Fundamentals of Inertial Navigation Systems and Aiding

413 Pages·2022·18.492 MB·English

by  Michael Braasch

Fundamentals of Inertial
Navigation Systems and
Aiding
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Fundamentals of Inertial
Navigation Systems and
Aiding
Michael Braasch
TheInstitutionofEngineeringandTechnology
PublishedbySciTechPublishing,animprintofTheInstitutionofEngineeringand
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Contents
Abouttheauthor xiii
Preface xv
1 Introductionandoverviewofinertialnavigationsystems 1
1.1 Asimplethoughtexperiment 1
1.2 Newton’sfirstandsecondlawsofmotion 3
1.3 Inertialreferenceframes 3
1.4 PositioningwithrespecttotheEarth 4
1.4.1 Asimpleexample 4
1.4.2 Goingbeyond 5
1.5 Guidance,navigationandcontrol 5
1.6 Aself-containedsystem? 6
1.7 Fourkeyprocessesininertialpositioning 7
1.7.1 Aslightlylesssimple2Dexample 8
1.7.2 Moregenerally... 8
1.8 Somehistoricalcontext 9
1.8.1 Deadreckoning 9
1.8.2 Keydevelopmentsinthehistoryofinertialnavigation 10
1.9 Modernnavigationsystems 13
1.10 INSandGNSS:madeforeachother 14
1.10.1 Thewayforward... 15
References 15
2 Referenceframesandthemathematicsofrotations 17
2.1 Earth-centeredinertialframe 17
2.2 Earth-centeredEarth-fixedframe 18
2.3 Locallylevelframes 19
2.4 WanderAzimuthframe 21
2.5 Vehicle-fixedbodyframe 21
2.6 Sensorframes 22
2.7 Themathematicsofrotations 22
2.8 Principleaxisrotationmatrices 24
2.9 Multiplerotationsinsuccession 26
2.10 Reversetransformation 27
2.11 Eulerangles:theaerospaceconvention 28
2.12 Quaternions 30
vi Fundamentalsofinertialnavigationsystemsandaiding
2.13 Additionalquaternionequations 33
References 33
3 Inertialprocessinginafixedreferenceframe 35
3.1 Introduction 35
3.2 Accelerationinaninertiallyfixedframe 35
3.3 Theaccelerometersensingequation 36
3.4 Thenavigationequationforafixedframe 39
3.5 Updatingthebody-to-inertialdirectioncosinematrix 41
3.6 Position/velocity/attitudeupdatinginaninertiallyfixedframe 46
3.7 Updatingthebody-to-inertialquaternion 48
References 49
4 Theimpactofpseudo-forcesinthevelocityupdate 51
4.1 Agraphicalderivationofthepseudo-forcesintheEarthframe 51
4.2 Inertialframemechanizationofthenavigationequation 56
4.3 Navigationframemechanizationofthenavigationequation 57
References 59
5 Earthrate,transportrate,gravity,andthevelocityupdate 61
5.1 Earthrateinthenavigationframe 61
5.2 Transportrateinthenavigationframe 63
5.3 Radiiofcurvature 65
5.4 Relationshipbetweentransportrateandpositionanglerates 67
5.5 Gravitymodels 69
5.6 Velocityupdate 71
References 72
6 Spatialrate,navigationframerotation,andpositionupdate 73
6.1 Navigation-frameupdate 73
6.2 Completebody-to-navupdate 75
6.3 Positionupdate 76
6.4 Derivationoftheearth-to-navDCM 76
6.5 Earth-to-navDCMupdate 77
6.6 Altitude 78
6.7 Overallstrapdowninertialnavigationprocessing 79
6.8 AppendixA:Spatial-ratevectorsimilaritytransformation 80
References 82
7 Thewanderazimuthmechanization 83
7.1 Introduction 83
7.2 Derivationofthehorizontalcomponentsoftransport
rateinthew-frame 84
7.3 Theearth-to-wanderdirectioncosinematrix 87
Contents vii
7.4 Alternateformofthehorizontalcomponentsoftransport
rateinthew-frame 88
7.5 Computationofheadingandnavigation-framevelocity
components 89
7.6 Earthrateconvertedtothewanderframe 90
References 91
8 Navigation-gradeinertialsensors 93
8.1 Introduction 93
8.2 Sagnacinterferometer 93
8.3 Ringlasergyro 96
8.4 Fiberopticgyro 99
8.5 Force-feedbackpendulousaccelerometers 100
8.6 Primaryinertialsensorerrors 102
8.7 Noiseandothereffects 104
8.7.1 Othererrorsources 105
References 106
9 Inertialerroranalysis 109
9.1 Introduction 109
9.2 TheSchulerpendulum 110
9.2.1 Pendulumbasics 110
9.2.2 BacktoSchuler 111
9.3 Schuleroscillationsinthehorizontalinertialchannels 113
9.4 Horizontalerroranalyses 115
9.4.1 Analysisofanaccelerometerbias 116
9.4.2 Analysisofagyrobias 119
9.5 Instabilityintheverticalinertialchannel 121
9.6 Sensornoiseandothererrors 125
References 128
10 Inertialerrorsimulation 131
10.1 Introduction 131
10.2 Northaccelerometerbias 131
10.3 Eastgyrobias 134
10.4 Verticalgyrobias 136
10.5 Azimuthinitializationerror 138
10.6 Velocityinitializationerror 138
10.7 Combinationoferrors 143
10.7.1 Whataboutnoise? 145
10.8 Long-durationflight 147
Reference 151
11 Inertialinitialization—PartA 153
11.1 Introduction 153
11.2 Initializationofposition 153
viii Fundamentalsofinertialnavigationsystemsandaiding
11.3 Initializationofvelocity 154
11.4 Initializationofattitude:leveling 154
11.4.1 Coarseleveling 154
11.5 Initializationofattitude:alignment/gyrocompassing 156
11.5.1 Coarsegyrocompassing 157
11.6 Acommentregardingrecursiveestimation 158
11.7 Conclusion 159
Reference 159
12 Introductiontointegrationandestimationtheory 161
12.1 Whyintegrate? 161
12.2 Ahighlevellookatanaircraftflightcontrolsystem 161
12.3 Navigationsystemrequirements 163
12.4 Casestudy:GNSSandINS 164
12.5 Introductiontoestimationtheory:optimality 165
12.6 Optimalestimationwithdataonly:least-squares 171
References 173
13 EstimationtheoryandintroductiontotheKalmanfilter 175
13.1 Casestudy:bucketfulofresistors 177
13.2 TheresistorbucketmeetstheKalmanfilter 178
13.3 Backtothebucket 180
13.4 SummaryoftheKalmanfilterforthebucketful
ofresistorsexample 182
13.5 Theoreticalperformancecomparison 183
13.6 MonteCarlosimulationsoftherecursiveleast
squaresestimator 184
13.7 MonteCarlosimulationsoftheKalmanfilter 186
13.7.1 Thedangerofmismodeling 186
References 191
14 ThemultivariateKalmanfilter 193
14.1 Asimpleradarexample 193
14.2 TheKalmanfilter“SystemEquation”and
“MeasurementEquation” 195
14.3 Covariancematrices 196
14.4 Systemandmeasurementnoisecovariancematrices 198
14.5 Estimationandpredictionerrorcovariancematrices 199
14.6 Thediscrete-timeKalmanfilter 199
14.7 Casestudy1:anagingresistor 201
14.8 Casestudy2:simpleradar 203
References 209