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Haltere Mediated Flight Stabilization in Diptera

Permanent Link: http://ufdc.ufl.edu/UFE0024793/00001

Material Information

Title: Haltere Mediated Flight Stabilization in Diptera Rate Decoupling, Sensory Encoding, and Control Realization
Physical Description: 1 online resource (148 p.)
Language: english
Creator: Thompson, Rhoe
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: campaniform, control, coriolis, df2, diptera, dynamics, encoding, flight, gyroscope, haltere, imu, saccade, sensilla, stabilization
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Insects of the order Diptera have a single pair of wings. The rear wings of Dipteran insects have evolved into organs that allow stabilizing control responses through sensing and encoding of body angular rate feedback. This dissertation documents research on the physical and physiological mechanisms that enable a pair of halteres to distinguish and encode three orthogonal components of the body rate vector. While the knowledge that the halteres play a role in flight stability has been accepted for centuries, the understanding of how the insect's very simple sensory structures are able to encode and decouple the orthogonal components of the rate vector has been lacking. The work described in this report furthers this understanding through modeling and simulation. First, a natural decoupling of the observable rate components has been identified that asserts proportionality of body rate components to averaged strain characteristics near the center of the haltere stroke. Second, a means of encoding and decoding the necessary rate information in a manner compatible with the insect's sensory structures and flight motor physiology has been identified and demonstrated. Finally, the ability of the proposed haltere model to stabilize fight in a 6DOF environment with competing behavioural objectives and randomly generated obstructions has been demonstrated.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Rhoe Thompson.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024793:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024793/00001

Material Information

Title: Haltere Mediated Flight Stabilization in Diptera Rate Decoupling, Sensory Encoding, and Control Realization
Physical Description: 1 online resource (148 p.)
Language: english
Creator: Thompson, Rhoe
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: campaniform, control, coriolis, df2, diptera, dynamics, encoding, flight, gyroscope, haltere, imu, saccade, sensilla, stabilization
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Insects of the order Diptera have a single pair of wings. The rear wings of Dipteran insects have evolved into organs that allow stabilizing control responses through sensing and encoding of body angular rate feedback. This dissertation documents research on the physical and physiological mechanisms that enable a pair of halteres to distinguish and encode three orthogonal components of the body rate vector. While the knowledge that the halteres play a role in flight stability has been accepted for centuries, the understanding of how the insect's very simple sensory structures are able to encode and decouple the orthogonal components of the rate vector has been lacking. The work described in this report furthers this understanding through modeling and simulation. First, a natural decoupling of the observable rate components has been identified that asserts proportionality of body rate components to averaged strain characteristics near the center of the haltere stroke. Second, a means of encoding and decoding the necessary rate information in a manner compatible with the insect's sensory structures and flight motor physiology has been identified and demonstrated. Finally, the ability of the proposed haltere model to stabilize fight in a 6DOF environment with competing behavioural objectives and randomly generated obstructions has been demonstrated.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Rhoe Thompson.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Dixon, Warren E.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024793:00001


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andchildren,Ryan,Rachel,andJessica 3

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Iwouldliketoacknowledgemycommitteechair,Dr.WarrenDixon,forhistimelywordsofencouragementandtheexamplehehasprovided.Hispassionforhisworkandprofessionalismhavebeenmotivationalfactorsinthesuccessofallofhisstudents.Iwouldalsoliketoacknowledgetheothermembersofmycommittee,Dr.CarlCrane,Dr.NormanFitz-Coy,Dr.DanielHahn,Mr.RicWehlingandMr.JohnnyEversfortakingthetimetoreviewmyworkandforprovidinginsightfulsuggestionsforimprovement.IwouldliketoacknowledgethemanydistinguishedprofessorsIhavelearnedfromovertheyearsattheUniversityofFlorida,oneofthemostmemorablebeingthelateProf.KnoxMillsaps.Tothedismayofsomestudents,Prof.Millsapswouldattimesspendmuchofhisclasstellingfascinatingstoriesabouthiscareerandthehistoricalgureshehadencountered.Hetoldusonce,\Learningisbestaccomplishedlateatnight,solvingproblemsinthecompanyofagoodtextbookandadesklamp."Ihavecometobelievethatthestorieshetoldtomotivatethedesiretolearnwereatleastasimportantastheexamplesheworkedontheboard.LearningfromProf.MillsapsandtheotheroutstandingmembersoftheUniversityofFloridafacultyhasbeenanhonor.IwouldliketothanktheAirForceResearchLaboratorywhichallowedmeapaidsabbaticaltopursuemyPhDstudiesandthenminimaldisturbancewhenIreturnedtocompletemydegree.Inparticular,IwouldliketothankMs.SandraLefstadforbeingsupportivewhenIbroughtuptheideaofleavingmyjobtogobacktoschool.Ingeneral,AFRLdeservesagreatdealofcreditforfosteringinnovationandallowingengineerstostudybiologicalsystemsinsearchofrevolutionarynewtechnologies.Finally,Imustacknowledgethemanyscientistswhohavelaidthefoundationforthisresearch.Theiringenuityandneverendingfascinationwithananimalasmalignedastheyhaveinspiredmeateverystepoftheway.Theirworkconvincesmethattherearemanyscientistswhounderstandthatloveforthisworld'severevolvingcreationismorethanapersonalpassion,itisalsoagiftbacktolife'ssource. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTIONANDPROBLEMSTATEMENT ................ 12 2HISTORICALSUMMARYOFHALTERERESEARCH ............. 16 2.1TheoriesfortheHaltere'sStabilizingInuence ................ 16 2.2SensoryPhysiology ............................... 19 2.3NeuralPathways ................................ 22 2.4WingandNeckMechanosensoryControlObservations ........... 24 2.4.1WingCompensatoryResponse ..................... 25 2.4.2HeadCompensatoryResponse ..................... 28 3HALTEREKINEMATICSANDDYNAMICS ................... 32 3.1Introduction ................................... 32 3.2Methods ..................................... 35 3.3Results ...................................... 37 3.3.1KinematicAssessment ......................... 37 3.3.2DynamicsEquationAllowingforOut-of-PlaneMotion ........ 41 3.3.3HaltereTrajectorySimulations ..................... 42 3.3.3.1Out-of-planestinessvariations ............... 42 3.3.3.2Dampingvariations ...................... 44 3.3.4AverageHalterePosition ........................ 44 3.3.5AnalysisofErrorsDuetoNon-Linearity ............... 48 3.4Discussion .................................... 51 3.4.1MechanoreceptiveEncoding ...................... 52 3.4.2MechanoreceptiveAveragingModality ................. 53 4RECONCILINGTHEPHYSICSWITHTHEPHYSIOLOGY .......... 57 4.1Introduction: .................................. 57 4.2MethodsandExperimentalProcedures .................... 59 4.2.1DynamicsModel ............................. 59 4.2.2CampaniformModel .......................... 60 4.2.3ModelingProcess ............................ 61 4.3Results ...................................... 61 5

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.......................... 61 4.3.2CampaniformResponseDecodingAlgorithm ............. 63 4.3.3AlgorithmApplication ......................... 68 4.4Discussion .................................... 68 4.4.1SignicanceofFindings ......................... 68 4.4.2DirectionalandTemporalSensitivityofCampaniformEncoding .. 72 4.4.3PhaseSensitivityofCampaniformAerents .............. 74 4.4.4PhysiologicalImplicationsofAerentDecoding ........... 75 4.4.5GeneralObservations .......................... 76 56DOFCONTROLREALIZATIONWITHHALTEREFEEDBACK ....... 78 5.1Introduction ................................... 78 5.26DOFSimulationMethods ........................... 80 5.2.1BehavioralDecisionLogic(Brain) ................... 82 5.2.1.1Longrangeactivation .................... 84 5.2.1.2Altitudecontrol ........................ 84 5.2.1.3Obstructionavoidance .................... 85 5.2.1.4Closerangetargetacquisition ................ 85 5.2.1.5Compositecontrolvector ................... 86 5.2.2KinematicandInertialCharacteristics ................. 87 5.2.3ControlLogic .............................. 87 5.2.4FlightEnvironment ........................... 90 5.3AnalysisResults ................................. 91 5.3.1AnalysisofTorqueCross-CouplingonInsectFlightStability .... 92 5.3.1.1Stabilityanalysiswithproportionalcontrol ......... 93 5.3.1.2Stabilityanalysiswithproportionalcrosscoupling ..... 94 5.3.1.3Stabilityanalysiswithdragandcrosscoupling ....... 95 5.3.1.4Stabilityanalysisconclusion ................. 97 5.3.2ContralateralVersusIpsilateralExpressionofWingControl ..... 98 5.3.2.1Wingkinematicswithcontralateralinuence ........ 99 5.3.2.2Ipsilateralhaltereinuencewithabductionasacontrolparameter ........................... 101 5.3.2.3Ipsilateralhaltereinuencewithaerodynamicwingmomentasacontrolparameter .................... 103 5.3.36DOFResultsComparingHaltereMeasurementswithTruth .... 104 5.3.46DOFObstructionAvoidanceandSaccade-LikeManeuvers ..... 108 5.4Discussion .................................... 110 5.5ConcludingRemarks .............................. 117 6CONCLUSIONS ................................... 118 APPENDIX ADERIVATIONOFTHEHALTEREDYNAMICSEQUATION .......... 123 BDERIVATIONOF6DOFEQUATIONSOFMOTION .............. 126 6

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......... 130 C.1ContralateralHaltereInuencewithAbductionasaControlParameter .. 130 C.2IpsilateralHaltereInuencewithAbductionasaControlParameter .... 134 C.3IpsilateralHaltereInuencewithAerodynamicWingMomentasaControlParameter .................................... 138 REFERENCES ....................................... 143 BIOGRAPHICALSKETCH ................................ 148 7

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Table page 3-1Notationassociatedwithkinematicanddynamicexpressions. .......... 38 5-1Nominalsimulationparametersforthesimulatedinsect. ............. 87 5-2Exampleofderivedsimulationcontrolparameters. ................. 89 5-3Parameterdenitionsforwingkinematiccontrolexpressions. ........... 100 B-1Denitionsofvariablesandsymbolsassociatedwith6DOFequationsofmotion. 126 8

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Figure page 1-1EnvironmentalScanningElectronMicrograph(ESEM)ofthehaltereoftherobbery,familyAsilidae. ................................. 13 1-2TheCoriolisforcechangessignasthevelocitycomponentperpendiculartotheratecomponentchangessign(~!~v). ........................ 14 2-1GeneralmorphologyofthehaltereonCalliphora. ................. 20 2-2Thecentralprojectionsofthehalterenerve. .................... 23 2-3Magnitudeofthereexiverollresponseoftheheadasafunctionofrollrate. .. 30 3-1Characteristiclocationsofthehalteresandtheirstrainsensors. ......... 33 3-2Halterevelocitysignchangeasafunctionofstrokekinematics. .......... 34 3-3Referenceframedenitionsforhaltereanalysis. .................. 36 3-4Halteretrajectoryinresponsetoverticalinputrate(1)withvariationinhalterenaturalfrequency. ................................... 43 3-5Halteretrajectoryinresponsetolateralinputrate(3)withvariationinhalterenaturalfrequency. ................................... 43 3-6Halteretrajectorieswithdampingratiosat10%ofcriticaland100%critical. .. 44 3-7Halteretrajectorieswiththeaveragedisplacementplottedasafunctionofstrokeangle. ......................................... 45 3-8Comparisonofsimpliedlinearmodelsofthehalterewiththefullnon-linearmodel. ......................................... 47 3-9Blockdiagramdescriptionoftheerroranalysisusedtocomparetrueandhalteremeasurementsofangularratecomponents. ..................... 49 3-10ErrorinestimatesofratecomponentsalongthebodyYaw,Pitch,andRollaxesforcaseRollRate=0. ................................ 50 3-11ErrorinestimatesofratecomponentsalongthebodyYaw,Pitch,andRollaxesforcaseRollRate=5rad/s. ............................. 50 3-12TheCoriolisforceinducedbypitchratehasbilateralsymmetry,butyawandrollhaveantisymmetricforces. ............................ 54 3-13Resultofsumminganddierencinglinearapproximationsofthestrainatthecenterofthehalterestroke. ............................. 55 9

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................................... 62 4-2Ratecomponentsexpressedinthebodyandhalterereferenceframes. ...... 64 4-3Halteretrajectoryoverthetimeperiodassociatedwith40haltereoscillations. 65 4-4Thesensillaresponsesaresummedateachintegrationtimesteptorepresentthenetmotorcontrolaerents. ........................... 66 4-5Thetimingofmodeledcampaniformsensillaresponsewithrespecttosignoftheratecomponents. ................................. 67 4-6Resultoftheratedecodingalgorithmappliedtothesimulatedoutputofthehalteresensilla. .................................... 69 4-7Unilateralresponseofthewingsresultsinforcesthatprovidenettorquesproportionaltooriginaldisturbances. ............................... 70 4-8Halterederivedrateestimatesundertheinuenceofamorerealistic\saw-tooth"motionprole. .................................... 71 5-1Non-linearresponseactivationlogicforthesimulatedinsect. ........... 83 5-2Attitudecontrollogicforsmallazimutherrors. ................... 88 5-3Nominal6DOFtestcaseshownfromatopviewperspective. ........... 90 5-4PlotdescribingtherangeoverwhichtheLyapunovderivativeisnegativedenite. 96 5-5Analysisgeometryforwingkinematicanalysis. .................. 99 5-6Measuredversusactualangularvelocitycomponenttimehistory. ........ 106 5-7Expandedviewoftruepitchrateandmodeledhaltereestimateofthepitchrate. 107 5-8Therelationshipbetweenstabilizingtorqueandangularvelocity. ......... 107 5-9Comparisonofightpathswithandwithouthalterefeedback. .......... 108 5-10ComparisonofEulerangleswithandwithouthalterefeedback. ......... 109 5-11Obstructionavoidanceresponsebasedondesiredrollangle. ........... 111 5-12Obstructionavoidanceresponsebasedonactualrollangle. ............ 112 5-13MeasuredsaccaderesultsfromSchilstraandVanHateren,1999. ......... 113 A-1Therelativeorientationofthereferenceframesassociatedwiththeequationofmotionderivation. .................................. 124 10

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InsectsoftheorderDipterahaveasinglepairofwings.TherearwingsofDipteraninsectshaveevolvedintoorgansthatallowstabilizingcontrolresponsesthroughsensingandencodingofbodyangularratefeedback.Thisdissertationdocumentsresearchonthephysicalandphysiologicalmechanismsthatenableapairofhalterestodistinguishandencodethreeorthogonalcomponentsofthebodyratevector.Whiletheknowledgethatthehalteresplayaroleinightstabilityhasbeenacceptedforcenturies,theunderstandingofhowinsect'sverysimplesensorystructuresareabletoencodeanddecoupletheorthogonalcomponentsoftheratevectorhasbeenlacking.Theworkdescribedinthisreportfurthersthisunderstandingthroughmodelingandsimulation.First,anaturaldecouplingoftheobservableratecomponentshasbeenidentiedthatassertsproportionalityofbodyratecomponentstoaveragedstraincharacteristicsnearthecenterofthehalterestroke.Second,ameansofencodinganddecodingthenecessaryrateinformationinamannercompatiblewiththeinsect'ssensorystructuresandightmotorphysiologyhasbeenidentiedanddemonstrated.Finally,theabilityoftheproposedhalteremodeltostabilizeightina6DOFenvironmentwithcompetingbehaviouralobjectivesandrandomlygeneratedobstructionshasbeendemonstrated. 11

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Insectshaveevolvedtheabilitytoperformstableightwithintheconstraintsofmanycompetingobjectivesthatinuencewingandbodyform.Opticalandinertialfeedbackformotioncontrolisexpressedingreatvarietyinbiologicalcontrolsystemarchitectures.TheorderDiptera,commonlyreferredtoas\ies,"hasdierentiatedoverhundredsofmillionsofyearsintoseveralhundredthousandspeciesthathaveonlytwowings.Inthedipterans,whichincludemanyofthemostdynamicyingformsofinsects,therearwingsevolvedinthemid-Triassicperiodintosmallclublikestructuresthatoscillateoutofphasebutatthesamefrequencyasthewingsduringight(Figure 1-1 ).Insomespecieshaltereoscillationalsooccursduringwalkingonthegroundwithoutwingmotion,indicatingthatthehalteresmightimpartstabilityduringothermodesoflocomotion.Thehaltereshavebeenlikenedtogyroscopes,providinginertialratefeedbacktobothwing,legandneckmusclefunctions. ThehistoryofhaltereresearchwillbereviewedindetailinChapter2.Thehaltereshavebeendocumentedasnecessaryforightstabilitysincetheearly18thcentury.Notuntiltheearly-tomid-twentiethcenturywassucientevidentiarydataandanalysisprovidedtoconvincethescienticcommunitythathalteresinuencedightstabilitythroughtheirroleasaninertialratefeedbacksensor.Whiledataassociatedwithneurologicalpathwaysandcompensatoryresponseshasbeenabundant,thedetailsaboutthemechanismbywhichthehalteresareabletodecouplethebodyratecomponentshasbeenlacking.Thelackofdetailsisinpartduetoanemphasisinthebiologicalcommunityonhighersystemlevelinteractionsbetweenthehaltereandthewingandneckcontrolstructures,andinpartduetothedicultyinuntanglingthesignalsassociatedwithhalteremechano-sensorycomplex.Atthebaseofthehalterearebetweenthreeandfourhundredindividualstrainsensors,calledcampaniformsensilla,onthesurfaceofthecuticle.Internaltothehingeareothersensorystructures,thechordotonalorgans, 12

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EnvironmentalScanningElectronMicrograph(ESEM)ofthehaltereoftherobbery,familyAsilidae.ImagecollectedatEglinAFB,July2008. thoughttobesensitiveinextensiononly.Asofthistime,verylittledetailedknowledgeexistsaboutthefunctionandpurposeofthemajorityofthesesensors.Compoundingtheproblemisthedicultydierentiatingindividualnervebersinthebundlethatleavesthehaltere.Typically,compoundextracellularpotentialsaremeasuredasopposedtomeasurementsknowntobeexpressedfromindividualsensilla,althoughmorerecentlytherehavebeenexamplesofmoredetailedmeasurements[ 1 ],[ 2 ]. Oneobjectiveofthisdissertationistoexpanduponthecurrentknowledgeofthemechanicsofthehaltere.Whileseveralbiologistshaveappliedkinematicanalysisofthehaltere,nonehavegonetothelevelofsimulatingthedynamicsofthehaltere.Thisresearchgoestothatlevelandasaresulthasuncoveredmechanismsthatmayexplainhowdipteransdecoupletworatecomponentsintheplaneofeachhaltere.Withtworatecomponentsfromtwohalteresanorthogonaltriadcanbeconstructedwhichexplainscompensatoryightandheadresponsestoorthogonalratedisturbances.Researchersintheearly20thcenturydiscovereduniquefrequencycharacteristicsassociatedwiththeCoriolisforcesactingonthehalteres(Figure 1-2 ).Fromthattimeforward,decoupling 13

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TheCoriolisforcechangessignasthevelocitycomponentperpendiculartotheratecomponentchangessign(~!~v). basedonfrequencycharacteristicshasbeenassumedbymostofthescienticcommunitywithoutfurthersubstantiation.Whilethefrequencycontentofthedrivingforcesmustberepresentedinthecontinuousresponseofthehalteres,thesensorymechanismsinvolvedarenotcontinuous.Thesensillaarethoughttoreatsomethresholdnearsaturationonlyinaunidirectionalsense,onceortwiceperstrokecycle.Therefore,thehalteresmayonlyprovideoutputwhenforcedoutofplaneinonedirection.Also,thereareindicationsthatthesignalsneartheendsofthestrokearedominatedbytheaccelerationsassociatedwithhalteremotionreversal.ThismeansthatthesmoothidealisticrepresentationsofFigure 1-2 arefarfromrepresentativeofthetruehaltereaerents.Inpart,thegoalofthisresearchwastolookforotherpossiblemeansbywhichratedecouplingcouldoccurthataredirectlycompatiblewiththehalteresensoryphysiology. Chapter3ofthisdissertationprovidesadetailedanalysisofthekinematicsanddynamicsofthehaltere,demonstratingthatanalternativedecouplingmechanismdoesexistifresponsesproportionaltoaveragedstrainandstrainratenearthecenterofthestrokeareavailable.TheviabilityofthisresultisdemonstratedinChapter3forcasesinvolvingconstantinertialrates,althoughmoregeneralconditionsarediscussedattheendofchapter.Chapter4attemptstoreconcilethephysicalmodeloftheidealhaltere 14

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RolandHengstenberg[ 3 ]conciselystatesthedecitinknowledgeregardingthehalterethatthisdissertationattemptstoaddress: Atpresent,thecentralprojectionsofthehalteresensillaareonlycoarselyknown,anditistotallyunknownhowCalliphoraextracts,fromthespatio-temporalpatternofexcitationofhalteresensilla,theinformationtocontrolitsheadandwingscorrectly. 15

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Theresearchonhalteresrelativetoightstabilityhasbeenlimitedtoafewdozenresearchersoverthelasttwocenturies.Theseresearchershavebeenpredominantlybiologistswithageneralprerequisitebackgroundinphysics,butlimitedformaltrainingincontrolsystemdesignandengineeringmechanics.Morerecently,withgreatlyincreasedinterestinroboticsandmicro-airvehiclestherehasbeenmoremulti-disciplinaryinvolvement,eventothepointofattemptstopatentandbuildmechanicalversionsofahaltere[ 4 ].Understandingofthehaltereanditssystemlevelimplicationsrequiresunderstandingofinsectphysiology,engineeringmechanics,andthefundamentalsofcontrolsystemdesign.Sincethehaltereisanintegralpartofasystemthatinvolveselectro-opticalsensingandaerodynamicactuation,knowledgeoftheconstraintsimposedbythesubjectsofthesedisciplinesisalsoquitehelpful.Followingisasummaryoftheliteratureinvolvinghalteres. 5 ].In1714,Derhamproposedthatthehaltereswereorganswhichprovidedmechanicalbalancesimilartothepoleofatightropewalker[ 6 ].Loew,in1858,assessedthemassofthehalterestobemuchtoosmalltoinuencetheequilibriumoftheybytheirownmovements[ 7 ].ThiscorroboratedtheobservationbySchelverin1802,thatremovalofasinglehaltereonlyhadasmallimpactonightstability[ 8 ].Twoothertheoriesfortheinuenceofthehaltereswereproposedthatwerequicklyrejected.In1878,JoussetdeBellesmeproposedthatthehalteresinuencethewingmotionthroughmechanicalinterferenceduringthedownstroke[ 9 ].Theanatomyoftheinsectswasshowntomakethisimpossible.Weinland,in1891,proposedthatthehalteresaremovedforeandafttochangethecenterofgravity,atheorywhichitappearshadalreadybeenexcludedasapossiblitybyLoew[ 7 ],[ 10 ].Buddenbrock,inpaperspublishedin1917and1919,ignored 16

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11 ],[ 12 ].Hisreasoningwaslatershowntobeinvalidasdescribedbelow. Anarticlewaspublishedin1938byG.FraenkelandJ.W.S.Pringle[ 13 ]thatrejectedtheassertionbyBuddenbrock[ 12 ].Thereasoningprovidedisasfollows:1)theightreexcanbeinducedinieswhetherornottheyhaveintacthalteres,2)anylossofspontaneityoftheightreexisatransientphenomenon,3)amplitudeandwingbeatafterablatingthehalteresisidenticaltonormalies,4)reductionofightmotorstimulationthroughfatiguedoesnotreducestability,buthaltereremovalcausescompletelackofcontrol,5)lossofcontrolisparticularlypronouncedinyaw,and6)attachingapieceofcottonthreadtotheabdomenalmosteliminatestheinstabilityresultingfromlossofthehalteres.ThefollowingyearFraenkelelaboratedconsiderablyonhisargumentsagainstBruddenbrock,addingfactssuchastheobservationofhaltereactivityduringwalkingandtheightinstabilityofabithoraxmutantofthefruitywhichhastwosetsofwings[ 5 ].Fraenkelfurtherproposedthetheorythatthehaltereswereinertialsensors,atheorywhichwassubsequentlyelaboratedonbyPringlein1948[ 5 ],[ 14 ].Brauns(1939)isalsoreportedtohaveestablishedacorrelationbetweenthedevelopmentofthehalteresandtheightcapabilityofspecies[ 15 ]. Pringlepublishedoneofthemostimportantmanuscriptsregardingthefunctionofhalteres,providingsucientevidencetoconvincethecommunityoftheirroleasinertialangularratesensorstothepresentday[ 14 ].Pringlenotonlyaddressedthekinematicsofthehaltere,butalsoobservedthedynamicsofhaltereoscillation,studiedthesensoryphysiologyofthehaltere,performedneurologicalexperiments,andmeasuredtheinuenceofthehaltereonightofinsectsusingashphotography.Whilethisworkwasquitecomprehensiveithaditsweaknesses.Primaryamongthesewasthelackofunderstandingofhowtwonon-coplanarhalterescanbeusedtodistinguishbetweenpitchandrollrates.Pringlecametotheconclusionthatthehaltereswereonlyusedtomeasureandstabilize 17

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16 ]basedonobservationsbyothersofcompensatoryresponseofthewingsandheadtopitchandrollrate[ 17 ]. Twopublications,onebySandemanaloneandaslightlyearlieronebySandemanandMarkllookedatcompensatoryresponsesofthewingsandtheheadtoforcesonthehaltere[ 18 ],[ 19 ].Thesepapersappearedtostartowiththemisconceptionthatthegyroscopicforcesthatinuencethehaltereareduetobodyangularaccelerationandtheresistanceofthehalteremasstothebody'sangularacceleration.Thispartialunderstandingoftheforcesinuencingthehaltereledtoexperimentalsetupsthatconrmedtheinuenceofyawangularaccelerationonthehaltere,butignoredtheimpactofCoriolisforcesthatareproportionaltothebody'sangularrate.Theseauthorsfoundthatthehaltereexpressesanelectricalpotentialonlywhenthehaltereisforcedrapidlyforward.Theydeducedfromtheirresultsthatthehaltereisusedduringrapidmaneuversinvolvinghighaccelerationsforwhichtheiesvisualinputisdisrupted.Thisconclusionappearstocompletelyignorethegeneralinstabilityofiesafterhaltereablation.Thisresultwassupersededbysubsequentworkwhereheadstabilizationwascorrelatedwithrollratewhichisatleastinpartinferredfromthehalteres[ 3 ],[ 20 ]{[ 22 ].TheworkbySandeman[ 18 ],[ 19 ],whichincludedtracingoftheneurologicalpathwaysofthehaltere,didaddsignicantlytounderstandingoftheipsi-andcontralateralpathwaysofthehaltere. Nalbachpublishedanumberofpapersrelatedtothemechanosensoryfunctionofthehalteres[ 23 ]{[ 25 ].Therstofthesepaperslookedattherelativemagnitudesoftheforcesactingonthehaltere,concludingtheCoriolisforceswerepredominantandthatsomeoftheconclusionsofSandemanwerenotwellfounded.Thesecondofthesepapersdemonstratedexperimentallythatthehaltereofayisinsensitivetothecomponentoftheratevectorperpendiculartotheplaneofmotionofthehaltere.The 18

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Anumberofothersignicantndingshaveoccurredthatmayexpandonthegeneraltheoryofhalterefunctionality.Inadetailedstudyofthehalterehingephysiology,Chan,etal.[ 26 ],discoveredthatthereare11controlmusclesassociatedwiththehaltere,twoofwhichreceivestrongactivationfromdirectionallysensitivevisualinterneurons.Thisdiscoveryopensupthepossibilitythatcertainightreexesmaybeinducedthroughmanipulationofthehaltereinafeedforwardmanner.Heidediscoveredthattherstbasalarmotorneuron(mnb1)controllingwingkinematicsreceivesaphasicimpulse,oncepercycle,fromthehalteresregardlessofinsectmotionduringight[ 27 ].ThisdiscoverymaylendsomecredibilitytothepropositionbyvonBuddenbrock[ 12 ]thatthehalteresacttonervouslystimulatethewingmotion.Heide'sndinghasledtosomecurrentdebateovertheinterpretationofpreviousightinstabilityresults.FayyazuddinandDickinsonfoundthattheconnectionbetweenthehaltereandmnb1consistsofbothafastelectricalmonosynapticimpulseandaslowchemicalsynapsethatattenuatesathighfrequency[ 28 ].TheirproposedmodelofthephasicstimulusfromthehalterewasthatithadnoeectonightcontroliftherewerenotCoriolisforcesactingonthehaltere. 29 ].Figure 2-1 ,fromHengstenberg[ 3 ],showsthegeneralcongurationofsensillaatthebaseofthehaltere.GnatzyassessedthetopologyofthecampaniformsensillaonthebodyofCalliphora[ 30 ].Hengenstenberg[ 3 ]summarizedcurrentknowledgeincludingsensoryhairsandchordotonalorgansasfollows: Thehaltereknobcarriesabout13innervatedhairs,thehalterebasecontains2chordotonalorgansandca.340campaniformsensilla,arrangedin5distinctelds.Thehalterenervecontains370-380sensorybres,allowingfor20-30brestobeattributedtothechordotonalorgans.Comparedtothewing, 19

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GeneralmorphologyofthehaltereonCalliphora.FromHengstenberg[ 3 ]. haltereshaveaboutthesamenumberofmechanoreceptors,butthetypedistributionissignicantlydierent. IntheassessmentofPringle,thecampaniformsensillaaretypicallyelongatedandaresensitivetocompressionintheelongateddirection[ 31 ].Pringlelatergaveadetaileddescriptionofthesensorsonthehaltere[ 14 ].Inadditiontocampaniformsensilla,thehaltereshavechordotonalorgansthatactasstretchsensors,sensitiveintension.Pringledeterminedthatthecampaniformsofthedorsalbasaleld(dF2)alongwiththelargechordotonalorganareorientedatroughly45degreesrelativetothehalterestalk,makingthempreferentialformeasuringbendingstrains.However,Hengstenberg[ 3 ]reported,basedontheexperimentalstudiesofThurmetal.[ 32 ]andStedtler[ 33 ],thatthedierenteldsdohavepreferreddirectionsofsensitivity,butthedirectionsarenotalwaysasexpected.Theventralscapaleld(vF2)respondsintherostral-ventraldirection,the 20

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Zill[ 34 ],whilestudyingthecochroach,Periplaneta,demonstratedthatthecampaniformsensillaareactuallysensitivetotensionalongthelongaxisofthecapofthesensilla.ThisisindirectoppositiontothedeductionofPringle[ 31 ].Note,however,thatPringlewasverycarefultoexplaininhispaperthathisreasoningwasbasedoncircumstantialevidence.Intheworkdescribedinthisdissertation,theassumptionthatthecampaniformsrespondintensionisappliedtothemodelsofthesensoryeld. Dickinson'smanuscriptontheencodingpropertiesofwingcampaniformssuggestswithcautionthatthewingcampaniformscouldnotfunctionecientlyasmagnitudedetectors[ 35 ].Hesuggeststhatthecampaniformswillsaturateatlowpowerandremainnearlyconstantwithincreasingstimulusstrength.Beyondacertainthresholdthecampaniformneuronewillrecordonlyoneeventduringeachwingbeat,thereforeactingasa\one-shot"detector.Increasesinstimulusstrengthareassumedtorecruitmorecampaniformneuronesasopposedtoincreasingtheoutputofanindividualneurone.Otherinsectssuchaslocustshavingmuchslowerwingbeatfrequencieshavecampaniformsthataresimilarbutdoactasmagnitudedetectors,ringcyclicburstsofactionpotentialsduringeachcyclewhosefrequencydependsonmagnitudeofwingtorsion.Sincethehaltereshaveevolutionarytraceabilitytothewings,thecampaniformsonthehalteresareprobablyalsoone-shotdetectorsringonceortwicepercycleandrecruitingmoresensillaasthestrainincreases. Elson,workingonthelocust(Schistocercagregaria)hindwingsensilla,determinedthatsupinationortwistingofthewingcausedaclearresponse,withtheinvolvedsensillaspikingatraisedfrequency[ 36 ],[ 37 ].Increasingtheamplitudeofsupinationraisedthe 21

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34 ],thatdirectionalsensitivityisassociatedwiththeellipticalshapeofthecuticularcapofthesensillum;compressionperpendicularto,ortensionparallelto,thelongitudinalaxisofthecapexcitethesensillum.Thesensillaonthelocustwingwereshowntorespondtodorsaloranteriorbendingofthewingandtoaxialstretching,consistentwiththereceptors'locations.Thesensillawereventralandposterioronthewingveinandtheorientationwasmorenearlylongitudinalthantransverserelativetothevein. Asstatedpreviously,littlehasbeendeterminedbythecitedauthorswithregardtothewaythattheangularratesareencodedbytheeldsofsensorsinthehalterebase.Anareathatisparticularlylackingintheliteratureistheroleofthechordotonalorgans,iftheyhaveone,intheencodingofangularrates. Pringlemeasuredtheresponseofthehalterenervetoavarietyofstimuliatthehaltere,distinguishingtheresponsesduetoin-planeandlateralforcesonthehaltere[ 14 ].Itwasnotuntillaterthatinvestigatorstracedthepathwaysthroughthethoracicganglion.SandemanandMarkl[ 18 ]producedFigure 2-2 whichshowstheresultofpreparations 22

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DiagramfromSandemanandMarkl(1980).Thecentralprojectionsofthehalterenerve(A),neckmotoneuronesandprosternalorgansensilla(B).Ck,neckconnectives;Nd,dorsalprothoracicnerve;Nf,frontalprothoracicnerve;Ni1,Ni2,Ni3,pro-,meso-andmetathoraciclegnerves;NaI,Nma,Naawingnerves;Nmes,lateralmesothoracicnerve;Nh,halterenerves;Nabd1,Nfa,abdominalnerves(terminologyanglicizedfromVater,1962). ofcobaltchorideafterinltrationthroughthehalterenerve.TheresultshowedthatthehaltereextendsthroughthegangliontothebrainwhichcanbeatleastpartiallyexplainedgivenChan's[ 26 ]ndingofvisualinputsintotwoofthehalterecontrolmuscles.Thenerveisalsoshowntoprojectintoneuropilesonboththeipsi-andcontralateralsidesoftheganglion.Whilestimulatingthehalteres,SandemanandMarklrecordedonlyipsilateralresponseinthewingnerves.Similarmeasurementsinthenecknervesshowedbothipsi-andcontralateralresponsesfromasinglehalterewiththelargerresponsebeingontheipsilateralside[ 18 ]. Thecentralprojectionofthehaltereeldsisnotparticulartoaspeciceldingeneral,accordingtoChanandDickinsonbasedonstudiesofCalliphoravicina[ 1 ].The 23

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FayyazuddinandDickinson[ 28 ]showedaconnectionbetweendF2andmnb1.Todemonstratethis,thedierenteldsofcampaniformsweresystematicallydisabled.OnlydF2wasseentohaveasignicantimpactontheb1motorneuron.Thecompoundpotentialwasobservedtoincludeafastelectricalresponsewithasynapticlatencyof~200microsecondsandaslowchemicalcomponentwhichdropsoatfrequenciesabove10Hz.Theb1musclehastypicallybeenshowntorespondwithin3-4msec[ 38 ].Thewingstrokeperiodfortheinsectbeingstudiedwas6-7msec.Theb1musclehasbeenshownbyanumberofauthorstoreasinglespikeduringanarrowphasebandduringeachwingstroke[ 27 ],[ 39 ],[ 40 ].Thisactstonicallytorecongurethewinghingeandthroughsmallchangesinthephasingofthesignalthewingdownstroketrajectoryismodied.Advancesinphasewereshowntoincreasethestrokeamplitudeandcausetheadduction(pullclosertothebody)ofthewingduringdownstroke.FayyazuddinandDickinsonalsonotethatmnb1receivesphasicinputsfromwingcampaniformsandthesearecriticaltotheirmodelofoverallcontroloftheb1muscle. 24

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16 ].Theheadisfreetomoveinthreeangulardegreesoffreedom,approximately+/-20degreesinpitchandyawinCalliphoraand+/-90degreesinroll[ 20 ].Boththewingsandtheheadarecontrolledthroughanumberofinputsbesidestheperceptionofegomotionwiththehalteres,e.g.,visual,windspeed,wingloading,proprioception.Infact,Hengstenbergdescribes8distinctinputstoheadrollcontrol,including4distinctvisualcuesalongwithselfmotion,wingload,gravity,andheadposture[ 22 ].Manyresearchershavestudiedinsectmotorcontrol,andthissectionisonlyintendedtogiveageneraloverviewoffunctionsassociatedwithhalteremechanosensoryfunction. 16 ].HisconclusionwasbasedontheworkofbothHollick[ 41 ]andFaust[ 17 ].Pringledocumentedthatblindinghadasignicantbutnottotaleectonthereactiontoarolldisturbance.Blindinghadlittleeectonthereexresponsetorotationsaroundyawandpitch.Thereexresponseswerespecicallydescribedintermsofpronationandsupinationofthewingsduringdownstrokeandupstroke.Extirpationofthehaltereseliminatedthereactions,indicatingthereactionswerepredominantlymediatedbythehaltere.Pringleproposedthatthenearnormalightofaywithasinglehalterewasadequatelyexplainedbythefactthatrollratemeasurementisgreatlysupplementedbyvisualmeasurementandyawrateisredundantlymeasuredbybothhalteres.Theabsenceofdistinctpitchratemeasurementfromthehaltereswasnotsucienttodestabilizethey. Heidepublishedacomprehensivereportonwingmotorcontrol[ 27 ].Thisreportisreferencedextensivelyinpartduetothedescriptionofaphasicsignalthatimpactsthebasalarwingcontrolmuscle(B1)eveninquiescentight.Heidesummarized,\Innon-rotatingieshaltereaerencesareinvolvedinthetimingofoutputimpulsessupplyingsteeringmuscles.Inaddition,inrotatingieshaltereaerencesgatetheoutputto 25

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DickinsondocumentedexperimentswiththepurposeofcharacterizingthehalteremediatedchangesinstrokeamplitudeandfrequencyinducedbyimposedbodyrotationinDrosophila[ 42 ].Thistestsetup,likethatofHengstenberg,wasabletorotatethevisualeldwiththeyinordertoisolatehaltere-inducedresponses.Whenpitchedforward(headdown),theiesincreasedboththestrokefrequencyandstrokeamplitudeoftheirtwowings.Backwardpitchingmotionelicitedadecreaseinamplitudeandfrequency.Duringfunctionalrollstotheleftandright,theiesincreasedthestrokeamplitudeofthewingonthesideofthebodythatwasrotatingdownward,anddecreasedtheamplitudeofthewingonthesidethatwasrotatingupward.Incontrasttothepitchresponse,therewaslittlemodulationofstrokefrequencyduringfunctionalroll.Oneresultofhaltereablationwasthatitsignicantlyincreasedthestrokefrequencyoftheyintheabsenceofanyrotationalstimuli.Ablationofonehaltereonlypartiallyinhibitedtheequilibriumreexes.Whileboththepitchandrollresponsesofthewingipsilateraltothe 26

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BalintandDickinsonfoundthat,bycorrelatingb1andb2spikeoccurrenceswithcycle-to-cyclechangesindownstrokewingtrajectorydeviation,thesetwomusclescouldaccountforalargeproportionoftheobservedvariationinwingtrajectoryduringvisuallyinducedsteeringreactions[ 43 ].However,complexinteractionsbetweenthesteeringmuscleswereseentoinuencestrokeparameters.Forexample,althoughb1andb2insummationcontributetodownstrokedeviation,incaseswhereneitherwereactivetherewasstillsignicantvariationindeviation,probablyduetotheantagonisticb3muscle.Theyspeculatedthatrecongurationofthewinghingeisinvolvedinchangingthe\mode"ofthewingstroke.Thesedierencesinmechanicaladvantagemayexplainthenon-linearrelationshipbetweenmuscleactivationandwingkinematics.Insummary,theyfoundthattheroleofanysinglemusclecannotbeconsideredintemporalorspatialisolationeitherfromitsprioractivityorfromtheactionofothersteeringmuscles. BalintandDickinsonidentiedthreeindependentlycontrolledfeaturesofthewingbeattrajectory{downstrokedeviation,dorsalamplitudeandmode.Modulationofeachofthesekinematicfeaturescorrespondedtobothactivityinadistinctsteeringmusclegroupandadistinctmanipulationoftheaerodynamicforcevector[ 44 ].Resultssuggestthatitistheabilitytomanipulatethecouplingamongaerodynamicallyrelevantkinematicparameters,ratherthantheabilitytocontroltheseparametersindependently,thatallowsCalliphoravicinatheexibilityofcontrolobservedinpreviousmeasurementsofitsdirectionalforceandmomentoutput.Itwasfoundthatthebasalarmusclesprimarilycontrolledliftandrollbyvaryingthedownstrokeforce,themusclesofpteralaeIIIandIcontrolledthrustandyawbychangingtheupstrokeforce,andanunknownmusclegroupcontrolledliftandrollbyvaryingtheupstrokeforceinclination.Duetothedependenceofmomentsontheinstantaneouspositionofthewing,rollandyawaremostsensitivetoforcesatmid-stroke,whereaspitchismostsensitivetoforcesduringstrokereversals. 27

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45 ],considerablewingdeformationwasobservedthroughthedurationoftheupstroke. BenderandDickinsonfoundthatalteringvisualfeedbackhadnosignicanteectonthedynamicsofsaccades,whereasincreasinganddecreasingtheamountofhaltere-mediatedfeedbackthroughhaltereinertialmodicationdecreasedandincreasedsaccadeamplitude,respectively[ 46 ].Inotherexperiments,aerodynamicsurfaceofthewingswerealteredsuchthattheieshadtoactivelymodifytheirwing-strokekinematicstomaintainstraightightonamagnetictether.Fliesexhibitsuchmodication,butthecontroliscompromisedinthedark,indicatingthatthevisualsystemdoesprovidefeedbackforightstabilityatlowerangularvelocities,towhichthehalteresystemislesssensitive.Resultssuggestthatthetimecourseofthesaccadeisdeterminedbyafeed-forwardmotorprogramthatisinuenced,butnotpreciselystructured,bymechanosensoryfeedback.Electrophysiologicalstudiesinrigidlytetherediessuggestedthatsaccadesarecausedbychangesinsteeringmuscleactivity,includingaburstofactionpotentialsinthesecondbasalarmuscle(b2)andaphaseadvanceoftherstbasalarmuscle(b1)[ 40 ].Thevisualsystemhasbandpassltercharacteristicsthatlargelysuppressitsresponsetorotationsabove600/s[ 21 ],[ 47 ];however,themajorityofthesaccadesBenderandDickinsonobservedinthemagneticallytetheredpreparationhavepeakvelocitiesbelowthisvalue,andexperimentalvisualrotationswereofaconstant500/s[ 48 ]. 19 ].Thiswasaccompaniedbydierential\feathering"(supinationorpronation)ofthewingssoanetyawtorquewouldbegenerated.Removalofthehaltereseliminatedboththeheadandwingresponses.Additionally,Sandeman 28

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18 ].Measurementsfromthemesothoracicwingnerveshowedtwolargeunitswhichrespondwiththesamelatencyastheneckresponse. RolandHengstenbergfocusedprimarilyonrollcompensationinhisexperimentalinvestigations[ 3 ],[ 20 ].Theearlierofthesemanuscripts[ 20 ]gaveagoodsummaryofoverallheadresponse:Calliphoraerythrocephalahasthecapabilitytoactivelycontrolheadpositionaboutallthreebodyaxes.Pitchandyawturnsaresmall(20),whilerollturnscanbeupto90.Fliesproducecompensatoryheadresponsewhilewalkingandying,butnotwhilestationary.Headresponseshavealatencyofapproximately5msandcanproduceangularvelocitiesofupto1000/s.Theheadcompensationtendstoundercompensatefordisturbances.Simultaneouslythebodyalsocompensatesfordisturbances.Hengstenbergproposedthattheheadandbodylikelyworkinconcerttobringvisualratedisturbancesdowntoacceptablelevels[ 3 ]. HengstenbergfoundthatmechanosensoryrollcontroloftheheadinCalliphoradependsonwhethertheinsectiswalkingorying[ 3 ].Whenwalking,theheadorientationisinuencedbygravityperceptioncomingfrommechanoreceptorsinthelegs.InightCalliphoradoesnotdirectlymakeuseofgravitytocontrolheadorientation.Whenying,mechanoreceptioninuencesonlyfastrotations.Virtuallynomechanoreceptiveresponsecouldbefoundatangularvelocitiesbelow50/s.Above50/stheresponseincreasestoamaximumatapproximately1500/s.Passiveattitudestabilizationandvisualmeansofcontrolarerequiredtomaintainanuprightightattitudeandheadorientation.Adierenceinaerodynamicalloadofthetwowings,probablymeasuredbysensillaatthewingbase,alsoelicitsatransientheadrollpartlycompensatingabankedattitude.Flieswithonehaltereremovedcoulddistinguishthedirectionofrollmotion.ItwasearliershownbyNalbachandHengstenbergthatusingonlyonehaltere,Calliphoracandistinguishyawfrompitch,butnotrollfrompitch[ 25 ]. 29

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FiguresfromHengstenberg(1988).a.)Magnitudeofthereexiverollresponseoftheheadinresponsetoainputbodyrollrate.Directionwasoppositedirectionofrollrate.b.)rollresponsewith(middle)andwithout(bottom)halteresrelativetotheinputbodyrate(top). NalbachandHengstenbergstudiedthethree-dimensionalnatureofcompensatoryreactionsoftheheadtoangularratestimulus[ 25 ].Theyobservedthatayrespondstoanegativebodyrolldisturbancewithapositiverollandanegativebodypitchresultsinapositiveheadpitch.However,inyawthereisacoupledreaction.Apositivebodyyawresultsinanegativeheadyawthatisaccompaniedbyapositiveheadroll.Similarly,apositiveheadyawreactionisaccompaniedbyanegativeheadroll.The 30

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31

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3-1 )arewellestablishedasorgansnecessaryforightstabilization.Themechanismbywhichthestabilizationoccurswasdebatedbetweenthe18thandthersthalfofthe20thcentury,withsomearguingthatthehalterewasa\stimulant"forightmotorfunction,othersclaimingthehalterefunctionedasaninertialbalancingsystem,andstillothersclaimingthehaltereswereagyroscopicsensorymechanism[ 5 ],[ 13 ].However,theworkofPringleprovidedarmbasisforviewingthehaltereasagyroscopicsensor,optimizedforsensitivitytoCoriolisforces[ 14 ].TheCoriolisforceoccurswhenanobjectwithmassandanitevelocityisconstrainedtomoveinaxedpathwithinareferenceframethatisrotating.TheCoriolisforceisproportionaltothereferenceframerotationrate,soifthroughanappropriatestrainsensortheforceismeasured,thenasignalisavailableforratedampinginastabilizingcontrolloop.Pringleinitiallydidnotrecognizetheabilityofthemechanicalcongurationofthehalterestodistinguishbetweenpitch(transverseaxis)androll(longitudinalaxis)rotationsandthereforeassumedtheuseofthehaltereswaslimitedtoyaw(verticalaxis)rotations.ThispositionwaslaterrecantedbasedonthendingsofFaustthatdemonstratedstabilizingwingreexesassociatedwithpitch,yawandrollinCalliphora[ 16 ],[ 17 ].ItwasnotuntilmuchlaterthatNalbachreviewedindetailthesignicanceofallforcesactingonthehalteresandelaboratedonthepotentialbenetsofnon-orthogonalityofthehalterepair[ 23 ],[ 24 ].Anumberofauthorshavedemonstratedthecompensatoryreactionsoftheheadandwingstoindependentcomponentsofthebodyratevector,therebydemonstratingtheroleofhalteresinbothimagestabilizationandattitudecontrol[ 17 ],[ 18 ],[ 20 ],[ 42 ],[ 46 ].Heideperformedextensiveresearchassociatedwiththehalterephasetuningoftherstbasalarmotorneuron(mnb1)incontrolofwingkinematics[ 27 ]. 32

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Characteristiclocationsofthehalteresandtheirstrainsensors.Fieldsofcampaniform(#ofsensilla):dF1:dorsal'Hickspapillae'(17),dF2:dorsalbasalplate(100),dF3:dorsalscapalplate(110),vF1:ventral'Hickspapillae'(10),vF2:ventralscapalplate(100).Notshownarethelargeandsmallchordotonalorgans. Thiswasfollowedbyanumberofrelatedstudiesofightmotorcontrol,e.g.,TuandDickinson[ 49 ],[ 50 ]. PringleandNalbachbothrecognizedthateachhaltere,duetoitslargeamplitudemotion,issensitivetotwoorthogonalratecomponentsinitsplaneofmotion,and,further,describethedistinctimpactsoftheverticalandhorizontalratecomponentsonthehaltere[ 14 ],[ 23 ].Theverticalratecomponent,xinFigure 3-2 ,generatesaforcewithtwicethefrequencycontentofthehorizontalcomponent.Thisknowledgehasbeenthebasisfortheacceptance,withincompleteunderstanding,oftheapparentabilityofiestodistinguishbetweenbodyratecomponents.Giventhehalterespecicratecomponents,variationsofbilateralsumminganddierencingallowconstructionofsignalsproportionaltocomponentsofthebodyratevectorinanybody-xeddirection.Studieshavedemonstratedthelocationandactivationofbothipsilateralandcontralateralneuralpathwaysbetweenthehalteresandthemusclesofthewingandneck[ 18 ],[ 28 ],[ 51 ].Furtherstudieshavedemonstratedvisualpathwaystothehalteresallowingspeculationof 33

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Asthehalterebeatsbackandforththevelocitycomponentperpendiculartoxchangessignattwicethefrequencyasthecomponentperpendiculartoy.ThisresultsinaCoriolisforcewithtwodistinctfrequencycomponents. afeedforwardprocessthatintroducesvirtualrate\errors"intothecontrolloopthroughthemanipulationofacomplexofmusclesatthebaseofthehaltere[ 26 ]. Theobjectiveofthecurrentresearchistousethetechniquesofengineeringmechanicstoanalyzethehaltereandtherebyestablishthefundamentalquantitiesrequiredforandmathematicallimitationsassociatedwithreconstructionofbodyratecomponents.Theonlyviablemechanismdocumentedsofarfordecouplingofthetwohaltereratecomponentshasbeenfrequencydemodulation[ 4 ].Incontrasttofrequencydemodulation,themethodsofthischapterdemonstratethepotentialtousestrainandstrainrateencodedbythemechanoreceptorsatthebaseofthehalteretomeasurethesametworatecomponents.Thismeasurementispossibleduetothenaturaldecouplingoftheratecomponentsatthecenterofthehalterestrokeandtheapproximatelylinearnatureofthegoverningequationofmotion.Anerroranalysisisalsoprovidedthatillustrateshowrelativeerrorsinherentinthemeasurementofallthreebodyratecomponentscanbeinferredduetothenonlinearitiesintheequationsofmotion.Theseresultshaveimplicationstointerpretationofpastexperimentalresultsandunderstandingofsensorystructuresassociatedwiththehalteres. 34

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42 ].Thisdescriptionofthegeometry(Figure 3-3 )isusedasastartingpointforthehaltereanalysisinthischapter.Thepredominantcharacteristicsofthebiologicalsystemusedinthecurrentdevelopmentaretheamplitudeofthehalterestrokeandthecongurationofthehaltereswithrespecttothemid-sagittalandtransverseplanesoftheybody.ThehalteresonDrosophilaoscillateinaplanethatistiltedbackroughlythirtydegreestowardthemid-sagittalplane.ThelinethatdenestheintersectionofthehalterestrokeplanewiththesagittalplaneisrotatedtowardtheheadbyapproximatelytwentydegreessothatatthetopofitsstrokethetipofthehaltereisinamoreanteriorpositionthanatthebottomofthestrokeasshowninFigure 3-3 .However,sincethelineofintersectionofthehaltereplanesis,forconvenience,usedtodenethebodyyawaxis^x3,thevalueofthisangleisarbitrary.Forthepurposeofthisstudytheintersectionofthehaltereplanesisassumedtobexedrelativetothebody.Thewingbeatfrequency,whichwasnominally215HzinthedatareportedbyDickinsonforDrosophila,variessignicantlybothwithinandbetweenspecies[ 42 ].Forthesakeofanalyticalconvenience,avalueof200Hzwasusedinsimulationswheregeneraleectsofout-of-planestinessanddampingimpactonthetrajectoryweresimulated. Theequationsofmotiondevelopedinthisstudyarenon-dimensionalanddescribethesystemintermsofitsnaturalfrequencyanddampingcoecient.Thecomponentofthehalteremotionintheprimaryplaneofoscillationisassumedtobedeterministicandpurelyharmonicasobservedinthebodyreferenceframe,oscillatingthrougharangeof+/-ninetydegrees.Dampingofout-of-planemotionisassumedtobeproportionaltotheangularrateoftheout-of-planemotionandthestinessproportionaltoout-of-planedisplacement.Thesourceofstinessisnotspecied,whetheritisduetotheresiliencyofthehalterestalkorthejointanditsassociatedmusculature.Thehalteremodelforout-of-planemotioncanbeconsideredasanequivalentmassattheradiusofgyrationof 35

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Referenceframedenitionsforthehalteres,bandc,andframexwhichdenestheroll(x1),pitch(x2)andyaw(x3)axis.Theanglebetaisarbitrarywiththisdenitionofreferenceframes. thehaltereonarigidmasslessstructurewithatorsionalspringanddamperatthebase.Theactualdynamicsandcontrolofthehalteremaybemuchmorecomplexandisanareaofongoingresearch.Chandescribeseightdirectcontrolmusclesatthebaseofthehalteresimilartothemusclesatthebaseofthewing[ 26 ].Thesemusclescouldpossiblynetunethekinematicsofthehaltere. Theequationsofmotionaregeneratedwithoutanysmallangleassumptionsforthepurposeofsimulatingthehalteretrajectoryundertheinuenceofconstantinertialbodyrates.Transientsarenotconsideredinthisphaseoftheresearchandonlythehaltereresponseundertheidealconditionsofconstantangularratewasexaminedtodrawpreliminaryconclusionsaboutthefundamentallimitationsofthehaltereorhalterepair.Thissteady-stateassumptionisequivalenttoassumingthatthebodyrateshave 36

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Finally,thecomponentofangularrotationofthehaltereinitsprimaryplaneisassumedtobesinusoidal.Thatis,theangularpositionofthehaltereinitsprimaryplaneofmotionisassumedtobe= 23 ].Thesinusoidalmodelprovidesananalyticallysimplerformthatcanbeusedtodevelopvalidconclusionsduetothesimilarsymmetryofmotionwithrespecttothecenterofthestroke. 3.3.1KinematicAssessment 3-3 showstherighthaltereandreferenceframedirectionsassociatedwiththehaltereandinertialspace.Inthefollowingsections,hattedvariablesrepresentunitvectorsthatdescribeorthogonaldirectionsfortherequiredreferenceframes.Leftsuperscriptsdescribewhichreferenceframethevectorquantityisobservedwithin.Rightsuperscriptsidentifythepointorreferenceframethequantitycharacterizes.NomenclatureissummarizedinTable 3-1 Thebodyangularratevectorrelativetotheinertialframe,e~!b=e~!x,isrepresentedintherighthalterereferenceframeas In( 3{1 ),i,aretheangularvelocitycomponentsand^biarethebodyxedunitvectorsasshowninFigure 3-3 .Theposition,velocityandaccelerationofapointmassattheradius 37

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Notationassociatedwithkinematicanddynamicexpressions. Intheseexpressions,0,1,and2refertoanarbitrarypointxedininertialspace,apointatthebaseofthehaltere,andapointattheradiusofgyrationofthehaltere,respectively.Therstaccelerationterm,e~a1,whichistheaccelerationofthebaseofthehalterewithrespecttotheinertialframe,isassumedtobesmall.Thesecondterm,b~a2whichrepresentsaccelerationofthehalteremassasobservedfromthebody,isentirelyintheplaneofthehaltere.Nalbachshowedthattheseprimaryaccelerationsintheplane-of-motionaremuchhigherthancontributionsassociatedwiththebodyangularrates,andtherefore,usefulinformationpertainingtothebodyratesisunlikelytobeascertainedfromin-planeforcemeasurements[ 23 ].Thelastterm,e!b~P12,whichinvolvestheangularaccelerationofthebody,wasalsoshownbyNalbachtobeafactorof5ormorelessthanthethird(Coriolis)termforsinusoidalbodyoscillationsunder50Hz. 38

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2(e!!bb!v2)=2r_[2sin()^b1 +(1sin()+3cos())^b22cos()^b3]e!!b(e!!b~P12)=r[(22sin()23sin()+12cos())^b1 +(12sin()+23cos())^b2+(13sin()21cos()22cos())^b3]: 3{5 )istheCoriolistermwhichgeneratesout-of-plane(^b2)forcecomponentsassociatedwiththein-planebodyrates.Thesecomponentsareproportionalto2_1and2_3.Theothercomponentsrepresentanin-planeaccelerationdirectedalongthestalkofthehaltereproportionalto2.Theexpressionin( 3{6 )describingthecentripetalaccelerations,alsogeneratesout-of-planeforcesonthehaltereproportionalto12and23.Therelativemagnitudesof_and2willdeterminethesignicanceofthesecentripetalterms.Errorsintroducedbythesetermsarequantiedsubsequently.Ifthecentripetaltermsaresmall,theout-of-planeforceonthehaltereshouldbepredominantlyduetotheCoriolistermandthereforewillbeassociatedwiththebodyratecomponentsthatareintheprimaryplaneofthehalteremotion.Anassumptionof( 3{5 )and( 3{6 )isthatthehaltereisinnitelyrigidanddoesnotdeectout-of-plane.ThisassumptionisthebasisforthepreviouskinematicanalysisofthehalterebyPringleandNalbachandisusefulfordevelopingintuitionregardingthepredominantforcesthatimpacttheproblem.Inthefollowingsection,thisassumptioniseliminatedtosimulatetheout-of-planemotion,orequivalentlythestrainsresultingfromthatmotion. IfthehalteresareassumedtomeasureforcesassociatedwiththeCoriolisaccelerations,themeasuredsignalsshouldbeproportionaltothein-planebodyratecomponents,1and3,asshownin( 3{5 ).IftwohalteresthatareinitiallyinacommonplanearerotatedoutoftheplanebyanangleasshowninFigure 3-3 ,thenallthreecomponentsofthebodyinertialratevectorcanbereconstructed.Thebodyratevectorrepresentedinthe 39

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where(W1;W2;W3)arethebodyroll,pitch,andyawrates,respectively. Therelationshipsbetweenthecomponentsofthebodyratevectorrepresentedinthebodyroll,pitch,yawframeandthecomponentsrepresentedintherighthaltereframe^bandthelefthaltereframe^care =c1^c1+c2^c2+c3^c3W1=b3+c3 (3{9) (3{10) Theimportanceofthesetransformationsisthattheyallowadirectcalculationofratecomponentsalongthebodyroll,pitch,andyawaxes,W1,W2,andW3,giventhetworatecomponentsthataremeasurableineachofthehalterereferenceframes.TheresearchonhalteresbyPringledidnotrecognizetheabilityoftheinsectphysiologytocombinetheoutputoftwohalteresandtherebydistinguishbetweenpitchandrollcomponentsofthebodyratevector[ 14 ].Pringleinitiallyassumedthatthehalteresrepresentedaredundantmeansofmeasuringyawrate.LaterexperimentalresultsbyFaustdemonstratedtheabilityofiestoreactindependentlytoeachofthebodyrates[ 17 ].NalbachalsopublishedanarticlethatexperimentallydemonstratedthebilateralcombinationofhalteremeasurementsinCalliphora[ 24 ].Therefore,withintheneuralarchitectureofdipteraninsectstheremaybeabasicrepresentationof( 3{9 )-( 3{11 ), 40

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+2!n_+!2n=_3sin()_1cos()_2cos()sin() (3{12) +2_[(3cos()+1sin())cos2()2cos()sin()]+(23cos2()+21sin2()22)cos()sin()+(23cos()+12sin())cos(2)+213cos()sin()cos()sin(): 3{12 ),isthedampingratio,and!nisthenaturalfrequencythatcharacterizestheout-of-planestinessandmasscharacteristicsofthehaltere.Inthisform,thehalterecanbesimulatedbyvaryingtheout-of-planenaturalfrequencyrelativetothehalterebeatfrequencyaswellasvaryingthehalteredampingcharacteristics.Again,thehalterestrokeangleisassumedtovarywithasimplecharacteristicmotion=sin(!ht),withtheangularfrequencyofthehaltere,!h=200Hz.Thederivationof( 3{12 )isdescribedintheappendix. Therelationshipdescribingthesingleaxissensitivityofamicro-electro-mechanical(MEMS)vibratingstructuregyroscopecanbefoundthroughsimplicationofthisexpression[ 52 ].Forthecasewherebothandaremuchlessthan1,dampingissmall,!2n>>A2!2h,and!2n>>!2h,( 3{12 )reducesto+!n=23_=23A!hcos(!ht):

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3-4 andFigure 3-5 showthetrajectoriesassociatedwithahaltereout-of-planenaturalfrequencyequaltoanddoublethebeatfrequencyof200Hz,respectively.Theplotsshowout-of-planedisplacementinradiansastheordinate,plottedagainstthestrokeangleofthehaltereastheabscissa.Ahalterestrokeangleof0hasthehaltereatthecenterofthestroke.The1inputgeneratestheexpectedfrequencydoubledsignalasthehalteresweepsthroughasemi-circulararccausingthevelocitycomponentperpendicularto1tochangesigntwice,thereforetheCoriolisforcechangessigntwice.Thehaltere 42

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Halteretrajectoriesfor!n=200Hz(left)and!n=400Hz(right).Halterebeatfrequency,!h=200Hz.Inputconditions1=10rad/s,2=3=0and=0.1. Figure3-5. Halteretrajectoriesfor!n=200Hz(left)and!n=400Hz(right).Halterebeatfrequency,!h=200Hz.Inputconditions3=10rad/s,1=2=0and=0.1. velocityperpendicularto3onlychangessignonce,givingnofrequencydoublingeect.Theangulardisplacementspeakatapproximatelyhalfadegreefortheconditionsshown.Whenthenaturalfrequencyissignicantlybelow200Hz,theout-of-planemotionisdriventoverylargeanglesandneverreachesasteadystatepattern. 43

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Halteretrajectorieswithdampingratiosat10%ofcritical(left)and100%critical(right)forbodyrateinputsof1=3=10rad/s.Plotsarebisectedbytheaveragedisplacementcurveasafunctionofhaltereposition. 3-6 forthecaseof!n=200Hzandinputbodyratesof1=3=10rad=s.Theseplotsdemonstratethesignicantimpactthatdampingvariations,whetherpassivelyoractivelyinduced,canhaveonthehalteretrajectory.Atlowdampinglevels,0:01,thetrajectoryneverreachedsteadystatewithinthefortyoscillation(0.2second)simulationtime(datanotshown). 3-6 .However,whentheaveragedisplacementisplottedseparatelyforthetworatecomponentsasinFigure 3-7 ,aninterestingcharacteristicemergesthatmayprovideinsightintoapossiblemechanismbywhichthebodyratesaredecoupledbytheinsect. Figure 3-7 demonstratesanaturaldecouplingofthebodyratecomponentsatthecenterofthehalterestroke.At=0,theaveragedmagnitudeoftheresponsedrivenby3iszeroandtheaveragedslopeoftheresponsedrivenby1iszero.Ifthegoverningdierentialequation( 3{12 )thatdescribesthemotionofthehaltereisapproximately 44

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Halteretrajectoriesfor1=10rad/s(left)and3=10rad/s(right)withtheaveragedisplacementplottedasafunctionofstrokeangle. linear,thenthenaltrajectoryofthehalterewouldsimplybethesuperpositionoftheresponseofthetwoplotsshown.Also,eachoftheseplotswouldscaleinproportiontothemagnitudeoftheassociatedbodyratesincetheCoriolisforcesdrivingthemotionareproportionaltotherespectivebodyrates.Therefore,bymeasuringtheslopeandthemagnitudeoftheresponsenearthepeakofthehalteretrajectory,andhavingtunedintheappropriateproportionalityconstants,thebodyratecomponentsintheplaneofthehalteremotioncouldbedirectlyobtained.Theseobservationssuggestthefollowinghypotheses: 1. Asystemwithhalteresinvokesaresponseproportionaltothemagnitudeoftheaveragedstrainatthecenterofthehalterestrokeandtakesadvantageoftheapproximatelinearityofthehalteredynamicstoestimate1(i.e.1isproportionaltotheaveragedmagnitudeofthestrainatthemiddleofthestroke). 2. Asystemwithhalteresinvokesaresponseproportionaltothemagnitudeoftheaveragedstrainrateatthecenterofthehalterestrokeandtakesadvantageoftheapproximatelinearityofthehalteredynamicstoestimate3(i.e.3isproportionaltotheaveragedmagnitudeofthestrainrateatthemiddleofthestroke). Notethatthetermstrainratecanrefertotwoquantitiesthatareproportionallyrelatedatthecenterofthehalterestroke.Becausetheangularaccelerationofthestroke 45

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dt=d" dd dt=Constsgn(d dt)d" d: 14 ].Thiswouldsupportthesuppositionthatthesensoryresponseofthehalteretowardthemiddleofthestrokeisofprimaryusebyinsects.TheproposedmethodofdeterminingthebodyratesismoredirectthanthatpatentedbyWuandWood[ 4 ].Intheirpatent,thefundamentalfrequencydoublingistakenadvantageofthroughademodulationschemetoseparatethetwosignalsanddeterminethedrivingforces.Themethodproposedheremaybedirectlyrealizableusingdiscretemeasurements,althoughitremainstobeproventhattheeldsofstrainmechanoreceptors(campaniformsensilla)existingatthebaseofthehaltereencodequantitiesproportionaltobothstrainandstrainrate. Thedescribedmechanismformeasuringthebodyratesrequiresthreecharacteristicsof( 3{12 ). 1. Linearity 2. Minimaldependenceontheout-of-planebodyrate2 Twoindependentforcingfunctionsproportionaltothein-planebodyratecomponents1and3 3{12 )canbereducedtoaformthatexpressesthedesiredcharacteristics, +2!n_+(!2n+_2)=2_3cos()+2_1sin():(3{13) 46

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ComparisonofthelinearsimplicationsrepresentedbyEq. 3{13 (red)andEq. 3{14 (green)withthenon-linearEq. 3{12 (blue).Resultsareshownfortwocases,W=(W1;W2;W3)=(10;10;10)ontheleftandW=(10;10;0)ontheright. If_2isfurtherassumedtobesmallcomparedto!2nthenasecondformthatsatisesthedesiredcharacteristicscanbefound, +2!n_+!2n=2_3cos()+2_1sin():(3{14) Thesecondform,shownin( 3{14 ),isintuitivesinceitisasimplespring-mass-damperdrivenbyCoriolisforces. Anopenquestioniswhethereither( 3{13 )or( 3{14 )isavalidapproximationofthefullnon-linearequation.Comparativesimulationswereperformedbetween( 3{12 ),( 3{13 )and( 3{14 ).TheclosenessofthetwodarkercurvesinFigure 3-8 demonstratethattherstformofthelinearapproximationsin( 3{13 )isanaccuraterepresentationofthehaltereresponse,unliketheresultsfrom( 3{14 )whichareplottedinthelightercolor.Since( 3{13 )isagoodapproximation,thenaturaldecouplingofthetrajectoriesisassumedtobeagenerallyvalidassumption. 47

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3{13 ).Simulationswereexecutedoverafullrangeofpitchandyawbodyrates(i.e.,20W220and20W320).AlthoughSchilstradescribedamaximumangularrateof2000deg/s(34rad/s)forCalliphoravicina,measurementsincludedintentionalsaccadicmaneuvers[ 53 ].Thelowerrateusedinthesesimulations(20rad/s)isconsideredasucientmaximumforrateerrorsincurredduringtypicalstabilizedight.Theseratesweretransformedintothereferenceframesforeachofthehalteresandthenthedynamicsforthehaltereweresimulatedusingthefullnonlinearmodelin( 3{12 ).Usingbestestimatesofthestrainrateandstrainmagnitudeproportionalityconstants(i.e.,constantsfoundtogivenearzeroerrorforanidealizedlinearmodel)thebodyratesinthehaltereframeswereestimated.Theestimatesfromthetwohaltereswerethencombinedusing( 3{9 )-( 3{11 )toreconstructanestimatefortheroll,pitchandyawratesinthebodyframe.Eachplotrepresentserrorsassociatedwith1681combinations(41*41)ofyawandpitchrateforaxedrollrate.TheerroristhedierencebetweentheexactinputbodyratesandtheestimatedbodyratesasdemonstratedinFigure 3-9 Figure 3-10 depictstheabsoluteerrorsforthepitch,yaw,androllcomponentsofthebodyratesforthecaseofcriticaldamping(=1)and400Hzout-of-planenaturalfrequency.Figure 3-11 showstheerrorsforthesamecase,butwithbodyrollrateof5rad/s. ThechangeincharacteristicsshowninFigure 3-11 canbeexplainedbyexaminingthegoverningequationofmotion( 3{12 ).Thetermsinvolving2,whichistheout-of-planeratecomponentandthecomponentmostcloselyalignedwiththebodyrollaxis,aresummarizedbelowafterassumingasmallout-of-planedisplacementangle, 48

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TheErrorAnalysiscomparedthetrueratecomponentsalongtheroll,pitch,andyawbodyaxiswiththosereconstructedusingtheproportionalassumptionsdescribedinthetext.Resultsarereportedasabsoluteerrorinrad/s.

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ErrorinestimatesofratecomponentsalongthebodyYaw,Pitch,andRollaxesforcaseRollRate=0.Conditionsvaryoverarangeof-20to20rad/sforthetrueyawandpitchrates.Thecolorindicatestheleveloferrorasindicatedonthecolorbartotherightofeachplot. Figure3-11. ErrorinestimatesofratecomponentsalongthebodyYaw,Pitch,andRollaxesforcaseRollRate=5rad/s.Conditionsvaryoverarangeof-20to20rad/sforthetrueyawandpitchrates. Sinceissmall,thelasttwotermsin( 3{15 )willdominate.Notethatcos()willalwaysbepositiveforallstrokeangles,,andwillbesymmetricaround=0.Therefore,theterminvolvingcos()willinuencethemagnitudeoftheout-of-planedisplacementat=0(i.e.,thetermwillinuencetheyawerror).Thetermisalsoproportionalto3,whichiscloselyalignedwiththebodypitchaxis.Therefore,rollcouplingwillintroduceerrorintheyawrateestimatethatisproportionaltothepitchrate.ThislinearrelationshipbetweenyawrateestimationerrorandpitchrateisexactlywhatisdepictedinthelefthandplotinFigure 3-11 .Similararguments,accountingfortheinuenceof 50

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3-11 .ThesimilarityofthethirdplotsinFigure 3-10 andFigure 3-11 indicatethattheerrorsfromthetwohalterescancel,leavingtherollestimateerrorunaectedbyrollrate. Simulationswereconstructedbasedontheassumedabilitytomeasureaveragedtrajectoryamplitudeandslope,orequivalentlystrainandstrainrate,atthecenterofthehalterestroke.Thesesimulationsquantifytheerrorassociatedwiththeassumedlinearityandassociatedratedecoupling.Simulationswereexecutedoverawiderangeofpitchandyawrate(-20to20rad/s)andwerepresentedfortworollratecases(0and5rad/s). 51

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Thehalteresensorstructureiscomposedofanitenumberofcampaniformsensilladistributedineldsatthebaseofthehaltere[ 30 ],[ 31 ].Thehomologstothehalteresensillaontheforewingsarethoughttoindividuallybepoormagnitudedetectorsduetotheirrapidsaturationandhighfrequencyfunctionality[ 35 ].Unlikecampaniformsensillaonlocustwingsthatoperateatlowerbeatfrequenciesandprovidecyclicactionpotentialswhosespikefrequencylikelydoescorrelatewellwithstrainmagnitude[ 36 ],thecampaniformsensillaonCalliphorawingsarethoughttorephasically,perhapsonceperstrokecycleatsaturation[ 35 ]. FayyazuddinandDickinsondocumentedresearchthatcharacterizedtheaerentstransmittedfromthebasalplatesensilla(dF2)andattemptedtoruleouttheothercampaniformeldsinipsilateralcontrolofthewingmuscles[ 28 ],[ 51 ].TheirconclusionwasthatdF2wasprimarilyresponsibleforsteeringmotorcontrolassociatedwithmnb1andthattheconnectionconsistedofbothafastmonosynapticelectricalcomponentandaslowchemicalcomponent.Thesestudiesdidnotreportanyattempttodistinguishbetweenstrainandstrainrateasparametersforwhichtheremaybedistinctproportionalsensitivity.AdditionalstudiesarerequiredtomeasurebilateralsignalcombinationandthepotentialroleofthesensillaofthechordotonalorganinthehalterewhichPringleestimatedwereorientedpreferentiallytomeasurebendingshear[ 14 ]. 52

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30 ].AssumingdF2isprimarilyresponsibleforCoriolissensing,theproposedhypothesisrequiresthatthetorquemotordistinguishesbetweenstrainrateandmagnitudeofstrainactivatingthesensoreld.Forexample,ashasbeenproposedfortheinsectwing[ 35 ],ifthemagnitudeofstrainisencodedthroughenlistmentofincreasednumbersofsensillawithindF2,thecompoundextracellularpotentialwouldincreasewithstrainmagnitude.Similarly,strainratemightbeencodedinthetimingofthedF2response,iftheincreasedstrainrateaectedphasingoftheresponse.Pringlereportedproportionalityofthetemporalphasingofthespikes,believedtobecomingfromthebasaleldandlargechordotonalorgans,tothemagnitudeofyawrate[ 14 ].FayyazuddinandDickinsonalsodemonstratedtheimpactofphasingofthesignalfromthehaltereonwingmuscleresponse,causingbothadduction/abductionandamplitudevariationinwingkinematics[ 51 ].Themonosynapticconnectionbetweenthehalteresandmnb1issucientlyfasttosynchronouslytransmitphasinginformation[ 51 ]. 3{9 )-( 3{11 ),canbeaccomplishedthroughanumberofmechanisms.Inadditiontodirectsignalsummation,lowpasslteringresultingfromtonicresponsemayalsoprovideasignalproportionaltotheaverage.Athirdpossibilityistoobtaintheendeectofadierenceorsummationofdrivesignalsthroughapplicationofopposingdrivemotorpairs.Forexample,commandinganincreasedwingstrokeamplitudeononesideandindependentlycommandingadecreaseontheothersideeectivelyprovidesthebilateralsummationofthetwocommandsintheformofarollmoment. Forpitchrateasdenedinthischapter,thedownstrokeandtheupstrokeresponsearesimultaneouslyexpressedbytheopposinghalteresduetothebilateralsymmetryofthesensoreldsandtheanti-symmetryoftheCoriolisforces(seeFigure 3-12 andFigure 53

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TheCoriolisforceinducedbypitchratehasbilateralsymmetry,butyawandrollhaveantisymmetricforces.Asaresult,summingleftandrighthaltereresponseallowsdirectdeterminationofpitchratewithoutpre-averaging. 3-13 ).Therefore,thestrainmagnitudefromtheupstrokeofthetwohalterescouldbesimultaneouslyencodedandcombinedtogenerateastabilizingtorqueproportionaltopitchrate.Bilateralprocessingisnotrequiredinthecaseoftheyawratecomponentduetothebilateralsymmetryofthehalteres(Figure 3-13 ).Eitherhalterecanprovideasignalproportionaltoyawratebyaveragingtheupstrokeanddownstrokestrainmagnitudeofthathaltere.FayyazuddinandDickinsonshowedbothaphasicandatoniccomponentbetweenthehalteresandmnb1[ 28 ].Thetonicresponsewassucientlyslowtoeectivelyaverageasignalatthewingbeatfrequency.Unambiguouslyrespondingtorollrateerrorswouldrequirebilateralsummationoftheaveragedstrainrateresponsefrombothhalteres.Assumingsomemeansofencodingstrainrate,rollcorrectioncouldbeaccomplishedbyipsilaterallytransmittingthesignalthroughasignalpathormuscleassociatedwithwingstrokeamplitudethatissucientlytonictoaveragethesignal.Thecombinedeectofthetwowingswouldthenbilaterallycombinetocreatethecorrectingrolltorque. Simulationresultsindicatethattheratedecouplingmechanismdescribedisfairlyinsensitivetothedetailsoftheencodingscheme.Forexample,whencontinuous,modeledstrainandstrainratesignalsarepassedthroughaweaklowpasslterandthenbilaterally 54

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Linearapproximationsofthestrainatthecenterofthehalterestroke.(mr,mp,my)representthemagnitudesoftheroll,pitchandyawratesofchangeofstrainwithrespecttostrokeangle.(br,bp,by)representthemagnitudesoftheroll,pitchandyawstrainwithrespecttostrokeangle.Theunilateralandbilateralprocessingrequiredtodecouplethecomponentsisclearlyseenbysumminganddierencingtherightandlefthaltereresponses. combinedaccordingto( 3{9 )-( 3{11 ),themeansignalstrackthetrueratecomponentswell.Thisindicatesthatencodingpreciselyatthecenterofthehalterestrokeisnotcritical.TheresultsofChapter4demonstratethefeasibilityofencodingandreconstructingthefullbodyratevectorusingonlydiscretecompressivestrainmagnitudemeasurementstodescribethesymmetricandantisymmetricaspectsofthehalteretrajectory.TheseresultsindicatethatwhiledirectencodingofstrainratewouldrepresentausefulsubmodalityofthedF2eld,itisnotnecessary.Chapter4willdevelopanddocumentamoredetailedmodelofthemechanoreceptorphysiologyandthetorquemotorsteeringcontrolmechanismsinordertofurtherestablishtheproposedmodelasabiologicallyplausiblemechanosensorymechanism. ThereportedsimulationresultsassumeaconstantangularrateandthereforeisolatetheimpactofCoriolisforcesfrombodyangularacceleration.Forthecaseoflowangularacceleration,theresultsimplythepotentialtodistinguishthecomponentsofthebodyratevector.Someauthorshaveentertainedthepossibilitythathalteresareprimarilyangularaccelerationsensorsusedforstabilizationafterextremesaccadicmaneuvers[ 18 ],[ 19 ].Incontradiction,Hengstenbergetal.laterdemonstratedadirectcorrelationbetweenangularratemagnitudeandcompensatoryresponse[ 20 ].Themechanicsdictatethatboth 55

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53 ].Assumingsimilaramplitudes,astheperiodofamaneuverapproachestheperiodofhalteremotion,theimpactofangularaccelerationandCoriolisforcewillapproachthesameorderofmagnitude.Whilethemodelgeneratedinthisreport( 3{12 )includesthebodyaccelerationterms,itwasoutsidethescopeofthischaptertofullyevaluatetheimpactofallpossiblekinematicscenariosonightstability. 56

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13 ],[ 54 ].Coriolisforcesaretheforcesinducedbyvelocityofamass,thehalteremassinthepresentcase,constrainedtomoveinarotatingbodyreferenceframe[ 14 ],[ 23 ].ThekinematicdescriptionthatleadstoCoriolissensitivitywasinitiallydocumentedbyPringle[ 14 ]andNalbach[ 23 ].Chapter3extendedthekinematicassessmentsofPringleandNalbachbyprovidingamorecompletesimulationofthedynamicsofthehaltere.Thedynamicsimulationofthehalteremotionledtodiscoverythattowithinaverygoodapproximationtheresponseofthehalteretothetworatecomponentsintheprimaryplaneofhaltereoscillationisdecoupledatthecenterofthestroke.Theaveragemagnitudeandrateofchangeofthehaltereout-of-planedisplacementatthecenterofthestrokeareindividuallyproportionaltothetworatecomponents.Thethirdratecomponent,perpendiculartotheprimaryplaneofhalteremotion,doesnothavearstorderimpactonout-of-planedeection.Theout-of-planecomponentintroducesnon-linearcouplingerrorsintotheprocessofmeasuringtheothertworatecomponentsaswasquantiedinChapter3. Whilethedescribeddynamicassessmentprovidedarmmathematicalbasisfortheabilityoftwohalterestocaptureandreconstructthreeorthogonalcomponentsofay'sinertialangularrate,itstoppedshortofreconcilingthekinematicsanddynamicswithwhatisunderstoodaboutthesensoryphysiologyoftheinsect.Thatreconciliationistheintentofthischapter.IntheassessmentofHengstenberg,whileprogresshasbeenmadeinunderstandingtheprojectionsofthehalteresensilla,ourbiggestdecitinunderstandingofthehaltereishowthestimulationofthehaltereisencodedbythesensilla[ 3 ].Theaerentpathwaysofthehalterehavebeenmappedoutthroughhistologicalanalysisusingdyellsofthehaltereprojections[ 1 ],[ 18 ].Thesestudiesshowedbothstrongipsilateral 57

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3 ],[ 20 ],[ 42 ].Thetemporalaccuracyoftheaerentsinthehalterenervehavebeenobservedtomaintainatightphaserelationshipwithinthewingbeatcycle[ 27 ],[ 51 ].Morerecently,FoxandDanielhaveshownwithsinglebermeasurementsthatindividualsensillahavealinearresponseouttofrequenciesmorethandoublethehalterebeatfrequency[ 2 ].AsshowninChapter3,thisissignicantsincetheyawcomponentofbodyrateresultsinaCoriolisforcethat,duetothelargeamplitudekinematicsofhalteremotion,hasafrequencyattwicethatofthehalterebeatfrequency. Thesensillaofthehaltereshavebeeninferred,fromcharacterizationofhomologouscampaniformsensillaonthewings,torespondindividuallyinverysimplefashion[ 35 ].Unlikethesensillaonthewingsoflocusts,whichrespondwithaslowlydecayingseriesofspikeswhosespikefrequencyincreasesasstimulusmagnitudeincreases[ 36 ],[ 37 ],thesensillaonthehalteresofhigherdipteransarethoughttoonlyreleaseasingleactionpotentialspikeonceortwiceduringasingleperiodofoscillation[ 35 ].Ashigherstrainsareintroduced,moresensillaareenlistedduetotheirspatialdistributionandtheassociatedspatialvariationofthestraineld.Therefore,theequivalencetoasensorwithalargedynamicrangeisensuredbyhavingasucientlylargenumberofidenticalsensilladistributeduniformlyoveraregionwherethestrainresponsehasasucientgradient.Someanalogiestothisformoffractionatedenlistmentofsimplesensorscanbefoundinthechordotonalorganinthelegsoflocusts[ 55 ]andthespindlecontrolledresponseofthemusclesinvertebrates[ 56 ].Thisappearstobeawidespreadcontrolstrategyinnaturalsystemswherelargenumbersofsimplehighlyintegratedsensorsareusedasopposedtofewhighlyrenedcomplexsensors. Theintentofthischapteristodemonstrate,usinganalyticalmodelsderivedfromcurrentunderstandingofhalterephysiology,thereconstructionofanarbitrarybodyratevector.ThischapterbuildsuponthepreviousdynamicmodelingofthehaltereinChapter 58

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30 ].Thisdemonstrationwilltherebyshedlightonwhatisthelargestdecitinunderstandingthefunctionofhalteres,theencodingoftheinertialratesbythesensoryeldsofthehaltereandtheassociateddecodingoftheratesintheformofreexivecontrolresponsebythewingcontrolstructures. 4.2.1DynamicsModel 3{12 )fromChapter3. Thein-planemotionofthehaltere,gamma,wasmodeledintwoforms,therstbeingasimplesinusoidwithanamplitudeof90degrees. Thesecondformwasmeanttoapproximatethemeasuredstrokecharacteristicswhichareknowntobemoreaccuratelyrepresentedbyatrianglewavewithmuchmoreuniformangularvelocityduringthemostofthestroke[ 23 ].Therstfewtermsofaseriesrepresentationwereused, 9sin(3!ht)+1 25sin(5!ht)1 49sin(7!ht)]1:0531:(4{2) 59

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42 ]. TherelationshipsbetweenthebodyratecomponentsexpressedinthebodyframeandthebodyratecomponentsmeasuredbythetwohalteresarethesameasthoseusedinChapter3, (4{3) (4{4) In( 4{3 )-( 4{5 ),thesubscriptsb1andb3representtheverticalandhorizontalratecomponentsintheplaneoftherighthaltere.Similarly,subscriptsc1andc3representtheverticalandhorizontalratecomponentsintheplaneofthelefthaltere.Theanglealphadenestherelationshipofthehalterebeatingplanerelativetoalineperpendiculartothemedialplane.Forthecurrentsimulation,thehalterebeatingplaneswerefoldedback30degreesrelativetothedescribedperpendicular. 30 ].Thecampaniformsensillaareassumedtofractionatetherangeofhalteredisplacementasanaturalconsequenceoftheirspatialdistributionoverthebaseofthehaltere.EachsensillumwasmodeledasifringaunitimpulsewhenthestrainatitslocationreachesastrainthresholdwhichwascommonforallsensillaindF2.However,eachsensillumwillreatauniquehalteredeectionlevelduetothespatialdistributionofstrainoverthebaseofthehaltere.Thedeectionlevelsatwhichtheindividualsensillarespondwereevenlydistributedbetweenhalteredisplacementsof0and0.001radians.Foranalternativerepresentation,thesethresholdscouldbedistributedtoapproximatethestochasticdistributionofthesensillum 60

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Thesensillawereassumedtorespondonlyintensionandonlyduringincreasingtensilemotion[ 34 ].AccordingtothemorphologicalstudiesofPringlethisshouldoccurduringposteriorbendingofthehaltere[ 14 ].Asthehalterepassesthroughthetensilethresholdduringrelaxationofthetensileload,reactivationwasnotallowed.Thesensillaweremodeledtoreasmanytimesduringthecycleastheypassedthroughthethresholdintheforwarddirection.Compressiveloadingassociatedwithnegativehalteredisplacementhadnoimpact. 4-1 4.3.1TestCaseDescription 61

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Functionalprocessesaremodeledtodemonstratetheencodinganddecodingofhalteresensoryresponse.(a)AngularratedisturbancesinduceCoriolisforcesthatcausestraininthehalterebase.(b)Compressivestrainsarerecordedby110binarysensilladistributedoverthehalterebase.(c)Physicalprocessesrepresentsimplesummationanddierencingthatdecoupletheorthogonalratecomponentsintheformofwingforces.(d)Bilateralsummationoftheforcesresultsinstabilizingreactionstopitch,yawandrollratedisturbances.

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Figure 4-2 showstherelationshipbetweentheratecomponentsexpressedintheconventionalbodyxedreferenceframeandtwobodyxedframesorientedpreferentiallyintheplanesoftherightandlefthalteres.ThetrajectoriesofthesimulatedhalteresareshowninFigure 4-3 withoneoscillationperiodemboldenedtoaidininterpretationofthemotion.Figure 4-4 showsthesimulatedextracellularpotential(cumulativecampaniformresponse)inrelationtothehalteremotion.TheevenlyspacedredlinesinFigure 4-4 representthepointintimewhenthehaltereisatthelowerendofthestroke.Eitheroneortwocompositespikesareseenpercycle. 4-5 demonstratestrajectoriesforverticalandhorizontalratecomponentsindividually.Anymeansthatallowformeasurementoftherelativelevelofsymmetryandasymmetryfromthecompositetrajectoryshouldbesucienttoobtainastabilizingcontrolsignal.Theslopecouldpossiblybeapproximatedthroughdirectmeasurementofaveragedstrainrateatthecenterofthestrokeorthroughdierencingofstrainmagnitudesmeasuredonbothsidesofthestrokecenter. Inthealgorithmattempted,theperiodofoscillationisdividedintotwohalvesandtheintegratedcampaniformresponseduringeachhalfisconsidered.Thesehalves 63

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Thesimulationprocesstakestheinertialratevectordenedinthebodyframeandrecastsitintothehaltereframes.Figured)denesthereferenceframesandthefrequencyofratevariationfortheopen-looptestcase.Therstthreeguresrepresenta)theratecomponentsinthebodyframe,b)theratecomponentsinareferenceframeattachedtotherighthaltere,andc)theratecomponentsintheframeattachedtothelefthaltere. correspondtotheup-strokeanddown-strokeofthehaltere.Thetwocompositecampaniformresponsesarethendierenced,providinganapproximationoftheasymmetryoraverageslopeofthetrajectory.Thesignofthisdierenceisindicativeofthesignofthehorizontalratecomponent.Theaverageorsummationofthetwocompositeresponsesafterremovingtheantisymmetricresponseisrepresentativeofthesymmetric(yaw)componentoftheresponse.Determiningthesignoftheyawcomponentismoreproblematicthanintheothercaseduetothefrequencydoublednatureofthehalteretrajectory.However,naturehasprovidedaratheruniquesolution.IftheemboldenedpartofthetopFigure 4-5 bisstudiedcloselyandcomparedtothebottomwherethesignoftheratehasbeenreversed,itcanbeseenthatthecenterofthecampaniformresponsesonthetwo 64

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Thedynamicsexpression,Equation(1),issolvedindividuallyforeachhaltere.Theresultingtrajectoriesarerepresentedfortheleftandrighthalteresover0.2secondsor40haltereoscillations. 65

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Thesensillaresponsesaresummedateachintegrationtimesteptorepresentthenetmotorcontrolaerentresponse.Theshortevenlyspacedmarksshowthebottomsofthehalterestrokeasatimingreference.Thecurrentmodelpredictsthecompoundresponseofthesensillatoappearasoneortwoburstsofsensillaspikesthatshowadistinctphaseshiftdependingonthesignoftheyawstimulus.

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Theresponseofthemodeledcampaniformsensilla(CS)alwaysoccursonthesamehalfofthedisplacementplane.a)Haltereresponsetoonlyalateralratecomponent.WhethertheCSresponseoccursonthedownstrokeorupstrokedeterminesthesignofthelateralrate.ThedierencebetweenupstrokeanddownstrokeCSresponseisrelatedtothemagnitudeoftherate.b)Haltereresponsetoonlyayawratecomponent.ThesignoftheyawratecomponentmustbedeterminedbyphaseinformationsinceupstrokeanddownstrokehavesimilarCSresponses.c)Correlationofphasechangeofthecompositeresponsewithsignofyawratebecomesapparentifresponseisplottedversustime. 67

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3=RupRdown 1=sgn()(Rup+Rdownabs(3)): Intheseexpressions,thevariablesaredenedasfollows: 1=yawratecomponent,3=lateralratecomponent,Rup=cumulativeupstrokecampaniformeldresponse,Rdown=cumulativedownstrokecampaniformeldresponse,sgn()=signoftheyawratecomponentasdeterminedbytherelativephaseofthecampaniformeldresponse. 4-6 ,showingapproximationsofthetwomeasurableratecomponentsintherighthalterereferenceframe.Figure 4-7 showsthereconstructionoftheroll,pitchandyawbodyratesfromthesehalteremeasurementsaccordingto( 4{3 ),( 4{4 )and( 4{5 ).Figure 4-8 showsthesameresultusingthetriangularwaveformof( 4{2 ). 4.4.1SignicanceofFindings 68

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Applicationofthemathematicaloperationsdescribedby( 4{6 )-( 4{7 )resultinsignalswhosemagnitudesroughlyrepresentthetwoorthogonalratecomponentsinuencingout-of-planedisplacementoftherighthaltere.a)Ratecomponentsresultingfromdecodingtheunidirectionalcampaniformresponses.b)Trueratecomponentsforcomparisonwiththedecodedapproximations. 69

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Unilateralresponseofthewingstohaltereaerentsresultsinforcesthatwhencombinedaccordingtotheconstructsof( 4{3 )-( 4{5 )providenettorquesproportionaltooriginaldisturbances.a)Sinusoidaltimehistoryofhalterestrokeangle.b)Decodedsignalsproportionaltoratecomponentsintherightandlefthalterereferenceframes. 70

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Halterederivedrateestimatesundertheinuenceofamorerealistic\saw-tooth"motionprole.a)Triangularorsaw-toothtimehistoryofhalterestrokeangle.b)Decodedsignalsproportionaltoratecomponentsintherightandlefthalterereferenceframes. 71

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4{3 )-( 4{5 )willresultinnettorquesthatareproportionaltothepitch,yawandrollratedisturbances. TheproposedmodelsatisestheobservedconstraintthatthecampaniformsensillafromdF2areprimarilyresponsibleforencodingratedisturbances[ 1 ],[ 18 ],[ 28 ].AsshowninChapter3,decouplingoftheratecomponentsdependsonmeasurementofsignalsproportionaltotheaveragedslopeofhalteredisplacementandtheaveragedmagnitudeofhalteredisplacementnearthecenterofthestroke.However,analogoustoanitedierenceapproximationofaderivative,iftheabilitytodierencestrainmagnitudesisallowed,directmeasurementoftheaveragedslopeofthehalteretrajectory,ortherelatedquantity{strainrate,isnotrequired.Theproposedmodelhasnofundamentalneedforadditionalinputbyothercampaniformeldsorbythechordotonalorganofthehalterebaseforratestabilization.Additionally,theencodingoftheratesbydF2couldbeassimpleasabinaryresponseataxedthreshold.AstheworkofinvestigatorssuchasFoxandDanielprogresses,morecomplicatedmodelsencompassingthedynamicresponsecharacteristicsandstatisticalvariationsinsensillaresponsecanbeincorporated[ 2 ].Thecurrentresultsdemonstratethatthesensillaonlyhavetoreactinthemostrudimentarywaytoencodesucientinformationtoreconstructstabilizingsignals.Thisdoesnotinanywayprecludethenecessityforfusionofinformationfromothersensormodalitiestoexplainthecomplexbehaviors,suchastargetpursuit,ofthelivingorganism. 29 ],[ 30 ],[ 57 ],[ 58 ].Pringleconcludedthroughself-proclaimedcircumstantialmorphologicalandbehavioralevidencethatthe 72

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31 ].Inhiscomprehensivepaperonhalteres,Pringlereiteratedhisconclusionsaboutcampaniformdirectionalsensitivityandsuggestedbasedontheirmorphologythatthepertinentsensillashouldonlyrespondduringforwardbendingofthehalteres[ 14 ].SandemanandMarklappearedtoconrmthispredictionwhentheyobservedthattheycouldonlymeasureanaerentresponseduringforwardaccelerationofthehaltereendknob[ 18 ].LaterexperimentsonPeriplanetabyZilletal.establishedconclusivelythatthecampaniformsonPeriplanetawerenotsensitivetocompressivestrainparalleltothelongaxisofthecampaniformcapbutinsteadtotensilestrain[ 34 ],[ 59 ],[ 60 ].Thedemonstratedmodelconformstothatnding,i.e.,astraineldwithatensilecomponentparalleltothelongaxisofthecapcausesacampaniformresponse.ApossibleexplanationoftheSandemanresults[ 18 ]wouldbethattheappliedforwardforceagainstthehaltereshaftresultedininertialforcesthatcausedthehalteretoactuallybendbackwardasifinuencedbyCoriolisforcescausingbackwardbending.Thesimulationresultsprovidedinthischapterestablishthathalteredisplacementmeasurements,encodedbyunidirectionalsensorsthatfractionatetherangewithsimplethresholdinducedresponses,aresucienttoencodeinertialangularrates.Thisresultwouldhavebeenqualitativelythesameregardlessoftheassumedorientationofthesensors. Themostrecentmeasurementsofhaltereaerentshavebeenmeasuredonsinglebersincloseproximitytothehalterebase[ 2 ].ThesemeasurementsdemonstratedthatonHolorusiatheaerentresponseaccuratelytracksthehaltereresponselinearlywellabovetwicetheoscillationfrequencyofthehalteres.Thiscorrespondswellwiththerequirementsofthecurrentresults.Thetimingoftheresponseofthemodeledcampaniformsdependsprimarilyonthetrajectoryofthehalteredisplacementwhichundercertaincombinationsofangularratemightbequitecomplicated.Theseparationofconsecutiveresponsesfromagivensensillumcouldingeneralbemuchlessthanhalf 73

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14 ].Pringlealsospeculatedthatyawrotationmagnitudecorrelateswithboththetimingofthesensillaburstandthenumberofsensillaenlistedintheresponse.Thetimingreferenceusedinthecurrentmodeltodeterminethesignofyawrateisbasedonaxedpointinthephaseoftheoscillationandnotonadedicatedsensoryresponse.Heideobservedastrongphaselockingeectbetweentheipsilateralhaltereandthemotorneuronoftherstbasalarmuscle(MNB1)andweakphaselockingbetweenthecontralateralhaltereandMNB1[ 27 ].HeidealsocorrelatedphaseofpotentialspikesinMNB1withwingamplitude[ 27 ].Anasymmetricchangeinwingbeatamplitudeataxedfrequencywouldbeexpectedtocausedierentialthrust,therebyinducingatorque.FayyazuddinandDickinsondemonstratedinCalliphorathatthehaltereaerentsweretightlycontrolledinphase[ 28 ].Intheirmodel,undernominalconditionswingaerentsentraintheringphaseofB1.Whentheinsectrotates,thephaseofthehaltereaerentshiftssucientlytoentrainMNB1,transientlyphaselockingandtherebyshiftingtheringoftheB1muscle[ 51 ].TheimportanceofphaseincontroloftheB1steeringmuscleisconsistentwiththeanalyticalndingthatdF2ringphasehasadirectinuenceoncontrolofyawrotationaldisturbances.Althoughithasnotbeendemonstratedexperimentally,acorrelationbetweenphaseandsignoftheyawrateisalogicalextension. 74

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4{6 )-( 4{7 )and( 4{3 )-( 4{5 ).Aspreviouslydescribed,thisprocesscanbebrokenintotwosteps,therstbeingipsilateralcontrolofwingsteeringmusclestogeneratetorquesrepresentativeof( 4{6 )-( 4{7 )and,thesecond,asimplesummationoftorquesinconformitywith( 4{3 )-( 4{5 ).Theexpressionsin( 4{6 )-( 4{7 )require: 1. Summationofupstrokeanddownstrokecompositeresponseofthecampaniformeld. 2. Dierencingtheupstrokeanddownstrokecompositeresponse. 3. Phasesensitivityofthecompositeresponsetodeterminethesignoftheyawmotion. 4. Dierencingof1)andthemagnitudeof2). Assumingtheexistenceofsuchmuscularandneurologicalconstructs,signalsproportionaltothetwoin-planeratecomponentsareavailable.Thesemustthenmanifestintheformof( 4{3 )-( 4{5 ).Forexample,aschemecouldbepostulatedusingindependentcontrolofsupination/pronationandamplitudestatesofthetwowingsandacommonabductionangletoallowforvedegreesoffreedomcontrollingallbuttheresiduallateralforcecomponent(personalobservation). Numerousmeansofsummationanddierencingofupstrokeanddownstrokeaerentscanbepostulated,involvingantagonisticmusclepairsorcompetinginhibitoryandexcitatoryinuences.Thefactthatthehaltereoscillationhasgenerallybeenobserved 75

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4{7 ).However,aforceincreasingtheamplitudeontheupstrokeandthesameforceappliedonthedownstrokecouldconceivablyhaveanantagonisticeect,fulllingtherequirementfordierencingin( 4{6 ).Anynumberofphysiologicalsolutionstotheproblemofaerentdecodingcouldbepostulatedgiventhelimitedexperimentalevidence.Itislikelythatdierentlevelsofevolutionaryrenementmayhaveresultedinarangeofimplementationvariationsamongspecies. 4-6 4-7 and 4-8 .Thisisduetothecomplexpatternsofmotioninducedbycombinationsofpitch,yawandrollrates.Intuitively,( 3{12 )caninducequitecomplexmotionundertheinuenceofnon-linearratecouplingterms,angularaccelerations,andthepredominantCoriolisaccelerations.CarefulinspectionofthecampaniformresponseinFigure 4-4 showsareaswhereonlyasingleburstofsensillapotentialsoccursinsteadofthemorecommonoccurrenceofoneontheupstrokeandoneonthedownstroke.Thesimplisticmodelemployedrequiresaspikeondownstroketodeterminethesignoftheyawrateestimate.Ifthereisnotanestimatethenthedefaultvaluefortheyawrateiszero.Figure 4-6 clearlyshowsduringperiodswhenonlyasinglespikeoccursduringtheoscillation,theestimateforyawrategoestozero.ThisisanalogoustotheresponseofthemotorneuronfortheB1musclestayingphaselockedtothephasicwingcampaniformresponseunlessthesignalfromthehalterefallswithintheappropriatephaseband[ 51 ].Amoresophisticatedalgorithmorabandlimitedresponseoftheindividualsensillamightmediatethiseectto 76

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4-7 and 4-8 isthatthegapsinresponseareminimizedinthecombinedresultoftwohalteres.Whenasinglecompositespikeislimitingtheestimatefromonehaltere,ingeneralthesameisnotoccurringattheotherhalterewhichhasasignicantlydierentorientationthantherst.Thereforetheseerrorsaremitigatedtosomeextentinthesummationofthereexiveresponsesmediatedbytwohalteres. Thecurrentmodelhasdemonstratedthefeasibilityofencodingandreconstructingastabilizingresponsetoarbitrarydisturbancesusingaeldofunidirectional,binarystrainsensorsatthebaseofthehaltere.Themodelhoweverislimitedbytherudimentaryrepresentationofthesensillaandlimitedunderstandingofhowthecampaniformaerentsaredecodedbythewingsteeringmechanisms.Futureworkwillattempttoprovidemorerealisticbandlimitationsonthesensillaandincludearepresentationofthehalteremodelina6degreeoffreedomightmodel.Theightsimulationenvironmentwillallowforinvestigationofreexiveresponsestopatterngeneratedmaneuversandthefusionofothersensormodalitiesintotheightcontrolstructure. 77

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Numerousauthorshavestudiedthekinematiccharacteristicsofywingmotionandtheresultingdynamicmotionofies[ 16 ],[ 61 ]{[ 65 ].Theightofinsectshasbeenofparticularinteresttothescienticcommunitybecauseofthestatementsbyearlyresearchersthatitwas\impossible"forinsectstoybasedontheapplicationofconventionalaerodynamicunderstanding.Thiswastrueduetothefactthatsteadystateaerodynamicmodelsareinadequatetopredicttheforcesony'swings.Insectsarenow 78

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66 ].Thecasualobserverconventionallythinksofarticulatedwingsininsectsmovingverticallyasistypicalinbirds.However,inanimalsthatyslowlyorhover,suchashummingbirdsandmanyinsects,thewingsmovepredominantlybackandforthinanalmosthorizontalplane,generatingthrustsucienttolifttheweightofthebody.Insomewaysthisissimilartotheightofahelicopter,generatingmainlyupwardthrustandmovingforwardbytiltingtheplaneofitsoscillatingwingsforward.Unlikehelicopterswhichgainsomestabilityfromtheangularmomentummaintainedbytherotatingcomponents,theangularmomentumofthetwowingsofinsectsnearlycancels.InsectsoftheorderDipteraaugmentightstabilityusingratefeedbackfromthehalterestoprovideactivedampingtotheirattitudecontrol. Nofewerthan18pairsofsynchronouscontrolmusclesnetunethetrajectoryofthewings[ 64 ].Inaddition,twoopposingsetsoflargemusclesinthethoraxareasynchronouslydrivenintoastateofstretch-activatedoscillationtopoweright[ 61 ].Whilecompleteunderstandingofthemechanicalprocessesassociatedwithinsectightisnotyetavailable,sometheorizethatenergyisconservedtosomeextentbystoringitcyclicallyinthemusclesasinsprings,addingpowerasnecessarytoreplacetheenergylostinthegenerationofaerodynamicpower[ 67 ].Somecontrolmusclesarethoughttoimpartphasicforcesonthewingduringthestrokecycle,andtheeectisthoughttobetonicinothers,providingcontinuoustensiontoinuenceparameterssuchaswingabductionangleorthedegreeofpronation[ 64 ]. Atightcontrolloopisknowntoexistbetweenthehalteresandthewings,withsignaltransportsucientlyaccuratetopreservephaseinformationassociatedwiththeresponseofthehalteres[ 28 ],[ 51 ].Thewingsareknowntobesensitivetothephaseofthecampaniformresponsesfromthebasaleld(dF2)inparticular.Whilethehalteresprovideratestabilization,stronglysupplementinganyrotationalaerodynamicdamping, 79

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3 ]. Obstructionavoidanceisasignicanttechnicalhurdleforsystemsthatintendtoyautonomouslyincomplexenvironments.Inadditiontobasicratestabilization,theresultsdescribedinthischapterareusedtodemonstratesynthesizedbehavioralresponseinthepresenceofrandomlygeneratedobstructions.Natureappearstohandlethisrequirementwellbyusingtheopticalmeasurementsprovidedbythecompoundeyestoinuencetheighttrajectory[ 68 ].Fliesarewellknownfortheirsaccadicmaneuverswheretheyrapidlyturnthroughalargeangle(10to90deg)inresponsetoabloomingvisualstimulus[ 53 ].Thesemaneuverstaketheirnamefromthemotionofthemammalianeyewhichiscontinuouslyjumpingfromonepointtoanother.Inthecaseofthey,thesemaneuverstaketheformofabank-to-turn,wheretheinsectrollsitsbodyinordertooptimallyapplyforcesandmomentstoreorientitsbodyandvelocityvectorawayfromaloomingobject.Totheextentpossiblewithoutdetailedsimulationofaerodynamicresponse,saccade-likemaneuversarecapturedinthedescribedsimulationstudy. 69 ].ThederivationofthethirteenrstorderdierentialequationsisincludedinAppendix2 80

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W8>>>><>>>>:FxbFybFzb9>>>>=>>>>;+8>>>><>>>>:rvqwpwruqupv9>>>>=>>>>; 2266666664exeyeze0ezeyeze0exeyexe03777777758>>>><>>>>:pqr9>>>>=>>>>;: Inthisformulation,(u;v;w)arethevelocitycomponentsofthevehiclerelativetotheinertialframeexpressedinthebodyframe,(p;q;r)aretherotationrateofthebodyrelativetotheinertialframeexpressedinthebodyframe,(x;y;z)arethepositioncoordinatesintheinertialframeand(e0;ex;ey;ez)denetheunitquaternionusedto 81

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70 ]. 5-1 .Inthecaseofcontrolresponsetoameasurederror,e.g.altitude,theactivationfunctionisrecastasahyperbolictangentwithanti-symmetryaroundthedesiredvalue.Inthecaseofaltitude,theresponsecanbethoughtofasaproportionalgainonerrorprovidingasignalwithscalableupperandlowerlimitsbetween1and-1. Theinputstotheneuralnetworkarerepresentativeofolfactoryandopticalsignals.Therateofgrowthofanobjectinthevisualeldcanprovideastimulusforobstructionavoidance,takingintoaccountbothvelocitytowardanobjectandtherangetotheobject.Theangularrateofgrowthofthecylindericalobstructionsisrepresentedas(Vperp 82

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Non-linearresponseactivationlogicforthesimulatedinsect.Anavigationsolutionisdeterminedbyolfactoryandopticalinputstonon-linearactivationfunctions.Errorsbetweendesiredandactualbodypositionandorientationdeterminecontrolforcesandtorques.

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5{5 )whoseinputisthedierencebetweenrange-to-targetandapresetactivationrange(RtargRthreshold).Thisresponseisequivalenttoathresholdedactivationcausedbyanacousticorolfactorysignalthatisfunctionallydependentonrangetothetarget.Theosetforactivationwasarbitrarilychosentobe15meters.ThethresholddierencerangeisgainedbyGLRtoaectthesensitivityoftheactivation.Theoutputoftheactivationfunctionismultipliedbyaunitvector,^ut,pointinghorizontallytowardthetarget. 1+exp(GLR(RtargRthreshold)) (5{5) 5{6 )withaninputosettothedesiredightaltitude.Theoutput,alt,variesfrom-1to1inawaythatisoppositeinsigntothealtitudeerrorinordertoelicitnegativefeedbackcontroltothedesiredaltitude.Thisresponserepresentsanapproximatelyproportionalcontrolgainwherethe 84

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1+exp(Galt(ZZdesired))1 (5{6) 5{8 ).Angulargrowthrateoftheobstructionisfoundbydividingthevelocitycomponentperpendiculartotheobstructionbytherangetotheobstruction.Theactivationrangesfrom0to1asthethresholdispassed.Thesensitivityoftheactivationcanbeadjustedwiththegain,Gobs.Thevalueismultipliedtimesahorizontalunitvector(^robs)pointingawayfromtheobstructionalongtheline-of-sight.Theresponsetotheclosestthreeobstructionsareaddedtogethersothatanaveragedresponsecanbefoundwhenincloseproximitytomorethanoneobstruction. 1+exp(Gobs(Vperp (5{8) 85

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5{10 ).Therelativevelocityisdecomposedintocomponentsparallelto( 5{11 )andperpendicularto( 5{12 )theline-of-sightvector(^rLOS).Aresponseproportionaltotheerrorisfoundbykeepingthecomponentoftherelativevelocityalignedwith^rLOSandchangingthesignofthecomponentperpendiculartotheline-of-sightvector.Afternormalizationthisprovidesaunitvectorusedtocommandattitudecontrol( 5{13 ). ~V=~Vfly~Vtar ^rLOS=~Ptar~Pfly ~Vperp=~V~Vparallel ^usr=sr~Vparallel~Vperp 5{7 ),obstructionavoidance( 5{9 ),andcloserange( 5{13 )responses,whereeachresponseisgainedusingconstantgains(G1;G2;G3)toestablishtherelativeimportanceoftheresponses.Boththeobstructionavoidanceandcloserangetargetpursuitresponsearegivenahigherprecedencethroughtheirgainsthanthelongrangeightcontrol.Thisoverwhelmsthetendencytomaintainaltitudewhileincloseproximitytothetarget. ^ucontrol=G1~uobs+G2^ulr+G3^usr 86

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Nominalsimulationparametersforthesimulatedinsect. 53 ].TheseparametersaresummarizedinTable 5-1 .Observedwindtunnelmaximumvelocity(2m/s)wasusedasopposedtothelowercagedobservation.Thesaccade,whichisarapidlateralchangeinightdirection,wasnotmodeledexplicitly,butwasinsteadallowedtooccurasanimplicitpartoftheobstructionavoidanceresponse. 5{14 ).Errorswithrespecttothisnavigationvectorareconvertedtoavectorforceandatorquecommandedtoorientthebody.Theresultingforceactsinthepresenceofaerodynamicdragandgravitytopushtheytowardtheobjective.Aerodynamicdrag-inducedforcesandtorquesareassumedtobequadraticfunctionsactingparalleltothevelocityandangularratevectors,respectively. 87

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Attitudecontrollogicisbasedonorientationerrorswithrespecttothedesiredazimuthdirection.Pitchandyawerrorsaredriventozero.Controlloopdampingisproportionaltotheangularratesmeasuredbythehalteres. ThebottomofFigure 5-2 providesadepictionofthesmallerrorcontrollogic.Thelogicrepresentedissimplyaproportional-derivative(PD)controller.Iftheazimutherrorisbelowauserdenedlimit,thenthethreerotationaldegreesoffreedomarecontrolledindependentlyinproportiontothemagnitudeoftheerrorintheorientationandthebodyrate.Thebodyratesareassumedtobemeasuredbythehalteresandthengainedandsummedalongwithgainedazimuth,elevation,androllerrorsmeasuredrelativetothedesirednavigationdirection.Theoverallobjectiveofthesimulationistotrackalongthenavigationvectorwithapitchandrollofzeroandthecommandedazimuth.Tocaptureareactionsimilartotheknownsaccadicresponseofinsects,asimplebank-to-turnmaneuverisalsoallowed.Iftheazimutherrorisabovetheuserdenedlimit,abank-to-turnmodeisentered.Thesimulatedinsectbanksinproportiontotheerroruptoamaximumbankangleandusesacombinedpitchandyawmaneuvertoreorient.Tofurthersimulatesaccadicbehavior,theinsectmaintainstheightpathassociatedwiththeobstructionavoidanceresponseforashorttime,oruntilanotherobstructioncausesaresponse,beforeresumingightbacktowardthetarget.Inbothskid-to-turnandbank-to-turnmodestheproportionalandderivativecomponentsaregainedandsentthroughhyperbolictangentfunctionsthathavebeenscaledtolimitthemaximumallowabletorques. 88

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Themaximumlinearforcesweredeterminedbytheidentiedmaximumlinearaccelerationsandthemassofthevehicle.Themaximumvelocitywasassumedlimitedbyadragforceproportionaltoairspeedsquared.Adragcoecientwasthereforeapproximatedastheratioofmaximumforcetomaximumvelocitysquared(Fmax=V2max).RepresentativesimulationparametersaresummarizedinTable 5-2 Table5-2. Exampleofderivedsimulationcontrolparameters. 89

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Nominal6DOFtestcaseshownfromatopviewperspective.Obstructionsareplacerandomlyintheformofverticalcylindersofrandomlyselecteddiameters.Theightpathbeginsonthegroundattheorigin,withthetargetplacedontheoppositeendofthearena. 5-3 90

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Thenaltwosections,5.3.3and5.3.4,areassociatedwithresultsfromthe6DOFsimulation.Thecontrolmodelinthe6DOFsimulationassumescontralateralcombinationofthehaltereoutputtoobtaintorquesproportionaltotheangularrates.Whileideallythesetorquescouldbeappliedexactly,somelevelofcrosscouplingwillexistwhichwillintroducetorquedisturbancesthatcorrelatewiththedesiredcontrolcommands.Therangeofcross-couplingoverwhichstabilitycanbeguaranteedisaddressedintherstsection.Thenextsectionaddressestheissueofcontralateralversusipsilateralfeedback.Asdescribedinthepreviouschapters,theliteratureindicatespredominantlyipsilateralfeedbacktotheightcontrolmuscles.Theimplicationofthenon-orthogonalityofthehalteresandthecontrolparameterschosenaredemonstrated.Thenaltwosectionsdescribetheresultsofthe6DOFsimulation,demonstratingtheabilityofthehalterestotrackdynamicangularvelocitiesusingtheencodingschemedescribedinChapter4andthedegreeofdelityatwhichsaccadeswerereplicatedwithrespecttothemeasureddataofSchlistraandVanHateren[ 53 ]. 91

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71 ]. Eulerequationsforarotatingrigidbodycanbedenedwithrespecttotheprincipalaxesas 5{15 )andatorqueinputthatincludescontroltorquesproportionaltothestateerror!.ThecharacteristicsoftheLyapunovfunctionderivativedeterminethenatureofthestabilityinthepresenceofcross-couplingandaerodynamicdrag. 92

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2!TJ!;and (5{16) _V=!TJ_! Intheseexpressions,Jisaconstantinertiamatrixand!isdenedwithrespecttoprincipalaxes,therefore, SubstitutionofJ_!from( 5{15 )intotheLyapunovderivativegives _V=!T266664(J2J3)!2!3(J3J1)!3!1(J1J2)!1!2377775+!Tu =!Tu: whichprovidesanegativedeniteresultwithnoextraneousforces.Basedon( 5{16 ),( 5{19 )and( 5{20 ),thefollowinginequalitycanbedeveloped: _V=!TK!

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5{16 )and( 5{21 )as!(t)!(0)ekmin =266664k1000k2000k3377775!+2666640c12c13c210c23c31c320377775!; _V=!TK!+!TC! =!T(KC)!: (5{24) _V!T266664kccckccck377775!: 94

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(5{26) 12c k33c k2>0:(5{27) TheplotinFigure 5-4 demonstratesthatpeakstabilityoccurswhenthereisnocross-couplingandprovidesevidencethataslongastheratioofthelargestmagnitudecrosscouplingtermtotheminimumgaineigenvalueislessthan0.5,globalexponentialstabilityshouldbeobtainable.Infact,thethreerootstothisexpressioncanbereadilyfoundat0.5and-1.0,with-1.0beingarepeatedrootwherealocalminimumoccurs. =Dj!j!: 95

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PlotdemonstratingtherangeoverwhichtheLyapunovderivativeisnegativedenite.IntherangewheretheprincipleminorsofthegainmatrixareallpositivetheLyapunovderivativeisnegativedenite. whereDisassumedtobeadiagonalmatrixofpositivecoecientsassociatedwitheachaxis.ThetermDj!jaddsstabilitytothesystem.TheconstantpositivediagonaltermsoftheKmatrixarenowsupplementedwithtermsthatareproportionaltothemagnitudeof!.Ifaconservative,boundingdenitionofthedragcoecientsisused,asdenedbydi=dmin=d,thentheappliedtorquesandtheassociatedLyapunovderivativecanbedenedas (5{30) 96

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Followingthesameprocessasbefore,for_Vtobenegativedenite,thenthegainandcross-couplingmatrixmustbepositivedeniterequiring (5{32) (k+dj!j)32c33(k+dj!j)c2>0: 12c k+dj!j33c k+dj!j2>0:(5{33) AplotofthisexpressionisnotprovidedsinceitwouldbeidenticaltothatFigure 5-4 ,withtheordinatedenedasc k+dj!jinsteadofc k. 97

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Figure 5-5 showstheassumedwingstrokegeometry.Forsimplicitytheaveragecenterofpressureisassumedtoexistinaplanethatincludesthecenterofgravityofthebody.Thenetforcegeneratedbythewingsactsthroughthecenterofpressureandisbrokenintotwocomponentsthatareindependentlycontrollable.Therstisinthex-direction,possiblygeneratedbywingpronation,andthesecondisinthez-direction,controlledwithwingamplitude.Forcesfromthetwowingsinthey-directionareassumedtobein 98

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Diagramdemonstratingthegeometriccongurationassociatedwithwingkinematicanalysis.Controlparametersaretherightandleftforcesinthex-direction(FRx;FLx),inthez-direction(FRz;FLz),andtheabductionangledeviationR;L.Theforcesactthroughtheaveragewingstrokecenterofpressure(CP)andtorquesarecalculatedwithrespecttothecenterofgravity(CG). oppositedirectionswithapproximatelyequalforcesothattheneteectonthebodyisnegligible.Thewingstrokeandtheabductionanglearedenedtobeinthexy-planeasiftheinsectwerehovering. 5-3 providesdenitionsofthesymbolsusedinthedescribedresults.Noteinreviewingthedescribedresultsthatthecontrolgains,(Kx;Ky;Kz),,L,w,and0areallconstants. Contralateralfeedbackallowsforcombinationoftheoutputofthetwohalterespriortocontrolresponse.Duetothenon-orthogonalorientationofthetwohalteres,thelateralratecomponentsmustbecombinedinaveryspecicmannertoreconstructpitchand 99

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Parameterdenitionsforwingkinematiccontrolexpressions. rollbodyrates.Forpitch,adierencingoperationmustbeperformedand,forroll,asummation.Therequiredgainsdependboththecontrolgainandtrigonometricfunctionsoftheanglethatdenesthehalterestrokeplane.Thecontralateralavailabilityofthesemeasurementseectivelyimpliestheavailabilityofthebodyrates(p;q;r)fordirectuseingenerationofthecontrolforcesandabductioncontroldeviation.Additionally,availabilityofthebodyratesallowsforsymmetricgenerationofcontrolcommandssothatthedesirednominalforcesareexactlymaintainedwhileachievingthenecessarycontroltorques.Theresultforthecontrolcommandsasafunctionofbodyrateandwinggeometryis FzdLcos(0) (5{34) FRz=Kxp (5{35) FRx=Kzr (5{36) FLx=FRx FLz=FRz: ThederivationoftheseexpressionsisincludedinAppendixC. 100

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(5{43) IntheseexpressionsFxdandFzdaretheforcesobtainedfromthecontrollogicdescribedbyFigure 5-1 andthenappliedinthebodyframeasiftherewerezeroattitudeerror.Thetorquesshownin( 5{39 )-( 5{41 )arethedesiredstabilizingresponseproportionaltotheratesexpressedinthebodyframe.SincethecontrolforcedeviationsassociatedwithrateerrorswererequiredtobeequalandoppositeforthederivationofSection C.1 ,equations( 5{39 )-( 5{44 )simplyconrmtheexpectedresult.Inthefollowingtwosections,wherecontralateralinuenceofthehalteresisnotassumed,therequirementforequalandoppositecontrolforceresponsescannotbeenforced. Theresultingexpressions,asderivedinAppendixC,forthecontrolparametersare 101

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FRx=C41 FLx=C41 FRz=C5b3 FLz=C5c3;where (5{50) (5{51) (5{54) Lookingatoneoftheexpressionsforthecontrolangle,twolimitingcasescanbeevaluated.First,ifthedenominatorterminvolvingtheliftforceislargewithrespecttotheterminvolvingthelateralratemeasurementthen Thecontrolangleisproportionaltothelateralratemeasurementandinverselyproportionaltotheliftforce.Ineect,thishasalreadybeenassumedwhenRwasassumedtobeasmallangle.Iftheliftforceweresmallthen Inthiscasethecontrolanglebecomesaconstant,dependentonthehalteresetbackangleandthecontrolgains.Sincethehalteresetbackangleisnotsmall,thesituationwherethecontrolforceislargerelativetotheliftforcewouldcontradictthesmallangleassumption. 102

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(5{59) Unlikethecasewherecontralateralfeedbackwasallowedandsymmetryofthecontrolforcesenforced,inthiscasethenetforceinthez-directionisnotexactlythedesiredforce.Therearecases,forexample,wheretheratevectorisunobservablebyoneofthenon-orthogonalhalteresbutobservablebytheother.Thiswouldresultinacontrolresponse,Fz,ononesideonly. ThecontrolparametersarederivedinAppendixC.Theexpressionsforcontrolforcesandmomentsaresummarizedhereas FRz=Kxb3 (5{64) FLz=Kxc3 (5{65) (5{66) 103

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(5{67) FLx=Kz1 (5{68) FRx=Kz1 Thesummedforcesandmomentsarethesameasthepreviouscase, (5{71) Onceagain,thelackofcontralateralinuenceeliminatesthepossibilityofcreatingasymmetryconstraintonthecontrolforcestoensurethenetforceinthez-directionisequaltothedesiredforce. 5{39 )-( 5{44 ).Dierencesbetweenactualanddesiredz-forcewhichshowedupin( 5{58 )-( 5{63 )and( 5{70 )-( 5{75 )wereassumedtobesmall.BasedontherobustnessdemonstratedintheLyapunovanalysis,theimplicationtostabilitywasassumedtobenegligible.Thehalteremodelconsistedofafullnon-linearrepresentationofbothhalteresattachedtothebodyoftheysuchthatthesetbackofthe 104

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Figure 5-6 demonstratesthedierencebetweentheactualandmeasuredbodyratesduringtheperiodbetween2to4seconds.Thistimeperiodcapturessomeofthemostdynamicightactivityofthesimulatedinsect.Themeasuredbodyratesarefoundbytakingtheoutputofbothhalteresand,afterdecodinginaccordancewith( 4{6 )and( 4{7 ),combiningtheminaccordancewith( 4{3 )through( 4{5 ).Thesemeasuredquantitiesarescaledusingapproximatecalibrationconstantsfoundbyforcingthesimulatedinsecttomovewithknownangularrates.Theinaccuracyinthisprocesscanbethoughtofasuncertaintyinthefeedbackcontrolgains.Thenon-observabilityofsmallscaleoscillationsdirectlyfollowsfromthediscretesamplingassociatedwithplacementoftheapproximately100sensillainthedF2eldonthehalteres.Figure 5-7 demonstratesthequantizationeectsandresultingmotionduringtheinitialpitchtransientintheight.LessthancriticaldampingatthelowerratesallowsfortheoscillationattheattitudeloopnaturalfrequencyobservedinFigure 5-7 Thehaltereseectivelyprovideaninnerratestabilizationloop,dampingthedynamicsassociatedwithtrackingthenavigationandend-gametargethomingsolution.Theactivedampingassociatedwiththehalteresissupplementedbypassiveaerodynamicdampingproportionaltothesquareoftheangularrate.Inthesimulationthesedampingcomponentswerecalibratedtoequallycontributetoadampingtorque,attheexperimentallydeterminedmaximumangularvelocity,thatequaledthemaximumtorqueachievablebythewings.Thisequivalenceatthemaximumangularraterequiresthat,duetoitsquadraticnature,thepassivedampingcomponentprovidesanincreasinglowerproportionofthedampingforceatlowerangularvelocities.Figure 5-8 demonstratestherelativeproportionsofthedampingforces. 105

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Angularvelocitycomponenttimehistorywithrespecttothebodyroll,pitch,andyawaxes(x,y,z).Thesearecommonlyreferredtoasp,q,andr,respectively.Thecurvesderivedfromhalteremeasurementsarearticiallyosetby10rad/s.Thetruevaluesappearmorecontinuous,whilethevaluesderivedfromhalteremeasurementsshowartifactsofsensillaquantization,cross-coupling,andhalteredynamiclimitations. 106

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Expandedviewoftruepitchrateandmodeledhaltereestimateofthepitchrate.Theinitialtransientoccursduetoastepchangeinstateerroratthebeginningofthesimulation.Theexpandedplotontherightdemonstrateshaltereerrorsduetobothdynamicsandencoding.Limitedmeasurementresolutionresultsinunobservableratedynamics. Figure5-8. Therelationshipbetweenstabilizingtorqueandangularvelocity.Bydesign,thecombinationofpassiveandactivedampingtorquesequalsthemaximumachievablewingtorqueat2000deg/s,constrainingthevehicletoremainbelowthatlevel. 107

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Comparisonofightpathswithandwithouthalterefeedback.Insetplotsshowthealtitude.Theplotontopshowsthetrajectorywithonlypassiveaerodynamicdamping.Theplotonthebottomshowsthetrajectorywithbothactiveandpassiveratestabilization. Atestcasewasexecutedwithonlytheassumedlevelofaerodynamicdampingtodemonstratethesignicanceoflossofactiveratefeedback.Thecomparison,withandwithouthalterefeedback,isshowninFigure 5-9 .Figure 5-10 representsthetranslationalandattitudetimehistory,bothwithandwithoutratedamping.Theresultsdemonstratebothamarkedlydierenttrajectoryresultingfromthetimingofthenon-linearobstructionavoidanceresponsesandthe"noisier"attitudedynamicsresultingfromanunderdampedresponseoftheattitudeloop. 108

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ComparisonofEulerangleswithout(left)andwith(right)halterefeedback.Halteredampingis50%ofcritical.Aero-dampingis50%ofcriticalat2000deg/s. wasimplemented.Sincethedesired,averagedeectofaerodynamicforcesisassumedwithoutadetailedmodeloftime-varyingaerodynamicforce,theabilitytocaptureallthesalientfeaturesofobstructionavoidanceislimited.The6DOFsimulationmodelsaerodynamicdrag,butdoesnotaccountfortheactualgeometryofthey.Thebasicbank-to-turnmodelrepresentationcommandsthey'srollorientationinproportiontomagnitudeoftheheadingerror,includingtheinuenceofobstructionavoidance.Toapproximatetheresultingeectofaerodynamicplanformalongwithactivereorientationtorques,acombinationofpitchandyawtorquesinthey'sbodyframeareapplied.Thesepitchandyawtorquesarecalculatedbasedontherollanglesothatthenettorquecausesareorientationinthehorizontalplaneiftherollangleisestimatedaccurately.Astheheadingerrordecreases,thecommandedrolldecreasesand,likewise,thereorientationtorquedecreases.Figures 5-11 and 5-12 demonstratetheobstructionavoidancemaneuverwithtwodierentassumptionforrollestimation.InFigure 5-11 theassumptionisthatthereorientationtorqueisintrinsicallyderivedasifitwereareexiveresponsebasedonthedesiredrollorientation.Thismodehasthepotentialtointroducelargeout-of-planemotionsduetolargeerrorsinrollorientation.Inthesecondgure,Figure 5-12 ,the 109

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ForcomparisontheworkofSchilstraandVanHaterenwasusedasshowninFigure 5-13 [ 53 ].Thisworkshowsaveragesforalargenumberofobservationsandincludesattitudedenedintermsofayaw-pitch-rollinertialtobodytransformationandangularvelocityandaccelerationdenedwithrespecttotheybodysystem. 65 ]withtheadditionalgeneralizationofipsilateralversuscontralateralfeedback.ThesespecicderivationsonlycoveredacoupleofthepossiblecontrolmechanismsdiscussedbyTaylor.Forexample,thecaseofusingabdominalmotiontoshiftthecenterofgravityinordertoaectpitchingmomentwasnotdirectlydiscussed.Observationsindicatethatthepredominantmechanismsmayvarymarkedlyfromspeciestospeciesandevenbetweenightregimesforagivenspecies[ 65 ].Thegeneralconclusionoftheanalysisprovidedshouldextendtoothermeansofgeneratingtorques.Thatis,thecontrolcanbeimplementedsothatthenetrotationalresponseofthebodyisthesamewhethersymmetricaltorquesaregeneratedbasedonbilateralcombinationofthehalterefeedback,orwhethercontrolresponseisbasedpurelyonipsilateralfeedback. Observationthattimingofthestrokereversalisamechanismofcontrolappearstobesupportedbycomparisonofresultsshownin( 5{45 )-( 5{50 )and( 5{64 )-( 5{69 ).Theresultbasedonwingabductiontostabilizethebody,( 5{45 )-( 5{50 ),requiresboththedesiredforceinthez-directionandthepitchrateasinputstoinuencethepitchcontrolmoment.Ifastabilizingpitchmomentcanbegeneratedthroughcontrolofstrokereversal,whilemaintainingtheliftforceroughlythroughthecenterofgravity,thenthesixcontrol 110

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Obstructionavoidanceresponsebasedondesiredrollangle.ThismethodofdeterminingbodytorquesleadstosignicantresponseofthepitchEulerangle.Theplotofaltitudealsoshowssignicantdeviationsfromlevelight. 111

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Obstructionavoidanceresponsebasedonactualrollangle.ThismethodofdeterminingbodytorquesleadstolowerdeviationofthepitchEuleranglefromnominal.Theplotofaltitudeshowsminimalvariationfromlevelight. 112

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MeasuredsaccaderesultsfromSchilstraandVanHateren,1999.Averagesofsaccadesfrom10iesshowingayawchange(yaw,pitch,rollEulerangles)totheleft,withamagnitudeof10-20(1217saccades,A,D,G),30-40(946saccades,B,E,H)and60-90(677saccades,C,F,I). 113

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5{64 )-( 5{69 ).Thiscouplingevidentin( 5{45 )-( 5{50 )isadirectconsequenceoftheuseofabductionangleasacontrolparameterandisevenevidentinthecasewherecontralateralinuenceisallowedtoensurecontrolsymmetryofcontrolforcemagnitudes.Ontheotherhand,theipsilateralcases,wherethereisalackofknowledgeofcontrolresponseontheotherside,leadstoerrorsinnetliftforceduetotheindependenceofthecontrolforces.Ideally,contralateralinuencewouldbeallowedtoensurecontrolforcesymmetryandamomentwouldbegeneratedduringstrokereversaltocontrolpitchmotion.Inthisway,allcontrolresponsesarepurelydecoupledand,duetothesymmetryofthecontrolforces,thedesiredverticalandhorizontalforcesonthebodyaremaintained. Theabovediscussionassumestheabilitytogeneratetherequiredcontrolforceswhencommanded,withouterrororcoupling.Thisisgenerallyunrealistic,especiallyintheviscous,timevaryingoweldassociatedwithappingwings.Ifthemusclesofthewingimpartaforcetoimpartanetpronation,onewouldexpectbotharesultingforceintheplaneofthestrokeandsomeresultingchangeintheout-of-planethrustbeinggeneratedbythewing.Directquanticationoftheseeectscanbeverydicultduetounsteadynatureoftheoweldandtheaero-elasticityofthewings.Theimplicationtoightstabilityofunavoidablecross-couplingintorquegenerationmechanismswasaddressedthroughaLyapunovstabilityanalysis[ 71 ].Theintentwastoinfera\rule-of-thumb"forwhatlevelofcross-couplingcanexistwhilestillguaranteeingstability.ALyapunovanalysisdoesnotingeneraltellwherethesystemisunstable,butinsteadcharacterizesandboundsaregionofstability.Thecaseforcontrolproportionaltothemeasuredrates,withcrosscouplingintotheotheraxes,showedthat,foranassumedrigidbody,thesystemwillbeasymptoticallystableaslongastheratioofthemaximumcrosscouplingtermtotheminimumcontrolgainislessthan0.5.Theadditionofviscousrotationaldragwasshowntoenhancestability.TheReynold'snumber,whichistheratioofinertial 114

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The6DOFsimulationcontainedbothbehavioralfeaturesandbasicfeedbackcontrollogic.Behaviorally,thesimulationusednon-linearactivationfunctionstomakedecisionswithregardtowhetherornottopursueatarget,whatmodetouseinmovingtowardthetarget,andwhethertoreacttoobstructions.Theobstructionavoidancereactionsinparticularallowedforinvestigationofsaccadicbehavior.Oneofthefundamentalobjectivesofthesimulationwastodeterminetheabilityofthehaltereswiththeassumedencodinganddecodingschemetoprovideadequatecontrolstability.Thestabilityanalysispreviouslydiscussedassumedperfectfeedbacksignals.Withnoiseandotherdisturbances,astablebutboundedperformancewouldbeexpected. ThegeneralcaseshowninFigure 5-3 providesnumerousobstaclestotraverse.Thesimulatedinsectsuccessfullynavigatesthroughtheobstacleswhetherwithfeedbackofperfectmeasurementsorwithfeedbackofhalteremeasurementsencodedbythemodeledsensilla.Thestabilityisboundedwiththehalteremeasurements,i.e.thebodyratesdonotconvergeasdesiredtozero.ThiscanreadilybeseeninFigure 5-7 wherethesolutionisstablebutitoscillateswithinaboundedregiondeterminedbythequantizationassociatedwithsensillasampling.Theerrorsfromthehaltereswereseentobelessthan10%ingeneral(Figure 5-6 ),however,duringlowleveldisturbances,asseenatthebeginningofFigure 5-7 ,theerrorsweremuchlargerandmoreerratic.Thisisexpectedduetotheupstroke/downstrokesummationanddierencinggoingonasrepresentedby( 4{6 and 4{7 ).Inaddition,thereconstructionofthebodyratescombinestheerrorfromthetwohalteres.Theseerrorsmaycanceloradd,dependingontheratecomponentbeingconstructedandthesignsoftheerrors.HengstenburgobservedthatinCalliphorathatthehaltereshavelittleinuencebelow50deg/sec,and,therefore,largedriftswouldbeexpectedifthehaltereswereusedtomaintainorientation[ 3 ].Similarly,Hengstenburgobservedthatthevisionsystemhaslimitedcapabilityathigherfrequencies.Optic 115

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Behavioralresponsesaretightlytiedtonon-linearactivation[ 72 ].Saccadeshavebeencharacterizedasobstructionavoidancemaneuversinies[ 66 ].InFigure 5-13 ,experimentalobservationsofsaccadesaredescribedbyanEulerangletransformationsequence.Denedinyaw-pitch-rollorder,thesaccadeinvolvesayawingmotionawayfromtheobstruction,accompaniedbyasimultaneousslightpitchupanddorsalrolltowardthecenteroftheturn.CaremustbetakenininterpretingthedatasincethedenedorderoftheEulerangletransformationstronglyinuencesthedescription.Afterthemaneuversettlesout,theendresultisavelocityvectoralonganewheading.Acharacteristicseeninthemeasureddataistheasymmetryintherollresponse.Therollresponsedecaysbacktozeromuchmoreslowlythantheinitialresponse.ThisisparticularlyevidentinFigure 5-13 cforlargesaccadeangles.Thesimulatedavoidancebehaviordidnotdemonstratethisasymmetry.Twopossibleexplanationsareprovidedforthisdierence.Therstdealswiththeapproximationofaerodynamicforcesinthesimulation.Inarealbank-to-turnmaneuver,aerodynamicforceswouldbemaximumduringthebeginningoftherollmaneuverwhenheadingerrorisatamaximum.Astheycomesaround,roll,whichisassumedtobedriveninproportiontoheadingerror,willdecreaseandtheywillgraduallyassumeanominalorientationatthenewheading.Inthesimulation,theyawtorquearoundtheinertialz-axisisappliedinproportiontotherollangleindependentofanyestimationoftheactualaerodynamicforces.Thisisstatedwithonecaveat,rotationaldragisincludedinthesimulation.Theselimitationsinhowaerodynamicforcesaremodeledarepossiblyafactorinthediscrepancy.Asecondpossibleexplanationwouldbethattherollresponseoftheymaybereexivereactiontoanactivationresponsethatrisesmuchmorerapidlythanitdecays.Inthesimulation,therollresponseisdrivenuniformlybytheerrorinheadingonceactivated.Eitherofthese 116

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5 ].Inthesimulation,eliminationofthehalteres,asshowninFigure 5-10 ,wasnotcatastrophic.Thisimpliesthattheaerodynamicdragwasperhapsoverestimatedinthesimulationandthehalteresmayhaveanevenmoresignicantinuencethanthechosencontrolgainsallowed.However,evenasmodeled,itisclearthatthehalteres,withtheirlimiteddynamicresponse,coarsequantization,andencodinglimitationsprovideasubstantialimprovementinightperformance. Thesimulationasimplementedhadnumerouslimitations.Oneofthemostglaringlimitationswasthelackoftrueaerodynamicmodels.Becauseofthis,controlschemesforappingwingscouldnotbedirectlyinvestigated.Neithercouldtheformoffeedback,ipsilateralversuscontralateral.TheLyapunovstabilityanalysis,includingderivationsofcontrolschemeswithandwithoutcontralateralinuence,wasmeanttoprovideadditionalcondenceinthesimulationresults.Fruitfulareasforfutureinvestigationincludetheadditionofaerodynamicmodelsandtheadditionofhigherdelityopticalsensingmodels.Theuseofcylindricalobstructionswasacompromisedictatedbythelackofacompoundeyemodel.Performancedemonstrationinanenvironmentwithmoregeneralobstructionsisdesiredandwillbepursuedinthefuture. 117

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Theworkembodiedinthisdissertationwasintendedtoestablishaworkingtheoryforhalterefunctionality.Inlinewiththisobjective,threemajortaskshavebeencompleted: 1. Themathematicalbasisforhalteredynamicresponsehasbeendocumented,anovelmechanismforratedecouplinghasbeenproposed,andperformancelimitationshavebeenanalyzed. 2. Amodelforthehaltere,compatiblewiththeinsectsensoryphysiology,hasbeendemonstrated,providingforthersttimeaplausibletheoryforhowangularratesareencodedbycampaniformsensilla. 3. Abehavioralightsimulationhasbeenconstructedthatdemonstratestheabilityofhalterestostabilizeightinthepresenceofdisturbancesandnon-linearlyactivatedbehavioralresponse. Atthebeginningofthisresearch,theliteratureconsistedofonlybasickinematicanalysisofthehaltereandobservationofthehaltere'sinuence.Understandingofthephysicalandmathematicalbasisbehindtheabilityofthehalteretoencodeanddecoupletheangularratesitrespondstowasverylimitedintheliterature.Priorkinematicanalysis,performedprimarilybyPringleandNalbach,wassucienttodeterminethebasicinertialforcesthatactonthehaltereasitoscillatesbackandforth[ 14 ],[ 23 ].Thatlevelofanalysisislimitedinitsabilitytoaddressthesensorymechanismsofthehalterewhicharerelatedtohalteredeectionand,therefore,directlytiedtotheinuenceofdampingandstinessinthehalterestructure.Additionaldecienciesintheliteraturewerealackoftreatmentofdecouplingmechanismsfortheforcesproportionaltothebodyratesandalackofanysubstantialtheoryforhowtheratecomponentsareencodedinthecampaniformsensilla.Theworkdescribedinthisdissertationaddressesthesedeciencies.Inaddition,itprovidesanalysiswithregardtoinsectightstabilityingeneralandtheimplicationsofipsilateralversusbilateralfeedbackintotheightcontrolsystem.Further,aproposedmodelforthehalterewithdynamiccrosscouplingerrorsandsensilla 118

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Chapter3performsafulldynamicassessmentofthehalterewithvariationsinstinessanddamping.Themodelofthehalterewasgeneralbydesign,representingthestructureinnon-dimensionaltermstoavoidanyconclusionstiedtoaspeciccongurationorspecies.Higherlevelphysicalunderstandingwasbeingsoughttoallowforafundamentaltheoryofthesensorymechanismsinvolved.Theresultsofthedynamicsimulationsledtothetheorythatbymeasuringtheaveragedslope,andtheaveragedmagnitudeofthehalteretrajectory,thetwoobservableratecomponentsintheplaneofhaltereoscillationcouldbedecoupled.Analternativearticulationofthisassertionisthat,relativetothestrokecenter,asymmetricresponseofthehaltereiscausedbytheyawrateandananti-symmetricresponseiscausedbythelateralratecomponent.Foraproportionalrelationshipbetweentheratecomponentsandthedescribedfeaturestoholdovertherangeofhaltereoutput,thegoverningequationsmustbeapproximatelylinear.Thislinearitywasdemonstratedmathematicallyandthroughparametricanalysisoverabroadrangeofbodyrates. Chapter4tookthemodelsdescribedinChapter3andaddedtothemamodelofthebedofcampaniformsensillathatexperimentalbiologistshavecorrelatedwithrateencodingandstimulationofwingcontrolmuscles.Thesesensillarespondinabinary,unidirectionalmannerandaredistributedoverthebaseofthehalteresoastofractionatethedynamicrangeofoutputusingthe100orsosensilla.Atheorywasproposedandsuccessfullydemonstratedusingmodelsandsimulationstoshowthattheseverysimplesensillacaninfactencodeboththesignandmagnitudeofthetworatecomponents.Inanamazingtwist,itistheverycharacteristicthatwewouldtrytoavoidasalimitation,theunidirectionalsensitivityofthesensilla,thatmakesthemuniquelysuitedforcapturingthesymmetryandasymmetry,andassociatedsignsoftheratecomponents.Natureis,bythistheory,recordingthesignsoftheratecomponentsusingthephasingoftheresponse 119

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Chapter5addressestheopen-looplimitationdescribedforChapter4.ThemodelsproposedinChapter3and4forhalteredynamicsandrateencodingwereincorporatedina6degree-of-freedom(6DOF)simulationofaDipteraninsect.Thissimulationwaswrittenspecicallyforthisresearchandallowsforcompetingbehavioralobjectivesstimulatedbynon-linearactivationfunctions.Thegoalwastodemonstratetheabilityoftheproposedhalteremodeltostabilizeightinthepresenceofdynamicobstructionavoidancereactions.Ifthemodelwerefunctionallyadequate,thesimulatedinsectwouldnavigatethroughacomplexenvironmentwithrandomlydistributedobstructions,usingmaneuversrepresentativeofthehighlydynamicsaccadesofies,andthenacquireandinterceptthedesiredtarget.Inordertofacilitatesaccade-likebehavior,abank-to-turnresponsewasprogrammedforlargeheadingerrors.Thesimulationperformedwell,demonstratingthatthehalteresasmodeled,withlimitedresolutionanderrorsassociatedwithcross-couplinganddynamicresponse,weresucienttoprovideratestabilizingfeedbackforthesimulatedinsect. Thesimulationdescribedhasanumberoflimitationsassociatedwithforcegeneration.Stroke-averagedforcesonthebodyareassumedtobegeneratedbythewingswithouterror.Aerodynamicsimulationofthecomplexunsteadyowassociatedwiththewingswasoutsidethescopeofthiseort.Becauseofthisassumption,aspecicmechanism 120

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Thesignicanceofthisworktoengineeringdesignisnotclear.BuildingahalterebasedinertialmeasurementunitisprobablynotadvantageousgiventhecurrentavailabilityoftwoaxisrategyrosonachipforapproximatelythirtyveUSdollars.Wemayhoweverlearnmoresubtlelessonsfromthewaynaturehasintegrateditssolution.Forexample,1)extremelysimplestrainsensors,highlyintegratedintothestructure,canprovidefeedbackoverawidedynamicrangebydistributingthemappropriately,aprocessreferredtoinbiologyasrangefractionation.2)Astructureoriginallydesignedforforcegenerationmaywithadaptationbeusedforratesensing.Thetworearwingsofieshaveadaptedforratesensitivity.Withafewstrainsensors,arotatingmotoronamicro-airvehiclemaybeabletoprovidebenecialratefeedbacktocreateastabilizedsensingplatform.3)Ratesensorsdonotneedtobesituatedonorthogonalaxes.Theplanesofhaltereoscillationareorientedapproximately120degreesapart,not90.Duetothesymmetryofthegeometry,thedierenceissimplyaconstantfactorfromthecontrolsystemperspective.4)Theuseandadvantageofsymmetryindesignsisunderutilized.Thesymmetryweseeeverywhereinnaturehasbeenmaintainedforreasonsassociatedwithcontroldesign.Thesymmetryofthehalteredynamicresponsecanbeusedsothatasingleoscillatingnon-lineardevicecanmeasuretwoindependentratecomponents. 121

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Thehalteresappeartobeonlyonemeansofmeasuringinertialmotionininsects.Dipteraninsectshavehalteres.Dragoniesappeartousetheirheadasasensitiveproofmasstosenseangularacceleration.Mothsutilizeantennavibrationstoprovideratefeedbackinawaythatisnotcompletelyunderstood[ 73 ].Aswecontinuetobuildsmallermoretightlyintegratedyingvehicles,therearemanylessonswecanlearnfromthevarieddesignsofnature.However,wemustbecarefultoassessthenaturalsolutionswechoose.Bothnatureandmanneedrobustsolutionsthatcandealwiththecomplexitiesanduncertaintiesofourrespectivemissions,butourdenitionsofsuccessarequitedierent.Innatureitissurvivalofthepopulationthatisimportantandthesuccessoftheindividualisoflittleconsequence.Inman'sdesignswestriveforveryhighperformanceoftheindividualforreasonsofcostandconsequence.Thedierencethattheserequirementsdictateintermsofrepeatabilityandpredictabilityislikelyquitesignicant. 122

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Theexpressionin( 3{12 )canbedeterminedbydeningtworeferenceframesinadditiontothebodyxedframe.Theseframesarerelatedbythestrokeangleandtheout-of-planedisplacementangle,asshowninFigure A-1 .Whentheseanglesarezero,thethreeframesareco-aligned.Theassociatedangularvelocitiesare Thepositionandvelocities,asobservedinthevariousreferenceframes,ofthemassattheendofthehaltere(Point2)are Theexpressionsleadingtotheaccelerationofthehaltererelativetotheinertialframeare Theexpressionin( A{10 )assumesthattheaccelerationofthebody(Point1)issmallrelativetotherelevanthaltereaccelerationterms.Thisresultsintheaccelerationofpoint 123

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Therelativeorientationofthereferenceframesassociatedwiththeequationofmotionderivationareshownabove.The"h"frameisrotatedbyanglewithrespectthe"b"frame.The"f"frameisrotatedbyanglewithrespecttothe"h"frame. 2withrespecttotheearth(inertial)frameinthe^f2directionas ^f2e~a2=r[_3sin()_1cos()_2cos()sin() (A{11) +2_[(3cos()+1sin())cos2()2cos()sin()]+(23cos2()+21sin2()22)cos()sin()+(23cos()+12sin())cos(2)+213cos()sin()cos()sin()]: 3{12 )isobtainedbytakingthedotproductoftheinertialforce,(me~a2),inthedirectionoftheout-of-planedeection(^f2)andthenaddingthe 124

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^f2(~Finertial+~Fdamping+~Fstiffness+~Fother)=0:(A{12) Inthisexpression,thedampingandstinessforcesareassumedtohavecomponentsinthedirectionof^f2asdescribedbelow.~Fotherdescribesanyotherexternalforcesthatoperateonthemassandisassumedtohaveanegligiblecomponentinthedirectionof^f2. Sincerincreasesinthenegative^f2direction,thestinessanddampingforcesweredenedas ^f2~Fdamping=rm2!n_ ^f2~Fstiffness=rm!2n: Theresultingexpressionwasdividedbytheproductoftheradiusofgyrationandmasstoputitinthenalnon-dimensionalformas 125

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ThefollowingderivationprovidestheequationsofmotionusedforthesimulationresultsprovidedinChapter5.Thederivationofthequaterniondynamicsintermsofthecurrentquaternionstateandthebodyratesisprovidedbyreference[ 69 ].DenitionsofthesymbolsusedinthederivationareshowninTable B-1 TableB-1. Denitionsofvariablesandsymbolsassociatedwith6DOFequationsofmotion. dt:derivativewithrespecttotheinertialframe[]b:vectorexpressedinthebodyframe:quaternionmultiplication:crossproduct dt=bdi~V dt+i~!bxi~V=bd dt266664uvw377775b+266664pqr377775b266664uvw377775b;idi~V dt=266664_u_v_w377775b+266664qwrvrupwpvqu377775b: 126

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dt=g W~F+g^k=g W266664FbxFbyFbz377775b+g1^k=g W266664FbxFbyFbz377775b+g266666664e0exeyez3777777752666666640001377777775266666664e0exeyez377777775idi~V dt=g W266664FbxFbyFbz377775b+g2666642(ezexeye0)2(e0ex+eyez)e20+e2ze2xe2y377775b Equatingtheseresultsgives W266664FbxFbyFbz377775+g2666642(ezexeye0)2(e0ex+eyez)e20+e2ze2xe2y377775+266664rvqwpwruqupv377775:(B{3) Thepositionintheinertialframeisfoundthroughintegrationofthevelocityaftertransformationintotheinertialframe, 127

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(B{5) dt([Ib]i~!b)=[Ib]bdi~!b WhereIb=266664IxxIxyIxzIxyIyyIyzIxzIyzIzz377775b: [Ib]bdi~!b 128

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2266666664exeyeze0ezeyeze0exeyexe0377777775266664pqr377775b:(B{9) See[ 69 ]foraderivationof( B{9 ).Equations( B{3 ),( B{4 ),( B{8 ),and( B{9 )makeupthirteenequationsthatwhenintegrateddenelinearandangularpositionandvelocitystatesofthebodyintermsofaquaterniontransformation. 129

PAGE 130

5-3 andFigure 5-4 foradenitionoftermsusedinthefollowingderivations.Thetorquesfromthewingsarethecrossproductofthepositionofthecenterofpressure,relativetothecenterofgravity,andthewingforce.Therefore,~TR=~RR~FR~TL=~RL~FL (C{1) (FRy(dLsin(0+R))FRx(w+Lcos(0+R)))^{z~TL=FLz(w+Lcos(0+L))^{xFLz(dLsin(0+L))^{y+ (C{2) 130

PAGE 131

C{1 )and( C{2 ),expressedintermsoftheabductiondeviationRandthetwostroke-averagedforcesFRxandFRyas (C{3) +(FLx(w+Lcos(0+L))FRx(w+Lcos(0+R)))Fx=FLx+FRxFy=FRy+FLy=0Fz=FRz+FLz:

PAGE 132

C{3 )gives (C{4) +(FLxFRx)(w+Lcos(0))+Lsin(0)(FRxRFLxL)Fx=FLx+FRxFy=FRy+FLy=0Fz=FRz+FLz: C{4 )gives 132

PAGE 133

(C{5) +Lsin(0)(Fxd C{6 )andthewingcontrolforcescanbefoundfromtheremainingtwoexpressionsin( C{6 )asKyq FzdLcos(0) 133

PAGE 134

C{3 )fromthepreviousderivation,Tx=FRz(w+Lcos(0+R))FLz(w+Lcos(0+L))Ty=(FRz(dLsin(0+R))+FLz(dLsin(0+L)))Tz=(FRy(dLsin(0+R))+FLy(dLsin(0+L)))+(FLx(w+Lcos(0+L))FRx(w+Lcos(0+R)))Fx=FLx+FRxFy=FRy+FLy=0 134

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135

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(C{7) FRx=Kz1 (C{8) FRz=Kxb3 C{7 )allowsforthesolutionofthedesiredabductionangles.Smallanglesareassumedfortheabductionangles,allowingthe 136

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C{3 )fromSection C.1 ,thesumoftheforcesandtorquesarerepresentedasTx=FRz(w+Lcos(0+R))FLz(w+Lcos(0+L))Ty=(FRz(dLsin(0+R))+FLz(dLsin(0+L)))Tz=(FRy(dLsin(0+R))+FLy(dLsin(0+L)))+(FLx(w+Lcos(0+L))FRx(w+Lcos(0+R)))Fx=FLx+FRxFy=FRy+FLy=0Fz=FRz+FLz:

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140

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142

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U.Thurm,A.Stedtler,andR.Foelix,\ReizwirksameVerformungenderTerminalstruktureneinesMechanorezeptors,"VerhDtschZoolGes,vol.67,pp.37{41,1974. [33] A.Stedtler,\DieReprasentationvonSchwingungenderFliegenhatlereindenReaktionenihrerMechanorezeptor-Felder,"Staatsexamens-Arbeit,UniversitatBochum,1974. [34] S.N.ZillandD.T.Moran,\Theexoskeletonandinsectproprioception.I.Responsesoftibialcampaniformsensillatoexternalandmuscle-generatedforcesintheamericancockroach,Periplanetaamericana,"JExpBiol,vol.91,pp.1{24,1981. [35] M.Dickinson,\Comparisonofencodingpropertiesofcampaniformsensillaontheywing,"JExpBiol,vol.151,pp.245{261,1990. [36] R.C.Elson,\Flightmotorneuronereexesdrivenbystrain-sensitivewingmechanoreceptorsinthelocust,"JCompPhysiolA,vol.161,pp.747{760,1987. [37] ||,\Interneuronalprocessingofinputsfromthecampaniformsensillaofthelocusthindwing,"JCompPhysiolA,vol.161,pp.761{776,1987. [38] A.MielkeandG.Heide,\EectsofarticiallygeneratedhalterenerveaerencesontheactivationoftheightsteeringmusclesinCalliphora.gene-brain-behavior,"inProceedingsofteh21stGottingenNeurobiologyConference,N.ElsnerandM.Heisenberg,Eds.Stuttgart:Thieme,1993,p.207. [39] M.Egelhaaf,\Visualaerencestoightsteeringmusclescontrollingoptomotorresponsesofthey,"JCompPhysiolA,vol.165,pp.719{730,1989. [40] G.HeideandK.G.Gotz,\OptomotorcontrolofcourseandaltitudeinDrosophilaisachievedbyatleastthreepairsofightsteeringmuscles,"JExpBiol,vol.199,pp.1711{1726,1996. [41] F.S.J.Hollick,\TheightofthedipterousyMuscinastabulans(Fallen),"PhilosTRoySocB,vol.230,pp.357{390,Nov.1940. [42] M.Dickinson,\Haltere-mediatedequilibriumreexesofthefruity,Drosophilamelanogaster,"PhilosTRoySocB,vol.354,pp.903{916,May1999. [43] C.N.BalintandM.H.Dickinson,\ThecorrelationbetweenwingkinematicsandsteeringmuscleactivityintheblowyCalliphoravicina,"JExpBiol,vol.204,pp.4213{4226,2001. [44] ||,\Neuromuscularcontrolofaerodynamicforcesandmomentsintheblowy,Calliphoravicina,"JExpBiol,vol.207,pp.3813{3838,2004. [45] A.R.Ennos,\ThekinematicsandaerodynamicsofthefreeightofsomeDiptera,"JExpBiol,vol.142,pp.49{85,1989. 145

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RhoeThompsonisanAerospaceEngineeremployedbytheAirForceResearchLaboratory(AFRL)atEglinAirForceBase.HereceivedhisBSandMEdegreesattheUniversityofFloridain1986and1992,respectively.InOctober1986hebeganworkingintheAirForceArmamentLaboratoryonmissiledefensetechnologydevelopmentassociatedwithguidanceandcontrolofkineticenergyinterceptors.Generalactivities,whilestillrelatedtomissiledefense,changedintheearly1990stobecomefocusedongroundtestingofinfraredguidancesystemsinhardware-in-the-loopenvironments.In2001,heassumedtheroleofChiefEngineeroftheKineticKillVehicleHardware-In-TheLoopSimulation(KHILS)facility,aninternationallyrenownedfacilityfortestingweaponsanddevelopmentoftesttechnologies.AftercompletingstudiesassociatedwiththedegreeofDoctorofPhilosophyinAerospaceEngineeringinAugust2009,heassumedtheresponsibilitiesofTeamLeaderforagroupofAFRLengineersstudyingbiologicalsystemsforapplicationtoinnovativesensingandcontrolsystemsassociatedwithautonomousairbornesystems. 148