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High Angle-of-Attack Flight Characteristics of a Small UAV with a Variable-Size Vertical Tail

Permanent Link: http://ufdc.ufl.edu/UFE0024449/00001

Material Information

Title: High Angle-of-Attack Flight Characteristics of a Small UAV with a Variable-Size Vertical Tail
Physical Description: 1 online resource (69 p.)
Language: english
Creator: Johnson, Baron
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: 3d, aircraft, alpha, angle, attack, baron, harrier, high, inverted, lind, mav, of, rc, rock, showtime, tail, uav, upright, vertical, wing
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The angle-of-attack parameter has a significant influence on the aerodynamics, flight dynamics, and handling qualities of aircraft. Flight at high angle-of-attack conditions enables many missions; however, the flight dynamics are challenging to model and are largely influenced by uncommanded and sometimes unpredictable motions. This study investigates the flight dynamics of a small UAV that is piloted in open air at high angle-of-attack conditions well beyond wing stall. Models are estimated from the flight data to indicate some characteristics of the flight dynamics. The lateral dynamics are linear and dominated by a traditional mode of roll convergence, while the longitudinal and directional dynamics exhibit nonlinearities and require high-order terms. The models, which are based upon responses to doublet perturbations, are used to predict steady-state high angle-of-attack flight with significantly smaller control inputs. Uncommanded oscillations are identified as the motions not predicted by the doublet-based models. Uncommanded oscillations about all three axes are observed with the most notable being about the roll axis, commonly called wing rock. This wing rock behavior of a small UAV with different vertical tail sizes and configurations is studied using time, frequency, and time-frequency analysis techniques. Wing rock is found to be a fairly narrow-band phenomenon, but with frequency variations in time. The wing rock behavior is found to be independent of vertical tail size but largely dependent on the vertical tail configuration; specifically, wing rock is pronounced with any size upright vertical tail but virtually non-existent with any size inverted vertical tail.
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 Baron Johnson.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024449:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024449/00001

Material Information

Title: High Angle-of-Attack Flight Characteristics of a Small UAV with a Variable-Size Vertical Tail
Physical Description: 1 online resource (69 p.)
Language: english
Creator: Johnson, Baron
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: 3d, aircraft, alpha, angle, attack, baron, harrier, high, inverted, lind, mav, of, rc, rock, showtime, tail, uav, upright, vertical, wing
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: The angle-of-attack parameter has a significant influence on the aerodynamics, flight dynamics, and handling qualities of aircraft. Flight at high angle-of-attack conditions enables many missions; however, the flight dynamics are challenging to model and are largely influenced by uncommanded and sometimes unpredictable motions. This study investigates the flight dynamics of a small UAV that is piloted in open air at high angle-of-attack conditions well beyond wing stall. Models are estimated from the flight data to indicate some characteristics of the flight dynamics. The lateral dynamics are linear and dominated by a traditional mode of roll convergence, while the longitudinal and directional dynamics exhibit nonlinearities and require high-order terms. The models, which are based upon responses to doublet perturbations, are used to predict steady-state high angle-of-attack flight with significantly smaller control inputs. Uncommanded oscillations are identified as the motions not predicted by the doublet-based models. Uncommanded oscillations about all three axes are observed with the most notable being about the roll axis, commonly called wing rock. This wing rock behavior of a small UAV with different vertical tail sizes and configurations is studied using time, frequency, and time-frequency analysis techniques. Wing rock is found to be a fairly narrow-band phenomenon, but with frequency variations in time. The wing rock behavior is found to be independent of vertical tail size but largely dependent on the vertical tail configuration; specifically, wing rock is pronounced with any size upright vertical tail but virtually non-existent with any size inverted vertical tail.
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 Baron Johnson.
Thesis: Thesis (M.S.)--University of Florida, 2009.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024449:00001


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IwouldliketoextendthankstoanumberofsupportiveandinspiringpeoplewithwhomI'vehadthehonor,privilege,andpleasuretoworkwithduringmytimeattheUniversityofFlorida.Firstofall,thankstomyadvisor,Dr.RickLind,forprovidingmetheopportunitytostudyunderhimduringgraduateschoolintheFlightControlLab.Hisguidance,suggestions,andcritiqueshaveproveninvaluable.ThankstoDr.PeterIfjuforoeringmethemostexcitingandrewardingjobIcouldhaveeverimaginedasanundergraduateintheMicroAirVehicleLab.AnumberoffellowstudentshavealsosupportedmeinmytimeatUF,eitherdirectlyorindirectly.ScottBowmanhasalwaysprovidedagreatdealofsupportandmadegreatcontributionstomyunderstandingofelectronicsystems,particularlyintheeldofshootingbottlerocketsfromRCplanes.Assistance,inspiration,andconstantamusementhasbeenprovidedbythecurrentmembersoftheFlightControlLab,listedinaveryparticularorder:DongTran,SankethBhat,Daniel'Tex'Grant,RobertLove,BrianRoberts,andRyanHurley.OthercollaboratorswhichhavemadestrongimpressionsuponmeovertheyearsincludeMujahidAbdulrahim,CarloFrancis,FrankBoria,DanClaxton,KyuhoLee,AdamWatts,JosCocquyt,andPatriciaMiller. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1HighAngle-of-AttackFlight .......................... 12 1.2Motivation .................................... 13 1.3PreviousResearch ................................ 13 1.4Contributions .................................. 16 2EXPERIMENTALSETUP ............................. 17 2.1Aircraft ..................................... 17 2.2VerticalTail ................................... 18 2.3Avionics ..................................... 19 3FLIGHTTESTING ................................. 25 4ANALYSISTECHNIQUES ............................. 28 4.1FourierTransform ................................ 28 4.2WaveletTransform ............................... 28 4.3Examples .................................... 30 4.3.1Example1:StationarySineWaveSignal ............... 30 4.3.2Example2:StationarySignalwithMultipleFrequencyComponents 31 4.3.3Example3:ChirpSignalwithIncreasingFrequency ......... 33 5SYSTEMIDENTIFICATION ............................ 36 5.1Procedure .................................... 36 5.2DoubletModeling ................................ 38 5.2.1Longitudinal ............................... 38 5.2.2Lateral .................................. 40 5.2.3Directional ................................ 41 5.3Steady-StateFlight ............................... 43 5.3.1Longitudinal ............................... 43 5.3.2Lateral .................................. 45 5.3.3Directional ................................ 47 5

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.............. 50 6.1UprightTail ................................... 50 6.2InvertedTail ................................... 52 6.3Parameterization ................................ 56 7SUMMARY ...................................... 62 7.1Recommendations ................................ 62 7.2Conclusion .................................... 63 REFERENCES ....................................... 65 BIOGRAPHICALSKETCH ................................ 69 6

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Table page 2-1MiniShowTimeSpecications ............................ 18 2-2MiniShowTimeComponents ............................ 18 2-3VerticalTailSpecications .............................. 19 2-4SizeandMassofAvionics .............................. 20 2-5IMU/FDRWiringSequence ............................. 22 2-6TechnicalSpecicationsofIMUSensors ...................... 23 2-7IMURawOutputMultipliersandResultingUnits ................. 24 7

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Figure page 1-1GraphicalRepresentationofAngle-of-Attack .................... 12 1-2F/A-18HARVandX-29 ............................... 14 1-3RQ-11BRavenSmallUnmannedAircraftSystem ................. 15 2-1MiniShowTime .................................... 17 2-2VerticalTails ..................................... 20 2-3InterchangeableVerticalTailMountedonFuselage ................ 21 2-4AirborneSensors:IMU,GPSReceiver,andFlightDataRecorder ........ 21 2-5AvionicsMountedUnderCanopy .......................... 22 2-6WiringDiagramofMiniShowTimeAircraftforDataCollection ......... 23 3-1MiniShowTimeinRepresentativeHighAngle-of-AttackFlight .......... 26 3-2MiniShowTimeinInvertedHighAngle-of-AttackFlight ............. 27 4-1RepresentativeWaveletWindowSizePattern ................... 29 4-2RepresentativeMorletMotherWavelet ....................... 30 4-3SignalforExample1:10rad/sSineWave ..................... 30 4-4FrequencyAnalysisofExample1Signal ...................... 31 4-53-DimensionalDepictionofExample1WaveletTransform ............ 32 4-6SignalforExample2:Summationof5and10rad/sSineWaves ......... 32 4-7FrequencyAnalysisofExample2Signal ...................... 33 4-8SignalforExample3:SinusoidalChirpwithIncreasingFrequency ........ 34 4-9ChirpFrequencyWithRespectToTime ...................... 34 4-10FrequencyAnalysisofExample3Signal ...................... 35 5-1ElevatorandPitchRateDuringDoublets ..................... 38 5-2IndividualContributionstoResponsefromLongitudinalModel ......... 39 5-3AileronandRollRateDuringDoublets ....................... 40 5-4IndividualContributionstoResponsefromLateralModel ............. 41 8

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....................... 42 5-6IndividualContributionstoResponsefromDirectionalModel .......... 43 5-7ElevatorandPitchRateDuringSteadyFlight ................... 44 5-8UncommandedPitchRate .............................. 44 5-9FFTofMeasuredPitchRate,UncommandedPitchRate,andElevatorInput .. 45 5-10AileronandRollRateDuringSteadyFlight .................... 46 5-11UncommandedRollRate ............................... 46 5-12FFTofMeasuredRollRate,UncommandedRollRate,andAileronInput ... 47 5-13RudderandYawRateDuringSteadyFlight .................... 47 5-14UncommandedYawRate .............................. 48 5-15FFTofMeasuredYawRate,UncommandedYawRate,andRudderInput ... 49 6-1TimeResponsesinUprightConguration ..................... 51 6-2FFTinUprightCongurationofRollRateandAileronDeection ........ 52 6-3WaveletTransformsofRollRateinUprightConguration ............ 53 6-4WaveletTransformsofAileronDeectioninUprightConguration ....... 54 6-5TimeResponsesinInvertedConguration ..................... 55 6-6FFTinInvertedCongurationofRollRateandAileronDeection ....... 56 6-7WaveletTransformsofRollRateinInvertedConguration ............ 57 6-8WaveletTransformsofAileronDeectioninInvertedConguration ....... 58 6-9MeanRollRatePeakMagnitudesandPeak-to-PeakFrequencies ......... 59 6-10UpperandLowerBoundsofWingRockFrequencyfromFFTsandWavelets .. 60 6-11MeanWingRockBandwidths ............................ 61 9

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1-1 ,whereangle-of-attackisrepresentedby,pitchangleisrepresentedby,andtheightpathinclinationisrepresentedby. Figure1-1. GraphicalRepresentationofAngle-of-Attack Angle-of-attackcanbedeterminedasafunctionofbody-framevelocities,asindicatedinEquation 1{1 ,wherevbxandvbzarebody-framevelocitiesinthexandzdirectionsrespectively. 12

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1 ).Highangle-of-attackightcouldalsoprovidegreaterobstacleavoidancecapabilitieswhenyinginurbanterrainsincetheturningradiusinhighangle-of-attackightisgreatlyreducedascomparedtothatofconventionalforwardight.Sensingcapabilitiescouldalsobeenhancedthroughtheuseofsensorpointingwithoutthecomplexityandweightofmorphingaircraftorgimbaledsensors( 2 ). 3 ; 4 ).AnovelcongurationfortheX-29usedforward-sweptwingsandacanardtodemonstrateightatthese 13

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5 ).Ineachcase,theightdynamicswerefoundtobechallengingtomodelbecauseoftheinuenceofaerodynamics.Additionalissues,suchaswingrock,werealsonotedinvaryinglevelsatvirtuallyallangle-of-attackconditionsinthepost-stallregime.TheF/A-18HARVandtheX-29canbeseeninFigure 1-2 ( 6 ; 7 ). Figure1-2. A)F/A-18HARV[ http://www.dfrc.nasa.gov/gallery/Photo/F-18HARV/ ,reprintedwithpermission]andB)X-29[ http://www.dfrc.nasa.gov/gallery/Photo/X-29/ ,reprintedwithpermission] TheightdynamicsassociatedwithsmallUAVsarereceivingsignicantattentioninthecommunityasaresultoftheirmissionpotential.Severalsmall,man-portablevehiclesfeatureadeep-stallshortlandingmodeinwhichtheelevatordeectsandthepowerisreducedwhiletheaircraftsteeplydescendsinanear-levelattitudeuntilitimpactstheground( 8 ; 9 ).TheRQ-11BRaven,showninFigure 1-3 ( 10 ),isanexampleofsuchaUAVthatfeaturesadeep-stalllandingmode.TheightdynamicsofsmallUAVshavealsobeenstudiedinhoveringmodestoenableautonomouscontrol( 11 { 13 ).SmallUAVsandremote-controlled(RC)aircraftarerapidlymaturinginightcapabilityforavarietyofmissionsincludingurbanoperations.Assuch,theabilitytooperateathighangle-of-attackconditionsisacriticalrequirementfortheseplatforms.Suchcapabilityisusuallyachievedthankstothecombinationofthrustgenerationandcontrolsurfaces.Theseaircraftoftenutilizefront-mountedpropellersandtractorpropulsionthatproducesairowoverthewingsandtail.Thelargecontrolsurfaces,which 14

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RQ-11BRavenSmallUnmannedAircraftSystem[ http://www.avinc.com/media gallery2.asp?id=224 ,reprintedwithpermission] constitute50%ormoreofthetailareaandmaydeect45ormore,arethusabletousethispropwashtomaintaincontrolauthority.Flightathighangle-of-attackconditionsisoftencharacterizedbywingrock.Thisphenomenonisdescribedasuncommandedself-inducedoscillationsprimarilyabouttherollaxis.Someresearchhasindicatedthat,althoughdominatedbyrollmotions,theuncommandedwingrockmayactuallybealightly-dampeddutchrollmotion( 14 { 16 ).Thesourceoftheuncommandedwingrockisnotcompletelyknownandseemstovarybyaircraftconguration.Someresearchhasledtothebeliefthatthewingrockphenomenonisalimitcycleoscillation(LCO)causedeitherbythelossofdynamicrolldampingathighangles-of-attack( 17 ; 18 )oranaerodynamichysteresiswhichgeneratesthespring-likeforcesrequiredtodrivetheLCO( 19 { 21 ).Thepresenceofsidesliphasbeenshowntohaveaneectonwingrockathighangles-of-attack,bothasacauseandmitigator( 20 ; 22 ; 23 ).Additionally,quiteafewstudieshavefoundtheuncommandedwingrockphenomenontobesomewhatunpredictableinnature,bothinmagnitudeandperiodicity( 22 ; 24 { 27 ).Someresearchhasdeterminedthephenomenontobecausedbyvorticesfromtheleadingedgesofthewings( 14 ; 28 ; 29 ),whileotherresearchhasdeterminedittobecausedlargelyorentirelybyvorticesgeneratedfromslenderforebodiesimpingingupon 15

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26 ; 28 ; 30 { 34 ).Thishasledtoresearchonthewingrockrelationshipwithandwithoutaverticaltailsurface( 35 ). 16

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2-1 ,isacommercially-availableo-the-shelfplatformthatiscommonlyusedbytheremote-control(RC)community. Figure2-1. MiniShowTime Thisaircraftisconstructedfromalightweightbalsawoodstructurethatallowsthewingspantobelargeincomparisonwiththevehicleweight.Thespecicplatformhasaweightofapproximately820galongwiththecharacteristicsgiveninTable 2-1 ( 36 ).TheaircraftwasoutttedwithconventionalRCcomponentsforcontrolandpropulsion,whicharelistedinTable 2-2 Thisaircraftisusedinthecommunityforaerobaticsbecauseofitsexcellentagilityandoutstandingcharacteristicsathighangle-of-attackconditions.Inparticular,theaircraftishighlycontrollableathighangle-of-attackconditionsasaresultoflow 17

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MiniShowTimeSpecications ParameterValue Wingspan1090mmLength1065mmWingArea26.7dm2FlyingWeight820-850gWingLoading30.7-31.8g/dm2FlightSpeed0-20m/s Table2-2. MiniShowTimeComponents ComponentManufacturerModel TransmitterSpektrumDX7ReceiverSpektrumAR6100eServos(4)JRDS285BatteryEliminatorCircuit(BEC)CastleCreationsCCBECElectronicSpeedControl(ESC)E-ite40-AmpBrushless(V2)MotorE-itePark480BLOutrunner,1020KvPropellerAPC12x6EBatteryThunderPower3S,11.1V,2100mAh 18

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2{1 ,whereSvrepresentsverticaltailarea,LvrepresentsthedistancefromtheCGoftheaircrafttotheaerodynamiccenteroftheverticaltail,Swrepresentswingarea,andbrepresentswingspan( 37 ). 2-3 .Therangeoftailsizesprovidesareasandverticaltailvolumecoecientsrangingfromapproximately50%to150%ofthestocktailsize.TheverticaltailsareshowninFigure 2-2 ,andatailmountedonthemodiedfuselageisshowninFigure 2-3 Table2-3. VerticalTailSpecications TailHeight(mm)Area(cm2)Vv 2-4 .Apairofsensorpackagesresultfromaninertialmeasurementunit(IMU)andglobalpositioningsystem(GPS).TheIMU,whichisaMEMSensenIMU,isaMEMS-basedunitwithtemperaturecompensationanddigitalI2Coutputof3-axis 19

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VerticalTails:Drawing(top)andActual(bottom) accelerations,angularrates,andmagneticux.TheGPS,whichisanEagleTreeExpanderModule,noteslocation,groundspeed,course,andUTCtimestampatarateof5Hz.Anadditionalightdatarecorder(FDR),whichistheEagleTreeSystemsFDRPro,logsthesensoroutputsalongwithbarometricaltitudeandservocommands.Thissystemisabletoobtainmorethan15minutesofdataatarateof25Hz.Thesesensorsarerelativelysmall,asnotedinTable 2-4 ( 38 ; 39 ),andineachcasetheweightisnearlynegligibleontheightdynamics. Table2-4. SizeandMassofAvionics UnitSize(mm)Mass(g) IMU46.5x22.9x13.920GPS36.0x43.0x13.023FDR50.0x35.0x17.022 20

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InterchangeableVerticalTailMountedonFuselage Figure2-4. AirborneSensors:IMU(left),GPSReceiver(middle),andFlightDataRecorder(right) 21

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2-5 Figure2-5. AvionicsMountedUnderCanopy:A)FDR,B)GPS,C)RCReceiver(UnderGPS),D)IMU,E)BEC AnadapterwasassembledtoconnecttheIMU,whichusesaHiroseHR306-pinconnector,totheFDR,whichusesa4-wireplug.TheconnectionsequenceisshowninTable 2-5 ( 38 ).Thecompletewiringdiagramoftheexperimentalaircraft,includingconventionalRCcomponents,isshowninFigure 2-6 .TheBECmustbeprogrammedtooutputavoltageintherangeof5.4-7.0VtoproperlypowertheIMU,FDR,andRCsystemssimultaneously( 38 ; 39 ). Table2-5. IMU/FDRWiringSequence HirosePortNo.I2CFunctionFDRWire 1SDAYellow2VDDRed3NotUsedNA4NotUsedNA5GNDWhite6SCLBrown 22

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WiringDiagramofMiniShowTimeAircraftforDataCollection Theseavionicshavebeendemonstratedashighlyaccuratewhencomparedtohigh-qualityavionics.Whencomparedtoahigh-qualityIMUwithlaser-ringgyros,thenIMUprovidedmeasurementsthatyieldedvelocityestimateswithinstandarddeviationsofapproximately0.2m=sonallaxesandattitudeestimateswithinstandarddeviationsofapproximately0.2inrollandpitchand0.35inheading( 40 ).ThemanufacturerspecicationsaregiveninTable 2-6 foreachofthesensors( 38 ). Table2-6. TechnicalSpecicationsofIMUSensors SensorDynamicRangeDigitalSensitivityOset/DriftNoise Gyro600o=s0:01831o=s1o=s0:56o=sAccelerometer5g1:5259e4g30mg4:87mgMagnetometer1:9Gauss5:79e5Gauss2700ppm=oC5:6e4Gauss

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2{2 ( 38 ). 32768!(2{2)BasedonthedynamicrangeofeachsensoraspresentedinTable 2-6 ,Equation 2{2 resultsinmultipliersoftherawsensoroutputandresultingunitsaspresentedinTable 2-7 Table2-7. IMURawOutputMultipliersandResultingUnits SensorMultiplierResultingUnits Gyro2.747e-2deg=sAccelerometer2.289e-4GMagnetometer8.698e-5Gauss

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3-1 .Aseriesofightswasperformedtocollectdataduringdoubletmaneuversandsteadyhighangle-of-attackightwiththestock(normal)verticaltail.Eachightbeganbytakingoandestablishingtheaircraftinhighangle-of-attackight.Acompletepassathighangle-of-attackwasrstperformedtoestablishthetrimpositionforallcontrolsurfaces.Eachsubsequentpassthenincludedthreedistinctactions:establishingthehighangle-of-attacktrimconditionatthedesiredheading,performingadoubletmaneuverwithacontrolsurface,andreestablishingthetrimconditionfortheremainderofthe 25

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MiniShowTimeinRepresentativeHighAngle-of-AttackFlight passlength.Eachightinvolveddoubletsofvaryingsizebyasinglecontrolsurface.Thelargestdoubletforeachcontrolwasconstrainedeitherbymaximumcontroldeectionsorbythepilot'sabilitytoquicklyreestablishstraightandleveltrimight.Adoubletwasperformedwithmaximumdeectionandseveralwereperformedwithprogressivelysmallerdeections.Aminimumofthreepasseswithdoubletmaneuverswereperformedforeachight.Aseriesofightswasthenperformedtocollectdataduringsteadyhighangle-of-attackightwithtendierentverticaltailcongurations:uprightandinvertedwitheachoftheveverticaltails.Invertedhighangle-of-attackwasperformedinasimilarfashiontouprightbutwithdown(positive)elevatordeectiontomaintainthepitchangle.Additionally,slightdierencesinruddertrimexistedbetweenuprightandinvertedduetothelargeyawmomentfromp-factorinhighangle-of-attackight.TheMiniShowTimeininvertedhighangle-of-attackightisshowninFigure 3-2 .Eachightconsistedofestablishingtheaircraftineitheruprightorinvertedhighangle-of-attackighttodetermineapproximatetrimconditions.Aminimumofvestraight,horizontalpassesinhighangle-of-attackightwereperformedwitheachtailconguration. 26

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MiniShowTimeinInvertedHighAngle-of-AttackFlight Anassistantwithastopwatchrecordedthetimesatwhicheachhighangle-of-attackpassbeganandended,aswellaswhendoubletswereperformed.Aftereachight,thedatafromtheFDRwasdownloadedtoacomputerwiththeUSBcableandtheFDR'sbuerwascleared.Thetrimconditionsfortheaircraftremainedconsistentfromighttoight.Theighttestingprocedureinvolved14ightswhichspannedfourdays. 27

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41 ).TheFouriertransformofadiscretesignal,suchasthatsampledfromacontinuoussignal,iscalledadiscreteFouriertransform(DFT).ThemostcommonmethodwithwhichaDFTisappliedisthroughafastFouriertransform(FFT),whichisaclassofecientalgorithmsthatcomputetheDFT( 41 ). 41 ).Thewavelettransformisawindowingtechniquewhichutilizeslongtimeintervalstoextractlowfrequencyinformationandshorttimeintervalstoextracthighfrequencyinformation.Thetime-frequencyplanebecomespartitionedintowindowsofconstantarea,asshowninFigure 4-1 ( 42 ),whichresultsinamultiresolutionanalysis. 28

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RepresentativeWaveletWindowSizePattern Withineachwindowthesignaliscomparedtoamother,ororiginal,waveletofagivenscaleandshifttodeterminealevelofcorrelation.Awaveletisalimiteddurationwaveformwhichbeginsat,endsat,andhasanaveragevalueofzero.Thescalealtersthefrequencyofthewaveletbystretchingorcompressingthemotherwaveletalongthetimeaxis.Theshiftaltersthelocationofthewaveletalongthetimeaxis.Therearemanydierenttypesofmotherwaveletswhichcanbeapplied,buttheMorletwaveletisthemostcommon.TheMorletwaveletisasinewavewhichislocalizedbyaGaussianenvelopeandcanbeseeninFigure 4-2 ( 41 ).Thewavelettransformdeterminesthecorrelationofasignaltothemotherwaveletofvariousscalesanddisplaysthecorrelationonatime-scaleplot.Thedominantfrequencyofawaveletcanbeapproximatedfromthescale.Awaveletwithalargescaleisstretchedintime,whichallowsittocorrelatewithlowfrequencysignals,andawaveletwithasmallscaleiscompressedintime,whichallowsittocorrelatewithhighfrequencysignals( 41 ).Theresultingwavelettransformationcanberepresentedeitherasa3-dimensionalplot,withaxesrepresentingtime,frequency,andcorrelationmagnitude,orasa2-dimensionalplotwithtimeandfrequencyaxesandintensityrepresentingcorrelationmagnitude.Highcorrelationataparticularfrequencyappearsonthewavelettransformationplotas 29

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RepresentativeMorletMotherWavelet alternatingpositiveandnegativecorrelationmagnitudesduetophaseshiftsbetweenthemotherwaveletandperiodicsignal. 4.3.1Example1:StationarySineWaveSignalTherstexampledemonstratestheFourierandwavelettransformsonastationarysinusoidalwavewithafrequencyof10rad/s.ThesignalisshowninFigure 4-3 Figure4-3. SignalforExample1:10rad/sSineWave 30

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4-4 .InthiscasetheFFTandwaveletrepresentationsbothprovidesimilarinformationaboutthesignal,asitisstationary. Figure4-4. FrequencyAnalysisofExample1Signal:FFT(left)andWavelet(right) ThewavelettransforminFigure 4-4 canbeviewedasa3-dimensionalplotinFigure 4-5 .Thecorrelationmagnitudeisdepictedalongthez-direction,alongwiththesametimeandfrequencyaxes.Itcanbeseenthatalong10rad/s,thewavelettransformindicatesalternatingstrongpositiveandnegativepeaks.Thisisduetothephaseshift;attimesthescaledmotherwaveletmatchesthesignalwellresultinginpositivecorrelation,andatothertimesitmatchesitwellinaninverserelationshipresultinginnegativecorrelation. 31

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3-DimensionalDepictionofExample1WaveletTransform summationofthe10rad/ssinewaveusedinexample1alongwithasecondstationarysinewavewithafrequencyof5rad/s.Thesignalforexample2isshowninFigure 4-6 Figure4-6. SignalforExample2:Summationof5and10rad/sSineWaves TheFouriertransformisappliedtothesignalwithanFFTalgorithm,andtheresultingfrequency-domainrepresentationdisplayspeaksatboth5and10rad/s.ThewavelettransformisalsoappliedtothesignalusingaMorletwavelet,andtheresulting 32

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4-7 .Inthiscase,theFFTandwaveletrepresentationsbothprovidesimilarfrequencyinformationaboutthesignal,asitisagainstationary.Bothtechniquesidentiedbothdominantfrequenciescontainedinthesignal. Figure4-7. FrequencyAnalysisofExample2Signal:FFT(left)andWavelet(right) 4-8 .ThechirpsignalshowninFigure 4-8 clearlyincreasesfrequencywithtime.ThefrequencymigrationwithrespecttotimeisshowninFigure 4-9 .TheFouriertransformisappliedtothesignalwithanFFTalgorithm,andtheresultingfrequency-domainrepresentationdisplayssimilarenergyacrossallfrequenciesbelowapproximately10rad/s.ThewavelettransformisalsoappliedtothesignalusingaMorletwavelet.Theresultingrepresentationdisplaysbandsofstrongcorrelationwhichinitiallyareatlowfrequencyandincreasewithtimeuntilreachingapproximately10rad/sattheendofthesample.TheFFTandwavelettransformrepresentationsoftheexample3signalareshowninFigure 4-10 33

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SignalforExample3:SinusoidalChirpwithIncreasingFrequency Figure4-9. ChirpFrequencyWithRespectToTime Inthiscase,thedierencebetweentheinformationthatFourierandwavelettransformsprovideiseasilyidentied.TheFFTidentiesthatthereisstrongcorrelationacrossarangeoffrequencies,butgivesnoindicationastowhetherthecorrelationsexistweaklyacrossalltimesorstronglyovershorttimesegments.Thewavelettransform,ontheotherhand,maintainsthetime-domaininformation.Itisclearlyseenthatatanyparticulartimeinstancethereisastrongcorrelationataparticularfrequencyandnotawiderangeoffrequencies.TheoverallrangeoffrequenciesidentiedbythewavelettransformcorrespondstotherangeoffrequenciesidentiedbytheFFT,butthewavelet 34

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FrequencyAnalysisofExample3Signal:FFT(left)andWavelet(right) transformalsoprovidesfrequencyinformationlocalizedintime.ItcanbeseenthatthecentersofthecorrelationpeaksofthewaveletinFigure 4-10 followthetrendofthechirpfrequencyasshowninFigure 4-9 35

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5{1 ,whereAisannxmmatrixofregressors,Xisanmx1vectorofunknowncoecients,Lisannx1vectorofmeasurements,andVisannx1vectorofresiduals,wherenisthenumberofsamplesandmisthenumberofstatesintheleast-squaresmodel( 43 ). 5{2 ( 43 ). 36

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43 ).Thisisaccomplishedthroughbackwardelimination,inwhicherroneousorinsignicantlysmallindividualcomponentsineachcompletemodelareremovediteratively.Ateachiterationthemodelisusedtosimulatetheoutputwiththesameinputsasthemodelis 37

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5.2.1LongitudinalAmodelofthelongitudinaldynamicsisgeneratedtorelatethepitchratetotheelevatorcommands.Suchamodelisestimatedtorelateasetofdoubletcommandstotheelevatorandtheresultingpitchrate.ThemeasurementsofpitchrateandthesimulatedvaluesfromthemodelareshowninFigure 5-1 alongwiththedoublets. Figure5-1. Elevator(left)andPitchRate(right)duringDoublets:Measured(|)andSimulated() ThemodelwithsimulatedresponseinFigure 5-1 isgiveninEquation 5{3 asadiscrete-timeequation,whereqispitchrateindeg=sandeiselevatordeectionin%.Thismodeldependsuponalinearcombinationoflaggedvaluesforelevatoranglecorrespondingto5and7previoustimesteps.Thecurrentvalueofpitchratealsodependsuponlaggedvaluesofpitchratecorrespondingtoaveragevaluesfrom2,4and6previoustimesteps. 38

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30:2077e(k6)+e(k7)+e(k8) 3+1:1516q(k1)+q(k2)+q(k3) 30:3956q(k3)+q(k4)+q(k5) 30:0215q(k5)+q(k6)+q(k7) 3 (5{3) ThecontributionsofeachtermfromEquation 5{3 totheresponseinFigure 5-1 isshowninFigure 5-2 .Thelargestcontributionresultedfromanegativepitchrateattimeofkresultingfromapositiveelevatordeectionattimeofk5;however,somehigher-orderdynamicsisalsopresentbecauseoftheneedtoalsoretainacontributionfromtheelevatordeectionattimeofk7.Thestatedynamicsareevidencedbythecontributionsfromseverallaggedvaluesofpitchrate.Apositivepitchrateattimeofk2generatesapositivecontributiontocurrentpitchratewhileapositivepitchrateateithertimeofk4ork6actuallygeneratesanegativecontributiontopitchrate.Suchdisparityispartlyduetoout-of-phasestatesfromashort-periodmodealthoughthedynamicsathighangle-of-attackdonotnecessarilyhavetraditionalmodes. Figure5-2. IndividualContributionstoResponsefromLongitudinalModel:Elevator(k-5)(|o),Elevator(k-7)(|),PitchRate(k-2)(...),PitchRate(k-4)(),PitchRate(k-6)(|.|.) 39

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5-1 ;however,theselongitudinaldynamicsareexpectedlydiculttomodelathighangle-of-attackconditions.Certainlytheaerodynamicsarenotnecessarilylinearnornite-dimensionalfunctionsofightcondition.Also,thelackofangle-of-attackmeasurementsdoesnotnecessarilylimitthedelity,giventhatatransferfunctionalwaysexistsbetweenaninputandanoutput,butthelackalmostcertainlylimitstheinterpretationoftheresultingmodel. 5-3 .Thedoubletsandresultingrollratesactuallyvarybyroughlyafactorof2betweentherstandthirdcommandsoarichsetofdataisavailableforthemodel. Figure5-3. Aileron(left)andRollRate(right)duringDoublets:Measured(|)andSimulated() ThemodelthatsimulatedtherollrateinresponsetothedoubletsinFigure 5-3 isgiveninEquation 5{4 ,wherearepresentsailerondeectionin%.Thismodelsimplygeneratestherollrateattimeofkfromabiastermalongwithaneandquadratictermsassociatedwithaverageaileronangleattimesofk5. 40

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38:42e52a(k4)+2a(k5)+2a(k6) 3 (5{4) Thismodelindicatesthelateraldynamicsaredominatedbyatraditionalmodeofrollconvergence.Theresponseisnearlylinearinaileronangle,asshowninFigure 5-4 ,sincethenonlinearcontributionisnegligible.Sucharesultissomewhatlogicalgiventhatanyeectsofhighangle-of-attackconditionswouldinuencetheaerodynamicsoflongitudinalmotionmuchmorethanthelateralmotion. Figure5-4. IndividualContributionstoResponsefromLateralModel:Aileron(|)andAileron2() 5-5 .ThemodelthatsimulatedtheyawrateinresponsetothedoubletsinFigure 5-5 isgiveninEquation 5{5 ,wherearepresentsailerondeectionin%,rrepresentsrudderdeectionin%,andprepresentsrollrateindeg=s.Thismodelrequiresmoretermstodescribethedynamicsthaneitherthelongitudinalmodelorlateralmodel.Essentially,the 41

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Rudder(left)andYawRate(right)DuringDoublets:Measured(|)andSimulated() directionaldynamicsareestimatedasbeinganonlinearfunctionofboththerudderandtheaileronalongwithbeinganonlinearfunctionoftherollrate. 3+0:00592r(k4)+2r(k5)+2r(k6) 30:5073a(k4)+a(k5)+a(k6) 30:03132a(k4)+2a(k5)+2a(k6) 3+0:0259p(k1)+p(k2)+p(k3) 30:0009p2(k1)+p2(k2)+p2(k3) 3 (5{5) ThecontributionsfromeachterminEquation 5{5 tothesimulatedresponseinFigure 5-5 areshowninFigure 5-6 .Theresponseisdominatedbythecontributionsfromtherudderwiththelineartermprovidingthesignicantportion.ThedirectionaldynamicsarechallengingtomodelasevidencedbytheinconsistentqualityofthetinFigure 5-5 despitethenonlineartermsinEquation 5{5 .Thenatureofthelinearcontributionsisnotconsistentgiventhatapositiveruddergeneratesnegativeyawrateandapositiveailerongeneratesnegativeyawrateasadverseyawbutapositiverollrategeneratesapositiveyawratetoimplysomeproverseyaw( 44 ).Eventhe 42

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IndividualContributionstoResponsefromDirectionalModel:Rudder(|o),Rudder2(|x),Aileron(),Aileron2(|.|.),RollRate(...),RollRate2(|) nonlinearitiesareinconsistentsinceanyruddergeneratesasmallpositivecontributiontoyawratewhileanyaileronorrollrategeneratesasmallnegativecontributiontoyawrate.Theissueofighttestingmustbeconsideredwhentryingtoevaluatethequalityofthemodelandanyassociatedinconsistencies.Inparticular,theinuenceofgustscanbeextremeonthedirectionaldynamicswhenyingathighangle-of-attackconditionsbutofcourseanygustexcitationisnotproperlyrepresentedinthemodel. 5.3.1LongitudinalFlightdataassociatedwithsteady-statehighangle-of-attackightisshowninFigure 5-7 fortheelevatorcommandsandassociatedvaluesofpitchrate.Inthiscase,theelevatorcommandsinFigure 5-7 areafactorof5lessthantheelevatorcommandsinFigure 5-1 toindicatethepilotismerelymovingthecontrolsurfacestomaintainightconditionandthusnotintroducingsignicantenergy. 43

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Elevator(left)andPitchRate(right)DuringSteadyFlight:Measured(|)andSimulated() ThecontributiontopitchratethatisnotpredictedbythemodelinEquation 5{3 isshowninFigure 5-8 .Clearlythemodelisnotabletoreproducethecompleteresponseindicatingtheelevatorisnotabletoaccountfortheentiretyofthemeasuredpitchrate. Figure5-8. UncommandedPitchRate ThisuncommandedestimateofpitchratefromFigure 5-8 isrepresentedinthefrequencydomaininFigure 5-9 fromaFouriertransform.Theenergyisconcentratedacrosslowfrequenciesbutapairofmodes,around3rad/sand6rad/s,isclearlyevident.Thesemodesmaycorrelatetoashort-periodmode;however,theyaremorelikelyassociatedwithsomeunmodeleddynamicofthehighangle-of-attackcondition. 44

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FFTofMeasuredPitchRate(...),UncommandedPitchRate(|),andElevatorInput() Figure 5-9 indicatesthatbetweenapproximately2and4rad/stheuncommandedpitchrateislargerthanthemeasuredpitchrate.Thisseemsimproper,astheuncommandedpitchrateisextractedfromthemeasuredpitchrateandshouldthereforebeasubsetofthemeasuredpitchrate.However,phaseshiftsbetweenthemodel-predictedpitchrateandthemeasuredpitchratecanresultinuncommandedpitchratesestimatedathighervaluesthanthemeasuredpitchrate. 5-10 alongwithassociatedaileroncommands.ThedeectionsoftheaileronarenearlyanorderofmagnitudelessthanthesizeofthedoubletsinFigure 5-3 toindicatethepilotisprovidingonlyminimalexcitation.SomeunexplaineddrifttonegativedeectionisclearlyevidentinFigure 5-10 ;however,thisdriftisextremelylow-frequencyandthuscanbedirectlyeliminatedintheanalysis.Theuncommandedportionoftheresponse,asdeterminedbysubtractingthesimulatedresponsetotheaileronfromthemeasuredresponse,isshowninFigure 5-11 .Thisportionisquitelargeinmagnitudeandactuallyappearsquiteperiodic. 45

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Aileron(left)andRollRate(right)DuringSteadyFlight:Measured(|)andSimulated() Figure5-11. UncommandedRollRate TheperiodicitynotedinFigure 5-11 isquantiedbyaFouriertransformonthatdatatoobtainthefrequency-domainrepresentationinFigure 5-12 .Thefrequency-domainrepresentationofthemeasuredrollrateandtheuncommandedrollrateareshownalongwiththeaileroncommand.Therollrateandtheaileronshowapeakat2.2rad/sindicatingthepilotisactivelycontrollingsomedynamicthatisaectingthemaintenanceofsteady-stateight;however,onlytherollrateshowsapeakat3.65rad/s.Thelargeamountofenergyatthisfrequencyisindicativeofasignicantlevelofwingrockathighangle-of-attackconditionsforthisaircraft. 46

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FFTofMeasuredRollRate(...),UncommandedRollRate(|),andAileronInput() 5-13 .ThisdataconsistsofsmallruddercommandstomaintaintheightconditionandtheassociatedyawrateswhichweremeasuredandsimulatedfromthemodelinEquation 5{5 Figure5-13. Rudder(left)andYawRate(right)DuringSteadyFlight:Measured(|)andSimulated() 47

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5-14 .Theyawratealsohassomeclearperiodicityandasubstantialmagnitudeindicatingsomeelementofsteady-stateightisnotcapturedbythemodelgeneratedfromdoublets. Figure5-14. UncommandedYawRate Thefrequency-domainrepresentationoftheyawrate,bothmeasuredanduncommanded,alongwithruddercommandsindicatestheperiodicity.Thisdata,asshowninFigure 5-15 ,hasanoticeablemodenear2.2rad/s.Boththerudderandtheyawratecontainthismode,asdidtheaileronandtherollrate,whichmaymeanitcorrelatestoaroll-yawcoupledmodethatthepilotisattemptingtodampoutandmaintaincondition.Figure 5-15 indicatesfrequencyregionswheretheuncommandedyawrateisoflargercorrelationthanthemeasuredyawrate,particularlybetweenapproximately2.5and5rad/sand6and10rad/s.Thisseemsimproper,astheuncommandedyawrateisextractedfromthemeasuredyawrateandshouldthereforebeasubsetofthemeasuredyawrate.However,phaseshiftsbetweenthemodel-predictedyawrateandthemeasuredyawratecanresultinuncommandedyawratesestimatedathighervaluesthanthemeasuredyawrate. 48

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FFTofMeasuredYawRate(...),UncommandedYawRate(|),andRudderInput() 49

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6-1 alongwiththeassociatedailerondeections.Theailerondeectionisdramaticallysmallerthantherollrateandthusdiculttodistinguish.Clearlytherollrateshowssomeamountofperiodicityamongallthetails;consequently,wingrockappearsforanysizeofthesetails.Also,themagnitudeoftherollrateshowsvariationduringtheresponse;however,thismagnitudeisactuallysomewhatconsistentdespitevariationsinthetailsize.Afrequency-domainrepresentationoftherollratesandailerondeectionsfromFigure 6-1 iscomputedandshowninFigure 6-2 usingaFouriertransform.Therollrateshowsaconsistentamountofenergyaround4rad/swhichcorrelateswiththeconsistentmagnitudeobservedinthetime-domainresponses.Verylittleenergyisintroducedbytheaileronaround4rad/s,indicatingthattheenergyobservedingure 6-2 representsuncommandedwingrock.Also,thispeakinenergyisactuallysomewhatbroadforeverytailandrangesfromapproximately2rad/sto6rad/sindicatingthewingrockisabroad-bandphenomenon.Atime-frequencyrepresentationiscomputedfortherollratetoinvestigatethetemporalnatureofanyinstantaneousfrequenciesinthewingrock.TheserepresentationsareshowninFigure 6-3 ascomputedbywavelettransformsusingaMorletwavelet.Thewingrockisevidentbythehighcorrelationsshownaround4rad/s;however,thisrepresentationisnotablydierentthanthefrequency-domaincharacterizationinFigure 6-2 .Thewingrockisshowntoactuallyhaveanarrowbandofenergywhen 50

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TimeResponsesinUprightCongurationforA)SmallerTail,B)SmallTail,C)NormalTail,D)BigTail,E)BiggerTail:RollRate(|)indeg/sandAileronDeection( ... )in%

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FFTinUprightCongurationofRollRate(left)andAileronDeection(right):SmallerTail({{),SmallTail({.{.),NormalTail(|),BigTail(...),BiggerTail({x{) localizedintimeusingFigure 6-3 .Thebroad-bandnatureobservedinFigure 6-2 resultsfromthevariationsobservedinFigure 6-3 inthecentralfrequencyofthatnarrowband.Thewavelettransformsoftheailerondeection,asshowninFigure 6-4 ,donotshowsignicantcorrelationwiththewingrock.Ineachcase,theailerondeectionsarepredominatelyatlowerfrequencies. 6-5 alongwiththeassociatedailerondeections.Therollratesareapproximatelyanorderofmagnitudegreaterthantheailerondeectionsalthoughbothappeartohaveperiodicity.Ineachcase,thismagnitudeandperiodicityarerelativelyconsistentdespitevariationstothetail.Thefrequency-domainrepresentations,showninFigure 6-6 ,agreewiththetime-domainanalysisofFigure 6-5 .Theenergyoftherollrateisaboutanorderofmagnitudegreaterthantheaileronforeachtail.Also,therollrateshowsaminorbroad-bandpeaknear4rad/salthoughtheenergyisnotexcessive.Awavelettransformisappliedtothetime-domaindatatogeneratetherepresentation,showninFigure 6-7 ,inthetime-frequencydomain.Theseplotsshowsomecorrelation 52

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WaveletTransformsofRollRateinUprightConguration:A)SmallerTail,B)SmallTail,C)NormalTail,D)BigTail,E)BiggerTail

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WaveletTransformsofAileronDeectioninUprightConguration:a)SmallerTail,b)SmallTail,c)NormalTail,d)BigTail,e)BiggerTail

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TimeResponsesinInvertedCongurationforA)SmallerTail,B)SmallTail,C)NormalTail,D)BigTail,E)BiggerTail:RollRate(|)andAileronDeection( ... )

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FFTinInvertedCongurationofRollRate(left)andAileronDeection(right):SmallerTail({{),SmallTail({.{.),NormalTail(|),BigTail(...),BiggerTail({x{) around4rad/s;however,themagnitudeofcorrelationisnotexcessivelyhighincomparisontothelowerfrequencies.ThislackofexcessivecorrelationagreeswiththeFouriertransformsinFigure 6-6 andisevidentforeachtail.Asimilartime-frequencyrepresentationoftheailerondata,ascomputedthroughwavelettransformandshowninFigure 6-8 ,indicatesastrongcorrelationtotherollratesthatareshowninFigure 6-7 .Inparticular,mostofthepeaksincorrelationforrollratearematchedbyapeakinaileronatthesametimeandfrequency.Examplesofsuchinput/outputmatchingcanbeseenwiththesmallertailatapproximately2.5to3rad/sbetween50and60sandwiththebiggertailatapproximately3rad/sbetween10and20sandatapproximately3to4rad/sbetween45and60s. 6-1 and 6-5 respectively,areanalyzed.PeakaveragesandfrequenciesareextractedforeachverticaltailsizeandshowninFigure 6-9 .Thetime-domainanalysisinFigure 6-9 indicatesthatwingrockfrequencyandmagnitudedoesnotdependuponthesizeoftheverticaltail.However,thewingrockmagnitudedoeschangebasedontheverticaltailconguration:theaveragewingrock 56

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WaveletTransformsofRollRateinInvertedConguration:A)SmallerTail,B)SmallTail,C)NormalTail,D)BigTail,E)BiggerTail

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WaveletTransformsofAileronDeectioninInvertedConguration:A)SmallerTail,B)SmallTail,C)NormalTail,D)BigTail,E)BiggerTail

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MeanRollRatePeakMagnitudes(|)andPeak-to-PeakFrequencies(...) peakmagnitudeintheinvertedightconditionisapproximatelyhalfofthatintheuprightightconditionacrossalltailsizes.Theaveragewingrockfrequencyisnearlyconstantatapproximately4rad/sforalltailsizesinbothuprightandinvertedhighangle-of-attackight,whichagreeswiththepeaksseenintheFouriertransformsinFigures 6-2 and 6-6 .ThedierenceinmagnitudeofuncommandedwingrockbetweenuprightandinvertedightislikelymorepronouncedthanFigure 6-9 indicates.Figures 6-3 and 6-4 showlittlecorrelationbetweenaileroninputandrollratefrequenciesintheuprightconguration.Figures 6-7 and 6-8 indicate,however,thatastrongcorrelationexistsbetweenrollrateandaileroninputfrequenciesatessentiallyalltimeswhenrollrateoscillationsareobservedintheinvertedconguration.Thisindicatesthatuncommandedwingrockisvirtuallynonexistentwhenintheinvertedconguration.TheupperandlowerboundsofwingrockfrequencyareextractedfromFigures 6-2 and 6-3 andareshowninFigure 6-10 .Onlytheuprightcongurationisanalyzedforsuchboundsbecauseonlyuncommandedwingrockisofinterest.TheupperandlowerboundsasdeterminedbyboththeFFTsandwaveletsarefairlyconsistentforalluprighttail 59

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Figure6-10. UpperandLowerBoundsofWingRockFrequencyfromFFTs(|)andWavelets(...) Thetime-frequencyrepresentationsshowninFigure 6-3 actuallyindicatethatthewingrockoscillationismuchnarrowerinbandwidthwhenlocalizedintimethanFigure 6-10 indicates.TheaveragebandwidthofthewingrockoscillationsatalltimesforeachtailareextractedfromthewaveletplotsandareshowninFigure 6-11 alongwiththebroadbandsextractedfromFigure 6-10 .ThebandwidthatmosttimesisnotablysmallerinFigure 6-11 thaninFigure 6-10 .Thisindicatesthatwingrockisactuallyafairlynarrow-bandphenomenon,butvariationsinfrequencyovertimewithinabroaderbandleadtotheappearanceofabroad-bandphenomenonwithmanyclassicalanalysistechniquessuchastheFFT. 60

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MeanWingRockBandwidths:FFT(|),WaveletBroadLimits(...),andWaveletNarrowLimits({{{) 61

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[1] Wright,K.,andLind,R.,\InvestigatingSensorEmplacementonVerticalSurfacesforaBiologically-InspiredMorphingDesignfromBats,"AIAAPaper2007-6488,Aug.2007. [2] Grant,D.,Abdulrahim,M.,andLind,R.,\FlightDynamicsofaMorphingAircraftUtilizingMultiple-JointWingSweep,"AIAAPaper2006-6505,Aug.2006. [3] Regenie,V.,Gatlin,D.,Kempel,R.,andMatheny,N.,\TheF-18HighAlphaResearchVehicle:AHigh-Angle-of-AttackTestbedAircraft,"NASATM104253,1992. [4] Ili,K.,andWang,K.,\FlightDeterminedSubsonicLongitudinalStabilityandControlDerivativesoftheF-18HighAngleofAttackResearchVehicle(HARV)WithThrustVectoring,"NASATP97-206539,1997. [5] Bauer,J.,Clarke,R.,andBurken,J.,\FlightTestoftheX-29AatHighAngleofAttack:FlightDynamicsandControls,"NASATP3537,1995. [6] NASADrydenF-18HighAlphaResearchVehicle(HARV)PhotoCollection .July26,2001.NationalAeronauticsandSpaceAdministration.Feb.23,2009< [7] NASADrydenX-29PhotoCollection .Feb.3,2000.NationalAeronauticsandSpaceAdministration.Feb.23,2009< [8] AeroVironmentUnmannedAircraftSystems,< [9] Quigley,M.,Barber,B.,Griths,S.,andGoodrich,M.,\TowardsReal-WorldSearchingwithFixed-WingMini-UAVs,"IEEE/RSJInternationalConferenceonIntelligentRobotsandSystems,Aug.2005. [10] AeroVironment,Inc.:MediaGallery .2009.AeroVironment,Inc.Feb.23,2009< gallery2.asp?id=224 [11] Frank,A.,McGrew,J.,Valenti,M.,Levine,D.,andHow,J.,\Hover,Transition,andLevelFlightControlDesignforaSingle-PropellerIndoorAirplane,"AIAAPaper2007-6318,Aug.2007. [12] Johnson,E.,Turbe,M.,Wu,A.,Kannan,S.,andNeidhoefer,J.,\FlightTestResultsofAutonomousFixed-WingUAVTransitionstoandfromStationaryHover,"AIAAPaper2006-6775,Aug.2006. [13] Gree,W.,andOh,P.,\AutonomousHoveringofaFixed-WingMicroAirVehicle,"IEEEInternationalConferenceonRoboticsandAutomation,May2006. [14] Hwang,C.,andPi,W.S.,\SomeObservationsontheMechanismofAircraftWingRock,"JournalofAircraft,Vol.16,No.6,pp.366-373,1979. 65

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Liebst,B.,Nolan,R.,\MethodforthePredictionoftheOnsetofWingRock,"JournalofAircraft,Vol.31,No.6,pp.1419-1421,1994. [16] Regenie,V.,Gatlin,D.,Kempel,R.,andMatheny,N.,\TheF-18HighAlphaResearchVehicle:AHigh-Angle-of-AttackTestbedAircraft,"NASATM104253,1992. [17] Ramnath,R.V.,andGo,T.H.,\AnAnalyticalApproachtotheAircraftWingRockDynamics,"AIAAPaper2001-4426,2001. [18] Go,T.H.,andRamnath,R.V.,\AnAnalysisoftheTwoDegree-of-FreedomWingRockonAdvancedAircraft,"JournalofGuidance,Control,andDynamics,Vol.25,No.2,pp.324-333,2002. [19] Schmidt,L.V.,\WingRockDuetoAerodynamicHysteresis,"JournalofAircraft,Vol.16,No.3,pp.129-133,1979. [20] Lie,F.A.P.,andGo,T.H.,\AnalysisofSingleDegree-of-FreedomWingRockduetoAerodynamicHysteresis,"AIAAPaper2007-6489,2007. [21] Abramov,N.,Goman,M.,Demenkov,M.,andKhabrov,A.,\Lateral-DirectionalAircraftDynamicsatHighIncidenceFlightwithAccountofUnsteadyAerodynamicEects,"AIAAPaper2005-6331,2005. [22] Bauer,J.,Clarke,R.,andBurken,J.,\FlightTestoftheX-29AatHighAngle-of-Attack:FlightDynamicsandControls,"NASATP3537,1995. [23] Liebst,B.S.,andDeWitt,B.R.,\WingRockSuppressionintheF-15Aircraft,"AIAAPaper1997-3719,1997. [24] Ili,K.W.,andWang,K.C.,\X-29ALateral-DirectionalStabilityandControlDerivativesExtractedfromHigh-Angle-of-AttackFlightData,"NASATP3664,1996. [25] Davison,M.T.,\AnExaminationofWingRockfortheF-15,"Master'sThesis,AirForceInst.ofTechnology,AFIT/GAE/ENY/92M-01,Feb.1992. [26] Brandon,J.M.,andNguyen,L.T.,\ExperimentalStudyofEectsofForebodyGeometryonHighAngle-of-AttackStability,"JournalofAircraft,Vol.25,No.7,pp.591-597,1988. [27] Meyer,R.R.,\OverviewoftheNASADrydenFlightResearchFacilityAeronauticalFlightProjects,"NASATM104254,1992. [28] Ericsson,L.E.,\VariousSourcesofWingRock,"JournalofAircraft,Vol.27,No.6,pp.488-494,1990. 66

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Walton,J.,andKatz,J.,\ApplicationofLeading-EdgeVortexManipulationstoReduceWingRockAmplitudes,"JournalofAircraft,Vol.30,No.4,pp.555-557,1993. [30] Ericcson,L.E.,\SourcesofHighAlphaVortexAsymmetryatZeroSideslip,"JournalofAircraft,Vol.29,No.6,pp.1086-1090,1992. [31] Ericsson,L.E.,\WingRockGeneratedbyForebodyVortices,"JournalofAircraft,Vol.26,No.2,pp.110-116,1989. [32] Ericsson,L.E.,\FurtherAnalysisofWingRockGeneratedbyForebodyVortices,"JournalofAircraft,Vol.26,No.22,pp.1098-1104,1989. [33] L.E.Ericsson,\EectofDeep-StalldynamicsonForebody-InducedWingRock,"AIAAPaper1996-3404,1996. [34] Alcorn,C.W.,Croom,M.A.,Francis,M.S.,andRoss,H.,\TheX-31Aircraft:AdvancesinAircraftAgilityandPerformance,"ProgressinAerospaceSciences,Vol.32,pp.377-413,1996. [35] Gilbert,W.P.,Nguyen,L.T.,andGera,J.,\ControlResearchintheNASAHigh-AlphaTechnologyProgram,"AGARDFluidDynamicsPanelSymposiumonAerodynamicsofCombatAircraftControlandGroundEects,1989. [36] MiniShowTime4DARFbyE-ite(EFL2500),HorizonHobby,Inc.,< [37] Smith,H.C.,TheIllustratedGuidetoAerodynamics,TABBooks,BlueRidgeSummit,PA,pp.228-229,1992. [38] nIMUDatasheetVersion2.7,MEMSense,LLC,< [39] InstructionManualfortheUSBFlightDataRecorderVersion3.7,EagleTreeSystems,LLC,< [40] Perry,J.,Mohamed,A.,Johnson,B.,andLind,R.,\EstimatingAngleofAttackandSideslipUnderHighDynamicsonSmallUAVs,"Proceedingsofthe21stInterna-tionalTechnicalMeetingoftheSatelliteDivisionoftheInstituteofNavigationIONGNSS2008,pp.1165-1173,2008. [41] Misiti,M.,Misiti,Y.,Oppenheim,G.,andPoggi,J.,WaveletToolboxUser'sGuide,Version2,TheMathWorks,Inc.,Natick,MA,Chap.1,6,2000. [42] Strang,G.,Nguyen,T.,WaveletsandFilterBanks,Wellesley-CambridgePress,Wellesley,MA,Chap.1,1996. [43] Klein,V.,andMorelli,E.A.,AircraftSystemIdenticationTheoryandPractice,AIAAEducationSeries,AIAA,Reston,VA,Chap.5,2006. 67

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Yechout,T.,IntroductiontoAircraftFlightMechanics,AIAAEducationSeries,AIAA,Reston,VA,pp.176-177,2003. 68

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BaronJohnson'saviationpursuitsbeganwithhisrstaerobaticridesafewdayspriortobeingborninMemphis,TNin1983.AftermovingtoOcala,FL,BaronsoloedanL23SuperBlaniksailplaneattheageof14,andthensoloedaPiperJ-3Cubonhis16thbirthday.Baronthenpursuedhisprivatepilotlicense,whichhereceivedattheageof17,andhiscommercialpilotlicense,whichhereceivedattheageof18,alongwithinstrumentandmulti-engineratings.Baronhasaccumulatedover600hoursofightexperience,includingveryuniqueopportunitiessuchasyinginpartofa60-shipformationofandservingasin-ightsafetyobserverintheEAAAirVentureairshow.AftergraduatingfromBelleviewHighSchoolin2002,BaronattendedtheUniversityofFloridainGainesville,FL.HegraduatedcumlaudewithaBachelorofScienceinaerospaceengineeringin2007,andreceivedaMasterofScienceinaerospaceengineeringin2009.Whilestudying,BaronhashadopportunitiestoworkonmanyexcitingUAVandMAVprogramswhileemployedbytheFloridaCooperativeFish&WildlifeResearchUnit,MicroAirVehicleLab,andFlightControlLab.WhilememberandpilotfortheUFMicroAirVehicleTeam,theteamcaptured4consecutivevictoriesattheInternationalMicroAirVehicleCompetition.BaronhasalsobeenanactiveRCaviatorduringhisyears.HisprimarypassionsinRCareIMACScaleAerobaticsandF3CHelicopterAerobatics.BaronhashadtheopportunitytoymanyRCdemonstrationsandcompetitions,andhaswon4nationalchampionshiptitlesalongwithotherhighnishesatinternationalcompetitions.Baronalsoauthoredachildren'sbooktitledHistoryTakesAWildRide,forwhichhewasawardedthekeytothecityofMemphisandappearedonNBC'sTheTodayShow.Baronhashadsomeincredibleexperiencesandopportunitiesinhislifeforwhichheisincrediblygrateful. 69