image aided inertial nav

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Tightly-CoupledImage-AidedInertialNavigationSystemviaaKalmanFilter

THESISMichaelG.Giebner,Captain,USAF

AFIT/GE/ENG/03-10

DEPARTMENTOFTHEAIRFORCEAIRUNIVERSITY

AIRFORCE INSTITUTEOFTECHNOLOGYWright-PattersonAirForceBase,Ohio

APPROVEDFORPUBLICRELEASE;DISTRIBUTIONUNLIMITED


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The views expressed in this thesis are those of the author and do not reflect theofficialpolicyorpositionoftheUnitedStatesAirForce,DepartmentofDefenseortheUnitedStatesGovernment.


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AFIT/GE/ENG/03-10


THESIS

PresentedtotheFacultyoftheDepartmentofElectricalandComputerEngineering

GraduateSchoolofEngineeringandManagementAirForceInstituteofTechnology

AirUniversity

InPartialFulfillmentofthe

RequirementsfortheDegreeofMasterofScienceinElectricalEngineering

MichaelG.Giebner,B.S.Captain,USAF

March,2003

Approvedforpublicrelease;distributionunlimited


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AcknowledgementsI would like to acknowledge those who have helped me in the completion of thisthesis. Thanks go to Major John Raquet for his expertise, guidance, and encouragementinhelpingmakethisthesispossible. Withoutlonghoursandhisincredibleexpertise in debugging Matlabc code, I would still be attempting to defend thisthesis. Thanks to Capt John Erickson for his donation of a Matlabc generatedKalman filter. I would also like to thank the other faculty members of the AFITElectricalEngineeringdepartment,Dr. PeterMaybeck,LtColMikelMiller,andDr.MeirPachterfortheirexpertacademicinstructionandtheirpatiencewhendealingwithstudents.

TherearenotenoughwordsintheworldtoexpressthegratitudeIfeeltowardmy AFIT Peeps. I was lucky enough to become friends with a great group offolkswhomadetheAFITexperiencenotonlybearable,butevendownrightfunattimes. You guys rock: Heather Crooks, my Aussie pal Matt Paps Papaphotis,GregGregariousHibbiddyDibbiddyHoffman,DaveCrash,BitterDaveGaray,ChristopherHamiltonandKate,ChristineEllering,ScottandBonnieBergren,MikeandGinaHarvey,RayBuzzandTheresaToth,JamesHaneyandBethHanley,andanyoneelseImanagedtoleaveout(sorryaboutthat).

ThanksgototheboysofUSAFTestPilotSchoolclass02A.ThereisnowayI would have made it through the 21 year AFIT/TPS program without your help

2during the year at Edwards. You guys are the best. A special thanks goes to thePeepingTalonflighttestteamfrom02A:MajBillAjaxPeris,MajJeanBilger(thecoolestWSO inFrance), MajCharlesDrEEvilHavasy,CaptStephenPhurterFrank,CaptRonBattaSchwing,andCaptCliftonMojoJanney.

MichaelG.Giebner

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TableofContentsPage

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiLi st of A bbr e v i at i ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-1

1.1 Bac k gr ound . . . . . . . . . . . . . . . . . . . . . . . . 1-11.1.1 InertialNavigationSystems . . . . . . . . . . 1-21.1.2 GPS Systems . . . . . . . . . . . . . . . . . . 1-31.1.3 IntegratedINS/GPSSystems . . . . . . . . . 1-41.1.4 CurrentTrendinNavigationSystems . . . . . 1-51.1.5 HowGPSDependenceAffectsAircraft . . . . 1-51.1.6 HowGPSDependenceAffectsMunitions . . . 1-8

1.2 Problem to be Solved . . . . . . . . . . . . . . . . . . . 1-91.2.1 Do Visual Measurements Enhance Navigation

Accuracy? . . . . . . . . . . . . . . . . . . . . 1-91.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-91.4 Literature Review . . . . . . . . . . . . . . . . . . . . 1-10

1.4.1 NavigationUsingImagingTechniques . . . . . 1-101.5 Methodology . . . . . . . . . . . . . . . . . . . . . . . 1-11

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Page1.6 Assumptions . . . . . . . . . . . . . . . . . . . . . . . 1-121.7 Materials and Equipment . . . . . . . . . . . . . . . . 1-12

1.7.1 AirForceFlightTestCenter . . . . . . . . . . 1-121.7.2 Matlabc . . . . . . . . . . . . . . . . . . . . 1-121.7.3 T-38A Aircraft . . . . . . . . . . . . . . . . . 1-131.7.4 GAINR . . . . . . . . . . . . . . . . . . . . . 1-131.7.5 Ashtech GPS Receiver . . . . . . . . . . . . . 1-141.7.6 Camera . . . . . . . . . . . . . . . . . . . . . 1-141.7.7 Camera Lenses . . . . . . . . . . . . . . . . . 1-141.7.8 V i de o T ape s . . . . . . . . . . . . . . . . . . . 1-141.7.9 TimeCodeGenerator/Inserter. . . . . . . . . 1-141.7.10 Video Monitor . . . . . . . . . . . . . . . . . . 1-15

1.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 1-152. Background Theory . . . . . . . . . . . . . . . . . . . . . . . . 2-1

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . 2-12.2 Kalman Filter Basics . . . . . . . . . . . . . . . . . . . 2-1

2.2.1 Kalman Filter . . . . . . . . . . . . . . . . . . 2-12.2.2 Kalman Filter Equations . . . . . . . . . . . . 2-12.2.3 Extended Kalman Filter . . . . . . . . . . . . 2-42.2.4 State Vector . . . . . . . . . . . . . . . . . . . 2-52.2.5 State Covariance Matrix . . . . . . . . . . . . 2-62.2.6 DynamicsDrivingNoiseCovarianceMatrix . 2-72.2.7 State Transition Matrix . . . . . . . . . . . . 2-72.2.8 Measurement Matrices . . . . . . . . . . . . . 2-72.2.9 Measurement Noise . . . . . . . . . . . . . . . 2-72.2.10 Kalman Filter Cycle . . . . . . . . . . . . . . 2-7

2.3 Arc Length Equation . . . . . . . . . . . . . . . . . . . 2-9v


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Page

2.4

2.5

2.6

2.3.1 MeridianRadiusofCurvature . . . . .2.3.2 TransverseRadiusofCurvature . . . .

Geometry . . . . . . . . . . . . . . . . . . . . .2.4.1 DerivativeofInverseTangentFunction

NavigationOrbitGeometryDetermination . . .2.5.1 Assumptions . . . . . . . . . . . . . . .

Summary . . . . . . . . . . . . . . . . . . . . .

3.

Methodology

. . . . . . . . . . . . . . . . . . . . . . . .

. . . . 2-9

. . . . 2-10

. . . . 2-10

. . . . 2-10

. . . . 2-12

. . . . 2-12

. . . . 2-12

. . . .

3-1

. . . . 3-1

. . . . 3-1

. . . . 3-2

. . . . 3-5

. . . . 3-5. . . 3-7

. . . . 3-8

. . . . 3-8

. . . . 3-13

. . . . 3-14

. . . . 3-15

3.13.23.33.4

3.5

3.63.7

Overview . . . . . . . . . . . . . . . . . . . . .Visual Measurements . . . . . . . . . . . . . . .VisualMeasurementGeneration. . . . . . . . .VisualMeasurementErrorEstimation . . . . .

3.4.1 GeneratingtheH(x) Matrix . . . . .VisualMeasurementGenerationusingaCamera

3.5.13.5.23.5.33.5.43.5.53.5.63.5.7

RequiredCameraSpecifications . . . .GenerateTargetLatitude/Longitude .GenerateMeasurementsfromanImageGenerateMeasurementEstimates . . .Camera-to-BodyDCMGeneration . .VisualMeasurementRMatrixGeneration . . 3-15CameraCalibrationProcedures . . . . . . . . 3-18

EstimationofAttitudeErrorStates. . . . . . . . . . . 3-22Summary . . . . . . . . . . . . . . . . . . . . . . . . . 3-23

4. Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . 4-14.1 Sortie Overview . . . . . . . . . . . . . . . . . . . . . . 4-14.2 General Filter Comments . . . . . . . . . . . . . . . . 4-6

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Page4.34.44.54.64.74.84.94.104.114.124.134.14

UnaidedandBaroFilterCases . . . . . .Different Cases . . . . . . . . . . . . . . .PerfectMeasurementCase . . . . . . . . .Nominal Case . . . . . . . . . . . . . . . .IndividualAxisPositionComparison . . .PositionStateErrorComparison . . . . .Velocity Comparison . . . . . . . . . . . .VelocityStateErrorEstimateComparisonVisual Residuals . . . . . . . . . . . . . .RSensitivity Analysis . . . . . . . . . . .

. . . . . . . 4-6

. . . . . . . 4-7

. . . . . . . 4-9

. . . . . . . 4-10

. . . . . . . 4-12

. . . . . . . 4-16

. . . . . . . 4-19

. . . . . . . 4-20

. . . . . . . 4-24

. . . . . . . 4-28ErrorCorrelationwithAngularMeasurements . . . . . 4-32Summary . . . . . . . . . . . . . . . . . . . . . . . . . 4-36

5. Conclusions and Recommendations . . . . . . . . . . . . . . . .5.15.2

AppendixA.A.1A.2A.3A.4A.5A.6A.7

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . 5-1Recommendations . . . . . . . . . . . . . . . . . . . . 5-1Flight Test Methodology . . . . . . . . . . . . . . . . . A-1Overview . . . . .SortieBreakdown .PriortoFlight . .Pre-flightandTaxiTakeoff . . . . . .

. . . . . . . . . . . . . . . . . . . . A-1

. . . . . . . . . . . . . . . . . . . . A-1

. . . . . . . . . . . . . . . . . . . . A-1

. . . . . . . . . . . . . . . . . . . . A-2

. . . . . . . . . . . . . . . . . . . . A-2Navigation Portion of Sortie . . . . . . . . . . . . . . . A-2Lessons Learned . . . . . . . . . . . . . . . . . . . . . A-3

A.7.1 Ground Test . . . . . . . . . . . . . . . . . . . A-3A.7.2 StationKeepingDuringCirclingNavigationMa

neuver . . . . . . . . . . . . . . . . . . . . . . A-4

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PageA.7.3 ThingsDoNotAlwaysWorkTheWayTheyAre

Supposed To Work . . . . . . . . . . . . . . . A-4A.7.4 SelectedTarget(s)BeingVisibleToCamera . A-5

A.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . A-6Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIB-1Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VITA-1

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Figure1.1.2.1.3.1.3.2.3.3.3.4.3.5.4.1.4.2.4.3.4.4.4.5.4.6.

4.7.4.8.4.9.4.10.4.11.4.12.

ListofFigures PagePEEPING TALON T-38 . . . . . . . . . . . . . . . . . . . . 1-13Kalman Filter Cycle . . . . . . . . . . . . . . . . . . . . . . . 2-8Sign Convention . . . . . . . . . . . . . . . . . . . . . . . . . 3-9D i s t a n c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-11Camera Calibration Target . . . . . . . . . . . . . . . . . . . 3-20Azimuth Error Surface . . . . . . . . . . . . . . . . . . . . . 3-21Ground Track . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2Altitude for Entire Sortie . . . . . . . . . . . . . . . . . . . . 4-2Ramp Area and Navigation Orbit . . . . . . . . . . . . . . . 4-4GPS Filter Navigation Orbit . . . . . . . . . . . . . . . . . . 4-5Navigation Orbit for Unaided Filter . . . . . . . . . . . . . . 4-53DPositionErrorforUnaidedFilter(top)andBaroFilter(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8Case13DPositionErrorforBaro/VisFilter . . . . . . . 4-9Case 1 3D Position Error for Vis Filter . . . . . . . . . . . 4-9Case 2 3D Position Error for Baro Filter . . . . . . . . . . 4-10Case23DPositionErrorforBaro/VisFilter . . . . . . . 4-11Case 2 3D Position Error for Vis Filter . . . . . . . . . . . 4-11NorthPositionErrorforUnaidedFilter(top),BaroFilter(middle),andBaro/VisFilter(bottom). (DashedLinesIndicate1-Filter-CompensatedCovariance) . . . . . . . . . . . . . . 4-13

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Figure Page4.13. EastPositionErrorforUnaidedFilter(top),BaroFilter(mid

dle),andBaro/VisFilter(bottom). (DashedLinesIndicate1-Filter-CompensatedCovariance) . . . . . . . . . . . . . . 4-14

4.14. DownPositionErrorforUnaidedFilter(top),BaroFilter(middle),andBaro/VisFilter(bottom). (DashedLinesIndicate1-Filter-CompensatedCovariance) . . . . . . . . . . . . . . 4-15

4.15. State 1 (East) Position Error for Unaided Filter (top), BaroFilter(middle),andBaro/VisFilter(bottom) . . . . . . . . . 4-17

4.16. State 2 (North) Position Error for Unaided Filter (top), BaroFilter(middle),andBaro/VisFilter(bottom) . . . . . . . . . 4-18

4.17. NorthVelocityErrorforUnaidedFilter(top),BaroFilter(middle),andBaro/VisFilter(bottom). (DashedLinesIndicate1-Filter-CompensatedCovariance) . . . . . . . . . . . . . . 4-21

4.18. EastVelocityErrorforUnaidedFilter(top),BaroFilter(middle),andBaro/VisFilter(bottom). (DashedLinesIndicate1-Filter-CompensatedCovariance) . . . . . . . . . . . . . . 4-22

4.19. DownVelocityErrorforUnaidedFilter(top),BaroFilter(middle),andBaro/VisFilter(bottom). (DashedLinesIndicate1-Filter-CompensatedCovariance) . . . . . . . . . . . . . . 4-23

4.20. State 4 (North) Velocity Error for Unaided Filter (top), BaroFilter(middle),andBaro/VisFilter(bottom) . . . . . . . . . 4-25

4.21. State 5 (East) Velocity Error for Unaided Filter (top), BaroFilter(middle),andBaro/VisFilter(bottom) . .

4.22. VisualResidualsfortheBaro/VisFilter . . . . .4.23.

3D

Position

Error

for

Cases

1through

4

.

.

.

.

.

4.24. 3D Position Error for All Cases . . . . . . . . . . 4.25. Case2CostvsBenefitforBaro/VisFilter . .4.26. NorthPositionErrorandAngularMeasurements4.27. EastPositionErrorandAngularMeasurements .4.28. NorthandEastPositionError . . . . . . . . . .

. . . . . . . 4-26

. . . . . . . 4-27

. . . . . . .

4-29

. . . . . . . 4-30

. . . . . . . 4-31

. . . . . . . 4-33

. . . . . . . 4-34

. . . . . . . 4-35

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Table3.1.3.2.4.1.4.2.4.3.A.1.

ListofTables PageNEDFrameElevationandAzimuthSense . . . . . . . . . . . 3-2CameraFrameElevationandAzimuthSense . . . . . . . . . 3-9Filter Configurations . . . . . . . . . . . . . . . . . . . . . . . 4-3Measurement Cases . . . . . . . . . . . . . . . . . . . . . . . 4-7Additional Measurement Cases . . . . . . . . . . . . . . . . . 4-28Modular Aircraft Components . . . . . . . . . . . . . . . . . A-4

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ListofAbbreviationsAbbreviation PageAFIT Air Force Institute of Technology . . . . . . . . . . . . . . . . . 1-1TPS USAF Test Pilot School . . . . . . . . . . . . . . . . . . . . . . . 1-1PVATPosition,Velocity,Attitude,andTime . . . . . . . . . . . . . . 1-2FLIR Forward-Looking Infrared . . . . . . . . . . . . . . . . . . . . . 1-3HUD Heads Up Display . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3LORAN Long-range Navigation . . . . . . . . . . . . . . . . . . . . . .

1-3

RADAR Radio Detection and Ranging . . . . . . . . . . . . . . . . . . 1-3TACAN Tactical Air Navigation . . . . . . . . . . . . . . . . . . . . . 1-3PVA Position, Velocity and Attitude . . . . . . . . . . . . . . . . . . . 1-4HF High-Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6ATC Air Traffic Control . . . . . . . . . . . . . . . . . . . . . . . . . . 1-6AWACSAirborneWarningandControlSystem . . . . . . . . . . . . . 1-7JDAM Joint Direct Attack Munition . . . . . . . . . . . . . . . . . . . 1-8JASSMJointAir-to-SurfaceStandoffMissile . . . . . . . . . . . . . . 1-8JSOW Joint Stand-Off Weapon . . . . . . . . . . . . . . . . . . . . . . 1-8AFFTC Air Force Flight Test Center . . . . . . . . . . . . . . . . . . . 1-12EKF Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . 2-4Rn Meridian Radius of Curvature . . . . . . . . . . . . . . . . . . . . . 2-9Re Transverse Radius of Curvature . . . . . . . . . . . . . . . . . . . . 2-10NED North - East - Down . . . . . . . . . . . . . . . . . . . . . . . . . 3-1DTED Digital Terrain Elevation Data . . . . . . . . . . . . . . . . . . 3-2DCM Direction Cosine Matrix . . . . . . . . . . . . . . . . . . . . . . 3-12SI Special Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . A-2RCA Radio Corporation of America . . . . . . . . . . . . . . . . . . . A-3

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ListofSymbolsSymbol PagexState Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5PState Covariance Matrix . . . . . . . . . . . . . . . . . . . . . . . . 2-6Qd Dynamics Driving Noise Matrix . . . . . . . . . . . . . . . . . . . 2-7State Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 2-7h(x) Nonlinear Measurement Matrix . . . . . . . . . . . . . . . . . . . 2-7H

Linearized Measurement Matrix . . . . . . . . . . . . . . . . . . . .

2-7

RMeasurement Noise Matrix . . . . . . . . . . . . . . . . . . . . . . . 2-7

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AFIT/GE/ENG/03-10Abstract

InertialnavigationsystemsandGPSsystemshaverevolutionizedtheworldofnavigation. Inertial systems are incapable of beingjammed and are the backboneofmostnavigationsystems. GPS ishighlyaccurateover longperiodsoftime,anditisanexcellentaidtoinertialnavigationsystems. However,asamilitaryforcewemust be prepared to deal with the denial of the GPS signal. This thesis seeks todetermine

if,

via

simulation,

it

is

viable

to

aid

an

INS

with

visual

measurements.

Visualmeasurementsrepresentasourceofdatathatisessentiallyincapableofbeing

jammed,andassuchtheycouldbehighlyvaluableforimprovingnavigationaccuracyinamilitaryenvironment.

The simulated visual measurements are two angles formed from the aircraftwith respect to a target on the ground. Only one target is incorporated into thisresearch. FivedifferentmeasurementcombinationswereincorporatedintoaKalmanfilterand comparedto each other overa six-minute circular navigation orbit. Themeasurement combinations included were with respect to the navigation orbit: 1)GPS signals throughout the orbit, 2) no measurements during the orbit, 3) simulatedbarometricmeasurementsduringtheorbit,4)simulatedbarometricandvisualmeasurementsduringtheorbit,5)andvisualmeasurementsonlyduringtheorbit.

ThevisualmeasurementswereshowntosignificantlyimprovenavigationaccuracyduringaGPSoutage,decreasingthetotal3-dimensionalerroraftersixminuteswithoutGPSfrom350mto50m(withvisualmeasurements). Thebarometric/visualmeasurementformulationwasthemostaccuratenon-GPScombinationtested,withthe pure visual measurement formulation being the next best visual measurementconfiguration. TheresultsobtainedindicatevisualmeasurementscaneffectivelyaidanINS.Ideasforfollow-onworkarealsopresented.

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1. IntroductionThis thesis is the culmination ofacourseof studyat the AirForce Institute

ofTechnology(AFIT) in navigation theory, alongwith a yearat USAF Test PilotSchool (TPS). The academic portion dealt with the incorporation of visual measurementsintoanavigationKalmanfilter. ThisworkwasperformedatAFIT.Theactual flight test took place within the TPS curriculum. The flight test (projectPEEPINGTALON [2])supportedtheAFITresearchandalsoa limitedevaluationof a camera system on the T-38 aircraft (the aircraft was specifically modified forthistest). Certaindifficultiesledtotheuseofsimulatedvisualmeasurementsratherthanmeasurementsfromthecollectedimages.1.1 Background

Thisresearchisintheareaofnavigation,whichistheartandscienceofmoving fromoneplacetoanother. Thetypeandqualityofsystem(s)useddeterminesthe accuracy with which the navigationoccurs. Many differentsystems have beenusedfornavigation. Twoofthesesystemsareconsideredmainstaysofmodernnavigation. Thesesystemsaretheinertialnavigationsystem(INS)[18]andtheGlobalPositioningSystem(GPS) [18:371]. Thisresearch investigatesthe incorporationofanewtypeofvisualmeasurementwiththeaforementionedtwonavigationsystems.TheINSandGPSsystemsaredescribedinlimiteddetailinSections1.1.1and1.1.2,whilethevisualmeasurementsandtheireffectsarediscussedinmuchgreaterdetail,astheyarethethrustofthisresearch.

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1.1.1 InertialNavigation Systems. Inertial systems sense specific forceand

rotation.

These

force

and

rotation

measurements

are

combined

with

knowledge

about the Earths rotation and gravitational field and can be used to determinevehicularmovement.

ThemostimportantcomponentsinanINSareaccelerometers,gyroscopesanda computer. Theaccelerometers andgyroscopesare sensorswhich providespecificforceandangularvelocitymeasurements,respectively. Thesemeasurementsareusedbythecomputertogeneratevehicleposition,velocity,attitudeandtime(PVAT)inachosencoordinatereferenceframebyusingthenecessarynavigationequationsforthat frame. The computer is used for required computations and to track vehiclemovement. Inertialsystemshavebecomevitalaircraftsystemstiedtotheefficientmission performance of civil and military aircraft. In many cases they would beunable to perform their missions at all without an INS. These inertial navigationsystemsareveryeffectiveandoffermanyadvantages.

One of the main benefits of inertial systems is their passive nature. Theyemitnodetectablesignatureandare incapableofbeingjammed. However, inertialsystemsrequireseveralkeyitemstobeeffective. Thefirstitemisaccuratepositioninformation during alignment. Any position error during alignment will grow overtime(andthereisalwaysapositionerrorbetweenthatenteredintotheINSandtheactualposition). Thisrateofgrowthisassociatedwiththequalityofthegyroscopes.Alargealignmenterrorwithhigh-qualitygyroscopesmightactuallybebetterthana small alignment error with low-quality gyroscopes. These systems also require alengthy alignment time. If both of these requirements are not met, even the mostaccurate INScanbe, orbecome, worthless. Thequalityof theaccelerometers andgyroscopesareanothersourceoferror. Eventhehighestqualitysensorsarepronetosomeerror. Alloftheseerrorsintegrate,andanINSsystemwilldriftawayfromitsactualpositionovertime. TheINSdriftproblembroughtabouttheneedtoupdatethe INS during long flights. The updates provided by these outside, additional

1-2


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measurements,areusedtoimproveINSaccuracybybounding,ordamping,theerrorgrowth

inherent

to

the

INS.

However,

some

of

these

measurements

are

cumbersome

toobtainornotalwaysavailable. Someofthesetypesofupdatemeasurementsare[6:293-294]:

1. Altitudesensors(a) BarometricAltimeter(b) RadarAltimeter

2. Forward-LookingInfrared(FLIR)3. GPS4. HeadsUpDisplay(HUD)5. Long-RangeNavigation(LORAN)6. RadioDetectionandRanging(RADAR)7. StarTrackers8. TacticalAirNavigation(TACAN)9. Overflyingaknowngeographicalpoint

Historically, new types of measurements have been integrated into the INSsolutionastheyhavebecameavailable. OneofthebestmeasurementsforaidinganINSwasprovidedbyGPS.

1.1.2 GPSSystems. TheGlobalPositioningSystem isasystemofsatellitesinmediumearthorbitthattransmitssignalstoreceiversontheEarthssurface,orflyingintheearthsatmosphere,oreveninspace. Thesereceiversarecapableofdeterminingtheirownpositionifatleastfoursatellitesarevisibletothatparticularreceiver. GPSsystemsdonotrequireanyalignmenttime(otherthanthatrequiredtolockontothesatellitesignals)sincetherearenointernalmechanicalcomponents.

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GPS ishighly accurate, doesnotdrift overtime, and isalmost continuously avail-able

for

many

typical

applications.

Therefore,

GPS

is

avery

promising

new

type

of

measurementforintegrationwithanINS.Thiswasmainlyduetotheerrorcharacteristics of the GPSbeing beneficial in limiting INSweaknesses (most notably theINSerrorsgrowingordriftingovertime).

1.1.3 Integrated INS/GPSSystems. It was seen above that INS systemshavestrengthsandweaknesses,asdoGPSsystems. INSsystemsareveryaccurateovershortperiodsoftime(i.e.,theyaresensitivetohighfrequencydynamics),andtheirerrorcharacteristicsgrowslowlyovertime. The INS iscapableofgeneratingvehicle body orientation (which is very important in aircraft and submarines forwhich outside visual references may be unavailable). GPS systems are not highlyaccurate over short periods of time, but they are over long periods of time. Theyarenotcapableofgeneratingvehiclebodyorientation(unlessmultipleantennasareusedalongwithsomeadditionalprocessing). Luckily,INSweaknessesarecompensatedforbyGPSstrengths,andGPSweaknessesarecompensatedbyINSstrenths.Asaresult, thesesystemsareoften integratedtogethertotakeadvantageofthesestrengths and reduce their individual weaknesses [6:342]. This is typically accomplishedviaaKalmanfilter(seeChapter2foradiscussiononKalmanfilters). Thereare different ways to integrate INS and GPS [13], but the method used in this re-search is the only method that will be described in this thesis. The method thatis used is a tightly-coupled integration. The GPS measurement is a pseudorangemeasurementfromeachvisiblesatellite. TheINSgeneratesitsownestimateofthissamepseudorangemeasurement. Thesetwomeasurementsaredifferencedtoobtaina measurement, which isthenthe input intothe errorstate Kalmanfilter. Thismethod of integration offers two advantages. First, even a single satellite can beused to update position, velocity and attitude (PVA), rather than the four beingrequiredfortheGPSitselftodetermineitsownposition(suchaGPS-alonesolutionisrequiredinanalternativeintegrationmethodknownasloose-coupling). Second,

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can negatively impact two important areas. First, lack of radar coverage over theworlds

oceans

causes

aircraft

deconfliction

to

be

implemented

by

assigning

aircraft

toflypre-determinedroutesatspecificaltitudes. Thisisperformedviavoiceordatacommunicationoverhigh-frequency(HF)radiowaves. Thereisnowaytodetermineif an airplane is where it is supposed to be, other than its own on-board systems.An aircraft could easily be misplaced on thejet route as it unknowingly reports afaultylocationtoairtrafficcontrol(ATC).Theaircraftcouldevenbecompletelyoffthejetroute. Thesecondarea impactedbyGPSsignal loss iscoastalpenetration.Aircraft

coasting

in

to

another

country

are

required

to

pass

over

certain

areas

so

theycanbecontrolledandde-conflictedwithotherairtraffic. AnaircraftusingonlyanINScouldeasilycoastinmilesfromtheintendedlocation.

1.1.5.2 HowGPS Loss Affects Tactical Airlift. Tactical airliftersdont often travel the same distances strategic airlifters do, but they suffer fromtheirownparticularnavigationproblems. AircraftliketheC-130andotherspecialoperations aircraft are often required to transit areas in a clandestine manner atnight,whileofferingminimalemissionstotheoutsideworld. Thisdifficultyissome-times compounded by having to perform this mission over featureless expanses ofwateror desert, whileretainingtheneedtoarriveataparticular locationwithouterror. INS-only systems cannot typically perform this type of mission ifthe flyingtimeapproachesevenonehour.

1.1.5.3 HowGPSLossAffectsTacticalFighters. Tacticalfightersareoftenrequiredtotakeoffatnight,flyoverenemyterritorywithoutbeingdetected,attacktargetsandreturnhomewithoutbeingshotdown. Aircraftofthistypeoftenflyunderhighg-forces,inextremeattitudes,andtheseflightconditionscanchangevery rapidly. Such an environment can make it even more difficult for an inertialsystemtooperateaccurately. PVATisextremelyimportanttothesetypesofaircraft,astheyareoftenrequiredtopassthis informationtomunitionsbeforetheycanbe

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releasedfromthehostaircraft. Errorsintheaircraftsnavigationsolutionnaturallytranslate

directly

to

aweapon.

1.1.5.4 HowGPS LossAffectsDatalink Systems. Datalink is be-

comingmuchmore predominant, andthis areaofferssomeveryuniquechallenges.Aircraftarebecomingcapableoftransmittingandreceivingveryusefulinformationwhilein-flight. Thisinformationcomesinmanyforms:

1. Wingmanlocation2. Radardatatoincluderadarlockandweaponfly-outinformation3. Targetlocation4. Downedpilotlocation5. Groundthreats6. Friendlyorneutralgroundpersonnel

Someofthis informationrequiresveryaccuratepositiondata; otherwise it isnot only useless, it can lead aircrew into potentially dangerous situations or geographicalareas. Oneparticularareaofconcernisthegeographiccorrelationofdata.This is an extremely complex task, and this must often occur between dissimilarplatforms. Accuratelypassingcoordinatesbetweenaircraftisonebasisforcorrelatingtwoseparatesourcesofdataintoone. Thiscorrelationcanincreasetheaccuracyoftheinformationknownaboutthetargetaircraft(position,velocity,heading,radarparameters, etc). Acertain amount of navigation error could causethe previouslymentioned correlation to not occur. For example, a hostile contact reported byan airbornewarningandcontrolsystem(AWACS)aircraft to afighteraircraftviadatalink might not properly resolve with the fighter aircrafts own radar detectionofthesamehostilecontact ifoneoftheaircrafthasnavigationerrors. Thiswouldcause both contacts to appear on the fighter aircraft radar scope. This can causea great deal of confusion, andcausethe fighteraircraftto waste time, energy, and

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evenweaponsonafalsetarget. Thisfalsetargetcouldevenpotentiallydrivetacticaldecisions.

There

are

many

other

similar

issues

related

to

the

datalink

area.

It

is

safe

tosaythatnavigationerrorscausesignificantproblemsfordatalinksystems.

1.1.6 HowGPSDependenceAffectsMunitions. ManyofthenewweaponsbeingdevelopedfortheUSAirForce(USAF)arealsoheavilyGPS-dependent. Theseweaponsfillamuch-needednichethatcannotbefilledbythehighlyaccurate laserguidedbombs(LGB)already inthe inventory. LGBsarenotall-weatherweapons.Thetargetmustbevisibletothelaserdesignator,andmustbevisibletotheweapona pre-determined time and distance from the target. This time and distance islargelydependentonthetypeofweaponandthegeometryoftheattack. ThenewGPSweaponsareall-weatherweaponsthatarenotrestrictedbycloudcover, highhumidity,orotherproblemsassociatedwithLGBs. Manyofthesenewsystemsarematedwithlow-costinertialsystems. Therefore,thesesystemsrequireeitherashortoperatingdurationtokeeptheINSerrorsfromdrifting,ortheymustbeintegratedwith GPS. The problem occurs when the flight duration is long and/or the GPSsignalislost. Insuchcases,theinertialerrorsgrowrapidlyuntilthePVAsolutionisnolongerusable. SomeofthesenewmunitionsaretheJointDirectAttackMunition(JDAM) [12], JointAir-to-SurfaceStandoffMissile(JASSM) [10]andJointStand-offWeapon(JSOW) [11]. Someoftheseweaponsareessentiallypointweapons, inthattheyneedtohitveryclosetothetargettobeeffective. A lossofGPSduringtheweaponfalltimecouldcausetheweapontobecompletelyineffectivebymissingthe target. This miss distance need not be very large for a point weapon to loseeffectiveness. Evensomeoftheareamunitionscouldpotentiallyhitfarenoughawayfromtheirintendedimpactpointtobeineffective. AnewtypeofvisualmeasurementmighthelpalleviatesomeoftheproblemsassociatedwithGPSsignallossbykeepingtheINSfromdriftingduringalongfalltime. Visualmeasurementscouldalsobeusedwith a target recognition system to refine the desired impact point as the weaponapproachesatarget.

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1.2 Problem tobeSolved1.2.1 DoVisualMeasurementsEnhanceNavigationAccuracy? Ithasbeen

shown that INS/GPS systems are vulnerable to loss of the GPS signal (especiallythosesystemsoutfittedwithlow-costinertialequipment). Manyofthesesystemsareequippedwith,orcouldbeequippedwith,visualsensorsofsometype. ThisvisualequipmentcouldbeusedtoprovidemeasurementstothenavigationKalmanfiltertoupdatethenavigationsolution. Thesemeasurementscould increaseaccuracyatalltimes(withandwithoutGPSbeingavailable),butcouldproveinvaluableduringtimesofGPSsystemoutages. Asystemofthistypecouldalsobeusedtodeterminetargetcoordinatespassively. Thisresearchseekstodeterminetheanswerstothesequestions.

Specifically,simulatedvisualmeasurementswillbeincorporatedintothenavigation Kalman filter, with and without GPS, and compared to a truth source todetermine if the inclusion of the visual measurements increases navigation systemperformance. Thesesimulatedmeasurementswillbegeneratedduringaportionofthesortiewhentheaircraftisorbitingaroundaparticulartargetarea. Thisissothesametargetareawillbeavailabletogeneratemeasurementsthatdonotdisappearfromview.

1.3 ScopeThisresearchisanavigationthesisandnotasignalprocessingone. Therefore,

noeffortwasmadetoobtainvisualmeasurementsviadigitaltechniques. Therearemethodsavailable,buttheycanbedifficultandtimeconsumingtoimplement. Someofthesemethodswillbebrieflyreviewed intheLiteratureReviewsection(Section1.4). Actual visual measurements were extracted even though simulated data wasused in the analysis. A simple hand/eye technique was used to capture this data.This data is available, and it will be incorporated into the filter at a later time.

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The data was post-processed, due to the increased burden imposed by real-timeprocessing.

1.4 LiteratureReviewThere is an abundance of research regarding navigation, motion estimation

using visual techniques, and a combination of the two topics. Much of the latterresearchislimitedtogroundnavigationorimageidentificationandisfundamentallydifferentthanthatproposed inthiswork. Also,muchoftheresearch is focusedonthe optical area only, and does not incorporate inertial systems, GPS or Kalmanfiltertheory. Thetermnavigation, as it isused inmanyofthesearticles, referstothe ability to move around in some autonomous fashion, but does not refer to thetypeofnavigationusedhere,i.e.,generatingPVATviaanaided-INS.

1.4.1 NavigationUsingImagingTechniques. Currenttheorybreaksmotionestimation intotwoparts [9,15],determineing inter-frame imagemotionandusingimagemotiontoestimatecameramotion. So,inter-frameimagemotionistypicallydetermined by either the optic flow constraint equation (Equation (1.1)) or sometypeofcorrespondence(matching)technique. Someestimateofmotionfromframeto frame is generated via the first step, and this information is used inthe secondsteptobackoutthemotionofthecamera. WhetherEquation(1.1)oramatchingtechniqueisused,theendprocessgeneratesmeasurementsthatcanbeincorporatedintoaKalmanfilter. Thisresearchwillimplementasimplisticmatchingtechnique.

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1.4.1.1 OpticFlowConstraintEquationTechniques. Thefirsttypeofinter-frame

motion

estimation

technique

relies

upon

solving

the

optic

flow

constraint

equation,Equation(1.1)[19]:

E =E (1.1)

twhere:EisintensityEisthegradientoftheintensity isobjectvelocityOptic flow is the apparent motion of brightnesspatterns inan image [1, 19]. Thismethod generates a motion vector for each element in the image. Some of themethods reviewed produced vectors that were wildly inaccurate and unusable dueto camera shock, vibration and inherent inadequacies in the optic flow estimationtechnique. Thesecondtypeof inter-framemotionestimationtechniquesarecorrespondencetechniques.

1.4.1.2 CorrespondenceBasedTechniques. Thistechniqueisamatchingtypeofsolution,anditalsousesapparentmotionofimagebrightnesspatterns.However, motion vectors are not generated for each image element. Objects areidentifiedinoneimageandlocatedagainintheprecedingimages. Thedisplacementbetweenobjectsovertimeisusedtoestimatetheinter-framevelocity,whetheritbetranslationaland/orrotationalvelocity.

1.5 MethodologyThe original idea was to use a simple technique on a sequence of images to

generateangularinformationforincorporationintotheKalmanfilter. Theaccuracyincreaseduetothesemeasurementswastobedeterminedexperimentallyusingdataobtained during the flight test phase of project PEEPING TALON at TPS. Thiswould be accomplished by comparing system accuracy with visual measurements

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Figure1.1 PEEPINGTALONT-381.7.3 T-38AAircraft. AT-38Aaircraft,tailnumber65-10325,wasmod

ified to support the PEEPING TALON test program. The test system includednavigation, video, and recording components. It consisted of a GPS-Aided Inertial Navigation Reference system (GAINR), an Ashtech GPS receiver, two model71A video cameras, a liquid crystal display (LCD) cockpit monitor, and a digitalrecordingsystem.

1.7.4 GAINR. TheGAINRsystemconsisted ofan EmbeddedGPS/INS(EGI),aPre-FlightPanel(PFP),aCockpitControlPanel(CCP),andanIntelligentFlash-memory Solid State Recorder (IFSSR). The GAINR recorded raw Inertial

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Measurement Unit (IMU) data (v and ) at a 256 Hz rate and a blended EGIposition

solution

at

a64

Hz

rate

to

an

internal

data

card.

1.7.5 AshtechGPSReceiver. The Ashtech Z-Surveyor was a 12-channel,

dualfrequencyGPSreceiverwithabuilt-inbatteryandremovablePCmemorycard.ThisreceiverwascapableofprovidingrawGPSsignaldataontheinternaldatacardforpost-processing. Thesedatawerestoredata2Hzrate. AsecondAshtechGPSreceiverwasusedtocollectstationarydatafordifferentialGPSdatapurposes.

1.7.6 Camera. Both the forward and side cameras were Model 71A BallAerospacecameras. Theywereruggedized all light level, high-performance, GatedIntensifiedChargeCoupledDevicemonochromecameras. Thesecamerasuseda3rdgeneration intensifiercoupledtoa768(H)x484(V)pixel interlinechargecoupleddevice imagesensor. Theanalogvideooutputcontained525 linesatafieldrateof60Hz,aframerateof30Hz,andwith2:1positiveinterlace. Thecamerasoperatedfrom28voltsdirectcurrentwithapowerconsumptionoflessthan6watts. Cameradimensionswere5x2.5x2inches(LxHxW),andeachweighed1.1pounds.

1.7.7 Camera Lenses. The collection of lenses used had three differentfields-of-view. Thesewerestandardtypevideocameralenses,inthattheywerenotoptimized for night video even though the PEEPING TALON test collected videounderlow-lightconditionsusinglow-lightcameras. TheairbornevideorecorderwasaSonyDigitalVideoCassetteRecorder,modelGV-D300.

1.7.8 VideoTapes. The tapes used in the airborne video recorder werePanasonicmini-DVME80/120cassettes. RecordingwasperformedintheSPrecordmodeasthiswasthehighestimageresolutionmode.

1.7.9 TimeCodeGenerator/Inserter. TheGPStimecodegenerator,Datum model 9390-3000, received the GPS signal from the externally mounted GPS

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antennaandconverted ittoadigital format forthevideotimecode inserter. Thevideo

time

code

inserter,

Datum

model

9559-835,

received

GPS

data

from

the

GPS

timecodegeneratorandthevideosignalfromtheselectedcamera. Thevideotimecodeinsertercombinedthesetwodatastreamstoprovideasingleoutput,whichwasavideopicturewithGPStimeinthebottomrighthandcornerofthevideoimage.ThisenabledthetestteamtodeterminethepreciseGPStimewheneach frameofvideowasrecorded.

1.7.10 VideoMonitor. The video monitor for the front seat was a 5-in.Transvideo International Rainbow II LCDMonitor. Themonitor wasmounted onthefrontcockpitglareshield,andwasrotatedrightapproximately40degrees(inaclockwisedirection). Thevideoselectorswitchlocatedinthefrontcockpitcontrolledwhichcamerawasdisplayedonthemonitorandrecordedonthevideorecorder.

1.8 SummaryThe previous was a broad overview of the work contained in the thesis ef

fort and the flight test to collect the data. Basic navigation and aiding conceptswere presented along with the problem at hand and the strategy for solving theproblem. WherethisthesisfitsintotheoverallAFITandTPSeducationprogramswasdiscussed,aswellasashort literaturereview,projectassumptions, andmate-rial/equipment intended foruse insolvingtheproblem. Thegroundwork isnow inplacetobeginwithprojectspecificsbeginningwithChapter2,BackgroundTheory.Thespecific design of thevisualmeasurementsare presented inChapter 3. Chapter4coverstheresultsobtainedduringtheresearchaswellasananalysisofthoseresults. Chapter5wrapsuptheresearchandlaysoutfollow-onworkthatcouldbeaccomplished. AppendixAcoverstheactualflighttestthatoccurredatUSAFTestPilotSchool.

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2.Background

Theory

2.1 Overview

This chapter covers theory that is referenced in Chapter 3. Kalman filterbasics are covered briefly, along with geometry relating to the shape of the Earth,thearclengthequation,othernecessarygeometryequations,andadescriptionofthetechniqueusedtogeneratetheflightprofile forthenavigationorbit. ThenotationusedthroughoutthisworkthatpertainstotheKalmanfilteristakenfromthethree-course Stochastic Estimation and Control sequence at AFIT which uses the textswritten by Dr Peter Maybeck. Vectors are bolded lower-case while matrices areboldedupper-case[6,7].

2.2 KalmanFilterBasics2.2.1 KalmanFilter. AKalman filtertakesadynamicmodelofthesys

tem,measurements,deterministicinputs,andaddsinstatisticaldescriptionsoftheuncertaintiesineachoftheseandblendsthemtogetherwithreal-worlddatatogeneratethebestpossibleestimateofcertainvariablesofinterest. LinearKalmanfilterequationsareusedandpresentedbelowwithabriefexplanation. Thisresearchen-counterssomenonlinearities inthemeasurements,andthemethodofdealingwiththis problem is also presented. Refer to [6, 7, 17] for a more lengthy derivation ofKalmanfilters.

2.2.2Kalman

Filter

Equations.

The

system

model

in

this

research

is

a

linear, time-invariant, discrete time system in the form of Equation (2.1), wherex(ti) is an (n x 1) state process vector,(ti, ti1) is an (n x n) system transitionmatrix,Gd(ti1) is an (n x s) noise input matrix, andwd(ti1) is an(sx 1) whiteGaussiannoisevector[3:2-2].

x(ti) =(ti, ti1)x(ti1) +Gd(ti1)wd(ti1) (2.1)2-1


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The discrete-time dynamics driving noise vectorwd(ti1) is assumed to be white,Gaussian,

with

amean

and

covariance

described

by

[3:2-3]:

E[wd(ti)]=0 (2.2)

Qd forti =tj

TE[wd(ti)wd(tj)]= (2.3) 0forti=tj

Nonlinear

measurements

are

available

and

are

modelled

by

Equation

(2.4),

where

z(ti)isan(mx1)measurementvector,h[x(ti), ti]isan(mx1)nonlinearmeasurementmodelvector,andv(ti)isan(mx1)whiteGaussianmeasurementnoisevector[3:2-6]

z(ti) =h[x(ti), ti] +v(ti) (2.4)Thediscrete-timenoisevectorv(ti)isassumedtobewhite,Gaussian,withameanandcovariancedescribedby[3:2-6]:

E[v(ti)]=0 (2.5)R(ti)forti =tj

E[v(ti)vT(tj)]= (2.6) 0forti=tj

Kalmanfilterequationscanbebrokenupintothreemajorcategories: initialcondition information, filterpropagation, and filter update. Equations for each of thesecategoriesarepresentedbelow.

2.2.2.1 KalmanFilter InitialConditionEquations. Equation (2.7)providesthebestinformationavailableabouteachofthestatesatthebeginningoffilteroperation[6:217],whileEquation(2.8)describestheinitialcovariancematrix,

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whichprovidesthebest informationavailable abouttheuncertainty ineachofthestates

at

the

beginning

of

filter

operation

[6:217].

x(t0) =E{x(t0)}=x0 (2.7)

P(t0) =E{[x(t0) x0]T}=P0 (2.8)x0][x(t0)2.2.2.2 KalmanFilterPropagationEquations. Equation(2.9)isused

topropagatethestatesforward intimefromthe initialconditionsorthe lastmeasurement update, whichever is the case. This continues until a measurement isavailableforincorporationintotheKalmanfilter[3:2-4].

x(ti ) =(ti, ti1) x(t+i1) (2.9)

Equation (2.10) is used to propagate the state covariance forward in time fromthe initial conditions or the last measurement update, whichever is the case. Thiscontinuesuntila measurement isavailable for incorporation intotheKalmanfilter[3:2-4].

P(ti ) =(ti, ti1)P(t+i1)T(ti, ti1) +Gd(ti1)Qd(ti1)GTd(ti1) (2.10)

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2.2.2.3 Kalman FilterUpdate Equations. The Kalman filter gainequation

(Equation

(2.11))

is

used

to

weight

ameasurement

for

incorporation

into

the

Kalmanfilter. Therelativeuncertaintybetweentheoptimalestimatesofwhatthestatesshouldbeandtheactualmeasurementsdeterminestheirrespectiveweighting[6:259].

K(ti) =P(ti )HT[ti, x(ti )]P(ti )HT[ti,x(ti )][H[ti, x(ti )]+R(ti)]1 (2.11)

wheretheH[ti,x(ti )]matrixisgeneratedby[3:2-10]:

H[ti,x(ti )]= h[x, ti] (2.12)x=x x(t

i )

The state update Equation (2.13) is used to update each of the states whenever ameasurementisavailable[3:2-10],whileEquation(2.14)isusedtoupdatethestatecovariancematrixeachtimeameasurementisavailable[3:2-10].

x(ti ), ti]] (2.13)x(t+i ) =x(ti) +K(ti)[zih[P(t+i ) =P(ti )K(ti)H[ti,x(ti )]P(ti ) (2.14)

2.2.3 ExtendedKalmanFilter. AnExtendedKalmanFilter(EKF) incorporates nonlinearities in the system model, the measurements, or both, and is anextensionoftheaforementionedKalmanfilter(hencethename). Thebasic idea isto linearize the nonlinear measurements, or dynamics models, about a redeclarednominaltrajectoryorpointatcertainintervalssolinearKalmanfilterconceptscanbeused. Thisisvalidifthenonlinearityisnottoogreatand/oriflinearizationerrorsremainsmall[7:41].

Asstatedpreviously,Kalmanfiltersworkbestwhenusingrawmeasurementsas opposed to measurements thatare pre-processed. Pre-processing measurements

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createstime-correlationinthemeasurementnoises,andthismustberesolved. Oneway

to

deal

with

this

problem

is

to

incorporate

measurements

with

enough

time

lapse

betweenepochsthattheassortedmeasurementnoisesarenolongertime-correlated.Anothermethodistomodelthetime-correlatednoisewithalinearsystemdrivenbywhiteGaussiannoise[6:183]. Otherwise,oneofthebasicKalmanfilterassumptionsisviolated,i.e.,alinearsystemdrivenbywhiteGaussiannoise[6:8].

2.2.4 StateVector. AKalmanfilterisusedtoestimatevariablesofinterest.These variables are contained in the (n x 1) dimension state vector (x) where thecaret symbol is called a hat and indicates the variable under the symbol is an

estimate and not the actual value. Therefore, the state vectorx is an estimate oftheactualstatevectorx. Theninthe(nx1)referencedaboveisthenumberofstatesbeingestimated. Thestatevectormaybetheactualvariablesof interest(adirect,ortotalstatefilter). Theseactualvariablesmightbepositionandvelocityofavehicle. Foranerrorstate(or indirect)filter,theerrors intheactualvariablesofinterestareestimated. Thesemightbetheerrorinpositionandvelocityofavehicle[6:294]. Navigation Kalmanfilterstypicallyuse an errorstate implementation dueto the high dynamic rates possible in aviation systems. These high dynamic rateswouldrequireaveryhighsamplerateforadirectstatefilter. However,theerrorsinthistypeofsystemgrowslowlyandthisallowsanerrorstateimplementationandamuchlowersamplerate[6:291-297].

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t+i1 ti t+i timeti1 ti

Figure2.1 KalmanFilterCyclefrequentlythanthat). Ifmeasurement incorporationonceperminuteproducesacceptablenavigationperformance,thenamuchfasterupdateratemaybeabandonedtoreduceunnecessarycomputationalloading.

Figure2.1showsatypicalKalmanfiltercycle[6:207]. Therearetwodifferenttimeepochsshown(ti1 isanarbitraryepochandti isoneepoch later). Thestateis represented at different times by the filled-in circles. The superscript - and +represent thetimejust before, andjust after, a measurement update, respectively.Thecircleatt+i1 representsthestatejustafteranupdateisperformedattimeti1.The

state

is

then

propagated

via

the

system

dynamic

equations

to

time

ti,

and

is

representedbyti indicatingthestateimmediatelypriortoupdateincorporation. Asbefore,thestateatt+i representsthestatejustafteranupdateisperformedattimeti. Thisrepresentsonlyonecycle;inrealitythisprocessisrepeatedmanytimes,buttheunderlyingconceptremainsunchanged.

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2.3 ArcLengthEquationThearclengthequationisusedrepeatedlyduringthisresearch. Thisequation

statesthatthelengthofanarc(onacircle)canbefoundbymultiplyingtheradiusby the angle (in radians). The radius used here is either the transverse radius ofcurvature(Re)orthemeridianradiusofcurvature(Rn)[16:305-306].

arclength=r (2.17)where:r=radius=angle(inradians)

2.3.1 MeridianRadius ofCurvature. The meridian radius of curvature(Rn)isusedtodeterminetheapproximateradiusoftheEarthatthegivenlatitudeforalineofconstantlongitude[18:54]:

R(1

e

2

)Rn = 3 (2.18)[1e2sin2(L)]2where:R=Earthsradiusattheequatore=EarthsellipticityL=AircraftLatitude

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2.3.2 TransverseRadiusofCurvature. Thetransverseradiusofcurvature(Re)

is

used

to

determine

an

equivalent

Earth

radius

at

a

given

line

of

constant

latitude[18:54]:Rcos(L)

Re = 1 (2.19)[1e2sin2(L)]2

where:R=Earthsradiusattheequatore=EarthsellipticityL=AircraftLatitude

ThedistancegeneratedbyusingRe inconjunctionwiththearclengthequationdoes not change when a change occurs in latitude. However, the conversion fromdistancetolongitudinalangle(degreesorradians)doeschangewhenachangeoccursinlatitude. Thisisduetolinesoflongitudeconvergingtogetherastheymoveawayfromtheequator. Thisexplainsthecos(L)terminthenumeratorofEquation(2.19).

2.4 GeometryBothtangentandinversetangentfunctionsareusedrepeatedlyinthisresearch.

Thepartialderivativeoftheinversetangentwillalsoberequired,andforthisreasonitwillbereviewedbelow.

2.4.1 DerivativeofInverseTangentFunction. Thepartialderivativeofaninverse tangent function is not a trivial calculation. However, it will be needed togenerate the visual measurementH matrix, and therefore it will be demonstratedbelow. The lower case variables a, b, and c are all functions of at least one ofthe states. In this formulation, the variable c represents the measurement to beincorporated,thevariablearepresentsdistance,andthevariablebrepresentsanotherdistance.

ac=tan1 (2.20)

b

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Takingthetangentofbothsidesoftheequationeliminatesthearctangentfunctionand

simplifies

the

right

side

of

the

equation.

a a

tan(c) =tan tan1 = (2.21)b b

Then take the partial derivative of both sides of the equation with respect to thestates(x).

atan(c)= (2.22)

x x bExpandingthelefthandsideofEquation(2.22)removesthetangentfunctionfromthe partial derivative which simplifies the equation and sets the equation up for asubstitutioninthenextstep:

c c atan(c) =sec2(c) = (1 +tan2(c)) = (2.23)

x x x x baRecallingfromEquation(2.21)thattan(c) =b,thisvalueissubstitutedintoEqua

tion(2.23),tosimplifythelefthandsidebyfurtherremovingthetangentfunction:a2c a2c a

1 + = 1 + = (2.24)b x b2 x x b

cThen,solveforthepartialderivativeofx:

c b2 a= (2.25)

x a2 +b2 x bFinally,afterexpandingthepartialderivativeportionontherightofEquation(2.25)

candcancellingtheappropriateterms,thefinalformforx isobtained:

a b a bc b2 x (b)(a) x x (b)(a) x (2.26)= =x a2 +b2 b2 a2 +b2

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2.5 NavigationOrbitGeometryDeterminationThe navigation orbit was designed so that a particular target area would be

withinthecameraFOVcontinuouslyforatenminuteperiod. Acircularorbitwasusedinanefforttomakethisaseasyaspossibleforthepilot.

2.5.1 Assumptions. Therewereseveralsimplifyingassumptionsmade:1. Nowind. Thiswassosimplegeometricequationscouldbeusedtodetermine

anapproximateairspeedandorbitsizeforthenavigationorbit.2. Aside-mountedcamerapointsouttherightsideoftheaircraft. Assuming it

pointsdowntherightwinglineisanapproximationsincethecamera-to-bodyframerotationisnotknown. Thisapproximationisaccurateenoughforflyingthenavigationorbitsincethepilotwillnotbeabletoflytheexactprofile.The first assumption caused some difficulties while the second did not. The

difficultiesencounteredwillbeaddressedlaterinthelessonslearnedsectionintheappendices.2.6 Summary

Thischaptercoveredsomebasictheorythat isneededinChapter3. Kalmanfilterbasicswerecoveredbriefly,alongwithgeometryrelatingtothearclengthequationandothernecessarygeometricrelationships. Adescriptionoftheassumptionspertainingtotheflightprofileforthenavigationorbitwasalsopresented.

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3. Methodology3.1 Overview

The thrust of this research is the incorporation of a newtype of visual measurement into a navigation Kalman filter. This measurement is formulated in twoparts,eachofwhichisanangle. Themeasurementsasimplementedinthisresearchweresimulated;however,amethodforgeneratingthemeasurementsusingacamerais presented at the end of this chapter. The chapter is broken up into five majorparts: 3.2)VisualMeasurements,3.3)VisualMeasurementGeneration,3.4)VisualMeasurementErrorEstimation,3.5)VisualMeasurementGenerationusingaCam-era,and3.6EstimationofAttitudeErrorStates. Theselasttwosectionsextendtheresearch to using an actual camera, even though one was not implemented in thisthesis.

3.2 VisualMeasurementsEach visual measurement is made up of two different parts azimuth and

elevationangles. Theseanglesaremeasuredfromtheaircrafttothetargetrelativeto the North - East - Down (NED) navigation frame. The angles used here arereferenced from the Down vector and not out the nose of the aircraft (the wayazimuthandelevationaretypicallyreferredto inanAir-to-Airradar). Thismeansapointonthegrounddirectlyundertheaircraftwouldbe0degreeselevationand0 degrees azimuth. Elevation and azimuth angles increase from 0 as the point ofinterest moves away from a point directly under the aircraft. The sense for thesemeasurementsislistedinTable3.1.

The previously mentioned NED navigation frame is a local-level frame. AsdefinedfortheNEDframe,elevationisalongalineofconstantlongitudeandazimuthisalongalineofconstantlatitude. Aircraftaltitudeisknown,andtargetaltitudeisassumedknown. Thisassumptionisconsideredreasonableinlightofdigitalterrain

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Table3.1 NEDFrameElevationandAzimuthSenseTarget

Direction

from

AircraftinNEDFrame Elevation Azimuth

North + n/aSouth - n/aEast n/a +West n/a -

elevationdata(DTED)beingreadilyavailable. Thisinformationisnotincorporatedintothisresearch,butthedataisavailableandcouldbeincludedatalaterdate.

3.3 VisualMeasurementGenerationThe visual measurements are formed according to the geometry between the

aircraftandthelocationoftheselectedtarget. Thevisualmeasurementsweregeneratedinthemannerdescribedbelow.

Atvisualmeasurement incorporationtime,avectorwasformedfromtheair-craftlocationtoaknowntargetlocation,expressedintheNEDframe. Thisknowntargetlocationwasusedwith,andwithout,errorsbeinginjectedinthetargetlocation(byaddingerrors intothestoredangle valuesastheyare used). Aircraftandtarget latitude, longitude and altitude were used to form this vector. No error isshown for tgtlat, tgtlon, or tgtalt. It is acknowledged that there would be errors inthese values, but this added complexity is not dealt with in this formulation. SeeChapter5forrecommendationsextendingtheresearchinthisarea.

Thedifferencebetweenthetarget latitudeandthe latitudeoftheaircraft(inradians)iscalculatedby:

lat=tgtlatacftlat lat (3.1)

where:latisthedifferencebetweenaircraftandtargetlatitude(radians)

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tgtlat isthetargetlatitude(radians)acftlat istheaircraftINS-reportedlatitude(radians) latisthefilter-estimatedaircraftlatitudeerror(radians)(acftlat + lat)isthebestestimateofthetruelatitudeoftheaircraft

Next,thedistancecorrelatingtothelatfromthepreviousstepiscalculatedusingthe arc length equation, with a radius defined by Equation (2.18). This will thenbe the distance from the aircraft to the target projected along a line of constantlongitude. This projected distance is the North component of a vector from theaircrafttothetargetintheNEDframe.

eldist =Rnlat=Rn(tgtlatacftlat lat) (3.2)

The difference between the target longitude and the longitude of the aircraft (inradians)iscalculatedby:

lon=tgtlonacftlonlon (3.3)

where:lonisthedifferencebetweenaircraftandtargetlongitude(radians)tgtlon isthetargetlongitude(radians)acftlon istheaircraftINS-reportedlongitude(radians)

lonisthefilter-estimatedaircraftlongitudeerror(radians)(acftlon +lon)isthebestestimateofthetruelongitudeoftheaircraft

Finally, the distance correlating to the lon from the previous step is determinedusing the arc length equation. This is the distance from the aircraft to the targetprojected along a line of constant latitude. This projected distance is the East

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componentofavectorfromtheaircrafttothetargetintheNEDframe.azdist =Relon=Re(tgtlonacftlonlon) (3.4)

Thedifferencebetweenthetargetaltitudeandthealtitudeoftheaircraftiscalculatedslightly differently in keeping with the sign convention listed in Table 3.1 for theNorthandEastdirectionsintheNEDframe. North ispositive,Eastispositive,soaccordingtotheright-handrule,Downispositive. Thiscalculationisgivenby:

altdist =acftalt +alttgtalt (3.5)

where:altdist isthedifferencebetweenaircraftandtargetaltitude(meters)acftalt istheaircraftINS-reportedaltitude(meters)altisthefilter-estimatedaircraftaltitudeerror(meters)tgtalt isthetargetaltitude(meters)(acftalt +alt)isthebestestimateofthetruealtitudeoftheaircraft

The aircraft-to-target vector in the NED frame is formed by the North, East, andDowncomponentsgeneratedinthepreviousthreesteps.

eldist

Rnlat

Rn(tgtlatacftlat lat)

n

n = azdist = Relon= Re(tgtlonacftlonlon) (3.6)

altdist altdist acftalt +alttgtalt

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Finally,theNEDframevectorisusedtodeterminetheelevationandazimuthanglesfrom

the

aircraft

to

the

target

via

Equations

(3.7)

and

Equations

(3.8):

Elevation=tan1 eldist (3.7)

altdist

Azimuth=tan1 azdist (3.8)altdist

3.4 VisualMeasurementErrorEstimationDuringfilteroperation,simulatedmeasurementsandfilterpredictionsofthose

samemeasurementsaredifferencedandusedasmeasurementsinthefilter. Thefilterpredictsthemeasurementsbytakingtheequationsusedtogeneratetheestimatedmeasurementsto formtheh(x)vector. Thenonlinearh()vector isthenusedtogeneratethelinearizedH(x)matrix. Thisisdonebytakingthepartialderivativeofh()withrespecttoeachstateinthefilter.

x)Matrix. The first row of theH(3.4.1 Generating theH( x) matrixcorrespondswiththeazimuthmeasurements,whilethesecondrowcorrespondswiththeelevationmeasurements. ThefullypopulatedH(x)willlooklikeFigure3.9.

az az 0 0 0 0 0 0 0 0 0

x) = azrow= lon 0 H( alt (3.9)el row 0 el el 0 0 0 0 0 0 0 0 0

lat alt

Generationofthefirstelementoftheazimuthrow isshownbelow inSection3.4.1.1. Equation(2.25) isrepeatedhere,as it isneededtogenerateeachofthe12componentsoftheH(x)matrixfirstrow:

a a btan1b x (b)(a) x= (3.10)

x a2 +b2

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3.4.1.1 AzimuthRowGeneration. TherightsideofEquation(3.11)must

be

calculated

with

respect

to

each

of

the

twelve

states

in

the

Kalman

filter

statematrix. Theendresultforazimuthwillpopulaterow1ofEquation(3.9). Theelementsthatarezeroareduetothepartialderivativesallbeingzero. This isduetotherightsideofEquation(3.11)notbeingafunctionofanyofthosestates.

tan1 azdistaltdist componentoftheH(First,the x)matrixwillbecalculated(elementlon

forrow1,column1). ThegenericequationisgivenbyEquation(3.11):

tan1 azdist azdist altdist

altdist statei(altdist)(azdist)

statei=statei az2 dist (3.11)dist +alt2

Thegenericequationbecomesmorespecificwhenthepartialderivativeistakenwithrespecttoaparticularstate:

tan1 azdist azdist (altdist)(azdist) altdist altdist lon lon

=lon az2 dist (3.12)dist +alt2

Theequation isthenbestbroken intotwoparts. Thepartialderivativeonthe leftsideofthenumeratorisshownhere:

azdist = [Re(tgtlonacftlonlon)]=Re (3.13) lon lonwhilethepartialderivativeontherightsideofthenumeratoris:

altdist = (acftalt +alttgtalt)=0 (3.14) lon lon

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Theresultsofthetwopartialderivativecalculationsarethensubstitutedback intoEquation

3.12,

which

can

now

be

solved

in

this

more

simplistic

form:

tan1 azdist altdist (Re)(acftalt +alttgtalt)(Re[tgtlonacftlonlon])(0)

= lon (Re[tgtlonacftlonlon])2 + (acftalt +alttgtalt)2

(3.15)whichreducesto:

tan1 azdistaltdist Re(acftalt +alttgtalt)

= (3.16)

lon

(Re[tgtlonacftlonlon])2 + (acftalt +alttgtalt)2TheremainingthreeelementsfromEquation3.9arecalculatedinalikeman

ner,andtheresultsareshownbelow. Intermediatecalculationsareomitted,becausetheyaresimilartothe tan1 azdist examplegivenabove.lon altdist

tan1 azdist altdist Re(tgtlonacftlonlon)= (3.17)

alt (Re[tgtlonacftlonlon])2 + (acftalt +alttgtalt)2

tan1 eldistaltdist Rn(acftalt +alttgtalt)

= (3.18)lat (Rn[tgtlatacftlat lat])2 + (acftalt +alttgtalt)2

altdist Rn(tgtlatacftlat tan1 eldist lat)= (3.19)

alt (Rn[tgtlatacftlat lat])2 + (acftalt +alttgtalt)23.5 VisualMeasurementGenerationusingaCamera

Section3.5 isnotactually implemented inthisthesis. It ispresentedheretosupportfollow-onwork.

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3.5.1 RequiredCamera Specifications. Certain camera specifications arerequired

to

relate

camera

frame

geometry

to

navigation

frame

geometry.

Camera

focallength(f)mustbeknown,andoneofthefollowingmustbeknowntodeterminethe width and height of the viewed image. This information is used to determinetheverticalandhorizontalpixelsize.

1. Verticalandhorizontalfield-of-view(f ovvc andf ovch respectively),wherethevandh subscriptsareforverticalandhorizontal,respectively. Thesuperscriptc indicatesthevariableisinthecameraframe.

2. Ifthefields-of-viewarenotknown,verticalandhorizontalcoverage(covvc andcovc respectively)canbeusedtogeneratetheseanglesby:h

covcc vf ovv = 2tan1 (3.20)2fcovccf ovh = 2tan1 h (3.21)2f

Then,theverticalandhorizontalpixelsizecanbecalculatedby:covcvpixelv = (3.22)

verpixelcountcovchpixelh = (3.23)

horpixelcount3.5.2 GenerateTargetLatitude/Longitude. Thecamerageometry isused

to translate target pixel location into a geometrical reference. This can then berotatedintothebodyframeandultimatelyintotheNEDnavigationframe.

Targets in an image are selected, and this selection generates pixel locationsfor eachtarget. These pixel locations are referenced from the center of the image.

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Thisvectorissho

n= nn

The camera frameframe. Thiscamera

Table

Figure3.1 SignConventionVectorn isgenerated inafashionsimilartothatshow inFigure3.2,andcanthenbeformedintoa3Dvectorinthecameraframebymultiplyingbytheproperpixelsize to get a distance in the camera frame. Knowledge of the camera focal length

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Figure3.2 Distancesprovidesthethirddimension. Equation(3.5.2)now formsavectortothetarget(s)inthecameraframe. Thisiscapturedbythefollowingequation:

(pixelv)(nx)

nc

x

c n = (pixelh)(ny)= nc (3.25) y f f

Cameraerrorsduetodeficiencies intheopticsmustnowberemovedsotheabovevectorpointstotheactualtargetlocationasopposedtotheskewedlocationshownonthephoto. Thecameraerrorsgeneratedduringthecameracalibrationprocedurediscussed in Section 3.5.7 are stored in an angular format, so vectornc is used togenerate theuncorrectedazimuth(azuncorr)(Equation(3.26)) andtheuncorrectedelevation (eluncorr) (Equation (3.27)). Figure 3.3 graphically demonstrates theseangles. Theyaretheazimuthandelevationangleswithopticalerrorsstillpresent.

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Figure3.3 Angles

nccazuncorr =tan1 y (3.26)f

ncxel

c=tan1 (3.27)uncorr f

These angles will become azimuth in the camera frame (azc) and elevation in thecameraframe(elc)oncetheerrorsareremovedviaEquations(3.28)and(3.29). Thevalues for the optical azimuth error (opticalazerror) and the optical elevation error(opticalelerror)arepre-computedduringthecameraalignmentprocedure(seeSection3.5.7).

c caz =azuncorropticalazerror (3.28)elc =elcuncorropticalelerror (3.29)

Thecorrectedazc andelc mustbeconvertedbacktoavectorsotargetinformationcanberotatedfromthecameraframe,throughthebodyframe(viaEquation(3.31)),

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tothenavigationframe(viaEquation(3.32)). (f)(tan(elc)) nc x

c y (VectorinCameraFrame) (3.30)n = (f)(tan(azc))= nc

f fThisisaccomplishedbypre-multiplyingthecameraframevectorbythecamera-to-bodydirectioncosinematrix(DCM)(Ccb)inEquation(3.31):

b cn =Ccbn (VectorinBodyFrame) (3.31)Thenthebodyframevectorispre-multipliedbythebody-to-navigationframeDCM(Cb

n)totransformthevectorintothenavigationframe:nn =Cbnnb (VectorinNavFrame) (3.32)

Theazn (Equation(3.33))andeln (Equation(3.34))anglescanthenbedeterminedfromthecomponentsofvectornn. TheseanglesareNEDframeangles:

nnazn =tan1 y (3.33)

nnz

nnxeln =tan1 (3.34)nnz

The next step is to determine target longitude and latitude. Longitude will becalculatedbyfirstdeterminingEarthradius(Re)alongalineofconstantlatitudeatthecurrentaircraftlatitude. Thenazn isusedtodeterminedistancealongalineofconstantlatitude(azdist)fromaircraftlongitudetotargetlongitude:

azdist = (altdist)tan(azn) (3.35)

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Then the arc length equation is used to determine the radian change in longitudefrom

the

aircraft

to

the

target:

azdist

lon= (3.36)Re

Targetlongitudecanthenbecalculatedbytherelationship:tgtlon = lon+acftlon +lon (3.37)

Asimilarprocedure isusedtodeterminetarget latitudebyfirstdeterminingEarthradius(Rn)alonga lineofconstant longitudeatthecurrentaircraft latitude. ThisisanalogouswithRe above,exceptthechangeinlatitudeisalongalineofconstantlongitudewhichisanellipseratherthanacircle. Useeln todeterminedistancealongalineofconstantlongitude(eldist)fromaircraftlatitudetotargetlatitude:

eldist = (altdist)tan(eln) (3.38)

eldist isthencombinedwiththearclengthequationtodeterminetheradianchangeinlatitudefromtheaircrafttothetarget:

eldistlat= (3.39)

Rn

Targetlatitudecanthenbedeterminedbytherelationship:tgtlat = lat+acftlat + lat (3.40)

3.5.3 GenerateMeasurementsfromanImage. TheprocedureforgeneratingmeasurementsfromanimageisexactlythesameasinSection3.5.2,steps3.5.2through 3.5.2. The remaining steps in Section 3.5.2 need not be accomplished to

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generatemeasurementsbecausetargetlatitude/longitudeisestimatedthefirsttimethe

target

was

incorporated.

3.5.4 GenerateMeasurement Estimates. The procedure for estimating

measurements is verysimilar to thatofSection3.5.2, except theprocedure is performed backwards. First, azn is determined from the estimate of target longitude

byre-arrangingEquation(3.37)togetthechangeinlongitudefromthecurrentINSlongitudetothetargetlongitude:

lon=tgtlonacftlonlon (3.41)

ThenEquation(3.36)isre-arrangedtogettheestimatedazdist:azdist =Relon (3.42)

Similarly,eln canbedeterminedfromtheestimateoftargetlatitudebyre-arrangingEquation (3.40) to getthe change in latitude fromthe current INS latitude tothetargetlatitude:

lat=tgtlatacftlat lat (3.43)ThenEquation(3.39)ismanipulatedtogettheestimatedeldist:

eldist =Rnlon (3.44)

The previously determined azdist and eldist can be combined with altitude data toformthevectortothetargetinthenavigationframe:

Rnlat nn eldist x

nn = Relon = nn = azdist (3.45) y altitude nn altdistz

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TheCb DCMisusedtorotatethenavigationframevectortothebodyframe:n nb =Cbnn (3.46)n

ThentheCcb DCMisusedtorotatethebodyframevectortothecameraframe: nc =Ccbnb (3.47)

cc Determine estimates of az and el in camera frame from the components of thecameraframevectornc:

nccaz =tan1 y (3.48)

nczc nc xel =tan1 (3.49)

ncz

3.5.5 Camera-to-BodyDCMGeneration. Thecamera-to-bodyDCM(Ccb)is generatedby trackingtargets withsurveyed coordinates. A vectortothe targetisgeneratedinthecameraframeandavectortothetarget isgenerated inthenavframe. The nav frame vector is rotated into the body frame. The two vectors arenowrelatedthroughCbc.

3.5.6 VisualMeasurementRMatrixGeneration. Generating the uncertainty in a measurement is somewhat arbitrary. It should be based on sound reasoningeventhoughthefinalvaluethatworksbestintheactualfiltermaybequitedifferent.

This

value

is

typically

found

by

tuning

the

Kalman

filter.

This

is

a

processinwhichdifferentuncertaintiesinthefilterareadjustedtogeneratethebestpossible filter performance. In a perfect world the measurements would have zeroerror. However,thisisnotthecasesoerrorsthatexistmustbeapproximated. Thisissotheuncertainty inthemeasurementscanbeusedtoweightthemeasurements

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properly. Thefollowingphenomenacauseerrorsinthevisualmeasurementsandwillbe

addressed

individually:

faulty

pixel

selection

and

optical

deficiencies.

Faultypixelselectionisduetotheinabilitytoselecttheexactsamepointona

targeteachtimethattargetisusedtogenerateameasurement. Anerrorofseveralpixels is realistic in such a procedure. This error will inject itself directly into themeasurements as an angle error. The vector generated in Section 3.5.2, Equation(3.24)willcontainthiserrorandinjectitdirectlyintoEquations(3.51)and(3.52).

(pixelv)(nx +nx) nxc +nc x nc = (pixelh)(ny +ny)= nc y (3.50) y +nc

f fnc +nc

c c cazuncorr =azuncorrtrue +azuncorr =tan1 ytrue y (3.51)fnc +ncxtrue xelc =elc +elc =tan1 (3.52)uncorr uncorrtrue uncorr f

Theworstcaseangleerror for faultypixelselecitonoccurswhentheselectedpixel location issupposedtobe intheexactcenterofthe imagebut isoffbysomeerroramount. Atypicalcaseisdemonstratedbelowusingcameraspecsforthecam-erainuseduringalgorithmdevelopmentatAFIT(notusedduringflighttest). Theuncorrected true azimuth (azc uncorrtrue)uncorrtrue) and uncorrected true elevation (elcareeachzerosincetheyaresupposedtobeintheimagecenter. Twopixelsoferroris assumed as a nominal case and 0.0074mm and 0.0073mm are the pixel sizes forazimuthandelevationpixels,respectively:

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c c c cazuncorr = (azuncorrtrue +azuncorr)azuncorrtrue = (3.53)nc +nc nc

=tan1 ytrue y tan1 ytrue (3.54)f f

=tan1 (0.00740)+(0.00742) tan1 0.00740 = (3.55)5.2 5.2

= 0.00280= (3.56)= 0.0028radianerror (3.57)

uncorr = (elc +elc uncorrtrue = (3.58)elc uncorrtrue uncorr)elcnc +nc ncxtrue x xtrue=tan1 tan1 = (3.59)

f f=tan1 (0.00730)+(0.00732) tan1 0.00730 = (3.60)

5.2 5.2= 0.00280= (3.61)= 0.0028radianerror (3.62)

Theerrorsincurredaboveinjectdirectlyintothenexttypeoferror,whichareopticaldeficiencies. Opticaldeficienciesareduetoabnormalitiesinthecameralens,imperfections in camera component alignment, software resident in the camera aswellasotherphysicalcameraphenomenon. AcalibrationprocedureisperformedinSection3.5.7,butthiscannotremovealltheerrorspresent. This isespeciallytruewhentheanglesgeneratedintheprevioussectionhaveerrorsduetoimproperpixelselection.

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cameraissetupaknowndistancefromacalibrationtarget(similartothatshowninFigure

3.4).

The

camera

must

be

carefully

aligned

so

its

focal

plane

is

parallel

to

the

targetorerrorswillbeinjectedintothefinalresultsincethiswillskewthegeometrybetweenthecameraandthetarget. Amis-alignmentofthissortessentiallychangesthe physical geometry of the setup. Since this error is not detected it will pollutetheerrorsurfacebeinggenerated. Thenapictureistakenofthecalibrationtarget.This picture is used to generate the azimuth and elevation angles to each elementin the calibration photo. These angles are then compared with the known anglesthat

are

generated

from

the

known

physical

geometry

between

the

camera

and

the

calibrationtarget. Theazerror andelerror aredefinedinEquations(3.71)and(3.72)respectively:

azerror =aztrueazmeas (3.71)elerror =eltrueelmeas (3.72)

These azimuth and elevation errors generate an error surface for the entirecamerafield-of-view. TheazimutherrorsurfacegeneratedatAFITduringdevelopment isshownbelow inFigure3.5. Thisfiguredescribestheerror inazimuthasafunctionofhorizontalandverticalposition inthe image. Thesurfaceappearsasaplanar image due to the sign convention described previously. The elevation errorsurfaceissimilarinappearance.

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0 5 10 15 20 25 30 350

5

10

15

20

25

30

Horizontal Distance in inches

VerticalDistanceininches

Camera Calibration Target

Figure3.4 CameraCalibrationTarget

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400

200

0

200

400

300

200

100

0

100

200

3000.04

0.03

0.02

0.01

0

0.01

0.02

0.03

0.04

0.05

Horizontal Distance (pixels)

Azimuth Error due to Camera Lens

Vertical Distance (pixels)

AzimuthError(radians)

Figure3.5 AzimuthErrorSurface

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3.6 EstimationofAttitudeErrorStatesThe method for estimating the attitude error states presented below is not

actuallyimplementedinthisthesis. Itispresentedheretosupportfollow-onwork.ThissectionissimilartoSection3.4,inthattheH(x)matrixmustbederived

and calculated. However, the attitude error terms must also be populated in thismatrix,inadditiontothefourtermsalreadydescribed.

Recall that the camera frame vectornc must be rotated through two framestogeneratenn:

nc =CbnCbcnc (3.73)The camera-to-body frame DCM (Cbc) is a fixed DCM in this research. However,the body-to-navigation frame DCM (Cnb) changes every epoch and contains errorsassociatedwiththreeofthestates. ThosestatesareNorthaxistilterror(),Eastaxistilterror(),andDownaxistilterror(). Thismeansthepartialderivativeof the measurements with respect to the states will have more elements that arenon-zero. ThoseelementsareshowninEquation(3.74)withthefirstthreecolumnsbeing the same as presented previously. The incorporation of a camera into thisresearch will certainly cause filter performance to suffer somewhat. How much isto be determined. However, the errors should be small enough, at least initially,thatvisualmeasurementsstillgeneratebetterfilterperformance. Thepointwherethe tilt errors grow too large for visual measurements to be effective is also to bedetermined.

az az 0 0 0 az az az 0 0 0H( alt x) = azrow= lon 0

el row 0 el el 0 0 0 el el el 0 0 0 alt lat

(3.74)TheactualformulationofthisnewH(x)isnotderivedhere,butitshouldbesimilarinderivationtothatfromSection3.4.

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3.7 SummaryThis chapter defined the visual measurements to be incorporated into the

Kalman filter and demonstrated the formulation of these angles. The method ofgeneratingvisualmeasurementerrorswascoveredindetail,aswellasthetheoryforincorporatingvisualmeasurementsfromanactualcamera. Theeffectsofthesimulatedmeasurementsasimplementedinthisthesisareanalyzedinthenextchapter.

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4. ResultsandAnalysis4.1 SortieOverview

SimulatedvisualmeasurementsweregeneratedduringKalmanfilteroperation.Elevationandazimuthanglesweregeneratedata2Hzrateduringthe6-minute27-second navigation portion from time 244995 to time 245382 (GPS week seconds).ThiswasaccomplishedbyrunningtheGPSfilter(seeTable4.1foranexplanationofthedifferentfiltertypes)andgeneratingazimuthandelevationanglesateachimageincorporationtime. Thisdatawasthenstored inadatafile for incorporation intotheKalmanfilterforanalysis.

The sortie chosen for analysis in this research was a daylight flight from Ed-wards AFB to the Channel Islands (off the coast of Los Angeles) and back. Anavigationorbitwasperformedrightaftertakeoff,withtheremainderoftheflightdedicated to the camera evaluation. The ground track for the sortie is shown inFigure 4.1, with the altitude displayed in Figure 4.2. These figures are given todemonstratethesortieprofilevisually. ThefilterdatashownhereincorporatedGPSmeasurementsonly,andnobaroorvisualmeasurementswereused. Thisdatawassaved and used as a truth source. There were other truth sources available (i.e.,Ashtech-only, EGI-only, and a differential GPS solution). The Ashtech-only andEGI-onlysolutionswere lesssmooththanthefilter-generatedtruthsolution. Theirsolutions showed positionjumps on the order of several meters, and for this reason they were discarded as potential truth sources. The differential GPS solutiongeneratedbytheairborneAshtechreceiverand thegroundbasedAshtechreceiverhad data dropouts that made this solution unsuitable. The filter-generated truthdata isnot ideal,but it isappropriatetodemonstratenavigationaccuracy increaseordecreaseasGPSmeasurementsareremoved,andbaroandvisualmeasurementstaketheirplace.

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120 119.5 119 118.5 118 117.533.6

33.8

34

34.2

34.4

34.6

34.8

35

35.2

35.4Position For Nav Orbit, GPS Filter

Longitude (degrees)

Latitude

(degrees)

GPS Measurements Only

NavigationOrbit

Channel Islands

2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.5

x 105

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Altitude For Entire Sortie (meters)

Time (GPS Week Seconds)

Altitude(meters)

GPS Measurements Only

Navigation Orbit

Figure4.2 AltitudeforEntireSortie4-2


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Table4.1 FilterConfigurationsFilterName FilterDescriptionGPSFilter GPSmeasurementsincorporatedbefore,

during,andafterthenavigationorbitUnaidedFilter Nomeasurementsincorporated

duringthenavigationorbitBaroFilter OnlyBaromeasurementsincorporated

duringthenavigationorbitBaro/VisFilter OnlyBaroandVisualmeasurements

incorporatedduringthenavigationorbitVisFilter OnlyVisualmeasurements

incorporatedduringthenavigationorbitUnfortunately,generatingsimulatedvisualmeasurementsusingtheGPSfilter

resultedinminorerrorsinthetruthpositionduetotimetaggingerrors(theGPSmeasurementswerenotsynchronouswiththesimulatedvisualmeasurements). Thisproducedan error of less than one meter fromthe truthpositionusedto generate visualmeasurements. This is acceptable since the best performing filter underanalysishasanapproximateaverageerrorof10meters(underoptimumconditions).There are five different filter configurations that will be discussed. Each filter hasGPSmeasurements incorporatedbeforeandafterthenavigationorbit. Theydifferonlyinthetype,orlackof,measurementsthatareincorporatedduringthenavigationorbit. TheyarelistedinTable4.1,alongwithhowtheywillbereferencedfromthispointforwardinthisthesis.

Figure4.1showsthestartofnavigation intheupperrightcorneralongwiththenavigationorbit. TheChannelIslandoverflightisinthelowerleftcorner. Figure4.2showsthealtitudefortheentiresortie(inmeters). Thefirstlevel-offatapproximately3200metersisthenavigationorbit. Thefigureclearlyshowssomealtitudedeviations during the orbit. This was due to difficulties in flying a constant orbit.Thesealtitudevariationswillbeinvestigatedingreaterdetaillaterinthissection.

Figure 4.3 highlights the navigation orbit area. This orbit was flown at approximately3,300metersabovethegroundandatanairspeedof154meters/second

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117.91 117.9 117.89 117.88 117.87 117.86

34.91

34.92

34.93

34.94

34.95

34.96

34.97

Position For Entire Sortie (degrees)

Longitude (degrees)

La

titude

(degrees)

FilterAshtechEGI

Aircraft Parking Spot

Navigation Orbit

Runway ApproachEnd

Runway Departure End

Figure4.3 RampAreaandNavigationOrbit(300knotsindicatedairspeed). Thisresultedinthenavigationorbitdiameterbeingroughly 5,721 meters (3.1 nautical miles). Figure 4.3 also shows the parking areafor the test aircraft, the ground taxi, both the takeoff and landing, as well as thenavigation orbit. The analysis will concentrate on the navigation orbit. Differentcombinationsofmeasurementswillbeincorporatedduringthisorbitandcomparedwitheachothertodeterminewhichfilteroperatesthebest.

TheorbitisshowninFigure4.4. ThedatawasgeneratedusingtheGPSfilterduringtheentireorbit. Thisisthetruthdata,againstwhichtheUnaidedfilter,theBarofilter,andtheBaro/Visfilternavigationorbitswillbecompared.

It is clear that the GPS filter performs much better than the Unaided filterfromacomparisonofFigures4.4and4.5,respectively. Itisalsoclearthatsomesort

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117.96 117.94 117.92 117.9 117.88 117.86 117.84 117.8234.92

34.93

34.94

34.95

34.96

34.97

34.98

34.99

Longitude (degrees)

L

atitude(

degrees)

Position For Nav Orbit, No Aiding

GPS Outage, No MeasurementsTruthNo Aiding

Figure4.5 NavigationOrbitforUnaidedFilter4-5


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ofaidingisrequiredduringthenavigationorbitGPSoutagetoimprovenavigationalaccuracy.

The

end

portion

of

the

Unaided

filter

orbit,

when

GPS

measurements

are

re-incorporated,iscauseforconcern. WithGPSmeasurementsagainavailable,theUnaided filter operates more poorlythan the Baro or Baro/Vis filter (not shown).Thelargerthedisparitybetweenthefilterestimatesandtheincorporatedmeasurements, the larger the filter gyrations until everything smoothes out. The UnaidedfiltershowsamuchlargerlateralfluctuationthantheBaroorBaro/Visfilters. Thereason forthis is unclearwhenonly looking atFigure 4.5, this is cause for furtherinvestigation.

4.2 GeneralFilterCommentsTheKalmanfilterusedinthisresearchisatwelve-statefilter. Therearesome

minor problems with the implementation that cause the filter to require aiding.These problems appear to reside in the vertical channel. Problems in the verticalchannelarenormalininertialnavigationsystems,butthisfilterimplementationappearstobeworsethannormal. Thesedifficultieswereallowedtoremain,asthefilterwouldbecontinuouslyaided insomefashion,andthenavigationaccuracy increaseordecreasecanstillbedeterminedwhenincorporatingvisualmeasurements. Theseproblemsareshown inFigure4.5wheretherewasnoaidingduringthenavigationorbit. TheUnaidedlateralpositionsolutionclearlydivergesfromthetrueposition.

GPSmeasurementsarere-incorporatedattheendofthe6 12-minutenavigation

orbit. The large lateral fluctuations that are visible at the end of the navigationorbitareduetothelargediscrepancybetweentheGPSmeasurementsandthefilterestimatesofposition.

4.3 UnaidedandBaroFilterCasesThe results of the filters without visual aiding (Unaided and Baro) will be

shownforcomparisonpurposeswiththeBaro/VisandVisfilters. Thethreedimen-

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sional(3D)errorforeachofthesetwofilters isshown inFigures4.6. TheUnaidedfilter

is

clearly

prone

to

very

large

errors.

However,

this

is

mostly

due

to

image aided inertial nav

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