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Polynomial Chaos Analysis of Micro Air Vehicles in Turbulence

Permanent Link: http://ufdc.ufl.edu/UFE0043683/00001

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Title: Polynomial Chaos Analysis of Micro Air Vehicles in Turbulence
Physical Description: 1 online resource (188 p.)
Language: english
Creator: Roberts, Brian C
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: chaos -- microairvehicle -- path -- polynomial -- turbulence -- uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Use of unmanned vehicles is expected to continue to grow. As the variety of missions conducted by unmanned vehicles increases, these vehicles will require greater precision while operating in environments with stochastic disturbances. Thus, understanding the nature of the impact of stochastic disturbances on the vehicles, developing methods to control vehicles in the presence of disturbances, and planning missions using the knowledge of the disturbances will be required for unmanned vehicles to reach their full potential. This dissertation presents a methodology for planning missions for unmanned vehicles operating in the presence of stochastic disturbances. The methodology is shown for a micro air vehicle flying in significant turbulence. Wind tunnel testing characterizes the effect of turbulence intensity on the open loop dynamics of a micro air vehicle. This knowledge and understanding of atmospheric turbulence is combined in the framework of polynomial chaos to make it possible to control and simulate micro air vehicle flight using known methods. Path evaluation strategies are developed to select mission profiles to reduce likelihoods of collision and increase likelihoods of sensory mission success.
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 Brian C Roberts.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043683:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043683/00001

Material Information

Title: Polynomial Chaos Analysis of Micro Air Vehicles in Turbulence
Physical Description: 1 online resource (188 p.)
Language: english
Creator: Roberts, Brian C
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: chaos -- microairvehicle -- path -- polynomial -- turbulence -- uncertainty
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Use of unmanned vehicles is expected to continue to grow. As the variety of missions conducted by unmanned vehicles increases, these vehicles will require greater precision while operating in environments with stochastic disturbances. Thus, understanding the nature of the impact of stochastic disturbances on the vehicles, developing methods to control vehicles in the presence of disturbances, and planning missions using the knowledge of the disturbances will be required for unmanned vehicles to reach their full potential. This dissertation presents a methodology for planning missions for unmanned vehicles operating in the presence of stochastic disturbances. The methodology is shown for a micro air vehicle flying in significant turbulence. Wind tunnel testing characterizes the effect of turbulence intensity on the open loop dynamics of a micro air vehicle. This knowledge and understanding of atmospheric turbulence is combined in the framework of polynomial chaos to make it possible to control and simulate micro air vehicle flight using known methods. Path evaluation strategies are developed to select mission profiles to reduce likelihoods of collision and increase likelihoods of sensory mission success.
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 Brian C Roberts.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0043683:00001


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POLYNOMIALCHAOSANALYSISOFMICROAIRVEHICLESINTURBULEN CE By BRIANCHRISTOPHERROBERTS ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c r 2012BrianChristopherRoberts 2

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Idedicatethistomyparents. Thebestofmeisareectionoftheloveandsupportyouhavegi ven. 3

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ACKNOWLEDGMENTS IwouldliketoacknowledgethesupportI'vereceivedfromth ethreementorsIhave hadinGainesville:RickLind,MrinalKumar,andSheldonCip riani.Ihavealsoreceived supporttomakethewindtunnelworkfrommyprojectpossible fromSimonWatkins andLawrenceUkeiley.Finally,Iwouldliketoacknowledgea llofthestudentsthatI haveworkedwithovermyveyearsingraduateschool.Bounci ngideasoffofeach otherandengaginginsometimesheateddiscussionshasgive nmethecreativityand understandingtocompleteaprojectofthismagnitude. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ..................................4 LISTOFTABLES ......................................8 LISTOFFIGURES .....................................9 ABSTRACT .........................................12 CHAPTER 1INTRODUCTION ...................................13 1.1Motivation ....................................13 1.2ProblemStatement ...............................15 1.3ResearchPlan .................................16 1.3.1DissertationOutline ..........................16 1.3.2Contributions ..............................17 1.3.3PublicationPlan .............................18 2TURBULENCE ....................................19 2.1TheNatureofTurbulence ...........................19 2.2AtmosphericTurbulence ............................23 2.3TurbulenceanditsImpactonAircraft .....................24 2.4ControlinthePresenceofTurbulence ....................26 3FLIGHTDYNAMICS .................................28 3.1AircraftAxisSystems .............................28 3.1.1Body-axisSystem ...........................28 3.1.2Stability-axisSystem ..........................28 3.1.3Wind-axisSystem ...........................30 3.1.4Earth-axisSystem ...........................30 3.2CoordinateTransformations ..........................30 3.2.1WindtoStabilityFrame .........................31 3.2.2StabilitytoBodyFrame .........................32 3.2.3EarthtoBodyFrame ..........................32 3.3NonlinearEquationsofMotion ........................35 3.3.1DynamicEquations ...........................35 3.3.1.1ForceEquations .......................36 3.3.1.2MomentEquations ......................38 3.3.2KinematicEquations ..........................40 3.3.2.1OrientationEquations ....................40 3.3.2.2PositionEquations ......................42 3.3.3TheEquationsCollected ........................43 5

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3.4LinearizedEquationsofMotion ........................44 4WINDTUNNELTESTING ..............................49 4.1StaticWindTunnelTesting ...........................50 4.1.1WindTunnelSetup ...........................50 4.1.2ExperimentalDesign ..........................51 4.1.3Results .................................52 4.1.4Analysis .................................61 4.1.5Conclusions ...............................65 4.2DynamicWindTunnelTesting .........................65 4.2.1WindTunnelSetup ...........................65 4.2.2ExperimentalDesign ..........................68 4.2.3Results .................................69 4.2.3.1SweepModel .........................70 4.2.3.2DihedralModel ........................72 5POLYNOMIALCHAOSTHEORY ..........................76 5.1Background ...................................76 5.2Theory ......................................78 5.3LinearSystem .................................84 5.4Difculties ....................................86 5.4.1NumericalIssues ............................87 5.4.2ModalInterpretation ..........................88 5.4.2.1InterpretingEigenvalues ..................89 5.4.2.2ClusteredEigenvalues ....................90 5.4.2.3InterpretingEigenvectors ..................90 5.5RelevantApplicationsofPolynomialChaos .................96 6AIRCRAFTMODELPARAMETRICINTURBULENCE ..............98 6.1ParameterizedModelDerivation .......................98 6.2ParameterizedGenMAVModel ........................106 6.3ModalAnalysisofParameterizedSystem ..................109 6.3.1EigenvalueAnalysis ..........................109 6.3.2ModeShapeAnalysis .........................110 6.4LinearizedModel ................................113 6.4.1LinearizationofModelwithRespecttoTurbulence .........113 6.4.2EigenvalueAnalysis ..........................115 6.4.3ModeShapeAnalysis .........................115 6.5PolynomialChaosModel ...........................117 6.5.1OrderofPCApproximation ......................118 6.5.2EigenvalueAnalysis ..........................120 6.5.3ModeShapeAnalysis .........................121 6.5.4AnalysisofShortPeriodModes ....................123 6.5.5AnalysisofPhugoidModes ......................125 6

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6.5.6ExampleSimulation ..........................127 6.5.7EffectsofUncertainParameterDistribution .............130 7STOCHASTICPATHEVALUATIONMETHODS ..................134 7.1ProbabilityBackground ............................134 7.2StochasticStateGenerationAlgorithm ....................137 7.3AlgorithmAppliedtoWaypointNavigationandCollision Avoidance ....145 7.4AlgorithmAppliedtoSensing .........................150 8EXAMPLESOFMAVSTOCHASTICPATHEVALUATION ............153 8.1AircraftModel ..................................153 8.2ControlDerivation ...............................155 8.3CollisionAvoidanceExample .........................160 8.4TargetSensingExample ............................166 9CONCLUSION ....................................172 9.1ResearchSummary ..............................172 9.2FutureWork ...................................173 REFERENCES .......................................175 BIOGRAPHICALSKETCH ................................188 7

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LISTOFTABLES Table page 1-1Completedandanticipatedresearchpublication .................18 4-1Normalizedrootmeansquaredeviationfor L ...................55 4-2Normalizedrootmeansquaredeviationfor M Y ..................55 4-3Normalizedrootmeansquaredeviationfor Y ..................55 4-4Normalizedrootmeansquaredeviationfor M X ..................55 4-5Normalizedrootmeansquaredeviationfor M Z ..................56 4-6Findingmeanslopeandstandarddeviation ....................58 4-7 L averagederivativesandstandarddeviations( N = ) ..............58 4-8 M Y averagederivativesandstandarddeviations( Nm = ) ............59 4-9 Y averagederivativesandstandarddeviations( N = ) ..............60 4-10 M X averagederivativesandstandarddeviations( Nm = ) .............60 4-11 M Z averagederivativesandstandarddeviations( Nm = ) .............61 4-12Experimentalturbulenceintensitiestested .....................67 4-13Naturalsweepmodel .................................71 4-14Codedsweepmodel .................................72 4-15Naturaldihedralmodel ................................74 4-16Codeddihedralmodel ................................74 5-1Commondistributionsandassociatedbasispolynomials .............79 6-1Magnitudesofstateuctuationsrelativeto forshortperiodmodes ......121 6-2Phaseleadofstatesrelativeto forshortperiodmodes .............121 6-3Magnitudesofstateuctuationsrelativeto forphugoidmodes .........122 6-4Phaseleadofstatesrelativeto forphugoidmodes ...............122 6-5Phaselagofstatesofexpandedsystem ......................127 8-1Probabilitiescalculatedforcollisionexampleofpath evaluationalgorithm ...166 8-2Probabilitiescalculatedforsensingexampleofpathev aluationalgorithm ....169 8

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LISTOFFIGURES Figure page 3-1Body-xedcoordinateframe .............................28 3-2Stabilitycoordinateframe ..............................29 3-3Windcoordinateframe ................................30 3-4Earth-xedcoordinateframe ............................31 3-5Rotationthrough ..................................33 3-6Rotationthrough ..................................34 3-7Rotationthrough ..................................34 4-1TurbulentsetupinRMITindustrialwindtunnel ...................51 4-2TestsetupinRMITindustrialwindtunnel .....................52 4-3Unsweptwingmodelinsmoothow ........................53 4-4Unsweptwingmodelinturbulentow .......................53 4-510 Sweptwingmodelinsmoothow .......................54 4-610 Sweptwingmodelinturbulentow .......................54 4-7Liftcurvesateachsideslipangle ..........................57 4-8Turbulencegeneratinggrids .............................66 4-9TestsetupinREEFlowspeedwindtunnel .....................68 4-10Liftandpitchcoefcientchangeswithturbulenceinte nsityinsweepmodel ...73 4-11Liftandpitchcoefcientchangeswithturbulenceinte nsityindihedralmodel ..75 5-1Exampleoftimeevolutionofmeanandvariancebounds .............92 5-2ComparisonofmeanandvarianceboundstoMonteCarlorun s .........92 5-3Mean,variance,andskewnessofstateunderonemodeofex pandedsystem .93 5-4Exampleofinitialuncertaintyinexpandedsystemmodal analysis ........94 5-5Variationinstateuncertaintybetweenmodesofexpande dsystem .......95 5-6Variationofmeansandvariancesbetweenmodesofexpand edsystem ....95 6-1Eigenvaluesofparameterizedsystem .......................109 9

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6-2Frequencyanddampingofmodesoftheparameterizedsyst em .........111 6-3Relativemagnitudesandphasesinshortperiodmode ..............111 6-4Relativemagnitudeandphaseinphugoid-divergentmode ............112 6-5Relativemagnitudeandphaseinphugoid-convergentmod e ...........113 6-6Eigenvaluesoflinearizedparametricsystem ...................115 6-7Frequencyanddampinginquadraticandlinearparametri csystems ......116 6-8Relativemagnitudesandphasesinshortperiodmode ..............116 6-9Relativemagnitudesandphasesinphugoid-divergentmo de ..........117 6-10Relativemagnitudesandphasesphugoid-convergentmo de ...........118 6-11Averagemodalcomponentmagnitudesusing 9 th orderPCE ...........119 6-12Averagemodalcomponentmagnitudesusing 5 th orderPCE ...........120 6-13EigenvaluesofPCexpandedsystem ........................120 6-14Meanvaluesofall4statesforoneshortperiodmode ..............123 6-15Meanandvarianceboundsoflongitudinalstatesforall 6shortperiodmodes .124 6-16Meanvaluesofstatesforonephugoidmode ...................125 6-17Meanandvarianceboundsoflongitudinalstatesforall 6phugoidmodes ...126 6-18Meanandvarianceboundsofallstatesforexamplesimul ation .........129 6-19Meanandvarianceboundsofallstatesforexamplesimul ation .........130 6-20Eigenvaluesofexpandedsystemusingtwodistribution softurbulenceintensity 131 6-21Effectofturbulenceintensitydistributiononphugoi dmodes ...........132 7-1Partitionofanexamplesamplespace .......................135 7-2Partitionofsamplespaceofmissionwithonewaypoint .............138 7-3Partitionofsamplespaceofmissionwithtwowaypoints .............138 7-4Partitionofsamplespaceofmissionwithoneno-yzone ............139 7-5Partitionofsamplespaceofmissionwithtwono-yzones ............140 7-6Visualizationofstochasticnatureofvehicle'spath ................141 7-7TimePDF .......................................145 10

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7-8XConditionalYPDF .................................146 7-9Probabilityofenteringrstno-yzonedemonstration ...............147 7-10Probabilityofenteringsecondno-yzonedemonstrati on ............147 7-11Probabilityofreachingnalpositiondemonstration ................148 7-12Normalizedpdfof y positionat x POI 1 conditionaluponconictwith NF 1 .....149 8-1BlockdiagramofLQRtrackingcontroller ......................157 8-2Environmentandvehiclemeanpath ........................161 8-3Probabilitydensityfunctionsatallpossiblecollisio ns ...............161 8-4Jointpdfof and y positionconditionalonaircraftat x POI 1 ............162 8-5Renormalizedjointpdfof and y positionconditionalonconictwith NF 1 ...163 8-6 NF 1 conictconditionallateralpositionpdfsat NF 2 NF 3 ,and RFP ......164 8-7 NF 2 conictconditionallateralpositionpdfsat NF 3 and RFP ..........164 8-8Lateralpositionpdfatdesirednalpositioncondition aluponconictwith NF 3 .165 8-9Environmentandvehiclemeanpath ........................167 8-10Longitudinalpositionsmostlikelytoresultinsucces sfulsensing ........168 8-11PDFofaircraftlateralpositionandrollangleat x POI 1 ...............168 8-12PDFofaircraftlateralpositionandrollangleat x POI 3 ...............170 8-13ConditionalPDFofaircraftlateralpositionandrolla ngleat x POI 4 ........170 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy POLYNOMIALCHAOSANALYSISOFMICROAIRVEHICLESINTURBULEN CE By BrianChristopherRoberts May2012 Chair:RichardC.Lind,Jr.Major:AerospaceEngineering Useofunmannedvehiclesisexpectedtocontinuetogrow.Ast hevarietyof missionsconductedbyunmannedvehiclesincreases,thesev ehicleswillrequire greaterprecisionwhileoperatinginenvironmentswithsto chasticdisturbances.Thus, understandingthenatureoftheimpactofstochasticdistur bancesonthevehicles, developingmethodstocontrolvehiclesinthepresenceofdi sturbances,andplanning missionsusingtheknowledgeofthedisturbanceswillbereq uiredforunmanned vehiclestoreachtheirfullpotential. Thisdissertationpresentsamethodologyforplanningmiss ionsforunmanned vehiclesoperatinginthepresenceofstochasticdisturban ces.Themethodology isshownforamicroairvehicleyinginsignicantturbulen ce.Windtunneltesting characterizestheeffectofturbulenceintensityontheope nloopdynamicsofamicroair vehicle.Thisknowledgeandunderstandingofatmospherict urbulenceiscombinedin theframeworkofpolynomialchaostomakeitpossibletocont rolandsimulatemicroair vehicleightusingknownmethods.Pathevaluationstrateg iesaredevelopedtoselect missionprolestoreducelikelihoodsofcollisionandincr easelikelihoodsofsensory missionsuccess. 12

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CHAPTER1 INTRODUCTION 1.1Motivation Unmannedsystemshaveseenincreasedresearchattentionan duseinmission applicationsinrecentyears.Advancesintechnologyhavem adeunmannedsystems bothmoreeffectiveinthemissionsforwhichtheywereorigi nallyintendedandopened upnewpossibilitiesinthediversityofmissionprolestha ttheycouldsuccessfully complete.Thesenewmissionprolesareleadingunmannedsy stemsintonew environmentswithcloserproximitytohumans,inbothmilit aryandcivilianapplications. Unmannedsystemsoffergreatadvantagesfordangerousmiss ions,suchasclose proximitysurveillanceandsensing. Theuseofunmannedsystemsinmilitaryapplicationsisexpe ctedtoincrease inboththenearandlong-termfuture[ 1 – 3 ].Asthemilitaryapplicationsexpand thecapabilitiesandprovetheviabilityofunmannedsystem s,theirusesincivilian applicationswillincreaseaswell,withsensingandsurvei llancechiefamongthem[ 4 – 6 ]. Theproliferationofunmannedsystemusewilllikelyleadto evenmoreapplicationsthat haveyettobeimagined. Newtechnologieswillalsoincreasetheeffectivenessofun mannedsystemsto performbothnovelandpreviouslyenvisionedmissions[ 7 ].Unmannedvehicledesigns arebecomingsmallerandlighter.Morphingtechnologyisma kingunmannedvehicles capableofmorecomplicatedmissionprolesandmaneuvers[ 8 – 11 ].Electroniccontrol systemsarebecomingsmaller,lighter,andcapableofhandl ingcomplexalgorithms tonavigateunmannedvehicles[ 12 – 14 ].Controltheoryisadvancingnewstrategies forcontrollingunmannedvehicles,suchascooperative,no nlinear,modelpredictive, fault-tolerant,androbustcontrol[ 15 – 19 ].Newmethodsofcontrolusingcombinationsof sensorinputalsoshowpromisetohelpnavigatebothaeriala ndterrestrialunmanned vehicles[ 20 – 26 ]. 13

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Theproliferationofunmannedsystemusewillresultinresu ltinunmannedsystems operatinginenvironmentsclosertohumansociety.Thesene wenvironmentscouldbe ruralorurbanbut,willundoubtedlybeclosertotheearth's surface,wheretargetsand missionsofinterestaremostlikelytoexist.Whethernavig atingeldstoexaminecrops fordiseaseandyieldorsearchingcitiesformissingperson s,theunmannedsystems ofthefuturewillbeaskedtoperformmissionsindenseobsta cleelds.Performing missionsintheseenvironmentsforbothterrestrialandaer ialvehiclesrequirestheability tonavigatebetweenobstacleswhilebeingsubjectedtolarg edisturbances.Ground vehiclesmustcompensateforpotholes,rocks,unevenwheel slippage,andotherterrain imperfections.Aerialvehiclesmusthandleuncertaintyin oncomingowintheformof turbulence. Attemptshavebeenmadeatsolvingtheproblemoflargedistu rbancerejection usingseveralmethods.Systemdesignforpassivedisturban cerejectionisasensible option,butisgenerallyaccompaniedbyalossoffastrespon setimesandmaneuverability[ 27 – 29 ].Somemethodsofcontrolhaveattemptedtouseenergyextra ctionandstorageto negatetheeffectsofdisturbances,orinsomecases,toderi vebenetsfromthem[ 30 ]. Stillothershaveattemptedtosolvethisproblembyimpleme ntingasensor-based aprioriknowledgesolution.Theconceptisthatiftheoncom ingdisturbancecanbe measured,thencontrolactuatorscouldrespondinanticipa tionofthedisturbance.They coulddeectinanticipationoftheuctuationsinwindspee dcomponents[ 31 32 ].While thismethodshowssomepromiseforlargeraircraft,theigh tregimeofMAVsisillsuited forsuchasolution.Instead,improvementsintheunderstan dingoftheinteractions betweenturbulenceandMAVightdynamicsandacontroldesi gnfocusedonreduction oftheturbulenteffectsonMAVsisthebestpathtoincreasin gthecapabilitiesofMAVsin turbulentenvironments. Thisthesisintendstodemonstrateamethodologyforcharac terizingthedynamics ofaunmannedvehicleinanenvironmentwithlargerandomdis turbances,designinga 14

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controlsystemthatismostlikelytominimizetheerrorinad esiredpathoftravel,and assigningcostfunctionsforstochasticpathevaluation.T hechaoticnatureofrandom disturbancescoupledwiththeneedforprecisemaneuvering intightspacesmakeit difculttocontrolunmannedvehiclesintheenvironmentsi nwhichtheywillbeused.A combinationofexperimentaltestingandapplicationofsto chasticcontrolconceptsare implementedtoachievesuitablecontrolrequirements.The methodologyisappliedtoa caseexampleofaMAVyinginaturbulentenvironment. 1.2ProblemStatement Amethodologyisshowninthethesistoaccountforturbulent effectsonMAVight. Thismethodologyconsistsoffoursteps. Therststepistounderstandtheeffectthatturbulencehas ontheopenloop dynamicsofaMAV.Windtunneltestingisundertakenthatinc ludesturbulenceintensity asaparameterthatimpactstheightdynamics.Theinteract ionsofturbulencewith aircraftstatesanddesignparametersisanalyzed. Thesecondstepinsolvingthevehicledynamicsistoframeth eproblemasa stochasticdynamicsprobleminaknownframework.Knowledg eofatmospheric turbulenceisfusedwiththeopenloopMAVdynamicalmodelto produceastochastic descriptionofvehicledynamicsinaturbulentenvironment .Theframeworkchosen isbasedinpolynomialchaostheory,whereinthestatesandu ncertainparameters areexpressedasaweightedsumofpolynomials.Theassigned weightsexpressthe system'sdeterministictimevariation,andthepolynomial sarefunctionsofarandom variable,andthus,expressthesystem'sprobabilisticvar iation. ThethirdstepistoimplementacontrollerontheMAVmodel.A polynomial chaosbasedoptimalregulatorisappliedtothedynamicmode lbecauseofitsability tominimizethevariationofthestatesinaleastsquaressen se,wheninthepresenceof uncertainty. 15

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Thefourthandnalstepistousepolynomialchaosmethodsto ndastatistical descriptionoftheaircraftstatesasitattemptstotravers eapre-denedpath.These statisticsarecombinedtocreateacostfunctionthatcould beusedtoevaluatepathson abasisoftime,distancetraveled,likelihoodofmissionfa ilure,sensingeffectiveness,or anyothermetricimportanttosuccessfulcompletionofanyo fthepossibleunmanned vehiclemissionsmentionedinSection 1.1 1.3ResearchPlan 1.3.1DissertationOutline Chapter 1 isanintroductiontothedissertationproject.Theproblem ismotivatedby theincreasinguseofMAV'sinturbulentregimes. Chapter 2 includesbackgroundinformationonturbulence.Adiscussi onof turbulencelengthscalesandspectraandtheirrelevanceto MAVightisimportantto theunderstandingofbothtestingandcontrollingMAV'sint urbulence.Pastresearchon modelingtheeffectofturbulenceonMAV'sandattemptstoco ntrolaircraftinturbulence isincluded. Chapter 3 derivesthedynamicsequationsthatareusedtodescribelin earized aircraftmotion.Thederivedequationsprovidethebasisfo rbothanalyzingMAVwind tunneldataandcontrollingthesimulatedMAV. Chapter 4 detailstheexperimentalsetup,results,andanalysisofwi ndtunneltesting toexaminetheeffectsofturbulenceofMAVightdynamics.T wosetsofexperiments areconducted.Therstsetofexperimentsxestheightang lesatconstantvalues,so theyarereferredtoasstatictests.Thesecondsetofexperi mentsincludestestswith time-varyingightangles,sotheyarereferredtoasdynami ctests. Chapter 5 includesbackgroundinformationonthepolynomialchaosex pansion methodofdescribingstochasticdynamicsystems.Theequat ionsofadynamicsystem arederivedinthepolynomialchaosframework,andprevious researchapplying 16

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polynomialchaosmethodstotheareasofturbulence,dynami cs,control,andpath planningarecoveredindetail. Chapter 6 ndsalinearightdynamicsmodeloftheformfoundinChapte r 3 that representsthetrendsfoundinthewindtunneltestinginCha pter 4 .Theightdynamics modelisthenlinearizedwithrespecttoturbulenceintensi tytomakeitamenabletothe polynomialchaostechniquesofChapter 5 ,andtheopenloopmodesofthesystemare analyzed. Chapter 7 providessomeprobabilitytheorybackgroundanddevelopst hepath analysisalgorithmstoevaluateavehicle-controller-pat hcombinationaccordingtoits probabilityofsuccessfullycompletingadesiredmission.1.3.2Contributions Thisprojectrequiressignicantcontributionsinseveral ofthestepsofthe methodologytoaccountforturbulenceinMAVight.Increas ingtheknowledgeof turbulence'seffectonMAVisaddressedbywindtunneltesti ngofaMAV.Thetestingof aMAVyieldsseveralcontributions. Inclusionofturbulenceintensityasaparameteraffecting MAVdynamics NonlinearcharacterizationofMAVlongitudinaldynamicsi npresenceofturbulence Thepolynomialchaosframeworkischosenasthebasisfordev elopingstrategies ofcontrolandsimulationofunmannedvehiclesinanenviron mentwithstochastic disturbances.Thispolynomialchaosframeworkisimproved byseveralcontributions. Fusionofknowledgeofatmosphericturbulencecharacteris ticsandtheresultsof turbulentwindtunneltesting Useofpolynomialchaosframeworkusingexternaldisturban cesasthesourceof stochasticityratherthanparametricuncertainty Applicationofpolynomialchaoscontroltotheproblemofco ntrolandsimulationof MAVinturbulence Pathplanningisarecentapplicationofpolynomialchaosme thods.Thestudyof pathevaluationmethodsinthisthesisyieldsseveralcontr ibutions. 17

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Pathevaluationinanenvironmentwithvaryinguncertainty inseverityofnavigable disturbances Polynomialchaosmethodofpathevaluationforlikelihoodo fcollision Polynomialchaosmethodofpathevaluationforsensorcond ence 1.3.3PublicationPlan ThecontributionsoutlinedinSubsection 1.3.2 areexpectedtoproducethatwillbe presentedatconferencesandpublishedinjournals.Someof theworkhasalreadybeen published.Asummaryofpublishedworkandanticipatedpubl icationsispresentedin Table 1-1 Table1-1.Completedandanticipatedresearchpublication TopicConferencePublication PterosaurInspiredMAVAIAAAFM2009[ 33 ]Bioinspiration&Biomimetics[ 34 ] GSA2008[ 35 ]Design&NatureV[ 36 ] WindTunnelTestingAIAAAFM2010[ 37 ] PolynomialChaosBasedFlightDynamics AIAAAFM2011[ 38 ] PolynomialChaosBasedPathPlanning AIAAGNC2012JournalofAircraft 18

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CHAPTER2 TURBULENCE Arudimentaryunderstandingofthenatureoftheturbulentm otionofuids andpreviousworkdoneintheintersectionsofturbulence,a ircraft,andcontrolsis essentialtotheworkdoneinthisthesis.Flowscharacteriz edasturbulentgenerally exhibitcomplexandhighlyerraticmotionthatdeesattemp tstobedescribedwith deterministicstatements.Turbulencehastheunfortunate distinctionofbeingone ofthemostfundamentallychallengingandcomplexareasofn aturalphenomena. HoraceLambisquotedassaying,“Iamanoldmannow,andwhenI dieandgoto heaventherearetwomattersonwhichIhopeforenlightenmen t.Oneisquantum electrodynamicsandtheotheristheturbulentmotionofui ds.AbouttheformerIam ratheroptimistic.”[ 39 ]Howeverdifculttounderstandturbulencemaybe,itisimp ortant tonotethatturbulentow,asopposedtolaminar,describes thevastmajorityofuid ow,includingatmosphericight[ 40 ]. 2.1TheNatureofTurbulence Turbulencedoesnotlenditselftoaprecisedenition.Itis partlyforthisreasonthat theterm'turbulent'canbeappliedjustaseasilyinthereal msofeconomicsorpolitics asitisusedinengineering.However,whenthetermisapplie dtouidow,several characteristicsareimplied[ 40 ]. Turbulentowsarerandom.Therandomnessofturbulencepar tlyarisesfrom theinabilitytondaclosed-formsolutiontothenonlinear partialdifferential Navier-Stokesequationsandtheinabilitytoperfectlyde neinitialconditionsinthe owbecausethedimensionalityistoogreat.Thus,takingad eterministicapproach inthepresenceofturbulenceisinherentlyawed. TurbulentowsoccuratlargeReynoldsnumbers.Reynoldsdi scoveredthat turbulentowsexhibitlargeinertialeffectsincompariso ntoviscousdamping effects.Forthisreasontheratioofinertialforcestovisc ousforcesisnamedin hishonor.Formostengineeringapplications,theReynolds numberoftheow ofinterestissufcientlylargetoinduceturbulence.Fort hisreason,turbulence hasbeenstudiedinrelationtosuchdiversetopicsascombus tion,forestry,blood circulation,andaquaticecosystems[ 41 – 44 ]. 19

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Turbulentowsarerotationalinthreedimensions.Turbule ntowincludesthe uctuationandmixingofthreedimensionalvortices.Itisi mpossibletolimit turbulencetofewerthanthreedimensions.Eveniftheiniti alvorticesproducing turbulencearetwodimensional,theviscosityoftheowwil lresultintheproduction vorticesinallthreedimensions. Turbulentowarediffusive.Turbulencemixesuidsatamuc hfasterratethan laminarow.Theresultistransferofmomentum,heat,andma ssatamuchfaster rate. Turbulentowsaredissipative.Theviscosityintheuidco nvertsthekineticenergy oftheuidmotionintointernalenergyoftheuid. Turbulentowsexhibittheircharacteristicsinarangeofs cales.Thevorticesin turbulentowrangeinsize,withthelargestlengthscalesg overnedbythelength oftheoweld,andthesmallestlengthscalesnearingthesi zeofmolecular interactionwithintheuid. Bradshawsummarizesmuchofthecharacteristicsofturbule ncerathersuccinctly: ”Turbulenceisathree-dimensionaltime-dependentmotion inwhichvortexstretching causesvelocityuctuationstospreadtoallwavelengthsbe tweenaminimum determinedbyviscousforcesandamaximumdeterminedbythe boundaryconditionsof theow.ItistheusualstateofuidmotionexceptatlowReyn oldsnumbers[ 45 ].” Thecharacteristicsofturbulentowarisefromthenatureo frelationshipsthat governuidow.TheNavier-Stokesequation,showninEquat ion 2–1 ,resultsfromthe applicationofNewton'slawsandtherelativelysafeassump tionofconservationofmass withintheow.ThenonlineartermonthelefthandsideofEqu ation 2–1 producesmuch oftheowcomplexitydescribedbythecharacteristicslist edabove. 20

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@ u @ t + ( u r ) u = r p + r 2 u where, u = owelementvelocityvector t = time(2–1) p = pressure = density = uidkinematicviscosity ThenonlinearityofNavier-Stokesalsoresultsintheclosu reproblemthatmakes turbulentuidowsuchadifcultproblem.Therstcharact eristicusedtodescribe turbulence,isitsrandomness.Intruth,itsmotionismoreq uasi-random,butbeing unabletosolvefortheclosedformsolution,engineerscann otdeterministicallydescribe theowquantities,andthusrefertouctuationsintheowa srandom.Theabsenceof adeterministicsolutionresultsinnumerousapproachesto describetheresultingow. Theoreticalattemptstodescribeturbulentowareverywel ldeveloped.After Reynolds,Richardsonproposedtheenergycascadethatprod ucestherangeofeddy sizesandenergylevels[ 46 ].KolmogorovadvancedRichardson'sideasandbecamethe rstresearchertousestatisticaldescriptionsofturbule ncetoadvanceitsfundamental understanding[ 47 ].Statisticalresearchintoturbulencecontinuestoadvan ce[ 48 – 50 ]. Kolmogorovproposedthatturbulenceislocallyisotropic, suchthatthesmallscale eddiesthattransferenergyintotheinternalenergyofthe uiditselfareisotropic,but thelargerscaleeddiesinuencedbytheowboundariesarea nisotropic[ 51 ].While otherresearchersproposedthestatisticalnatureofturbu lencenearthesametime, Komogoroviswidelycreditedwithpioneeringastatistical approachtodescribing turbulence.Kolmogorovproposedaspectralenergylaw,kno wnasKolmogorov's Two-ThirdsLaw,thatpredictedthattheexpectedsquareddi fferenceinvelocitybetween 21

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twopointsinanisotropicturbulenteldisproportionalto thedistance, r ,betweenthe twopointsraisedtothepower2/3.Thisrelationshipisstat edinEquation 2–2 h [ v ( r )] 2 i/ r 2 3 (2–2) Theoreticaldescriptionscontinuetoadvanceintheircomp lexityandabilityto explainthenatureofturbulence[ 52 – 57 ]. Numericalmethodstodescribeturbulentowconstituteamo rerecentdevelopment intheunderstandingofturbulence.Multiplenumericaltec hniqueshavebeenemployed. Largeeddysimulationltersoutthesmallscalesinturbule nceandsolvestheNavier-Stokes equationsforthelongerlengthscales[ 58 ].TheFiniteElementMethod(FEM)hasbeen adaptedfromstructuralanalysistobeappliedtouids[ 59 ].Othermethods,suchas stress-closureandlattice-basedalgorithmshaveprovent obeefcientandaccurate algorithms[ 60 – 62 ]. NumericalapproachescontinuetoimproveinusefulnessasM oore'slawstays true.Moore'sLawisapredictionmadebyGordonMoore,arese archeratIntel,that thenumberoftransistorsthatwillbeputonasinglecompute rchipwoulddoubleevery twoyears[ 63 ].Theimplicationisthatcomputingpowerwouldcontinueto getcheaper, orconversely,thatcomputerswouldgrowevermorepowerful .Thissecondimplication allowsmorecomplicated(read:accurate)algorithmssimul atingturbulentowtobe runoncommerciallyavailablecomputers.Ascomputational powerincreases,the computationaluiddynamics(CFD)codescanusegreaterpre cision,calculatingthe velocity,pressure,density,andstresseswithcloseragre ementtoexperimentalresults inbothtime-averagedandtransientanalysis.Providedtha ttheassumptionsinherent intheCFDcodesarecorrect,theresultshouldbeanincreasi ngavailabilityofaccurate numericalowsolutions.Directnumericalsimulation(DNS )methodsleveragethe improvedspeedofcomputers[ 64 – 66 ]. 22

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However,theseCFDcodesshouldnotbereliedupontoprovide asolutionby themselves.Numericalmethodswillneverbeabletofullyde scribeasystem,because justliketheirtheoreticalcounterparts,theysimplyprov ideasolutionofasystemmodel. Duetothecomplexityofturbulenceandtheinabilityoftheo reticalandnumerical methodstoprovideperfectmodels,experimentalmethodsar eaverypopularmethod forgaininginsighttoturbulence.Richardsoncomplemente dhistheoreticalworkon turbulencewithexperimentalmeasurementsofatmospheric turbulence[ 46 ].Itis understoodthattheplacementofprobesintoaowtomeasure velocities,changes theshapeandvelocitiesoftheowitself,makingmeasureme ntofturbulentows difculttoobtain.Photographyofturbulentowthathasbe enseededwithsmokeor oilhasbeenalongusedmethodforunderstandingtheshapesa ndcharacteristicsof turbulentow,butisnotusefulforexactmeasurements.Par ticlevelocimetrytechniques revolutionizedtheexperimentalstudyofturbulence,allo wingforstudiesintothenature ofturbulence[ 67 68 ]. Experiencehasshownthatnoneofthethreeapproachescanbe reliedonsolely andcompletely.Asaresult,modernstudiestendtousemulti pleapproachestoeither validatetheresultsofonestudyorattempttoexplaintheca usesoftheresultsof anotherstudy.Numericalandexperimentaltechniquescanb eusedtovalidatethe predictionsofatheoreticalmodel[ 69 – 74 ].Numericaltechniquescanbeusedto explainindetailtheinteractionsthatareproducingthere sultsseeninanexperimental study[ 75 76 ]. 2.2AtmosphericTurbulence Atmosphericturbulencehasbeenasomewhattangentialtopi cofresearchtothe overallunderstandingofturbulence.Theowcannotbeassu medtobelaminarat lowaltitudes,asiscommonwhendesigningaircraftandthei rcontrollers.Instead,the turbulencethatispresentmustbetakenintoaccount.Atlow altitude,thisturbulenceis describedastheatmosphericboundarylayer(ABL).TheABLi stheatmosphericregion 23

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thatrangesfromgroundleveltoanywherebetween100 m and1000 m inaltitude,andis characterizedbyturbulenceandlowermeanwindspeedsduet othedragbetweenthe highvelocityowsathighaltitudesandthesurfaceoftheea rth.Theturbulenceisonly increasedbythepresenceoflargeobstacles,suchasbuildi ngs,movingvehicles,etc.in urbanareas. Atmosphericturbulencewasrstofconcerntostructuralen gineersbuilding skyscrapersandbridges.Asaresult,thevastmajorityofat mosphericwinddata wastakenusinglargeanemometerswithlargespacinginbetw eenthematgreat heights[ 77 78 ].TheEngineeringSciencesDataUnitpublishedaseriesofr eports thatcharacterizedwinddowntowithinafewmetersofground level,butstillwith lowspatialresolution[ 79 – 81 ].Laterworkcharacterizedtheturbulencethatimpacts groundvehicles,producingacousticvibrationsandperfor mancedegradation[ 82 83 ]. Someworkexaminedthecorrelationbetweenturbulenceatmu ltiplepoints[ 84 ].Some numericalturbulencemodelshavebeendevisedtosimulatea tmosphericturbulencefor simulationpurposes[ 85 ].Unfortunately,muchofthisresearchhasnotbeenconduct ed atthealtitudes,orspatialandtemporalresolutionrequir edtomeasuretheturbulence frequencies,intensities,andlengthscalesthatimpactMA Vight.Onlyrecentlyhasthe natureofthisturbulenceanditseffectonMAVsbeguntobech aracterized[ 86 ]. 2.3TurbulenceanditsImpactonAircraft ThenatureoftheturbulencethataffectsMAVsisnottheonly gapintheaircraft community'sknowledgebase.Additionally,theimpactoftu rbulenceonMAVs,and howtomitigateorexploititseffectsisanunderstudiedpro blemtothispointthat mustbesolvedtomakemanyMAVmissionspossible[ 87 88 ].Thesmallmass,low altitudes,andlowairspeedsofMAVsmeanthatatmospherict urbulencehasamuch greaterimpactforthesevehiclesthantheirlarger,faster ,highercounterparts.Forthis reasonpreviousworkonthenonlineareffectsofturbulence ontheloadsandmodal characteristicsofaircraft[ 89 90 ]canbeusedonlyasaguide,notastheanswerto 24

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theeffectsofturbulenceonMAVs.Infact,previousresearc hhasnotedthevarietyof qualitativeaerodynamiceffectsthatturbulencecanhaveo nairpassingoveraMAV wing,dependingontheintensity,lengthscales,andfreque ncyoftheturbulence[ 91 ]. Someworkisattemptingtorestricttheproblembycharacter izingtheresponseto “worst-case”turbulenceproles;thus,reducingtheorder oftheproblem[ 92 ].More appliedworktakestheapproachthatrapiddevelopmentandt esting,inessencea largescaletrial-and-errorapproach,ofMAVswithdiffere ntcharacteristicsisthebest methodtondthemechanismsthatwillcontrolormitigateth eeffectofturbulenceon thevehicles[ 93 ]. Asensor-basedsolutiontotheproblemofturbulenceimpact ingaircrafthasbeen implementedonlargescaleaircraftandshowssomepromise[ 31 32 ].Thiswork implementsanairborneLIDARairspeedmeasurementsystemt o“see”theoncoming ow.Hopefullyinthefuturethisinformationwillallowthe controlsurfacestomoveina synchronizedmannerwiththeturbulencetoregulatetheloa ductuationsacrossthe wingsoflargeaircraft.However,thissystemseemsunfeasi bleforuseinMAVsbecause theoncomingowtoaMAVisnotalwaysinthesamedirection.I nahighspeedaircraft, theowthatitwillpassthroughislocatedinitsdirectiono ftravel.MAVsyslowenough thatalargegustwillresultintheoncomingowcomingfroma verylargeangleof sidesliporangleofattack.Thus,effectivelyimplementin gtheLIDARsystemonaMAV wouldrequiresensorcoverageoveraverylargeconeofthesp acearoundthevehicle. Producingasystemwithsuchlargesensorcoveragewouldmak ethesystemmore complicatedandincreaseitsweightpenalty,acrucialcons iderationinthedesignof MAVs. Oneofthemostimportanttopicsinthestudyofturbulenceis theconceptof turbulencescales.Asmentionedpreviously,Richardson's ideasoftheenergycascade andKolmogorov'stheoriesaboutthenatureofthesmallscal esofturbulencelaidthe groundworkforthiswayofthinkingaboutturbulence.Thela rgerscalesofturbulence 25

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tendtoconservemomentumwhilethesmallerscalesofturbul enceproducethe energytransferintointernalenergyofuid[ 39 ].Thescalesofturbulenceinvolvedin experimentshaveenormousimplicationsintheforcesprodu ced[ 94 ]. Whenappliedtoaircraft,therangeofscalesinturbulencec anbebrokenintothree broadcategories.Thesmallestscales,whereenergytransf erandchemicaldiffusion takeplace,areonlyafewordersofmagnitudeabovethesizeo ftheuidmolecules themselves,andhavelittleeffectontheforcesonbodyinth eow[ 94 ].Thelargest scalesareontheorderofmagnitudeofmanymeterslong.Anai rcraftmovesthrough vorticesofthesescalesslowlyenoughthatstandardcontro lmethodscanhandle thechanges.Whenlookingattheforcesproducedbyturbulen ceonanaircraft,the importantscalesaretheonesaroundtheorderofmagnitudeo ftheaircraft[ 94 ]. Giventhecommunity'sknowledgebaseatthemoment,themost effectiveapproach tocontrollingMAVsinturbulenceisrsttounderstanditse ffectsonthedynamics andexamineaircraftdesignparametersasamethodforameli oratingtheseeffects. Three-dimensionalisotropicturbulenceproducesuctuat ionsinboththeangleof attackandtheangleofsideslip,andtheirderivativesaswe ll.Furthercomplicatingthe problemisthattheseanglesthatarecommonlyusedtodescri beaircraftightconditions arenolongerconstantacrosstheentirevehicle;rather,th eyvarybothspanwise andchordwiseandcanevenbedifferentbetweenthewingandt hecontrolsurfaces. Theseuctuationscouldenhancetheimportanceoftermstha tareoftensmall,oreven ignored,contributionsincommonderivationsofaircrafts tatematrices,eg. C L C n or C m q ,ortheycouldevenproducesignicantnonlinearitiesthat mightpreventlinear controlsystemsfrombeingappliedtoMAVsinturbulentcond itions. 2.4ControlinthePresenceofTurbulence Theinteractionsbetweenmanyturbulentowparametersand aircraftgeometry parametersmakeacompleteunderstandingofMAVightintur bulentowdifcultto obtain.Additionally,theunpredictablenatureofturbule ncemakescontrolverydifcult. 26

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Themostcommonmethodofcontrolinthepresenceofturbulen ceisthedesign oftheopenloopdynamicsinherentintheaircraftdesignpro cess.Passivestability, achievedinmanyaircraftbyplacingtheverticalandhorizo ntaltailsaftofthecenter ofmassandplacingthecenterofmassforwardoftheneutralp oint,allowsanaircraft torecoverfromthedisturbanceinputofturbulenceandrest oreitselftoitsnominal operatingcondition.Passivemethodsarestillbeingadvan cedforbothcontrolin turbulenceandotherowcontrolapplications[ 95 96 ]. Activecontrolmethodsaredividedintotwoapproaches:new typesofowcontrol actuationandnewalgorithmsofcomputationalcontrol.Flo wcontrolactuationcan involveapsorothertypesofmovablepartstoimposebounda ryconditionsonthe ow[ 97 – 99 ].Otherowcontroltechniquesusenozzlesdesignedtoinpu tenergy intotheowatspecicratesandlocations[ 100 ].Newalgorithmsdesignedforuse inturbulencerangefromlineartononlinearandadaptiveco ntrolmethods[ 101 – 103 ]. Someworksimplyattemptstomitigatetheeffectsofturbule nce[ 88 ];whileother workbuildsacontrollerforextractingenergyoutofthever ticalgustsforperformance improvement[ 104 ]. Thisthesisintendstoleveragetheprobabilisticnatureof turbulenceandapply stochasticcontrolmethodstoreducetheimpactofturbulen ceonaircraftmotion.These controlmethodsarediscussedinChapter 5 andshowpromisetobeabletobeapplied tocontrolinthepresenceofturbulence. 27

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CHAPTER3 FLIGHTDYNAMICS 3.1AircraftAxisSystems Therearefourmainaircraftaxissystemsusedtodescribeve hicledynamics. Therstsystemisxedinthereferenceframeoftheaircraft andiscalledthe“body axis”system.Thesecondsystemisxedinthereferencefram eoftheoncomingow projectedandiscalledthe“windaxis”system.Thethirdaxi ssystemisanintermediate systemthatrelatesthebodyaxissystemtothewindaxissyst emandiscalledthe “stability-axis”system.Thenalaxissystemisxedwithr especttotheearthandis calledthe“earthaxis”system.3.1.1Body-axisSystem Thebody-axissystemisusedtodenetheaircraftmotion.Th ebody-axissystem isdenedbyplacingtheoriginattheaircraftcenterofmass ,thex-axis, ^ x B directed throughthenoseoftheaircraft,andthez-axis, ^ z B ,directedthroughthebottomofthe aircraft.Therighthandruledictatesthatthey-axis, ^ y B ,willpointouttherightwing (foraconventionalaircraft).Thecoordinatesystemissho wnonanexampleaircraftin Figure 3-1 Figure3-1.Body-xedcoordinateframe3.1.2Stability-axisSystem Thestability-axissystemisusedtorelatetheeffectsofth eoncomingowtothe aircraftmotion.Theoriginofthestability-axissystemis locatedattheaircraftcenter 28

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ofmass,butthex-axis, ^ x S ,pointsinthedirectionofprojectionofthewindontothe ^ x B ^ z B -plane.Thus,they-axisofthestability-axissystem, ^ y S ,iscoincidentalwith ^ y B .The angleofrotationthatwouldalign ^ x S with ^ x B and ^ z S with ^ z B isdenedastheangleof attackandisdenotedbythesymbol ,where 2 R .Figure 3-2 detailstherotation betweenthebody-axissystemandthestability-axissystem Figure3-2.Stabilitycoordinateframe Theaerodynamicforcesinthestability-axissystemaregiv enconventionalnames. Theaerodynamicforceinthenegative ^ x S directioniscalledthedragandisrepresented bythevariable, D .Theaerodynamicforceinthepositive ^ y S directioniscalledthe sideforceandisrepresentedbythevariable, Y .Theaerodynamicforceinthenegative ^ z S directioniscalledtheliftandrepresentedbythevariable L .Theserelationsare showninEquation 3–1 266664 D Y L 377775 S 266664 F Aero x F Aero y F Aero z 377775 S (3–1) where, F Aero i = aerodynamicforcein i direction for i = x y z 29

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3.1.3Wind-axisSystem Thewind-axissystemalsohasitsoriginattheaircraftcent erofmass,butits x-axis, ^ x W ,isdirectedintotheoncomingow.Thewind-axissystemisr elatedtothe stability-axissystembyacoordinatesystemrotation.The body-axissystemisrotated bytheangleofsideslip, ,about ^ z s .Figure 3-3 showstherelationshipbetweenthe wind-axisandthestability-axissystem. Figure3-3.Windcoordinateframe3.1.4Earth-axisSystem Theearth-axissystemisusedtoincludegravitationaleffe ctsontheaircraftmotion. Theoriginoftheearth-axissystemislocatedatthesurface oftheearthandthez-axis, ^ z E ,pointstowardthecenteroftheearth.Theexactdirectiono fthex-andy-axes, ^ x E and ^ y E ,isarbitraryaslongastheyformanorthogonalsetwith ^ z E .Figure 3-4 illustrates thedifferencesinbothrotationandtranslationbetweenth eearth-axissystemandthe body-axissystem. 3.2CoordinateTransformations Mathematicalrelationsareestablishedbetweentheaxissy stemsofSection 3.1 torelatethedynamiceffectsontheaircraftduetoforcesin otheraxissystems.These relationsarestatedinacoordinaterotationframeworkwhe rebyonevectorcanbe multipliedbyanorthonormalmatrixtondthevectorinthec oordinatesofasecondaxis 30

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Figure3-4.Earth-xedcoordinateframesystem.ThismathematicalrelationisstatedinEquation 3–2 ,where X istheoriginal vector, X 0 istherotatedvector,andtherotationmatrix R relatesthetwo. X 0 = RX (3–2) where, X X 0 2 R n R 2 R nxn 3.2.1WindtoStabilityFrame Therelationbetweenthewind-axissystemandthestability -axissystemisarotation about ^ z W byanangle .Thisangleisimportantwhenexaminingthelateraldynamic s oftheaircraft.Therelationoftheaerodynamicforcesinth etwosystemsisshownin Equation 3–3 31

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266664 F Aero x F Aero y F Aero z 377775 S = 266664 cos sin 0 sin cos 0 001 377775 266664 F Aero x F Aero y F Aero z 377775 W (3–3) 3.2.2StabilitytoBodyFrame Therelationbetweenthestability-axissystemandthebody -axissystemisa rotationabout ^ y S bytheangleofattack, .Thisangleisimportantwhenexaminingthe longitudinaldynamicsoftheaircraft.Therelationofthea erodyamicforcesinthetwo systemsisshowninEquation 3–4 266664 F Aero x F Aero y F Aero z 377775 B = 266664 cos 0 sin 010 sin 0cos 377775 266664 D Y L 377775 S (3–4) 3.2.3EarthtoBodyFrame Therelationbetweentheearth-xedframeandthebody-xed frameisarotation followedbyatranslation,butthetranslationisunnecessa ryforthepurposesofrelating forcesbetweenthetwoframes.So,rotationsneedtobedene dthattransformthe orientationoftheearth-xedframetothatofthebody-xed frame.Theserotationscan bedenedinmultipleways,buttheconventional3-2-1trans formationisdescribedhere. Byconvention,thistransformationrequirestherotationa boutthez-axistobedonerst, followedbyarotationaboutthetransformedy-axis,andna lly,arotationaboutthenew transformedx-axis.Theanglesusedintherotationsarecal ledtheEuleranglesand areconventionallydenedsuchthat:theyawangle, ,istheangleofrotationabout thez-axis,thepitchangle, ,istherotationaboutthey-axis,andtherollangle, ,isthe rotationaboutthex-axis. 32

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Intheprocessoftransformationbetweentheearth-xedfra meandthebody-xed frame,intermediatecoordinatesystems, 1 and 2 ,willbedened.Theseareusedsimply forexplanatorypurposesandwillnotbeusedagain. Therstrotationisdoneabout ^ z E byangle ,asshowninFigure 3-5 Figure3-5.Rotationthrough Thisrotationtransformstheaxesintheearth-xedframeto axesinintermediate frame 1 byapplyingEquation 3–5 266664 ^ x ^ y ^ z 377775 1 = R 3 ( ) 266664 ^ x ^ y ^ z 377775 E = 266664 cos sin 0 sin cos 0 001 377775 266664 ^ x ^ y ^ z 377775 E (3–5) Thesecondrotationisdoneabout ^ y 1 byangle ,asshowninFigure 3-6 Thisrotationtransformstheaxesinintermediateframe 1 toaxesinintermediate frame 2 byapplyingEquation 3–6 266664 ^ x ^ y ^ z 377775 2 = R 2 ( ) 266664 ^ x ^ y ^ z 377775 1 = 266664 cos 0 sin 010 sin 0cos 377775 266664 ^ x ^ y ^ z 377775 1 (3–6) Thethirdrotationisdoneabout ^ x 2 byangle ,asshowninFigure 3-7 33

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Figure3-6.Rotationthrough Figure3-7.Rotationthrough Therotationtransformstheaxesinintermediateframe 2 toaxesinthebody-xed framebyapplyingEquation 3–7 266664 ^ x ^ y ^ z 377775 B = R 1 ( ) 266664 ^ x ^ y ^ z 377775 2 = 266664 1000cos sin 0 sin cos 377775 266664 ^ x ^ y ^ z 377775 2 (3–7) Equation 3–8 showshowtherotationscanbecombinedtoprovideaonestep processthatorientstheearth-xedaxeswiththebody-xed axes. 34

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266664 ^ x ^ y ^ z 377775 B = R 1 ( ) R 2 ( ) R 3 ( ) 266664 ^ x ^ y ^ z 377775 E (3–8) Thus,anyvectorinearth-xedcoordinates, v E ,canbeexpressedinbody-xed coordinatesas v B byusingtherelationshipinEquation 3–9 v B = 266664 cos cos sin sin cos cos sin cos sin cos +sin sin cos sin sin sin sin +cos cos cos sin sin sin cos sin sin cos cos cos 377775 v E (3–9) Forexample,thegravitationalforceisexpressedinbothea rth-xedandbody-xed coordinatesinEquation 3–10 F Grav = 266664 00 mg 377775 E = 266664 mg sin mg sin cos mg cos sin 377775 B (3–10) 3.3NonlinearEquationsofMotion 3.3.1DynamicEquations TherigidbodyequationsofmotionarederivedfromNewton's secondlaw,which holdstrueininertialreferenceframes.Theonlyinertialr eferenceframethatisdened inSection 3.2 istheearth-xedreferenceframe;so,thedynamicequation sare derivedintheearth-xedreferenceframe.Newton'ssecond lawisappliedtoforces inEquation 3–11 andequatesthesumofexternalforcestothetimerateofchan ge ofmomentumofthebody.WhenappliedtomomentsinEquation 3–12 ,Newton's secondlawequatesthesumofexternalmomentstothetimerat eofchangeofangular momentum. 35

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X F = d dt ( m v ) E (3–11) X M = d dt H E (3–12) 3.3.1.1ForceEquations Fortheaerospacesystemsinvestigatedinthisresearch,th emassoftheaircraft canbeassumedtobeconstant.Thisassumptionpermitsasimp licationofEquation 3–11 expressedinEquation 3–13 X F = m d dt v E = m a E (3–13) NotethattoapplyEquation 3–13 ,theaccelerationintheearth-xedframe, a E must beknown.Thedifcultyisthattheaccelerationsareoftenm easuredinthebody-xed frame;thus,theaccelerationintheearth-xedframemustb efoundusingthetransport theorem,whichisstatedinEquation 3–14 .Thetransporttheoremrelatesthetimerate ofchangeofavectorinonereferenceframetothetimerateof changeofthatvectorin anotherreferenceframeusingtheangularvelocitybetween thetworeferenceframes, 1 2 d b dt 1 = d b dt 2 + 1 2 b (3–14) Notethattwopropertiesholdtrueforangularratevectors, ,permittingconversion betweenmanydifferentreferenceframes. 1 2 = 2 1 (3–15) 1 n = 1 2 + 2 3 +...+ n 1 n (3–16) 36

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Thevelocityofanaircraftintheearth-xedreferencefram ecanbeexpressedin body-xedcoordinates,andisdenedinEquation 3–17 v E B = u ^ i B + v ^ j B + w ^ k B = 266664 u v w 377775 B (3–17) Theangularvelocitybetweentheearth-xedandbody-xedr eferenceframescan beexpressedinbody-xedcoordinates,andisdenedinEqua tion 3–18 E B = p ^ i B + q ^ j B + r ^ k B = 266664 p q r 377775 B (3–18) Tondtheaccelerationofanaircraftintheearth-xedrefe renceframe,the transporttheoremisappliedtotheearth-andbody-xedref erenceframesinEquation 3–19 a E B = d v E B dt E = d v E B dt B + E B v E B (3–19) TherightsideofEquation 3–19 canbereducedtosolvefortheaccelerationofthe aircraftasviewedbyanobserverinanearth-xedreference frame,butexpressedin body-xedcoordinates.Theresultingrelationshipisshow ninEquation 3–20 a EB = 266664 u + qw rv v + ru pw w + pv qu 377775 B (3–20) NowthattherighthandsideofEquation 3–13 hasbeensolved,thelefthandside mustbedened.Itisassumedthattheonlythreetypesofforc esthatactontheaircraft aregravitational,aerodyamic,andthrustforces;thus,re sultinginasumofexternal forcesthattakestheformshowninEquation 3–21 37

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X F = 266664 F x F y F z 377775 B = 266664 F Grav x + F Aero x + F Thrust x F Grav y + F Aero y + F Thrust y F Grav z + F Aero z + F Thrust z 377775 B (3–21) Recallthegravitationalforcesexpressedinthebody-xed coordinatesfrom Equation 3–10 andtheaerodynamicforcesexpressedinthebody-xedcoord inates fromEquation 3–4 .WhenthoserelationshipsandEquation 3–20 areappliedto Equation 3–13 ,threeequationsresult,asshowninEquations 3–22 3–24 m (_ u + qw rv )= mg sin +( D cos + L sin )+ F Thrust x (3–22) m (_ v + ru pw )= mg sin cos + Y + F Thrust y (3–23) m (_ w + pv qu )= mg cos cos +( D sin L cos )+ F Thrust z (3–24) 3.3.1.2MomentEquations ReferringbacktoEquation 3–12 ,theangularmomentumintheearth-xed referenceframeexpressedinbody-xedcoordinates, H E B ,istheproductoftheinertia tensor, I B ,andtheangularvelocityvector, E B .Thisrelationshipisexpressedin Equation 3–25 H E B = I B E B (3–25) Theinertiatensor, I B ,canbedenedasinEquation 3–26 I B = 266664 I xx I xy I xz I yx I yy I yz I xz I yz I zz 377775 B (3–26) Thus,usingthedenitionsinEquation 3–26 andEquation 3–18 ,theangular momentumcanbeexpressedasinEquation 3–27 38

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H E B = 266664 I xx I xy I xz I yx I yy I yz I xz I yz I zz 377775 B 266664 P Q R 377775 B = 266664 pI x qI xy rI xz qI y rI z pI xy rI z pI xz qI yz 377775 B (3–27) Notethattheangularmomentumisexpressedinbody-xedcoo rdinates;so,the transporttheoreminEquation 3–14 willneedtobeappliedtondtherateofchangeof theangularmomentumintheearth-xedreferenceframe.The transporttheorem,as appliedtotheangularmomentum,isstatedinEquation 3–28 d H E B dt E = d H E B dt B + E B H E B (3–28) ThersttermontherighthandsideofEquation 3–28 issimpletocalculateandis furthersimpliedbyassumingthatthemomentofinertia, I B ,isconstant.Theresultis expressedinEquation 3–29 d H E B dt B = 266664 pI x qI xy rI xz qI y rI yz pI xy rI z pI xz qI yz 377775 B (3–29) ThesecondtermontherightsideofEquation 3–28 isexpressedinEquation 3–30 E B H E B = 266664 qrI z qpI xz q 2 I yz qrI y + r 2 I yz + rpI xy rpI x qrI xy r 2 I xz rpI z + p 2 I xz + qpI yz pqI y rpI yz p 2 I xy pqI x + q 2 I xy + rqI xz 377775 (3–30) CombiningtheresultsofEquations 3–28 3–29 ,and 3–30 ,thetimerateofchange ofangularmomentuminanearth-xedreferenceframecanbee xpressedasin Equation 3–31 39

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d dt H E = d H E B dt E = 266664 pI x qI xy rI xz + qrI z qpI xz q 2 I yz qrI y + r 2 I yz + rpI xy qI y rI yz pI xy + rpI x qrI xy r 2 I xz rpI z + p 2 I xz + qpI yz rI z pI xz qI yz + pqI y rpI yz p 2 I xy pqI x + q 2 I xy + rqI xz 377775 B (3–31) NowthattherighthandsideofEquation 3–12 hasbeensolved,thelefthandside mustbedened.Thesumofexternalmomentthatactonanaircr aftisdenedsimplyas threemomentsthatactabouteachoftheprincipalbodyaxes; thus,resultinginasumof externalmomentsthattakestheformshowninEquation 3–32 X M = 266664 M x M y M z 377775 B = 266664 L M N 377775 B = 266664 L Aero + L Thrust M Aero + M Thrust N Aero + N Thrust 377775 B (3–32) Equations 3–32 and 3–31 canbesubstitutedintoEquation 3–12 toyieldEquations 3–33 3–35 thatexpresstherotationaldynamicsofanaircraftinight L =_ pI x qrI y + qrI z +( pr q ) I xy ( pq +_ r ) I xz +( r 2 q 2 ) I yz (3–33) M = prI x +_ qI y prI z ( qr p ) I xy +( p 2 r 2 ) I xz +( pq r ) I yz (3–34) N = pqI x + pqI y +_ rI z +( q 2 p 2 ) I xy +( qr p ) I xz ( pr +_ q ) I yz (3–35) 3.3.2KinematicEquations Equations 3–22 3–24 and 3–33 3–35 donotcompletelydescribetheaircraft dynamics.Sixmoreequationsarerequiredtofullydescribe theaircraftdynamics;these sixequationsareprovidedbytheaircraftkinematics.3.3.2.1OrientationEquations Threeequationscanbefoundbyrecognizingthattherotatio nofthebody-xed framerelativetotheearth-xedframeinbody-xedcoordin atescanbeexpressedin 40

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anothercoordinatesystem.ThisrelationisstatedinEquat ion 3–36 andisderivedfrom thefactthatavectorexpressedinonecoordinatesystemcan beexpressedinanyother coordinatesystem. B E B = B E (3–36) ExpandingthetwotermsofEquation 3–36 usingthecoordinatesystemsof Section 3.2 yieldsEquation 3–37 p ^ i B +_ q ^j B +_ r ^ k B = ^ i 2 + ^j 1 + ^ k E (3–37) Notethatthex-axisofthebodyframeissharedbyreferencef rame2,they-axis ofreferenceframe1issharedbyreferenceframe2,andthezaxisoftheearth-xed referenceframeissharedbyreferenceframe,soEquation 3–37 canberewrittenas Equation 3–38 p ^ i B + q ^j B + r ^ k B = ^ i B + ^j 2 + ^ k 1 (3–38) ThecoordinatetransformationsofSection 3.2 canbeusedtoshowrelations betweenthecoordinatesintherighthandsideofEquation 3–38 andbodyxed coordinates.TheserelationsareshowninEquations 3–39 and 3–40 ^ j 2 =cos ^j B +sin ^ k B (3–39) ^ k 1 = sin ^ i B +sin cos ^j B +cos cos ^ k B (3–40) TheserelationsofEquations 3–39 and 3–40 aresubstitutedintoEquation 3–37 to producethreeequations,expressedinEquations 3–41 3–43 41

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p = sin (3–41) q = cos + cos sin (3–42) r = cos cos sin (3–43) Equations 3–41 3–43 canberewrittenwiththeEuleranglesontheleftsideofthe equations,asseeninEquations 3–44 3–46 = p + q (sin + r cos )tan (3–44) = q cos r sin (3–45) =( q sin + r cos )sec (3–46) 3.3.2.2PositionEquations Thenalthreeequationsdescribethepositionoftheaircra ftinanearth-xed referenceframebutexpressedinbody-xedcoordinates,as denedinEquation 3–47 266664 x y z 377775 EB = 266664 dx / dt dy / dt dz / dt 377775 EB (3–47) InEquation 3–48 therightsideofEquation 3–47 isexpressedasthematrixproduct ofthevelocitiesinthebody-xedreferenceframeandthein verseofthetransformation matrixusedinEquation 3–9 266664 dx / dt dy / dt dz / dt 377775 EB = 266664 cos cos sin sin cos cos sin cos sin cos +sin sin cos sin sin sin sin cos cos cos sin sin +sin cos sin sin cos cos cos 377775 266664 u v w 377775 B (3–48) 42

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Thus,thethreeequationsdescribingthevelocityinaneart h-xedreferenceframe result.TheseequationsareseparatedandstatedinEquatio ns 3–49 3–51 x E B = u cos cos + v (sin sin cos cos sin )+ w (cos sin cos +sin sin ) (3–49) y E B = u cos sin + v (sin sin sin +cos cos )+ w (cos sin sin sin cos ) (3–50) z E B = u sin + v (sin cos )+ w cos cos (3–51) 3.3.3TheEquationsCollected Thenonlinearaircraftequationsofmotioncanbecollected intoaformalset,as showninEquation 3–52 43

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m (_ u + qw rv )= mg sin +( D cos + L sin )+ F Thrust x m (_ v + ru pw )= mg sin cos + Y + F Thrust y m (_ w + pv qu )= mg cos cos +( D sin L cos )+ F Thrust z L =_ pI x qrI y + qrI z +( pr q ) I xy ( pq +_ r ) I xz +( r 2 q 2 ) I yz M = prI x +_ qI y prI z ( qr p ) I xy +( p 2 r 2 ) I xz +( pq r ) I yz N = pqI x + pqI y +_ rI z +( q 2 p 2 ) I xy +( qr p ) I xz ( pr +_ q ) I yz = p + q (sin + r cos )tan (3–52) = q cos r sin =( q sin + r cos )sec x E B = u cos cos + v (sin sin cos cos sin )+ w (cos sin cos +sin sin ) y E B = u cos sin + v (sin sin sin cos cos )+ w (cos sin sin +sin cos ) z E B = u sin + v (sin cos )+ w cos cos 3.4LinearizedEquationsofMotion ThenonlinearequationsinEquation 3–52 arecomplicatedandhighlycoupled. Often,thecoupledandnonlineartermsaredominatedbytheu ncoupledlinearterms; thus,alinearsetofequationscanbeamuchmorepowerfultoo linpredictingand controllingthemotionofanaircraft.Theequationsarelin earizedusingsmall-disturbance theory,wherebyastandardoperatingconditionforallstat esisgiven,andanydeviations aboutthatoperatingconditionareassumedtobesmall.Ther esultsofapplying equationsobtainedusingthisassumptionwilldeteriorate asthetrueoperatingcondition deviatesfromtheassumedstandardoperationcondition. ThestatesandmomentsofEquation 3–52 areexpressedasasumofareference value, ( ) o ,andaperturbation, ( ) ,inEquation 3–53 44

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u = u o + u p = p o + p x = x o + x M = M o + M L = L o + L v = v o + v q = q o + q y = y o + y N = N o + N D = D o + D w = w o + w r = r o + r z = z o + z L = L o + L Y = Y o + Y (3–53) Themostcommonreferenceconditiontolinearizeaboutisth atofsteadylevelight. Equation 3–54 showsthereferenceconditionsthatcanassumedtobezeroin steady levelightforaleft-rightsymmetricaircraft. v o = w o = p o = q o = r o = o = o =0 (3–54) Additionally,asmallangleassumptionwillbemadethatthe longitudinalvelocity, u o isequaltothereferenceightspeed.Trigonometricidenti ties,showninEquation 3–55 canbeappliedwhensubstitutingEquation 3–53 intoEquation 3–52 sin( o + )=sin o cos +cos o sin =sin o + cos o cos( o + )=cos o cos sin o sin =cos o sin o (3–55) TheresultofcombiningEquations 3–52 3–55 andeliminatingallhigherorder termsisshowninEquation 3–56 45

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( D cos + L sin )+ F Thrust x mg (sin o + cos o )= m u Y + F Thrust y + mg cos o = m ( v + u o r ) ( D sin L cos )+ F Thrust z + mg (cos o sin o )= m ( w u o q ) L = I x p I zx r M = I y q N = I zx p + I z r o + = q o + = p + r tan o o + = r sec o x E o + x E =( u o + u )cos o u o sin o + w sin o y E o + y E = u o cos o + v z E o + z E = ( u o + u )sin o u o cos o + w cos o (3–56) IfallofthedisturbancesinEquation 3–56 aresettozero,thentheresulting Equation 3–57 showstheequalitiesofthereferenceightcondition. X o mg sin o =0 Y o =0 Z o + mg cos o =0 L o = M o + N o =0 x E o = u o cos o y E o =0 z E o = u o sin o (3–57) Equation 3–57 issubstitutedintoEquation 3–56 ,sothatthelinearizedmotion equationscanberewrittenasEquation 3–58 46

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_ u = x m g cos o v = y m + g cos o u o r w = z m g sin o + u o q L = I x p I zx r M = I y q N = I zx p + I z r = q = p + r tan o = r sec o x E = u cos o u o sin o + w sin o y E = u o cos o v z E = u sin o u o cos o + w cos o (3–58) Theperturbationtermsofaerodynamicforcesandmomentsin Equation 3–58 can beexpressedasaTaylorseriesexpansion.Forexample,thep erturbationtermofthe aircraftrollmomentcanbeexpressedasinEquation 3–59 L = @ L @ u u + @ L @ v v + @ L @ w w + @ L @ q q + @ L @ p p + @ L @ r r + @ L @ a a + @ L @ r r + @ L @ e e (3–59) ThepartialderivativesinthisrstorderTaylorseriesexp ansionformthebasisofa linearanalysisofaircraftdynamics.Thesetermsrepresen tthesensitivityoftheaircraft tochangesintheaircraftstates.Importantpartialderiva tivetermsinclude: L (the sensitivityoflifttochangesinangleofattack), M Y (thesensitivityofpitchmomentto changesinangleofattack,whichisindicativeofthelongit udinalstabilityoftheaircraft), M X (thesensitivityofrollmomenttochangesintheangleofsid eslip,whichisindicative oftherollstabilityoftheaircraft), M Z (thesensitivityofyawmomenttochangesin 47

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theangleofsideslip,whichisindicativeoftheyawstabili tyoftheaircraft),and Y (the sensitivityofsideforcetochangesintheangleofsideslip ). 48

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CHAPTER4 WINDTUNNELTESTING Itiswellunderstoodfromthepreviouschaptersthatneithe raircraftnorturbulence arestrictlylineardynamicalsystems.Forsomepurposesli nearitycanbeasufcient approximation.Itisinthisframeofmindthattraditionall inearightcontrolhastreated turbulentuctuationsasperturbationswithrespecttoatr imstate.Doingsoassumes thatsmallpositivechangesinthevelocitycomponentswill haveasymmetriceffect withrespecttosmallnegativechanges.However,theresear chconductedherein hypothesizesthatthisassumptionisincorrect. Windtunneltestingisconductedtounderstandtheinteract ionsofturbulenceand microairvehicle(MAV)ightdynamics.Twosetsoftestsare conducted;eachina differentresearchlocationusingdifferentvehicles,tes tmatrices,andmeasurement procedures.Bothtestsusegridstogeneratetheturbulence thatimpactstheaircraft dynamics. Therstsetoftestsarereferredtoasstatictestsbecauset heykeepthevehicle stationaryintheowandexamineonlytherelationshipsbet weentheightangles, turbulenceintensity,andsymmetricwingsweepangle.Inth eseteststhemodelis assumedtohavelineardynamics,andthestaticstabilityco efcientsaretimeaveraged duringlongtestrunsunderbothsmoothandturbulentowcon ditions.Thestatic stabilityderivativesarehypothesizedtodiffersignica ntlybetweenbothsmoothand turbulentow,provingthatthepresenceofturbulenceintr oducesnewtermsinthe equationsrepresentingtheightdynamicsofMAVs.Totestt hishypothesis,theforces andmomentsarefoundatvaryinganglesofattackandsidesli pinowsofdifferent turbulenceintensities,andtheforceandmomentderivativ eswithrespecttothoseight anglesarecalculated. Thesecondsetoftestsarereferredtoasdynamictestsbecau setheyincreasethe numberofparametersinthetestmatrixbyincludingpitchin gandplungingmaneuvers. 49

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Thesecondsetoftestsalsoincludetheeffectsofsymmetric wingdihedral.Inthese tests,themodelisassumedtohavenonlineardynamicsofaqu adraticform.In otherwords,allstatesorparametersthatareincludedinth emodelhavealllinear, quadratic,andassociatedcross-couplingtermsincluded. Ofparticularimportanceare thecross-couplingtermsbetweenturbulenceintensityand otherstates.Thesetests intendtoshownewmathematicaltermstorepresenttheeffec tofturbulenceonMAV dynamics. 4.1StaticWindTunnelTesting 4.1.1WindTunnelSetup StaticwindtunneldataisfoundusingtheRoyalMelbourneIn stituteofTechnology (RMIT)IndustrialWindTunnel(IWT).TheIWTisa2 m x3 m closedjet,closedtest sectionwindtunnel. Turbulenceisproducedbyasetofgrids,seeninFigure 4-1 ,placedupstreamofthe contractiontothetunneltestsection.Themodelismounted neartheexpansionendof thetestsection(9 m fromthegrids,whichisgreaterthan10timesthewidthofthe grid elements)toallowtheturbulencetobecomewellmixedandho mogeneous.Thewind tunnelhasaninherentturbulenceintensityof1.2%atthemo delwhenthegridsarenot installed,andaturbulencelevelof7.4%whenthegridsarei nstalled[ 105 ].Thesegrids havebeenshowntoproduceturbulencesimilartothatpresen tintheABL[ 105 ]. AsshowninFigure 4-2 ,theMAVismounteddownthewindtunnelcenterline.The modelisattachedtoaverticalmountingrod,whichattaches tothe6degreeoffreedom JR3100M40Aforcebalance.Thebottomofthebalanceisattac hedtoastingthatbolts tothewindtunneloor.Apitottubeislocatedjustbelowthe rightwingleadingedge. Experimentsareconductedwiththe“Flash”commerciallyav ailableexpanded polypropylene(EPP)foamMAV.Themodelhasanaft-placedta il,midwingdesign,and fuselagewitharectangularcross-section.Thefactorysta ndardwingsarereplacedwith hotwirecutEPPfoamwings.ThewingsuseaNACA2410airfoila ndaredesignedto 50

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Figure4-1.TurbulentsetupinRMITindustrialwindtunnel( PhotocourtesyofBrian Roberts) holdasmanyparametersconstantaspossible,whilevarying theleadingedgesweep. Wingspan(0.8 m ),wingarea(0.2 m 2 ),rootchordlength(0.3 m ),taperratio(0.67),and aspectratio(0.8)areallheldconstant.Thepropellerandm otorarealsoremovedto eliminatepropellerdownwash,butthepushrodsandservosu sedforightcontrolare leftonthemodel. Thefoammodelshowsnoticeablevibrationinturbulentow. Asaresult,the wingtipsandcontrolsurfacesarestiffenedusingwireinbo thsmoothandturbulentow tests.Copperwireisusedtoconnect:1)thewingtips(atthe quarterchordpoint)to thefuselagetoreducewingbending,2)eachsideoftherudde rtothetopandbottom surfacesoftheelevatortoreducecontrolsurfacedeectio ns,and3)thewingtipstothe tipsofthehorizontaltailtoreducefuselagetwisting.4.1.2ExperimentalDesign Thetestmatrixtondtheaircraftstabilityderivativesin cludes5anglesofattack (-5 ,0 ,5 ,10 ,15 )at4anglesofsideslip(0 ,-3 ,-6 ,-9 ).Theanglesofattackare measuredusinganinclinometer,andtheanglesofsideslipa remeasuredbydropping aplumblinefromthetailandmeasuringthedistancefromthe centerlineofthewind 51

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Figure4-2.TestsetupinRMITindustrialwindtunnel(Photo scourtesyofBrianRoberts) tunnel.Thissetof20datapointsismeasuredonbothaswept( 10 atleadingedge) andunswept(0 atleadingedge)wingunderbothsmoothandturbulentwindtu nnel conditions.Thewindtunnelconditionsarecharacterizedb ypreviousresearchwith smoothowat1.2%turbulenceandtheturbulentowtestinga t7.4%turbulence[ 105 ]. Thetestprocedureisdesignedtoyieldaccurateaveragefor cesandmomentsin thepresenceofanobservedsensordrift.Thebalanceistare datthestartofeachtest run,followedbyanimmediaterampingofthewindtunnelaver ageairspeedto8.3 m/s Avesecondaveragedmeasurementistaken,followedbyatwo minuteaveraged measurement,andthenanothervesecondmeasurement.Atth ispointthepower tothewindtunnelturbineisturnedoffandanalvesecondm easurementistaken. Thethreeshortmeasurementsarethenusedtondtheratesof driftandtheaverage amountofdriftpresentduringthetwominutetest.Oncethet wominutemeasurement hasbeencorrectedfordrift,itisreadytobeplotted.4.1.3Results Thethreeforcesandthreemomentsareshownforbothsweptan dunsweptwings inbothsmoothow(1.2%turbulenceintensity)andturbulen tow(7.4%turbulence intensity)inFigures 4-3 4-6 52

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-5 0 5 10 15 -10 -5 0 0 0.5 1 1.5 2 2.5 Beta (deg) Alpha (deg) Drag (N) -5 0 5 10 15 -10 -5 0 -0.5 0 0.5 1 1.5 Beta (deg) Alpha (deg) Sideforce (N) -5 0 5 10 15 -10 -5 0 -5 0 5 10 Alpha (deg) Beta (deg)Lift (N) -5 0 5 10 15 -10 -5 0 -0.1 0 0.1 0.2 0.3 Beta (deg) Alpha (deg) Mx (Nm) -5 0 5 10 15 -10 -5 0 -0.4 -0.2 0 0.2 0.4 0.6 Beta (deg) Alpha (deg) My (Nm) -5 0 5 10 15 -10 -5 0 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Beta (deg) Alpha (deg) Mz (Nm) Figure4-3.Unsweptwingmodelinsmoothow -5 0 5 10 15 -10 -5 0 0.5 1 1.5 2 2.5 3 3.5 Beta (deg) Alpha (deg) Drag (N) -5 0 5 10 15 -10 -5 0 -0.5 0 0.5 1 1.5 2 Beta (deg) Alpha (deg) Sideforce (N) -5 0 5 10 15 -10 -5 0 -5 0 5 10 15 Alpha (deg) Beta (deg)Lift (N) -5 0 5 10 15 -10 -5 0 -0.1 0 0.1 0.2 0.3 0.4 Beta (deg) Alpha (deg) Mx (Nm) -5 0 5 10 15 -10 -5 0 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Beta (deg) Alpha (deg) My (Nm) -5 0 5 10 15 -10 -5 0 -0.4 -0.3 -0.2 -0.1 0 0.1 Beta (deg) Alpha (deg) Mz (Nm) Figure4-4.Unsweptwingmodelinturbulentow Linearizationofightsystemsaboutanoperationpointisa nacceptedmethodto gaininsighttotheightdynamics;however,thesystemneed stobeprovenlinearabout thedesignpoint.So,linearregressionisperformedoneach subsetofdata,andthe validityofusingalinearmodelforitsdynamicsisevaluate dusingthenormalizedroot meansquaredeviationoftheexperimentaldatafromthemode l'spredicteddatavalues. Thenormalizedrootmeansquaredeviationisastandardmetr icusedtojudgethe 53

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-5 0 5 10 15 -10 -5 0 0.5 1 1.5 2 2.5 3 3.5 Beta (deg) Alpha (deg) Drag (N) -5 0 5 10 15 -10 -5 0 -0.5 0 0.5 1 1.5 2 Beta (deg) Alpha (deg) Sideforce (N) -5 0 5 10 15 -10 -5 0 -5 0 5 10 15 Alpha (deg) Beta (deg)Lift (N) -5 0 5 10 15 -10 -5 0 -0.1 0 0.1 0.2 0.3 0.4 Beta (deg) Alpha (deg) Mx (Nm) -5 0 5 10 15 -10 -5 0 -0.5 0 0.5 Beta (deg) Alpha (deg) My (Nm) -5 0 5 10 15 -10 -5 0 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Beta (deg) Alpha (deg) Mz (Nm) Figure4-5.10 Sweptwingmodelinsmoothow -5 0 5 10 15 -10 -5 0 0.5 1 1.5 2 2.5 3 3.5 Beta (deg) Alpha (deg) Drag (N) -5 0 5 10 15 -10 -5 0 -0.5 0 0.5 1 1.5 2 Beta (deg) Alpha (deg) Sideforce (N) -5 0 5 10 15 -10 -5 0 -5 0 5 10 15 Alpha (deg) Beta (deg)Lift (N) -5 0 5 10 15 -10 -5 0 -0.1 0 0.1 0.2 0.3 0.4 Beta (deg) Alpha (deg) Mx (Nm) -5 0 5 10 15 -10 -5 0 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Beta (deg) Alpha (deg) My (Nm) -5 0 5 10 15 -10 -5 0 -0.4 -0.3 -0.2 -0.1 0 0.1 Beta (deg) Alpha (deg) Mz (Nm) Figure4-6.10 Sweptwingmodelinturbulentow qualityoftforamodel[].Theformulaforthenormalizedro otmeansquaredeviationis giveninEquation 4–1 NRMSD ( exp model )= q P ni =1 ( exp i model i ) 2 n exp max exp min (4–1) Thenormalizedrootmeansquaredeviationsforselectmomen tandforce derivativesareshowninTables 4-1 4-5 .Analyzingnormalizedrootmeansquare 54

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deviationscanberatherarbitrary,butthevastmajorityof valuesarebelow5%,andthe valuesunderhighvaluesofturbulenceshownosignicantin creases,whichindicates thatthelinearforceandmomentderivativesunderturbulen cearejustasusefulasthose takeninlowturbulence.Table4-1.Normalizedrootmeansquaredeviationfor L TestConditions =0 =3 =6 =9 =15 LE =0 I =1.2%2.99%3.96%4.48%3.47% LE =10 I =1.2%3.55%12.18%3.04%3.78% LE =0 I =7.4%3.35%2.70%4.32%3.89% LE =10 I =7.4%2.37%3.07%3.90%2.71% Table4-2.Normalizedrootmeansquaredeviationfor M Y TestConditions =0 =3 =6 =9 =15 LE =0 I =1.2%7.63%15.80%15.41%17.39% LE =10 I =1.2%7.99%8.41%5.64%8.26% LE =0 I =7.4%8.63%7.85%13.72%7.68% LE =10 I =7.4%15.45%10.46%13.41%17.13% Table4-3.Normalizedrootmeansquaredeviationfor Y TestConditions =-5 =0 =5 =10 =15 LE =0 I =1.2%4.97%4.43%4.19%1.94%3.23% LE =10 I =1.2%2.28%1.68%5.10%2.44%3.30% LE =0 I =7.4%2.43%1.59%1.88%3.02%3.38% LE =10 I =7.4%3.82%3.59%5.17%3.22%3.50% Table4-4.Normalizedrootmeansquaredeviationfor M X TestConditions =-5 =0 =5 =10 =15 LE =0 I =1.2%3.60%1.81%3.07%1.56%5.99% LE =10 I =1.2%2.70%3.87%4.38%4.01%8.96% LE =0 I =7.4%2.04%2.20%2.85%4.47%4.29% LE =10 I =7.4%8.39%3.67%7.15%4.58%4.56% Theliftforceandpitchmomentshowastronglinearcorrelat ionwiththeangle ofattack( ),andthesideforce,rollmoment,andyawmomentshowastron glinear 55

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Table4-5.Normalizedrootmeansquaredeviationfor M Z TestConditions =-5 =0 =5 =10 =15 LE =0 I =1.2%5.11%7.32%4.86%2.34%5.51% LE =10 I =1.2%3.24%1.88%5.76%3.32%5.24% LE =0 I =7.4%3.26%2.64%0.40%4.94%4.71% LE =10 I =7.4%3.98%2.02%7.36%5.04%4.85% correlationwiththeangleofsideslip( ).Thedatademonstratesstronglineartrends, sotheslopesoftheselineartendenciesarecomparedoveram ountsofsweepand turbulence.Theslopesarefoundateachightangle,averag ed,andshownwiththeir standarddeviation.Forexample,thederivativeofliftwit hrespecttoangleofattackis foundbyrunningalinearcorrelation,intheleastsquaress ense,onthedataateachof thefouranglesofsideslip,asshowninFigure 4-7 usingthedatafromtheunsweptwing insmoothow. 56

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-5 0 5 10 15 -2 0 2 4 6 8 10 Alpha (deg)Lift (N) -5 0 5 10 15 -2 0 2 4 6 8 10 Alpha (deg)Lift (N) -5 0 5 10 15 -2 0 2 4 6 8 10 Alpha (deg)Lift (N) -5 0 5 10 15 -4 -2 0 2 4 6 8 10 Alpha (deg)Lift (N) Figure4-7.Liftcurvesateachsideslipangle(solidlineforexperimentaldata,dashedlineforlinearre gression, =0topleft, =-3top right, =-6bottomleft, =-9bottomright) 57

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Thefourslopesareaveragedtogiveameanslopeforthesurfa ceandthestandard deviationisalsofound,asshowninTable 4-6 Table4-6.Findingmeanslopeandstandarddeviation LiftCurveSlope( N = ) 0 0.5210 -3 0.5439 -6 0.5111 -9 0.5496 Mean0.5134 0.0183 Themeanslopesandstandarddeviationsforbothforcesandm omentsunderall testconditionsareshowninTables 4-7 – 4-11 Table 4-7 showstheliftcurveslope.TheliftcurveslopeisdenedinE quation 4–2 L = @ L @ (4–2) Table4-7. L averagederivativesandstandarddeviations( N = ) 1.2%Turbulence7.4%Turbulence 0 Sweep0.531 0.0180.722 0.024 10 Sweep0.660 0.0550.679 0.031 AccordingtoTable 4-7 ,intheabsenceofturbulence,theadditionof10 wing sweepproducesa24%increaseintheproductionofliftdueto anincreaseintheangle ofattack.Inthepresenceofturbulence,adecreaseof6.0%i ntheliftcurveslopeis observedasthewingisswept;however,thisdifferenceinme ansisnotlargeenough tobeoutsidetheboundsoftheone-sigmarangesofthetwoval ues.Thepresenceof turbulenceproduceda36%increaseintheproductionoflift duetoanincreaseinthe angleofattackforanunsweptwing,butproducedaninsigni cant(2.8%andnotoutside theboundsofaone-sigmarange)increaseinliftproduction forasweptwing. 58

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Table 4-8 showsthepitchingmomentcurveslope.Thepitchingmomentc urve slope, M Y ,isalsoreferredtoasthelongitudinalstaticstabilityde rivativeisdenedin Equation 4–3 .Anegativevalueindicatesthatitisstaticallystableabo utthelateralaxis. M Y = @ M Y @ (4–3) Table4-8. M Y averagederivativesandstandarddeviations( Nm = ) 1.2%Turbulence7.4%Turbulence 0 Sweep-0.0247 0.0025-0.0191 0.0036 10 Sweep-0.0402 0.0022-0.0200 0.0057 AccordingtoTable 4-8 ,intheabsenceofturbulence,theadditionof10 wing sweepproducesa63%increaseintheproductionofanegative pitchmomentdueto anincreaseintheangleofattack.Inthepresenceofturbule nce,thisincreaseisonly 4.7%(adifferencethatlieswithintheone-sigmabounds).T hisobservedtrenddoes notagreewithsomeconventionalexpectationsthatsweepba ckofthewingschanges thedownwashonthetail,andthus,reduceslongitudinalsta bility[ 106 ].However,itis possiblethatthemarginalchangeinsweepangleinthetests isnotenoughtocreatethe signicantincreaseinthedownwashonthetailthatisassoc iatedwithsweepinduced reductionoflongitudinalstability.So,thedominanteffe ctseenisanaftwardmovement ofthecenterofliftandresultinglongitudinalstabilityi mprovements.Theaddedsweep movesthecenterofliftfurtheraftofthecenterofgravity, increasingthestaticmargin andimprovingthelongitudinalstability.Thepresenceoft urbulenceproducesa23% decreaseintheproductionofanegativepitchmomentduetoa nincreaseintheangleof attackforanunsweptwing.Thisdecreaseliesjustwithinth eone-sigmarangesforthe M Y slopes. Table 4-9 showsthesideforcecurveslope.Thesideforcecurveslopei sdenedin Equation 4–4 59

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Y = @ Y @ (4–4) Table4-9. Y averagederivativesandstandarddeviations( N = ) 1.2%Turbulence7.4%Turbulence 0 Sweep-0.149 0.011-0.200 0.0093 10 Sweep-0.164 0.023-0.205 0.014 AccordingtoTable 4-9 ,intheabsenceofturbulence,theadditionof10 wing sweepproducesa10%increaseintheproductionofsideforce duetoanincreaseinthe sideslipangle.Inthepresenceofturbulence,thisincreas eisonly2.5%.Thepresence ofturbulenceproducesa34%increaseintheproductionofsi deforceduetoasideslip angleforanunsweptwing.However,itshouldbenotedthatth eone-sigmaranges(the rangeofvalueswithinonestandarddeviationofthemean)fo rthe Y slopesoverlap eachotherwhentheturbulenceintensityisheldconstant;t hus,indicatingthatwing sweepmayhaveanegligibleimpacton Y inbothsmoothandturbulentow. Table 4-10 showstherollmomentcurveslope.Therollmomentcurveslop e, M X ,isalsoreferredtoastherollmomentstaticstabilityderi vative,andisdenedin Equation 4–5 .Anegativevalueindicatesthatthevehicleisstaticallys tableaboutthe longitudinalaxis. M X = @ M X @ (4–5) Table4-10. M X averagederivativesandstandarddeviations( Nm = ) 1.2%Turbulence7.4%Turbulence 0 Sweep-0.0326 0.0036-0.0432 0.0040 10 Sweep-0.0358 0.0049-0.0442 0.0037 AccordingtoTable 4-10 ,intheabsenceofturbulence,theadditionof10 wing sweepproducesa9.8%increaseintheproductionofanegativ erollingmomentdue toanincreaseinthesideslipangle.Thechangeinthemean M X isnotoutsidethe 60

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boundsofaone-sigmarange.Thepresenceofturbulenceprod ucesa32%increasein theproductionofanegativerollingmomentduetoanincreas einthesideslipangleon anunsweptwing,anda23%increaseona10 sweptwing. Table 4-11 showstheyawmomentcurveslope.Theyawmomentcurveslope, M Z ,isalsoreferredtoastheyawmomentstaticstabilityderiv ative,andisdenedin Equation 4–6 .Apositivevalueindicatesthatthevehicleisstaticallys tableaboutthe verticalaxis. M Z = @ M Z @ (4–6) Table4-11. M Z averagederivativesandstandarddeviations( Nm = ) 1.2%Turbulence7.4%Turbulence 0 Sweep0.027 0.00240.038 0.0028 10 Sweep0.030 0.00510.038 0.0027 AccordingtoTable 4-11 ,intheabsenceofturbulence,theadditionof10 wing sweepproducesa11%increaseintheproductionofayawmomen tduetoanincrease intheangleofsideslip.Inthepresenceofturbulence,noch angeisseenin M Z as sweepchanges.Neitherinturbulence,norinsmoothowisas ignicant(outsideof one-sigmarange)changeseenwithwingsweep.Thepresenceo fturbulenceproduces a41%increaseintheproductionofyawmomentduetoanincrea seintheangleof sideslipforanunsweptwing.4.1.4Analysis Thecausefortheincreaseintheliftcurveslopewithwingsw eepissomewhat counterintuitive.Infact,thesectionliftcoefcientise xpectedtodecreasewith increasingsweepduetothereductioninthemagnitudeofthe velocitythatisperpendicular tothewing[ 107 ].Ifthewingisassumedtobehavelikeabeam(theexuralaxi sis alwaysparalleltothesweepaxis),thenstaticaeroelastic effectsalsopredictareduction intheliftcurveslopewithincreasingsweep,becausewingb endingforasweptback 61

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wingresultsinareducedangleofincidence[ 108 ].However,thewingsusedfortesting mighthavetoolowofanaspectratiotoapplytheresultsof“b eam-model”wings.The wingvibrationseeninthetestscouldcauseanincreaseinth eliftcoefcient,because theeffectiveangleofincidencevariesthroughthepitchin gandtwistingmovement[ 108 ]. Theplacementofthecopperwireatthewingquarterchordmig htalsoincreasethe effectiveangleofincidence,becausetheexuralaxisfora sweptwingliesaftofthe midchord[ 108 ]. Theimprovementinlongitudinalstaticstabilityderivati ve, M Y ,inthestatictestscan beattributedtotheincreaseintheliftcurveslopeandmove mentofthecenterofliftas thewingissweptbackwards.Asthecenterofliftofthewing, x wing ,movesbackwards, theneutralpoint, x np ,becomesmovesbackwardsaswell.Thisrelationshipisappa rent fromEquation 4–7 ,whichassumessmallangles,thattheaircraftisinatrimst ate,and thatfuselageeffectsarenegligible. ( L wing + L tail + D wing + D tail ) x np = L wing x wing + D wing y wing + L tail x tail + D tail y tail (4–7) Thereisarelationshipbetweentheneutralpoint, M Y ,and L ,showninEquation 4–8 Therelationshipindicatesthatastheneutralpointmovesb ackwards,andlongitudinal staticstabilityincreases.Equation 4–8 alsoindicatesthatif L increases,then M Y increasesproportionately. M Y = x np L (4–8) Thelongitudinalpositionoftheneutralpointonboththesw eptandunsweptwings iscalculatedaccordingtoEquation 4–9 x np = M Y L (4–9) 62

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ThecalculationsinEquation 4–10 applytheliftcurveslopeandpitchmomentslope dataforsmoothowinTables 4-7 and 4-8 toshowthepositionoftheneutralpointofthe unsweptwing. x np unswept = 0.0247 0.531 (4–10) =0.0465 m ThecalculationsinEquation 4–11 applytheliftcurveslopeandpitchmomentslope dataforsmoothowinTables 4-7 and 4-8 toshowthepositionoftheneutralpointofthe sweptwing. x np swept = 0.0402 0.660 (4–11) =0.0610 m Equation 4–12 showsthatifthedataisassumedtobecorrect,thentheneutr al pointwouldhavemoved1.5 cm x np = x np swept x np unswept =0.0610 0.0465 (4–12) =0.0145 m Thus,thedramaticincreasein M Y couldbecreatedbythecombinationofincrease intheliftcurveslopeandamovementoftheneutralpointof1 .5 cm Itisimpossibletotelltheexactquantityofmovementofthe neutralpointbecause nomeasurementsaremadeoftheforcesofthewingortailinde pendently.However, calculationsshowthattheaircraftneutralpointcouldhav emovedupto3.2 cm 63

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GeometricpropertiesleadtothederivationofEquations 4–13 tondthelongitudinal shiftofthemeanaerodynamicchordwhenwingisswept. x MAC =sin( LE ) b 2 c root +2 c tip 3( c root + c tip ) (4–13) AgeometriccalculationinEquation 4–14 usingthedataofthewingusedintesting ndsthesolution,whichshowsthata10 wingsweepmovesthemeanaerodynamic chordby3.2 cm x MAC =sin(10 ) 0.8 2 0.3+2 0.2 3(0.3+0.2) (4–14) =0.0324 m Therefore,thedramaticincreaseinthe M Y makesphysicalsense. Thechangesseenin Y withwingsweeparenotsignicantenoughtobelievethat thereisaphysicalphenomenoncreatingthosechanges.Wing sweepisnotexpectedto beanoteworthycontributortothecreationofsideforcebec ausetheprimarycreatorof sideforceistheverticaltail[ 109 ]. Theimprovementintherollmomentstaticstabilityderivat iveisanticipatedby theevidenceofpreviousaircraftdesignexperience[ 109 ].Theadditionofwingsweep createsagreaterimbalanceinthespeedoftheowperpendic ulartotheleftandright wings.Forpositivesideslip,theresultisthecreationofm oreliftontherightwinganda rollingmomentthatrestorestrimmedightwithnosideslip Theincreaseintheyawmomentstaticstabilityderivativei sexpectedwithwing sweepback[ 107 ].Similarlytotheimprovementintherollmomentstaticsta bility derivative,thewingsweepproducesanimbalanceinthespee doftheowperpendicular totheleftandrightwings.Forpositivesideslip,theresul tisgreaterliftontheright, producinggreaterinduceddragontherightwing.Thedragim balancecreatesa restoringmoment. 64

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4.1.5Conclusions Inalllinearanalysesthechangeinturbulenceproducesala rgerpercentagechange intheightderivativethantheadditionofsweeptoamodeli nsmoothow.Theother trendfromthedataworthnotingistheeffectofturbulenceo nthedifferencebetween derivativesofsweptandunsweptwings.Insmoothow,thede rivativesbetweenswept andunsweptwingsshowsomedifferences.Thesweptwingwasm orestableinall threemomentsthanitsunsweptcounterpart.Whilethistren dremainsunderturbulent conditions,thedifferencebetweenthederivativesdecrea sestonegligibleamounts. Thus,itappearsthatthepresenceofturbulencemayreducet heeffectsofchangesin wingsweeponstaticstabilityderivatives. 4.2DynamicWindTunnelTesting 4.2.1WindTunnelSetup MoredetailedtestingisconductedattheUniversityofFlor idaResearchand EngineeringEducationFacility(REEF)LowSpeedWindTunne l(LSWT)toaddmore parameterstotheexperimentalspace.TheLSWTisanopenjet ,opentestsectionwind tunnelwitha1 m x1 m testsection. TheaircraftusedintestingistheUSAFGenericMAV(GenMAV) ,athoroughly investigated“baselineMAV”intendedforexactlythetypeo fgeometrictradestudies performedherein[ 110 ].Theaircrafthasaconventionaltail,ahighwing,arectan gular fuselage,andathincamberedairfoil.Theunsweptwinghasa wingspanof0.610 m and arootchordof0.127 m .Themotorandpropellerareabsentfromthemodel,soalltes ts areconductedunpowered.However,allservosandpushrodsu sedincontrolarelefton themodel.Multiplewingsaremanufacturedandcutattheroo tchordtoachievevarious sweepangles.Wingdihedralissetbyattachingasmallalumi numrectangletothe topsurfaceboththeleftandrighthalvesofthewing.Thealu minumisbenttoachieve variousdihedralangles,andwingdihedralismeasuredusin gadigitalinclinometerafter applicationtothewing. 65

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Themodelusedforturbulentanalysisusesawingconstructe dof6layersof bi-directionalcarbonberina0 -0 -45 -45 -0 -0 pattern(where0 plyhasbers runningspanwiseandchordwise),asopposedtothelighter2 -layerwingusedforight. Thepurposeoftheincreasedstiffnessistoreducetheeffec tsofexibilityinthewing. Toaccomodatetestinginturbulence,asystemofwoodendowe lrodgridsis designedandinstalled.Twogridsareimplementedinthiste sting,examplesofwhichare showninFigure 4-8 Figure4-8.Turbulencegeneratinggrids(Photoscourtesyo fBrianRoberts) Grid-generatedturbulenceisaneffectivemethodofturbul enceproductioninwind tunneltesting[ 111 ].Designingthegridsrequiresthebalancingofcompetingo bjectives. Turbulencelengthscalesgrowastheowgetsfurtherbeyond thegrid,buttheintensity oftheturbulencedecreases[ 40 ].Makingthegridelementslargerandincreasingthe ratiooftheareathatisblockedtothetotalareaoftheowcr osssectionincreasethe turbulencelengthscalesandintensity,butproduceturbul encethatdoesnotbecome well-mixedandhomogeneousuntilmuchfurtherbeyondthegr id. Theprimarydesignconcernforthistestingisproducingtur bulencelengthscales closertotheatmosphericturbulencelengthscales,sotheg ridsareplaceasfar 66

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upstreamfromthemodelaspossible.Theresultisthatthegr idsareplacedatthe exitofthecontractionsectionatadistanceof1.34 m fromthemodel. Thesecondconcernistoensurethattheturbulenceishomoge neous.Sothe maximumgridelementsizeischosentobe5.1 cm ,thusensuringthatthedistance betweenthegridsandthemodelisatleast26timeslongertha nthegridelements, whichhasbeenshowntoproducehomogeneousturbulence[ 112 ].Theresultisthat Grid1hasameshwidthof5.1 cm andaroddiameterof0.95 cm andGrid2hasa meshwidthof5.1 cm andaroddiameterof1.9 cm Previousresearchhascharacterizedtheturbulencethatca nbeexpectedtobe producedbygridsofthissize.Theturbulenceintegralleng thscaleisexpectedto beoftheorderof0.5 m ,whichisnearlythesizeoftheGenMAVwingspan[ 113 ]. Theanisotropymeasureisexpectedtobebetween0.9and1.1i nboththelateraland verticaldirections,with1.0beingtheidealcasewithnegl igibleisotropiccharacteristics[ 112 ]. Thus,theturbulencecanbeexpectedtoberelativelywell-m ixed.Theturbulence intensityforthegivenmeshwidthsisexpectedtobeatabout 3.2%[ 112 ].Acalibrated hot-wireanemometerisusedtondexperimentalturbulence intensitiesproducedbythe gridsintheLSWT,showninTable 4-12 Table4-12.Experimentalturbulenceintensitiestested MeshUsedTurbulenceIntensity None0.89%Grid12.71%Grid23.85% Theexperimentalturbulenceintensitiesareonthesameord erastheexpected turbulenceintensities.Notethatthereisabaselinelevel ofturbulenceinthetunneltest section. AsshowninFigure 4-9 ,thetestmodelisattachedtoahorizontalmountingrodthat connectstoaJR330E12A4forcebalance.Thebalanceisattac hedtoamotioncontrol rodthatconnectstothetworoboticarms.Theopentestsecti onfacilitatestheuseof 67

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thissystemofroboticarmsthatmanipulatethemodelinpitc h-plungemaneuvers.The armscanmoveupordownindependentlyandatdifferentspeed sbyapairoflinear electricmotorslocatedbelowtheairowofthetestsection .Thetwomotorspermitthe productionofawiderangeofboth and speeds.Dynamicpressureistakenusinga pitottubeplacedbelowtheleftwing. Figure4-9.TestsetupinREEFlowspeedwindtunnel(Photosc ourtesyofBrian Roberts) 4.2.2ExperimentalDesign Thegoaloftheexperimentistondtheinteractionsbetween statevariables,basic designparameters,andturbulenceintensitywithregardto thelongitudinaldynamicsof theGenMAV.Testingisconductedatvaryinganglesofattack ,ratesofchangeofangle ofattack,andpitchrates.Thus,thereare6variablesthata rechangedtocapturethe fulllongitudinaldynamicsoftheMAVinturbulentow:turb ulenceintensity( I = u rms / U testedat0.89%,2.71%,and3.85%),leadingedgesweepangle ( LE ,testedat0 ,15 and30 ),dihedralangle( ,testedat0 ,5 ,and10 ),angleofattack( ,testedat-10 0 ,and10 ),rateofchangeofangleofattack( ,testedat0 = s ,10 = s ,and20 = s ),and pitchrate( ,testedat0 = s ,-10 = s ,and-20 = s ).Alltestsarerunatanaveragespeedof 13 m/s Testsaredividedintoanalysisofsweepanddihedralangles toreducethesizeof thetestmatrix.Thegoalistotthedynamicsintoamodelsuc hthattheerrorbetween 68

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themathematicalmodelandtheexperimentalresultsismini mized.Anexampleofsuch amodelttingisgiveninEquation 4–15 y exp = f ( x 1 x 2 ,..., x n )+ where, y exp = experimentalresult(4–15) x i = experimentalparameter = error Themodelisassumedtobequadratictondallrstandsecond ordereffects andcouplingterms.Themodelisassumedtobequadratictoal lowinvestigationinto couplingbetweenturbulence,anyofthestateuctuations, oranyoftheMAVdesign parameters.Anexampleofaquadraticmodelforathreeparam etertestisgivenin Equation 4–16 f ( x 1 x 2 x 3 )= c 0 + c 1 x 1 + c 2 x 2 + c 3 x 3 + c 4 x 2 1 + c 5 x 2 2 + c 6 x 2 3 + c 7 x 1 x 2 + c 8 x 1 x 3 + c 9 x 2 x 3 (4–16) where, c i 2 R Thefulltestmatrixisstilltoolargetotesteverycombinat orialcondition;therefore, aresponsesurfacemethodisusedtodetermineasamplingoft hetestspacethatwill produceaquadraticmodelofthedynamics[ 114 ].Thetestconditionsarechosento produceaface-centeredcentralcompositedesign.Thistyp eofdesignusespoints locatedatthecenterandattheextremesofparameterranges tondthemostaccurate modelusingtheleastnumberofexperiments[ 114 ]. 4.2.3Results Attemptsaremadetotthedatatosuchamodel,butnotallter msarefoundtobe signicant.Thequadraticregressionmodelsareformedint heleastsquaressense,but 69

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arethenadjustedusinganalysisofthecoefcientofdeterm ination( R 2 ),theadjusted coefcientofdetermination( R 2 ),andp-valuestoeliminateinsignicantterms.The coefcientofdeterminationmeasureshowwellthemodeldes cribesthevariationin data.Theadjustedcorrelationcoefcientindicateswheth ertheadditionofanewterm inthemodelproducesanimprovementintthatisanybettert hancouldhappenby randomchance.Thep-valueisusedtoindicatehowlikelyiti sthatthemodelwould produceanoutcomeasextremeasacertaindatapoint. Theresultingmodelscanbeanalyzedthrougheitheranatura lmodeloracoded model.Inanaturalmodel,thedynamiccoefcientsarefound intermsoftheunitsof thecorrespondingstate.Forexample, C L isnormalizedsuchthatmultiplyingby (in degrees)createsacoefcientofliftterm.Inacodedmodel, thedynamiccoefcientsare relatedtotherangeofthestatethatwasvaried.Forexample isallowedtorangefrom -10 to10 ,butisnormalizedtocreatearangefrom-1to1,and C L isthennormalized tothatrange.If is5 thenthecoded C L shouldbemultipliedby0.5toobtainthelift contributionfrom .Thecodedmodelallowsthemostsignicanttermstobeident ied moreeasilybecauseitisnotrelatedtotheunitsofthestate s.Thedisadvantagetothe codedmodelisthatthecoefcientslosesomeoftheirphysic alsignicance.So,the codedmodelisusedtocomparetherelativeimportanceofter ms,whilethenatural modelisusedforcalculationofforceandmomentcoefcient s. 4.2.3.1SweepModel Thevehicledynamicsaremodeledasaquadraticfunctionoft hestatesand congurationalongwithturbulenceintensity.Thestatesa retheangleofattackwithits rateofchangeandthepitchanglewithitsrateofchange.The congurationissimply thesweepangle.Theforcecoefcientsforbothliftanddrag andthepitch-moment coefcientareeachmodeledassuchaquadratic.Thereprese ntativeexpressionfor thepitchmomentisstatedexplicitlyinEquation 4–17 withtheforceexpressionsusinga similardependency. 70

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C M = C M o + C M + C M LE LE + C M + C M + C M I I + C M LE LE + C M + C M + C M I I + C M LE LE + C M LE LE + C M LE I LE I (4–17) + C M _ + C M I I + C M I I + C M 2 2 + C M 2 LE 2LE + C M 2 2 + C M 2 2 + C M I 2 I 2 Theactualdependencyoftheaerodynamicsonthestatesandc ongurationalong withturbulenceintensityisdemonstratedtobenonlinear; however,severalcoefcients fromtheformofEquation 4–17 areshowntobenegligible.Inparticular,thecoupling termsbetweentheratesofchangeofstatesareevidentinthe aerodynamicsbutthe couplingtermsbetweentheturbulenceandsweeparenotcrit icaltodescribingthe aerodynamics.Theremainingtermswhicharenumericallysi gnicantintheanalysisof theexperimentaldataaregiveninTable 4-13 Table4-13.Naturalsweepmodel C L C D C M 0 th orderterm6.60 e 1 2.07 e 2 -2.23 e 1 ( )7.23 e 2 4.86 e 3 -4.35 e 2 LE ( )-4.82 e 3 2.72 e 3 -3.48 e 3 ( = s )-2.04 e 2 6.93 e 5 1.24 e 2 ( = s )8.06 e 3 2.53 e 4 -5.80 e 3 I 1.45 e 1 -4.24 e 4 1.05 e 2 LE crossterm-3.06 e 4 0.000.00 crossterm-5.85 e 5 -2.24 e 4 4.42 e 4 crossterm4.44 e 5 -5.07 e 5 -2.09 e 4 crossterm-6.65 e 4 1.62 e 5 4.57 e 4 2 term0.001.16 e 3 0.00 2LE term0.00-8.79 e 5 0.00 I 2 term-3.26 e 2 0.000.00 Theeffectofturbulenceontheaerodynamicsissomewhatmor eevidentwhen consideringacodedmodel.Thecoefcientsforsuchamodelr esultbyscalingtheterms inTable 4-13 toobtainthevaluesinTable 4-14 .Inthiscase,thecodedmodelshows 71

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thattheeffectofturbulenceisactuallyonthesameorderof magnitudeastheeffectof angleofattack.Table4-14.Codedsweepmodel C L C D C M 0 th orderterm5.31 e 1 3.73 e 2 -1.14 e 1 6.27 e 1 3.13 e 2 -3.70 e 1 LE -7.24 e 2 1.31 e 3 -5.22 e 2 -1.37 e 1 -9.25 e 4 7.86 e 2 1.40 e 2 4.14 e 3 -1.23 e 3 I -1.37 e 2 -6.27 e 4 1.55 e 2 LE crossterm-4.58 e 2 0.000.00 crossterm-5.85 e 3 -2.24 e 2 4.42 e 2 crossterm4.44 e 2 -5.07 e 3 -2.09 e 2 crossterm-6.65 e 2 1.62 e 3 4.57 e 2 2 term0.001.16 e 1 0.00 2LE term0.00-1.98 e 2 0.00 I 2 term-7.15 e 2 0.000.00 Theaerodynamicsaredemonstratedtohaveaquadraticdepen dencyonthe turbulenceintensity.ThismodelofEquation 4–17 alongwiththecoefcientsof Table 4-13 arecombinedtohighlightthisdependency.Theliftandpitc hmomentthat resultfromavehiclewithidenticalstatesandconguratio n( =5 =0 = s =0 = s LE =30 )butvariedlevelsofturbulenceareshowninFigure 4-10 .Thepitchmoment isrelativelylinearandmonotonicallyincreasesasthetur bulenceincreasesbuttheliftis noticeablyquadraticwiththemaximumliftoccurringfortu rbulencebetween2%and3%. 4.2.3.2DihedralModel Thevehicledynamicsaremodeledasaquadraticfunctionoft hestatesand congurationalongwithturbulenceintensity.Thestatesa retheangleofattackwithits rateofchangeandthepitchanglewithitsrateofchange.The dihedralangledenes theconguration.Theforcecoefcientsforbothliftanddr agandthepitch-moment coefcientareeachmodeledassuchaquadratic.Thereprese ntativeexpressionforthe pitchmomentisstatedexplicitlyinEquation 4–18 withtheforceexpressionsfollowinga similarform. 72

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0% 1% 2% 3% 4% 5% 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Turbulence IntensityC L 0% 1% 2% 3% 4% 5% -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Turbulence IntensityC M Figure4-10.Liftandpitchcoefcientchangeswithturbule nceintensityinsweepmodel C M = C M o + C M + C M + C M + C M + C M I I + C M + C M + C M + C M I I + C M + C M + C M I I (4–18) + C M _ + C M I I + C M I I + C M 2 2 + C M 2 2 + C M 2 2 + C M 2 2 + C M I 2 I 2 Thedataanalysisprocessfoundthatsomeofthetermsinclud edinthequadratic modelofEquation 4–18 arenegligible.Thecouplingtermsbetweenthestatesplaya roleintheforceandmomentcreation,butcouplingbetweent urbulenceanddihedral angleisnotvitaltodescribingthedynamics.Thetermsthat arenumericallysignicant intheexperimentaldataareshowninTable 4-15 Therelativemagnitudeoftheeffectofturbulenceontheaer odynamicsismore obviouswhenexaminingacodedmodel.Thecodedmodelcoefc ientsinTable 4-16 are scaledversionsoftheircounterpartsinTable 4-15 .Theeffectofturbulenceprovestobe justassignicanttotheproductionofliftandpitchingmom entasangleofattackorthe rateofchangeofangleofattack. Justasinthesweepmodel,theturbulenceintensitycreates aquadraticeffecton theproductionofliftandalineareffectonthepitchmoment .ThemodelofEquation 4–18 iscombinedwiththecoefcientsofTable 4-15 toshowthedependencyonliftand 73

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Table4-15.Naturaldihedralmodel C L C D C M 0 th orderterm4.64 e 1 5.55 e 3 -2.45 e 1 ( )7.14 e 2 5.72 e 3 -3.91 e 2 ( )2.56 e 3 9.16 e 3 6.43 e 4 ( = s )-2.04 e 2 2.71 e 4 1.06 e 2 ( = s )7.06 e 3 1.35 e 4 -1.51 e 2 I 3.91 e 1 -1.83 e 4 1.81 e 2 crossterm1.05 e 4 -2.16 e 4 4.83 e 4 crossterm5.37 e 4 -5.14 e 5 -2.37 e 4 crossterm-6.11 e 4 2.85 e 5 4.90 e 4 I crossterm0.000.002.83 e 3 2 term0.001.31 e 3 0.00 2 term0.00-7.82 e 4 0.00 I 2 term-8.25 e 2 0.000.00 Table4-16.Codeddihedralmodel C L C D C M 0 th orderterm7.27 e 1 2.99 e 2 -5.80 e 2 6.71 e 1 4.07 e 2 -3.19 e 1 -1.28 e 2 6.67 e 3 3.21 e 3 -1.42 e 1 -1.45 e 4 5.70 e 2 9.41 e 3 4.20 e 3 -3.46 e 2 I -8.12 e 4 -2.70 e 4 -1.51 e 2 crossterm1.05 e 2 -2.16 e 2 4.83 e 2 crossterm5.37 e 2 -5.14 e 3 -2.37 e 2 crossterm-6.11 e 2 2.85 e 3 4.90 e 2 I crossterm0.000.004.19 e 2 2 term0.001.31 e 1 0.00 2 term0.00-1.96 e 2 0.00 I 2 term-1.81 e 1 0.000.00 pitchmomentastheturbulenceintensityvariesonamodelwi thidenticalstatesand conguration( =5 =0 = s =0 = s =5 ).Thisdependencyisshownin Figure 4-11 .Theliftisstronglyquadraticwiththemaximumliftoccurr ingforturbulence between2%and3%,whilethepitchmomentislinearwiththeno se-downmoment beingreducedastheturbulenceincreases. 74

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0% 1% 2% 3% 4% 5% 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Turbulence IntensityC L 0% 1% 2% 3% 4% 5% -0.45 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 Turbulence IntensityC M Figure4-11.Liftandpitchcoefcientchangeswithturbule nceintensityindihedralmodel 75

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CHAPTER5 POLYNOMIALCHAOSTHEORY PolynomialChaosExpansion(PCE)isthenameforboththepro cessandthe productofamathematicalprocedurethatpermitsthetransf ormationofastochastic systemintoadeterministicsystemexpressedinanexpanded setofvariables.The procedureisinitiatedbyrepresentingthesystemasafunct ionofasetofrandom variables, i 's,withassumeddistributions.Thenthesystemequationsa reprojected intothesubspaceofthebasisfunctionsofthoseassumeddis tributions.Theresultis asystemofexpandedorder,butexpressedincompletelydete rministicvariables.By producingadeterministicsystem,manyconventionaltechn iquesofsimulationcanbe applied,aswellastheapplicationofstandardcontroltech niquestoachievedesired performance. TheresultsfromsimulationofasystemusingPCEanalysisha vebeenshowntobe remarkablysimilartothoseproducedbyMonteCarlosimulat ion(MCS).Theadvantage ofPCEisthedramaticreductionincomputationalcostwhenc omparedtoMCS,aswell astheimprovedusestocontrollerdesign. PCEhasbeenappliedintheeldsofmechanical,aerospace,e lectricalengineering astheneedforgreaterprecisioninsimulationandcontrolh asbeenhinderedbythe constraintsofuncertaintyinmodelknowledgeandrepresen tation. 5.1Background Polynomialchaosaroseoutofthenonlinearfunctionalwork thatVolterradidto createtheequationsthatbearhisname.Volterraworkedonn onlinearfunctionalsas ageneralizationofthepowerseriesinthe1880's[ 115 ].Hisideasremainedrelatively unuseduntilthe1930's,whenWienerfoundtheVolterra'sfu nctionalscouldbeusedto explaintherandommovementofaparticle,alsoknowasBrown ianmotion.Wiener introducedthemathematicalconceptofaHomogeneousChaos torepresenta homogenousmediumandtohelpexplainthestatisticalnatur eofthemovementofa 76

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particleinthemedium[ 116 ].Inessence,thesenonlinearfunctionalscanbeusedasthe kerneltorepresentanonlinear,inthiscasestochastic,op erator.Astochasticoperator, beingonetypeofnonlinearoperator,couldthenbeapproxim atedusinganiteseriesof nonlinearfunctionals.Wiener'scontributionstothestud yofBrownianmotionisnotedby thenamingoftheintegralsthatdescribetherandommotioni nhishonor. Atthispointthenonlinearfunctionalthatwasusedtorepre sentthestochastic systemwasnotwell-dened.CameronandMartintransformed theWiener'sintegrals tocreateanorthonormalsetofoperatorsthatcouldbeusedt oconstructtheWiener space[ 117 ].TheFourier-HermitefunctionalsthatCameronandMartin proposedwould bethemostefcientsetoffunctionalstousetorepresentBr ownianmotion,whichis aGaussiansystem.Wienerusedhistheoriesandthecontribu tionsofCameronand Martininapplicationstorandomphenomenaasvariedasthem easurementofbrain wavesonmagnetictapeandtheperiodicityofasteroidsinou rsolarsystem[ 118 ]. AfterWiener,thetheoryofpolynomialchaosmovedforwards lowlyasvarious mathematicianssoughttoexpandhisideasintomoregeneral izedequationsand functionals.KarhunengeneralizedaPCEtoaKarhunen-Loev eexpansionthatpermits theuseofnon-stationaryGaussianprocesses[ 119 ].Ogurafoundtheorthogonal functionalsthatwoulddescribeaPoissonprocessandSegal landKailathfurther generalizedthefunctionalstoadmitallLvyprocesses[ 120 121 ].XiuandKarniadakis createdtheWiener-Askeyschemeofhypergeometricpolynom ialsthatcanrepresent stochasticprocesseswithavarietyofdistributions. GhanemandSpanosareresponsibleformuchoftherecentappl icationofPCEto uncertaintyinengineeringprocesses.Ghanem'sPhD.thesi sadvancedtheuseofPCE ondiscreteelements[ 122 ].Thetwoofthemappliedtheirworktoseveralproblems, rangingfromstructuralreliabilitytoseawavesimulation [ 123 – 125 ].Theysummarized theirworkbypublishingabooktosummarizetheconceptsand illustratethepossible applicationsofPCEtoniteelementanalysis[ 126 ].Bothproceededtojoinwithmany 77

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otherresearchersintheapplicationofPCEtospecicprobl ems,suchasanalyzing structuraldynamicswithparameteruncertainty[ 127 ]. 5.2Theory Arandomvariable, ,isafunctionoftheoutcomeofanexperiment, ,whose resultwillchangeeachinstancethatitisconducted.Inthe applicationofPCEto turbulence,theexperimentisthemeasurementofaoweld, whichisassumedtobea functionoftime,andtherandomvariablecanbeanyfunction ofthosemeasurements. So, isexpressedasafunctionoftimeinEquation 5–1 = ( ( t )) (5–1) where 2 R AfunctionofarandomvariableisshowninEquation 5–2 x = g ( ( ( t ))) (5–2) where x 2 R g : R 7! R Thisfunctioncanberepresentedasasumofkernelfunctions i ,asinEquation 5–3 x = 1 X i =0 x i ( t ) i ( ( )) (5–3) where x i ( t ) 2 R Oftenthesefunctionsarepolynomialsofaknownseries.The selectionofthe polynomialsisarbitraryinthedemonstrationofthetheory ,buthasanimpactonthe numericalperformanceofimplementationofthemethod.Con vergenceisquickestwhen 78

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usingasystemofpolynomialsthatareorthogonalwithrespe cttotheprobabilitydensity functionoftherandomvariable.Itispossibletousemultip letypesofpolynomialsif randomvariablesofdifferentrandomprocessesaffectthes amedynamicsystem[ 126 ]. Atableofmanyofthecommontypesofdistributionsandtheir associatedpolynomialsis showninTable 5.2 Table5-1.Commondistributionsandassociatedbasispolyn omials ProbabilityDistributionSetofOrthogonalPolynomials GaussianHermiteUniformLegendreBetaJacobiGammaLaguerre TheapplicationofPCEtodynamicalsystemsbeginswiththee xpressionofthe systemintermsofasumofthekernelfunctions, .Adeterministicdynamicalsystem with n statesand m controleffectorscanbewritteninstatespaceformasEquat ion 5–4 x = f ( x c u t ) (5–4) where x 2 R n u 2 R m c 2 R q x ( t = t o )= x o where x isthestatevector, c isavectorofparameters, u isthecontrolvector, t representstime,and f isavectoroffunctions. Inpractice,nosuchsystemcanbewrittenwithoutsomelevel ofuncertainty associatedwithsomepartofthesystemrepresentation.Unm odeleduncertaintyis duetodynamiceffectsthatarenotrepresentedintheequati onsofEquation 5–4 .An exampleofthistypeofuncertaintyisanonlinearsystemtha tisrepresentedasalinear systemofequations.Thelinearequationsmayormaynotrepr esentthedynamics 79

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sufcientlywelltoproducethedesiredoutcome,butitwill notexactlydescribethe stateevolutionofthesystem.Parametricuncertaintyisth etermusedforuncertainty inthecoefcients, c i .Thisuncertaintyisdifferentfromunmodeleduncertainty inthat thecorrecttermasafunctionofstatesandtimeispresentin theequations,butthe factorthatdeterminesthemagnitudeoftheeffectonthedyn amicsmaynotbemodeled exactlyorcorrectly.Uncertaintyintheinitialcondition s, x o ,canalsoproduceuncertainty inthestateevolution.ThestrengthofPCEisitsabilitytoa ccountfortheeffectsof parametricuncertainty. Torepresenttheeffectsofuncertainty,thecomponentsofE quation 5–4 arewritten asfunctionsoftherandomvariable ,whichisafunctionoftheexperiment ,in Equations 5–5a 5–5d x = x ( t ( )) (5–5a) x =_ x ( t ( )) (5–5b) c = c ( ( )) (5–5c) u = u ( t ( )) (5–5d) Therandomvariable isassumedtohaveastationaryprobabilitydensityfunctio n (pdf), f ( ) .Givenastationarypdf,therandomvariable canbedescribedbybasis functions, i ,thatareorthogonaltoeachotherwithrespecttothegivenp df.The applicationofPCEtodynamicalsystemsbeginswiththeexpr essionofthesystem intermsofasumofthesekernelfunctions,showninEquation 5–6 .Forsimplicityin 80

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understanding,Equation 5–6 andallthefollowingequationsarederivedforaonestate system.AderivationofPCEforamultiplestatesystemfollo wsthesamesteps. x ( t )= 1 X i =0 x i ( t ) i ( ( )) (5–6a) x ( t )= 1 X i =0 x i ( t ) i ( ( )) (5–6b) c ( )= 1 X i =0 c i i ( ( )) (5–6c) u ( t )= 1 X i =0 u i ( t ) i ( ( )) (5–6d) Equation 5–6 separatesthedependenceontimefromthestochasticdepend ence ontherandomvariable .Thekernelfunctions, i ,accountforthedependenceonthe randomvariable andthedeterministicvariables, x i ,accountforthetimeevolutionof thesystem'sstates. Duetocomputationallimitations,thesumsofthekernelfun ctionsinEquation 5–6 mustbetruncatedatachosenorder, p ,resultinginEquation 5–7 x ( t )= p X i =0 x i ( t ) i ( ( )) (5–7a) x ( t )= p X i =0 x i ( t ) i ( ( )) (5–7b) c ( )= p X i =0 c i i ( ( )) (5–7c) u ( t )= p X i =0 u i ( t ) i ( ( )) (5–7d) TherepresentationsofEquation 5–6 aresubstitutedintothedynamicsofEquation 5–4 81

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p X i =0 x i ( t ) i ( ( ))= f ( p X i =0 x i ( t ) i ( ( )), t p X i =0 c i i ( ( )), p X i =0 u i ( t ) i ( ( ))) (5–8) NowthedynamicsofEquation 5–8 aresolvedinanaveragesenseoverthedomain ofthekernelfunctions, i .ThissolutionisobtainedbyperformingaGalerkinproject ion onEquation 5–8 .TheGalerkinprojectionndsthedeterministicvariables ( x i u i ,and c i )thatminimizethedistancebetweentheapproximatedsumso fEquation 5–7 andthe kernelfunctions, i TheGalerkinprojectionisessentiallyaninnerproductbet weentwofunctions,and assuch,willberepresentedby h i ,asshowninEquation 5–9 h x y i = Z D xyg ( ) d ( ) (5–9) Notethattheuseoftheorthogonalfunctionals, i ,resultsinmanyzerotermsdue totheorthogonalitypropertyshowninEquation 5–10 h i j i =0 if i 6 = j (5–10) So,toexecutetheGalerkinprojectiononthesystemdynamic s,Equation 5–8 is multipliedby k toobtainEquation 5–11 p X i =0 x i ( t ) i ( ( )) k ( ( ))= f ( p X i =0 x i ( t ) i ( ( )), t p X i =0 c i i ( ( )), p X i =0 u i ( t ) i ( ( ))) k ( ( )) (5–11) TheGalerkinprojectionofEquation 5–8 istakenwithrespecttotheassumed probabilitydensityfunction.Thus,Equation 5–11 isintegratedwithrespecttothegiven pdf,andresultsinasetofordinarydifferentialequations representedbyEquation 5–12 82

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_ x k ( t )= h f ( p X i =0 x i ( t ) i ( ( )), t p X i =0 c i i ( ( )), p X i =0 u i ( t ) i ( ( ))), k i (5–12) TherightsideofEquation 5–12 canberewrittenintermsofthestochasticvariables x i c i ,and u i x k ( t )= f 0 ( X t C U ) (5–13) where, X =[ x 1 x 2 ,..., x p ] | C =[ c 1 c 2 ,..., c p ] | U =[ u 1 u 2 ,..., u p ] | Thisnewfunction f 0 iscomposedofdeterministictimedependentvariables, x i c i u i .Thus,thetimehistoryofthisstochasticdynamicsystemis solved,yieldingthe probabilisticdistributionofthephysicalsystematevery time, t Thepolynomialchaostransformationallowsthestatistica ldistributionofasystem's behaviortobesolvedusingonesimulationofadeterministi csystem,ratherthan requiringanarbitrarilylargenumberofsolutionsofaprob abilisticsystemtobefound inanattempttoapproximatethestatisticaldistribution, asisdoneinaMonteCarlo simulation.Thestatisticaldistributionisproducedbyn dingthedesiredexpectationsof thestate. Theexpectationofacontinuousrandomvariableisfoundbyi ntegratingaweighting oftherandomvariableoveritsdomain.Theweightingusedwi llbetheprobability densityfunctionoftherandomvariableitself,asinEquati on 5–14 83

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E [ ]= Z D xf ( x ) dx (5–14) where, f ( x )= pdfof Alternatively,theexpectationofafunctionofarandomvar iablecanbefoundby integratingwithrespecttotheprobabilitydensityfuncti onofthebaserandomvariable, asshowninEquation 5–15 E [ x ( )]= Z D x ( ) f ( ) d (5–15) where, f ( )= pdfof Thestatesandcontrol,beingrepresentedasafunctionofth erandomvariable canbeincorporatedintoEquation 5–15 Thedesiredmomentsofthestatesandcontrolarefoundbyasi milarprocess. Then,aprobabilitydensityfunctioncanbeconstructedtha tsatisesthecalculated moments.Forexample,theexpectationof x x 2 x 3 x 4 and x 5 couldbecomputed,and thenaprobabilitydensityfunctionthatsatisesthoseexp ectationscanbefound. 5.3LinearSystem TheapplicationofPCtolinearsystemsisaspecialcaseofth eexpansionprocess showninSection 5.2 .Theexpansionprocessbeginswiththedenitionofalinear system,asshowninEquation 5–16 84

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_ x = Ax + Bu where, x 2 R n (5–16) u 2 R m A 2 R n n B 2 R n m TousethePCframework,thedynamicsmatrices A and B areprojectedontobasis functions, i uptotheorder, p .Theseprojectionsarecompletedinthreesteps.First,a newsetofvariablesarecreatedthataredenedinEquation 5–17 a ij k = h A ij ( ), k ( ) i h ( ) 2 i where, i j 2 [1, n ] and k 2 [0, p ] b ij k = h B ij ( ), k ( ) i h ( ) 2 i (5–17) where, i 2 [1, n ] j 2 [1, m ] ,and k 2 [0, p ] e ijk = h i j k i h 2i i where, i j k 2 [0, p ] Second,asetofmatrices, k ,aredenedaccordingtoEquation 5–18 k = 266666664 ^ e 1 k 1 ^ e 1 k 2 ^ e 1 kp ^ e 1 k 1 ^ e 2 k 2 ^ e 2 kp ... ... ... ^ e 1 k 1 ^ e 1 k 1 ^ e pkp 377777775 (5–18) Third,thematricesforthePCEsystemareassembledbyEquat ion 5–19 85

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A = 266666664 A 11 A 12 A 1n A 21 A 22 A 2n ... ... ... A n1 A n2 A nn 377777775 B = 266666664 B 11 B 12 B 1m B 21 B 22 B 2m ... ... ... B n1 B n2 B nm 377777775 (5–19) where, A ij = p X k =0 a ij k k B ij = p X k =0 b ij k k NowthePCEsystemdynamicsarebackintothestandardlinear systemformatof Equation 5–16 ,exceptnowthestateandcontrolvectorsarelarger.Thenew dynamics areshowninEquation 5–20 x = Ax + Bu where, x 2 R ( n p ) (5–20) u 2 R ( m p ) A 2 R ( n p ) ( n p ) B 2 R ( n p ) ( m p ) 5.4Difculties ItshouldberecognizedthattheapplicationofPCEtoadynam icalsystemisnot withoutsomecomplications.IntheprocessofcreatingtheP CEsystem,somenumerical 86

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issuescanarise.AfterproducingthePCEsystem,thelossof physicalsignicanceof thevariablesaddsdifcultytoamodalanalysis.5.4.1NumericalIssues TheprimarydifcultyinapplyingPCEisachievingconverge nceinthehigherorder terms.WhilePCEapproximationsshowmean-squareconverge nce,implyingthatthe meanandvarianceconvergetotheirtruevaluesastheordero fthePCapproximation increases,theapproximationsofhigherordertermscannot beguaranteed[ 128 ].Other complicationsalsoappearintheconvergencepropertiesof PCE's.Theapproximation maynotimproveastheorderofthetermsusedincreases,orth eimprovementsmaybe minimal.Additionally,theapproximationsofstationary, non-Gaussianprocessesmaynot bestationarythemselves[ 128 ]. AnothersourceofnumericalerrorinPCEisthetruncationof termswhenmultiplications ofrandomvariablesareperformed[ 129 ].Whenonerandomvariableismultipliedby anotherinthePCEframework,theproductpossessestermsof higherorderthanthose possessedbytheoriginalrandomvariables.Tworandomvari ablesareapproximatedin thePCEframework,asshowninEquation 5–21 ^ a = p X i =0 a i i ( ( )) ^ b = p X j =0 b j j ( ( )) (5–21) Theirproduct, c = ab ,isrepresentedbyitsownPCEapproximation,asshownin Equation 5–22 ^ c = p X k =0 c k k ( ( )) (5–22) ThePCEapproximationofrandomvariable c canberewrittenasinEquation 5–23 ^ c = p X i =0 a i i ( ( )) p X j =0 b j j ( ( )) (5–23) 87

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ItshouldbeapparentthattheproductinEquation 5–23 willhavehigherorder termsin thantheexpansionsof ^ a and ^ b bythemselves.Asaresult,theproductmust betruncated.Thecoefcientsofthetruncatedproductaref oundbyusingaGalerkin projectionmethodtondthebestapproximationfortheprod uctgiventheorderofthe PCE'sbeingmultiplied.Theresultisthatthe c k = p X i =0 p X j =0 C ijk a i b j (5–24) where, C ijk = h i ( ( )) j ( ( )), k ( ( )) i h k ( ( )), k ( ( )) i Furthercomplicationsarisewhenattemptingtoevaluateno n-polynomialfunctions ofPCE'sofrandomvariables[ 129 ].However,theresearchproceedingfromPCEtheory willbecontainedtopolynomialfunctionsofrandomvariabl es. Thespectralanalysisprocessthattransformstheexpectat ionoftheintegerpowers ofarandomvariabletoaprobabilitydensityfunctionhasnu mericalissuesofitsown. 5.4.2ModalInterpretation OnedifcultyinusingPCEisthelossofdirectphysicalmean ingtothestatesofthe expandedsystem.Asystemwith n statesthathasbeenexpandedusingpolynomials uptotheorder p nowhas n ( p +1) states.Theresultisthatmodalinterpretation becomesdifcult.Themodesoftheexpandedsystemaremodes ofthestatesthathave beenmappedontopolynomialsoftheuncertainparameter.In thissense,wecansay thatthemodesoftheexpandedsystemarenotphysicalmodes, butrathermodesofthe uncertaintyofthephysicalstates. Additionally,thenumberofmodesincreasesduetotheincre aseinnumberof states.Thesemodesoftheexpandedsystemmaybeunstableev enthoughthephysical systemisstable.Themodesoftheexpandedsystemmaybeosci llatory,convergent,or divergent,eventhoughthephysicalsystemonlyhadmodesof onetype.Thesemodes couldbejusttheresultofnumericaltruncationperformedi nthePCE,ortheycouldbe 88

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theresultoftheuncertainparametervaryingenoughtodram aticallyalterthemodes ofthephysicalsystem.Allofthisconfusioncomesoutofthe creationoftheexpanded, deterministicsystem,andmakesinterpretationofthesyst emdynamicsdifcult. 5.4.2.1InterpretingEigenvalues Interpretingtheeigenvaluesoftheexpandedsystemissimi lartothestandard interpretation.Whenasinglemodeisexcitedinastandardd eterministiclinear dynamicalsystem,thevalueofaphysicalstategrows,decay s,oroscillatesaccordingto Equation 5–25 x ( t )= e t x o (5–25) where, = eigenvalueofmode x o = initialconditions Thestatesoftheexpandedsystemwillactinthesameway.The senon-physical stateswillgrow,decay,oroscillateaccordingtothesamem athematicallawof Equation 5–25 .However,itcannotbestatedthatthemeanoranyotherstati stical descriptorofthephysicalstatewillgrowordecayaccordin gtothetimeconstantthat governsthestatesoftheexpandedsystem.Anystatisticald escriptionofthephysical statewillbeafunctionofthenonphysicalstatesoftheexpa ndedsystem,andso,their growthordecaywillbeafunctionofthegrowthordecayofthe nonphysicalstates.The exactrelationshipbetweenthegrowthordecayofanystatis ticaldescriptorsandthe eigenvaluesoftheexpandedsystemwilldependonthepolyno mialbasisthatischosen fortheexpansion. Forexample,ifasystemhasastate, x ,evaluatedattime, t ,thatisexpandedin thehermitepolynomialbasisuptoorder p foranuncertainparameterwithastandard Gaussiandistribution,thentheexpectationof x 2 isshowninEquation 5–26 89

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E [ x ( t ) 2 ]= p X i =0 x i ( t ) 2 (5–26) Itisknownthatforagivenmodeeachofthe x i ( t ) willgrowordecayinmagnitude exponentially,buttheexpectationofthesecondmomentwil lgrowordecayasafunction ofeach x i ( t ) 5.4.2.2ClusteredEigenvalues Ifthevariationintheuncertainparameterissmallenough, orthesensitivityof thesystemtovariationsintheparameterissmallenough,th entheeigenvaluesofthe expandedsystemmayberemarkablyclosetotheeigenvalueso ftheoriginalsystem. Indeed,iftheprobabilitydensityfunctionoftheuncertai nparameteristheDiracdelta function,thentheexpandedsystemwouldbeexpectedtohave repeatedeigenvaluesat theeigenvaluesofthephysicalsystem. Analysisofclusteredeigenvaluesasvariationsofasingle physicalmodeofthe systemcouldbealogicalwayofanalyzingthemodesoftheexp andedsystem.Care mustbetakentonotonlyensurethattheeigenvaluesindicat easimilarbehavior betweenthemodeswithclusteredeigenvalues,butalsotoen surethatthevariationin theeigenvectorsofthemodesalsorepresentasimilarbehav ior. 5.4.2.3InterpretingEigenvectors Interpretingtheeigenvectorsoftheexpandedsystemismuc hmoredifcult thaninterpretingtheeigenvectorsofthephysicalsystem. Asimplephase-magnitude interpretationoftheeigenvectorsoftheexpandedsystemd oesnotnecessarilyprovide anyphysicalmeaning,becausethosestatesarenon-physica l.However,dependingon theuser'schoiceofpolynomialbasis,someinformationabo utthephysicalstatesmay arisefromaphase-magnitudeanalysis. Forexample,ifatwo-statesystemhasstates x and y ,evaluatedattime t ,thatare expandeduptoorder p ,thentheexpandedsystemwillhavestates x 0 x 1 ,..., x p and y 0 y 1 ,..., y p .Iftheuncertainparameterinthesystemfollowsastandard Gaussian 90

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distributionandthesystemisexpandedinthehermitepolyn omialbasis,thenthemeans for x ( t ) and y ( t ) areasshowninEquation 5–27 E [ x ( t )]= x 0 ( t ) (5–27) E [ y ( t )]= y 0 ( t ) Consequently,aphase-magnitudeanalysisoftheexpandeds ystemmodeswillyield relativephaseandmagnituderelationshipsbetweenthemea nsofstates x ( t ) and y ( t ) Thus,somephysicalmeaningcanbelearnedaboutthemagnitu deandphasedifference betweenthemeansofthephysicalstatesinthisspecialcase .However,ingeneralthis willnotbetrue.Thus,modesneedtobeexaminedbothintheph ase-magnitudesense andfromtheperspectiveofthetime-evolutionofstatistic alinformationofthephysical states. Forexample,ananalysisofaparticularmodecouldentailco mputingthetime evolutionofmean,variance,skewness,kurtosis,etc.fora llthephysicalstates.Note thatonlythemeanandvarianceareguaranteedtoconvergewh enusingPCE,butsome higherorderanalysiscouldbetrustedifpropercareistake n. Inmanycases,aplotofmeanplusorminusthreestandarddevi ationswillgive usefulinformationaboutthemodalevolutionofthephysica lstates.Anexampleofsuch aplotisshowninFigure 5-1 Notethatnocondenceboundsareguaranteedusingsuchinfo rmation.A statementofcondenceboundswouldrequirethatthestatei sknowntobeGaussian. EveniftheuncertainparameterfollowsaGaussiandistribu tion,itcannotbeassumed thatthestatesofthesystemarealsoGaussian.Theaboveexa mpleisoverlaidonthe statetimehistoriesof100MonteCarlorunsofaPCEsystemth atusesaGaussian distributionfortheuncertainparameter.Theresultissho wninFigure 5-2 91

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0 10 20 30 40 50 60 -6 -4 -2 0 2 4 Time (s)U (m/s) Mean Variance Bounds Figure5-1.Exampleoftimeevolutionofmeanandvariancebo unds 0 10 20 30 40 50 60 -8 -6 -4 -2 0 2 4 6 Time (s)U (m/s) Mean Variance Bounds MC Runs Figure5-2.ComparisonofmeanandvarianceboundstoMonteC arloruns Ifthestate u wereGaussian,thenitwouldbeexpectedthat99.7%oftheMon te Carlorunswouldliewithintheboundsdenedfromthestatem eanandvariance,yet threeofthe100MonteCarlorunslieoutsidethesebounds.So ,itisquiteclearthat 92

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eventhoughtheuncertainparameterwasGaussian,thestate softhesystemarenot Gaussian. Forsuchanexample,furtherinsightmaybegleanedfromexam iningtheskewness metriccalculatedfromthestatesoftheexpandedsystem.Th emean,variance,and skewnessmetricsofthestate u areshowninFigure 5-3 0 10 20 30 40 -15 -10 -5 0 5 Time (s)U (m/s) Mean Variance Skewness Figure5-3.Mean,variance,andskewnessofstateunderonem odeofexpanded system Usingskewnesscouldhelpthecontroldesignerconstructbo undsontheexpected responsethatmorecloselyresembletheMonteCarlosimulat ions,butfornow,the analysiswillrestrictitselftoplotsofthemeanandvarian cebounds.Someimportant characteristicsofsuchplotsareexplainedbelow. Asmentionedpreviously,themodesoftheexpandedsystemca nbeunderstood betterasbeingmodesoftheuncertaintyofaphysicalstate. Figure 5-4 showsan exampleofthetimeevolutionofthemeanandvarianceofasin glestateduetoagiven modefromanexamplesystem. Thisguredemonstratesthatinitialuncertaintyinthesta teisnon-zero.Thus,the informationfromthisgurecannotbeinterpretedastheevo lutionofthephysicalstate 93

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 -50 0 50 100 150 Time (s)Q (deg/s) Mean Variance Bounds Figure5-4.Exampleofinitialuncertaintyinexpandedsyst emmodalanalysis overtimegivenanyinitialstartingposition.Alogicalnex tthoughtwouldbetoshiftthe plotforwardorbackwardintimetoapointsuchthattheiniti alvariancemightbezero,as mayhappeninsomecases,butthisisnotguaranteed. Figure 5-5 showsanexampleofthe3boundsofasinglestateundertwodifferent modes,Mode7andMode11,ofanexampleexpandedsystem. InMode7thevarianceseemstoapproachzeroduringeachcycl e,butMode11 showsalargedifferencebetweenthe3boundsofthephysicalstate, q .Thus,itis notpossible,usingthismethod,todifferentiatebetweent heresponseduetoinitial uncertaintyinthestate,andtheresponseduetouncertaint yinthesystem. Instead,qualitativeconclusionscanbedrawnregardingth emagnitudeofvariation inthemeanandthevariance.Somemodeswillcontainenergya lmostentirelyinthe uncertaintyofthephysicalstate,whileotherswillhaveen ergypredominantlyinthe meanofthephysicalstate.Figure 5-6 showsthemeanandvarianceofasinglestate undertwodifferentmodes,Mode3andMode6,ofanexampleexp andedsystem. Notethatthetimeevolutionofthe stateinMode3containsverylittlechangein thevariancecomparedtolargechangesinthemean,whileMod e6showstheopposite. 94

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0 10 20 30 40 -20 -15 -10 -5 0 5 10 15 20 Time (s)Q Variance Bounds (deg/s) Mode 7 Mode 11 Figure5-5.Exampleofvariationinstateuncertaintybetwe enmodesofexpanded system 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1 -0.5 0 0.5 1 1.5 Time (s)Theta (deg) Mode 3 Mean Mode 3 Variance Mode 6 Mean Mode 6 Variance Figure5-6.Exampleofvariationofmeansandvariancesbetw eenmodesofexpanded system Whensimulatingasystemwithlargesensormeasurementunce rtainty,the feedbackcanexpressthisuncertaintyandexcitemodessuch asMode6thatwillyield largeuncertaintyinthefuturestates.Ifstatemeasuremen tscanbetrusted,thenthe modesrelatedtolargeuncertaintyvariations,likeMode6, wouldbelargelymitigated, 95

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whilethemodeswithalotofenergyinthemeanofthestate,li keMode3,wouldbe moreimportant. 5.5RelevantApplicationsofPolynomialChaos TheideaofusingPCEtohelpwiththeunderstandingofturbul enceisasoldas PCEitself.InhisseminalpaperonHomogeneousChaos,Wiene rcitesthepotential applicationofhistheoriestoaturbulenceeld[ 116 ].However,notmuchapplicationof Wiener'stheorieswasuseduntilrecently.GhanemandSpano susedPCEtoassistin aniteelementanalysisofBrownianmotion[ 130 ].Xiu,Karniadakis,andothersbegan usingPCEintheearlypartofthelastdecadetohelpwithcomp utationaluiddynamics (CFD)programs[ 131 132 ].KarniadakisalsoworkedwithWantoanalyzethevalidity oflong-termpolynomialchaosCFDsimulations[ 133 ].WaltersleveragedPCEtocreate astochasticcompressibleEulerandNavier-Stokessolvert oreducethevariationseen betweennumericalCFDandexperimentalresults[ 134 135 ].Jardakbeganapplying PCEtothethermodynamicrelationswithinows[ 136 ].Thesepapersallseektoexploit PCE'sabilitytocalculateprobabilisticdescriptionsof owparameterswithfarless computationalcostthanaMonteCarloSimulation.Veryrece ntresearchhasusedPCE tomodelnotjusttheparametersofowvariables,buttheaer odynamiceffectsofthat owonobjects.PettitusedPCEtomodeltheliftcoefciento naatplatesubjectedto gusts[ 137 ]. Polynomialchaostheoryhasalsoshownpromisetobeanewmet hodofdealing withuncertaintyfromanengineeringdynamicsandcontrolp erspective.PCE'suses incontrolexplodedrecentlysincethepublicationofapape rin2006byHoverand TriantafyllouthatproposedtheuseofPCEinstabilityandc ontrolapplications[ 138 ]. PCEhassincebeenusedtoanalyzethedynamiceffectsofunce rtaintyineverything fromseismicprocessestoelectricpowergridsandchemical processes[ 139 – 141 ]. MontidesignedPCEbasedcontrolmethodsandappliedthecon troltothedesignofnew powerconverters[ 142 143 ].MontialsousedPCE'stodesignnewobserversystems 96

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andlowpasslters[ 144 145 ].Fisherconductedstabilityanalysisoflinearsystems underuncertaintyandderivednewformsofstochasticlinea rquadraticregulation fromPCE's[ 146 147 ].Controlofnonlinearsystemshasalsobeenconductedusin g PCE[ 148 ]. Sofar,verylittleresearchhasbeendonetousePCEintherea lmsofpathplanning andtrajectoryoptimization.Kewlanihasappliedpolynomi alchaostoground-based robotmobility[ 149 150 ].HeappliesPCE'stothedynamicsandusesthemtoproduce polynomialresponsesurfaceplotstoattempttondacloser tooptimalresult[ 151 ]. ThevastmajorityofhisworkusedDubinspathstoattemptton avigatearoundknown environment[ 152 ].OnlyveryrecentlyhaveKewlaniandIshigamabeguntocons ider terrainnavigationwithoutaprioriknowledge[ 153 ].Additionally,Kewlanihasnotused thestatisticaldistributionsofthestatestoalterthenod ecosts;rather,hehasused condenceintervalsofthestatestocheckforpossiblecoll isions[ 154 ].Fisherhas alsobeguntoexamineoptimalpathplanningunderuncertain ty.Fisherndsoptimal solutionsforminimumexpectationandminimumvariancepat hsunderdynamic uncertainty[ 155 ]. ThePCEframeworkhasprovedtobefruitfulforbothrepresen tingtherandom natureofturbulentowsandopenloopdynamicsanalysisand controllerformulation.It ispreciselythesestrengthsofPCEthatareleveragedinthi sthesis.However,beforethe PCEtheorycanbeapplied,therelationshipbetweentheow eldturbulenceandmicro airvehicledynamicsmustbeunderstood. 97

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CHAPTER6 AIRCRAFTMODELPARAMETRICINTURBULENCE Alongitudinalightdynamicsmodelisderivedthatisparam eterizedintermsof theturbulenceintensityinordertoanalyzetheeffectsoft urbulenceintensityonthe motionofanaircraft.Themodesofthissystemareanalyzedt ondnotableeffectsas turbulenceintensityincreases. 6.1ParameterizedModelDerivation Thelift,drag,andpitchingmomentcoefcientsareexpress edintermsofthe relevanttermsfoundinthesweepmodelofthedynamicwindtu nneltestingof Section 4.2 andanadditionaltermfortheeffectoftheelevatorisadded .Thisterm fortheeffectoftheelevatorontheforcesandmomentsarise soutofaTaylorseries expansion.TheseexpressionsareshowninEquations 6–1 6–3 C L = C L o + C L + C L LE LE + C L + C L + C L I I + C L LE LE + C L (6–1) + C L + C L _ + C L I 2 I 2 + C L e e C D = C D o + C D + C D LE LE + C D + C D + C D I I + C D + C D (6–2) + C D _ + C D 2 2 + C D 2 LE 2LE + C D e e C M = C M o + C M + C M LE LE + C M + C M + C M I I + C M + C M (6–3) + C M _ + C M e e Amodelisderivedassuminganaircraftwithnowingsweep,so thoseterms includingwingsweepareeliminated.Also,Equations 3–38 3–40 demonstratethat = q when =0 .Thus,all termswillbereplacedwith q ,duetothefactthat alongitudinalmodelisbeingconstructedthatassumesnono nzerovaluesinthe lateral-directionalstates.Usingthisinformation,Equa tions 6–1 6–3 arerewritten, replacingstateswiththeperturbationequationsshowninE quations 3–53 and 3–54 TheresultingequationsaregiveninEquations 6–4 6–6 98

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C L =( C L o + C L o + C L e e o + C L I I + C L I 2 I 2 )+ C L (6–4) +( C L q + C L q o ) q +( C L + C L o ) + C L e e C D =( C D o + C D o + C D 2 2 o + C D e e o + C D I I ) +( C D +2 C D 2 o ) +( C D q + C D q o ) q (6–5) +( C D + C D o ) + C D elev e C M =( C M o + C M o + C M e e o + C M I I )+ C M (6–6) +( C M q + C M q o ) q +( C M + C M o ) + C M e e ThetermsinEquations 6–4 6–6 thatarenotmultipliedbystateperturbationsare redenedinEquations 6–7 6–9 C L trim = C L o + C L o + C L e e o + C L I I + C L I 2 I 2 (6–7) C D trim = C D o + C D o + C D 2 2 o + C D e e o + C D I I (6–8) C M trim = C M o + C M o + C M e e o + C M I I (6–9) Thelift,drag,andaerodynamicpitchingmomentarerelated totheirrespective coefcientsbyEquations 6–10 6–12 L = C L 1 2 V 2 S ref (6–10) D = C D 1 2 V 2 S ref (6–11) M Aero = C M 1 2 V 2 S ref c ref (6–12) NotethatthedynamicpressureisinherentineachofEquatio ns 6–10 6–12 .The dynamicpressure, Q ,isdenedinEquation 6–13 99

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Q = 1 2 V 2 (6–13) TheaircraftvelocitycomponentofEquations 6–10 6–12 V 2 ,canberewrittenasin Equation 6–14 usingtheorthogonalvelocitycomponentsofEquation 3–17 V 2 = u 2 + v 2 + w 2 (6–14) NowEquations 3–53 and 3–54 aresubstitutedintoEquation 6–14 ,resultingin Equation 6–15 V 2 =( u o + u ) 2 +( v o + v ) 2 +( w o + w ) 2 =( u 2 o + w 2 o )+2 u o u +2 w o w = V 2 1 +2 u o u +2 w o w (6–15) where V 1 = p u 2 o + v 2 o + w 2 o Equations 6–4 6–9 and 6–15 aresubstitutedintoEquations 6–10 6–12 .The resultingequationsareshowninEquations 6–16 6–18 L = QS ref C L trim + u o S ref C L trim u + w o S ref C L trim w + QS ref C L (6–16) + QS ref ( C L q + C L q o ) q + QS ref ( C L + C L o ) + QS ref C L elev e D = QS ref C D trim + u o S ref C D trim u + w o S ref C D trim w + QS ref ( C D +2 C D 2 o ) + QS ref ( C D q + C D q o ) q (6–17) + QS ref ( C D + C D o ) + QS ref C D elev e M Aero = QS ref c ref C M trim + u o S ref c ref C M trim u + w o S ref c ref C M trim w + QS ref c ref C M + QS ref c ref ( C M q + C M q o ) q (6–18) + QS ref c ref ( C M + C M o ) + QS ref c ref C M elev e 100

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TheperturbationtermsinEquations 3–53 and 3–54 aresubstitutedintothe longitudinaldynamicsinEquation 3–52 ,resultinginEquations 6–19 6–21 m u + mw o q = L (sin o +cos o ) D (cos o sin o ) (6–19) mg (sin o +cos o )+ F Thrust x m w mu o q = L (cos o sin o ) D (sin o cos o ) (6–20) + mg (cos o sin o )+ F Thrust z I y q = M Aero + M Thrust (6–21) Equations 6–16 6–18 aresubstitutedintoEquations 6–19 6–21 .Termsare collected,resultingintwosetsofequations.Therstsete stablishtherequirementsfor trimmedightandareshowninEquations 6–22 6–24 0= QS ref C L trim sin o QS ref C D trim cos o mg sin o + F Thrust x (6–22) 0= QS ref C L trim cos o QS ref C D trim sin o + mg cos o + F Thrust z (6–23) 0= M Thrust + QS ref c ref C M trim (6–24) Thesecondsetofequationsshowthelinearizedperturbatio ndynamicsofthe aircraftinEquations 6–25 6–27 101

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m u = u o S ref ( C L trim sin o C D trim cos o ) u + w o S ref ( C L trim sin o C D trim cos o ) w + QS ref sin o ( C L + C D trim )+cos o ( C L trim C D C D 2 o ) (6–25) + mw o + QS ref sin o ( C L q + C L q o ) cos o ( C D q + C D q o ) q mg cos o + QSref ( C L elev sin o C D elev cos o ) e + QS ref sin o ( C L + C L o ) cos o ( C D + C D o ) m w = u o S ref ( C L trim cos o + C D trim sin o ) u w o S ref ( C L trim cos o + C D trim sin o ) w + QS ref cos o ( C D trim C L )+sin o ( C L trim C D 2 C D 2 o ) (6–26) + mu o QS ref cos o ( C L q + C L q o )+sin o ( C D q + C D q o ) q mg sin o QS ref ( C L elev cos o + C D elev sin o ) e QS ref cos o ( C L + C L o )+sin o ( C D + C D o ) I y q = u o S ref c ref C M trim u + w o S ref c ref C M trim w + QS ref c ref C M + QS ref c ref ( C M q + C M q ) q + QS ref c ref C M elev e (6–27) + QS ref c ref ( C M + C M ) NotefromChapter 3 thattheangleofattack, ,isrelatedtotheverticalwind velocity, w ,throughtherelationshipshowninEquation 6–28 ,andisalsorelatedtothe longitudinalwindvelocity, u ,throughtherelationshipshowninEquation 6–29 w = V 1 sin (6–28) u = V 1 cos (6–29) Thisrelationshipcanbeusedtoproducethemathematicalre lationshipsbetween and w ,and and w ,respectively.TheserelationshipsareshowninEquations 6–30 102

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and 6–31 ,aftersubstitutingintheperturbationrelationshipsofE quation 3–53 and makingsmallangleassumptions w = w o + V 1 sin( o + ) = w o + V 1 (sin o cos +cos o sin ) = w o + V 1 sin o + V 1 cos o = u o (6–30) d dt w = d dt ( V 1 sin ) d dt ( w o + w )= V 1 d dt sin( o + ) w = V 1 cos( o + ) = V 1 (cos o cos sin o sin ) = V 1 cos o V 1 sin o = u o (6–31) Equations 6–30 and 6–31 aresubstitutedintoEquation 6–26 toproducethenal statespaceequationfortheverticalvelocitydynamics,sh owninEquation 6–32 103

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_ w = 1 m + QS ref u o h cos o ( C L + C L o )+sin o ( C D + C D o ) i u o S ref C L trim cos o + C D trim sin o u + QS ref u o h cos o ( C D trim C L )+sin o ( C L trim C D 2 C D 2 o ) i w o S ref h C L trim cos o + C D trim sin o i w (6–32) + mu o QS ref h cos o ( C L q + C L q o )+sin o ( C D q + C D q o ) i q mg sin o QS ref C L elev cos o + C D elev sin o e # Theremaininglongitudinaldynamicsequations,Equations 6–33 and 6–34 are createdbysubstitutingEquations 6–30 6–32 intoEquations 6–25 and 6–27 104

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_ u = u o S ref C L trim sin o C D trim cos o sin o ( C L + C L o ) cos o ( C D + C D o ) C L trim cos o + C D trim sin o mu o QS ref +cos o C L + C L o +sin o C D + C D o u + w o S ref C L trim sin o C D trim cos o sin o ( C L + C L o ) cos o ( C D + C D o ) C L trim cos o + C D trim sin o mu o QS ref +cos o C L + C L o +sin o C D + C D o + QS ref u o sin o C L + C D trim +cos o C L trim C D C D 2 o + sin o ( C L + C L o ) cos o ( C D + C D o ) mu o QS ref +cos o C L + C L o +sin o C D + C D o cos o ( C D trim C L )+sin o ( C L trim C D 2 C D 2 ) mu o QS ref +cos o C L + C L o +sin o C D + C D o w (6–33) + mw o + QS ref sin o C L q + C L q o cos o C D q + C D q o + h sin o C L + C L o cos o C D + C D o i mu o + QS ref h cos o C L + C L o +sin o C D + C D o i h mu o QS ref cos o [ C L q + C L q o ]+sin o [ C D q + C D q o ] i mu o + QS ref h cos o C L + C L o +sin o C D + C D o i q mg cos o + sin o sin o ( C L + C L o ) cos o ( C D + C D ) mu o QS ref +cos o C L + C L o +sin o C D + C D o + QS ref C L elev sin o C D elev cos o sin o ( C L + C L o ) cos o ( C D + C D o ) C L elev cos o + C D elev sin o mu o QS ref +cos o C L + C L o +sin o C D + C D o e 105

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_ q = u o S ref c ref I y C M trim QS ref u o C M + C M o C L trim cos o + C D trim sin o m + QS ref u o cos o ( C L + C L o )+sin o ( C D + C D o ) u + w o S ref c ref C M trim + QS ref c ref u o C M + QS 2 ref c ref u o C M + C M o Q u 2 o cos o ( C D trim C L )+sin o ( C L trim C D 2 C D 2 o ) w o C L trim cos o + C D trim sin o m + QS ref u o cos o ( C L + C L o )+sin o ( C D + C D o ) w (6–34) + QS ref c ref I y C M q + C M q o + QS ref u o C M + C M o mu o QS ref cos o ( C L q + C L q o ) sin o ( C D q + C D q o ) m + QS ref u o cos o ( C L + C L o )+sin o ( C D + C D o ) q QS ref c ref u o I y C M + C M o mg sin o m + QS ref u o cos o ( C L + C L o )+sin o ( C D + C D o ) + QS ref c ref I y C M elev QS ref u o C M + C M o C L elev cos o + C D elev sin o m + QS ref u o cos o ( C L + C L o )+sin o ( C D + C D o ) e 6.2ParameterizedGenMAVModel Equations 6–32 6–34 inconjunctionwiththecontinueduseof = q produce asemi-completefourstatelinearmodelforthelongitudina ldynamicsofamicroair vehicleinturbulence.Unfortunatelythismodelisincompl etebecausetheexperimental datawastakenwiththepurposeofexaminingwingdesignpara metersasamethodto mitigatetheeffectsofturbulence,ratherthantoconstruc tacompletedynamicmodel oftheaircraft.Asaresult,thedataismissingsomeinforma tionneededtoconstructa completemodelofthelongitudinaldynamicsoftheGenMAV. So,theprogramlesusedtodoacomputeranalysisoftheaero dynamicsofthe GenMAVareobtainedandanalyzedusingtheAthenaVortexLat tice(AVL)program underightconditions(angleofattack,velocity,Reynold snumber,etc.)similartothose 106

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usedinthewindtunneltesting.AVLisavortex-latticebase daerodynamicprediction codethatestimatestheaerodynamicsofbothwingsandslend erbodies,suchasthe fuselage[ 156 ].Thislow-ordercodemakesassumptionsthattheowisinco mpressible andinviscid;however,itiswidelyusedinthecommunityand isparticularlyaccuratefor analyzingmicroairvehicleswiththinwings[ 27 157 – 164 ]. AVLcomputestheopenloopaircraftdynamicstotthemodels howninEquation 6–35 Forsimplicity,the willbedroppedfromall termsfromnowon.Itshouldbe assumedthatallstatesindynamicequationsrefertopertur bationsaboutatrim condition. 266666664 u w q 377777775 = 266666664 A (1,1) A (1,2) A (1,3) 9.81 A (2,1) A (2,2) A (2,3)0 A (3,1) A (3,2) A (3,3)0 0010 377777775 266666664 u w q 377777775 + 266666664 B (1,1) B (2,1) B (3,1) 0 377777775 e (6–35) TheresultingmodelfromAVLisshowninEquation 6–36 266666664 u w q 377777775 = 266666664 0.02320.9007 2.515 9.81 1.0483 2.901012.310 0.7266 17.93 6.2220 0010 377777775 266666664 u w q 377777775 + 266666664 0.0105 0.1302 3.798 0 377777775 e (6–36) Atthispoint,thereisacompletenumericalmodelfromEquat ion 6–36 thathasno parameterizationwithrespecttoturbulence,andanincomp leteexperimentalmodel fromSection 6.1 thatincludesturbulenceintensityasamodelparameter.To combine theinformationfromtheexperimentalandnumericalmodels ,thetheoreticaldenition ofthelinearaircraftdynamicsisderived,includingturbu lenceintensityasaparameter. Thistheoreticaldenitionshouldmatchthelineardynamic smatrixthatAVLhasyielded, whenturbulenceintensityissettozero.So,theightderiv ativesinthetheoretical 107

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denitionofthelineardynamicsmatrixaresettovaluestha twouldproducetheAVL systemmatrixforaturbulenceintensityofzero.Then,ther atiosbetweenthecomputed ightderivativesandtheexperimentalightderivativesa refoundandthecoefcient thatmultipliestheeffectoftheturbulenceintensitypara meterisscaledaccordingtothat ratio. Forexample,theelementofthestandardaircraftlongitudi naldynamicsmatrix inthesecondrowandsecondcolumnrelatestheforceintheve rticaldirectiontothe verticalvelocitycomponentexperiencedbytheaircraft.I facrosscouplingtermbetween w and I hadbeenfound,thatelementcouldbederivedtobedenedacc ordingto Equation 6–37 A (2,2)= k ( C Z w + C Z wI I + C Z wI 2 I 2 ) (6–37) where, k 2 R isaproductofphysicalconstants Thenthevaluefor A (2,2) givenbyAVLissubstitutedintothelefthandsideof Equation 6–37 andtheturbulenceintensityissettozero.Thus, C Z w iscomputedand comparedtothe C Z w foundbywindtunneltesting.Ifthe C Z w fromtheAVLderivationis halfthesizeofthe C Z w foundbywindtunneltesting,thenthe C Z wI and C Z wI 2 foundby windtunneltestingarealsoscaledbyone-half.Thus,there lativeeffectofturbulence intensityismaintained.Theparameterizedequationfor A (2,2) isnowused. Inthiswaytheeffectsoftheturbulenceintensityhavebeen scaledtomatchthe forcecoefcientsfoundbythenumericalprogramandinclud edinthesystemdynamics. Theresultisalinearmodelofthelongitudinaldynamicsoft heGenMAVthathasbeen parameterizedwithrespecttoturbulenceintensityandmat chesthemodelproducedby acomputationaluiddynamicscodewhentheturbulenceisse ttozero. TheresultingmodelisshowninEquation 6–38 108

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266666664 u w q 377777775 = 266666664 0.0232 0.00608 I +0.00135 I 2 0.901+0.193 I 0.0434 I 2 2.52 9.81 1.05 0.2249 I +0.0506 I 2 2.90 0.444 I +0.0985 I 2 12.30 0.727 0.0357 I +0.000345 I 2 17.9+0.568 I 0.00270 I 2 6.220 0010 377777775 266666664 u w q 377777775 + 266666664 0.0105 0.13023.7980 0 377777775 e (6–38) 6.3ModalAnalysisofParameterizedSystem Aninitialmodalanalysisisconductedontheparametrizeda ircraftdynamics toevaluatetheeffectsthatachangeinturbulenceintensit yhasontheGenMAV's longitudinalmodes.6.3.1EigenvalueAnalysis TheeigenvaluesoftheparametricsystemareplottedinFigu re 6-1 asturbulence intensityvariesfrom0%to10%. -6 -4 -2 0 2 -15 -10 -5 0 5 10 15 Real AxisImaginary Axis -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Real AxisImaginary Axis Figure6-1.Eigenvaluesofparameterizedsystem(alleigen valuesshownonleft, zoomedinonphugoideigenvaluesonright) 109

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ThelefthandsideofFigure 6-1 showsthattheshortperiodmodeeigenvalues begintotheleftbeforemovingbacktotherightasturbulenc eintensityincreases.Thus, thedampingratioincreasesslightlyforsmalllevelsoftur bulencebutdecreasesmuch moredramaticallyasturbulencelevelsgetlarger.Meanwhi le,theshortperiodmode eigenvaluesconsistentlymoveclosertotherealaxis,indi catingthatthefrequencyof shortperiodoscillationsgetsslightlysmallerasturbule ncelevelsincreases. Whenexaminingthephugoideigenvaluesaloneintherightha ndsideofFigure 6-1 itbecomesclearthatthechangesinthephugoideigenvalues arelargerproportional totheoriginaleigenvalue;thus,theeffectonthemodalcha racteristicswillbegreater. Figure 6-1 evenshowsthatwithhighenoughturbulenceintensity(7.4% accordingto thismodel),theoscillatoryphugoidmodewouldbreakdowni ntooneconvergentmode andonedivergentmode.Thismovementinthepolesofthephug oidmodesuggests thatforsmalllevelsofturbulence,thephugoidmodedampin gratioandfrequencywould increase,butasturbulenceintensitycontinuestoincreas e,bothdampingratioand frequencywoulddecreaseuntilitbreaksintotworealmodes ,onestableandtheother unstable.ThesetrendsareshownclearlyinFigure 6-2 .Thisndingshouldbetested furthertomakeanyrmconclusions,becausethewindtunnel testinguponwhichthis modelisbasedonlytestedturbulenceintensitiesupto4%.6.3.2ModeShapeAnalysis Notonlydothepolesofthelongitudinalsystemchangeastur bulenceintensity increases,buttheshapeofthemodeschangeaswell.Themagn itudeandphaseofthe statesrelativetothepitchangle, ,ineachmodeareshowninthefollowinggures. Therelativemagnitudeandphaseofthestatesintheshortpe riodmodeareshown inFigure 6-3 Figure 6-3 showsthattheshapeoftheshortperiodmodeunderturbulenc eremains similartotheshapeofthemodeintheabsenceofturbulence. Thepitchratedominates whileleadingthepitchangleoscillationsbyjustover90 o .Theangleofattackvariesby 110

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0 2 4 6 8 10 -1 -0.5 0 0.5 1 Turbulence Intensity (%)Damping Ratio 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 Turbulence Intensity (%)Frequency (rad/s) Figure6-2.Frequencyanddampingofmodesoftheparameteri zedsystem(shortperiod -dashed,phugoidthatbecomesunstable-solid,phugoidtha tbecomes stable-dash-dot) 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 Turbulence Intensity (%)Relative Magnitude 0 2 4 6 8 10 0 50 100 150 200 Turbulence Intensity (%)Relative Phase (deg) Figure6-3.Relativemagnitudeandphaseofstatesinshortp eriodmodeof parameterizedsystem( u component-solid, component-dashed, q component-dash-dot) nearlyexactlythesameamountasthepitchangleandstaysne arlyinphasewiththe pitchangle.Thelongitudinalvelocitystaysatnearlythee xactsamemagnitudeuntilthe turbulencebeginstogetverylarge,andshowssomemovement tobeclosertobeingin phasewiththepitchangle.However,thelongitudinalveloc itycomponentstillremains closertobeing180 o outofphaseuntiltheturbulenceintensitygetsquitelarge Therelativemagnitudeandphaseofthestatesinthephugoid modethatbecomes andivergentmodeareshowninFigure 6-4 111

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0 2 4 6 8 10 -10 0 10 20 30 40 50 60 70 80 Turbulence Intensity (%)Relative Magnitude 0 2 4 6 8 10 0 50 100 150 200 Turbulence Intensity (%)Relative Phase (deg) Figure6-4.Relativemagnitudeandphaseofstatesinphugoi dmodethatbecomesan divergentmodeoftheparameterizedsystem( u component-solid, component-dashed, q component-dash-dot) Figure 6-4 showsthatuntilthephugoidmodedisappearsandbecomestwo exponentialmodes,theshapeofthemodestayslargelyconst ant.Undersmall variationsinturbulenceintensity,therelativemagnitud esofallcomponentsremain roughlythesame,withvariationsinlongitudinalvelocity dominating,whilepitchangle andpitchratestayroughlyofthesamemagnitude,andangleo fattackvariations areminimal.Additionally,bothpitchrateandlongitudina lvariationsleadpitchangle oscillationsbyabout90 o whileturbulenceintensityvariationsremainsmall.Theon ly signicantvariationinthemodeshapeofthephugoidmodeun dersmallturbulence levelsisthephasebywhichtheangleofattackleadsthepitc hrate.However,as previouslymentioned,therelativemagnitudeoftheangleo fattackisverysmall,so thischangeinrelativephasewouldnotsignicantlyaltert heaircraft'smotionwhenthis modeisexcited.Whenturbulenceintensitybecomessufcie ntlylarge,theuctuationsin thelongitudinalvelocitycometodominatetoanevengreate rdegree. Whentheturbulenceintensitybecomeslargeenoughastoeli minatethephugoid mode,theresultingdivergentmodedemonstratesthesamere lativemagnitudes amongstthestatesasthephugoidmode.However,thelongitu dinalvelocityandangleof attackcomponentsvaryintheoppositedirectionasthepitc hangleandpitchrate. 112

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Therelativemagnitudeandphaseofthestatesinthephugoid modethatbecomes aconvergentmodeareshowninFigure 6-5 0 2 4 6 8 10 -10 0 10 20 30 40 50 60 70 80 Turbulence Intensity (%)Relative Magnitude 0 2 4 6 8 10 0 50 100 150 200 Turbulence Intensity (%)Relative Phase (deg) Figure6-5.Relativemagnitudeandphaseofstatesinphugoi dmodethatbecomesa convergentmodeoftheparameterizedsystem( u component-solid, component-dashed, q component-dash-dot) Figure 6-5 yieldsthesameinformationonthephugoidmodeasFigure 6-4 However,theconvergentmodethatresultsfromreachingsuf cientlyhighturbulence levelsdemonstratespitchratevariationsofoppositesign fromthoseofthelongitudinal velocity,angleofattack,andpitchangle. 6.4LinearizedModel ThemodelderivedfromthewindtunneldataandshowninEquat ion 6–38 has componentsoftheforceandmomentderivativesthatarequad raticwithrespectto turbulenceintensity.Suchamodelisdifculttoanalyzeus ingthepolynomialchaos methodsuponwhichtheremainderofresearchisbased.Asare sult,thismodelismade linearwithrespecttoturbulenceintensityanddemonstrat edtobeagoodapproximation oftheoriginallyderivedmodel.6.4.1LinearizationofModelwithRespecttoTurbulence Tolinearizethemodel,aturbulenceintensityof3.5%ischo sentobethenominal conditionaboutwhichtheforceandmomentderivativesarel inearized.Thisvalueis chosenbecausewhenusedinconjunctionwithastandarddevi ationof1%turbulenceto 113

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beusedinlateranalysis,theresultingrangeofturbulence levelsismostlycontained withinthetestedrangeofturbulencelevelsfromChapter 4 andavoidsnegative turbulenceintensitylevels,whichwouldviolatemathemat icallaws.Additionally,this levelofturbulenceintensityliesinthemiddleoftherange overwhichthelinearized modelcanbetrusted,0-7%. So,turbulenceintensityisgiventheformpreviouslyseeni nperturbationdynamics andshowninEquation 6–39 I =3.5+ ^ I (6–39) Equation 6–39 issubstitutedintotheparametricmodelfromEquation 6–38 and ^ I 2 termsareignored.Theresultingsystemisnotonlylinearin thatitsformcanbe expressedinvectorandmatrixformat,butnowtheforceandm omentderivatives arelinearwithrespecttoturbulenceintensity.Thislinea rizedsystemisshownin Equation 6–40 266666664 u w q 377777775 = 266666664 0.02798+0.003352 ^ I 1.045 0.1107 ^ I 2.515 9.81 1.216+0.1292 ^ I 3.250+0.2448 ^ I 12.310 0.6060 0.03324 ^ I 15.97+0.5494 ^ I 6.2220 0010 377777775 266666664 u w q 377777775 + 266666664 0.0105 0.1302 13.7980 0 377777775 e (6–40) ThelinearizedmodelofEquation 6–40 needstobecomparedtothequadratic parametricmodelofEquation 6–38 toensurethattheydemonstratesimilareffects withrespecttovariationsinturbulenceintensity.Compar isonsaremadebetweenthe 114

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eigenvaluesandthemodeshapesofthemodesofeachoftheses ystemstoshowthe rangethroughwhichthelinearizedmodelagreeswiththequa draticmodel. 6.4.2EigenvalueAnalysis Theeigenvaluesofthelinearizedparametricsystemareplo ttedinFigure 6-6 as turbulenceintensityvariesfrom0%to10%. -6 -4 -2 0 2 -15 -10 -5 0 5 10 15 Real AxisImaginary Axis -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Real AxisImaginary Axis Figure6-6.Eigenvaluesoflinearizedparametricsystem(a lleigenvaluesshownonleft, zoomedinonphugoideigenvaluesonright) Theeigenvaluesofthelinearizedsystem,showninFigure 6-6 ,followsimilartrends tothoseseeninthequadraticmodel,showninFigure 6-1 .However,thesimilarities betweenthedampingratioandfrequencyofthelongitudinal modesofthequadraticand linearparametricsystemsarebestshowninFigure 6-7 ItisapparentfromFigure 6-7 thatthelinearizedsystemisinagreementwiththe quadraticsystemonthefrequencyanddampingratiosofthel ongitudinalmodesuntil turbulenceintensityexceeds7%andthephugoidmodenearst hepointatwhichitsplits intotwoexponentialmodes.6.4.3ModeShapeAnalysis Therelativemagnitudesandphasesofthelongitudinalstat esintheshortperiod modeareshowninFigure 6-8 .JustasinSubsection 6.3.2 ,allmagnitudesandphases aremaderelativetothe state. 115

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0 2 4 6 8 10 -1 -0.5 0 0.5 1 Turbulence Intensity (%)Damping Ratio 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 Turbulence Intensity (%)Frequency (rad/s) Figure6-7.Frequencyanddampingofmodesofthequadratica ndlinearparametric systems(quadratic-solid,linear-dashed) 0 2 4 6 8 10 0 2 4 6 8 10 12 14 16 Turbulence Intensity (%)Relative Magnitude 0 2 4 6 8 10 0 50 100 150 200 Turbulence Intensity (%)Relative Phase (deg) Figure6-8.Relativemagnitudeandphaseofstatesinshortp eriodmodeofthe parameterizedsystem(quadratic-solid,linear-dash-dot ) TherelativemagnitudesandphasesinFigure 6-8 showagreatdealofsimilarityin theshortperiodmodeshapebetweenthelinearizedandquadr aticparametricmodels. Themodelsareinclosestagreementforturbulenceintensit iesoflessthan7%. Therelativemagnitudesandphasesofthelongitudinalstat esinthephugoidmode thatbecomesandivergentmodeareshowninFigure 6-9 Figure 6-9 showsagreatdealofsimilarityinthemagnitudesofthe and q componentsofthephugoidmodebetweenthelinearizedandqu adraticparametric models.Remarkableconsistencybetweenthetwomodelsisal sopresentintherelative 116

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0 2 4 6 8 10 -10 0 10 20 30 40 50 60 70 80 Turbulence Intensity (%)Relative Magnitude 0 2 4 6 8 10 0 50 100 150 200 Turbulence Intensity (%)Relative Phase (deg) Figure6-9.Relativemagnitudeandphaseofstatesinphugoi dmodethatbecomesan divergentmodeoftheparameterizedsystem(quadratic-sol id,lineardash-dot) phasesofthe q and u components.Thesignicantdifferencesbetweenthetwomod els manifestthemselvesintherelativemagnitudeofthe u componentandtherelativephase ofthe component.JustasinSubsection 6.3.2 ,differencesintherelativephaseof theangleofattackcanbeignoredbecausethemagnitudeofva riationsintheangle ofattackissosmallinthephugoidmode.Thedifferencesint hemagnitudeofthe variationsoflongitudinalvelocitygrowlargewhenthephu goidmodeofthequadratic systemnearsthepointatwhichitsplitsintotwoexponentia lmodes.So,withintherange of0-7%turbulence,thelinearizedsystemapproximatesthe changesseeninthemodes ofthequadraticsystem. Therelativemagnitudesandphasesofthelongitudinalstat esinthephugoidmode thatbecomesaconvergentmodeareshowninFigure 6-10 Figure 6-10 yieldsthesameinsightthatcamefromFigure 6-9 6.5PolynomialChaosModel Nowthatthelinearizedparametricmodelhasbeenveried,t hisvariationin dynamicscanbeapproximatedusingthepolynomialchaosthe orydetailedinChapter 5 117

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0 2 4 6 8 10 -10 0 10 20 30 40 50 60 70 80 Turbulence Intensity (%)Relative Magnitude 0 2 4 6 8 10 0 50 100 150 200 Turbulence Intensity (%)Relative Phase (deg) Figure6-10.Relativemagnitudeandphaseofstatesinphugo idmodethatbecomesa convergentmodeoftheparameterizedsystem(quadratic-so lid,lineardash-dot) 6.5.1OrderofPCApproximation BeforethemodesofthePCsystemcanbeanalyzed,someveric ationmustbe donetoensurethatthePCsystemisimplementedcorrectly.R ecallfromSubsection 5.4.1 thatconvergenceforhigherordertermsisnotguaranteed. So,theparametricdynamicsofEquation 6–40 isexpandedbytheprocessshown inSection 5.3 usingHermitepolynomialsuptothe9thorder.Asaresultoft hisprocess, thePCEsystemhas40statesandexhibits20oscillatorymode s.Withineachmode therst10statesrelatetothe u state,thesecondgroupof10statesrelatetothe w state,thethirdgroupof10statesrelatetothe q state,andthenal10statesrelateto the state.Thersttermrelatedtothe u stateisthe 0 th ordertermofthe u state.The magnitudeofthistermisaverageacrossall10modes.Thispr ocessisrepeatedforthe followingtermsrelatedtothe u stateandforallthetermsrelatedtotheotherphysical states.Thisprocessyieldstherelativemagnitudeofthemo dalcomponentsaveraged acrossall20modes.TheresultisshowninFigure 6-11 Figure 6-11 showsthattheaveragerelativemagnitudeofthemodalcompo nents steadilydeclineswithincreasingorderoncetheorderpass esfour.Thistrendindicates thatthereismuchlessvariationinthehigherordertermsof thePCEsystemcompared 118

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0 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Polynomial OrderAverage Relative Magnitude Figure6-11.Averagerelativemagnitudeofmodalcomponent swhenusingninthorder PCapproximation tothelowerorderterms;signifyingthatthePCEsystemhasc onvergedandiscapturing themuchoftheimportantstatisticaluctuationsinthephy sicalstates. Additionally,itisnotedfromFigure 6-11 thathigherordertermsaremuchsmaller thanthelowerorderterms.Thisfeatureindicatesthatther eislittleactivityinthehigher ordermomentsofthestatevariables,andthatalowerordera pproximationcould beusedtocapturemostoftheinformationthatPCoffers.Asa result,afthorder approximationisusedforalllateranalysis.Figure 6-12 showstheaveragerelative magnitudeofthemodalcomponentswhenusingafthorderPCa pproximation. ItisclearfromFigure 6-12 thatthehigherordertermshavesufcientlydecreasedin magnitudetoconcludethatthisapproximationhasalsoconv ergedandcanbeexpected tocapturemostoftheuctuationinthemomentsofthestates oftheparametricsystem. ThisfthorderPCEexpansionisusedforanalysisofaPCEexp andedsystemtoreduce thecomputationtimesandsimplifyanalysis. 119

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0 1 2 3 4 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Polynomial OrderAverage Relative Magnitude Figure6-12.Averagerelativemagnitudeofmodalcomponent swhenusingfthorder PCapproximation 6.5.2EigenvalueAnalysis Theeigenvaluesoftheexpandedsystemareplottedoverthee igenvaluesofthe parametricsysteminFigure 6-13 -6 -4 -2 0 2 -15 -10 -5 0 5 10 15 Real AxisImaginary Axis -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 Real AxisImaginary Axis Figure6-13.EigenvaluesofPCexpandedsystem(shortperio dmodesareshownon theleft,andthephugoidmodesareshownontheright) Theeigenvaluesoftheexpandedsystemcanbeseparatedinto twogroupsthat clusteraroundtheeigenvaluesoftheparametricsystem.As aresult,themodesthat haveeigenvaluesneartheshortperiodeigenvalueswillber eferredtoasshortperiod 120

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modes,andthesameforthephugoidmodes.Themodes1-6aresh ortperiodmodes andmodes7-12arephugoidmodes.6.5.3ModeShapeAnalysis Themagnitudeandphaseoftheeigenvectorsalsosupportthe notionofgrouping themodesinsuchawaythatcomparesthemtotheoriginalphys icalmodes. Therelativemagnitudeoftheshortperiodeigenvectorsoft heparameterized systemarenormalizedwithrespecttothestate andshowninTable 6-1 .This informationisalsocomparedtotherelativemagnitudeof0t horderstochasticstates ( u 0 w 0 ,and q 0 )inModes1-6oftheexpandedsysteminTable 6-1 ParameterizedSystem ExpandedSystem State ShortPeriodMode Mode1Mode2Mode3Mode4Mode5Mode6 u 2.427 2.4082.3942.3752.3852.3752.376 0.9891 0.98640.98370.97550.98140.97880.9771 q 14.71 14.3313.9512.8613.6613.3713.08 Table6-1.Magnitudesofstateuctuationsrelativeto forshortperiodmodes Therelativephaseoftheshortperiodeigenvectorsofthepa rameterizedsystem arenormalizedwithrespecttothestate andshowninTable 6-2 .Thisinformationis alsocomparedtotherelativephaseof0thorderstochastics tatesinModes1-6ofthe expandedsysteminTable 6-2 ParameterizedSystem ExpandedSystem State ShortPeriodMode Mode1Mode2Mode3Mode4Mode5Mode6 u 186.6 o 183.6 o 180.4 o 170.9 o 178.0 o 175.4 o 172.9 o 11.79 o 10.90 o 10.00 o 7.298 o 9.296 o 8.580 o 7.857 o q 108.3 o 108.2 o 108.1 o 107.8 o 108.0 o 108.0 o 107.9 o Table6-2.Phaseleadofstatesrelativeto forshortperiodmodes Tables 6-1 and 6-2 showthatthereisagreatdealofsimilaritybetweenthemode shapesoftheparameterizedsystemandthe0thorderstateso ftheexpandedsystem inboththerelativemagnitudesandphases.Fromthisinform ationwecouldanticipate thattheexpectedshortperiodresponseofthesysteminthep resenceofuncertainty 121

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inturbulenceintensitywillbeverysimilartotheresponse seenwithoutuncertaintyin turbulenceintensity. Therelativemagnitudeofthephugoideigenvectorsofthepa rameterizedsystem arenormalizedwithrespecttothestate andshowninTable 6-3 .Thisinformation isalsocomparedtotherelativemagnitudeof0thorderstoch asticstatesinModes 7-12oftheexpandedsysteminTable 6-3 .Modes8and10doexhibitsomesignicant deviationsintherelativemagnitudesofthe u and q states.However,therelative magnitudesforallstatesarethesameorderofmagnitudeina llofthemodes. ParameterizedSystem ExpandedSystem State PhugoidMode Mode7Mode8Mode9Mode10Mode11Mode12 u 9.921 10.6315.8211.4614.2412.1613.00 0.0042 0.00440.01640.00550.01220.00700.0090 q 0.9846 0.91950.61870.85320.68720.80410.7524 Table6-3.Magnitudesofstateuctuationsrelativeto forphugoidmodes Therelativephaseofthephugoideigenvectorsoftheparame terizedsystemare normalizedwithrespecttothestate andshowninTable 6-4 .Thisinformationis alsocomparedtotherelativephaseof0thorderstochastics tatesinModes1-6ofthe expandedsysteminTable 6-4 .Therelativephaseofthe statevariessignicantly betweenthemodes,butthemagnitudeofuctuationsinthe state,seeninTable 6-3 aresosmallcomparedtothoseoftheotherstatesastorender thephasedifferences inconsequential.Therelativephasesofthe u and q statesareremarkablysimilar amongstthemodes. ParameterizedSystem ExpandedSystem State PhugoidMode Mode7Mode8Mode9Mode10Mode11Mode12 u 98.49 o 97.74 o 94.39 o 96.96 o 95.09 o 96.41 o 95.85 o 8.655 o 33.34 o 85.51 o 55.63 o 81.43 o 67.12 o 75.22 o q 97.68 o 97.18 o 95.00 o 96.71 o 95.53 o 96.34 o 95.95 o Table6-4.Phaseleadofstatesrelativeto forphugoidmodes 122

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Tables 6-3 and 6-4 showthatthereisasomesimilaritybetweenthemodeshapes oftheparameterizedsystemandthe0thorderstatesoftheex pandedsysteminboth therelativemagnitudesandphases.Fromthisinformation, wecouldanticipatethat theexpectedphugoidresponseoftheaircraftinthepresenc eofuncertaintyinthe turbulenceintensitywilllooksimilartotheresponseofth eaircraftintheabsenceof uncertainty.However,itisdifculttoseefromthisinform ationexactlyhowtheresponse inthepresenceofuncertaintywilldifferfromtheresponse intheabsenceofuncertainty. 6.5.4AnalysisofShortPeriodModes Thestandardaircraftshortperiodmodeisdominatedbyener gyinthe q ,and states,soitisexpectedthatasimulationofoneoftheshort periodmodesofthe expandedsystemwillshowmostofitsuctuationinthesesta tes.Figure 6-14 showsthe meantimeresponseofallfourphysicalstateswhenoneofthe sixshortperiodmodesof theexpandedsystemisexcited. 0 0.5 1 1.5 -15 -10 -5 0 5 Time (s) u (m/s) alpha (deg) q (deg/s) theta (deg) Figure6-14.Meanvaluesofall4statesforoneshortperiodm ode Figure 6-14 showsthattheshortperiodmodesofexpandedsystemaresimi lar totheshortperiodmodeoftheparameterizedsysteminthema gnitudeofthestate 123

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variations.Theotherveshortperiodmodesshowsimilartr endstoFigure 6-14 .The meanandvarianceboundsofallfourphysicalstatesinallsi xshortperiodmodesofthe expandedsystemareshowninFigure 6-15 0 0.5 1 1.5 -0.5 0 0.5 Time (s)u (m/s) Mean Variance Bounds 0 0.5 1 1.5 -10 -5 0 5 10 Time (s)alpha (deg) Mean Variance Bounds 0 0.5 1 1.5 -200 -100 0 100 200 Time (s)q (deg/s) Mean Variance Bounds 0 0.5 1 1.5 -10 -5 0 5 10 Time (s)theta (deg) Mean Variance Bounds Figure6-15.Meanandvarianceboundsoflongitudinalstate sforall6shortperiod modes Allsixofthesemodesinallthreedominantstatesshowagrea tdealofphase similaritybetweenthevarianceandmean.Thevariancereac hesaminimumatthe sametimeasthemeancrosseszero.So,itappearsthatevenun deruncertainty inturbulenceintensity,thephysicalstatesrelatedtothe shortperiodmodecanbe predictedwithagreatdealofprecisionatcertaintimeinte rvals,specically,thezero crossingsofeachstate. 124

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6.5.5AnalysisofPhugoidModes Thestandardaircraftphugoidmodeisdominatedbyenergyin the u q ,and states,soitisexpectedthatasimulationofoneofthephugo idmodesoftheexpanded systemwillshowmostofitsuctuationinthesestates.Figu re 6-16 showsthevariation inthemeanvaluesofallfourstatesforoneofthesixphugoid modesoftheexpanded system. 0 10 20 30 40 50 60 -1 -0.5 0 0.5 1 1.5 Time (s) u (m/s) alpha (deg) q (deg/s) theta (deg) Figure6-16.Meanvaluesofstatesforonephugoidmode Figure 6-16 showsthatthephugoidmodesofexpandedsystemaresimilart othe phugoidmodeoftheparameterizedsysteminthemagnitudeof thestatevariations.The othervephugoidmodesshowsimilartrendstoFigure 6-16 .Themeanandvariance boundsofallphysicalstatesinallsixofthephugoidmodeso ftheexpandedsystemare showninFigure 6-17 Itisonceagainnotedisthatthevarianceseemstoreachamin imumwheneverthe meanofthestatecrosseszero;however,thevariancedoesno tgetasclosetozerofor somemodesasforothers.ThiscanbenotedinFigure 6-17 .Thatgureshowsthetime evolutionofthevarianceboundsonthepitchangleunderall ofthephugoidmodes.If 125

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0 10 20 30 40 50 60 -6 -4 -2 0 2 4 6 Time (s)u (m/s) Mean Variance Bounds 0 10 20 30 40 50 60 -0.2 -0.1 0 0.1 0.2 0.3 Time (s)alpha (deg) Mean Variance Bounds 0 10 20 30 40 50 60 -20 -15 -10 -5 0 5 10 15 20 Time (s)q (deg/s) Mean Variance Bounds 0 10 20 30 40 50 60 -20 -15 -10 -5 0 5 10 15 20 Time (s)theta (deg) Mean Variance Bounds Figure6-17.Meanandvarianceboundsoflongitudinalstate sforall6phugoidmodes Equations 5–26 and 5–27 areappliedtotheaircraftsystem,thenthemeanofthepitch angleisthe19thstateoftheexpandedsystem(fromhereonth isstateisreferredtoas 0 ),andthevarianceofthepitchrateisthesumofthesquareso fthe20ththrough24th statesofthesystem(fromhereonthesestatesarereferredt oas 1 through 5 ).Thus, ananalysisofthephasedifferencebetweenthesestatesoft heexpandedsystemyields insightastowhythevariancereachesaminimum,andinsomec asesseemstogoto zeroatthesametimeasthemean.Thephaseofthestates 0 through 5 forModes10 and11areshowninTable 6-5 Table 6-5 showsthatthereasonthevarianceisaminimumwhenthemeanc rosses zeroisbecauseallofthestatesrelatedtothephysicalstat e arenearlyinphaseor 180 o outofphase.Thus,theywouldallcrosszeroatthesametime, causingaminimum 126

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State Mode10Mode11 0 00 1 -0.5 o 183.6 o 2 -3.4 o 180.2 o 3 180.3 o 145.7 o 4 179.7 o -2.1 o 5 177.3 o 181.3 o Table6-5.Phaselagofstatesofexpandedsysteminthevarianceofstate .ThereasonthatMode11doesnothavethevariancegetas smallasinMode10isbecause 3 inMode11is145 o outofphasewith 0 ,soitislarge enoughtoyieldasignicantvariancewhenthemeanofthesta teiszero. 6.5.6ExampleSimulation Themodalanalysisoftheprecedingsectionshasyieldedsom einsightintothe modesoftheopenloopresponseoftheGenMAVunderuncertain tyintheturbulence intensity,butitisunclearhowthesemodesinteractwithon eanother.Tounderstandthe modalinteractions,anexamplesimulationisrunwithiniti alconditionsthatwillexciteall ofthemodesofthesystem. Thedatapresentedintherestofthissubsectionallrelatet oasimulationofthe expandedsystemconductedusingtheinitialconditionssho wninEquation 6–41 Theseinitialconditionsareaspecialcaseinthattheyrepr esentaninitialstatewithno uncertainty;theinitialphysicalstatesareknownexactly andareequaltothe0thorder statesintheexpandedsystem. 127

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u 0 =2 m = s u 1 ,..., u 5 =0 m = s w 0 =1 m = s w 1 ,..., w 5 =0 m = s (6–41) q 0 = 0.2 rad = s q 1 ,..., q 5 =0 rad = s 0 =0.1 rad 1 ,..., 5 =0 rad Figure ?? showsthetimeevolutionofthemeanandvarianceboundsofal lfour statesgiventheinitialconditionsofEquation 6–41 ThemostinterestingfeaturetonoteinFigure 6-18 isthegrowthandconsequential decayinstateuncertainty.Themagnitudeoftheuncertaint yinthestatesismost pronouncedinthephysicalstates u and .Itisinterestingtonotethatdespitethefact thatformanyofthemodesoftheexpandedsystem,thevarianc edecreasedtonearly zerowhenthemeanstatescrossedzero;yetinthissimulatio n,thestateuncertainty doesnotapproachzerountilsufcienttimehaspassedtoall owallthestatesofthe expandedsystemtodecaytozero.Thisexampledemonstrates howthemodescombine invaryingmagnitudesandatvaryingrelativephaseanglest oproducethetimeevolution ofthesystemstateuncertainty. Figure 6-18 showsthelongtermeffectsofuncertaintyinturbulenceint ensityon thelongitudinalstatesoftheGenMAV,buttheshorttermeff ectsarenotclearfromthis gure.So,Figure 6-19 showsthetimeevolutionofthemeanandvarianceboundsofal l fourstatesgiventhesameinitialconditions,butonlyshow sthetimehistoryoftherst fewseconds.NotethattheshortperiodmodalanalysisofSub section 6.5.4 showeda 128

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0 20 40 60 80 100 -3 -2 -1 0 1 2 3 Time (s)u (m/s) 0 20 40 60 80 100 -2 -1 0 1 2 3 4 5 Time (s)alpha (deg) 0 20 40 60 80 100 -40 -30 -20 -10 0 10 20 30 Time (s)q (deg/s) 0 20 40 60 80 100 -10 -5 0 5 10 15 Time (s)theta (deg) Figure6-18.Meanandvarianceboundsofallstatesforexamp lesimulation settlingtimeofaroundonesecond,soFigure 6-19 showsthreesecondsoftheexample simulationtoobservethetransitionfromshortperiodtoph ugoiddynamics. Thecontrastinthetimeevolutionofthestateuncertaintie sbetweenthedifferent statesisremarkable.Theangleofattack, ,showsalmostnouncertainty,yetthe states u and showlargeuncertaintygrowthwithintherstsecond.Thepi tchrate, q isperhapsthemostinterestingstatetoobserve,becauseit suncertaintybeginstogrow beforedecayingtonearlyzero,yetthenbeginstogrowagain Thesetrendsarelikelyexplainedbyobservationsmadeprev iouslythatthephugoid modesaremoresubstantiallyaffectedbytheuncertaintyin theturbulenceintensity.The angleofattack, ,playsalmostnoroleinthephugoidmodes,andanyuncertain tyin duetotheshortperiodmodesisminimalanddecaysquickly.T hepitchrate, q ,plays 129

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0 1 2 3 4 -2 -1 0 1 2 3 Time (s)u (m/s) 0 1 2 3 4 -2 -1 0 1 2 3 4 5 Time (s)alpha (deg) 0 1 2 3 4 -40 -30 -20 -10 0 10 20 30 Time (s)q (deg/s) 0 1 2 3 4 -5 0 5 10 15 Time (s)theta (deg) Figure6-19.Meanandvarianceboundsofallstatesforexamp lesimulation amuchmoresignicantroleintheshortperiodmodes,yetsti llhasapresenceinthe phugoidmodes.Sotheuncertaintyin q duetotheshortperiodmodesisobserved, decaysquickly,andthenuncertaintyduetothephugoidmode sbeginstogrowasthe phugoiddynamicsdominatethesystem.Thestates u and arelargecomponentsof thephugoidmode,sotheuncertaintyintheturbulenceinten sitypropagatesthroughthe phugoidmodesoftheexpandedsystemandresultsinlargegro wthintheuncertaintyof thesestates.6.5.7EffectsofUncertainParameterDistribution TheresultsshownsofarinSection 6.5 haveassumedaGaussiandistributionfor theturbulenceintensitywithameanof3.5%turbulenceanda standarddeviationof1% turbulence.However,thisdistributionisnotobtainedfro manyscienticmethod;rather, 130

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itisassumedusingpreviousworkonatmosphericturbulence asaguide.Therefore,it wouldbeinterestingtondwhateffectswouldresultfromac hangeintheprobability densityfunctionoftheatmosphericturbulence. Anewprobabilitydensityfunctionisassumedforturbulenc eintensity,withamean of3.5%turbulenceandastandarddeviationof2%turbulence .ThesamePCEprocess asbeforeisperformedontheparameterizedsystemusingthe newprobabilitydensity functionforturbulenceintensity.Figure 6-20 showstheeigenvaluesoftheexpanded systemundertwodifferentdistributionsofuncertaintywi threspecttoturbulence intensity. -4.5 -4 -3.5 -3 -15 -10 -5 0 5 10 15 Real AxisImaginary Axis Std Dev = 1 Std Dev = 2 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 -1 -0.5 0 0.5 1 Real AxisImaginary Axis Std Dev = 1 Std Dev = 2 Figure6-20.Eigenvaluesofexpandedsystemusingtwodistr ibutionsofturbulence intensity ItisapparentfromFigure 6-20 thattheeigenvaluesoftheshortperiodmodes arerelativelyunaffectedbytheincreaseinvarianceofthe turbulenceintensity.Some eigenvaluesmovedasmallamountinthedirectionofthenega tiverealaxisunder thewiderdistribution,butthechangerelativetothelocat ionoftheeigenvaluesunder thetighterdistributionissmall.Theeigenvaluesoftheph ugoidmodemovedquite signicantlyinthepresenceofthewiderdistributionoftu rbulenceintensity.Thenew uncertainparameterdistributionpushedoneoftheeigenva luepairstotherealaxis,thus splittingitintoaconvergentandadivergentmode.Thisocc urrencetswithFigure 6-1 whichsuggestedthatiftheturbulenceintensitygrowslarg eenough,thephugoid 131

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modewouldbesplitintoaconvergentandadivergentmode.It appearsthatthewider distributionofturbulenceintensityhasbroughtthateffe ctintotheexpandedsystem. Thephugoidmodesoftheexpandedsystemunderthewiderdist ributionare nowanalyzed.Theconvergentanddivergentmodeswillnotbe examinedduetothe previouslymentioneddoubtastotheveracityofthesemodes .Figure 6-21 showsthe longitudinalstatesforthevestablephugoidmodespresen tintheexpandedsystem underthewiderdistributionofturbulenceintensity. 0 10 20 30 40 50 60 -6 -4 -2 0 2 4 6 Time (s)u (m/s) Mean Variance Bounds 0 10 20 30 40 50 60 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 Time (s)alpha (deg) Mean Variance Bounds 0 10 20 30 40 50 60 -20 -15 -10 -5 0 5 10 15 20 Time (s)q (deg/s) Mean Variance Bounds 0 10 20 30 40 50 60 -20 -15 -10 -5 0 5 10 15 20 Time (s)theta (deg) Mean Variance Bounds Figure6-21.Meanandvarianceboundsoflongitudinalstate sforallstablephugoid modes,givenawiderdistributionofturbulenceintensity Figure 6-21 showsasimilarmixofratiosofmeanandvarianceresponseth at appearedinFigure 6-17 .However,thephasedifferencesbetweenthemodeshave becomeevenmorepronounced,andthefrequenciesatwhichth eyoscillatehave 132

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becomeevenmorevaried.Oneofthephugoidmodesofthisexpa ndedsystemshowsa verylowfrequencyofoscillationandverylowdampingratio inthevarianceofthestates. Thisdataindicatesthatuncertaintywithrespecttoturbul enceintensitystronglyaffects thephugoidmodeandresultsinstrongeruncertaintypropag atingthroughthephugoid modeasthevarianceofturbulenceintensityincreases. 133

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CHAPTER7 STOCHASTICPATHEVALUATIONMETHODS Thischapterintroducestechniquesthatcanbeusedtoapply polynomialchaos (PC)theorytopathevaluationstrategies.Thebasicconcep tistousetheinformation thatPCanalysisprovideswiththeprinciplesofconditiona lprobabilitytoderivethe probabilityofmissionsuccesswithadenedvehicle,contr oller,environment,and desiredpath.ThetechniquesareexplainedingeneralinSec tion 7.2 suchthatthey canbeappliedtomanydenitionsofmissionsuccess.Thetec hniquesareapplicable aslongasmissionsuccessisdenedassomemathematicalcom binationofmission statesandtime,suchasobstacleavoidance,sensorcoverag e,orwaypointnavigation. Sections 7.3 and 7.4 showexamplesofthealgorithmbeingappliedtodenedmissi on successmetrics. 7.1ProbabilityBackground Constructingthepathevaluationalgorithmrequiressomeb ackgroundinformation intoprobabilityandsettheory. Aprobabilityspaceisdenedusingthreeparts:asamplespa ce,asetofevents, andthecorrespondingprobabilitiesoftheeventsoccurrin g.Forthisreason,aprobability spaceissaidtobedenedaprobability“triple”, ( n F P ) .Thesamplespace, n ,isthe setofallpossibleoutcomes. Theunionoftwosets A 1 and A 2 isdenedinEquation 7–1 A 1 [ A 2 = allelementscontainedineither A 1 or A 2 (7–1) Theintersectionoftwosets A 1 and A 2 isdenedinEquation 7–2 A 1 \ A 2 = allelementscontainedinboth A 1 and A 2 (7–2) Thecomplementofset A isdenedinEquation 7–3 134

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A c = allelementsin n notcontainedin A (7–3) Therelativecomplementofset A withrespecttoset B isdenedinEquation 7–4 A B = allelementsin A notcontainedin B (7–4) Givenasamplespace n ,apartitionof n isdenedinEquation 7–5 partitionof n is f A i : i 2 I g suchthat [ i A i = n (7–5) where, A i \ A j = ; for i 6 = j AnexampleofasamplespaceisshowninFigure 7-1 A B C Figure7-1.Partitionofanexamplesamplespace ApartitionofthissamplespacecouldbewrittenasshowninE quation 7–6 n = A [ ( B C ) [ ( C B ) [ ( B \ C ) [ ( A [ B [ C ) c (7–6) Theprobabilityofaneventoccurringisanormfunctionthat canrangeinvaluefrom 0to1.Forexample,inFigure 7-1 ,theprobabilityofeventCoccurring, P ( C ) ,wouldtake onapositivevaluesomewherebetween0and1. 135

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Probabilitiescanalsobeconditioneduponothereventsocc urring.Theprobabilityof event A 1 occurringconditionedonevent A 2 occurringisdenedbyEquation 7–7 P ( A 1 j A 2 )= P ( A 1 \ A 2 ) P ( A 2 ) (7–7) Forexample,inFigure 7-1 ,theprobabilityofeventCoccurring, P ( C ) ,wouldbe smallerthantheprobabilityofeventCoccurringcondition edonthefactthateventB hadoccurred, P ( C j B ) .Additionally,itcanbeseenthattheprobabilityofeventC occurringconditionedonthefactthateventAhadoccurred, P ( C j A ) ,wouldbeequalto 0,because P ( A \ C ) isequalto0. Partitionsandconditionalprobabilitiescanbeusedincon certthroughthelaw oftotalprobability.Thelawofconditionalprobabilityst atesthatif f A i : i 2 I g forms apartitionof n ,thentheprobabilityofevent B occurringcanbeexpressedasin Equation 7–8 P ( B )= X i 2 I P ( B \ A i ) (7–8) Probabilitydistributionfunctions(PDFs)areanothermet hodofexpressingthe probabilityofaneventoccurring.Ifarandomvariable, ,cantakeoneofanycountable numberofrealvalues,thenthePDFof F ,isdenedinEquation 7–9 F ( x )= P ( = x ) (7–9) Thus,aPDFisausefuldescriptionoftheprobabilityofdisc reteeventsoccurring. Cumulativedistributionfunctions(CDFs)providesimilar informationtoPDFsalthoughin aslightlydifferentform,andcanbeusedforbothdiscretea ndcontinuousdistributions. ACDF, G ,isdenedaccordingtoEquation 7–10 G ( x )= P ( x ) (7–10) 136

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IfarandomvariablehasaCDFthatiscontinuousanddifferen tiable,thenit alsopossessesaprobabilitydensityfunction(pdf).Apdf, f ,isdenedaccordingto Equation 7–11 f ( x )= @ G ( x ) @ x (7–11) NotethatapdfonlyexistsiftherandomvariableexhibitsaC DFthatisboth continuousdifferentiable,andthattheexistenceofapdfi mpliesthattheprobabilityof therandomvariabletakinganysingularvalueisnegligible .Equation 7–12 demonstrates thisfact. P ( = x )=0 (7–12) if 9 f ( x ) 7.2StochasticStateGenerationAlgorithm Thealgorithmtogeneratestochasticstateinformationtha tisamenabletopath evaluationconsistsofcombiningthePCexpandeddynamicsi nformation,whichis explainedinChapter 6 ,withapartitionofthesamplespaceofthegivenmission. Forexample,ifthegoalofamissionistoreachanalpositio n(RFP)whilepassing throughonewaypointregion( WP 1 ),thepartitionofthesamplespaceofthismission wouldlooklikeFigure 7-2 Theprobabilityofmissionsuccessforthisexamplewouldbe foundusingEquation 7–13 P ( success )= P ( RFP \ WP 1) (7–13) = P ( RFP j WP 1) P ( WP 1) 137

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Reach Final Position (RFP) Pass through Waypoint 1 (WP1) Figure7-2.Partitionofsamplespaceofmissionwithoneway point (hatchedareaindicatessetofresultswithmissionsuccess ) Iftheenvironmentinquestionhastwowaypointregions( WP 1 and WP 2 )andthe vehiclehasanonzeroprobabilityofpassingthroughboth,t henthepartitionofthe samplespacewouldlooklikeFigure 7-3 Reach Final Position (RFP) Pass through Waypoint 1 (WP1) Pass through Waypoint 2 (WP2) Figure7-3.Partitionofsamplespaceofmissionwithtwoway points (hatchedareaindicatessetofresultswithmissionsuccess ) Theprobabilityofmissionsuccessforthisexamplewouldbe foundusingEquation 7–14 138

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P ( success )= P ( RFP \ WP 2 \ WP 1) = P ( RFP j WP 2 \ WP 1) P ( WP 2 \ WP 1) (7–14) = P ( RFP j WP 2 \ WP 1) P ( WP 2 j WP 1) P ( WP 1) Ifthewaypointregionsfromtheprevioustwoexamplesareno -yzonesinstead, thenmissionsuccesswouldbedescribedasreachingthenal positionwhileavoiding passingthroughanyno-yzone.Ifthereisonlyoneno-yzon e,thentheresulting partitionwouldappearasshowninFigure 7-4 Reach Final Position (RFP) Pass throughNoy Zone 1 (NF1) Figure7-4.Partitionofsamplespaceofmissionwithonenoyzone (hatchedareaindicatessetofresultswithmissionsuccess ) Theprobabilityofmissionsuccesswithoneno-yzone( NF 1 )wouldbedescribed usingEquation 7–15 P ( success )= P ( RFP ) P ( RFP \ NF 1) (7–15) = P ( RFP ) P ( RFP j NF 1) P ( NF 1) Ifthereisonlyoneno-yzone,thentheresultingpartition wouldappearasshown inFigure 7-5 139

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Reach Final Position (RFP) Pass throughNo-€y Zone 1 (NF1) Pass throughNo-€y Zone 2 (NF2) Figure7-5.Partitionofsamplespaceofmissionwithtwonoyzones (hatchedareaindicatessetofresultswithmissionsuccess ) Theprobabilityofmissionsuccesswithtwono-yzones( NF 1 and NF 2 )wouldbe describedusingEquation 7–16 P ( success )= P ( RFP ) P ( RFP \ NF 1) P ( RFP \ NF 2)+ P ( RFP \ NF 2 \ NF 1) = P ( RFP ) P ( RFP j NF 1) P ( NF 1) P ( RFP j NF 2) P ( NF 2) (7–16) + P ( RFP j NF 2 \ NF 1) P ( NF 2 j NF 1) P ( NF 1) Thus,theformulationofamissionsuccessprobabilitymetr icrequiresthe calculationofconditionalprobabilities.Theseconditio nalprobabilitiesarefoundusing analgorithmthatexploitsPCEdynamicsinanovelway.Thisa lgorithmndsnotonly theprobabilitythatafunctionofvehiclestatesliewithin givenbounds,butalsothe probabilitydensityfunctions(pdf's)ofthestatescondit ionedonthefactthatthefunction ofvehiclestateslieswithinthosebounds.Thesepdf'saret henusedasinitialconditions tondprobabilitiesoflatereventsoccurring,yieldingth econditionalprobabilitiesneeded tocalculatethemissionsuccessprobabilitymetric. Thestatisticsofthevehicleandthetrajectoryofitsstate sneedtobefoundto calculatetheneededconditionalprobabilities.Thepatho fthevehiclecanbepicturedas 140

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showninFigure 7-6 .Thevehiclefollowsameanpathingeneralbutateachpointo nits pathitsactuallocationisstochasticanddescribedbyapdf Figure7-6.Visualizationofstochasticnatureofvehicle' spath Additionaldenitionsmustbemadetondtheneededpdfsfor thepathevaluation algorithm.First,a“zoneofinterest”(ZOI)isdenedasase tinmulti-dimensional spacesuchthatthevehicle'spositionrelativetosaidseti nuencesmissionsuccess. AZOIcanbeanareainphysicalspace,suchasanobstacle,noyzone,orwaypoint. AZOIcanalsoexistinhigherdimensionalspace.Forexample ,anaircraftangleor velocitymayneedtobewithinaspeciedrangewhileanaircr aftisinagivenpositionto successfullysenseatarget. Second,a“positionofinterest”(POI)isdenedasapointin spacewhen/where thelikelihoodofthemissionsuccessrelevantfunctionofs ystemstatesismostlikely toliewithinthesetdenedbyitsrespectiveZOI.Forexampl e,inamissionwithone waypointsuchasshowninFigure 7-2 ,thetimehistoryofofthejointpdfoflongitudinal position, x ,andlateralposition, y ,canbereconstructed,andthePOIcouldbedened asthelongitudinalposition, x POI ,atwhichthevehicle'spositionismostlikelytolie withintheboundsofwaypoint1.Ofcourse x POI wouldhavetoliewithinthelongitudinal 141

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positionrangeofwaypoint1.ThePOIcouldalsobechosenast helateralposition, y POI ,atwhichthevehicle'spositionismostlikelytoliewithin theboundsofwaypoint 1.AchoicebetweenalateraloralongitudinalpositionasaP OIdependsonwhich directionistravelingasitcrossestherespectivewaypoin t.APOIcouldevenbedened asatime, t POI ,ratherthanaphysicallocation.WhatisrequiredofaPOIis thatthe aircraftisguaranteedtohavethatstatecrossthatthresho ldvalue.Ifthemissionis time-independent,thenitwouldmakemoresensetochooseap hysicallocationforthe POI,becausetheprobabilityofthevehiclepositionlyingw ithinwaypoint1couldbe signicantlylargeatawiderangeoftimes, t .Thus,theprobabilitythatthevehiclewould passthroughwaypoint1atsomepointinitspathcouldbesign icantlygreaterthanthe probabilitythatitwouldliewithinwaypoint1attime, t POI Ifthelengthscalesofthevehicledynamicsarelessthanthe lengthoftheZOI,and x POI and x POI + bothliewithinthelongitudinalboundsofwaypoint1,theni tisquite likelythatvehicletrajectoriesthatliewithinwaypoint1 at x POI + wouldalsoliewithin waypoint1at x POI .Thus,theprobabilityofintersectingwaypoint1, P ( WP 1) ,being writtenasaconditionalprobability,showninEquation 7–17 P ( WP 1) P ( y 1 y y 2 j x = x POI ) (7–17) Findingtheprobabilityconditionedonafunctionofthephy sicalstatesrequires additionalstepsbecausethestochasticinformationprovi dedbyPCbaseddynamicsis allrelevantforagiventime, t .Theseadditionalstepswilldifferdependingonthenature andthetimehistoryofthefunctionofphysicalstatesusedt oconditiontheprobability. So,anexamplewillbeusedtodemonstratehowsuchaconditio nalprobabilitywouldbe found. Inthisexample,thelongitudinalposition, x ,isusedtoconditionthelateralposition, y .So,ndingaCDFforthelateralpositionconditionedonthe longitudinalposition consistsofndingtheprobabilityshowninEquation 7–18 142

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G y j x = P ( y y 1 j x = x POI ) (7–18) Equation 7–18 showsthatamappingmustbefoundfrom x to y .PCanalysis yieldsmappingsfromtimetoboth x and y .So,themappingfromtimeto x isreversed, creatingamappingfrom x totime,andthencombinedwiththemappingfromtimeto y ; thus,amappingisachievedfrom x to y .Reversingthemappingfromtimeto x requires someassumptions.Itisnecessarytoassumethat P ( x x POI j t = t i ) ismonotonically increasingwithtime, P ( x x POI j t = t 1 )=0 ,and P ( x x POI j t = t n )=1 Usingtheseassumptions,if P ( x x POI j t = t i ) isfoundatevery t i ,thosedatapoints formacumulativedistributionfunctionofthetimeatwhich thevehiclepasses x POI Thistimeinstantishereafterreferredtoasthecrossingti me, t c .Whiletimeitselfisnot arandomvariable, t c isarandomvariable.Thus,aPDFof t c isdenedaccordingto Equation 7–19 P ( t c = t i ) P ( t = t i j x = x POI ) (7–19) P ( x x POI j t = t i ) P ( x x POI j t = t i 1 ) Thus,theprocessofreversingthemappingfromtimeto x ,andcombiningitwiththe mappingfromtimeto y canbecompletedusingthestepsshowninAlgorithm 7.1 Algorithm7.1. (MomentofInterestConditionalPDFGenerationAlgorithm) Given jointpdfsatrangeoftimes, f xy ( t i ) ,for i =1to n xmarginalpdfatrangeoftimes, f x ( t i ) ,for i =1to n Find P ( x x POI j t = t i ) ateachtime, t i Find P ( t c = t i ) accordingtoEquation 7–19 Use f xy ( t i ) tond f y j x ( t i ) P ( y y 1 j x = x POI )= n X i =1 P ( y y 1 j x = x POI \ t = t i ) P ( t c = t i ) 143

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Thus,aPDFofthelateralpositionofthevehiclegiventhatt helongitudinalposition is x POI isobtained. NowthebackgroundhasbeenlaidtopresenttheStochasticPa thEvaluation Algorithm,showninAlgorithm 7.2 Algorithm7.2. (StochasticPathEvaluationAlgorithm) Given closedloopplant P uncertainparameter associatedwith P polynomialchaoskernelfunctions, i 's physicalstateinitialconditions, x IC missionstategoals(waypoints,no-yzones,sensingrequi rements,etc.) desiredstatetrajectory missionsuccessfunction, g ( x ) Project P and x IC ontoPCkernelfunctionstoproduce P PC and x IC PC Simulate P PC tondhistoryofexpandedsystemstates Identifyallpositionsofinterest(POI's)Beginloop: Findmomentsof g ( x ) atagivenPOI Reconstructpdfof g ( x ) andjointpdfsof g ( x ) andphysicalstates, x Findprobabilityof g ( x ) lyingwithingivenbounds Normalizesegmentsofjointpdfswhere g ( x ) lieswithingivenbounds Projectjointpdfsegmentsonto i 'stoproduceconditionalinitialconditions Simulate P PC undernewinitialconditions RepeatloopusingnextPOI CalculatenalmissionsuccessprobabilityNoaccuracyguaranteesaremadewithregardtothemissionsu ccessprobabilities, becausegreateraccuracymayormaynotbeattainablebyusin ghigherordermoments 144

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inthecalculationofstateprobabilitydensityfunctions( pdfs).Instead,themission designershouldanalyzetheestimatesofhigherordermomen tsandmakeajudgement astowhethertotrustthepdfsthatusethem. 7.3AlgorithmAppliedtoWaypointNavigationandCollision Avoidance WhenAlgorithm 7.2 isappliedtowaypointnavigationandcollisionavoidance, the importantfunctionsofstates, g ( x ) ,arethecoordinatesoftheaircraftthataredeemed relevant.Figures 7-7 7-11 showtheapplicationofAlgorithm 7.2 toanenvironmentthat hasadesirednalposition,twono-yzones,andacommanded pathdirectlyeast.As such,missionsuccessprobabilityiscalculatedusingEqua tion 7–16 .Thedesirednal positionandno-yzonesaredenedintwodimensions,sothe altitudeoftheaircraftis notshownasitisnotrelevant.Theaircraftisgivenaniniti alpositionattheoriginwithno uncertaintyandinitialstateswithrandommeansanddistri butions. ThePDFofcrossingtimethatisdenedbyEquation 7–19 ,iscalculatedwithregard tothePOIoftherstno-yzoneandshowninFigure 7-7 0 5 10 15 20 25 30 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Crossing TimeProbability Distribution Figure7-7.TimePDF 145

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ThePDFofcrossingtimefromFigure 7-7 isusedthecalculatethepdfofyposition conditionedonthexpositionbeingequaltoPOIoftherstno -yzone.Thispdfis showninFigure 7-8 -10 -5 0 5 10 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Y PositionProbability Density Figure7-8.XConditionalYPDF Figure 7-9 showsthatthePOIwithregardtotherstno-yzoneisatthev ery backofthearea,soitisatthispointthattheaircraftismos tlikelytohaveenteredthe rstno-yzone.Thepdfoftheaircraft'slateralpositionf romFigure 7-8 isoverlaidon Figure! 7-9 .Theareaofoverlapbetweenthepdfandno-yzoneyieldsthe probability thattheaircraftenterstheno-yzoneatthisPOI,represen tedby P ( NF 1) inEquation 7–16 Thesameprocessisrepeatedforthesecondno-yzoneandthe desirednal position.Figure 7-10 showsthatthePOIwithregardtothesecondno-yzoneisatth e verybackofthearea,soitisatthispointthattheaircrafti smostlikelytohaveentered thesecondno-yzone,andthepdfofitsypositionisasshown .Theareaofoverlap betweenthepdfandno-yzoneyieldstheprobabilitythatth eaircraftenterstheno-y zoneatthisPOI,representedby P ( NF 2) inEquation 7–16 Figure 7-11 showsthatthePOIwithregardtothedesirednalpositionis atthevery frontofthearea,soitisatthispointthattheaircraftismo stlikelytohaveenteredthe 146

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0 5 10 15 20 25 -10 -5 0 5 10 X PositionY Position Figure7-9.Probabilityofenteringrstno-yzonedemonst ration 0 5 10 15 20 25 -10 -5 0 5 10 X PositionY Position Figure7-10.Probabilityofenteringsecondno-yzonedemo nstration 147

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area,andthepdfofitsypositionisasshown.Theareaofover lapbetweenthepdfand desirednalpositionyieldstheprobabilitythattheaircr aftenterstheareaatthisPOI, representedby P ( RFP ) inEquation 7–16 0 5 10 15 20 25 -10 -5 0 5 10 X PositionY Position Figure7-11.Probabilityofreachingnalpositiondemonst ration TocompletethecalculationofEquation 7–16 ,theconditionalprobabilities P ( RFP j NF 1) P ( RFP j NF 2) P ( NF 2 j NF 1) ,and P ( RFP j NF 2 \ NF 1) mustbecalculated. Theseconditionalprobabilitiesarefoundusingthesamepr ocessthatisusedto generatethenon-conditionalprobabilities P ( NF 1) P ( NF 2) ,and P ( RFP ) .Theonly differenceisthatthecalculationoftheconditionalproba bilitiesrequiresdifferentinitial conditions.Theinitialconditionsusedinthisprocessare generatedfromjointPDF's ofthestatesconditionedontheeventthattheconstraintso nrelevantfunctionsofthe states, g ( x ) ,aresatised.Figure 7-12 showsthepdfofthelateralpositionconditioned ontheeventthat x = x POI and y min y y max ;whichisequivalentto f y j x = x POI \ y min y y max 148

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0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Y PositionProbability Density Figure7-12.Normalizedpdfof y positionat x POI 1 conditionaluponconictwith NF 1 NotethatFigure 7-12 isnothingmorethanthepdfshowninFigure 7-8 onlywith thetailsremovedbeyondthelimitsoftherstno-yzoneand renormalizedtohavean integralof1whenintegratedwithrespectto y position. ApdfliketheoneshowninFigure 7-12 isgeneratedforeveryaircraftpositionand state.Forthepositions,apdfthatmatchestheconditional pdfofFigure 7-12 ofuptoa desirednumberofmomentsiscreatedandusedastheinitialp ositionoftheaircraft.For thestates,pdfsareconvertedintoexpandedstatesbysetti ngthemomentsofthepdf equaltothemomentsfoundusingtheexpandedstates.Forexa mple,ifphysicalstate v isexpandedtothe3rdorder,thentherst4momentsofthephy sicalstate v arefound usingEquation 7–20 .Notethatthe inEquation 7–20 arenotindicativeofrollangle, butarethebasispolynomialsdiscussedinChapter 5 149

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E [ v ]= E [ v 0 0 + v 1 1 + v 2 2 + v 3 3 ] E [ v 2 ]= E [( v 0 0 + v 1 1 + v 2 2 + v 3 3 ) 2 ] (7–20) E [ v 3 ]= E [( v 0 0 + v 1 1 + v 2 2 + v 3 3 ) 3 ] E [ v 4 ]= E [( v 0 0 + v 1 1 + v 2 2 + v 3 3 ) 4 ] Equation 7–20 hasfourequationsandfourunknowns,thusallowingtheexac t calculationofthestatesoftheexpandedsystemthatyieldt hepdfsofthephysicalstates conditionedontheeventthattherelevantstateboundsares atised.Theseexpanded statesareusedasinitialconditionsintheanalysistondt heconditionalprobabilities. Figure ?? showsthedistributionof y positionatthenalpositionpositionofinterest conditionedontheeventthattheaircraftenterstherstno -yzone.Theinitial x position isgivenasthe x POI oftherstno-yzone.Theinitial y positionisgivenasthepdfof y positionconditionedontheeventthat x = x POI and y min y y max ;whichisequivalent to f y j x = x POI \ y min y y max .Allinitialstatesarecalculatedusingthemethoddescrib edinthe paragraphaboveandEquation 7–20 7.4AlgorithmAppliedtoSensing WhenAlgorithm 7.2 isappliedtosensingmissions,theimportantfunctionsof states, g ( x ) ,areacombinationofaircraftcoordinatesandstatesthata redeemed relevant.Theserelevantstatesandcoordinateswilldepen donthetypeofsensorused. Forexample,theperformanceofachemical“sniffer”mighto nlybeaffectedbyproximity tothetarget,inwhichcaseonlytheaircraftcoordinateswo uldbeimportant.However, theperformanceofothersensors,suchascameras,alsodepe ndonotheraircraft states,suchasincidenceangle,eldofviewangle,andimag eplanevelocities[ 165 166 ].Theaircraftstatesthataredeemedrelevantmustbeevalu atedtodetermineifthe aircraftwillsuccessfullycompleteitssensingmission. 150

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ThecorestructureofAlgorithm 7.2 doesnotneedtobechangedtoallowforpath analysisofasensingmission.However,someadditionalste psintheexecutionofthe programmayberequiredtoaccountfortheadditionalrestri ctionsonwhatisconsidered a“successful”mission.Forexample,ifcameraisdownwardl ooking,thenthesuccessful sensingofatargetrequiresalongitudinalposition, x ,alateralposition, y ,andaroll angle, ,thatallliewithingivenbounds.InSection 7.3 ,aPDFoftheaircraft's x and y positionisfoundateachtime, t .ThisPDFisequivalentto f x y ( t ) .ThisPDFisthen manipulatedusingAlgorithm 7.1 tondthePDFof y positionconditionedontheevent thatthe x positionisatthepositionofinterest x POI ,or f y j x = x POI .Toincorporatethe restrictiononrollangle, ,aPDFoftheaircraft's x and y positionandrollanglemust befoundatalltimes, t .ThisPDFisequivalentto f x y ( t ) .Then,inaprocessvery similartoAlgorithm 7.1 ,thisPDFisconvertedintoaPDFofthe y positionandrollangle conditionedontheeventthatthe x positionisatthepositionofinterest x POI ,or f y j x = x POI TheareaunderthecurveofthisPDFthatlieswithinthegiven rangesof y positionand rollangleistheprobabilitythatthetargetwouldbesucces sfullysensed. Thecalculationofconditionalinitialconditionsisalsos lightlycomplicatedbythe additionofnewstatethatmustsatisfyboundsforthemissio ntobecalled“successful”. TheexampleofSection 7.3 showsthegenerationofthePDFofstate v conditionedon theeventthat x = x POI and y min y y max ;whichisequivalentto f v j x = x POI \ y min y y max Iftherollangle, isaddedtothepertinentstates,thentheinitialcondition sforcases whentheconstraintsaresatisedmustalsotakeintoaccoun t .So,therelevantPDFof state v becomes f v j x = x POI \ y min y y max \ min max So,theadditionoffurtherstateconstraintstoensuresucc essfultargetsensing requiresadditionaldimensionsinthejointdistributiono frelevantstates.These additionaldimensionscanbeaddedbycalculatingthemomen tsoftheaddedrelevant statesandthecross-productmomentsbetweentheaddedande xistingstates.To continuewiththeaboveexample,if f x y ( t ) istobeconstructedwithonlythersttwo 151

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moments,thenthenecessarymomentswouldbe E [ x ] E [ y ] E [ x 2 ] E [ y 2 ] ,and E [ xy ] Theadditionoftheconstraintonrollangle, ,togenerate f x y ( t ) wouldalsorequire thecalculationof E [ ] E [ 2 ] E [ x ] ,and E [ y ] .Thecalculationoftheseexpectationsis easytoperform,however,theconstructionofthePDFinhigh erdimensionscanbecome computationallyexpensive.Socareshouldbetakentoonlyi ncludeconstraintson statesthatarerelevantandthathaveasignicantlikeliho odofbeingviolated. 152

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CHAPTER8 EXAMPLESOFMAVSTOCHASTICPATHEVALUATION ThepathanalysismethoddescribedinChapter 7 isusedtoevaluateapath, controller,aircraftcombinationformissioneffectivene ssgiventwodifferenttypesof missions:collisionavoidanceandtargetsensing.Thesele ctionoftheaircraftand controlleristreatedinSections 8.1 and 8.2 .Theapplicationoftheaircraftmodeland controllerareappliedtothetwomissiontypesinSections 8.3 and 8.4 8.1AircraftModel TheaircraftmodelfromChapter 6 isusedasthebasisforthemodeltodemonstrate thepotentialofthepathanalysismethod.Notethattheairc raftmodelinEquation 6–40 containsonlythestatesnecessaryforalongitudinalanaly sisoftheaircraft.Toanalyze theaircraftinthelateral-directionalaxes,afullstatea ircraftmodelmustbederived. TheGenMAVisonceagainusedtoprovideanaircraftmodeltod emonstratean analysistool.TheAVLmodeloftheGenMAVisevaluatedinthe samewayasdescribed inSection 6.1 .Thistimetheightdynamicsofthelateral-directionalst atesareincluded. TheresultingfullstateaircraftdynamicsgivenbyAVLissh owninEquation 8–1 153

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26666666666666666666666664 u w q v p r 37777777777777777777777775 = 26666666666666666666666664 0.02320.9007 2.515 9.8100000 1.0483 2.901012.31000000 0.7266 17.93 6.222000000 0010000000000 0.21872.549 12.709.810 0000 38.93 23.69 15.2700 0000 0.49042.402 1.18900 000001000000000100 37777777777777777777777775 26666666666666666666666664 u w q v p r 37777777777777777777777775 + 26666666666666666666666664 0.0105 8.346 10 10 5.807 10 11 0.1302 6.761 10 9 4.106 10 11 3.7985.724 10 9 2.266 10 9 000 4.170 10 10 4.926 10 3 2.521 10 2 9.448 10 8 2.871 0.1903 1.722 10 8 0.26070.7774 000000 37777777777777777777777775 266664 elev ail rudd 377775 (8–1) TheprocessinSection 6.1 convertstheupperlefthandcornerofthestate dynamicsmatrixofEquation 8–1 intoamodelparametricandlinearwithrespectto turbulenceintensity.Thatderivationusestheresultsofw indtunneltestingatdifferent turbulencelevels.However,nowindtunneltestinghasyetb eenconductedtodiscover theeffectsofturbulenceintensityonthelateral-directi onalightdynamicsofMAVs.As such,aneducatedguessismadetoprovideamodelforthepurp osesofdemonstrating thepathanalysistechniquesdevelopedinChapter 7 ItisnotedinEquation 6–40 thattheturbulencedependenttermsariseinthe columnsthataremultipliedbytheightvelocities, u ,and w .Alsonotethat,whereit 154

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appears,therstorderterminturbulenceisroughlyone-te nththesizeofthezeroth ordertermandoftheoppositesign.Thesetwopropertiesare usedtocreateamodel thatincorporatesaneffectofturbulenceinthelateral-di rectionalstatesthatissimilarto theeffectseenbythelongitudinalstates.Thismodelissho wninEquation 8–2 Theparameter ^ I isassumedtohaveameanof0%andastandarddeviationof1%. Thisphysicalsystemisthenexpandedusingtheprocessshow ninSection 5.3 using Hermitepolynomialsuptothefthorder.Theresultingsyst emhas54statesand18 controlinputs,andisusedtosimulatethestatisticsofthe physicalaircraftstates. 8.2ControlDerivation ToevaluatethelinearparametricightdynamicsofEquatio n 8–2 inapathanalysis exampleacontrollermustbederivedforthesystem.However ,thecontrollerisrst derivedusingthenon-parametricsystemshowninEquation 8–1 andthenadaptedto useontheparametricsystem.Thisdecisionismadewiththei ntenttoshowtheeffects ofturbulenceonasystemthatisdesignedinignoranceofthe presenceandeffectsof turbulenceonboththeaircraftandthecontroller. Thelinearquadratictrackingcontroller,alsoknownasanL QRtracker,isselected asthetypeofcontrollertouseduetoitseaseofdesignandab ilitytofollowadened trajectory.TheLQRtrackerforthissystemisdesignedtore gulateallstatesexcept headingangle, ,ie.tokeepallstatesbut asclosetozeroaspossible.The stateiscomparedtothedesiredheadingangleandthecontro lisdesignedtodrive thatdifferencetozero.Ablockdiagramschemeofthistypeo fcontrollerisshownin Figure 8-1 155

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26666666666666666666666664 u w q v p r 37777777777777777777777775 = 26666666666666666666666664 0.02798+0.003352 ^ I 1.045 0.1107 ^ I 2.515 9.8100000 1.216+0.1292 ^ I 3.250+0.2448 ^ I 12.31000000 0.6060 0.03324 ^ I 15.97+0.5494 ^ I 6.222000000 0010000000000 0.2187+0.0219 ^ I 2.549 12.709.810 0000 38.93+3.893 ^ I 23.69 15.2700 0000 0.4904+0.0490 ^ I 2.402 1.18900 000001000000000100 37777777777777777777777775 26666666666666666666666664 u w q v p r 37777777777777777777777775 + 26666666666666666666666664 0.0105 8.346 10 10 5.807 10 11 0.1302 6.761 10 9 4.106 10 11 3.7985.724 10 9 2.266 10 9 000 4.170 10 10 4.926 10 3 2.521 10 2 9.448 10 8 2.871 0.1903 1.722 10 8 0.26070.7774 000000 37777777777777777777777775 266664 elev ail rudd 377775 (8–2)156

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R K P K ++ r y Figure8-1.BlockdiagramofLQRtrackingcontroller Tocalculatethegains K and K foruseinFigure 8-1 ,thestandardightdynamics mustbeconvertedtoasupplementeddynamicsthatincludest heintegralofthestate inthestatevector.Thestandardightdynamicsareshownin Equation 8–1 andare representedinshorthandbyEquation 8–3 x = Ax + Bu (8–3) ThestatevectorforthesupplementeddynamicsisshowninEq uation 8–4 x supp = 264 R x 375 (8–4) Thus,thedynamicsofthesupplementedsystemareasshownin Equation 8–5 x supp = A supp x supp + B supp u where, A supp = 266664 0 01 ... A 0 377775 (8–5) B supp = 264 000 B 375 157

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ThemathematicalformulationoftheLQRtrackerisshowninE quation 8–6 .The LQRtrackerisdesignedtominimizetheintegralofthe2-nor mofthesupplemented statevectorandcontrolactuationoverinnitetime. min Z 1 0 x T supp ( t ) Qx supp ( t )+ u T ( t ) Ru ( t ) (8–6) where, Q and R areuser-dened InEquation 8–6 ,the Q and R matricesareweightingmatricesthataredened bythecontroldesignertopenalizecertainstatesoractuat ion.Thechoiceof Q and R affectsthecontrolgains K and K ofFigure 8-1 .Thecontrolgainsarisefromthe solutiontotheContinuous-timeAlgebraicRiccatiEquatio n(CARE),asshownin Equation 8–7 K supp = R 1 B T supp X where, K = rstcolumnof K supp (8–7) K = remainingcolumnsof K supp X isthesolutionto A Tsupp X + XA supp XB supp R 1 B T supp X + Q =0 NotethatimplementationoftheLQRformulationrequiresth atallstatesbeboth observableandcontrollable.Theactuationinherentinthe aircraftmakesallstates controllable,andfullstatefeedbackisassumedmakingall statesobservableaswell. The Q and R matriceschosenforuseinthisexampleareshowninEquation 8–8 158

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Q = 266666666664 1.500 0 010 0 ... ... 0 010 0 000 377777777775 (8–8) R = 266664 100010001 377775 Thesechoicesfor Q and R producecontrolgainsthatareshowninEquation 8–9 K = 266664 2.169 10 8 0.3502 1.174 377775 K = 266664 0.1181 0.37370.41495.325 5.548 10 9 5.225 10 9 1.303 10 9 4.707 10 8 5.053 10 9 7.239 10 9 2.120 10 9 9.983 10 8 (8–9) 4.959 10 9 6.299 10 9 5.356 10 8 2.060 10 8 6.847 10 8 2.008 10 2 8.013 10 2 0.2433 0.8898 1.1402 0.1723 0.13222.1211.7553.748 377775 Thesecontrolgainsaredesignedforuseonthephysicalsyst em.Toapplythese controlgainstothePCsystem,thePCsystemcontrolgains, K PC and K PC aredened asthekroneckerproductofthephysicalsystemcontrolgain sandtheidentitymatrix. ThesedenitionsareshowninEquation 8–10 159

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K PC = K n I p +1 (8–10) K PC = K n I p +1 8.3CollisionAvoidanceExample AnexampleisshowninthisSectiontodemonstratehowAlgori thm 7.2 isapplied toaxedobstacleeldcollisionavoidanceproblem.Anenvi ronmentisdenedwitha setofobstaclesandadesirednalposition.Asaresult,the aircraftmustavoidpassing throughanyoftheobstaclesandstillpassthroughthedesir ednalpositiontocomplete asuccessfulmission.Adesiredtrajectoryisdenedusinga ninitialpositionattheorigin andaninputsignalthatgivesadesiredheadingangle.Theai rcraft'sinitialpositionis settobetheoriginwithnouncertainty.Theinitialstateso ftheaircraftaregivenrandom distributionsbyassigningthestatesoftheexpandedsyste mtobenumberselectedbya randomnumbergenerator. Theenvironmentandtrajectoryinformationisdesigned,an dtheclosedloop dynamicsaresimulated.Theenvironmentandtheaircraftme anpaththatresultsfrom asetofrandominitialconditionsisshowninFigure 8-2 .Theboxfurthesttotherightin Figure 8-2 representsthedesirednalpositionoftheaircraft,while therestoftheboxes representobstaclesintheenvironment.Toensurethatthee ntirepdfoftheaircraft's longitudinalpositionpassesthedesirednalposition,th esimulationextendsthemean vehiclepathbeyondthedesirednalposition. Asimplescreeningisrunbasedonthemeanandvarianceofthe aircraftposition tondwhichoftheobjectsofinterestintheenvironmentact uallycreatethepotential forconict.Thisscreeningprocesshelpstoeliminateunne cessarycalculationsby ndingthatonly3obstaclesandthedesirednalpositionha veausersetthreshold ofprobabilityforconictwiththeaircraft'spath.Thus,t hetworemainingobstaclesare ignoredfortheremainderofthecalculations.Additionall y,thisscreeningprocessuses 160

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0 200 400 600 800 -200 -100 0 100 200 300 400 X PositionY Position Figure8-2.EnvironmentandvehiclemeanpathAlgorithm 7.1 tocalculatethelongitudinalpositionthatismostlikelyt oresultinconict foreachregionofinterest, x POI i .Basedontherandominitialconditions,Algorithm 7.2 is usedtocalculatethepdfofaircraftlateralpositionwheni tslongitudinalpositionreaches each x POI .ThesepdfsareoverlaidontheenvironmentandmeanpathinF igure 8-3 0 200 400 600 800 -200 -100 0 100 200 300 400 X PositionY Position Figure8-3.Probabilitydensityfunctionsatallpossiblec ollisions 161

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So,withthreepotentialobstaclesandadesirednalpositi on,theequationusedto calculatetheprobabilityofasuccessfulmissionisexpres sedinEquation 8–11 P ( success )= P ( RFP ) P ( RFP \ NF 1) P ( RFP \ NF 2) P ( RFP \ NF 3) + P ( RFP \ NF 2 \ NF 1)+ P ( RFP \ NF 3 \ NF 1) + P ( RFP \ NF 3 \ NF 2) P ( RFP \ NF 3 \ NF 2 \ NF 1) = P ( RFP ) P ( RFP j NF 1) P ( NF 1) P ( RFP j NF 2) P ( NF 2) (8–11) P ( RFP j NF 3) P ( NF 3)+ P ( RFP j NF 2 \ NF 1) P ( NF 2 j NF 1) P ( NF 1) + P ( RFP j NF 3 \ NF 1) P ( NF 3 j NF 1) P ( NF 1) + P ( RFP j NF 3 \ NF 2) P ( NF 3 j NF 2) P ( NF 2) P ( RFP j NF 3 \ NF 2 \ NF 1) P ( NF 3 j NF 2 \ NF 1) P ( NF 2 j NF 1) P ( NF 1) ThepdfsfromFigure 8-3 areusedtocalculate P ( RFP ) P ( NF 1) P ( NF 2) ,and P ( NF 3) .Then,jointpdfsofthestatesandaircraftpositionaregen eratedconditional ontheaircrafthavingalongitudinalpositionof x POI 1 .Anexampleofoneofthesepdfsis showninFigure 8-4 Figure8-4.Jointpdfof and y positionconditionalonaircraftat x POI 1 162

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Segmentsofthesejointpdfsareremovedandnormalizedtoha veanintegralof one.Thus,ajointpdfof and y conditionalontheaircraftbeinginconictwiththerst obstacleisproduced.ThisnormalizedpdfisshowninFigure 8-5 Figure8-5.Renormalizedjointpdfof and y positionconditionalonconictwith NF 1 Expectationsaretakenaboutthesepdfstoyieldthemoments oftheaircraftstates conditionalonconictwithagivenobstacle.Themomentsar ethenusedtogenerate theinitialexpandedstatesforthenextsimulation.Thissi mulationnowrepresentsthe aircraftmovementconditionalontheaircrafthavingpasse dthroughtherstobstacle.It isunderstoodthatinthephysicalworld,iftheaircraftpos itionenteredthisrange,then theaircraftwouldhavecrashedandthemissionwouldbeover .So,itisbettertothink oftheobstaclesassimplyzonesofconict,thusallowingth econditionalprobabilities tobecalculated.Themeanpathofthevehicleandthepdfsofi tslateralpositionatall subsequent x POI i conditionalontheaircraftpassingthroughtherstzoneof conictare showninFigure 8-6 ThepdfsfromFigure 8-6 areusedtocalculate P ( RFP j NF 1) P ( NF 2 j NF 1) ,and P ( NF 3 j NF 1) 163

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200 300 400 500 600 700 -100 0 100 200 300 X PositionY Position Figure8-6.Lateralpositionpdfsat NF 2 NF 3 ,anddesirednalpositionconditional uponconictwith NF 1 Thesameprocessisrepeatedtondthemeanpathandlateralp ositionpdfs at x POI i conditionalontheaircraftpassingthroughthesecondzone ofconict.This informationisshowninFigure 8-7 200 300 400 500 600 700 -100 0 100 200 300 X PositionY Position Figure8-7.Lateralpositionpdfat NF 3 anddesirednalpositionconditionalupon conictwith NF 2 ThepdfsfromFigure 8-7 areusedtocalculate P ( RFP j NF 2) and P ( NF 3 j NF 2) 164

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Thesameprocessisrepeatedtondthemeanpathandlateralp ositionpdfs at x POI i conditionalontheaircraftpassingthroughthethirdzoneo fconict.This informationisshowninFigure 8-8 200 300 400 500 600 700 -100 0 100 200 300 X PositionY Position Figure8-8.Lateralpositionpdfatdesirednalpositionco nditionaluponconictwith NF 3 ThepdffromFigure 8-8 isusedtocalculate P ( RFP j NF 3) Now,theonlytermsfromEquation 8–11 thatareleftunknownare, P ( RFP j NF 2 \ NF 1) P ( RFP j NF 3 \ NF 1) P ( RFP j NF 3 \ NF 2) P ( RFP j NF 3 \ NF 2 \ NF 1) and P ( NF 3 j NF 2 \ NF 1) .Theseprobabilitiesarefoundbyiteratingthesameproces s asusedtondtheotherconditionalprobabilities.Forexam ple,tond P ( NF 3 j NF 2 \ NF 1) ,thestateandpositionpdfsat x POI 2 arecalculatedfromthesimulationshown inFigure 8-6 ,andthesepdfsareusedtoinitializeanewsimulationthatw illyieldall probabilitiesconditionalupontheaircraftyingthrough bothofthersttwozonesof conict.Thisprocessisrepeatedtondtheremainingunkno wnprobabilities. ThisexampleyieldstheprobabilitiesshowninTable 8.3 .Thenalmissionsuccess probabilityiscalculatedandshownatthebottomofTable 8.3 165

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Table8-1.Probabilitiescalculatedforcollisionexample ofpathevaluationalgorithm P ( NF 1) 0.0561 P ( NF 2) 0.3566 P ( NF 3) 0.0543 P ( RFP ) 0.8603 P ( NF 2 j NF 1) 0.0000 P ( NF 3 j NF 1) 0.0000 P ( RFP j NF 1) 0.0000 P ( NF 3 j NF 2) 0.0622 P ( RFP j NF 2) 0.8471 P ( RFP j NF 3) 0.0000 P ( NF 3 j NF 2 \ NF 1) 0.0000 P ( RFP j NF 2 \ NF 1) 0.0000 P ( RFP j NF 3 \ NF 1) 0.0000 P ( RFP j NF 3 \ NF 2) 0.0000 P ( RFP j NF 3 \ NF 2 \ NF 1) 0.0000 P ( success ) 0.5582 8.4TargetSensingExample AnexampleisshowninthisSectiontodemonstratehowAlgori thm 7.2 isapplied toaxedsensingtargetproblem.Theenvironmentisdenedw ithasetofdesired targets.Inthisexample,theaircraftisassumedtohaveado wnwardpointingcamera. Asaresult,theaircraftmustpassoverthetargetswhilemai ntainingarollanglewithin adenedrangeinordertocompleteasuccessfulmission.Ift herollangleistoogreat ineitherdirection,thenthetargetbeneaththeaircraftwi llnolongerbewithinthe camera'seldofview.Infutureapplicationstheallowable rollanglerangecouldbe denedintelligentlyusingthesensor'seldofviewanglea ndtheaircraft'saltitude,but fordemonstrationpurposestheallowableaircraftrollang lewillrangefrom-0.2to+0.2 radians(-11.5to+11.5degrees). Theenvironmentandtrajectoryinformationisdesigned,an dtheclosedloop dynamicsaresimulated.Theenvironmentandtheaircraftme anpaththatresultsfrom asetofrandominitialconditionsisshowninFigure 8-9 .ThefourboxesinFigure 8-9 representsensingtargetsintheenvironment.Toensuretha ttheentirepdfofthe 166

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aircraft'slongitudinalpositionpassesthenalsensingt arget,thesimulationextendsthe meanvehiclepathbeyondthenalsensingtarget. 0 200 400 600 800 1000 1200 -400 -200 0 200 400 X PositionY Position Figure8-9.Environmentandvehiclemeanpath Asimplescreeningisrunbasedonthemeanandvarianceofthe aircraftposition tondthelongitudinalpositionthatismostlikelytoresul tineffectivesensingforeach target.Theselongitudinalpositions, x POI i ,areindicatedontheenvironmentshownin Figure 8-10 Withfourtargetstobesensedtheequationusedtocalculate theprobabilityofa successfulmissionisexpressedinEquation 8–12 P ( success )= P ( S 4 \ S 3 \ S 2 \ S 1) (8–12) = P ( S 4 j S 3 \ S 2 \ S 1) P ( S 3 j S 2 \ S 1) P ( S 2 j S 1) P ( S 1) Itisunnecessarytocalculatealloftherelatedconditiona lprobabilitiesthatare neededinthecollisionavoidanceexampleofSection 8.3 inordertondtheprobability ofcompletemissionsuccessofEquation 8–12 .However,afullconditionalprobability analysisyieldssomeadditionalinformationthataidsthem issiondesigner. 167

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0 200 400 600 800 1000 1200 -400 -200 0 200 400 Y PositionX Position Figure8-10.Longitudinalpositionsmostlikelytoresulti nsuccessfulsensing So,afullconditionalprobabilityanalysisisundertaken. AjointPDFofthelateral positionandrollangleisfoundateachofthelongitudinalp ositionsidentiedin Figure 8-10 .Figure 8-11 showsthePDFofthelateralpositionandrollangleconditio ned ontheaircrafthavingalongitudinalpositionof x POI 1 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -10 -5 0 5 10 Roll Angle (deg)Y Position (m) Figure8-11.PDFofaircraftlateralpositionandrollangle at x POI 1 (3-Dplotonleft, contourplotonright) NotethatthescalesinFigure 8-11 staywithintheboundsgivenfortherollangle andonlyslightlyextendbeyondthelateralpositionbounds ofsensingtarget1.Hence,it makessensethattheprobabilitycalculatedforsensingtar get1issocloseto1. 168

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JointPDFsarecreatedateach x POI bothindependentlyandconditionedupon successfulsensingofpriortargets.Thefullsetofconditi onalprobabilitiesisshownin Table 8.4 .Thenalmissionsuccessprobabilityiscalculatedandsho wnatthebottomof Table 8.4 Table8-2.Probabilitiescalculatedforsensingexampleof pathevaluationalgorithm P ( S 1) 0.9826 P ( S 2) 0.9363 P ( S 3) 0.3551 P ( S 4) 0.9999 P ( S 2 j S 1) 0.9412 P ( S 3 j S 1) 0.2854 P ( S 4 j S 1) 1.0000 P ( S 3 j S 2) 0.2651 P ( S 4 j S 2) 1.0000 P ( S 4 j S 3) 0.0000 P ( S 3 j S 2 \ S 1) 0.2726 P ( S 4 j S 2 \ S 1) 1.0000 P ( S 4 j S 3 \ S 1) 0.0000 P ( S 4 j S 3 \ S 2) 0.0000 P ( S 4 j S 3 \ S 2 \ S 1) 0.0000 P ( success ) 0.0000 ThisexampleshowsthatAlgorithm 7.2 yieldsnotonlyanalprobabilityofmission success,butalsoprovidesinformationastowhichpartsoft hemissionaremoreorless likelytobefullled.FromtheprobabilitiesshowninTable 8.4 ,theprobabilityofcomplete missionsuccessisnegligible.However,thedifcultyprev entingmissionsuccessis clearlythesensingoftarget3. Figure 8-12 showsthePDFofaircraftlateralpositionandrollanglecon ditionalupon theaircraftlongitudinalpositionbeing x POI 3 .Thelateralpositionoftheaircraftlieslargely withintheboundsofsensingtarget3(-83 m to-93 m ).Therollangleoftheaircraftis likelytobeoutsidetheboundsgivenforsuccessfulsensorp ointingattarget3;however, aportionofthejointPDFsatisesboththeboundsonrollang leandtheboundson lateralposition. 169

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-40 -20 0 20 40 60 -105 -100 -95 -90 -85 -80 -75 -70 -65 Roll Angle (deg)Y Position (m) Figure8-12.PDFofaircraftlateralpositionandrollangle at x POI 3 (3-Dplotonleft, contourplotonright) Figure 8-13 showsthePDFofaircraftlateralpositionandrollanglecon ditionalupon successfulsensingoftarget3andtheaircraftlongitudina lpositionbeing x POI 4 .Note thattheboundsonrollanglearesatised,buttheboundsonl ateralpositionatsensing target4(-10 m to+10 m )arenot.Thisindicatesthatiftheaircrafthastherequire droll angletosuccessfullysensetarget3,thenitisunlikelytha titwillendupinasatisfactory lateralpositiontosensetarget4.So,whileitispossiblef ortheaircrafttosuccessfully sensetarget3,Figure 8-13 demonstratesthatishighlyunlikelythattheaircraftwill successfullysensetarget4,ifithassuccessfullysensedt arget3. -2 0 2 4 -29 -28 -27 -26 -25 -24 -23 Roll Angle (deg)Y Position (m) Figure8-13.PDFofaircraftlateralpositionandrollangle at x POI 4 conditionalupon successfulsensingoftarget3(3-Dplotonleft,contourplo tonright) 170

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Iftherestrictiononsensingtarget3iseliminated,thenth eaircraftwouldhavea highlikelihoodofmissionsuccess.Theprobabilityofsens ingtargets1,2,and4would becalculatedbyEquation 8–13 P ( partialsuccess )= P ( S 4 \ S 2 \ S 1) = P ( S 4 j S 2 \ S 1) P ( S 2 j S 1) P ( S 1) (8–13) =0.9248 Giventhisinformation,themissiondesignercanchangethe pathwiththeintention ofmakingitmorelikelytosuccessfullysensetarget3asapa rtoftheentiremission; ratherthancompletelythrowouttheentirepathandstartov er. 171

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CHAPTER9 CONCLUSION 9.1ResearchSummary Thisresearchattemptstoconnectadvancementsinaerodyna mics,ightmechanics, dynamics,andcontroltoproduceausefulmethodologyforan alyzingtheeffectsof turbulenceonmicro-airvehiclemissions. Thewindtunneltestingfoundthatturbulenceintensityhas signicanteffectson theightdynamicsthatcanbeaccountedforbyaddingparame trictermsintotheight dynamicsequations.Itisimportanttonotethatturbulence intensityisarootmean squareaveragemeasure.Thus,theparametrizationofthemo delwillonlyaccountfor therootmeansquareaverageightdynamics;theturbulence intensityparametricmodel willnotaccountfortheinstantaneousdisturbancescreate dbythegustsandvortices withintheturbulentow.Rather,thesedisturbanceswills tillneedtobeattenuatedby somecombinationofastablevehicleand/orstabilizingcon troller. Whenthedynamicsarelinearized,theseturbulenceintensi typarametricterms provideanewmethodofunderstandingtheightmodesofamic ro-airvehiclein turbulence.Viewingtheightmodesofthepolynomialchaos expandedversionofthe parametricightdynamicsyieldsinsightintohowturbulen ceintensityaffectsthestability oftheaircraftaswellastheshapeoftheaircraftmodes. Thepolynomialchaosexpandedightdynamicsmodelcanalso beusedto simulateastatisticaldescriptionoftheaircraft'sight path.Thisstatisticaldescription canbeusedtoevaluatethevehicle,controller,andpathfor itssuitabilityforaproposed missionthatincludespointsofinterestsuchas,waypoints ,no-yzones,ordesired sensingtargets.Thispathanalysismethodprovidesanintu itivemeasureofthe probabilityofmissionsuccess,aswellasinformationthat couldbeusedtojudge thesensitivityofthemission'ssuccesstoeachmissionpoi ntofinterest. 172

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Anovelalgorithmhasbeenderivedtogeneratethestatistic aldescriptionofthe aircraft'sightpath.Thetruevalueofthisalgorithmlies initsabilitytogenerateboth timerangedependentandtimeindependentstatisticsofthe aircraft'sstatesandposition fromthetime-specicstatesthatareyieldedbypolynomial chaosdynamicssimulations. 9.2FutureWork Theadvancementsmadeinthisthesisopenuppossibilitiest omoreresearch. Furtherwindtunneltestingtoanalyzetheeffectsofturbul enceonaircraftdynamicsand improvementstothepathanalysistoolareeasilywithinrea ch. Thewindtunneltestingconductedforthisresearchprovide sjustanintroductory glimpseintotheeffectsofturbulenceonaircraftdynamics .Thetestingexaminedthe effectsofturbulenceintensity,butdidnotinvestigateth eimpactofturbulencelength scalesortimescales,whicharenotedinChapter 2 tobeasignicantfactorinthe effectofturbulenceonaircraftdynamics.Thelengthscale swoulddeterminethephase differenceinturbulenceuctuationsseenbydifferentpar tsoftheaircraft.Forexample, theleftwingmayseedifferentowfromtherightwingproduc ingroll,orthehorizontal tailcouldseedifferentowfromthewingproducingpitch.T hetimescaleswouldinteract withtheaircraftmodestoeitheractinphaseandexacerbate oscillatorymodes,or actoutofphasetomitigatethosemodes.Additionally,thel evelsofturbulencethat aretestedarewithinarelativelysmallrangeof0.89%to3.8 5%.However,research hasshownthatturbulenceintensitiescanreachmuchhigher levelsintheregimesin whichmicro-airvehiclesaredesignedtooperate[ 86 ].Windtunneltestingcouldalso beexpandedtoincludemultiplevelocitiesandlateraldyna micstobeabletoderivea completeaircraftdynamicsmodelfromthewindtunneltesti ngalone. Therearesomepartstothepathanalysistoolthatcouldbeim proved.Themost signicantdrawbacktothetoolisthecomputationtime.Ate verytimesteptheprogram mustreconstructmanymultivariateprobabilitydensityfu nctionsandperformmultivariate 173

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functioninterpolations.Withsomeintelligentdesign,th eprogramcouldberedesignedto eliminatetheneedperformmanyofthesecalculations. Ifgreateraccuracyisdesiredfromthepathanalysistool,t henmanyofthe calculationsusedintheprogramcouldusemorethantherst twomoments.The foundationofthepathanalysisalgorithmpermitshigheror derapproximations,andthe initialconditionsofthestatesthatareconditionaloncon ictdomatchtothehighest ordercalculated.But,tosavecomputationtime,thecumula tivedistributionfunctions andprobabilitydensityfunctionsofpositionswereconstr uctedassuminganormal distributionandusingonlythersttwomoments. Finally,amethodcouldbedevisedtotakeintoaccountforbo ththeinstantaneous disturbancesofgustandvorteximpingementsontheaircraf tandtheroot-mean-square averagedeffectsofturbulenceintensitytoprovideastati sticalsimulationtoolthatwould bemoreindicativeofthetruevehicleightpath.Thiscould beachievedbycombininga turbulencemodel,suchastheDrydenwindturbulencemodel, withamovingaverageof theturbulenceintensityproducedbythemodel.Themovinga veragecouldbeusedina gainschedulingmannertoaltertheightdynamics,whileth einstantaneousvorticesare accountedforasanexternaldisturbancestothemodel. 174

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BIOGRAPHICALSKETCH BrianRobertsearnedhisPh.D.inaerospaceengineeringatt heUniversityof Florida,GainesvilleinMay2012.Priortocompletinghisdo ctorate,Brianearnedhis BachelorofSciencewithhonorsinaerospaceengineeringat theUniversityofMaryland, CollegeParkinMay2007.HeimmediatelymovedtoGainesvill ewherehebegan workingwithDr.RickLind.In2009,Mr.Robertsearnedhisma stersdegreestudying novelcongurationsformicro-airvehicleagilitybyusing pterosaursasamodelfor morphinganddesign.Inthesummerof2009,Mr.Robertswasaw ardedanNSFEAPSI granttotraveltoMelbourne,AustraliatoworkwithDr.Simo nWatkinsinhisresearchon ightdynamicsofMAVsinturbulence.Hewilllooktondways tousehisknowledgeof controlsystemsandstochasticdynamicsasapositiveinue nceonhiscommunityand theworld. 188