproblem.39(a)(b)(c)Figure2.17:Comparisonofnon-dimensionalizedparametersoftheroboticactuatedbypairedpectoralwithdifferentxibilitiesversusdifferentfrequenciesforthecaseofz=2:(a)CalculatedefyandStrouhalnumber,(b)dimensionlessvelocityandReynoldsnumber,and(c)forwardswimmingvelocityincm/sandefy.40NotethattheStrouhalnumbersshownherearehigherthan1,whicharefarfromtheStrouhalnumbersofbiological(typicallyintherangeof0.25-0.5[35,95,96]).ThereasonisthattheroboticusedinthisstudyisdrivensolelybythepectoralwhichresultsinrelativelylowforwardswimmingspeedsandthuslargeStrouhalnumbers.2.8ConclusionThegoalofthischapterwastostudytheimpactofthepectoralxibilityontheroboticper-formanceandmechanicalefy.Weintroducedanoveldynamicmodelforaroboticac-tuatedwithpairedpectoralwheretheismodeledasmultiplerigidelementsjoinedthroughtorsionalspringsanddampersandbladeelementtheoryisusedtocalculatethehydrodynamicforcesontheelements.Thedynamicmodelwasvalidatedthroughexperimentsconductedonaroboticwheretheswimmingandturningperformanceoftherobotwasmeasuredforwithdifferentxibilities(twoxible,onerigid).Themodelwasthenusedextensivelytoevaluatetheimpactsofstiffness,frequency,andpower/recoverystrokespeedratioonrobotswim-mingspeedandmechanicalefy.Itisfoundthatwithmorexibilityarethewinnerintermsofroboticforwardvelocityperformanceinlowerfrequenciesandforhigherfrequencies,morerigidoutperformthexibleAsfortheeffectofzonroboticperformance,wefoundthatmorexiblehavelowercriticalzeta,whichcorrespondstothemaximumswimmingspeedachievableforroboticInotherwords,thereisanoptimalrecoverystrokespeed,whichresultsinthebestperformanceofroboticforagivenpowerstroke.Theanalysisrevealstheintricatetrade-offbetweenobjectives(swimmingspeedversusefy)andsupportstheuseofthepresentedmodelformulti-objectivedesignofmorphologyandcontrol.41Chapter3DESIGNANDMODELINGOFFLEXIBLEPASSIVEROWINGJOINTFORROBOTICFISHPECTORALFINS3.1IntroductionOneoftheswimmingmodesthataliveoftenusesinmaneuveringandassistivepropulsionisthefilabriformflswimmingmode,inwhichtheoscillatesitspairedpectoraltogeneratethrust[5,11].Previousworksdoneonaroboticactuatedbypairedpectoralincludebothrigidpectoral[35,44,55,81,93]andxibleorwithcontrolledcurvature[21,32,62,83].AsillustratedinFigure3.1,pectoralmotionscangenerallybeintothreemodesbasedontheaxisofrotation,rowing,feathering,andThefeatheringmotionrepresentsFigure3.1:Typesofpectoralmotion(Adaptedfrom[81]).Therotationaxesfortherowing,feathering,andmotionsarevertical,transverse,andlongitudinal,respectively.42rotationaboutthetransverseaxis,andinroboticfeatheringpectoralhaveoftenbeenusedasbowplanestocontrolthediveandascentoftherobots[44,97,98].Themotioninvolvesrotationaboutthelongitudinalaxis,whichhasbeenusedinseveralroboticmantaraysinvolvingexpandedxiblepectoral[99,100].Finally,therowingmotioninvolvesrotationabouttheverticalaxis,whichcanbeutilizedforanumberofin-planelocomotionandmaneuveringtasks,suchasforwardswimming,sidewayswimming,andturning[101].Thecycleintherowingmotioninvolvesapowerstroke,wheretherotatestowardthebackoftherobotandgainsthrustviatheinduced-dragontheandarecoverystroke,wheretherotatesbacktowardthefrontofthebodyandgetsreadyforthenextcycle.Inordertogenerateanetthrustovereachcycle,thehastobeactuateddifferentlyinthepowerandrecoverystrokes.Forexample,onecanactuatethe(much)fasterinthepowerstrokethanintherecoverystroke[21].Thedownsideofthisapproach,however,isthattherobotwilldecelerateandlosemomentumduringtheextendedrecoverystrokeandtheresultingrobotmotionisslow.Analternativeapproachistofeatherthetoreduceitseffectiveareaandthusdragduringtherecoverystroke[27,35,101].Thelatter,however,entailstheneedofoneadditionalactuatorforeachpectoralwhichincreasesthesize,weight,andcomplexityoftheandtheoverallrobot.Thecontributionofthischapteristheproposalandmodelingofaxible,passivejointforapectoralthatenablesnetthrustgenerationundersymmetricactuationofasinglerowingactuatorinpowerandrecoverystrokes.Theproposeddesignhasreducedcomplexityandcostcomparingtotheapproachadoptingactivefeathering,andasdemonstratedlaterinthepaper,itresultsinsuperiorswimmingperformancecomparingtothecaseofasingleactuatorwitharigidlinkanddifferentialpower/recoveryactuation.Thexiblejointallowsthepectoraltosweepbackpassivelyduringtherecoverystroke,whilefollowingthemotionprescribedbytheactuator43duringthepowerstroke.Consequently,theexperienceslessdragintherecoverystrokethaninthepowerstroke,resultinginanetthrust.Toanalyzetherobotlocomotionperformance,adynamicmodelisdevelopedforthejointandstructureandforaroboticequippedwithapairofsuchpectoralThismodelisthenvalidatedbyperformingexperimentsonafree-swimmingroboticExperimentsarealsoconductedtocomparetherobotperformanceusingthexiblejointwiththecasewherearigidjointisused.Themodelisfurtherexploitedtoinvestigatetheeffectoflengthandstiffnessofthexiblejointontheroboticswimmingperformanceatdifferentfrequencies.Jointstructuresofdifferentlengthandstiffnessvaluesareprototypedwithamulti-material3Dprintertothemodelanalysis.Finally,themechanicalefyforagivenxiblejointdesigniscomputed,which,alongwiththeswimmingperformanceanalysis,offersaninstrumentaltoolformulti-objectiveoptimizationofthejointanditsoperatingfrequency.Theremainderofthischapterisorganizedasfollows.Thedesignandprototypingofthex-iblerowingjointaredescribedindetailinSection3.2.Section3.3presentsthedynamicmodelforthejointstructurealongwiththemodelforroboticadoptingsuchjointsforpectoralBladeelementtheoryisusedtocalculatethehydrodynamicforcesonthepectoralandthexiblerowingjointismodeledasapairoftorsionalspringanddamper.InSection3.4,experi-mentalresultsareprovidedtosupportthemodelinganalysis.TheeffectofthexiblejointlengthandstiffnessisinvestigatedinSection3.5.ThemechanicalefyofroboticadoptingagivendesignofthepectoraljointisderivedandexplorednumericallyinSection3.6.Finally,concludingremarksareprovidedinSection3.7.44(a)(b)Figure3.2:Illustrationofthemotionofthepectoralwithxiblerowingjoints(topview):(a)Powerstroke,(b)recoverystroke.Thexiblejointsaremarkedwithreddots.3.2JointDesignThissectionisdedicatedtodescribingthedesignandprototypingoftheproposedxiblerowingjoint.Eachpectoralmovesbackandforthutilizingaservomotorasthesourceofactuation.Inlocomotion,themaintargetistomaximizetheoverallthrustforceandminimizethehy-drodynamicdragforceintherecoverystroke[5].Tomeetthisgoalforaroboticthexiblerowingjointisdesignedsuchthatthepectoralmaintainsthemotionprescribedbytheservointhepowerstroke,toproducethemaximumthrust,whilesweepingbackpassivelyalongthebodyintherecoverystroke,tominimizethedragforceontheFigure3.2(a)and3.2(b)illustratethemotionofthepectoralwiththexiblerowingjointsduringthepowerandrecoverystrokes,respectively.Onecanseethatinthiscasetheplanestaysverticalthroughoutthestrokecycleandthustheresultinghydrodynamicforceisrestrictedtothehorizontalplane.SolidWorkssoft-wareisusedtodesignthepassivejoints,whichisshowninFigure3.3.Theentirejointassemblyconsistsoffourparts:(1)arigidservoarmconnectionthatwilltotheservoarm,(2)ame-chanicalstopperrigidlyattachedtotheservoarmconnection,(3)amount(rigid)withaslitfor45(a)(b)(c)Figure3.3:TheproposedxiblerowingjointdesignedinSolidWorkssoftware:(a)Duringthepowerstroke,themechanicalstopperpreventsthefromsweepingforwardpassively(whichwouldreducethrust),(b)duringtherecoverystroke,thebendsbackpassivelyunderthehydro-dynamicforces,whichreducesthedragontheandthusontherobot,and(c)3D-printedrowingpassivejointassembledontheroboticattachingthepectoraland(4)apieceofxiblematerialwitharectangularshape,servingasthejointbetweentheservoarmconnectionandthemount.Thestopperisdesignedtopreventthepectoralfromsweepingforwardpassivelyandletitfollowtheprescribedservomotion,duringthepowerstroke,asillustratedinFigures3.2(a)and3.3(a),whileallowingthetosweepbackpassivelyduringtherecoverystroke,asillustratedinFigures3.2(b)and3.3(b).Thejointisprototypedusingamulti-material3Dprinter(Connex350fromObjet).Theprinteriscapableofsimultaneouslyjettingrigidandxiblematerials,sotheentirejointstructureisprintedseamlesslyasasinglepiece,asshowninFigure3.3(c).AlltherigidpartsareprintedwithRGD835(VeroWhitePlus).Twodifferentmaterials,FLX980(TangoBlackPlus),whichisthemostxiblematerialfromtheprinter,andDM9850(DigitalMaterial9850),whichisstillxiblebutstifferthantheformer,areexploredforthexiblepartofthejointstructure.Otherthandifferentstiffnessesforthexiblepart,weaimtoinvestigatetheeffectofjointdimensiononthe46Table3.1:offourdifferentxiblerowingjoints.JointnameFlexiblepartFlexiblepartmateriallength(mm)JR1FLX9800.5JR2FLX9801JR3FLX9801.5JR4DM98500.5performanceoftheaswell.Todoso,fourdifferentjointsareprinted,threeusingFLX980andoneusingDM9850.Allthejointshaveeddepthandthicknessof10mmand1mm,respectively,toensurethejointssurvivethroughextensiveexperiments.Table3.1summarizestheofallfourjoints.3.3DynamicModelingOneofthemainfociofthisworkistoanalyzeandcomparethepassivejointmechanismwithatraditionalrigidjoint.Forthispurpose,wehavedevelopedadynamicmodelforroboticpropelledwithpectoralforthecaseinvolvingxible,passiverowingjoints.Thethattheroboticoperatesinisconsideredtobeinviscidandincompressible.Therobotisassumedtohavearigidbodywithapairofrigidpectoralwhicharecoupledtotheactuatorarmsthroughtheproposedxiblejoints.Whileonecanincorporateanactivecaudalfortheroboticaswedidforourprototypereportedinthispaper,itsmodelingandstudyareoutsidethescopeofthiswork.Thebladeelementtheory[5]isusedtoevaluatethehydrodynamicforcesgeneratedbythepectoral47Figure3.4:Topviewoftheroboticactuatedbypectoralinaplanarmotion.3.3.1RigidBodyDynamicsTomodeltheroboticmotionproperly,somecoordinatesystemsneedtobeAsil-lustratedinFigure3.4,theinertialcoordinatesystemisdenotedwith[X;Y;Z],andtheedcoordinatesystemisrepresentedby[x;y;z],withthecorrespondingunitvectorsdenotedby[‹i;‹j;‹k],whichisattachedtothecenterofmassoftheroboticHerethex-axisisalongthebody'slongitudinalaxispointingtothehead,thez-axisisperpendiculartothex-axisandpointsupward,andthey-axisisautomaticallyformedbytheright-handorthonormalprinciple.Wedenoteby~rcp=cp‹jthevectorpointingfromroboticcenterofmasstothebaseofthepectoralser-vomotor(pointA0).PointA1isthebaseofthepectoralWeuse‹mand‹ntodenotetheunitvectorsparallelandperpendicular,respectively,toeachpectoralwheresubscriptsrandlareusedtodenotetherightandleftrespectively.Theroboticisconsideredtobeneutrallybuoyant.LetVC=[VCx;VCy;VCz]Tdenotethevelocityvectoroftheroboticintheed48coordinates,whereVCx,VCy,andVCzarethesurge,sway,andheavecomponents,respectively.Ontheotherhand,wC=[wCx;wCy;wCz]Tdenotestheedangularvelocityvectorofthebody,wherewCx,wCy,andwCzaretheroll,pitch,andyawcomponents,respectively.Weuseg1andg2,alongwithsubscriptsrandl,todenotetheangleoftheservo,andangleofeachpectoralwithrespecttothex-axis,respectively.Theangleofattackforthebodyisdenotedasb,whichistheanglebetweenthex-directionoftheedcoordinatesystemandthevelocityvectorVC.Finally,letydenotetheanglebetweenthex-axisandX-axis.Therigidbodydynamicsintheedcoordinatesarerepresentedas[102]264m00I375264VCwC375+264wCmVCwCIwC375=264ft375;(3.1)wheremisthemassmatrix(incorporatingboththeactualrobotmassandtheaddedmass,whichiscalculatedconsideringanellipsoidacceleratinginthe[63]),Iistheinertiamatrix(includingboththeactualandaddedinertias),f=[fx;fy;fz]Trepresentstheexternalhydrodynamicforces,t=[tx;ty;tz]Trepresentstheexternalmoments,appliedtothecenterofmassoftheroboticandfifldenotesthevectorproduct.Inthispaperwefocusontheplanarmotionfortheroboticsoithasthreedegreesoffreedom,namely,surge(VCx),sway(VCy),andyaw(wCz).Wefurtherassumethatthebodyissymmetricwithrespecttothexz-plane,thepectoralmoveinthexy-plane,andthez-axisoftheedframeisparalleltotheZ-axisoftheinertialframe.Theinertialcouplingsbetween49thesethreestatesareassumedtobenegligible[28],whichEq.(3.1)to(mbmax)VCx=(mbmay)VCywCz+fx;(3.2)(mbmay)VCy=(mbmax)VCxwCz+fy;(3.3)(IzIaz)wCz=tz;(3.4)wherembistheroboticmass,maxandmayaretheaddedmasscomponentsalongthexandydirectionsoftheedcoordinates,respectively.Izistherobotinertiaaboutthez-axis,andIazistheaddedinertiaoftherobotaboutthesameaxis.Thevariablesfx,fyandtzdenotetheexternalhydrodynamicforcesandmomentexertedonthebody,whicharedescribedasfx=FhxFDcosb+FLsinb;(3.5)fy=FhyFDsinbFLcosb;(3.6)tz=Mhz+MD;(3.7)whereFhx,FhyandMhzarethehydrodynamicforcesandmomenttransmittedtothebodybythepectoralandthecalculationprocedureisaddressedindetailinSection3.3.2.FD,FL,andMDarethebodydrag,lift,andmoment,respectively.Theseforcesandmomentareexpressedas[28,29,44]FD=12rV2CSACD;(3.8)FL=12rV2CSACLb;(3.9)MD=CMw2Czsgn(wCz);(3.10)50Figure3.5:Illustrationofarigid,rectangularpectoralanditsparametersandvariables.whereVCisthelinearvelocitymagnitudeoftheroboticbody,VC=qV2Cx+V2Cy,risthemassdensityofwater,SAisthewettedareaofthebody,CD,CLandCMarethedimensionlessdrag,lift,anddampingmomentcoefrespectively,andsgn(:)isthesignumfunction.Finally,thekinematicsoftheroboticisdescribedas[29],X=VCxcosyVCysiny;(3.11)Y=VCycosy+VCxsiny;(3.12)y=wCz:(3.13)3.3.2HydrodynamicForcesfromPectoralFinswithFlexibleRowingJointsInthissubsection,wepresentthedetailedmodelforcomputingthehydrodynamicforcesgeneratedbythepectoralFirstweintroducethebladeelementtheorythatisusedtoevaluatethehydro-dynamicforcesandmomentforagivenmovement.Wethendescribethedynamicmodelofthepectoralundertheproposedxiblejoints,whichenablethecomputationofthecorrespondinghydrodynamicforcesandmomentforaprescribedservomotion.513.3.2.1BladeElementTheoryFollowing[5],thebladeelementtheoryisusedtoevaluatethehydrodynamicforcesonthepectoralForeaseofcalculations,thepectoralisconsideredtoberectangularwithspanlengthSandchordlength(depth)C,asillustratedinFigure3.5.Thefollowingcalculationusestheleftpectoralasanexample,butitwillextendtriviallytotherightpectoralTherelationshipbetweentheunitvectors‹mand‹n,andtheedcoordinatesisgivenby‹m=cosg2l‹i+sing2l‹j;(3.14)‹n=sing2l‹i+cosg2l‹j:(3.15)Thehydrodynamicforcesonthepectoralhavespan-wiseandnormalcomponents.Sincethepectoralareconsideredtohavepurerowingmotioninthiswork,theanglebetweenthepectoralandthewislarge,whichresultsinaverysmallspan-wiseforce,whicharisesfromfriction,andcanbeneglected[103].Inbladeelementtheory,thenormalforcedFn(s;t)iscalculatedoneachbladeelement,ds,attimetdFn(s;t)=12CnrCj~vp(s;t)j2ds;(3.16)where~vp(s;t)isthevelocityofeachbladeelementofthepectoralasaresultofboththerobotbodymotionandthepectoralmotion,andCnisthenormalforcecoefwhichdependsontheangleofattackofeacharbitraryblade,a(s;t).Utilizingamodelempiricallyevaluatedforinsectwingsandassumingthatitsvalidityholdsunderwater[63],Cn=3:4sina.ThedetailsoncalculatingtheangleofattackfortheispresentedinSection3.3.2.2.Thetotalhydrodynamicforceactingoneachpectoraliscalculatedbyintegratingtheforcedensityalongthespanlength52(a)(b)Figure3.6:Dynamicofthepectoralwithxiblerowingjointduring:(a)Powerstroke,and(b)recoverystroke.A1representsthexiblejoint.oftheFn(t)=ZS0dFn(s;t):(3.17)3.3.2.2ModelingoftheFlexibleJointThemotionoftheinbothpowerandrecoverystrokesshouldbeknown,inordertoutilizebladeelementtheorytocalculatethehydrodynamicforces.Todoso,thexiblerowingjointismodeledasacoupleoftorsionalspringanddamper,wheretheparametersarederivedfromthepropertiesofthexiblepartanditsdimensions.Weconsidertheservoarmandtherigidpectoralastwolinks,whichareconnectedbythexiblerowingjoint.Wedenotetheanglesmadebytheandsecondlinkswithrespecttothex-axisasg1andg2,respectively.AsillustratedinFigure3.6(a),duringthepowerstroke,theangleg1isdictatedbytheservoandthefollowstheprescribedmotionoftheservoarm,resultinging2=g1,sothetrajectoryofthepectoralisfullyknown.Ontheotherhand,fortherecoverystroke,themotionofeachpointontherigidisdeterminedbythehydrodynamicinteractions,asshowninFigure3.6(b).Therefore,weneedtotheangleofthesecondlink,g2,inordertocomputethemotionofeachpointonthe53RefertoFigure3.4.VelocitiesofthepointA0(baseoftheservomotor)andpointA1(baseofthepectoralintheinertialframecanbeexpressedas~vA0(t)=VCxcpwCz‹i+VCy‹j;(3.18)~vA1(t)=~vA0l1(g1+wCz)sing1‹i+l1(g1+wCz)cosg1‹j;(3.19)wherecpisthedistancefromthebodycentertopointA0,andl1isthelengthoftheservoarm.Thevelocityateachpointsalongthepectoralis~vp(s;t)=~vA1s(g2+wCz)sing2‹i+s(g2+wCz)cosg2‹j=vpx‹i+vpy‹j:(3.20)Theangleofattackofeachbladeelementiscalculatedviatana=<vp(s;t);‹n><vp(s;t);‹m>=vpxsing2+vpycosg2vpxcosg2+vpysing2;(3.21)where<;>denotestheinnerproduct,vpxandvpyarethevelocityofthepectoralinxandydirection,respectively.Thetotalforceactingontherigidpectoralis~F2=~Fn~FA1=mpd~vp(s;t)dts=S2;(3.22)where~FA1representstheforceappliedbytherigidpectoralontheservoarm,andmpistheeffectivemassoftherigid(whichcontainsthemassandtheaddedmass,wheretheadded54massiscalculatedbaseonarigidplatemovinginthewater)[104].Themomentoftherigidrelativetoitspivotpoint(pointA1)isevaluatedas~Mn=ZS0s‹mdFn:(3.23)Notethat~Mnisafunctionofg2andg2.Themomentproducedbythetorsionalspringanddamper(namely,thexiblejointitself)isevaluatedas~M(S+D)=[KS(g1g2)+KD(g1g2)]‹k;(3.24)whereKSandKDarethespringanddampercoefusedtomodelthexiblerowingjoint.ThetotalmomentequationoftherigidrelativetopointA1iswrittenas~M2=~Mn+~M(S+D)=Ip(¨g2+wCz);(3.25)whereIpistheeffectiveinertiaoftherigid(whichcontainstheinertiaandtheaddedinertia,andiscalculatedbaseonarigidplatemovinginthewater)and¨g2istheangularaccelerationofthesecondlink.BysolvingEq.(3.25),whichisasecond-orderequationforg2,thedynamicsofthepectoralwithaxiblejointintherecoverystrokeisfullydescribed.Thehydrodynamicforcetransmittedtotheservoarmcanbeobtainedas~FA1=Fn‹nmpd~vp(s;t)dt,whered~vp(s;t)dtcanbeevaluatedonceg2andg2aresolvedfromEq.(3.25).Thetotalforceexertedbythearmontherobotbodyis~Fh=Fhx‹i+Fhy‹j=~FA1:(3.26)55Themomentappliedbytheonthebodyisrepresentedas~Mh=Mhz‹k=cp‹j~FA1:(3.27)BysubstitutingEqs.(3.26),(3.27)intoEqs.(3.5)-(3.7),thedynamicsoftheroboticutilizingxiblerowingjointsarefullydescribed.Thepresentedmodelappliestothecasewheretheroboticisfree-swimming.Thecou-pledbodyandmotionsintroducecomplexityinevaluatingthehy-drodynamicforceandmoment.Alternatively,onecouldassumeananchoredrobotbodywhenevaluatingtheforceandmoment,asoftenadoptedintheliteratureforsimilarprob-lems[2,19,29].Whilethelatteralsoadoptedinthesimulationpartofthispaper,introducesmodelingerror,theerroristypicallyacceptableconsideringthemuchlargervelocitycomparingtothevelocityoftherobotitself.3.4ExperimentalModelValidation3.4.1RoboticFishPrototypeandExperimentalSetupToevaluatetheproposedxiblerowingjointmechanismandvalidatethepresenteddynamicmodel,weconductexperimentsonafree-swimmingroboticprototype.Therobotisdesignedtoswimonthesurfaceandisslightlypositivebuoyantwithabout15%ofitsheightabovewater(asopposedtotheneutrallybuoyantassumptionfordynamicmodeling).Duetotherelativelyslowpectorallocomotion,theeffectoftankwallsandsurfacewavesiscontemplatedtobenegligibleinthiswork.ThebodyoftheroboticisdesignedinSolidWorksand3D-printed.BoththedesignschematicandtheactualprototypeareshowninFigure3.7.Thisprototypeis56(a)(b)Figure3.7:Roboticprototype:(a)DesignedSolidWorksmodel;(b)3D-printedroboticbodyalongwithmountedabout15cmlong,8cmhighand4.6cmwidewithoutthepectoralandcaudalandweighscloseto0.3kg.TheroboticutilizesaprominimicrocontrollerboardfromArduinotorealizethecontrolofthethreeservos.Apowerconverterprintedcircuitboard(PCB)isdesignedforthisroboticThreewaterproofservos(Traxxas2065)areutilizedtoactuatethealthoughtailactuationisnotincludedinthisstudy.Theservomotorsareprogrammedtorotateeachpectoralaccordingtog1(t)=gAsin(wgt)+90;(3.28)withgAandwgdenotingtheamplitude(indeg)andtheangularfrequencyofactuation,re-spectively.Theactualpectoralaremadeofapolypropylenesheetwith0.5mmthicknessandYoung'smodulusofapproximately2GPa,whichisconsideredtobealmostrigid.Theexperimentsareconductedinatankthatmeasures2feetwide,6feetlong,and2feetdeep.ThetankisequippedwithamotioncapturesystemfromNaturalPoint,whichcontainsfourOptitrackFlex13camerasalongwiththeMotivesoftwaretocapturethemotionoftheroboticTheexperimentalsetupisshowninFigure3.8.Twotypesofexperiments,forwardswimmingand57(a)(b)Figure3.8:Experimentalsetup:(a)Schematic;(b)actual.turning,areperformedtoevaluatethedynamicmodel.Welettheroboticswimforsometime(approximately30seconds)toreachthesteady-statemotion,andthenvideo-tapeitsswimming.Forexample,intheforwardswimmingcasewerecordthetimeittakesfortherobottoswimadistanceof50cm.Theexperimentforeachsettingisrepeated10times.Finally,weanalyzethecapturedvideostoextractthesteady-statespeedfortheforwardswimming,andturningradiusandperiodfortheturningmotion.3.4.2ParameterTheparametersforthedynamicmodelaremeasureddirectlyorcalculatedbasedonmeasure-mentsandarelistedinTable3.2.Thebodyinertiaaboutz-axisisevaluatedasIz=15mb(a2+c2),wherea=Bodylength2andc=Bodywidth2arethesemi-axislengths[28].Eventhoughtheroboticbody(withallitsinternalcomponents)isnothomogeneous,laterexperimentalresultsshowthattheaforementionedinertiaformulaproducesasatisfactoryapproximationtothereality.Thewetsurfacearea,addedmasses,andaddedinertiaarecalculatedconsideringanprolatespheroidacceleratinginthe[28,88].58Table3.2:ModelParametersComponentParameterValueUnitBodyMass(mb)0.295KgInertia(Iz)4:26104Kg=m2max0.095Kgmay0.1794KgIaz2:7105Kg=m2Wetsurfacearea(SA)0.0325m2Dragcoef.(CD)0.42ŒLiftcoef.(CL)4.86ŒMomentcoef.(CM)7:6104Kg=m2FinLength(S)0.043mDepth(C)0.025mServoarmlenght(l1)0.01mEffectivemass(mp)0.0194KgEffectiveinertia(Ip)3:49106Kg=m2Distancefrombodycentercptoservobase,0:025mWaterdensity(r)1000Kg=m3TheroboticdragandliftcoefCD,CL,andCM,areempiricallyusingthedatacollectedwhentheroboticisequippedwithrigidjointsforthepectoralWiththerigidjoints,thepowerstrokeandrecoverystrokeneedtohavedifferentspeeds,inordertoproduceanetthrust[21].ThisratioisdenotedasPR(PowerstrokespeedRecoverystrokespeed),whichisequalto1forthesymmetricpattern.Inthispaper,experimentsareconductedforthecasesofPR=2,3,4,and5.TheexperimentalresultsforbothforwardandturningswimmingmotionsoftheroboticwithPR=2areusedinthebodyparameterTurningisrealizedbyactuatingonepectoralonly.Inparticular,theseparametersaretunedtomatchtheforwardvelocity,turningradius,andturningperiodobtainedinsimulationwiththeexperimentalmeasurementwhentwodifferentpowerstrokespeedsareused,completingthepowerstrokein0.5sand0.3s,respectively.TheamplitudeissettogA=25deg.TheresultingcoefareCD=0:42,CL=4:86,andCM=7:6104Kg=m2.Theseparametersarethenusedinindependentmodelvalidationforall59othercasesusingthexiblerowingjoint.AmongalltherowingjointsmentionedinTable3.1,jointfiJR1flresultsinthehighestfor-wardvelocity.Withoutthelossofgenerality,thiscaseischosentoillustratethemodelvalidationperformance.ToidentifythespringanddampercoefKSandKDaretunedtomatchtheforwardswimmingvelocityoftherobotichobtainedinsimulationwiththeexperimentalmea-surementsforfrequenciesof0.75Hz,1Hzand1.5Hz.ThecoefareKS=6:34104NmandKD=9:98105Nms.Theseparametersarethenusedformodelvalidationforvariousothercasesinvolvingthesamejoint.3.4.3ComparisonbetweenFlexibleandRigidJointsBeforepresentingthemodelvalidationresults,wecomparetheperformanceofxiblerowingjointswiththatofarigidjoint(wherethepectoralareconnectedtotheservoswitharigidconnection).Fortherigidjointcase,wehavethedifferentpowerandrecoverystrokespeeds,asmentionedinSection3.4.2,sothattheroboticcanhaveanetthrust.Figure3.9(a)showstheexperimentalresultsontheforwardswimmingvelocitiesoftherigidjointcasewithPR=1,2,3,4,5,thecaseofthexiblefeatheringjointfrom[86],andthexiblerowingjointJR1,overdifferentpowerstroketimes.Figure3.9(b)presentstheseresultsintermsoftheeffectivefrequencies.Thefrequencymeans1T,whereTdenotestheperiodofeachcycle(powerandrecoverystrokecombined)andtheservosareprogrammedtorunuptothelimitof200/sec.ThismaximumspeedcorrespondstotherightmostpointineachcurveinFigure3.9(b).FromFigure3.9,wecanconcludethat,overall,thevelocityperformanceofthexiblerowingpassivejointoutperformstherigidjointcase.60(a)(b)Figure3.9:Experimentalresultsoftheforwardswimmingvelocityversus(a)powerstroketime,and(b)effectiveactuationfrequency,forthecasesofrigidjointandxiblerowingjoint.3.4.4DynamicModelValidation3.4.4.1DynamiccharacteristicsofpectoralBeforepresentingexperimentalresultsthatvalidatethedynamicmodel,wepresentsimulationresultsbasedontheexperimentallymodel,toshedinsightintothedynamiccharacteris-ticsofthepectoralwithxiblejoints,aswellastheireffectsontheroboticbody.Intheinterestofbrevity,wehaveonlyincludedtheplotsforonecase(jointJR1withfrequencyof1Hz,wherebothareactuated).Figure3.10(a)showsthetimehistoryofthepectoraland61servoarmanglesinonebeatcycle.Itisinterestingtonotethat,whilethepectoralanglefollowscloselythemotorshaftangleduringmuchofthepowerstroke,thefidetachmentflstartsshortlyaftertheservopassesthe90duringthepowerstroke,whichisduetotheinertialeffectofthewhentheservoarmstartsdecelerating.Similarly,thedifferencebetweenthetwoanglesshrinksdowntozerobeforetherecoverystrokeends.Figure3.10(b)showstheangleofattackinpowerandrecoverystrokesofonebeatcycle,assumingthattheroboticbodyisanchored.Figure3.10(c)showsthetotalforceexertedonthebodybypectoralinthex-direction(Fhx).Notethatthemeanvalueofthepositivethrustisapproximately4timeslargerthanthemeanvalueofnegativethrust.Thetotalhydrodynamicforceinthey-direction(Fhy)andthetotalhydrodynamicmoment(Mhz)arezerointhiscaseduetotheleft-rightsymmetryinpairedpectoralFinally,Figure3.10(d)showsthesurgevelocityoftherobotic(VCx)fromthesimulation.Again,thesway(VCy)andyaw(wCz)componentsoftheroboticvelocitiesarezeroduetothesymmetryinItcanbeseenthat,startingatrest,therobottakesapproximately11secondstoreachthesteady-state.3.4.4.2AnchoredexperimentsTovalidatetheproposeddynamicmodel,twosetsofexperimentsareconductedontheroboticDuringthesetofexperiments,theroboticbodyisanchoredusingabracketandtheangleofthepectoral(g2)ismeasuredwithrespectto‹i,therobot'sheadingdirection.Themotionofthepectoraliscapturedfromabove,usingaCasioExilim(EX-FH25)high-speedcameraat40framespersecond.Figure3.11comparesthemeasuredmaximumvaluesoftherowingangleduringtherecoverystokeinbothsimulationandexperimentsatdifferentfrequencies,whentheroboticbodyisanchored.Itcanbeseenthatthemodelisabletocapturetherowinganglewellforallfrequenciesupto1.75Hz.Forthecaseof2Hz,thenoticeable62(a)(b)(c)(d)Figure3.10:Withfrequencyof1Hz:(a)Variationofthepectoralandservoarmangleforonemovementcycle,(b)Variationofthepectoralangleofattackforonemovementcycle,(c)Variationofthetotalhydrodynamicforceexertedtoroboticbodyinx-direction(Fhx)versussimulationtime,(d)Variationofroboticvelocityinx-direction(VCx)versussimulationtime.discrepancybetweenthemodelpredictionandthemeasurementislikelycausedbytheconstraintsinfabrication,weretheactualpectoralanglegoesbeyondtheservoangleinthepowerstroke,(g16=g2),duetothelargerhydrodynamicloadingonthepectoralFigure3.12comparesthemeasuredtime-dependentpectoralangle(g2)duringtherecoverystrokeandthecorrespondingmodelpredictionforthecaseof1Hzactuation.Here,weshowtheframesevery0:1secduringtherecoverystroke.Overallthereisagoodmatchbetweenthemodelpredictionandexperimentalmeasurement.Thepredictionerrorisslightlylargeratthebeginningandtheendofthecycle,whichisattributedtothetransitionfrom/tothepowerstroke,wherethemechanicalstopperisineffect.63Figure3.11:Comparisonbetweenmodelprediction(whitedashedline)andexperimentalmea-surement(bluesolidline)ofthemaximumrowingangleduringtherecoverystroke,withfrequenciesof(a)0.75Hz,(b)1Hz,(c)1.25Hz,(d)1.5Hz,(e)1.75Hz,and(f)2Hz.Theblackverticallineindicatestheroboticheadingdirection,thegreendottedlineshowstheservoarmdirectionandtherightpectoralisshown.3.4.4.3Free-swimmingexperimentsForthesecondsetofexperiments,theroboticswimsfreelyinthetank,includingbothforwardswimmingandturningthatareenabledwiththepectoralutilizingthexiblerowingjoints.Figure3.13showsthecomparisonbetweenmodelpredictionandexperimentalmeasurementoftheforwardswimvelocityatdifferentfrequencies.Figures3.14and3.15showsimilarcom-parisonsontheturningradiusandturningperiod.FromFigure3.14,theturningperioddropswiththefrequency,whichisexpected.ThesimulationresultsinFigure3.15suggestthattheturningradiushasnegligibledependenceonthefrequency,whichissupportedbytheexperimentalresults,wherethemeanvaluesofthemeasuredradiusstayaround23-24cmacrossallfrequencies.ThediscrepancybetweenthesimulationandexperimentalresultsinFigure3.15islargelyattributedto64Figure3.12:Comparisonbetweenexperimentalmeasurementofthetime-dependentrecoverystrokeanglewithmodelpredictions.Thepectoralbeatsat1Hz.Thebluesolidlineandwhitedashedlineimplytheexperimentalmeasurementandmodelprediction,respectively,andthegreendottedlineshowstheservoarmdirection.thechallengeinmeasuringpreciselytheturningradiusinexperimentsŒtherobotdoesnottrackclosedorbitsforeachturn,whichcouldbeduetothedisturbancesfromtheinteractionsbetweenthewaterandtankwalls.TheresultsofFigures3.13-3.15showthattheproposedmodelisabletocapturethemotionoftheroboticwithxiblerowingjointsverywell.Inparticular,forthetestedfrequencyrange,theforwardswimmingvelocityincreaseswiththefrequency.Intheturningcase,theturningperiod(thetimeittakestocompleteoneturn)dropswiththeincreasingfrequency,whichmatchesone'sintuition.3.5EffectofFlexibleJointLengthandStiffnessInthissection,weinvestigatetheimpactoftwodesignparametersforthexiblejoint,itslengthandstiffness,whichwillallowfurthervalidationoftheproposedmodelanddemonstrateitspo-tentialusefordesignoptimization.Asdescribedin[91],thetorsionalspringconstantofaxible65Figure3.13:Caseofrowingjoint(JR1):Comparisonbetweenthemodel-predictedandmeasuredforwardswimmingspeed,fordifferentfrequencies.Figure3.14:Caseofrowingjoint(JR1):Comparisonbetweenthemodel-predictedandmeasuredturningperiod,fordifferentfrequencies.materialcanbeevaluatedasKS=Edh312l;(3.29)wherehisthethickness,listhelength,disthewidth(depth),andEistheYoung'smodulusofthexiblematerial.ThedampercoefKDcanbeevaluatedasKD=kKS,wherekisaproportionalconstant.Sothespringconstantchangeswithbothdimensionandstiffnessofthexiblepartofthepassivejoint.66Figure3.15:Caseofrowingjoint(JR1):Comparisonbetweenthemodel-predictedandmeasuredturningradius,fordifferentfrequencies.WehavechosenthreedifferentvaluesforthelengthofthexiblerowingjointmadeofFLX980material,0.5mm,1mmand1.5mm(JointsJR1,JR2,andJR3inTable3.1).ThespringanddamperconstantsforJR2andJR3arecalculatedusingEq.(3.29),wheretheYoung'smodulus(E)andkvaluesarekeptthesameastheonesderivedfromtheparametersKSandKDforJR1.Figure3.16showsthemodelpredictionandexperimentalresultsontheforwardswimmingveloc-ityatdifferentfrequencies,forallthreejoints.ThejointJR1hasthebestperformanceamongthethreejointsforhigherfrequencies.Forlowerfrequencies(upto1.25Hz),jointJR3(mostxibleamongthethree)hasabetterperformance.Wecanseethatthemodelisabletoeffectivelycapturethejointlength-dependenceoftheforwardswimmingvelocityforallcases.Whiletheexperimentallimitfortheactuationfrequencyis2Hz,wehaveextendedthesimulationresultstofrequencyof3Hzinordertocapturetheoptimalfrequencyofeachjoint.Theforwardswimmingspeedwilldropafterreachingthisoptimalfrequency.Finally,wecomparetheperformanceofxiblejointswithdifferentmaterialstiffness.Twoxiblejointswithidenticaldimensions,JR1madeofFLX980andJR4madeofDM9850(stiffer),areusedinthecomparison.ThespringanddampercoefforJR4aretobeKS=67Figure3.16:Modelpredictionandexperimentalmeasurementoftheforwardswimmingvelocityoftheroboticwiththeuseofthreexiblejoints(allmadeofFLX980)withdifferentlengths.Figure3.17:Modelpredictionandexperimentalresultsoftheforwardswimmingvelocityoftheroboticwiththeuseoftwoxiblejointswithdifferentstiffnessvalues.4:38103NmandKD=9:34104NmsusingthesamemethoddescribedinSection3.4.2,andareusedformodelpredictionsforallothercases.Figure3.17showsthecomparisonofforwardswimmingspeedbetweenthetwocases.Again,itcanbeseenthatthereisagoodmatchbetweenmodelpredictionsandexperimentaldata.Forthelowerfrequencies,thejointJR1outperformsJR4,whileJR4isthewinnerforthehigherfrequencies.Againwehaveextendedthesimulationresultstohigherfrequenciestobettercapturetheperformancetrendofthejoints.683.6MechanicalRobotefy,astheratioofusefulworkforpropulsionovertotalconsumedenergy,isofgreatrelevancetopracticaloperationoftherobot.Mechanicalworkdonebytothesurroundingwater,energyusedforpoweringelectronics,electricallosses,andfrictionallosses,amongothers,allcontributedtothetotalconsumedenergy.Mechanicalworkisarguablythemostsourceofenergyexpenditure,andtherefore,itisimportanttounderstandhowthedesignofpectoraljointsthemechanicalefyoftherobot.Inthissection,weusethevalidateddynamicmodeltoevaluatethepropulsiveefyoftheroboticswimmingwithpectoralthatusethexiblerowingjoints.Themechanicalefyduringsteady-stateswimmingiscalculatedas[5,92,93]h=WbWT;(3.30)whereWbistheamountofusefulworkneededtopropeltheroboticandWTisthetotalworkdonebythepectoralduringeachcycle.Wecallthisthemechanicalefysinceitdoesnotconsiderotherenergylosses,suchastheelectricalpowerusedforrunningtheelectronicsorfrictionallossesinmotorsandgears.TheusefulworkWb1canbecalculatedasfollows[92,93],Wb1=Zt0+T0t0FThrust(t)VC(t)dt(3.31)whereFThrustisthex-componentofthetotalhydrodynamicforceexertedontheroboticbody(Fhx),VC(t)isthevelocityoftheroboticbodyprojectedintothexdirection,andT0denotethetotaldurationofeachcycle.Inthispaperwetakeanalternativeapproachthatusesproductofthemeanthrustandthemeanvelocityatthesteady-state.Whentheroboticisatthesteady-stateandcruiseswithaconstantspeedVCmean,itsmeanthrustisbalancedbyits69Table3.3:ComparisonbetweenthetwomethodsofcomputingWb.Frequency(Hz)Wb1Wb2Wb1Wb2Wb2(%)0.750.1120.1141.7510.18870.19392.681.250.28260.28882.151.50.37870.39053.021.750.42810.43832.3320.42330.43913.6(mean)dragforce,andthusFT=12rV2CmeanSACD:(3.32)whichresultsinthefollowingexpressionofWb2Wb2=12rV3CmeanSACDT0:(3.33)AsshowninTable3.3,thevaluesofWbcomputedwiththesetwomethodsareactuallyveryclosetoeachother(witherrorlessthan4%).GiventhatthesecondmethodofevaluatingWb2ignorestheoscillatorynatureofthethrustandvelocityandisthussimpler,itisadoptedintheefyanalysisfortheremainderofthispaper.ThetotalworkdonebythepairedpectoralWT,isobtainedasWT=2Zt0+T0t0maxf0;ZS0dFn(s;t)~vp(s;t)gdt=2Zt0+T0t0maxf0;ZS012CnrCj~vp(s;t)j2~vp(s;t)dsgdt;(3.34)wheret0representsthebeginningofacycle,fifldenotestheinnerproduct.Notethat70Figure3.18:Calculatedmechanicalefyandforwardvelocityofdifferentxiblerowingjointsatdifferentfrequencies.Figure3.19:Calculatedmechanicalefyversusfrequencyandspringconstantofthexiblejoint.atsometimeinstantst,theinstantaneousmechanicalpowerexertedbypectoralonwatercouldbenegative;however,sincetheservoscannotreclaimthisenergyfromwater,wetreattheinstantaneouspoweratsuchataszero,whichexplainstheoperatormaxf0;ginEq.(3.34).Notethatevenatthesteady-state,theactualvelocityisnotaconstant;instead,itperiodicallyaroundsomevalue.Therefore,VCmeaninEq.(3.32)isevaluatedbythedistancetraveledoverNcycles(N=10)dividedbyNT0.71(a)(b)Figure3.20:Comparisonofnon-dimensionalizedparametersforjointsJR1,JR2,andJR3atdif-ferentfrequencies:(a)CalculatedmechanicalefyandStrouhalnumber,(b)Dimen-sionlessvelocityandReynoldsnumber.Figure3.18showsthecalculatedefy,alongwiththecorrespondingswimmingvelocity,forthejointsJR1,JR2andJR3.TheefyofthejointJR1ishigherthantheothertwoandoveralltheefyishigherforlowerfrequencies.Figure3.18revealsinterestingtrade-offbetweenthespeedperformanceandmechanicalefy.Inparticular,foragivenjointdesign,withahigherfrequency,thespeedishigherbutatthecostoflowerefy.Figure3.19showstheefycurveversusdifferentfrequenciesandspringconstantvalues(kS).Thisshowsthattheroboticperformsmoreefinlowern-beatfrequencieswithstifferxiblerowingjointsuptoacertainoptimalstiffness(KSˇ7104Nm).Foranyjoint72stifferormorexiblethanthisoptimalamount,theefystartstodrop.Overall,Figures3.18and3.19indicatethattheoptimizationofthexiblejointpresentsaninteresting,multi-objectivedesignproblemthatinvolvesconsiderationofthejointstiffness,dimension,andthefrequencyofoperation.Theproposeddynamicmodelinthispapershowspromiseinaddressingtheoptimaldesignproblem.Figure3.20providesacomparisonofthenon-dimensionalizedparametersforjointsJR1,JR2,andJR3atdifferentfrequencies.Thenon-dimensionalizedparametersconsideredincludetheReynoldsnumberRe,theStrouhalnumberSt,andthedimensionlessvelocityVDL.RecalltheReynoldsnumberRe=VCmeanLn,whereVCmeanistheswimmingspeedoftherobot,Listheroboticlength,andnisthekinematicviscosityofwater.TheroboticlengthL=0.15mandn=106m2=sareusedinthecalculation.TheStrouhalnumberisasSt=fAVCmean,wherefisthefrequency,AisthemaximumexcursionofthetrailingedgeforpectoralandVCmeanistheswimmingspeedoftherobot.WeuseA=2SsingA,whereSisthepectoralspanlengthandgAistheappingamplitude[27,35].ThedimensionlessvelocityisasVDL=VCmeanfL[105].ItcanbeseeninFigure3.20(a)thattheefyshowsclearinversecorrelationwiththeStrouhalnumber.Forexample,JR1demonstratesthehighestmechanicalefyamongthethreejointsandhasthelowestStrouhalnumber.Foreachjoint,theStrouhalnumberincreaseswhiletheefydropswhenthefrequencyincreases.NotethattheStrouhalnumbersforbio-logicalareusuallyintherangeof0.25-0.5[35,95,96].TheStrouhalnumberspresentedherearehigherthan1andthusbeyondthebiologicalrange.ThereasonisthattheroboticusedinthisstudyispropelledpurelybythepectoralwhichresultsinlowspeedsandhigherStrouhalnumbers.Notethat,fromFigure3.20(a),whentheefyoftheroboticgetshigher,theStrouhalnumbergetsclosertothebiologicalrange.Ontheotherhand,Figure3.20(b)shows73Table3.4:Comparisonofnon-dimensionalizedparameters.RoboticStReVDLThiswork-JR11.2742730.19Thiswork-JR21.3540440.18Thiswork-JR31.3241160.183[35]withgA=302.9559250.042[35]withgA=452.9687370.0621[35]withgA=607.7643500.031[21]6.059000.04[86]-JF12.5517440.078[34]0.94100000.16thattheroboticdemonstratesthehighestdimensionlessvelocitywhentheReynoldsnumberisatthelowerend.ComparingFigure3.20(a)and3.20(b)alsosuggeststhatthereisapositive(negative,resp.)correlationbetweenthemechanicalefy(theStrouhalnumber,resp.)andthedimensionlessvelocity,whichisexpectedgiventheoftheStrouhalnumberandthedimensionlessvelocity.Wehavealsocomparedourresultswiththeresults(allactuatedat1Hz)fromseveralpectoralroboticreportedintheliterature,asseeninTable3.4.Fromthetable,itcanbeseenthattheStrouhalnumbersachievedinthisworkaregenerallyclosertothebiologicalrangethanwhatwasachievedinotherreportedwork(withtheexceptionof[34],whichisslightlylowerthanourresults).Thedimensionlessvelocitiesachievedinthisworkarealsothehighestamongallcases.Thesecomparisonsprovidestrongsupportfortheeffectivenessoftheproposedapproach.743.7ConclusionWhilebiologicalusesophisticatedpectoralkinematicstoachievesuperiorswimmingandmaneuveringperformance[18,106],thegoalofthisworkistoachievesoundperformanceforroboticpectoralwithsimplestructureandsimplecontrol.Inparticular,wehaveproposedanovelxible,passivejointforrowingpectoralinroboticandpresentedadynamicmodelfortheroboticequippedwithsuchpectoralmechanisms.Thexiblejointenablesthepectoraltobendbackpassivelyalongthebodyduringtherecoverystroke,tominimizethedragforce,whilemaintainingtheprescribedmotionoftheactuatorduringthepowerstroke.Thisdesigneliminatestheneedtohavedifferentactuationspeedsforpowerandrecoverystrokes.BladeelementtheoryisusedtoevaluatethehydrodynamicforcesonthepectoralThexiblejointismodeledasapairoftorsionalspringanddamper.Tovalidatethedynamicmodel,wehaveconductedexperimentsinvolvingbothawheretheroboticisanchoredandthebendinganglesaremeasured,andafree-swimmingwhereforwardswimmingspeedsandturningradii/periodsatdifferentfrequenciesaremeasured.Theperformanceoftheproposedjointisalsocomparedwithatraditionalrigidjoint,toshowtheeffectivenessofthisdesign.TheresultsshowedadrasticimprovementintheperformanceoftheroboticMultiplexiblerowingjointsareusedintheexperimentstoexaminetheofthexiblejoint'slengthandstiffnessontheroboticperformance,andtheexperimentaldatamatchthemodelpredictionswellinallcases,whichfurthersupportstheutilityofthepresentedmodelindesignoptimization.Finally,withtheaforementionedmodel,wehavenumericallyevaluatedthemechanicalefyoftheroboticandexploreitsdependenceonthexiblejointstiffnessandtheoperatingfrequency.75Chapter4BIO-INSPIREDFLEXIBLEJOINTSWITHPASSIVEFEATHERINGFORROBOTICFISHPECTORALFINS4.1IntroductionDevelopmentofrobotichasbeeninspiredbyuniquecharacteristicsofswimminginliveandotheraquaticanimals,suchasagility,maneuverability,andefy[2Œ4,28,30,31,95,107Œ115].Roboticchangetheirbodyshapeordifferenttogeneratepropulsion[5,27,39,51,101,116Œ119].Accordingto[11],basedonthepropulsorsthatuse,theirlocomotioncanbedividedintotwomaincategories:median/pairedpropulsion,andbody/caudalpropulsion.Inthiswork,weconsiderthecasewherearoboticoscillatesitspairedpectoraltogeneratethrust.Thepectoralpropulsionprovidesgoodmaneuverabilityandstabilityforrobotic[80].TherearesomestudiesdealingwithroboticpropelledbypairedpectoralMostoftheearlyinvestigationsemployedrigidpectoralthatweremotor-driventoproducedifferentmotions[32,35,44,55].Severalrecentstudiesinvestigatedtheimpactofxiblepectoralonroboticperformance[21,62].Inordertogenerateanetthrust,therearetypicallytwostrategies.Thestrategyinvolvestheuseofmultipleactuatorsforeachpectoraltoprovide76(a)(b)Figure4.1:schematicsofthedrag-basedlabriformswimmingmode,usingxiblefeatheringjoints(topview):(a)Powerstroke,(b)recoverystroke.Thexiblefeatheringjointisshownwitharedcircle.combinationsofdifferentdegreesoffreedom,namelyrowing,featheringandwheretheaxesofrotationarevertical,transverse,andlongitudinal,respectively.Althoughthisstrategyenablesthemimickingoflivepectoralmotion,itresultsinlargesizeandhighenergyconsumptionforrobotic[35,55].Analternativeactuationstrategyistouseasingleactuatorpertomaintainthesmallrobotsize,butemploydifferentpowerandrecoverystrokespeedstominimizethedragforceduringtherecoverystroke.However,thismethodtendstoslowdowntheintheextendedrecoverystrokeperiod[21].Thisissuewasaddressedin[84],wheretheauthorsproposedadesignofapassivejointfortherowingmotion,whichenablesthepectoraltosweepbackpassively(alongthesamerowingaxis)inordertominimizethedragforceduringtherecoverystroke.Inthisstudy,tomorepreciselymimicdrag-basedlabriformswimmingmodeoflive[11],wecombinetwodifferentpectoralmotions,rowingandfeathering,realizedwithonlyasingleactuatorpern,asillustratedinFigure4.1.Asdiscussedin[120,121],arealrarelymoves77itspectoralbyanexclusiverowingorfeatheringmovement;instead,itusesacombinationofthesemotionstomoveforward.Thecontributionofthispaperisthedesignandmodelingofaxible,passivelyfeatheringjointthatenablestherobotictomimicthedrag-basedlabriformswimmingmode.Here,thepectoralmotionisdividedintotwophases,namely,powerandrecoverystrokes.Duringthepowerstroke,themechanicalstoppersofthedesignedjointsallowthepairedtomovebackwardwithrespecttothebody,followingaprescribedrowingmotion.Thiswouldinduceadragforceoppositetothemovingdirectionofthepointingintheforwarddirection.Intherecoverystroke,thepectoralnfeatherspassivelywhilefollowingtheactuatedrowingmotion,whicheffectivelyreducesthedragforceontheThemechanismofthejointsandhowthestoppersworkineachcyclearedescribedindetailinSection4.2.Theproposedjointreducesthecostandcomplexityofthemotion,comparingtoadoptinganactivefeathering[35,55].Thedynamicmodelofthepectoralisdevelopedbasedonbladeelementtheory[5],wherethejointismodeledasapairoftorsionalspringanddamper.Withtheconsiderationofthecom-binedrowingandfeatheringmotions,the3Dhydrodynamicforcesarecapturedinthemodel.ThemodelisthenvalidatedbyconductingdifferentexperimentsonaroboticTheperformanceoftheroboticutilizingthexiblefeatheringjointisalsocomparedwiththecasewheredif-ferentialactuationduringpower/recoverystrokesisadoptedalongwithatraditionalrigidjoint.Theeffectofthedepthandstiffnessofthexiblejointisfurtherinvestigatedusingthedynamicmodel,whichisalsovalidatedwithexperiments.Finally,themechanicalefyoftheroboticiscomputedforxiblefeatheringjointsfordifferentspringconstantsandoperatingfrequen-cies,whichprovidesinsightthatisusefulinoptimizingthejointdesignandthefrequencyregimeofTheremainderofthispaperisorganizedasfollows.InSection4.2thedesignandprototyping78oftheproposedxiblejointaredescribedindetail.ThedynamicmodelofthejointalongwiththemodelforroboticadoptingsuchjointsispresentedinSection4.3.InSection4.4theexperimentalsetupisdescribedandexperimentalresultsareprovidedalongwiththesimulationresultstovalidatethedynamicmodel.Section4.5isfocusedontheeffectofjointdepthandstiffness.Section4.6addressesthecalculationofthemechanicalefyoftheroboticadoptingthexiblejoint.Finally,concludingremarksareprovidedinSection4.7.4.2DesignofFlexibleFeatheringJointThissectioncoversthedetailsofdesignandprototypingofthexiblefeatheringjoint.Asmen-tionedearlier,eachpectoralfollowsarowingmotionprescribedbytheservomotor,whichactuatestheproximalendofthesymmetricallyduringthepowerandrecoverystrokes.Ourprimarygoalistominimizethedragforceduringtherecoverystroke,byaddinganotherdegreeoffreedomtothepectoralwithoututilizinganyadditionalactuator.Toaccomplishthisgoal,axiblefeatheringjointisdesignedtoenablethepectoralfeatherpassivelywhenitisrowedbackduringtherecoverystroke.Thismodeofswimmingiscalleddrag-basedlabriformswim,andisillustratedinFigure4.1.Inparticular,thepectoralmaintainstheservo-prescribedrowingmotionduringthepowerstroke,toproduceamaximumnetthrust,whileitrotatespassivelyalongthetransverseaxis(feathers)duringtherecoverystroke,toreducethehydrodynamicdragontheTheproposedfeatheringjointdesignisshowninFigure4.2.Theentirejointmechanismcon-sistsofarigidservoarmconnectorthatconnectsthewholestructuretotheservomotor,amechanicalstopper,amountandarectangularxiblepiece(showninblackinFigure4.2(a)and(b)),servingasthefeatheringjoint,whichconnectsthemountstructuretotheservoarm79(a)(b)(c)Figure4.2:Theproposedxiblefeatheringjoint:(a)Duringthepowerstroke,themechanicalstopperpreventsthefromfeathering,(b)duringtherecoverystroke,therotatesandtendstoalignwiththehorizontalsurface,toreducethedragforceontheand(c)3D-printedfeatheringpassivejointassembledontheroboticconnector.Duringthepowerstroke,themechanicalstopperenablethepectoraltomaintaintherowingmotionprescribedbytheservomotor,asshowninFigure4.1(a)andFigure4.2(a),whileduringtherecoverystroke,thexiblejointenablesthetofeatherpassivelyandreducethehydrodynamicdragforce,asshowninFigure4.1(b)andFigure4.2(b).Flexiblefeatheringjointsareprototypedusingamulti-material3Dprinter(Connex350fromObject),whichiscapableofsimultaneouslyjettingrigidandxiblematerials,resultinginseam-lessintegrationofthepliableandrigidcomponentsofthexiblejointmechanism,asshowninFigure4.2(c).Alltherigidparts(servoarmconnectorandmount)areprintedwiththematerialRGD835(VeroWhitePlus).Twodifferentxiblematerials,FLX980(TangeBlackPlus),whichisthemostxiblematerialsupportedbytheprinter,andDM9850(DigitalMaterial9850),whichisstifferthanFLX980butstillxibleenough,areexploredforthexiblepartofthefeatheringjointstructure.Otherthandifferentmaterials,itisalsoourgoaltoinvestigatetheimpactofjointdimensionsonthepropulsionperformance.Forthispurpose,atotaloffourjointsareprinted,threeusingFLX980andoneusingDM9850.Alljointshavewidthof4mmandthicknessof1mm,to80ensureadequatestrengthforsurvivingthroughextensiveexperiments.ThethreeFLX980jointshavedifferentvaluesfortheirdepth,0.5mm,1mmand1.5mm,whiletheDM9850jointhasadepthof0.5mm.Herethejointdepthreferstotheextentofthegapbetweenthetopandbottomrigidelementsonthesideoppositetothemechanicalstopper.Thegapisnegligibleonthestopperside.Thefourjoints,withtheirdifferentcombinationsofmaterialsanddepths,enableacompactsetofexperimentsforvalidatingtheproposeddynamicmodelandrevealingdesigntrade-offs.Thejointsarereferencedasfollows.JointfiJF1fl,withFLX980asthexiblematerialanddepthof0.5mm,jointfiJF2flwithFLX980asthexiblematerialanddepthof1mm,jointfiJF3flwithFLX980asthexiblematerialanddepthof1.5mm,and,jointfiJF4flwithDM9850asthexiblematerialanddepthof0.5mm.4.3DynamicModelofFin-ActuatedRoboticFishIncorporat-ingtheFlexibleFeatheringJoint4.3.1HydrodynamicForcesontheFinInthissection,wedescribetheuseofbladeelementtheoryinrepresentingthehydrodynamicforceontheforagivenmovementpattern,whichisdeterminedbythe(yettosolve)dy-namicsofthexiblejoint,namely,thefeatheringdynamics.Thehydrodynamicforceisthenincorporatedintothedynamicmodelforthefeatheringmotion,whichiscapturedviaapairoftorsionalspringanddamper.Finally,thetotalhydrodynamicforcesandmomentsresultingfromthemechanismareusedtodevelopthedynamicmodelfortheroboticpropelledbytheAdaptedfrom[5],thebladeelementtheoryisusedtoevaluatethehydrodynamicforcesonthepectoralForallthesecalculations,weassumeananchoredroboticbody.Thisassumption81Figure4.3:Topviewoftheroboticactuatedbypectoralinplanarmotion.isoftenadoptedintheliteratureforsimilarproblems[2,19,29].Whilethisintro-ducesmodelingerror,theresultingerroristypicallyacceptableconsideringthemuchlargervelocitycomparingtothevelocityoftheroboticitself.Foreaseofcalculation,thepectoralisconsideredtoberigidandrectangularwithspanlengthSandchordlength(depth)C.Figure4.3showsthetopviewofaroboticconsistingofarigidbodyandpairedpectoral[X;Y;Z]Tindicatestheinertialcoordinatesystemand[x;y;z]Trepresentstheedcoordinatesystem,withcorrespondingunitvectors[‹i;‹j;‹k],whichisattachedtothecenterofmassoftheroboticWeuse[‹m;‹n;‹p]todenotetheunitvectorsofthepectoralcoordinatesystem,wheresubscriptsrandlareusedtorepresentrightandleftrespectively.Here‹misparalleland‹nisperpendiculartothepectoraland‹pisautomaticallyformedbytheright-handorthonormalprinciple.Thenotation~rcp=cp‹jdenotesthevectorpointingfromtheroboticcenterofmasstothepectoralservomotorbase(pointA0).82Figure4.4:Illustrationofarigid,rectangularpectoralanditsparameters,duringthepowerstroke.Wedividethepectoralmovementcycleintopowerandrecoverystrokes,andstudyeachseparately.Duringthepowerstroke,thepectoralundergoesarowingmotionprescribedbytheservomotor;therefore,theplanestaysverticalandthehydrodynamicforcesarerestrictedtothehorizontalplane,asshowninFigure4.4.Here,allthecalculationsaredonefortheleftpectoralwhichcanbeextendedtotherightinastraightforwardmanner.Duringthepowerstroke,therelationbetweentheorthonormalunitvectors[‹m;‹n;‹p]andtheedcoordinatesystemisgivenby‹m=cosg‹i+sing‹j;(4.1)‹n=sing‹i+cosg‹j;(4.2)‹p=‹k:(4.3)wheregistheprescribedangleoftheservoarmwithrespecttothebodyheading‹i.Inbladeelementtheory,thehydrodynamicforcedFhp(s;t)oneachbladeelement,ds,83attimet,iscalculatedasdFhp(s;t)=12Cn(a(s;t))rj~v2p(s;t)jCds‹n;(4.4)whererdenotesthewaterdensity,~vp(s;t)isthevelocityofeachbladeelementofthepectoralandCnisthenormalforcecoefwhichisdependentontheangleofattackoftheblade,a(s;t).Here,weconsiderCn=3:4sina(s;t),byutilizinganempiricallyevaluatedmodelforinsectwingwhichwasusedforarobotic[122]androbotic[63].Eventhoughinsects(orroboticinsects)inairwhileroboticswiminwater,theassociateddynamicswillhavesimilarbehavioriftheirReynoldsnumbersareclose.Inparticular,theReynoldsnumberoftheroboticinthisworkisattheorderof103,whichisclosetotheReynoldsnumberreportedin[122]fortherobotic(30-1000).Thevelocityofeachelement,~vp(s;t),isexpressedas~vp(s;t)=vpx‹i+vpy‹j=n(l1+s)gsingo‹i+n(l1+s)gcosgo‹j;(4.5)wherel1isthelengthoftheservoarm.Theangleofattackofeachbladeelementcanbeevaluatedviatana=<vp(s;t);‹n><vp(s;t);‹m>=vpxsing+vpycosgvpxcosg+vpysing;(4.6)where<;>denotestheinnerproduct.Withtheanchoredbodyassumption,itiseasytoverifythattheangleofattackis90.Thetotalhydrodynamicforceactingoneachpectoraliscalculatedbyintegratingtheforce84Figure4.5:Illustrationofarigid,rectangularpectoralanditsparameters,duringtherecoverystroke.densityalongthespanlengthofthe~Fhp(t)=ZS0dFhp(s;t):(4.7)Ontheotherhand,duringtherecoverystroke,thepectoralundergoesa3Dmotion.Wemodifythebladeelementtheory,sothatwehavebladesinbothspanandchordlengthoftheresultingin2Delements,whichweusetoevaluatethehydrodynamicforces.TheparametersduringtherecoverystrokeareshowninFigure4.5,whereListhefeatheringanglethatweneedtoinordertofullyknowthepectoraldynamics.NotethatthefeatheringangleL=0duringthepowerstroke.Therelationshipbetweenthepectoralcoordinatesystemandtheedcoordinatesystemisasfollows‹m=cosg‹i+sing‹j+0‹k;(4.8)‹n=singcosL‹icosgcosL‹jsinL‹k;(4.9)‹p=singsinL‹i+cosgsinL‹jcosL‹k:(4.10)85whereListhefeatheringanglewithrespectto‹k.Thebladeelementtheoryisrevisedtoevaluatethehydrodynamicforcesona2DelementofthepectoralThehydrodynamicdragforceproducedbyeachelementdcdsduringtherecoverystrokeisevaluatedasdFhP(c;s;t)=12Cn(a(c;s;t))rj~v2p(c;s;t)jdcds‹evp;(4.11)where‹evpisaunitvectorinthedirectionof~vp(c;s;t),a(c;s;t)=atan<vp(c;s;t);‹n><vp(c;s;t);‹m>istheangleofattack,and~vp(c;s;t)isthevelocityofeachelementdcds,andisrepresentedas~vp(c;s;t)=vpx‹i+vpy‹j+vpz‹k=n(l1+s)gsingcgcosgsinLcLsingcosLo‹i+n(l1+s)gcosgcgsingsinL+cLcosgcosLo‹jncLsinLo‹k(4.12)WenotethatthereisnotationabuseassociatedwithdFhp,a,and~vp,whichdependonlyonsandtin(4.4)butdependons,c,andtin(4.11),andhopetheirmeaningswillbeclearfromthecontext.Thetotalhydrodynamicforceisevaluatedbyintegratingtheforcedensityoverthesurfaceofthepectoral~FhP(t)=ZS0ZC0dFhP(c;s;t):(4.13)4.3.2SolvingtheFeatheringDynamicsDuringthepowerstroke,therigidfollowstheservomotion(L=0),andthecorrespondinghydrodynamicforceonthecanbeevaluatedgiventheservomotion.Ontheotherhand,during86therecoverystroke,theevaluationofthehydrodynamicforce(Eq.(4.11))requiresknowingthefeatheringangleL,whichhastobesolvedforthroughthedynamicsequationforthefeatheringjoint.Thetotalforceactingontherigidisrepresentedas~F2=~FhP~FA1=mpd~vp(c;s;t)dts=S2;c=C2;(4.14)where~FhPisthehydrodynamicforceontherigid(calculatedbasedontheequationspresentedinSection4.3.1),~FA1representstheforceappliedbytherigid(throughthejoint)ontheservoarm,andmpistheeffectivemassoftherigidpectoralwhichcontainsthemassandtheaddedmass(wheretheaddedmassiscalculatedbaseonarigidplatemovinginthewater).SinceweneedtothefeatheringangleoftheL,theprojectionofthehydrodynamicforcein‹ndirectionproducesthecorrespondingmoment.Themomentoftherigidrelativetoitspivotpointisevaluatedas~MhP(t)=ZS0ZC0c‹pdFhP(c;s;t):(4.15)Here~MhPisafunctionofLandL.Themomentproducedbythexiblefeatheringjoint,whichismodeledasapairoftorsionalspringanddamper,isevaluatedas~M(S+D)=(KSLKDL)‹m;(4.16)whereKSandKDarethespringanddampercoefusedtomodelthexiblefeatheringjoint.87Thetotalmomentequationoftherigidrelativetoitspivotpointoffeatheringiswrittenas~M2=~MhP+~M(S+D)=Ip¨L;(4.17)whereIpistheeffectiveinertiaoftherigid(whichcontainstheinertiaandtheaddedinertia,andiscalculatedbaseonarigidplatemovinginthewater)and¨Listheangularaccelerationofthein‹m-direction.BysolvingEq.(4.17),thedynamicsofthepectoralwithaxiblefeatheringjointduringtherecoverystrokeisfullydescribed.4.3.3HydrodynamicForcesandMomentsontheRoboticFishThehydrodynamicforcetransmittedtotheservoarmcanbeobtainedas~FA1=~FhPmpd~vp(c;s;t)dts=S2;c=C2:(4.18)Thetotalforceexertedbythearmontherobotbodyis~Fh=Fhx‹i+Fhy‹j=~FA1:(4.19)Themomentappliedbytheonthebodyisrepresentedas~Mh=Mhz‹k=cp‹j~FA1:(4.20)OtherthanhydrodynamicforcesandmomenttransmittedfromthepectoraltheroboticbodyexperiencesdragforceFD,liftforceFL,anddragmomentMD,whichcanberepresented88as[28,29,44]FD=12rV2CSACD;(4.21)FL=12rV2CSACLb;(4.22)MD=CMw2Czsgn(wCz);(4.23)whereVCisthelinearvelocitymagnitudeoftheroboticbody,wCzistheangularvelocityofthebodyaboutthez-axis,risthemassdensityofwater,SAisthewettedsurfaceareaforthebody,bistheangleofattackofthebody,formedbythedirectionofbodyvelocityvectorwithrespecttothex-axis.CD,CLandCMarethedimensionlessdragforce,liftforce,anddampingdragmomentcoefrespectively,andsgn(:)isthesignumfunction.4.3.4Rigid-BodyDynamicsofaPectoralFin-actuatedRoboticFishUnder-goingPlanarMotionThedynamicequationsofrigidbodyundergoingplanarmotionintheedcoordinatesarerepresentedas[20,85,102](mbmax)VCx=(mbmay)VCywCz+fx;(4.24)(mbmay)VCy=(mbmax)VCxwCz+fy;(4.25)(IzIaz)wCz=tz;(4.26)wherembistheroboticactualmass,maxandmayrepresenttheaddedmasseffectsalongthexandydirectionsofthebodyedcoordinates,respectively.IzistherobotinertiaandIazistheaddedinertiaoftherobotaboutthez-axis.Thevariablesfx,fyandtzindicatetheexternal89hydrodynamicforcesandmomentexertedonthebodycenterofmass,whichareinducedbythepectoralmotionandtheinteractionoftheroboticbodywiththesurroundingwhichcanbedescribedasfx=FhxFDcosb+FLsinb;(4.27)fy=FhyFDsinbFLcosb;(4.28)tz=Mhz+MD;(4.29)Finally,thekinematicequationsfortherobotintheinertialcoordinatesystemaredescribedas[29],X=VCxcosyVCysiny;(4.30)Y=VCycosy+VCxsiny;(4.31)y=wCz:(4.32)whereydenotetheanglebetweenthex-axisandX-axis.4.4ExperimentalResults4.4.1RoboticFishPrototypeandExperimentalSetupExperimentsareperformedtostudytheperformanceofaroboticwithxiblefeatheringjointandvalidatetheproposedmathematicalmodel.TheroboticbodyisdesignedinSolidWorkssoftwareand3D-printed,asshowninFigure4.6.Thebodyisabout15cmlong,8cmhighand4.6cmwidewithoutthepectoralandcaudalandweighscloseto0.3kg.AnArduinopro90Figure4.6:3D-printedroboticprototypealongwithmountedminimicrocontrollerboardisincorporatedintherobottorealizethecontrolofservos.Therobotbodyalsohousesapowerconverterprinted-circuitboardwithvoltageregulatorsforthemotorandelectronics.ThemotorsusedforactuationofthepectoralandcaudalareTraxxas2065waterproofservoswithmaximumspeedof200/sec.Althoughtherobotiscapableofmovingitscaudaltailactuationisnotincludedinthisstudy.Theservomotorsareprogrammedtorotateeachpectoralaccordingtog(t)=gAsin(wgt)+90;(4.33)wheregAistheamplitudeindegreesandwgdenotestheangularfrequencyofThepectoralaremadeofalightplasticmaterial(polypropylene)thathas0.5mmthicknesswithYoung'smodulusofapproximately2GPa,whichisconsideredtobealmostrigid.AsshowninFigure4.7,theexperimentsareconductedinawatertankthatmeasures2feetwide,6feetlong,and2feetdeep.ThetankisequippedwithamotioncapturesystemfromNatu-ralPoint,whichcontainsfourOptitrackFlex13camerasalongwithMotivesoftwaretocapturethe91Figure4.7:Experimentalsetupforfree-swimmingroboticmotionofroboticTwodifferentexperimentsareconductedtoevaluatetheproposeddynamicmodel.First,therobotichisstudiedwhenthebodyisanchoredtomeasurethefeatheringangle,andsecond,free-swimmingoftheroboticisruntomeasuretheforwardswimmingvelocity,turningradius,andturningperiod.Allthemeasurementsaredoneapproximately30secondsaftertherobotinitiatedswimmingtoensurethatithasreachedsteady-statemotion.Theexperimentforeachsettingisrepeated10times.Attheend,thecapturedvideosareanalyzedbytheMotivesoftwaretoextractthesteady-statespeedfortheforwardswimming,andtheturningradiusandperiodfortheturningmotion.4.4.2ParameterTheparametersofthemathematicalmodelareeithermeasureddirectlyorexperimentallyasfollows:Thebodyinertiaaboutz-axisisevaluatedasIz=15mb(a2+c2),wherea=Bodylength2andc=Bodywidth2aresemi-axislengthsofthebody[28].Theaddedmasses,addedinertiaand92Table4.1:ModelParameters.ComponentParameterValueUnitBodyMass(mb)0.295KgInertia(Iz)4:26104Kg=m2max0.095Kgmay0.1794KgIaz2:7105Kg=m2Wetsurfacearea(SA)0.0325m2Dragcoef.(CD)0.42ŒLiftcoef.(CL)4.86ŒMomentcoef.(CM)7:6104Kg=m2FinLength(S)0.035mDepth(C)0.02mServoarmlength(l1)0.01mEffectivemass(mp)0.0166KgEffectiveinertia(Ip)3:32106Kg=m2Distancefrombodycentertoservobase,cp0:025mWaterdensity(r)1000Kg:m3wettedsurfacearecalculatedbyapproximatingtherobotbodyasanprolatespheroidacceleratinginthe[28,88].TheparametersusedinsimulationsarelistedinTable4.1.TheroboticdragandliftcoefCD,CL,andCM,areiempiricallyusingthecollecteddatafromtheroboticequippedwithrigidjointsforthepectoralWithrigidjoints,weneedtohavedifferentpowerandrecoverystrokespeedstoproduceanetthrust[21].ThisratioisindicatedasPR(PowerstrokespeedRecoverystrokespeed),whichisequalto1forthesymmetricHere,weexperimentwiththecasesofPR=2,3,4,and5.TheexperimentalresultsforbothforwardandturningswimmingmotionsoftheroboticwithPR=2areusedtoidentifythebodyparameters,whereturningisrealizedbyactuatingonepectoralonly.CD,CL,andCMaretunedtomatchtheforwardvelocity,turningradius,andturningperiodobtainedinsimulationwiththeexperimentalmeasurementwhenthepowerstrokeiscompletedin0.5sand0.3s,respectively.93Thebeatamplitudeissetto25.ThecoefareCD=0:42,CL=4:86,andCM=7:6104Kg=m2.Theseparametersarethenusedinindependentmodelvalidationforallothercasesusingthexiblefeatheringjoint.AmongallthefeatheringjointsmentionedinSection4.2,jointfiJF1flresultsinthehighestforwardvelocityinthetestedfrequencyrange.So,withoutthelossofgenerality,thisjointischosentoperformthemodelvalidation.Toidentifythespringanddampercoefforthisjoint,KSandKDaretunedtomatchtheforwardswimmingvelocityoftheroboticobtainedinsimulationwiththeexperimentalmeasurementsforbeatfrequenciesof1Hz,1.5Hzand2Hz.ThecoefareasKS=1:31104NmandKD=4:64105Nms.Theseparametersarethenusedformodelvalidationofvariousothercasesinvolvingthesamefeatheringjoint.4.4.3ComparisonbetweenFlexibleFeatheringandRigidJointsFirst,weprovideacomparisonontheforwardswimmingvelocityoftheroboticwiththexiblefeatheringjoint,withthatofarigidjoint.Here,rigidjointreferstoarigidconnectionbetweentheservoarmsandthepectoralFortherigidjointcase,inordertohaveanetthrust,weusedifferentpowerandrecoverystrokespeeds,introducedinSection4.4.2.Figure4.8(a)providestheexperimentalresultsonforwardswimmingvelocitywiththerigidjoint,wherePR=1,2,3,4,5,andwiththexiblefeatheringjointfiJF1fl,overdifferentpowerstroketimes.Figure4.8(b)presentsthesameresultsintermsoftheeffectivebeatfrequency.Here,theeffectivebeatfrequencyiscalculatedas1T,whereTistheperiodofeachbeatcycle,combiningbothpowerandrecoverystrokes.Theservosareprogrammedtorunuptothelimitof200/sec,whichreferstotherightmostpointineachcurveofFigure4.8(b).FromFigure4.8,onecanconcludethat,theperformanceofthexiblefeatheringjointoutperformstherigidjointcaseathigherfrequencies94(a)(b)Figure4.8:Experimentalresultsoftheforwardswimmingvelocityintermsof(a)thepowerstroketime,and(b)theeffectiveactuationfrequency,forthecasesofrigidjointandxiblefeatheringjointfiJF1fl.(1:3Hzandabove).Forlowerbeatfrequencies,therigidjointcasesoutperformthexiblefeatheringjoint.Notethattherelationshipbetweenthefrequencyandtheswimmingspeedisalmostlinearuptoathresholdvalueforthefrequency,whichisobservednaturallyin[123].954.4.4DynamicModelValidationThissubsectiondescribestheexperimentscarriedoutontheroboticwithxiblefeatheringjoint,tovalidatetheproposedmathematicalmodel.Twokindsofexperimentsareperformedinstillwaterforvalidationpurposes.Forthesetofexperiments,therobotbodyisedusingabracket.Thepectoralareactuatedwithg(t)=25sin(wgt)+90.Themotionoftherightpectoralistrackedfromtheside(xzplane),usingaCasioExilim(EX-FH25)high-speedcamera,recordingat40framespersecond.Thevideosarethenprocessedandthemaximumfeatheringanglewithrespectto‹kismeasuredandcomparedtothosepredictedbythemodel.Figure4.9showsthemaximumfeatheringangleduringtherecoverystroke,inbothsimulationandexperimentsatdifferentfrequencies.Themodelisabletocapturethemaximumfeatheringanglewellforallfrequenciesupto1.5Hz.Forhigherfrequencies,thediscrepancybetweenthemodelpredictionandthemeasurementstartstogrow.Thiscanbeattributedtotheconstraintofthefabrication,whichimposesalimitationonthefeatheringangleofthejoint.Forthesecondsetofexperiments,theroboticisallowedtoswimfreelyinthetank.Bothforwardswimmingandturningareenabledwiththepectoralincorporatingthexiblefeather-ingjoints.Figure4.10showstheexperimentalandsimulationresultswheretheforwardswimmingvelocitiesoftheroboticareplottedatdifferentfrequencies.Theforwardswimmingvelocitiesoftheroboticisreportedbothincm/secandBL/secscales.Figures4.11and4.12showsimilarcomparisonsontheturningradiusandturningperiodofafree-swimmingroboticTheresultsofFigures4.10-4.12showthattheproposedmodelisabletocapturethemotionoftheroboticwithxiblefeatheringjointsverywell.Inparticular,forthetestedfrequencyrange,theforwardswimmingvelocityincreaseswiththefrequency.Intheturningcase,theturn-ingperiod(thetimeittakestocompleteoneturn)dropswiththeincreasingfrequency,96Figure4.9:Comparisonbetweenmodelprediction(whitedashedline)andexperimentalmeasure-ment(bluesolidline)ofthemaximumfeatheringangleduringtherecoverystroke,withbeatfrequenciesof(a)0.75Hz,(b)1Hz,(c)1.25Hz,(d)1.5Hz,(e)1.75Hz,and(f)2Hz.Theyellowsolidlineindicatesthe‹kdirection.Figure4.10:Caseoffeatheringjoint(JF1):Comparisonbetweenthemodel-predictedandmea-suredforwardswimmingspeed,fordifferentbeatfrequencies.Theforwardswimmingvelocityisreportedincm/secscaleonthelefty-axisandinBL/secontherighty-axis.97Figure4.11:Caseoffeatheringjoint(JF1):Comparisonbetweenthemodel-predictedandmea-suredturningperiod,fordifferentbeatfrequencies.Figure4.12:Caseoffeatheringjoint(JF1):Comparisonbetweenthemodel-predictedandmea-suredturningradius,fordifferentbeatfrequencies.whichmatcheswithone'sintuition,andtheturningradiusincreaseswithfrequency.984.5EffectofFlexibleJointDepthandStiffnessHere,westudytheeffectofdifferentparametersofthexiblefeatheringjointonitsperformance.Asdescribedin[91],thestiffnessofthetorsionalspringconstantisevaluatedasKS=Edh312l;(4.34)wherehisthethickness,listhelength(whichcorrespondstothedepthinthecaseoftheproposedxiblejoint),disthewidth,andEistheYoung'smodulusofthexiblematerialusedforthepassivejoint.ThedampercoefKDisevaluatedasKD=kKS,wherekisaproportionalconstant.Keepingthewidthandthicknessofthejointconstant,thespringcoefcanbevariedbychangingthedepth(l)andstiffness(E)ofthexiblejoint.Thisstudywillletusfurthervalidatetheproposedmathematicalmodelandprovidesusefulinformationonthejointoptimization.WechoosethreedifferentdepthforthexiblefeatheringjointmadeofFLX980material,0.5mm,1mmand1.5mm(JointsJF1,JF2,andJF3).ThespringanddamperconstantsforJF2andJF3arecalculatedusingEq.(4.34),wheretheYoung'smodulus(E)andkvaluesarekeptthesameastheonesforJF1.Figure4.13showsthemodelpredictionandexperimentalresultsofforwardswimmingvelocityatdifferentfrequencies,fordifferentxiblefeatheringjointlengths.ThejointJF1(leastxibleamongthethree)hasthebestperformanceamongthethreejointsinthehigherfrequencies(higherthan1.75Hz).Forlowerfrequencies,jointJF3(mostxibleamongthethree)outperformstheothertwo.Sowecanconcludethatthemorexiblejointperformsbetteratlowerfrequencies,whilethestifferjointhasabetterperformanceathigherfrequencies.Wecanseethatthemodelisabletocapturethejointdepth-dependenceoftheforwardswimmingvelocityeffectivelyforallthreecases.Here,theexperimentallimitfortheactuationfrequencyis2Hz,sowehaveextendedthesimulationresultstofrequencyof99Figure4.13:Modelpredictionandexperimentalmeasurementoftheforwardswimmingvelocityoftheroboticusingthreedifferentxiblefeatheringjoints,madeofFLX980,withdifferentdepths.Theforwardswimmingvelocityisreportedincm/secscaleonthelefty-axisandinBL/secontherighty-axis.3Hzinordertocapturetheperformancetrendofeachjoint.Theforwardswimmingspeedwilldropafterreachinganoptimalfrequencyforeachcase.Finally,weinvestigatetheeffectofchangingthestiffness(E)ofthejointontheroboticperformance.Here,wechoosetwoxiblefeatheringjointswiththesamedimension,oneusingFLX980asthexiblematerial,jointJF1,andtheotherusingDM9850asthexiblematerial,jointJF4.ThespringanddampercoefforJF4aretobeKS=0:0018NmandKD=0:0064NmsusingthesamemethoddescribedinSection4.4.2,andarekeptthesameformodelpredictionofallothercasesusingthesamejoint.ThecomparisonofforwardswimmingvelocityusingthesetwojointsarereportedinFigure4.14.Itcanbeseenthatthereisgoodmatchbetweenthemodelpredictionandexperimentaldata.Overall,thejointJF1outperformsJF4atlowerfrequencies,whilethejointJF4startstooutperformjointJF1athigherfrequencies.Again,wehaveextendedthemodelpredictionresultstocapturetheperformanceofthejointsathigherfrequencies.100Figure4.14:Modelpredictionandexperimentalresultsoftheforwardswimmingvelocityoftheroboticwiththeuseoftwoxiblefeatheringjointswithdifferentstiffnessvalues.Theforwardswimmingvelocityisreportedincm/secscaleonthelefty-axisandinBL/secontherighty-axis.4.6MechanicalInthissection,wecalculatethepropulsiveefyoftheroboticswimmingwiththexiblefeatheringjointforthepectoralTheefyduringthesteady-stateswimmingiscalculatedas[5]h=WbWT;(4.35)whereWbistheamountofusefulworkneededtopropeltheroboticandWTisthetotalworkdonebythepectoralforeachcycle.Thisefyiscalledmechanicalefy,sincetheenergylosses,suchasfrictionallossesorthepowerusedtorunthemotors,arenotcon-sideredinthecalculations.Duringsteady-stateswimming,whentherobotswimswithaconstantspeedVCmean,thedragforceactingonthebodyisbalancedbythethrustforceFT.SowehaveFT=12rV2CmeanSACD:(4.36)101Sotheusefulpropulsivepoweriscalculatedbymultiplyingthrustforce,FT,bytheconstantspeed,VCmean,resultingintheusefulworkWb=FTVCmeanT0=12rV3CmeanSACDT0;(4.37)whereT0=Tp+TRdenotesthetotaldurationofeachcycle,consistsofdurationofpowerandrecoverystrokes,whereTP=TR=T02.Notethatevenatthesteadystate,theactualvelocityisnotaconstant;instead,itperiodicallyaroundsomevalue.Therefore,VCmeaninEq.(4.36)isevaluatedbythedistancetraveledoverNcycles(forexample,N=10)dividedbyNT0.ThetotalworkdonebythepairedpectoralWT,isobtainedasWT=2Zt0+TPt0max(0;ZS0dFhp(s;t)~vp(s;t))dt+2Zt0+TP+TRt0+TPmax(0;ZS0ZC0dFhp(c;s;t)~vp(c;s;t))dt=2Zt0+TPt0max(0;ZS012Cn(a(s;t))rj~vp(s;t)j2~vp(s;t)ds)dt+2Zt0+TP+TRt0+TPmax(0;ZS0ZC012Cn(a(c;s;t))rj~vp(c;s;t)j2~vp(c;s;t)dcds)dt;(4.38)where,t0denotesthebeginningofacycleandfifldenotestheinnerproduct.Notethatatsometimeinstantst,theinstantaneousmechanicalpowerexertedbypectoralonwatercouldbenegative;however,sincetheservoscannotreclaimthisenergyfromwater,wetreattheinstantaneouspoweratsuchtaszero,whichexplainstheoperatormaxf0;ginEq.(4.38).Figure4.15showstheresultsforcalculatedmechanicalefy,alongwiththecorrespond-ingforwardswimmingvelocity,forjointsJF1,JF2andJF3.JointJF3hashighestefyatlowerfrequenciesandjointJF1isthemostefathigherfrequencies.Eachjointhasamaxi-102Figure4.15:Calculatedmechanicalefy(blue)andforwardswimmingvelocity(red)fordifferentxiblefeatheringjointsatdifferentbeatfrequencies.mumefyatacertainfrequency.Sowecanconcludethatamorexiblejoint(JF3)ismoreefatlowerfrequenciesandalessxiblejoint(JF1)ismoreefathigherfrequencies.Figure4.16(a)showstheefycurveversusdifferentfrequenciesandspringconstantvalues(KS).Thisshowsthattheroboticperformsmoreefinlowerfrequencieswithmorexiblefeatheringjointsuptoacertainoptimalstiffness.Figure4.16(b)showsthespringconstantofthefeatheringjointsthathavemaximummechanicalefyindifferentfrequencies.Fromthiswecanconcludethat,themorexiblefeatheringjointsareperformingmoreefatlowerfrequencies,whilethestifferjointsactmoreefathigherfrequencies.Notethat,thereisanoptimalpointforthemaximumefyamongallthefeatheringjoints.FromFigure4.16(c),onecanseethereisanoptimalspringconstantforthemaximumefy.Foranyjointstifferormorexiblethanthisoptimalamount,theefystartstodrop.Notethatasimilartrendisobservedin[124Œ127].Overall,Figures4.15and4.16indicatethattheoptimizationofthexiblejointpresentsaninteresting,multi-objectivedesignproblemthatinvolvesconsiderationofthejointstiffness,dimension,andthefrequencyofoperation.Theproposeddynamicmodelinthispapershowspromiseinaddressingtheoptimal103(a)(b)(c)Figure4.16:Mechanicalefy:(a)Calculatedmechanicalefyversusatfrequencyandspringconstantofthexiblefeatheringjoint,(b)Springconstantofthefeatheringjointwithmaximumefyversusfrequency,(c)Maximumefyversusspringconstant.designproblem.Table4.2presentsthemechanicalefyandStrouhalnumberforjointsJF1,JF2,andJF3.HeretheStrouhalnumberoftheroboticiscalculatedasSt=fAVCmean(4.39)wherefisthefrequency,AistheamplitudeforpectoralandVCmeanisthe104Table4.2:MechanicalefyversustheStrouhalnumber.Frequency(Hz)JointJF1JointJF2JointJF3EfyStrouhalEfyStrouhalEfyStrouhal(%)number(%)number(%)number0.75113.7911233.5436462.38431172.5447282.4366592.13831.25222.3680382.2194642.04301.5272.3653482.1865432.20051.75292.2947362.3189302.57972322.2798222.3611202.87662.25232.3688192.3437173.05332.5152.3857152.4653153.50512.75132.4653122.5583104.00763102.479092.756284.4154averageswimmingspeed.HeretheamplitudeA=2Ssing,whereSisthepectoralspanlengthandgistheangularamplitudeof[27,35].Weobserveconsistent(negative)correlationbetweentheefyandtheStrouhalnumber.Inparticular,foreachjoint,atthebeatfrequencywheretheefyachievesthemaximum,thecorrespondingStrouhalnumberisthelowest.NotethattheStrouhalnumberforbiologicalisusuallyintherangeof0.05-0.6,andthenumberspresentedherearebiggerthanthatrange.Thereasonisthattheroboticusedinthisstudyswimsforwardwithitspectoralalone,whichresultsinrelativelyslowspeedsandthusrelativelyhighStrouhalnumberscomparingtoitsbiologicalcounterparts.FromTable4.2,therobotictendstohavehighermechanicalefywhenitsStrouhalnumbergetsclosertotherangeforbiologicaldata.1054.7ConclusionInthisstudy,wehaveproposedanoveldesignforaxiblepassivejoint,whichenablesthepectoraltomovesimilartothedrag-basedlabriformswimmingmode.Adynamicmodelispresentedforaroboticpropelledbyapairofrigidpectoralconnectedtotheactuatorsviatheproposedxiblefeatheringjoints.Thejointenablesthepectoraltobeactuatedsym-metricallytorowforpowerandrecoverystrokes,whileprovidingfeatheringaboutthetransverseaxisduringtherecoverystroketominimizethedragforce.Thecombinedrowingandfeatheringresultsin3Dmovementofthepectoralwhichneedstobecapturedproperlyinthemodeling.Thebladeelementtheoryisusedtoevaluatethehydrodynamicforcesonthepectoralduringbothpowerandrecoverystrokes.Thexiblefeatheringjointismodeledasapairoftorsionalspringanddamper.Acompletedynamicmodelforaroboticincorporatingtheproposedjointsisalsopresented.Tovalidatetheproposeddynamicmodel,wehavemeasuredthefeatheringangleofananchoredroboticalongwiththeforwardvelocity,turningradiusandperiodoftherobotduringfreeswimming,andcomparedthosetothemodelpredictions.Multiplexiblefeatheringjointshavebeenexploredtostudytheeffectofdepthandstiffnessofthexiblepart.Theme-chanicalefyoftheroboticinforwardswimmingisexplorednumerically,tounderstandthetrade-offsinthejointdesignandoperationfrequency.106Chapter5DESIGNANDDYNAMICMODELINGOFELECTRORHEOLOGICALFLUID-BASEDVARIABLE-STIFFNESSFINFORROBOTICFISHFishpropelinwaterbymovingdifferentordeformingthebody[16,17],whichhasinspiredthedevelopmentofroboticthataccomplishlocomotioninwaysthatemulatethoseofbiological[9,86,87,128Œ130].Comparedwithrigidxibleandjointsintroduceadditionaldynamicbehaviorthatcanbeexploitedtoenhanceroboticperformance[21,29,58,84,85,117,131Œ135].Althoughpassivecompliantcouldresultinmoreefswimming,theoptimalxibilitychangeswithparameterssuchasfrequencyoramplitude[86,87,94,136].Forexample,withanincreasedfrequencyandamplitude,theoptimalstiffnesstendstoincrease[135,137,138].TheconnectionbetweenpropulsorstiffnessandswimmingperformancehasalsobeenstudiedforbiologicalTheliveadjustthestiffnessofthe/bodytocompletedifferenttasks[38,42,43].Thediscussionsaboveindicatethatitisofinteresttoactivelytunethestiffnessforroboticaccordingtoswimmingconditions,andtherehasbeensomelimitedworkreportedinthisarea107overthepastfewyears[68,69,139].Ziegleretal.[139]introducedatail-actuatedroboticwherethetailwascapableofchangingitselasticity,whichwasrealizedbyactivelyinsertingadditionalfoilsintothetailorremovingthesefoilsfromtheusingtwoservomotors.Parketal.[68]designedawithavariable-stiffnessmechanism,whichwasrealizedbycompressingacompliantmaterialtoincreasethestiffness.Thedesignedtailconsistedofsixrigidplates,alsousedasthebackboneoftheTwotendonswereusedfordrivingthetailandtwoothertendonswereusedtochangeitsstiffness.Inparticular,whenthelattertendonswerepulled,thevariablestiffnessstructurewouldbecompressedandresultinanincreaseinaxialstiffness.Nakabayashietal.[69]developedaroboticwithavariablestiffnessmechanismusingavariableeffective-lengthspringmechanism,achievedbyalteringthelengthofarigidplate,whichresultedinchangingtheeffective-lengthofthespringandhencethestiffness.Allthesereportedmechanismswerebulkyandcomplex.Inthiswork,weproposeacompactmechanismforstiffnesstuningusingelectrorheological(ER)Thisconsistsofabaseliquid(usuallysiliconeoil)withsuspendedpolymerparti-cles.AnERexperienceschangesinrheologicalpropertiesinthepresenceofanelectricgoingfromtheliquidphasetoasolidgelphaseastheelectricincreases.Inparticular,thepar-ticlesalignwiththeelectricline,resultinginchangesinviscosity,yieldstress,andsomeotherpropertiesoftheTheresponsetimeoftheERisintheorderofmilliseconds,whichprovidesafastsolutionforstiffnesstuning.ERhavearangeofengineeringapplications,suchasshockabsorbers[140],brakesandclutchsystems[141],andvibrationcontrol[142Œ144].Theproposedstiffness-tuningconsistsofanERurethanerubber,withembeddedcoppersheetsaselectrodes.Adynamicmodelfortheispresented,whichisderivedusingtheHamilton'sprincipleanduseslarge-amplitudeelongated-bodytheorytocapturethehydrodynamicforcesontheTheequationsofmotionareobtainedthroughaprocedure108andsolvednumerically.Experimentsareconductedonaprototypetoexaminetheperformanceofstiffness-tuning,identifymodelparameters(including,inparticular,thecomplexshearmodu-lusoftheERunderdifferentelectricandthehydrodynamicfunction),andvalidatetheproposeddynamicmodel.First,foragivenelectricpassivedampedvibrationofthexibleinairismeasuredtoextractthenaturalfrequencyanddampingratio,whichareusedsubsequentlytoidentifythecomplexshearmodulusoftheERNext,asimilarprocedureisrepeatedinwatertoidentifythecomplexhydrodynamiccoefofthexibleFinally,thebehaviorofthebase-actuatedoscillationisstudied,inananchoredroboticbodysetup,tovalidatetheproposeddynamicmodel.,goodmatchbetweenthemeasuredbeamshapeandtipandtheirmodelpredictionsindicatetheefcacyofthemodel.Theexperimentsalsodemonstratethescapabilityinmodulatingstiffness.Forexample,whentheelectricisincreasedfrom0V/mto1:5106V/m,thesnaturalfrequencyincreasesfrom8.1Hzto10.1Hzinair(25%change),andfrom3.6Hzto5.1Hzinwater(40%change).Theorganizationoftheremainderofthischapterisasfollows.First,thefabricationprocedureoftheERispresentedinSection5.1.InSection5.2,thedynamicmodelisdescribed.TheexperimentalresultsareprovidedinSection5.3,wheretheeffectofchangingtheelectriconthestiffnessisstudiedandtheproposeddynamicmodelisvalidated.Finally,concludingremarksareprovidedinSection5.4.5.1FabricationProcedure5.1.1MaterialsTheERusedinthisstudyisLID-3354DfromSmartTechnologyLtd.,WestMidlands,Eng-land.Thisconsistsof37.5%ofsub-45mmpolymerparticlesinadensity-matchedsilicone109Figure5.1:3D-printedmoldsusedtoprototypethevariablestiffnessoil.ThedensityoftheERis1460Kg=m3andtheviscosityis110mPasat30C.Twothincopperfoils(Copper110,99.9%pure,fromBasicCopper,Carbondale,IL,USA)withdimen-sionsof15mm8mm0.035mmareusedastheelectrodes.Thexibleencasingismadeofx10,whichisaliquidurethanerubberfromSmooth-OnInc.,Macungie,PA,USA.Thisrubberhasadensityof1000Kg=m3.5.1.2ManufacturingProcedureTheprototypingofthevariable-stiffnessisdoneinmultiplestages.First,threedifferentmoldsaredesignedand3D-printedfortheencasing:Bottomhalf,tophalf,andthatforassemblingthetwohalves.Theactual3D-printedmoldsareshowninFig.5.1.Themoldsaredesignedtomakeawithdimensionsof65mm20mm4mm.Thebottomandtopmoldshaveasmalldentinthemtosecuretheelectrodesinplace.Agapof1.5mmisformedbetweenthetwoelectrodes,wheretheERwillbeinjected.Inthestep,thecopperelectrodesarecutandplacedinthedesigneddentineachmold,andthexmixtureispouredoverthem.Thepartsareplacedinavacuumchamberfordegassing(29flHgofvacuumfor5minutes),andissettocurefor24110(a)(b)(c)Figure5.2:Prototypingthevariable-stiffness(a)Thehalvesofthewithelectrodesincorpo-rated;(b)degassingthepartsinavacuumoven;(c)product:hollowwithwiresattachedtotheelectrodes.Figure5.3:Prototypedvariable-stiffnesswithERhours.Next,weremovethetwoprototypedhalvesofthefromthemolds,attachawiretoeachelectrode,putbothinthethirdmold,pourxmixture,degas,andletitcureforanother24hours.Theresultingprototypeisahollow,xiblewiththeelectrodesandwiresincorporated.ThedescribedprototypingstepsareillustratedinFig.5.2.Finally,a3D-printedrigidclampingpartisattachedtothewireendoftheandtheERisinjectedfromtheposteriorendofthebeamintothehollowgapbetweenthetwohalvesandtheinjectionholesaresealedafterward.Fig.5.3showsaERprototype.1115.2DynamicModelfortheVariable-StiffnessFinAlthoughthepresentedfabricationprocedureinSectionIIcanbeusedtomakedifferentvariable-stiffness(suchascaudalpectoralamongothers),thisstudyisfocusedontheuseforthecaudal(tail).Tosimplifythemodelingprocedure,weconsidertheroboticbodytobeanchored,sothecalculationsofthebodydynamicsarenotcoveredinthispaper.Themotionofthevariable-stiffnesscaudalwithERismodeledusingHamilton'sprinciple.Theresultingequationsofmotionarecomplexandhighlynonlinear,soelementmethodisusedtonumericallysolvetheequations.5.2.1EvaluationofHydrodynamicForceUsingLighthill'sLarge-AmplitudeElongated-BodyTheoryThehydrodynamicforceonthevariable-stiffnessisevaluatedusingLighthill'slarge-amplitudeelongated-bodytheory,whichwasdevelopedtostudythecarangiformswimmingmodeofa[70].AsillustratedinFig.5.4,[X;Y;Z]Tistheinertialframe,[xt;yt;zt]Taretheedcoordi-nate(originatthebaseofthetail),and(‹m;‹n)aretheunitvectorstangentialandperpendiculartothexiblerespectively.TheroboticisassumedtohaveaplanarmotionintheXY-plane.Weassumethatthewaterfarfromtheroboticbodyisatrest,andtheextensibilityofthecaudalisnegligible.TheLagrangiancoordinatesindicatesapointonthexibletailanditsdistancefromthebaseofthewhichvariesfrom0toL(lengthofthecaudalThelocationofeachpointontheattimet,intheinertialframe,isdenotedas(x(s;t);y(s;t)).Theinextensibilityassumptionisexpressedas ¶x¶s!2+ ¶y¶s!2=1:(5.1)112Figure5.4:Planarviewoftheroboticandthedetailedillustrationofthexibletailcoordinatesystem.Thetangential(‹m)andnormal(‹n)unitvectorsarerepresentedas‹m= ¶x¶s;¶y¶s!;(5.2)‹n= ¶y¶s;¶x¶s!:(5.3)Thevelocityvectorofthecaudal~Vt=(¶x=¶t;¶y=¶t)hastangentialandnormalcomponents,representedrespectivelyby,Vtm=<~Vt;‹m>=¶x¶t¶x¶s+¶y¶t¶y¶s;(5.4)Vtn=<~Vt;‹n>=¶y¶t¶x¶s¶x¶t¶y¶s;(5.5)113where<;>denotestheinnerproductofthevectors.Finally,thehydrodynamicforcedensityexperiencedbythecaudalduetotheadded-masseffect,fors<Lisobtainedby~fh(s)=mvGddtVtn‹n;(5.6)wheremvdenotesthevirtualmassofthetailandiscloseto14prb2,withrindicatingthedensityofwaterandbshowingthedepthofthecaudalinzdirection,andGisthecomplexhydrodynamiccoefforthebeam[39,145],whichislaterthroughexperiments.Ats=L,thereisaconcentratedforcecalculatedas~FhL=hmvVtnVtm‹n12mvV2tn‹mis=L:(5.7)5.2.2DynamicModelingoftheVariable-StiffnessCaudalFinFilledwithERFluidWeneedtheinformationaboutthemovementofthexibletail,particularlyitsnormalandtan-gentialvelocitycomponentsateachpoint,todeterminethehydrodynamicforcesandmomentsontheForthispurpose,wefollow[144,146Œ148]anduseHamilton'sprincipletodeveloptheequationsofmotionofthexibletail.Thevariable-stiffnesshasave-layerstructure.ItconsistsofanERcorewithtwocopperfoillayersaroundit,whichisfurtherencasedbythexiblerubberonbothsides.Drivenbyaservomotor,thecanoscillateatitsbasewiththeoscillationanglegivenbyq(t)=qAsin(wqt);(5.8)114(a)(b)Figure5.5:SchematicforactuationoftheERcaudal(a)Topview,(b)sideview.Figure5.6:ERreactiontotheelectricwhereqAistheamplitude(indegree)andwqistheangularfrequencyoftheactuation.Fig.5.5showstheactuationschematicforthewhichhasalengthofL,thicknessofh,anddepthofb.5.2.2.1ERFluidAnERisatypeofsmartthatchangesrheologicalpropertiesinthepresenceofanelectricTypicallyanERconsistsofanon-polarliquidwithdielectricparticlessuspendedinit.AsillustratedinFig.5.6,thechangesstatefromliquid(intheabsenceofanelectrictoasolidgelform(inthepresenceofastrongelectricwhenthesuspendedparticlesarealignedwiththelinesoftheelectric[149,150].Wheninamulti-layerbeamthefunctionsinitspre-yield(static)regimeandbehavesasaviscoelasticmaterial[151,152].Therefore,itspropertycanbemodeledwithaparametercalledcomplexshearmodulus(G).Thelinearrelationshipbetweenshearstress,115t,andshearstrain,g,isasfollows:t=Gg(5.9)whereGisafunctionoftheelectricandconsistsofastoragemodulus(G0)andalossmodulus(G00)[144]:G=G0+iG00(5.10)5.2.2.2Multi-layerBeamTomodelthexiblewithERsomekinematicrelationshipsneedtobede-Wemakeafewassumptionstosimplifythesekinematicrelationships:ThereisnoslippagebetweentheERlayerandtheelectrodes;thetransversedisplacement(alongtheyt-axis)isthesameforallthelayers;thereisnonormalstressintheERlayerandthereisnoshearstrainintheelectrodesorintherubberlayers;therubberandcopperlayersarebondedperfectly,resultinginthesamedisplacementinlongitudinaldirection.Therefore,theshearstrain,g,andthelongitudinaloftheERlayer,u3,isexpressedasfollows[147,148]g=u1u5h3+hh3¶w¶x;(5.11)u3=u1+u52+(h1+h2)(h4+h5)4¶w¶x;(5.12)whereuk(k=1;;5)arethelongitudinaldisplacementsofthemid-planeofthekthlayerwith(u1=u2)and(u4=u5),wisthetransversedisplacementofthebeam,hk(k=1;;5)isthethicknessofthekthlayer,withh=h1=2+h2+h3+h4+h5=2,and¶w¶xistheangle.AschematicofaportionoftheERxibleinaisshowninFig.5.7.116Figure5.7:SchematicsoftheERxibleThegoverningequationsofmotionforthevariablestiffnessxiblewithERareobtainedusingtheHamilton'sprinciple,whichisdescribedasfollowsZt2t1d(TU+W)dt=0;(5.13)whereTisthekineticenergy,Uisthepotentialenergy,Wistheworkdonebyexternalloads,anddisthevariationaloperatorthroughoutthexiblecaudalThekineticenergyisdeterminedasT=12ZL05åk=1rkAkrTkrkdx+12Jq2;(5.14)whererkisthedensityofthekthlayer,Akisthecross-sectionalareaofthekthlayer(Ak=bkhk),wherek=1;;5;LdenotesthelengthoftheandJisthemomentofinertiaassociatedwith117theservomotoractuation.Thevelocityvectorrkisasrk=(ukwq)‹xt+(xq+ukq+w)‹yt;(5.15)where‹xtand‹ytaretheunitvectorsinthextandytdirections,respectively.ThepotentialenergyiscalculatedasfollowsU=12ZL0"åk=1;2;4;5EkAk(¶uk¶x)2+EkIk(¶2w¶x2)2+GA3g2+mbq2[12(L2x2)+r(Lx)](¶w¶x)2#dx;(5.16)whereEkistheYoung'smodulusofthekthlayer,Ikisthemomentofinertiaofthekthlayer,misthetotalmassofthexisthedistancefromedendoftheandristheservoarmlength.Thetotalworkdonebytheexternalforces(servoarmandhydrodynamicforces)areasfollowsW=tq+ZL0fhwdx+FhLxu3+FhLywx=L;(5.17)wheretistherotationaltorquefromtheservomotorandthehydrodynamicforcesfhandFhLaredeterminedbaseonLighthill'slarge-amplitudeelongated-bodytheory.BysubstitutingEqs.(5.14),(5.16),and(5.17)intoEq.(5.13),weobtainthedynamicequations118ofmotion:(r1A1+r2A2+14r3A3)¨u1+14r3A3¨u52(r1A1+r2A2+12r3A3)w¨q3(r1A1+r2A2+12r3A3)wq(r1A1+r2A2)(r+x+u1)q212r3A3(r+x+u1+u52)q2+12(E1A1+E2A2)¶¶u1¶u1¶x2+GA3h23(u1u5)=0;(5.18)(14r3A3+r4A4+r5A5)¨u5+14r3A3¨u12(12r3A3+r4A4+r5A5)w¨q3(12r3A3+r4A4+r5A5)wq(r4A4+r5A5)(r+x+u5)q212r3A3(r+x+u1+u52)q2+12(E4A4+E5A5)¶¶u5¶u5¶x2GA3h23(u1u5)=0;(5.19)119(r1A1+r2A2+r3A3+r4A4+r5A5)¨w+(r1A1+r2A2)(x+r+u1)¨q+(r4A4+r5A5)(x+r+u5)¨q+r3A3(x+r+u1+u52)¨q+3(r1A1+r2A2)u1q+3(r4A4+r5A5)u5q+r3A3(u1+u52)q(r1A1+r2A2+r3A3+r4A4+r5A5)wq2+12(E1I1+E2I2+E4I4+E5I5)¶¶w¶2w¶x22+12mbq2h12(L2x2)+r(Lx)i¶¶w¶w¶x2+GA3h2h23¶¶w¶w¶x2fhw+FhLxu3+FhLywx=L=0:(5.20)5.2.2.3DiscretizationusingFiniteElementMethod(FEM)Theequationsofmotion(Eqs.(5.18),(5.19),and(5.20)),whicharecoupledandnonlinear,aresolvedusingelementmethod[146Œ148].Eachelementconsistsoftwonodes,andeachnodehasfourdegreesoffreedom.Nodedisplacementsofelementiareformedbythefollowingvectorwhichisboundedbetweennodesjandk(withk=j+1):qi=hu1ju5jwj¶wj¶xu1ku5kwk¶wk¶xiT:(5.21)Aschematicofthexiblenelementsandnodes,andthedetailsofoneelementareshowninFig.5.8.120(a)(b)Figure5.8:SchematicofthexibleinFEM:(a)oftheelementandnodenum-bers,(b)elementi,itstwonodes,anddegreesoffreedom.Thevector,representedintermsofthenodevector,isgivenby[u1u3u5w¶w¶xg]T=[N1N2N3N4N5N6]Tqi;(5.22)whereFEMshapefunctionsN1,N2,N3,N4,N5andN6areasN1=[1x000x000];(5.23)N2=12(N1+N3);(5.24)N3=[01x000x00];(5.25)N4=[0013x2+2x3(x2x2+x3)Li(5.26)003x22x3(x2+x3)Li];(5.27)N5=[¶N4¶x];(5.28)N6=[N1N3h3+hh3N5];(5.29)wherex=xLi,withLibeingthelengthofeachelement.ApplyingtheFEMpresentedinEq.(5.22)totheHamilton'sprinciple,weobtainthedynamic121equationofmotionattheelementallevel:Mi¨qi+2qCiqi+Kiqi=Fi;(5.30)whereMiistheelementmassmatrix,Ciistheelementgyroscopiceffectmatrix,Kiistheelementstiffnessmatrix,andFiistheelementforcematrix,whichareformedasfollows:Mi=ZLi05åk=1hrkAk(NTkNk+NT4N4)idx;(5.31)Ci=ZLi05åk=1hrkAk(NTkN4NkNT4)idx+ZLi0G00A3NT6N6dx;(5.32)Ki=Ki1+q2(Ki2Mi)¨qCi;(5.33)whereKi1=ZLi0åk=1;2;4;5h(EkAkNTk;xNk;x+EkIkNT4;xxN4;xx)idx+ZLi0G0A3NT6N6dx;(5.34)Ki2=12ZLi05åk=1rkAkL2(xi+x)2NT5N5dx+rZLi05åk=1rkAkL(xi+x)NT5N5dx;(5.35)122Fi=ZLi0n5åk=1rkAkq2(r+xi+x)NTk5åk=1rkAk¨q2(r+xi+x)NT4+fhNT4odx+FhLxu3+FhLywi=N:(5.36)Theequationofmotion,resultingfromastandardFEMassemblyprocedure,isformedasfollowsM¨q+2qCq+Kq=F;(5.37)withu1,u5,w,and¶w¶xbeingzeroatthebaseofthebeam.5.3ExperimentalResults5.3.1ofShearModulusValuesThedetailedspeofthedesignedbeamarepresentedinSection5.1.OtherthanthosephysicaltheERhasacomplexshearmodulus,G,thatneedstobeidenti-forthesystemateachelectriclevel.Todoso,asetofexperimentsareconductedontheprototypedbeam.TheERbeamisclampedtoastandardprecisiondovetailZ-axisstage(ZDTLS80,MisumiUSA)viaacustom-made3D-printedplatform.Alasersensor(BaumerElectric,OADM20I6441/A14F)isattachedtoastandardprecisiondovetailXY-axisstage(XY-DTS90,MisumiUSA).Thetwostagesareedonasetupplate(MisumiUSA),sothatthelaserispointedatthecentertipofthexibleThelasersensoroutputiscapturedbyadSPACEsystem(RTI1104,dSPACE).Ahighvoltagegeneratorpowermodule(Inputvoltage:3V,outputvoltage:7kV,Sunkee,China)isusedtogeneratethedesiredelectricFortheexperiment,the123(a)(b)Figure5.9:Experimentalsetupforobservingpassive,dampedvibrationoftheERxibleunderdifferentelectric(a)Schematic,(b)actual.beamismanuallyaround1cmatthetipandthenreleased,sothedisplacementofthetipisrecordedusingthelasersensorasthebeamoscillates.Theexperimentisrepeated20timesforeachelectricvalue.TheexperimentalsetupisshowninFig.5.9.Fromtherecordedsignals,thenaturalfrequencyandthedampingratioofthebeamforthegivenelectricareextracted,whicharesubsequentlyusedtoidentifythecorrespondingG.Thedampingratio,z,iscalculatedfromthetwoconsecutivepeaksoftherecordedsignal(XcandXc+1,respectively),asfollows:z=vuuut12plnXcXc+12:(5.38)Thenaturalfrequency,fn,isobtainedfromthetimeinstancesoftwoconsecutivepeaks(TcandTc+1,respectively)oftherecordedsignal,asfollows:fn=1(Tc+1Tc)p1z2:(5.39)ToidentifytheparametersG0andG00,weuseexhaustivesearchinrange[0,50000]forG0and[0,124(a)(b)Figure5.10:parametersunderdifferentelectricintheair.(a)Dampingratio,(b)naturalfrequency.5000]forG00.,westartwithacoarsegridandevaluatedtheresultingnormalizederrorbyJ="fSimnfExpnfExpn2+zSimzExpzExp2#(5.40)AftertheminimumvalueforJ,wesearchinagridaroundthecorrespondingG0andG00.Thisprocedureisrepeateduntiltheerrordropsbellow5%forbothzandfn.Fig.5.10(a)and(b)showtheempiricaldampingratio,z,andnaturalfrequency,fn,fordifferentelectricrespectively.Thedampingratioincreasesfrom0.09to0.25(170%change)andthenaturalfre-quencyincreasesfrom8.1Hzto10.1Hz(25%change),whentheelectricisincreasedfrom0V/mto1:5106V/m.Onecanseehowincreasingtheelectricaffectstheperformanceofthebeam.Intheabsenceoftheelectrictheisatitsmostxibleandthushasthelowestnaturalfrequency.Byincreasingtheelectricthenaturalfrequencyanddampingratioofthexibleincreases,anditbecomesstiffer.125(a)(b)Figure5.11:complexshearmodulusvaluesandthecurveunderdifferentelectric(a)Storagemodulus(G0),(b)lossmodulus(G00).ThestoragemodulusG0andthelossmodulusG00aretobeG0=103(0:053E3f0:043E2f+0:295Ef+0:588);(5.41)G00=1:33E2f+33:8Ef+74:8;(5.42)whereEfistheelectricappliedtotheERFigs.5.11showsthecomplexshearmodulusvaluesandthecurveunderdifferentelectricMatlabcommandpolyfitisusedtoathirdorderpolynomialtothestoragemodulusandasecondorderpolynomialtothelossmodulus.Theseparametersareusedthroughoutsimulationofallotherexper-imentalsettings(passivedampedoscillationsandbase-actuatedrotations)discussedintherestofthissection.Fig.5.12showsboththesimulationandexperimentaldataonthetipdisplacementofERxiblefordifferentelectricvalues.126(a)(b)(c)(d)(e)(f)(g)Figure5.12:Asampleofexperimentallymeasureddampedoscillationsofthebeamtip,alongwithsimulationresultscorrespondingtothematchedparameters,foreachappliedelectric(a)Ef=0Vm,(b)Ef=0:25106Vm,(c)Ef=0:5106Vm,(d)Ef=0:75106Vm,(e)Ef=1106Vm,(f)Ef=1:25106Vm,(g)Ef=1:5106Vm.5.3.2ofHydrodynamicAsimilarsetofexperiments,butwiththeproposedstiffness-tuningsubmersedinwater,areconductedtoidentifythecomplexhydrodynamiccoefG,inEq.(5.6).Theexperimental127Figure5.13:Experimentalsetupformeasuringdampedoscillationsofthexibleinwater.(a)(b)Figure5.14:parametersunderdifferentelectricinwater.(a)Dampingratio,(b)naturalfrequency.setupisshowninFig.5.13.Thebeamismanuallyaround1cmatthetipandthenreleasedtogeneratepassive,dampedoscillations,whicharerecordedusingthelasersensor.Theexperimentisrepeated20timesforeachelectricvalue.Thenaturalfrequencyandthedampingratioareextractedfromtherecordedsignal,andarematchedtotheonesfromthesimulationtoidentifythecomplexhydrodynamiccoefFig.5.14(a)and(b)showtheempiricaldampingratio,z,andnaturalfrequency,fn,fordifferentelectricrespectively,whenthebeamisinwater.The128dampingratioincreasesfrom0.18to0.42(130%change)andthenaturalfrequencyincreasesfrom3.64Hzto5.12Hz(40%change),whentheelectricisincreasedfrom0V/mto1:5106V/m.Notethatthenaturalfrequencyanddampingratiodecreasewhenthebeamissubmersedinwater(versusthecasewhenitisinair).ThesameprocedureasdescribedinSection5.3.1isappliedtocomplexhydrodynamiccoefG=G0+iG00,withrange[0,10]forbothG0andG00.ThevaluesofGareclosedtoeachotherforallthetestedelectricvalues;therefore,weconsiderthemeanvalueamongalltheelectrictobeourcomplexhydrodynamiccoefThiscomplexhydrodynamiccoefG,istobeG=4:31+i3:84(5.43)NotethatthecomplexshearmodulususedinthesesimulationsisasdiscussedinSection5.3.1.Fig.5.15showsthecomparisonofexperimentallyobserveddampedoscillationswiththesim-ulatedonesusingthehydrodynamiccoefwherewecanseethattheymatchwell.5.3.3BaseActuationExperimentsforModelValidationThelastsetofexperimentsinvolveactuatingthexibleinwateronitsbasewithaservomotor,emulatingtheofananchoredroboticitstail.Thepurposeistoexaminethestiff-tuningbehaviorinthisnewsettingandtovalidatethedynamicmodelwithindependentexperiments.Thexibleisactuatedwithawaterproofservo(HS-5086WPfromHitec),edinthetank,andthemotionoftheisrecordedfromaboveusingahigh-speedcamera(CasioExilimEX-FH25)at120frames/s.TheexperimentalsetupisshowninFig.5.16.Fig.5.17comparesthemeasuredtime-dependentbeamshapesandthosepredictedbythedy-129(a)(b)(c)(d)(e)(f)(g)Figure5.15:Passive,dampedvibrationoftheERxiblebeaminwater:(a)Ef=0Vm,(b)Ef=0:25106Vm,(c)Ef=0:5106Vm,(d)Ef=0:75106Vm,(e)Ef=1106Vm,(f)Ef=1:25106Vm,(g)Ef=1:5106Vmnamicmodelforamplitudeof22.5andfrequencyof2Hz,fortwodifferentelectric0V/m(Fig.5.17(a)and(b))and1:5106V/m(Fig.5.17(c)and(d)).Tobebrief,weshowtheandlastframesofeachhalf-cycle.Itcanbeseenthat,withanincreasingelectricthebecomesstifferandasaresult,thetipdisplacementgetssmaller,andgoodagreementbetween130Figure5.16:Experimentalsetupformeasuringtheshapeunderbaseactuation(topview).theexperimentalandsimulationresultsprovidesvalidationoftheproposeddynamicmodel.Thecomparisonofthesimulatedtime-dependentERxiblebeamtipdisplacementsforthetwoelectricvaluesisfurthershowninFig.5.17(e).Finally,weconductsimilarexperimentswithfrequencyof4Hz,whichisclosetothenaturalfrequencyoftheinwater.Toachievethisfrequency,theamplitudeoftheservomotionisreducedtoqA=10,inorderfortheservotounderitsactuationlimit.Thehigh-speedcameraissettorecordat240frames/s.Fig.5.18showsthecomparisonbetweenex-perimentalmeasurementofthetime-dependentshapeswithmodelpredictions.Overall,themodel-predictedbeamshapetrajectoriesmatchwellwiththemodelpredictions.Thediscrepancybetweenthemodelpredictionandtheexperimentalmeasurementcanbeattributedtosomelimita-tionsintheprototypingprocess.First,thewasnotperfectlysymmetricandtherewasasmalldifferencebetweenliquidurethanerubberthicknessesontwosides.Therewasnoprecisecontrolontheamountoftherubberthatleakedonthesurfaceofthecopperelectrodesduringthepro-totypingprocedure,whichresultedinanotherfactorinbreakingthesymmetry.Inaddition,the131(a)(b)(c)(d)(e)Figure5.17:Base-actuatedstiffness-tunablewithamplitudeof22:5andfrequencyof2Hz:(a)-(b)Comparisonbetweenexperimentalmeasurementofthetime-dependentbeamshapeswithmodelpredictions,wheretheelectricappliedtothebeamisEf=0Vm,(c)-(d)comparisonbetweenexperimentalmeasurementofthetime-dependentbeamshapeswithmodelpredictions,wheretheelectricappliedtothebeamisEf=1:5106Vm,wherethesolidblacklinecorrespondstotheservoarmdirectionandtheyellowdashedlinerepresentstheshapebythedynamicmodel,and(e)comparisonbetweenthesimulatedtipdisplacementsunderthetwodifferentelectricbondingagentusedtosealthetwohalvesofthebeamtogetherwasnotconsideredinthemodelingprocedure.Thisbondingagentresultedinastifferbeamthanexpected.5.4ConclusionThegoalofthisworkwastointroduceanovelcompactmechanismforaxiblewithactivelytunablestiffness.ThiswasachievedbyembeddingERinxiblerubber,anditwasdemon-stratedinexperimentsthatthestiffnesscouldbetunedatafastspeed.Adynamicmodelderived132(a)(b)(c)(d)(e)(f)Figure5.18:Base-actuatedstiffness-tunablewithamplitudeof10andfrequencyof4Hz:(a)-(c)Comparisonbetweenexperimentalmeasurementofthetime-dependentbeamshapeswithmodelpredictions,wheretheelectricappliedtothebeamisEf=0Vm,(d)-(f)comparisonbetweenexperimentalmeasurementofthetime-dependentbeamshapeswithmodelpredictions,wheretheelectricappliedtothebeamisEf=1:5106Vm,wherethesolidblacklinecor-respondstotheservoarmdirectionandtheyellowdashedlinerepresentstheshapebythedynamicmodel.usingHamilton'sprinciplewasproposedtocapturethesbehavior,wherethehydrodynamicforceonthewasincorporated.Themodelparameterswerewithpassiveoscillationexperimentsinairandinwater,andthemodelwasfurthervalidatedwithbase-actuationexperi-mentsemulatingasetting.133Chapter6CONCLUSIONSANDFUTUREWORK6.1ConcludingRemarksThroughoutthisdissertation,wehaveinvestigatedthedesign,developmentanddynamicmodelingofxibleelmentsofaroboticandstudiedtheeffectofxibilityonitsperformance.First,theswimmingperformanceandmechanicalefyofxiblepectoralconnectedtoactuatorshaftsviarigidlinks,arestudied,whereitisfoundthatxibledemonstrateadvan-tagesoverrigidinspeedandefyatrelativelylowtfrequencies,whiletherigidoutperformthexibleathigherfrequencies.Thepresentedmodeloffersapromisingtoolforthedesignofxibilityandswimminggait,toachievespeedandefyobjectivesfortheroboticSecond,axiblepassiverowingjointisproposed.Thisnoveljointenablesthepectoraltoperformasymmetricrowingmotion,asopposedtothetraditionalrigidjoint,whereoneneedsafasterpowerstrokeandslowerrecoverystrokespeedtohaveanetthrust(thismech-anismwasusedinChapter2).Thisjointallowsthepectoraltosweepbackpassivelyduringtherecoverystrokewhileitfollowstheprescribedmotionoftheactuatorduringthepowerstroke,whichresultsinnetthrustevenundersymmetricactuationforpowerandrecoverystrokes.Thexibilityofthejointismodeledasapairoftorsionalspringanddamper,andthebladeelementtheoryisusedtoevaluatethehydrodynamicforces.Thedynamicmodelofaroboticequipped134withsuchjointsisdevelopedandvalidatedthroughextensiveexperiments.Motivatedbytheneedfordesignoptimization,themodelisfurtherutilizedtoinvestigatetheofthejointlengthandstiffnessontherobotlocomotionperformanceandefy.Third,analternativexiblejointforpectoralisalsoproposed,whichenablesthepec-toraltooperateprimarilyintherowingmode,whileundergoingpassivefeatheringduringtherecoverystroketoreducehydrodynamicdragontheHere,thepectoralundergoesa3Dmo-tion,whichresultsincalculationofhydrodynamicforcesin3Dspace.Adynamicmodel,vexperimentally,isdevelopedtoexaminethetrade-offbetweenswimmingspeedandmechanicalefyinthedesign.Finally,weinvestigatexiblewithactivelytunablestiffness,enabledbyelectrorheolog-ical(ER)Thetunablestiffnesscanbeusedinoptimizingtheroboticspeedormaneu-verabilityindifferentoperatingregimes.FinswithtunablestiffnessareprototypedwithERenclosedbetweenlayersofliquidurethanerubberx10).Freeoscillationandbase-excitedoscillationbehaviorsofthearemeasuredunderwaterwhendifferentelectricareappliedfortheERwhicharesubsequentlyusedtodevelopadynamicmodelforthestiffness-tunable6.2FutureWorkTheinvestigationontheeffectofxibilityontheperformanceofaroboticcanbeextendedfurther.Forthexiblepassivejoint,weconsideredrigid,rectangularpectoralhowever,xiblepectoralwithdifferentsizesandshapesareworthfurtherinvestigation.Anotherinter-estingproblemistocombinethexiblepassiverowingjointandxiblepassivefeatheringjointdesigns,andexploretheresultingperformance.Interactionsofxiblecaudalandpectoral135presentinterestingchallenges,whichcanbestudiedfurther.Forthestiffness-tunablethefabricationprocedureforthecanbesothatthelimitationsdiscussedinSection5.3.3areaddressed.Also,thestiffness-tunableneedstobeintegratedwithafree-swimmingrobotictoinvestigatetheactiveonboardcontrolofstiffnesswhentherobotoperatesindifferentregimes(fast/slowspeeds,forexample),foroptimizationoftheswimmingefy.136REFERENCES137REFERENCES[1]C.C.Lindsey,fi1-Form,function,andlocomotoryhabitsinflinFishPhysiology,ser.Locomotion,W.S.HoarandD.J.Randall,Eds.AcademicPress,1978,vol.7,pp.1Œ100.[2]P.ValdiviayAlvaradoandK.Youcef-Toumi,fiDesignofmachineswithcompliantbodiesforbiomimeticlocomotioninliquidenvironments,flJournalofDynamicSystems,Measure-ment,andControl,vol.128,no.1,pp.3Œ13,Sep.2005.[3]T.IchikizakiandI.Yamamoto,fiDevelopmentofroboticwithvariousswimmingfunc-tions,flinSymposiumonUnderwaterTechnology(UT)andWorkshoponUseofSubmarineCablesandRelatedTechnologies,Tokyo,Japan,Apr.2007,pp.378Œ383.[4]J.Yu,M.Tan,S.Wang,andE.Chen,fiDevelopmentofabiomimeticroboticanditscontrolalgorithm,flIEEETransactionsonSystems,Man,andCybernetics,PartB(Cyber-netics),vol.34,no.4,pp.1798Œ1810,Aug.2004.[5]R.W.Blake,FishLocomotion.Cambridge:CambridgeUniversityPress,May1983.[6]H.Wang,fiDesignandimplementationofabiomimeticroboticflMastersThesis,Con-cordiaUniversity,Montreal,Quebec,Canada,2009.[7]P.R.Bandyopadhyay,D.N.Beal,andA.Menozzi,fiBioroboticinsightsintohowanimalsswim,flJournalofExperimentalBiology,vol.211,no.2,pp.206Œ214,Jan.2008.[8]X.Tan,D.Kim,N.Usher,D.Laboy,J.Jackson,A.Kapetanovic,J.Rapai,B.Sabadus,andX.Zhou,fiAnautonomousroboticformobilesensing,flinIEEE/RSJInternationalConferenceonIn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