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Modeling and Dynamic Analysis of a Multi-Joint Morphing Aircraft

Permanent Link: http://ufdc.ufl.edu/UFE0024477/00001

Material Information

Title: Modeling and Dynamic Analysis of a Multi-Joint Morphing Aircraft
Physical Description: 1 online resource (89 p.)
Language: english
Creator: Grant, Daniel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: daniel, grant, lind, morphing, uav
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: Morphing, which changes the shape and configuration of an aircraft, is being adopted to expand mission capabilities of aircraft. The introduction of biological-inspired morphing is particularly attractive in that highly-agile birds present examples of desired shapes and configurations. A previous study adopted such morphing by designing a multiple-joint wing that represented the shoulder and elbow of a bird. The resulting variable-gull aircraft could rotate the wing section vertically at these joints to alter the flight dynamics. This paper extends that multiple-joint concept to allow a variable-sweep wing with independent inboard and outboard sections. The aircraft is designed and analyzed to demonstrate the range of flight dynamics which result from the morphing. In particular, the vehicle is shown to have enhanced crosswind rejection which is a certainly critical metric for the urban environments in which these aircraft are anticipated to operate. Mission capability can be enabled by morphing an aircraft to optimize its aerodynamics and associated flight dynamics for each maneuver. Such optimization often consider the steady-state behavior of the configuration; however, the transient behavior must also be analyzed. In particular, the time-varying inertias have an effect on the flight dynamics that can adversely affect mission performance if not properly compensated. These inertia terms cause coupling between the longitudinal and lateral-directional dynamics even for maneuvers around trim. A simulation of a variable-sweep aircraft undergoing a symmetric morphing for an altitude change shows a noticeable lateral translation in the flight path because of the induced asymmetry.
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 Daniel Grant.
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: UFE0024477:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024477/00001

Material Information

Title: Modeling and Dynamic Analysis of a Multi-Joint Morphing Aircraft
Physical Description: 1 online resource (89 p.)
Language: english
Creator: Grant, Daniel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: daniel, grant, lind, morphing, uav
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: Morphing, which changes the shape and configuration of an aircraft, is being adopted to expand mission capabilities of aircraft. The introduction of biological-inspired morphing is particularly attractive in that highly-agile birds present examples of desired shapes and configurations. A previous study adopted such morphing by designing a multiple-joint wing that represented the shoulder and elbow of a bird. The resulting variable-gull aircraft could rotate the wing section vertically at these joints to alter the flight dynamics. This paper extends that multiple-joint concept to allow a variable-sweep wing with independent inboard and outboard sections. The aircraft is designed and analyzed to demonstrate the range of flight dynamics which result from the morphing. In particular, the vehicle is shown to have enhanced crosswind rejection which is a certainly critical metric for the urban environments in which these aircraft are anticipated to operate. Mission capability can be enabled by morphing an aircraft to optimize its aerodynamics and associated flight dynamics for each maneuver. Such optimization often consider the steady-state behavior of the configuration; however, the transient behavior must also be analyzed. In particular, the time-varying inertias have an effect on the flight dynamics that can adversely affect mission performance if not properly compensated. These inertia terms cause coupling between the longitudinal and lateral-directional dynamics even for maneuvers around trim. A simulation of a variable-sweep aircraft undergoing a symmetric morphing for an altitude change shows a noticeable lateral translation in the flight path because of the induced asymmetry.
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 Daniel Grant.
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: UFE0024477:00001


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IwouldrstliketoacknowledgetheUniversityofFloridaandUnitedStatesAirForceforsupportingmyambitionandgivingmetheopportunitytoconductsuchresearch.ThanksshouldbegiventoDr.WarrenDixon,Dr.PeterIfju,andDr.OscarCrisalleforprovidingdirectionandservingasmycommitteemembers.IwouldalsoliketothankmyseniorlabfellowsDr.MujahidAbdulrahim,Dr.AdamWatkins,Dr.RyanCausey,Dr.JosephKehoeandDr.SeanRegisfordfortheirpatienceandguidancewhilementoringme.MuchthanksisgiventomycolleaguesSankethBhat,BrianRoberts,RobertLove,BaronJohnson,RyanHurley,DongTranandStevenSorelyfortheirsupport,inspiration,anddeterminationtorenemypersonalskillsandcommunication.IwouldliketoextendmysincerestthanksandgratitudetoDr.RickLindforhiseortsinsupportingmyeducation,guidanceinacademia,andprovidingmewithaninvaluableopportunitytoachievesuccess.Withoutthecontinuoussupportandunconditionalloveofmyfamilyandfriends,noneofthisworkwouldhavebeenpossible.Lastandmostimportant,IwouldliketothankmylovingwifeforbeingtheshininglightbehindallthatIdo. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 13 1.1Motivation .................................... 13 1.2ProblemDescription .............................. 14 1.3ProblemStatement ............................... 17 1.4ThesisOverview ................................. 17 2EQUATIONSOFMOTION ............................. 19 2.1Aircraft-AxisSystem .............................. 19 2.1.1Body-AxisSystem ............................ 19 2.1.2Stability-AxisSystem .......................... 19 2.1.3Earth-AxisSystem ........................... 20 2.2CoordinateTransformations .......................... 21 2.2.1EarthtoBodyFrame .......................... 21 2.2.2StabilitytoBodyFrame ........................ 23 2.3NonlinearEquationsofMotion ........................ 24 2.3.1DynamicEquations ........................... 24 2.3.1.1Forceequations ........................ 24 2.3.1.2Momentequations ...................... 29 2.3.2KinematicEquations .......................... 33 2.3.2.1Orientationequations .................... 33 2.3.2.2Positionequations ...................... 34 2.3.3TheEquationsCollected ........................ 35 2.4LinearizedEquationsofMotion ........................ 36 2.5Examples .................................... 39 2.5.1Linearization ............................... 40 2.5.2AsymmetricMorphing ......................... 43 2.5.3SymmetricConguration ........................ 44 3EXAMPLEOFVARIABLESWEEPAIRCRAFT ................. 45 3.1Design ...................................... 45 3.1.1BiologicalInspiration .......................... 45 3.1.2MechanicalDesign ............................ 46 5

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..................................... 50 3.2.1ComputationalTools .......................... 50 3.2.2SweepDetermination .......................... 52 3.3AerodynamicProperties ............................ 52 3.3.1SymmetricCongurations ....................... 52 3.3.1.1Aerodynamiccoecients ................... 53 3.3.1.2Modaldynamics ....................... 55 3.3.2AsymmetricCongurations ....................... 57 3.3.2.1Flightdynamics ........................ 60 3.3.2.2Modalcharacterization .................... 61 3.3.2.3Crosswindrejection ...................... 63 3.4DynamicProperties ............................... 65 3.4.1MissionScenario ............................. 65 3.4.1.1Divemaneuver ........................ 65 3.4.1.2Turnmaneuver ........................ 66 3.4.2MassDistribution ............................ 66 3.4.3ManeuverAssumptions ......................... 67 3.4.4DiveManuever ............................. 68 3.4.4.1Modeling ........................... 68 3.4.4.2Altitudecontroller ...................... 70 3.4.4.3Time-varyingdynamics .................... 71 3.4.4.4Simulation ........................... 72 3.4.4.5Missionevaluation ...................... 75 3.4.4.6Eectsoftime-varyinginertia ................ 75 3.4.5CoordinatedTurnManeuver ...................... 76 3.4.5.1Modeling ........................... 76 3.4.5.2Turncontroller ........................ 77 3.4.5.3Time-varyingdynamics .................... 78 3.4.5.4Simulation ........................... 79 3.4.5.5Missionevaluation ...................... 82 3.4.5.6Eectsoftime-varyinginertia ................ 83 4CONCLUSION .................................... 85 REFERENCES ....................................... 86 BIOGRAPHICALSKETCH ................................ 89 6

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Table page 3-1Referenceparametersforsymmetricsweep ..................... 49 3-2Setofeigenvalues ................................... 61 3-3Timeconstantsofnon-oscillatorymodes ...................... 62 3-4Modeshapesofnon-oscillatorymodes ....................... 62 3-5Modalpropertiesofoscillatorymodes ........................ 62 3-6Modeshapesofoscillatorymodes .......................... 63 3-7Massdistribution ................................... 67 3-8Inertialmasscharacteristics ............................. 68 7

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Figure page 1-1Surveillancemissionthroughanurbanenvironment ................ 13 1-2Vision-basedpathplanning ............................. 14 1-3Readinessformissioncapability ........................... 15 1-4MorphingMAVs ................................... 15 1-5Modeltostate-spaceowchart ........................... 18 2-1Body-xedcoordinateframe ............................. 19 2-2Stabilitycoordinateframe .............................. 20 2-3Earth-xedcoordinateframe ............................. 20 2-4Rotationthrough 21 2-5Rotationthrough 22 2-6Rotationthrough 23 2-7Asymmetriccongurations .............................. 43 2-8Symmetriccongurations .............................. 44 3-1Picturesofseagulls .................................. 46 3-2Jointsonwing .................................... 47 3-3Floatingelbowjoint ................................. 47 3-4Feather-likeelements ................................. 48 3-5Trackandrunnersystem ............................... 48 3-6Underwingsparstructure .............................. 49 3-7Modelingoftheliftvectors ............................. 50 3-8Modelingofthetrailinglegvectors ......................... 51 3-9Sweepcongurations ................................. 52 3-10Sweepangles ..................................... 53 3-11Variationofliftwithangleofattackforsymmetricsweep ............. 53 3-12Variationofpitchmomentwithangleofattackforsymmetricsweep ....... 54 8

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........... 54 3-14Variationofyawmomentwithangleofsideslipforsymmetricsweep ....... 55 3-15Numberofunstablepolesoflongitudinaldynamicsforsymmetricsweep ..... 56 3-16Numberofunstablepolesoflateral-directionaldynamicsforSymmetricSweep 56 3-17Numberofoscillatorypolesforlongitudinaldynamicswithsymmetricsweep .. 57 3-18Numberofoscillatorypolesforlateral-directionaldynamicswithsymmetricsweep 57 3-19Variationofliftwithangleofattackforasymmetricsweep ............ 58 3-20Variationofpitchmomentwithangleofattackforasymmetricsweep ...... 58 3-21Variationofrollmomentwithrollrateforasymmetricsweep ........... 59 3-22Variationofyawmomentwithangleofsideslipforasymmetricsweep ...... 59 3-23Variationofcoupledaerodynamicsforasymmetricsweep ............. 60 3-24Numberofunstablepolesfordynamicswithasymmetricsweep ......... 60 3-25Numberofoscillatorypolesfordynamicswithasymmetricsweep ........ 61 3-26Eectiveanglesofsideslip .............................. 64 3-27Maximumangleofsideslipatwhichaircraftcantrim ............... 64 3-28Pointmasslocations ................................. 68 3-29Symmetricvelocityprolebasedonconstantthrustmorphing .......... 69 3-30Closed-loopblockdiagram .............................. 71 3-31Plantmodelwithtrimlogic ............................. 72 3-32Symmetricmorphingschedule ............................ 73 3-33Diveresponse ..................................... 73 3-34Longitudinalstates .................................. 74 3-35Elevatorresponse ................................... 74 3-36Simulateddivemaneuver ............................... 75 3-37Eectsofinertiaondiveperformance ........................ 76 3-38Asymmetricvelocityprolebasedonconstantthrustmorphing ......... 77 3-39Open-loopblockdiagram ............................... 78 9

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............................. 79 3-41Asymmetricmorphingschedule ........................... 80 3-42Eectsofmorphingonturnperformance ...................... 81 3-43Lateralperturbationstates .............................. 81 3-44Directionalperturbationstates ........................... 82 3-45Longitudinalperturbationstates .......................... 82 3-46Simulatedturnmaneuver .............................. 83 3-47Eectsofinertiaonturningperformance ...................... 84 10

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1-1 Figure1-1. Surveillancemissionthroughanurbanenvironment Agilityisincreasinglyrequiredforthesevehiclesasthemissiontasksconsidertheightconditionsassociatedwithurbanenvironments.Theclosespacingofobstacleswillrequireavehiclethatcanturnsharplyinasmallradiusbutyetloiterandcruise.Thewindsaroundtheseobstaclessignicantlyvaryindirectionwhichwillrequirethevehicletoincurlargeanglesofsidesliptomaintainsensorpointing.Thedurationforwhichthesensorismaintainedonthetargetiscrucialtocompletingmissionobjectivessuchaslaser-basedswathmapping[ 1 ]andvision-basedpathplanning[ 2 ],asshowninFig 1-2 13

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Vision-basedpathplanning Suchdisparaterequirementsplaceconstraintsonthedesignwithinwhichasinglevehiclecannotlie.Therefore,morphingisbeingincorporatedtoenablemulti-rolecapabilitiesofasinglevehicle.Essentially,thevehiclechangesshapebyalteringparameters,suchasspanorcamber,duringight.Theresultingrangeofcongurationswillhaveanassociatedrangeofightdynamicsand,consequently,maneuvering. 1-3 .Modernavionics,andtheirrespectivesub-systems,haverecentlymadelargeadvancesinthereductionoftheiroverallweightandsize.Asaresult,MAVsarebeginningtobeoutttedwithmoresophisticatedsensorpackagesandcontrolsystems.Itshouldbenotedhowever,thatevenwiththeseadvances,thelargerUAVisstillsuperiortotheMAVintermsofbeingmissioncapable.Duetothefactofthistechnologicalgap,ithasbeensuggestedthataviancharacteristics,andtheireectivebenets,bestudied. 14

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Readinessformissioncapability TwoMAVsutilizingtheconceptofwingmorphing,orshapechanging,(acommoncharacteristicofavianight)canbeshowninFig. 1-4 BFigure1-4. MorphingMAVs:A)capableofhorizontalmorphingB)capableofverticalmorphing TheightdynamicsoftheaircraftshowninFig. 1-4 aresomewhatuniqueanddierentfromthosedenedforsymmetricxed-wingight.Withtheintroductionofmorphing,afewpreviouslymadeassumptionsmustnowbereconsidered.Itshouldbenotedthatmorphingchangesthesystemfromtime-invarianttotime-varying,andasaresult,introducesnewinertialtermsintothedynamics. 15

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3 ].Movingmasswasalsoincludedinthedynamicsofavehiclewithasolarsailthatcouldmoveforcontrolpurposes[ 4 ].Thedynamicsandassociatedtime-varyinginertiawasmodeledforatwo-vehicleformationinwhichaCoulombtethercontrolledtherelativedistanceandmassdistribution[ 5 ].Anotherstudyoptimizedadesignforatwo-vehicleformationwithaexibleappendagewhosemotionalteredtheinertiaproperties[ 6 ].Theinuenceofthrusters,whichexpendmassthroughactivationandthusvarytheinertia,wasinvestigatedusingaformulationoffeedbackandfeedforwardtocanceltheeects[ 7 ].Thetime-varyinginertiaduetothrusterswascoupledwitheectsofuidsloshinginanotherexaminationofspacecraftdynamics[ 8 ].Thesetraditionalcauseshavealsobeenexaminedwithrespecttotheireectsonaircraftalthoughnotnecessarilytothesamedegreeasspacecraft.Fuelburnisoftenneglectedsinceitstimeconstantisslowerthantheightdynamicsofmanyaircraft;however,thateectsontime-varyinginertiawereshownforthecaseofaerialrefuelinginwhichmasswasrapidlytransferredfromthetankertotherecipient[ 9 ].Onasmallerscale,thedynamicsofaapping-wingmicroairvehiclewerestudiedbynotingtheeectofwingmotion[ 10 ].Theintroductionofmorphing,orshape-changingactuation,toanaircraftwillaltertheshapeandmassdistributionofthevehicle,andasaresult,producetime-varying 16

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11 ],rangeandendurance[ 12 ],costandlogistics[ 13 ],actuatorenergy[ 14 ],maneuverability[ 15 ]andairfoilrequirements[ 16 ].Additionally,aeroelasticeectshavebeenoftenstudiedrelativetomaximumrollrate[ 17 18 19 ]andactuatorloads[ 20 ].Morphinghasalsobeenintroducedtomicroairvehiclesforthepurposeofmanueveringcontrol[ 21 22 ].Specically,anaircraftisdesignedthatusesindependentwing-sweep,asshowninFig. 1-4 ,ofinboardandoutboardsectionsonboththerightandleftwings[ 23 ].Thataircraftisshowntousethemorphingforalteringtheaerodynamicsandachieveperformancemetricsrelatedtosensorpointing.Thewingsareabletosweepontheorderofasecond;consequently,thetemporalnatureofthemorphingmustbeconsidered. 17

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1-5 ,isdesignedtoillustratetheprocessforwhichthemorphingcongurationismodeledandthenconvertedintoastate-spacesystem.Thecorrespondingstate-spacesystemsarethenusedtosimulatebasiclineartime-invariant(quasi-static)controlledmaneuverssuchasdivingandturning. Figure1-5. Modeltostate-spaceowchart 18

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24 25 ]. 2-1 Figure2-1. Body-xedcoordinateframe 2-2 .Theresulting^xSaxispointsinthedirectionoftheprojectionoftherelativewindontothexzplaneoftheaircraft.The^ySaxisisoutoftherightwingcoincidentwith^yB,whilethe^zSaxispointsdownwardinthedirectioncompletingthevectorsetdescribedbytheright-handruleandshowninFigure 2-2 19

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Stabilitycoordinateframe 2-3 Figure2-3. Earth-xedcoordinateframe 20

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2.2.1EarthtoBodyFrameAvectormaybetransformedfromtheEarth-xedframeintothebody-xedframebythreeconsectutiveconsecutiverotationsaboutthez-axis,y-axis,andx-axis,respectively.TraditionalightmechanicsdenetheanglesthroughwhichthesecoordinateframesarerelativelyrotatedastheEulerangles.TheEuleranglesareexpressedasyaw,pitch,androll,anditisimportanttonotethataparticularsequenceofEuleranglerotationsisunique[ 26 ].Thefollowingillustratesthetransformationofavector,PE,intheEarth-xedframe,asdenedinEquation 2{1 ,intothebody-xedframe. PE=x^iE+y^jE+z^kE=266664XEYEZE377775(2{1)Therstrotationisthroughtheangleaboutthevector^kE,asshowninFigure 2-4 Figure2-4. Rotationthrough 2{2 21

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2-5 Figure2-5. Rotationthrough 2{3 2-6 .Therotationabout^i00bytheangle,,isreferredtoasR1(),whereR1()istheshort-handnotationusedtodescribetherotationmatrixdenedinEquation 2{4 22

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Rotationthrough 2{5 PB=R1()R2()R3()PE(2{5)Forexample,Earth-xedgravityforcescanbeexpressedinthebody-xedcoordinatesystembyimplementingEquation 2{5 ontheEarth-xedweightvector FGE=26666400mg377775E=266664mgsinmgsincosmgcossin377775B(2{6) 23

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2{7 2.3.1DynamicEquationsTherigidbodyequationsofmotionareobtainedformNewton'ssecondlaw,whichstatesthatthesummationofallexternalforcesactingonabodyisequaltothetimerateofchangeofthemomentumofthebody;andthesummationoftheexternalmomentsactingonthebodyisequaltothetimerateofchangeofthemomentofmomentum(angularmomentum).ThetimeratesofchangeoflinearandangularmomentumarerelativetoaninertialorNewtonianreferenceframe(Earth-xedframe).Newton'ssecondlawcanbeexpressedbythevectorsdenedinEquations 2{8 2{9 dt(mv)E(2{8) dtHE(2{9) 2{8 .Theseassumptionsincludethattheaircraftbeconsideredarigidbody,andthatthemassoftheaircraftremainconstant.Asaresultofassumingthemasstobeconstant,themassterm,seeninEquation 2{8 ,canbemovedoutsideofthetimederivativeandredenedasEquation 2{10 24

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dtvE=maE(2{10)Theaccelerationofanaircraftisnormallymeasuredinthebody-xedframe,therefore,touseEquation 2{10 ,thebody-xedaccelerationmustbetransformedintotheEarth-xedframe.Itisnotedthatsincethetransformationinvolvesarotationofcoordinates,anyvectorinthebody-xedframecanbecalculatedintheEarth-xedframebyusingthetransporttheorem[ 27 ],asseeninEquation 2{11 .Itshouldalsobenotedthatforthefollowingderivations,allofthevectorswillbeexpressedinthebody-xedcoordinatesystem,denotedbythesubscript,B,unlessotherwisestated.Forexample,thevelocityvectorasseenbyanobserverintheEarth-xedframe,expressedinthebody-xedcoordinatesystem,willbedontedasvEB. 2{12 2{13 25

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2{12 iscalledthebody-denedvelocity.Thebody-denedvelocityvector,vBB,issimplythetimederivativeofthepositionvector,r,asseeninEquation 2{14 .Notethatthisisonlythecasewhenthepositionvectorisdescribedbythebody-xedcoordinatesystem. 2{12 ,isdescribedbythecrossproductbetweentheEarth-denedvelocityvector,vEBandatermcalledtheangularvelocityvector.Theangularvelocityvector,E!B,describestheangularvelocityofreferenceframeB(body-xedframe)asviewedbyanobserverinreferenceframeE(Earthxedframe),representedinthebody-xedcoordinatesystem.Theangularveloctyvectorcanbeconvenientlyrewrittenanddened,asseeninEquation 2{15 2{16 26

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27 ],whichisshowninEquation 2{17 2{18 2{19 2{19 isdenedbythebody-xedcoordinatesystem,thenonlythetimederivativeneedstobetaken.Ifthistermisnotdenedbythebody-xedcoordinatesystem,thenthetransporttheoremmustbeappliedtocompensateforthecoordinatechange.TherighthandsideofEquation 2{19 cannowbesolvedforbyrstcomputingthecrossproductbetweentheEarth-denedangularvelocityvector,B!E,andtheEarth-denedvelocityvector,vEB.Thecrossproductisthenaddedtothebody-denedaccelerationvector,aBB.Itshouldnotedthatthebody-denedaccelerationisnotsimplythesecondderivativeoftheofthebody-denedposition.Thistermisdenedastherate 27

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2{20 ,representstheaccelerationoftheaircraftasseenbyanobserverintheinertially-xedEarthframe,representedinthebody-xedcoordinatesystem. 2{8 ,itisrstassumedthatonlythemostsignicantforcesaectthemotionoftheaircraft.Theappliedforcescanthenbebrokendownintovectorcomponentsandarrangedinamannersuchthattheyaredenedbythevector,FEB,asseeninEquation 2{21 FEB=266664FxFyFz377775EB=266664FGx+FAxFGy+FAyFGz+FAz377775EB=XF(2{21)Aresultingsetoffull-order,nonlinearforceequations,asseeninEquation 2{22 ,canbederivedbyinsertingEquations 2{20 2{21 intoEquation 2{8 andrecallingthattheappliedforcesaregivenbyEquations 2{6 2{7 28

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2{2 thatNewton'ssecondlawissatisedbyequatingthesummationofmomentstothetotalrateofchangeofthemomentofmomentum(angularmomentum).ThesamerelationshipusedpreviouslytodenetheEarth-denedaccelerationcanbeusedtodenetheEarth-denedangularmomentum.Itshouldbeagainnotedthatforthefollowingderivations,allofthevectorswillbeexpressedinthebody-xedcoordinatesystem,denotedbythesubscript,B,unlessotherwisestated.Forexample,theangularmomentumvectorasseenbyanobserverintheEarth-xedframe,expressedinthebody-xedcoordinatesystem,willbedontedasHEB.Traditionally,theangularmomentumiscomputedfrommeasurementstakeninthebody-xedcoordinatesysytem.InordertoproperlydescribetheEarth-denedangularmomentumvector,itmustbersttransformedintotheinertially-xedEarthframe.Thistransformationcanbeaccomplishedbythetransporttheorem,asseeninEquation 2{23 2{24 H=I!(2{24)TheangularmomentumdenedinEquation 2{24 canbemaderelavanttoanaircraftbydeningtheinertialtensorintheaircraft'sbody-xedcoordinatesystem,asseeninEquation 2{25 ,andrecallingthattheEarth-denedangularvelocityvectorwaspreviouslydenedinEquation 2{15 29

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2{15 and 2{25 canbeinsertedintoEquation 2{24 toproducetheEarth-denedangularmomentumvector,asseeninEquation 2{26 2{27 2{27 intothelefthandsideofEquation 2{26 andcarryingoutthematrixmultiplicationontherighthandside,resultsintheEarth-denedvectornotationoftheaircraft'sangularmomentum,asseeninEquation 2{28 2{29 2{31 .Therefore,Ixindicatestheresistancetorotationaboutthex-axis(relativelydened).Theproductsofinertiaaredescribedasindicatorstothesymmetryoftheaircraft,asdenedinEquations 2{32 2{34 30

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Duetothefactthatthemasseswereassumedtobepointmasses,theintegralsinEquations 2{29 2{34 canbereducedto: (2{35) (2{36) (2{37) (2{38) (2{39) (2{40) ThecorrespondinginertialratescanbecalculatedbytakingthetimederivativeofEquations 2{35 2{40 ,asshowninEquationrefeqrates. 31

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2{23 isdescribedastherateofchangeoftheEarth-denedangularmomentumvectorasseenbyanoberverinthebody-xedcoordinatesystem,representedinthebody-xedcoordinatesystem.ThistermcanbefoundbysimplytakingthetimederivativeofEquation 2{28 ,asseeninEquation 2{42 2{23 canbefoundbytakingthecrossproductbetweenthepreviouslydenedangularmomentumvector,HEB,asseeninEquation 2{28 ,andtheangularvelocityvector,!EB,asseeninEquation 2{19 .Thistermcanbethentemporarilydenedas,HT,andrewrittenintheformshownbyEquation 2{43 2{23 cannowbesolvedforintermsoftheEarth-denedangularmomentumvector,HEB,byinsertingEquations 2{42 and 2{43 intoEquation 2{23 ,asseeninEquation 2{44 32

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2{44 canbeputintovectornotation,suchthatthevectorcomponentsoftheEarth-denedangularmomentumvector,HEB,aredenedbyindividualmomentterms,asshownbyEquation 2{45 .` 2{45 intoEquation 2{44 andthenequatingsides,asseeninEquation 2{46 33

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2{47 !B=p^i+q^j+r^k=_i+_j+_k(2{47)Toequatebothsidesofequation 2{24 ,itisnecessarythatbothvectorsarerepresentedinthesamecoordinateframe.Thus,bytransformingtheEuleranglesintothebody-xedcoordinatesystem,threenonlinearbodyrateequationscanbewritten,asseeninEquation 2{48 2{49 _=p+q(sin+rcos)tan_=qcosrsin_=(qsin+rcos)sec(2{49) 2{50 34

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2{5 ,inreverseorder,tothebody-denedvelocities,asseeninEquation 2{51 2{51 andthenequatingbothsides,asseeninEquation 2{52 _xEB=uBcoscos+vB(sinsincoscossin)+wB(cossincos+sinsin)_yEB=uBcossin+vB(sinsinsincoscos)+wB(cossinsin+sincos)_zEB=uBsin+vB(sincos)+wBcoscos(2{52)IntegratingEquation 2{52 yieldstheairplane'spositionrelativetotheinertially-xedreferenceframe. 2{53 35

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2{54 36

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2{55 2{56 sin(0+)=sin0cos+cos0sin:=sin0+cos0cos(0+)=cos0cossin0sin:=cos0sin0(2{56)Ageneralsetoflinearizedmotionequationsmaybeobtained,asshowninEquation 2{57 ,byapplyingthesmall-disturbancetheory,combinedwiththepreviouslymadeassumptions,tothenonlinearsetofmotionequations,givenbyequation 2{53 ,andretainingonlytherstorderterms. 37

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2{57 aresetequaltozero,thentheresultingsetoflinearizedmotionequationsarerepresentativeofthosedenedforreferenceight.Iftheassumptionismadethattheaircraftisatitsreferenceightcondition,thedisturbancequantitiesareconsiderednegligibleandthereforesetequaltozero.Applyingthisassumptiontoequation 2{57 ,itisseenthatasetofequationsisdeveloped,asseeninEquation 2{58 ,whichcanbeusedtoeliminateallofthereferenceforcesandmomentsfoundinEquation 2{57 38

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2{58 canbesustitutedbackintoEquation 2{57 ,suchthattheresultinglinearizedmotionequationscanberewrittenanddened,asseeninEquation 2{59 _u=x mgcos0_v=y m+gcos0u0r_w=z mgsin0+u0qL=Ix_pIzx_rM=Iy_qN=Izx_q_=q_=p+rtan0_=rsec0_xE=ucos0u0sin0+wsin0_yE=u0cos0+v_zE=usin0u0cos0+wcos0(2{59)TheperturbationtermsrepresentaerodynamicforcesandmomentsthatcanbeexpressedbymeansofaTaylorseriesexpansion.TheTaylorseriesexpansionmaycontainallofthemotionvariables,butisnormallyreducedtoonlythesignicanttermsrelevanttothatpaticularforceormoment.Forexample,theTaylorseriesexpansionforthechangeinrollmoment,L,maybeexpressedasafunctionofthemoments,forcesandcontrolsurfacedeections,asseeninEquation 2{60 L=@L @uu+@L @vv+@L @ww+@L @qq+@L @pp+@L @rr+@L @aa+@L @rr+@L @ee(2{60) 39

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2{46 .Thenonlinearmomentequationswillrstbelinearizedandthenreducedaccordingtocongurationandaerodynamicassumptions. 2{46 ,thesmall-disturbancetheoryisapplied,thusresultingintheperturbationequationsshowninEquation 2{61 2{61 canberearrangedinsuchamannerthatitisexpressedasadierentialequation,asseeninEquation 2{62 40

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2{62 canbefurtherreduced,asshownbyEquation 2{63 2{63 intermsof_p;_qand_rresultsinthethreeequationsshowninEquation 2{64 _p=Ppp+Pqq+Prr+PLL+PMM+PNN D_q=Qpp+Qqq+Qrr+QLL+QMM+QNN D_r=Rpp+Rqq+Rrr+RLL+RMM+RNN D(2{64)ThecoecientsoftheperturbationtermsinEquation 2{63 areexpressedasfunctionsoftheinertialmoments,productsandrates,asseeninEquation 2{65 41

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2{64 isalsoexpressedasafunctionoftheinertialmoments,products,andrates,asseeninEquation 2{66 2{67 42

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@uu+@L @vv+@L @ww+@L @qq+@L @pp+@L @rr+@L @aa+@L @rr+@L @eeM=@M @uu+@M @vv+@M @ww+@M @qq+@M @pp+@M @rr+@M @aa+@M @rr+@M @eeN=@N @uu+@N @vv+@N @ww+@N @qq+@N @pp+@N @rr+@N @aa+@N @rr+@N @ee(2{67)ThefullyexpandedsetoflinearizedmomentequationscanbeobtainedbyinsertingEquations 2{63 and 2{67 intoequation 2{63 andmultiplyingouttheterms. Asymmetriccongurations Theasymmetriccaseassumesthatthereisnosymmetrytakenwithrespecttotheaircraft'scenterofgravity.Itisalsoassumedthattheaircraftisactivelymorphing,andtherefore,theinertialratesareretained.Themomentequationsspecictotheasymmetricmorphingcase,havepreviouslybeendenedandareshowninEquations 2{63 2{67 .Ifitisdeterminedthattheaircraftnolongermorphs,theinertialratesgotozeroandequation 2{63 canbefurtherreduced,asseeninEquation 2{68 43

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Symmetriccongurations 2{69 2{69 canbereducedtothemomentequations,asseeninEquations 2{59 and 2{70 44

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3.1.1BiologicalInspirationBiologically-inspiredapproachesformorphingarequiteappropriateforminiatureairvehiclesgiventheirsimilaritytobirdsinsizeandairspeed.OptimaldesignsofsuchvehiclesaredicultgiventheuncertaintiesassociatedwithlowReynolds'numbers[ 28 ];[ 29 ];however,adoptingshapesfrombiologicalsystemshasgeneratedsomeeectivedesigns.Obviouslyaerodynamicsareanimportantfeatureofmanybiologicalsystems[ 30 ]asdemonstratedbytestinginwindtunnels[ 31 ].Theconceptsfromaviansystemshavebeenstudiedforightbyconsideringpitching[ 32 ],expandablespan[ 33 ],two-jointsweep[ 34 ]andevenhigh-frequencyapping[ 35 ].Ineachcase,thestudyshowedtheeciencyandperformanceofthebiologicalconceptbutwereunabletorealizetheconceptthroughanactualightvehicle.Theseagullisalogicalchoicefromwhichtoderivebiologicalinspirationsinceitissoadeptatagileyinginwindyconditions.Suchbirdsareroutinelyseentrackingboats,divingtocatchprey,andlandingonbuoysdespiteheavywindsandstronggustsfromdierentdirections.Themissionsenvisionedforaminiatureairvehiclerequireasimilarsetofabilities;therefore,abiomimeticapproachiswarranted.Theskeletalstructureoftheseagullisacriticalcomponentthatenablesightcapability.Inparticular,thejointsattheshoulderandelbowareusedtorotatethewingsandconsequentlyaltertheightdynamics.Suchrotation,asseeninFig. 3-1 ,causesdisplacementinbothverticalandhorizontaldirectionswhichcorrelatestowingdihedralandwingsweep.Thewings,asshowninFig. 3-1 ,willusuallyvarythesweepbetweentheinboardandoutboard.Thevariationresultsfromtheindependentactuationabouttheshoulderand 45

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Picturesofseagulls elbowtovarythehorizontalrotation.Thismorphingprovidesavarietyofchangesintheightcharacteristicssuchasstability,divespeed,andturnradius.Also,thewingsareshowninFig. 3-1 tovarythesweepbetweenrightandleftwingsalongwiththeinboardandoutboard.Thisvariationutilizes4degreesoffreedomresultingfromindependentactuationofshoulderandelbowoneachwing.Thismorphingenablesseveralmaneuversrelatedtohoming,rolling,andrejectingcrosswinds.Emphasisisplacedontherelationshipbetweenwingsweepandmaneuvers.Thesweepisalreadyadesignparameterwhoseeectsonaerodynamicshavebeenstudiedfortraditionalaircraft;however,thestudyofbirdsprovidesadditionalinsightintotheperformancethatmaybeachievableusingindependentmulti-jointsweep.Inthiscase,thecorrelationsbetweensweepanddiveareaugmentedwithcorrelationsbetweensweepandagilityforbothturningandtrimming. 36 ].Thebasicconstructionusesskeletalmembersofaprepregnated,bi-directionalcarbonberweavealongwithrip-stopnylon.Thefuselageandwingsareentirelyconstructedoftheweavewhilethetailfeaturescarbonsparscoveredwithnylon.Theresultingstructureisdurablebutlightweight.Thewingsactuallyconsistofseparatesectionswhichareconnectedtothefuselageandeachotherthroughasystemofsparsandjoints.Thesejoints,asshowninFig. 3-2 46

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Figure3-2. Jointsonwing Itisnotedthatconventionalaileroncontrolsurfacesareomittedfromtheaircraft'snaldesign.Thisfeatureisadirectresultofspan-wiseinconsistenciescreatedbythedynamicrangeofmorphingcongurations.Therefore,theelbowjointsaredesignedinsuchamannerthattheyallowbothhorizontalsweepandrollingtwist.Thismotionisaccomplishedbycreatingaoatingjointthatcloselymimicsthevariousrangesofmotionattainablebyanautomobile'suniversaljoint,asshownifFig. 3-3 Figure3-3. Floatingelbowjoint 47

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3-4 ,withinthejoint. Figure3-4. Feather-likeelements[ http://www.kidwings.com/bodyparts/feathers/graphics/wings/senegalparrotsmall.jpg Thesestructuresretractontoeachotherunderthewingwhenboththeinboardandoutboardaresweptback.Conversely,theycreateafan-likecoveracrosstheensuinggapwhentheinboardissweptbackandtheoutboardissweptforward.Thecontractionandexpansionofthesurfaceareacreatedbythesestructuresissmoothlymaintainedbyatractandrunnersystemimplementedontheouterregionsofeachmember,asseeninFigure 3-5 Figure3-5. Trackandrunnersystem 48

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3-6 Figure3-6. Underwingsparstructure Overall,thevehiclehasaresultingweightof596gandafuselagelengthof48cm.Thereferenceparameters,suchasspanandchord,dependonthesweepconguration.ArepresentativesetoftheseparametersaregiveninTable 3-1 foralimitedsetofsymmetriccongurationsinwhichtheleftandrightwingshaveidenticalsweep. Table3-1. Referenceparametersforsymmetricsweep InboardOutboardReferenceReferenceReference(deg)(deg)Span(cm)Chord(cm)Area(cm) -15-3066.1714.681028.11-10-2073.9713.121003.45-5-1078.8112.38976.250080.3911.84947.1151078.6111.62916.68102073.6111.69885.66153065.7212.13854.76 49

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3.2.1ComputationalToolsAerodynamicsolutionsforthree-dimensionalwingsofanyshapeorsizecanbecalculatedbyusingavortex-latticemodel.Assumingtheowtobeincompressibleandinviscid,thewingismodeledasasetofliftingpanelswitheachcontainingasinglehorse-shoevortex.Bothspan-wiseandchord-wisevariationinliftcanbemodeledasasetofstepchangesfromonepaneltothenext,asshowninFig 3-7 Figure3-7. Modelingoftheliftvectors Locatedatthepanelquarter-chordpositionisaboundvortex,whichshedstwotrailingvortexlines.Therequiredstrengthoftheboundvortexoneachpanelwillneedtobecalculatedbyapplyingasurface-owboundarycondition.Thisboundaryconditionstatesthereiszeroownormaltothesurfaceofthewing.Foreachpanelthisconditionisappliedatthethree-quarter-chordpositionalongthecenterlineofthepanel.Thenormalvelocityismadeupofafreestreamcomponentandaninducedowcomponent.Thisinducedcomponentisafunctionofstrengthsofallvortexpanelsonthewing.Thus,foreachpanelanequationcanbesetupwhichisalinearcombinationoftheeectivestrengthsproducedfromallpanel.Bysolvingtheseequations,onecanproduceamodelthateectivelydescribestheaerodynamicqualitiesandcontrollabilityofanaircraft.Athenavortex-lattice[ 37 ](AVL)isavortex-latticemodelthatisbestsuitedforaerodynamiccongurationswhichconsistmainlyofthinliftingsurfacesatsmallanglesofattackandsideslip.Thesesurfacesandtheirtrailingwakesarerepresentedassingle-layer 50

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3-8 Figure3-8. Modelingofthetrailinglegvectors Athena-vortex-latticeprovidesthecapabilitytoalsomodelslenderbodiessuchasfuselagesandnacellesviasource-doubletlaments.Theresultingforceandmomentpredictionsareconsistentwithslender-bodytheory,buttheaerodynamicsaregenerallychallengingtocompute,thereforethemodelingofbodiesshouldbedonewithcaution.Ifafuselageisexpectedtohavelittleinuenceontheaerodynamicloads,itshouldbeleftoutoftheAVLmodelentirely.Thisexclusionofthebodyisprescribedtoavoidpotentialinaccuraciesfromenteringtheoverallmodel.Athenavortex-latticeassumesquasi-steadyow,whichallowsunsteadyvorticitysheddingtobeneglected.Moreprecisely,itassumesthelimitofsmallreducedfrequency,whichmeansthatanyoscillatorymotion(e.g.,inpitch)mustbeslowenoughsothattheperiodofoscillationismuchlongerthanthetimeittakestheowtotraverseanairfoilchord.Thisassumptionisvalidforvirtuallyanyexpectedightmaneuver.Also,theroll,pitch,andyawratesusedinthecomputationsmustbeslowenoughsothattheresultingrelativeowanglesaresmall,asjudgedbythedimensionlessrotationrateparameters. 51

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3-9 todemonstratetherange. Figure3-9. Sweepcongurations Acoordinatesystemisdenedtofacilitatetheproperdescriptionofeachconguration.Sweepanglesassociatedwiththeinboardsectionsaredenotedas1fortherightwingand3fortheleftwing,whileoutboardsectionsuse2fortherightwingand4fortheleftwing.Theseangles,asshowninFig. 3-10 ,aredenedsuchthatpositivevaluesindicateabackwardsweep.Also,eachangleisdescribedrelativetotheright-sidereferencelinethatisperpendiculartothefuselagereference. 3.3.1SymmetricCongurationsTheaerodynamicsareevaluatedforthesymmetriccongurationsinwhichthesweepoftherightwingisequivalenttothesweepoftheleftwing.Inthiscase,theonlydegreesoffreedomaretheinboardandoutboardangleswhicharesharedbyeachwing.Theaerodynamicsarecomputedusingavortex-latticemethodthatisdesignedtoconsiderthin 52

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Sweepangles airfoils[ 37 ].Asetofrepresentativedataispresentedthatisparticularlyinformativewithrespecttothemaneuversanticipatedforthisclassofvehicle. 3-11 forarangeofsweepcongurations.ThedatashowsthattheaircraftobtainsitshighestCLalongaridgelinecorrelatingtoequalbutoppositesweepofinboardandoutboardsections.Conversely,thisderivativedecreasessignicantlyforcongurationsofinboardandoutboardbeingbothsweptbackorbothsweptforward.Assuch,theliftismoredependentonangleofattackbyutilizingtheadditionaldegreeprovidedbytheelbowtoopposethesweepoftheshoulder. Figure3-11. Variationofliftwithangleofattackforsymmetricsweep Anotherlongitudinalparameter,Cm,isshowninFig. 3-12 forthesweepcongurations.Thisparameterisdirectlyindicativeofthestaticstability;consequently,thepositive 53

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Figure3-12. Variationofpitchmomentwithangleofattackforsymmetricsweep Thedamping-in-rollderivative,towhichClpiscommonlyreferred,isshowninFig. 3-13 forthecongurationspace.Arollratecausesvariationsinangleofattackalongthespanofthewingwhichcreatesarollingmoment.Thisderivativeisnegativeforallsweepcongurations,withthelargestvaluedmagnitudesoccurringinregionscorrespondingtocongurationswithequalbutoppositesweepoftheinboardandoutboardsections.Themagnitudedecreasesforcongurationswithinboardandoutboardbeingbothforwardsweptorbothbackwardsweptwhichsuggestsapotentialitytoauto-rotateorspin. Figure3-13. Variationofrollmomentwithrollrateforsymmetricsweep 54

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3-14 forallsymmetriccongurations.Thisdatarelatesthederivativeofyawmomentwithangleofsideslipwhosepositivitydemonstratesthestabilitycondition.Thestabilityisincreasedasthebackwardsweepincreasesbecausethestabilizingcontributionsofthefuselageandverticaltaildominateasthewingloseseectiveness. Figure3-14. Variationofyawmomentwithangleofsideslipforsymmetricsweep 41 ],asgiveninFig. 1-5 .Theselinearizedmodelshavedecoupledstatesthatallowseparateanalysisoflongitudinaldynamicsandlateral-directionaldynamics.Modelsarecomputedforeverysymmetriccongurationintherangeofsweepanglestoindicatethevariedstabilityproperties.Thelongitudinaldynamicsarestable,asshowninFig. 3-15 ,forthemajorityofobtainablecongurations.Largevaluesofforwardsweepfortheinboardrequirealargevalueofbackwardsweepfortheoutboardtomaintainstability.Thesweepoftheoutboardsectionisallowedtodecreaseastheinboarddecreasesitsforwardsweep.Eventually,thevehiclecanremainstabledespiteasmallvalueofforwardsweepfortheoutboardaslongastheinboardhasalargevalueofbackwardsweep.Stabilityofthelateral-directionaldynamics,asshowninFig. 3-16 ,isachievedforasmallsetofcongurations.Theonlyregionofstabilitycorrespondstocongurations 55

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Numberofunstablepolesoflongitudinaldynamicsforsymmetricsweep withlargevaluesofbackwardsweepofbothinboardandoutboard.Theoneunstablepole,showninFig. 3-16 ,correspondstoaclassicallydenedspiralmodethatiscommonlyfoundtobeunstablewithalargetimeconstant. Figure3-16. Numberofunstablepolesoflateral-directionaldynamicsforSymmetricSweep SomemodalpropertiesofthelongitudinaldynamicsarepresentedinFig. 3-17 toindicatethenumberofcomplexpoles.Eachpairofpolesrelatestoanoscillatorymodesoresponsecharacteristicscanbedirectlyinferred.Inthiscase,thevehicledemonstratesaclassicalsetofphugoidandshort-periodmodesforthemajorityofcongurationsincludingallthosewithbackwardsweepoftheoutboardsections.Thephugoidmodeislostastheoutboardsectionsincreaseinforwardsweepuntileventuallyeventheshort-periodmodeislostforlargevaluesofforwardsweepfortheoutboard.Itcanbesaidthatthe 56

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3-15 ,isdirectlyrelatedtothelossofbothoscillatorymodes,asshowninFig. 3-17 ,orviceversa. Figure3-17. Numberofoscillatorypolesforlongitudinaldynamicswithsymmetricsweep ThenumberofoscillatorypolesisshowninFig. 3-18 forthelateral-directionaldynamics.Itisseenthatvehicleretainstwo-oscillatorypolesregardlessofthesweepconguration.Therefore,itcanbeinferredthatthevehiclehasaclassicdutchrollmodeforallcongurations.Itcanbesaidthattheintroductionofunstablepoles,asshowninFig. 3-16 ,isnotcausedbyachangeinmodenature. Figure3-18. Numberofoscillatorypolesforlateral-directionaldynamicswithsymmetricsweep 57

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3-19 ,andmoment,showninFig. 3-20 ,canbecomparedwithFig. 3-11 andFig. 3-12 ,respectively.Theclearsimilaritybetweenthesymmetricandasymmetricvaluesindicatessomerelationshipbetweenthecongurationscanbeinferred.Inparticular,thevariationcausedbysweepingbackasinglewingaresimilarinnaturetothevariationcausedbysweepingbackbothwings.Themagnitudeissmallerwhensweepingbackthesinglewingsosomelossofeciencyissuggested;however,thestabilityderivativesdisplaythesameshapeforeachsituation. Figure3-19. Variationofliftwithangleofattackforasymmetricsweep Figure3-20. Variationofpitchmomentwithangleofattackforasymmetricsweep 58

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3-21 withFig. 3-13 alongwithvariationofyawmomentwithangleofsideslipshowninFig. 3-22 andFig. 3-14 forasymmetricandsymmetriccongurations.Again,thevariationsaresimilarinnatureforeachsetofcongurationssuggestingasimilarityinowphysicsbutalossofeciencyintheeect. Figure3-21. Variationofrollmomentwithrollrateforasymmetricsweep Figure3-22. Variationofyawmomentwithangleofsideslipforasymmetricsweep Anadditionalsetofaerodynamicparametersarecomputedfortheasymmetriccongurationsthatarenullforthesymmetriccongurations.Theseparameters,whichareshowninFig. 3-23 ,representthecouplingbetweenlongitudinaldynamicsandlateral-directionaldynamics.Thedatashowsthatsweepingtheleftwingcausesadramaticincreaseinmagnitudeoftheseparameters.Sucharesultisexpectedsincetheseparametersreecttheasymmetrythatincreaseswithsweep. 59

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Variationofcoupledaerodynamicsforasymmetricsweep 3-24 ,foranycongurationofasymmetricsweep.Thesystemisshowntohaveoneunstablepoleforthemajorityofcongurationsandthreeunstablepolesforasmallregion.Thesmallregionisindicatedbyalargeforwardsweepoftheinboardsectionandaequaldisplacementofsweeparoundtheoutboardneutralposition. Figure3-24. Numberofunstablepolesfordynamicswithasymmetricsweep ThenumberofoscillatorymodesispresentedinFig. 3-25 todemonstratesomepropertiesofthevehiclemotion.Inthiscase,thevehiclehastheclassical3oscillatorymodesformostcongurationswhentheoutboardhasneutralorbackwardsweep.Asthe 60

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Figure3-25. Numberofoscillatorypolesfordynamicswithasymmetricsweep 3-2 ,indicatesevenstablepolesandoneunstablepolewhichisadistributionexpectedfromFig. 3-24 Table3-2. Setofeigenvalues Eigenvalues -17.59426.401i-37.202-2.78413.394i-0.1920.685i0.0610 FromTable 3-2 ,itcanbeseenthatthedynamicshavetwonon-oscillatorymodes.Thetimeconstantsofthesemodes,asshowninTable 3-3 ,indicatestheonemodehasastableconvergenceandtheotherhasanunstabledivergence.Thestableconvergenceisatleasttwoordersofmagnitudefasterthantheunstabledivergence. 61

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Timeconstantsofnon-oscillatorymodes ModeEigenvalueTimeConstant 1-37.200.026920.061-16.393 Theightmotionassociatedwitheachofthesemodesisdeterminedbythemodeshapes.SuchshapesaregiveninTable 3-4 todescribetherelativevalueofeachstateduringtheresponse.Theconvergenttermischaracterizedbymostlyrollratewithminorcontributionsfromangleofattack,rollangleandpitchrate.Thismodeissimilarinnaturetotheclassicallydenedrollmode.Thedivergenttermischaracterizedbyfullycoupledmotioninwhichtherollangleisvaryingalongwithprimarilytheyawrate,butalsotheforwardvelocity.Thismodeissimilarinnaturetotheclassicallydenedunstablespiralmode. Table3-4. Modeshapesofnon-oscillatorymodes statemode1mode2 forwardvelocity-0.0007-0.1324angleofattack-0.03980.0163pitchrate-0.05380.0003pitchangle-0.00140.0053angleofsideslip-0.00030.0100rollrate0.99710.0557yawrate-0.02310.3811rollangle-0.02680.9131 Table 3-2 alsoindicatesthattheightdynamicshavethreeoscillatorymodes.ThevaluesofnaturalfrequencyanddampingaregiveninTable 3-5 foreachmode.Themodewiththelowestnaturalfrequencyisunstablewhiletheothermodesarestable. Table3-5. Modalpropertiesofoscillatorymodes ModeEigenvalueFrequency(rad/s)Damping 3-17.59426.401i31.730.5464-2.78413.394i13.680.2035-0.1920.685i0.710.270 62

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3-6 ,showsthatmodes3and5areprimarilydominatedbylongitudinalmotionwithonlyasmallcouplingtothelateral-directionalmotion,whilemode4isthedirectopposite.Suchmotionisnotentirelyunexpectedsinceeventhesymmetriccongurationshadoscillatorymodesaectingboththelongitudinalandlateral-directionaldynamics.Assuch,mode3haspropertieswithsomesimilaritytoashort-periodmode,whilemode4haspropertiessimilartoadutch-rollmodeandmode5similartoaphugoidmode. Table3-6. Modeshapesofoscillatorymodes Mode3Mode4Mode5statemagnitudephase(deg)magnitudephase(deg)magnitudephase(deg) forwardvelocity0.02-42.60.0195.30.990.0angleofattack0.49-102.60.0536.10.08-179.1pitchrate0.620.00.0787.70.050.2pitchangle0.02-123.70.005-14.00.07-105.5angleofsideslip0.0002-61.20.0781.30.0002-20.3rollrate0.61-28.50.24-66.50.0051.5yawrate0.02-129.10.970.00.003-82.8rollangle0.02-152.20.02-168.30.007-104.2 63

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3-26 Figure3-26. Eectiveanglesofsideslip Theangleofsideslipatwhichtheaircraftcantrimisanindicatoroftheamountofcrosswindinwhichtheaircraftcanmaintainsensorpointing.Arepresentativedemonstration,showninFig. 3-27 ,presentsthemaximumpositivevaluesforangleofsideslipatwhichtheaircraftcantrim.Thewingsareconstrainedinthisdemonstrationsuchthatinboardandoutboardanglesareidenticalwhichlimitsthedegreesoffreedomandfacilitatespresentation.Also,eachconditioncorrespondstothelargestangleofsideslipatwhichtheaircraftcantrimgivendeectionlimitsof15degfortherudderandelevatoralongwithaileron. Figure3-27. Maximumangleofsideslipatwhichaircraftcantrim 64

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3-27 demonstratesthatwingsweepisbenecialforsensorpointing.Specically,aforward30degsweepoftheleftwingandabackward30degsweepoftherightwingallowsanangleofsideslipof44degtobemaintained.Thismaximumangledecreasesastheleftwingdecreasesitsforwardsweepandtherightwingdecreasesitsbackwardsweep.Thevehicleiseventuallyunabletotrimatanypositiveangleofsideslipwhenthebothwingsaresweptbackward. 3.4.1MissionScenarioThevariablewing-sweepvehicleisdesignedforsurveillanceinurbanoperations.Inparticular,itisdesignedtoallowightinconstrainedareaswithlimitedairspace.Arepresentativemissionisplacingasensorintoawindowonabuildingwhichisclosetootherbuildings. 23 ]. 65

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38 ]andtextbooks[ 41 39 40 ].Theattitudeandvelocityequationsdealwithkinematicsoftheaircraftininertialreferenceframessoarenotaectedbymorphing.Theforceequationsrelatetheaerodynamicforcestogravitationalandthrustforcessoalsoarenotaectedbymorphing.Conversely,themomentequationsrelatetheaerodynamicmomentstothemassdistributionoftheaircraftsoareverymuchaectedbymorphing.Anelementalbreakdownoftheaircraft'smassdistributionallowsformoreaccurateinertialmomentsandratestobecomputed.Thesemassesarerepresentedaspointmassesandarelocatedatsomedistancefromtheaircraft'scenterofgravity.Thecenterofgravityisafunctionofwingmorph;therefore,ittranslatesalongthethree-dimensionalbody-axisaccordingly.Themagnitudeofthistranslationalchangeisfoundtobenegligible 66

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3-28 .NotethatinFigure 3-28 ,thecoordinatesystemisuniquetothemodelingprogramandisorientedoppositetotheearlierdenedaircraftbody-axis.ThisgureisaccompaniedbyTable 3-7 ,whichliststhemassesforeachsection.ThesemassesarethenusedtocalculatetheindividualinertialmomentslistedinTable 3-8 Table3-7. Individualpointmasses 1302955015884545 67

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Pointmasslocations Table3-8. Characteristicsofelementsgivenascentroidposition(in)andmomentsofinertia(gin2) elementSymbolXYZIxxIyyIzzIxyIxzIyz relatetheunsteadyaerodynamicsbutwillincludethedominantsteady-stateaerodynamics.Thetime-varyinginertiasarethenintroducedtoaccountforthemorphingusingtheexpressionsderivedinSection 2.3.1.2 3.4.4.1ModelingThevehicleisconstrainedinthismaneuvertosimplifythecongurationspace.Thephysicalvehiclehas4degrees-of-freedominthateachinboardandoutboardcansweepindependentlyontheleftandrightsides;however,thissimulationconstrainstheleftandrightwingstosymmetricsweepwiththeoutboardhavingtwiceasmuchsweepastheinboard.Thisconstraintlimitsthesystemtoasingledegree-of-freedomwhichisappropriateforthelongitudinalnatureofthealtitudechangeassociatedwiththemission. 68

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3-29 ,torelateeachmodel. Figure3-29. Symmetricvelocityprolebasedonconstantthrustmorphing:0deg( {2{ ),5deg( {/{ ),10deg( {{ ),15deg( {{ ),20deg( {.{ ),25deg( {{ ),30deg( {4{ ) Figure 3-29 relatestheoveralldragtovelocitywhereeachtrendline(designatedbylinetype),representsamorphingconguration.AconstantthrustoftwoNewtonsischosenand,thusalineisdrawntorepresentthisvalue.Itisshownthatasthewingsaremorphedfurtherback,theintersectionatwhichthetrendlinescrosstheconstantthrustlinesteadilyincreases.ThispointofintersectionrepresentsthevelocitynecessarytomaintaintwoNewtonsofdragatthatconguration.Overall,Figure 3-29 suggeststhatbysweepingthewingsback,highervelocitiescanbeattained,whilemaintainingaconstantdrag(equaltothrustintrimmedcases).Thistrendisasensibleresultduetothefactthatlesssurfaceareaisexposedtotheoncomingairowwhilethewingsaremorphedbackward. 69

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42 ]usingafeedbackelement,K,andafeedforwardelement,k,alongwithanintegrator.Actually,thedesignisbasedonshort-perioddynamicstoavoidthepolesandzeroesassociatedwiththephugoiddynamics.Theguaranteeofnosteady-stateerrorinresponsetoastepcommandisonlyassociatedwiththeshort-periodmodelbutthefull-orderdynamicsusedinthesimulationstillshowanacceptableresponse.Anouter-loopcontrollerisderivedtoaecterrorinaltitude.Thedierencebetweencommandedaltitudeandmeasuredaltitudeisprocessedthrougharst-orderlter,F,toshapethephaseproperties[ 24 ].Theresultingsignalissomewhatindicativeofapitchcommandsoitsderivativeisusedasapitch-ratecommandfortheinner-loopcontroller. 70

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3-30 ,incorporatesthevariouselementsusingappropriatefeedbackinthemulti-looparchitecture.Anadditionalfeedforwardelement,Z,isincludedinthedesigntoaectthepitchduringtheresponse. a 6 ? Closed-loopblockdiagram 3-30 ,isthussomewhatcomplicatedinitsimplementation.Adiscretizedtypeofmorphingisutilizedinthesimulationtorepresentthesymmetricwing-sweepvariations.ThephysicalaircraftshowninFigure 3-9 hasasweepthatvariesasacontinuousfunctionoftime;however,adiscretizedversionofthismorphingsimpliesthesimulationwithoutlossofgenerality.Inthiscase,thedynamicsassumea5-degreesweepcanbeaccomplishedinstantaneouslybuteach5-degreeincrementmustbeseparatedbysomeminimumtime.Astandardstate-spaceformulationisusedtorepresenttheplantasshowninFigure 3-31 .Theplantdynamics,whichvarywithmorphingpositionandrate,aregivenbythequadrupleoffA;B;C;Dg. 71

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-y Plantmodelwithtrimlogic TheelementofXisusedtoaccountforthechangeintrimconditionsforeachconguration.Considerthatatanypointintime,t,thetotalstatevalue,x(t),isactuallytheadditionofatrimvalue,xo(t),andperturbation,x(t),suchthatx(t)=xo(t)+x(t).Thistotalstatemustremaincontinuousdespitethemorphingeventhoughthetrimvalue,xo(t),willchange.Sincetheplantmodelusesstateperturbationsasafeedback,thenthelogicofXissuchthatx(t+t)=xo(t)+x(t)xo(t+t)forsomeintegrationstep-sizeoft.TheelementofUissimilarinnaturetoX.Inthiscase,theelementisusedtoreectthevariationsinelevatorpositionassociatedwithtrim. 3-32 ,isvariedforsimulationsbasedonfastmorphingorslowmorphing.Thealtitudevariation,showninFigure 3-33 ,isthefundamentalmetricusedtoevaluateperformance.Inthiscase,themorphingvehiclesareabletochangetherequiredaltitudeandreturntostraight-and-levelightwithin2.2s.Thefastandslowmorphingaresimilarfortherstsecondbutthenbegintodivergesomewhat.Thefastmorphinghas 72

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Symmetricmorphingschedule:morphingcongurationforfastmorphing( | ),slowmorphing( reachedafullyswept-backcongurationatthistimesoitsresponseahassomewhatlargerovershootthanthatseenwiththeslowmorphing.Thisobservationisadirectresultofthefastmorphingcaseacquiringahighervelocitysoonerandretainingitforalongerperiodoftime.Twoothercasesareconsiderwheremorphingisnotutilized.Thesecasesincludedivemaneuversperformedwiththenominalstraight-wingandfullysweptcongurations.ItisseeninFig. 3-33 thatthenominalcongurationreachesthethedesiredaltitudetheslowestbuthasthesmallestovershoot,whilethefullysweptcongurationreachesthedesiredaltitudemuchfaster,yethasthelargestovershoot. Figure3-33. Diveresponse:altitudeinresponsetofastmorphing( | ),slowmorphing( ThestatesassociatedwithpitchareshowninFigure 3-34 inresponsetothealtitudecommandwhilemorphing.Theslowmorphing,incomparisontothefastmorphing,incurs 73

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3-33 Figure3-34. Longitudinalstates:pitchangle(left)andpitchrate(right)inresponsetofastmorphing( | ),slowmorphing( Thelateral-directionalstatesarefoundnottovarywithalongitudinaldivemaneuver.Thisresultissomewhatexpected,duetothefactthattheaircraftissymmetricallymorphingandthusthecross-couplinginertiasarecancelledoutduetosymmetry.Finally,theelevatorangleisshowninFigure 3-35 todemonstratetheresponsedoesnotincurexcessiveactuation.Themorphingcertainlyinuencestheelevatorinthatthecontroleectivenessdecreasesasthewingsaresweptback.Assuch,theslowmorphinghastheslowestspeedbutalsousesthesmallestrotationofelevator. Figure3-35. Elevatorresponse:altitudeinresponsetofastmorphing( | ),slowmorphing( 74

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3-33 ;however,ifconstrainedtoacertainspaceandtime,theresultingightpathsassociatedwithmorphingmissthewindowandintersectthesideofthebuilding,asseeninFigure 3-36 .Thisfailuredirectlyresultsfromtheinabilitytoaccountfortime-varyingeectsinthecontrollerdesign.NotethatinFigure 3-36 thecoordinatesarebasedonanEarth-xedframewithaxesdenedbyX,Y,andH. Figure3-36. Simulateddivemaneuver:altitudeinresponsetofastmorphing( | ),slowmorphing( 3-37 .Itshouldbenotedthattheinertialrates,asshowninEquationrefeq41,arecalculatedfromwingscontaininglessthanfteenpercentoftheaircraft'soverallmass.Whenthemassesofthewingsegmentsareincreased(orthetimetomorphisdecreased,asshownbythefastmorphinginFig. 3-37 ),thenthecontributionsfromtheinertialratesbecomemoreprominentinthe 75

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Figure3-37. Eectsofinertiaondiveperformance:altitudeinresponsetofastmorphing( | ),slowmorphing( Dierentapproachesmaybetakentoaltertheightcontroller,buteachhasit'sownchallenges.Acompensatortoregulatethepitchisdicultbecausethegainsmustbeoptimizedtothedynamicsbutthosedynamicsarerapidlychangingduringthemorphing.Alternatively,atrackingcontrollercanbeusedafterthemaneuverwhenthedynamicsareknownandxedbutthevehiclemustre-hometowardstheoriginalwaypointorre-locatethewindowusingvisionfeedbackandthencomputeanewtrajectory. 3.4.5.1ModelingThevehicleisconstrainedinthismanuevertoagainsimplifythecongurationspace.Thismaneuverconstrainstheleftandrightwingstoequal-but-oppositeasymmetricsweepwiththeoutboardhavingnorelativesweepcomparedtotheinboard.Thisconstraint,likethedivemaneuver,limitsthesystemtoasingledegree-of-freedomwhichisappropriateforthelateral-directionalnatureofthechangeinturnperformanceassociatedwiththemission. 76

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3-38 ,torelateeachmodel. Figure3-38. Asymmetricvelocityprolebasedonconstantthrustmorphing:30deg( {2{ ),25deg( {/{ ),20deg( {{ ),15deg( {{ ),10deg( {.{ ),5deg( {{ ),0deg( {4{ ) Fig. 3-38 relatestheoveralldragtovelocitywhereeachtrendlinerepresentsamorphingconguration.AconstantthrustoftwoNewtonsisagainchosenasthedesiredvalue.Itisshownthatasthewingsareasymmetricallymorphed(equalbutopposite,wheretheangleismeasuredwithrespecttotherightwing),theintersectionatwhichthetrendlinescrosstheconstantthrustlinesteadilydecreases.ThispointofintersectionrepresentsthevelocitynecessarytomaintaintwoNewtonsofdragatthatconguration.Overall,Fig. 3-38 suggeststhatbyasymmetricallysweepingthewingsinanequalbutoppositemanner,slowervelocities(inabankedturn)areattainedwhilemaintainingaconstantdrag(equaltothrustintrimmedcases).Thevelocitiesfoundateachintersectionarethenusedascorrespondingtrimvelocitieswhenperformingthecoordinatedturnmaneuver. 77

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3-39 u ?uwqprFigure3-39. Open-loopblockdiagram 3-39 ,isslightlydierentfromtheplantmodelderivedforthedivemaneuverinFigure 3-30 ,itisstillsomewhatcomplicatedinitsimplementation.Adiscretizedtypeofmorphingisagainutilizedinthesimulationtorepresenttheasymmetricwing-sweepvariations.Theturnsimulationisidenticaltothedivesimulation 78

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3-40 .Theplantdynamics,whichvarywithmorphingpositionandrate,aregivenbythequadrupleoffA;B;C;Dg. u -y Plantmodelwithtrimlogic TheelementofXisagainusedtoaccountforthechangeintrimconditionsforeachconguration.Thesameconsideration(asthatforthedivesimulation)ismadetoaccountforthecontinuityofthetotalstatedespitethemorphing,aswellasthefeedbackofperturbations.TheelementsA,EandRaresimilarinnaturetoU,asseeninFigure 3-31 .Inthiscase,theelementsareusedtoreectthevariationsinaileron,elevator,andrudderposition,respectively,associatedwithtrim. 79

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3-41 ,isvariedforeachsimulationbasedonfastmorphingorslowmorphing. Figure3-41. Asymmetricmorphingschedule:morphingcongurationforfastmorphing( | ),slowmorphing( Theturnvariation,showninFigure 3-42 ,isthefundamentalmetricusedtoevaluateperformance.Inthiscase,themorphingvehiclesareabletochangetherequiredturningradiusandreturntostraight-and-levelightata270degreeheadingchange.Thefastandslowmorphingaresimilarduringtherstandlastsectionsofthemaneuverbutvarysomewhatthroughouttheactualturn.Theresponseofthemorphingcasescloselyresemblethatofthenon-morpedsweptresponseduetothefactthemorphingcasesreachthefullysweptcongurationatanearliertime.Therefore,themorphingresponsescontainalargerportionproducedfromthesweptwingcongurationthanthatoftheslowmorphingresponsewhichtakeslongertoachievethesameconguration.ThelateralperturbationstatesareshowninFigure 3-43 astherollangleandrollrate.Itisseenthatthefastermorphinginboththerollangleandrateperturbationsachievehighermagnitudes.Therollangledisplaysthistrendthroughouttheentiremaneuver,whiletherollrateonlyhaslargermagnitudesatthetransientregions. 80

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B CFigure3-42. Eectsofmorphingonturnperformance:turninresponsetofastmorphing( | ),slowmorphing( Figure3-43. Lateralperturbationstates:rollangle(left)androllrate(right)inresponsetofastmorphing( | ),slowmorphing( ThedirectionalperturbationstatesinFigure 3-44 indicatetheyawrateandsideslipangle.Itisnoticedthatthefastermorphingincurslargerperturbationsinbothdirectionalstatesthroughouttheentiremaneuver. 81

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Directionalperturbationstates:yawrate(left)andsideslipangle(right)inresponsetofastmorphing( | ),slowmorphing( Itisnotedthattheturnmaneuverrequirestheaircrafttomorphasymmetrically,andthereforeresultsincoupleddynamics.ThestateperturbationsassociatedwithpitchareshowninFigure 3-45 inresponsetothesimplecontrolsurfacecommandwhilemorphing.Thefastmorphing,incomparisontotheslowmorphing,incursagreaterpitchrateperturbationatbothtransientregions,yetbehavesverysimilarduringthesteadystateregion.Incontrast,thepitchangleperturbationassociatedwithfastmorphinghasahighermagnitudethroughouttheentiremaneuver. Figure3-45. Longitudinalperturbationstates:pitchangle(left)andpitchrate(right)inresponsetofastmorphing( | ),slowmorphing( 82

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3-46 .Thisfailuredirectlyresultsagainfromtheinabilitytoaccountfortime-varyingeectsinthecontrollerdesign.NotethatinFigure 3-36 ,thecoordinatesarebasedonanEarth-xedframewithaxesdenedbyX,Y,andH. Figure3-46. Simulatedturnmaneuver:turninresponsetofastmorphing( | ),slowmorphing( | ),xedstraight( TheeectsofmorphingonthetrajectoryoftheightpathcanbetterbeseeninFigs. 3.4.5.4 3.4.5.4 83

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3-47 ,thanthatseenfordiveinthepreviousmission,asseeninFigure 3-37 Figure3-47. Eectsofinertiaonturningperformance:turninresponsetofastmorphing( | ),slowmorphing( Theightcontrollerforthismissionalsoneedstobealteredtocompensateforthetime-varyingparametersassociatedwithmorphing.Simplyintroducingacompensatortoregulatethelateral-directionalstatesischallengingbecausethegainsmustbeoptimizedtothedynamicsbutthosedynamicsarerapidlychangingduringthemorphing.Itispointedoutthatthedynamicsarenowcoupled(duetoasymmetricmorphing)andthereforethecontrollermustalsocompensateforthelongitudinalstates.Atrackingcontrollermayalsobeusedbutthesamedicultiesarisethatwerepointedoutwiththedivemaneuver 84

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[1] Slatton,K.C.,W.E.Carter,R.L.Shrestha,andW.Dietrich(2007),AirborneLaserSwathMapping:Achievingtheresolutionandaccuracyrequiredforgeosurcialresearch,Geophys.Res.Lett.,34,L23S10,doi:10.1029/2007GL031939. [2] J.J.Kehoe,J.Grzywna,R.Causey,R.Lind,M.NechybaandA.Kurdila,"ManeuveringandTrackingforaMicroAirVehicleusingVision-BasedFeedback,"SAEWorldofAviationCongress,November2004. [3] Spenny,C.H.andWilliams,T.E.,\LibrationalInstabilityofRigidSpaceStationduetoTranslationofInternalMass,"JournalofGuidance,ControlandDynamics,Vol.14,No.1,January-February1991,pp.31-35. [4] Wie,B.,\SolarSailAttitudeControlandDynamics,Part2,"JournalofGuidance,ControlandDynamics,Vol.27,No.4,July-August2004,pp.536-544. [5] Natarajan,A.andSchaub,H.,\LinearDynamicsandStabilityAnalysisofaTwo-CraftCoulombTetherFormation,"JournalofGuidance,ControlandDynam-ics,Vol.29,No.4,July-August2006,pp.831-838. [6] Oliver,R.I,andAsokanthan,S.F,\Control/StructureIntegratedDesignforFlexibleSpacecraftundergoingOn-OrbitManeuvers,"JournalofGuidance,ControlandDynamics,Vol.20,No.2,March-April1997,pp.313-319. [7] Thurman,S.W.andFlashner,H.,\RobustDigitalAutopilotDesignforSpacecraftEquippedwithPulse-OperatedThrusters,"JournalofGuidance,ControlandDynamics,Vol.19,No.5,September-October1996,pp.1047-1055. [8] Hill,D.E.,Baumgarten,J.R.andMiller,J.T.,\DynamicSimulationofSpin-StabilizedSpacecraftwithSloshingFluidStores,"JournalofGuidance,ControlandDynamics,Vol.11,No.6,November-December1988,pp.597-599. [9] Venkataramanan,S.andDogan,A.,\DynamicEectsofTrailingVortexwithTurbulenceandTime-VaryingInertiainAerialRefueling,"AIAA-2004-4945. [10] Vest,M.S.andKatz,J.,\AerodynamicStudyofaFlapping-WingMicroUAV,"AIAA-99-0994. [11] Bowman,J.,\AordabilityComparisonofCurrentandAdaptiveandMultifunctionalAirVehicleSystems,"AIAA-2003-1713. [12] Gano,S.E.andRenaud,J.E.,\OptimizedUnmannedAerialVehiclewithWingMorphingforExtendedRangeandEndurance,"AIAA-2002-5668. [13] Bowman,J.,SandersB.andWeisshar,T.,\EvaluatingtheImpactofMorphingTechnologiesonAircraftPerformance,"AIAA-2002-1631. 86

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Prock,B.C.,Weisshaar,T.A.andCrossley,W.A.,\MorphingAirfoilShapeChangeOptimizationwithMinimumActuatorEnergyasanObjective,"AIAA-2002-5401. [15] Rusnell,M.T.,Gano,S.E.,Perez,V.M.,Renaud,J.E.andBatill,S.M.,\MorphingUAVParetoCurveShiftforEnhancedPerformance,"AIAA-2004-1882. [16] Secanell,M.,Suleman,A.andGamboa,P.,\DesignofaMorphingAirfoilforaLightUnmannedAerialVehicleusingHigh-FidelityAerodynamicShapeOptimization,"AIAA-2005-1891. [17] Khot,N.S.,Eastep,F.E.andKolonay,R.M.,\MethodforEnhancementoftheRollingManeuverofaFlexibleWing,"JournalofAircraft,Vol.34,No.5,September-October1997,pp.673-678. [18] Gern,F.H.,Inman,D.J.andKapania,R.K.,\StructuralandAeroelasticModelingofGeneralPlanformWingswithMorphingAirfoils,"AIAAJournal,Vol.40,No.4,April2002,pp.628-637. [19] Bae,J.,Seigler,T.M.,Inman,D.J.andLee,I.,\AerodynamicandAeroelasticConsiderationsofaVariable-SpanMorphingWing,"JournalofAircraft,Vol.42,No.2,March-April2005,pp.528-534. [20] Love,M.H.,Zink,P.S.,Stroud,R.L.,Bye,D.R.andChase,C.,\ImpactofActuationConceptsonMorphingAircraftStructures,"AIAA-2004-1724. [21] Abdulrahim,M.andLind,R.,\FlightTestingandResponseCharacteristicsofaVariableGull-WingMorphingAircraft,"JournalofAircraft,inreview. [22] Abdurahim,M.andLind,R.,\ControlandSimulationofaMulti-RoleMorphingMicroAirVehicle,"JournalofGuidance,ControlandDynamics,inreview. [23] Grant,D.T.,Abdulrahim,M.andLind,R.,\FlightDynamicsofaMorphingAircraftutilizingIndependentMultiple-JointWingSweep,"AIAAAtmosphericFlightMechanicsConference,Keystone,CO,August2006,AIAA-2006-6505. [24] Stevens,B.L.andLewisL.L.,AircraftControlandSimulation,2nded.JohnWiley&Sons,Inc.,2003 [25] Yechout,T.R.,Morris,S.L.,Bossert,D.E.andHallgrenW.F.,IntroductiontoAir-craftFlightMechanics:Performance,StaticStabilityDynamicStability,AndClas-sicalFeedbackControl,AIAAEducationalSeries,AmericanInstituteofAeronauticsandAstronautics,Inc.,2003 [26] CraneIII,C.D.andDuy,J.,KinematicAnalysisofRobotManipulators,CambridgeUniv.Press,Cambridge,NY,1998,pp.4-17. [27] Rao,A.V.,DynamicsofParticlesandRigidBodies:ASystematicApproach,CambridgeUniv.Press,Cambridge,NY,2006,pp.42-51. 87

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Shyy,W.,Berg,M.andLjungqvist,D.,\FlappingandFlexibleWingsforBiologicalandMicroAirVehicles,"ProgressinAerospaceSciences,Vol.35,No.5,1999,pp.455-506. [29] Arning,R.K.andSassen,S.,\FlightControlofMicroAirVehicles,"AIAA-2004-4911. [30] Norberg,U.M.L.,\Structure,FormandFunctionofFlightinEngineeringandtheLivingWorld,"JournalofMorphology,No.252,2002,pp.52-81. [31] Lazos,B.S.,\BiologicallyInspiredFixed-WingCongurationStudies,"JournalofAircraft,Vol.42,No.5,September-October2005,pp.1089-1098. [32] Kim,J.andKoratkar,N.,\EectofUnsteadyBladePitchingMotiononAerodynamicPerformanceofMicrorotorcraft,"JournalofAircraft,Vol.42,No.4,July-August2005,pp.874-881. [33] Heryawan,Y.,Park,H.C.,Goo,N.S.,Yoon,K.J.andByun,Y.H.,\DesignandDemonstrationofaSmallExpandableMorphingWing,"ProceedingsofSPIE-Volume5764,May2005,pp.224-231. [34] Liu,T.,Kuykendoll,K.,Rhew,R.andJones,S.,\AvianWings,"AIAA-2004-2186. [35] Raney,D.L.andSlominski,E.C.,\MechanizationandControlConceptsforBiologicallyInspiredMicroAirVehicles,"AIAA-2003-5345. [36] Abdulrahim,M.,Garcia,H.andLind,R.,\FlightCharacteristicsofShapingtheMembraneWingofaMicroAirVehicle,"JournalofAircraft,Vol.42,No.1,January-February2005,pp.131-137. [37] Drela,M.andYoungren,H.,\AVL-AerodynamicAnalysis,TrimCalculation,DynamicStabilityAnalysis,AircraftCongurationDevelopment,"AthenaVortexLattice,v.3.15,http://raphael.mit.edu/avl/. [38] Duke,E.L.,Antoniewicz,R.F.andKrambeer,K.D.,DerivationandDenitionofaLinearAircraftModel,NASA-RP-1207,August1988. [39] Nelson,R.C.,FlightStabilityandAutomaticControl,TheMcGraw-HillCompanies,Boston,MA,1998. [40] Roskam,J.,AirplaneFlightDynamicsandAutomaticFlightControls,DARcorporation,Lawrence,KS,2001. [41] Etkin,B.andReidL.D.,DynamicsofFlightStabilityandControl,3rded.JohnWiley&Sons,Inc.,1996 [42] Ogata,K.,ModernControlEngineering,Prentice-Hall,UpperSaddleRiver,NJ,2002,pp.843-909. 88

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DanielThurmondGrantwasborninGainesville,Floridain1983.Asachild,heoftenfoundhimselfintriguedbythemechanicaloperationofremote-controlledtoys.Whennotlostinapileofnutsandbolts,hefoundenjoymentinbuildingmodelairplaneswithhisfather.Thecombinationofthesetwohobbieseventuallyledhimtodevelopastrongpassionforaeronauticalengineering.DanielgraduatedfromSantaFeHighSchoolin2002,afterwhichheenrolledattheUniversityofFlorida.WhileanundergraduateattheUniversityofFlorida,DanieljoinedtheMicro-AerialVehicle(MAV)Laboratory,wherehelearnedhowtobuildandysmallUnmannedAerialVehicles(UAVs).Thishands-onexperienceallowedDanieltoco-leadaDesign,Build,Fly(DBF)teamthatplacedninthataninternationalAIAAstudentcompetitionheldinWichita,Kansas.Asasenior,DanieljoinedtheFlightControlLabattheUniversityofFlorida,wherehedesigned,builtandewamulti-jointed,wing-sweepingMAV.Danielpresentedhismorphingworkatthe2006Guidance,Navigation,andControlConferenceinKeystone,Colorado,andasaresult,wontheAtmosphericFlightMechanicsstudentbestpaperaward.Upongraduatingin2006,DanielenrolledintograduateschoolattheUniversityofFlorida.Inthespringof2009,DanielreceivedhisMaster'sdegreeinaerospaceengineering,andcontinuestoworktowardshisPh.D.intheeldoftime-varyingmorphingsystems.AmajorhighlightofDaniel'sgraduatecareerthusfar,wastheopportunityforhimtopresenthisworkforCongressduringtheeveningofaPresidentialStateoftheUnionAddress. 89