Preface to the Fourth Edition xxv
Acknowledgments xxix
About the Authors xxx
Acronyms xxxi
About the Companion Website xxxix
1 Introduction 1
1.1 Navigation 1
1.1.1 Navigation-RelatedTechnologies 1
1.1.2 NavigationModes 2
1.2 GNSS Overview 3
1.2.1 GPS 4
1.2.1.1 GPSOrbits 4
1.2.1.2 Legacy GPS Signals 4
1.2.1.3 ModernizationofGPS 6
1.2.2 Global Orbiting Navigation Satellite System (GLONASS) 6
1.2.2.1 GLONASS Orbits 6
1.2.2.2 GLONASS Signals 6
1.2.2.3 Modernized GLONASS 7
1.2.3 Galileo 7
1.2.3.1 Galileo Navigation Services 7
1.2.3.2 Galileo Signal Characteristics 8
1.2.4 BeiDou 9
1.2.4.1 BeiDouSatellites 10
1.2.4.2 Frequency 10
1.2.5 RegionalSatelliteSystems 10
1.2.5.1 QZSS 10
1.2.5.2 NAVIC 10
1.3 Inertial Navigation Overview 10
1.3.1 History 11
1.3.1.1 TheoreticalFoundations 11
1.3.1.2 Development Challenges: Then and Now 12
1.3.2 DevelopmentResults 12
1.3.2.1 Inertial Sensors 12
1.3.2.2 SensorAttitudeControl 14
1.3.2.3 Initialization 15
1.3.2.4 Integrating Acceleration and Velocity 15
1.3.2.5 Accounting for Gravity 15
1.4 GNSS/INS Integration Overview 16
1.4.1 TheRoleofKalmanFiltering 16
1.4.2 Implementation 17
Problems 17
References 18
2 Fundamentals of Satellite Navigation Systems 21
2.1 ChapterFocus 21
2.2 Satellite Navigation Systems Considerations 21
2.2.1 Systems Other than GNSS 21
2.2.2 Comparison Criteria 22
2.3 Satellite Navigation 22
2.3.1 GNSS Orbits 23
2.3.2 Navigation Solution (Two-Dimensional Example) 25
2.3.2.1 Symmetric Solution Using Two Transmitters on Land 25
2.3.2.2 Navigation Solution Procedure 27
2.3.3 User Solution and Dilution of Precision (DOP) 28
2.3.4 ExampleCalculationofDOPs 32
2.3.4.1 Four Satellites 32
2.4 Time and GPS 33
2.4.1 Coordinated Universal Time (UTC) Generation 33
2.4.2 GPS System Time 33
2.4.3 ReceiverComputationofUTC 34
2.5 Example: User Position Calculations with No Errors 35
2.5.1 User Position Calculations 35
2.5.1.1 PositionCalculations 35
2.5.2 User Velocity Calculations 37
Problems 39
References 41
3 Fundamentals of Inertial Navigation 43
3.1 ChapterFocus 43
3.2 Terminology 44
3.3 Inertial Sensor Technologies 50
3.3.1 Gyroscopes 50
3.3.1.1 MomentumWheelGyroscopes(MWGs) 50
3.3.1.2 Coriolis Vibratory Gyroscopes (CVGs) 51
3.3.1.3 OpticalGyroscopes(RLGsandFOGs) 53
3.3.2 Accelerometers 53
3.3.2.1 Mass-spring Designs 53
3.3.2.2 Pendulous Integrating Gyroscopic Accelerometers (PIGA) 54
3.3.2.3 Electromagnetic 54
3.3.2.4 Electrostatic 55
3.3.3 Sensor Errors 55
3.3.3.1 Additive Output Noise 55
3.3.3.2 Input-output Errors 55
3.3.3.3 Error Compensation 56
3.3.4 Inertial Sensor Assembly (ISA) Calibration 57
3.3.4.1 ISA Calibration Parameters 58
3.3.4.2 Calibration Parameter Drift 59
3.3.5 Carouseling and Indexing 60
3.4 Inertial Navigation Models 60
3.4.1 GeoidModels 61
3.4.2 Terrestrial Navigation Coordinates 61
3.4.3 Earth Rotation Model 63
3.4.4 Gravity Models 63
3.4.4.1 Gravitational Potential 63
3.4.4.2 Gravitational Acceleration 64
3.4.4.3 Equipotential Surfaces 64
3.4.4.4 Longitude and Latitude Rates 64
3.4.5 Attitude Models 68
3.4.5.1 Coordinate Transformation Matrices and Rotation Vectors 69
3.4.5.2 Attitude Dynamics 69
3.5 Initializing The Navigation Solution 70
3.5.1 Initialization from an Earth-fixed Stationary State 70
3.5.1.1 AccelerometerRecalibration 70
3.5.1.2 InitializingPositionandVelocity 70
3.5.1.3 InitializingISAAttitude 70
3.5.1.4 Gyrocompass Alignment Accuracy 71
3.5.2 Initialization on the Move 73
3.5.2.1 Transfer Alignment 73
3.5.2.2 Initializing Using GNSS 73
3.6 Propagating The Navigation Solution 73
3.6.1 AttitudePropagation 73
3.6.1.1 StrapdownAttitudePropagation 73
3.6.1.2 QuaternionImplementation 78
3.6.1.3 Direction Cosines Implementation 79
3.6.1.4 MATLAB® Implementations 80
3.6.1.5 GimbalAttitudeImplementations 80
3.6.2 Position and Velocity Propagation 82
3.6.2.1 VerticalChannelInstability 82
3.6.2.2 Strapdown Navigation Propagation 82
3.6.2.3 Gimbaled Navigation Propagation 84
3.7 Testing and Evaluation 86
3.7.1 LaboratoryTesting 86
3.7.2 Field Testing 86
3.7.3 Performance Qualification Testing 87
3.7.3.1 CEPandNauticalMiles 87
3.7.3.2 Free Inertial Performance 87
3.8 Summary 89
3.8.1 FurtherReading 89
Problems 90
References 92
4 GNSS Signal Structure, Characteristics, and Information Utilization 93
4.1 Legacy GPS Signal Components, Purposes, and Properties 93
4.1.1 Signal Models for the Legacy GPS Signals 94
4.1.2 NavigationDataFormat 98
4.1.2.1 Z-Count 99
4.1.2.2 GPSWeekNumber(WN) 101
4.1.2.3 InformationbySubframe 101
4.1.3 GPSSatellitePositionCalculations 102
4.1.3.1 Ephemeris Data Reference Time Step and Transit Time Correction 103
4.1.3.2 True, Eccentric, and Mean Anomaly 105
4.1.3.3 Kepler’s Equation for the Eccentric Anomaly 106
4.1.3.4 Satellite Time Corrections 107
4.1.4 C/A-Code and Its Properties 108
4.1.4.1 TemporalStructure 109
4.1.4.2 Autocorrelation Function 110
4.1.4.3 PowerSpectrum 111
4.1.4.4 Despreading of the Signal Spectrum 111
4.1.4.5 RoleofDespreadinginInterferenceSuppression 113
4.1.4.6 Cross-correlation Function 114
4.1.5 P(Y)-CodeandItsProperties 115
4.1.5.1 P-Code Characteristics 115
4.1.5.2 Y-Code 116
4.1.6 L1 and L2 Carriers 116
4.1.6.1 Dual-FrequencyOperation 116
4.1.7 TransmittedPowerLevels 117
4.1.8 FreeSpaceandOtherLossFactors 117
4.1.9 ReceivedSignalPower 118
4.2 ModernizationofGPS 118
4.2.1 BenefitsfromGPSModernization 119
4.2.2 ElementsoftheModernizedGPS 120
4.2.3 L2 Civil Signal (L2C) 122
4.2.4 L5 Signal 123
4.2.5 M-Code 125
4.2.6 L1C Signal 126
4.2.7 GPS Satellite Blocks 128
4.2.8 GPS Ground Control Segment 129
4.3 GLONASS Signal Structure and Characteristics 129
4.3.1 FrequencyDivisionMultipleAccess(FDMA)Signals 130
4.3.1.1 Carrier Components 130
4.3.1.2 SpreadingCodesandModulation 130
4.3.1.3 NavigationDataFormat 131
4.3.1.4 Satellite Families 131
4.3.2 CDMA Modernization 131
4.4 Galileo 132
4.4.1 ConstellationandLevelsofServices 132
4.4.2 Navigation Data and Signals 132
4.5 BeiDou 134
4.6 QZSS 135
4.7 IRNSS/NAVIC 138
Problems 138
References 141
5 GNSS Antenna Design and Analysis 145
5.1 Applications 145
5.2 GNSS Antenna Performance Characteristics 145
5.2.1 SizeandCost 145
5.2.2 Frequency and Bandwidth Coverage 146
5.2.3 Radiation Pattern Characteristics 147
5.2.4 Antenna Polarization and Axial Ratio 149
5.2.5 Directivity, Efficiency, and Gain of a GNSS Antenna 152
5.2.6 Antenna Impedance, Standing Wave Ratio, and Return Loss
5.2.7 Antenna Bandwidth 154
5.2.8 Antenna Noise Figure 155
5.3 Computational Electromagnetic Models (CEMs) for GNSS Antenna Design 157
5.4 GNSS Antenna Technologies 159
5.4.1 Dipole-BasedGNSSAntennas 159
5.4.2 GNSSPatchAntennas 160
5.4.2.1 Edge-Fed, LP, Single-Frequency GNSS Patch Antenna 161
5.4.2.2 Probe-Fed, LP, Single-Frequency GNSS Patch Antenna 163
5.4.2.3 Dual Probe-Fed, RHCP, Single-Frequency GNSS Patch Antenna 165
5.4.2.4 Single Probe-Fed, RHCP, Single-Frequency GNSS Patch Antenna 165
5.4.2.5 Dual Probe-Fed, RHCP, Multifrequency GNSS Patch Antenna 168
5.4.3 Survey-Grade/Reference GNSS Antennas 169
5.4.3.1 ChokeRing-BasedGNSSAntennas 169
5.4.3.2 AdvancedPlanner-BasedGNSSAntennas 171
5.5 Principles of Adaptable Phased-Array Antennas 173
5.5.1 Digital Beamforming Adaptive Antenna Array Formulations 176
5.5.2 STAP 179
5.5.3 SFAP 179
5.5.4 ConfigurationsofAdaptablePhased-ArrayAntennas 179
5.5.5 Relative Merits of Adaptable Phased-Array Antennas 180
5.6 Application Calibration/Compensation Considerations 181
Problems 183
References 184
6 GNSS Receiver Design and Analysis 189
6.1 ReceiverDesignChoices 189
6.1.1 Global Navigation Satellite System (GNSS) Application to Be Supported 189
6.1.2 SingleorMultifrequencySupport 189
6.1.2.1 Dual-FrequencyIonosphereCorrection 190
6.1.2.2 Improved Carrier Phase Ambiguity Resolution in High-Accuracy Differential Positioning 190
6.1.3 NumberofChannels 191
6.1.4 Code Selections 191
6.1.5 DifferentialCapability 192
6.1.5.1 Corrections Formats 193
6.1.6 Aiding Inputs 194
6.2 Receiver Architecture 195
6.2.1 RadioFrequency(RF)FrontEnd 195
6.2.2 Frequency Down-Conversion and IF Amplification 197
6.2.2.1 SNR 198
6.2.3 Analog-to-Digital Conversion and Automatic Gain Control 199
6.2.4 Baseband Signal Processing 200
6.3 Signal Acquisition and Tracking 200
6.3.1 Hypothesize About the User Location 201
6.3.2 Hypothesize About Which GNSS Satellites Are Visible 201
6.3.3 Signal Doppler Estimation 202
6.3.4 Search for Signal in Frequency and Code Phase 202
6.3.4.1 SequentialSearchinginCodeDelay 205
6.3.4.2 Sequential Searching in Frequency 205
6.3.4.3 Frequency Search Strategy 206
6.3.4.4 Parallel and Hybrid Search Methods 206
6.3.5 Signal Detection and Confirmation 207
6.3.5.1 Detection Confirmation 207
6.3.5.2 Coordination of Frequency Tuning and Code Chipping Rate 209
6.3.6 CodeTrackingLoop 210
6.3.6.1 Code Loop Bandwidth Considerations 214
6.3.6.2 Coherent Versus Noncoherent Code Tracking 214
6.3.7 Carrier Phase Tracking Loops 215
6.3.7.1 PLL Capture Range 217
6.3.7.2 PLL Order 218
6.3.7.3 Use of Frequency-Lock Loops (FLLs) for Carrier Capture 218
6.3.8 BitSynchronization 219
6.3.9 DataBitDemodulation 219
6.4 Extraction of Information for User Solution 220
6.4.1 Signal Transmission Time Information 220
6.4.2 Ephemeris Data for Satellite Position and Velocity 221
6.4.3 Pseudorange Measurements Formulation Using Code Phase 221
6.4.3.1 Pseudorange Positioning Equations 222
6.4.4 Measurements Using Carrier Phase 223
6.4.5 Carrier Doppler Measurement 225
6.4.6 Integrated Doppler Measurements 226
6.5 Theoretical Considerations in Pseudorange, Carrier Phase, and Frequency Estimations 228
6.5.1 TheoreticalErrorBoundsforCodePhaseMeasurement 229
6.5.2 Theoretical Error Bounds for Carrier Phase Measurements 230
6.5.3 Theoretical Error Bounds for Frequency Measurement 231
6.6 High-SensitivityA-GPSSystems 232
6.6.1 How Assisting Data Improves Receiver Performance 233
6.6.1.1 ReductionofFrequencyUncertainty 233
6.6.1.2 DeterminationofAccurateTime 234
6.6.1.3 TransmissionofSatelliteEphemerisData 235
6.6.1.4 ProvisionofApproximateClientLocation 236
6.6.1.5 Transmission of the Demodulated Navigation Bit Stream 236
6.6.1.6 Server-ProvidedLocation 237
6.6.2 Factors Affecting High-Sensitivity Receivers 237
6.6.2.1 Antenna and Low-Noise RF Design 238
6.6.2.2 Degradation due to Signal Phase Variations 238
6.6.2.3 Signal Processing Losses 238
6.6.2.4 Multipath Fading 239
6.6.2.5 Susceptibility to Interference and Strong Signals 239
6.6.2.6 The Problem of Time Synchronization 239
6.6.2.7 Difficulties in Reliable Sensitivity Assessment 239
6.7 Software-Defined Radio (SDR) Approach 239
6.8 Pseudolite Considerations 240
Problems 242
References 244
7 GNSS Measurement Errors 249
7.1 SourceofGNSSMeasurementErrors 249
7.2 Ionospheric Propagation Errors 249
7.2.1 IonosphericDelayModel 251
7.2.2 GNSS SBAS Ionospheric Algorithms 253
7.2.2.1 L1 L2 Receiver and Satellite Bias and Ionospheric Delay Estimations for GPS 255
7.2.2.2 Kalman Filter 257
7.2.2.3 Selection of Q and R 259
7.2.2.4 Calculation of Ionospheric Delay Using Pseudoranges 261
7.3 Tropospheric Propagation Errors 262
7.4 The Multipath Problem 263
7.4.1 How Multipath Causes Ranging Errors 264
7.5 MethodsofMultipathMitigation 266
7.5.1 Spatial Processing Techniques 266
7.5.1.1 AntennaLocationStrategy 266
7.5.1.2 GroundPlaneAntennas 266
7.5.1.3 DirectiveAntennaArrays 267
7.5.1.4 Long-Term Signal Observation 267
7.5.2 Time-Domain Processing 269
7.5.2.1 Narrow-Correlator Technology (1990-1993) 269
7.5.2.2 Leading-Edge Techniques 270
7.5.2.3 Correlation Function Shape-Based Methods 271
7.5.2.4 Modified Correlator Reference Waveforms 271
7.5.3 Multipath Mitigation Technology (MMT) 271
7.5.3.1 Description 271
7.5.3.2 Maximum-Likelihood (ML) Multipath Estimation 272
7.5.3.3 The Two-Path ML Estimator (MLE) 272
7.5.3.4 AsymptoticPropertiesofMLEstimators 273
7.5.3.5 The MMT Multipath Mitigation Algorithm 274
7.5.3.6 The MMT Baseband Signal Model 274
7.5.3.7 Baseband Signal Vectors 274
7.5.3.8 The Log-Likelihood Function 275
7.5.3.9 Secondary-Path Amplitude Constraint 277
7.5.3.10 Signal Compression 277
7.5.3.11 PropertiesoftheCompressedSignal 279
7.5.3.12 TheCompressionTheorem 280
7.5.4 Performance of Time-Domain Methods 281
7.5.4.1 Ranging with the C/A-Code 281
7.5.4.2 Carrier Phase Ranging 282
7.5.4.3 Testing Receiver Multipath Performance 282
7.6 Theoretical Limits for Multipath Mitigation 283
7.6.1 Estimation-TheoreticMethods 283
7.6.1.1 OptimalityCriteria 284
7.6.2 Minimum Mean-Squared Error (MMSE) Estimator 284
7.6.3 Multipath Modeling Errors 284
7.7 Ephemeris Data Errors 285
7.8 Onboard Clock Errors 285
7.9 Receiver Clock Errors 286
7.10 ErrorBudgets 287
Problems 289
References 291
8 Differential GNSS 293
8.1 Introduction 293
8.2 Descriptions of Local-Area Differential GNSS (LADGNSS), Wide-Area Differential GNSS (WADGNSS), and Space-Based Augmentation System (SBAS) 294
8.2.1 LADGNSS 294
8.2.2 WADGNSS 294
8.2.3 SBAS 294
8.2.3.1 Wide-Area Augmentation System (WAAS) 294
8.2.3.2 European Global Navigation Overlay System (EGNOS) 298
8.2.3.3 Other SBAS 298
8.3 GEO with L1L5 Signals 299
8.3.1 GEO Uplink Subsystem (GUS) Control Loop Overview 302
8.3.1.1 IonosphericKalmanFilters 302
8.3.1.2 RangeKalmanFilter 304
8.3.1.3 CodeControlFunction 304
8.3.1.4 FrequencyControlFunction 305
8.3.1.5 L1L5 Bias Estimation Function 305
8.3.1.6 Code-CarrierCoherence 306
8.3.1.7 CarrierFrequencyStability 307
8.4 GUS Clock Steering Algorithm 307
8.4.1 ReceiverClockErrorDetermination 309
8.4.2 Clock Steering Control Law 311
8.5 GEO Orbit Determination (OD) 312
8.5.1 ODCovarianceAnalysis 313
8.6 Ground-Based Augmentation System (GBAS) 318
8.6.1 Local-AreaAugmentationSystem(LAAS) 318
8.6.2 Joint Precision Approach and Landing System (ALS) 318
8.6.3 Enhanced Long-Range Navigation (eLORAN) 319
8.7 Measurement/Relative-Based DGNSS 320
8.7.1 CodeDifferentialMeasurements 320
8.7.1.1 Single-Difference Observations 321
8.7.1.2 Double-Difference Observations 321
8.7.2 Carrier Phase Differential Measurements 322
8.7.2.1 Single-Difference Observations 322
8.7.2.2 Double-Difference Observations 323
8.7.2.3 Triple-Difference Observations 323
8.7.2.4 Combinations of Code and Carrier Phase Observations 324
8.7.3 Positioning Using Double-Difference Measurements 324
8.7.3.1 Code-Based Positioning 324
8.7.3.2 Carrier Phase-Based Positioning 325
8.7.3.3 Real-Time Processing Versus Postprocessing 325
8.8 GNSS Precise Point Positioning Services and Products 325
8.8.1 The International GNSS Service (IGS) 325
8.8.2 Continuously Operating Reference Stations (CORSs) 326
8.8.3 GPS Inferred Positioning System (GIPSY) and Orbit Analysis Simulation Software (OASIS) 326
8.8.4 Scripps Coordinate Update Tool (SCOUT) 327
8.8.5 The Online Positioning User Service (OPUS) 327
8.8.6 Australia’s Online GPS Processing System (AUPOS) 328
8.8.7 National Resources Canada (NRCan) 328
Problems 328
References 328
9 GNSS and GEO Signal Integrity 331
9.1 Introduction 331
9.1.1 Range Comparison Method 332
9.1.2 Least-SquaresMethod 332
9.1.3 ParityMethod 334
9.2 SBAS and GBAS Integrity Design 334
9.2.1 SBASErrorSourcesandIntegrity^reats 336
9.2.2 GNSS-Associated Errors 337
9.2.2.1 GNSS Clock Error 337
9.2.2.2 GNSS Ephemeris Error 338
9.2.2.3 GNSS Code and Carrier Incoherence 338
9.2.2.4 GNSS Signal Distortion 338
9.2.2.5 GNSS L1L2 Bias 338
9.2.2.6 Environment Errors: Ionosphere 339
9.2.2.7 Environment Errors: Troposphere 339
9.2.3 GEO-Associated Errors 339
9.2.3.1 GEO Code and Carrier Incoherence 339
9.2.3.2 GEO-Associated Environment Errors: Ionosphere 340
9.2.3.3 GEO-Associated Environment Errors: Troposphere 340
9.2.4 Receiver and Measurement Processing Errors 340
9.2.4.1 ReceiverMeasurementError 340
9.2.4.2 Intercard Bias 340
9.2.4.3 Multipath 341
9.2.4.4 L1L2 Bias 341
9.2.4.5 Receiver Clock Error 341
9.2.4.6 Measurement Processing Unpack/Pack Corruption 341
9.2.5 Estimation Errors 341
9.2.5.1 Reference Time Offset Estimation Error 341
9.2.5.2 Clock Estimation Error 342
9.2.5.3 Ephemeris Correction Error 342
9.2.5.4 L1 L2 Wide-Area Reference Equipment (WRE) and GPS Satellite Bias Estimation Error 342
9.2.6 Integrity-Bound Associated Errors 342
9.2.6.1 Ionospheric Modeling Errors 343
9.2.6.2 Fringe Area Ephemeris Error 343
9.2.6.3 Small-Sigma Errors 343
9.2.6.4 Missed Message: Old but Active Data (OBAD) 343
9.2.6.5 Time to Alarm (TTA) Exceeded 343
9.2.7 GEO Uplink Errors 343
9.2.7.1 GEO Uplink System Fails to Receive SBAS Message 343
9.2.8 MitigationofIntegrityThreats 344
9.2.8.1 MitigationofGNSSAssociatedErrors 344
9.2.8.2 Mitigation of GEO-Associated Errors 346
9.2.8.3 Mitigationof Receiverand Measurement Processing Errors 347
9.2.8.4 Mitigationof Estimation Errors 348
9.2.8.5 Mitigation ofIntegrity-Bound-Associated Errors 348
9.3 SBAS Example 349
9.4 Summary 351
9.5 Future: GIC 351
Problems 352
References 352
10 Kalman Filtering 355
10.1 ChapterFocus 355
10.2 Frequently Asked Questions 356
10.3 Notation 360
10.3.1 RealVectorsandMatrices 360
10.3.1.1 Notation 360
10.3.1.2 Vectorand Matrix Properties 361
10.3.2 Probability Essentials 363
10.3.2.1 Basic Concepts 363
10.3.2.2 Linearity of the Expectancy Operator E(·) 364
10.3.2.3 Means and Covariances of Linearly Transformed Variates 365
10.3.3 Discrete Time Notation 365
10.3.3.1 Subscripting 365
10.3.3.2 A Priori and A Posteriori Values 365
10.3.3.3 Allowing for Testing and Rejecting Measurements 365
10.4 Kalman Filter Genesis 366
10.4.1 MeasurementUpdate(Corrector) 366
10.4.1.1 Linear Least Mean Squares Estimation: Gauss to Kalman 367
10.4.1.2 Kalman Measurement Update Equations 373
10.4.2 Time Update (Predictor) 373
10.4.2.1 Continuous-Time Dynamics 373
10.4.2.2 Discrete-Time Dynamics 377
10.4.3 Basic Kalman Filter Equations 378
10.4.4 The Time-Invariant Case 378
10.4.5 ObservabilityandStabilityIssues 378
10.5 Alternative Implementations 380
10.5.1 Implementation Issues 380
10.5.2 Conventional Implementation Improvements 381
10.5.2.1 MeasurementDecorrelationbyDiagonalization 381
10.5.2.2 Exploiting Symmetry 382
10.5.2.3 Information Filter 382
10.5.2.4 Sigma Rho Filtering 383
10.5.3 James E. Potter (1937-2005) and Square Root Filtering 383
10.5.4 Square Root Matrix Manipulation Methods 384
10.5.4.1 Cholesky Decomposition 384
10.5.4.2 Modified Cholesky Decomposition 385
10.5.4.3 Nonuniqueness of Matrix Square Roots 386
10.5.4.4 Triangularization by QR Decomposition 386
10.5.4.5 Householder Triangularization 386
10.5.5 Alternative Square Root Filter Implementations 386
10.5.5.1 PotterImplementation 386
10.5.5.2 Carlson “Fast Triangular" Square Root Filter 387
10.5.5.3 Bierman-Thornton UD Filter 387
10.5.5.4 Unscented Square Root Filter 388
10.5.5.5 Square Root Information Filter (SRIF) 388
10.6 NonlinearApproximations 388
10.6.1 LinearApproximationErrors 389
10.6.2 Adaptive Kalman Filtering 392
10.6.3 Taylor-Maclauren Series Approximations 392
10.6.3.1 First-Order: Extended Kalman Filter 393
10.6.3.2 Second-Order: Bass-Norum-Schwartz Filter 393
10.6.4 Trajectory Perturbation Modeling 393
10.6.5 Structured Sampling Methods 394
10.6.5.1 Sigma-PointFilters 395
10.6.5.2 Particle Filters 396
10.6.5.3 The Unscented Kalman Filter 396
10.7 Diagnostics and Monitoring 397
10.7.1 CovarianceMatrixDiagnostics 397
10.7.1.1 SymmetryControl 398
10.7.1.2 Eigenanalysis 398
10.7.1.3 Conditioning 398
10.7.2 Innovations Monitoring 398
10.7.2.1 Kalman Filter Innovations 398
10.7.2.2 Information-Weighted Innovations Monitoring 399
10.8 GNSS-Only Navigation 401
10.8.1 GNSSDynamicModels 402
10.8.1.1 ReceiverClockBiasDynamics 402
10.8.1.2 Discrete Time Models 403
10.8.1.3 ExponentiallyCorrelatedRandomProcesses 403
10.8.1.4 Host Vehicle Dynamics for Standalone GNSS Navigation 403
10.8.1.5 Point Mass Dynamic Models 404
10.8.2 GNSS Measurement Models 406
10.8.2.1 Measurement Event Timing 406
10.8.2.2 Pseudoranges 407
10.8.2.3 Time and Distance Correlation 407
10.8.2.4 Measurement Sensitivity Matrix 408
10.8.2.5 Noise Model 408
10.9 Summary 410
Problems 412
References 414
11 Inertial Navigation Error Analysis 419
11.1 Chapter Focus 419
11.2 Errors in the Navigation Solution 420
11.2.1 Navigation Error Variables 421
11.2.2 Coordinates Used for INS Error Analysis 421
11.2.3 Model Variables and Parameters 421
11.2.3.1 INS Orientation Variables and Errors 421
11.2.4 Dynamic Coupling Mechanisms 427
11.3 Navigation Error Dynamics 430
11.3.1 Error Dynamics Due to Velocity Integration 431
11.3.2 Error Dynamics Due to Gravity Miscalculations 432
11.3.2.1 INSGravityModeling 432
11.3.2.2 NavigationErrorModelforGravityCalculations 432
11.3.3 Error Dynamics Due to Coriolis Acceleration 433
11.3.4 Error Dynamics Due to Centrifugal Acceleration 434
11.3.5 Error Dynamics Due to Earthrate Leveling 435
11.3.6 Error Dynamics Due to Velocity Leveling 436
11.3.7 Error Dynamics Due to Acceleration and IMU Alignment Errors 437
11.3.8 CompositeModelfromAllEffects 438
11.3.9 VerticalNavigationInstability 439
11.3.9.1 AltimeterAiding 442
11.3.9.2 Using GNSS for Vertical Channel Stabilization 444
11.3.10 SchulerOscillations 444
11.3.10.1 SchulerOscillationswithCoriolisCoupling 445
11.3.11 Core Model Validation and Tuning 445
11.3.11.1 HorizontalInertialNavigationModel 446
11.4 Inertial Sensor Noise Propagation 447
11.4.1 1/f Noise 447
11.4.2 WhiteNoise 447
11.4.3 HorizontalCEPRateVersusSensorNoise 449
11.5 Sensor Compensation Errors 450
11.5.1 Sensor Compensation Error Models 450
11.5.1.1 Exponentially Correlated Parameter Drift Models 452
11.5.1.2 Dynamic Coupling into Navigation Errors 453
11.5.1.3 AugmentedDynamicCoefficientMatrix 454
11.5.2 Carouseling and Indexing 456
11.6 Chapter Summary 456
11.6.1 Further Reading 457
Problems 458
References 459
12 GNSS/INS Integration 461
12.1 Chapter Focus 461
12.2 NewApplicationOpportunities 462
12.2.1 Integration Advantages 462
12.2.1.1 Exploiting Complementary Error Characteristics 462
12.2.1.2 Cancelling Vulnerabilities 463
12.2.2 Enabling New Capabilities 463
12.2.2.1 Real-Time Inertial Sensor Error Compensation 463
12.2.2.2 INS Initialization on the Move 463
12.2.2.3 Antenna Switching 464
12.2.2.4 Antenna-INS Offsets 464
12.2.3 Economic Factors 464
12.2.3.1 EconomiesofScale 464
12.2.3.2 Implementation Tradeoffs 465
12.3 Integrated Navigation Models 468
12.3.1 CommonNavigationModels 468
12.3.2 GNSSErrorModels 470
12.3.2.1 GNSS Time Synchronization 470
12.3.2.2 Receiver Clock Error Model 470
12.3.2.3 Propagation Delay 472
12.3.2.4 Pseudorange Measurement Noise 473
12.3.3 INS Error Models 473
12.3.3.1 Navigation Error Model 473
12.3.3.2 Sensor Compensation Errors 473
12.3.4 GNSS/INS Error Model 474
12.3.4.1 StateVariables 474
12.3.4.2 Numbers of State Variables 474
12.3.4.3 Dynamic Coefficient Matrix 475
12.3.4.4 Process Noise Covariance 475
12.3.4.5 Measurement Sensitivities 476
12.4 Performance Analysis 476
12.4.1 r^e Influence of Trajectories 476
12.4.2 Performance Metrics 477
12.4.2.1 Application-Dependent Performance Metrics 477
12.4.2.2 General-Purpose Metrics 478
12.4.2.3 Mean Squared Error Metrics 478
12.4.2.4 Probabilistic Metrics 479
12.4.3 Dynamic Simulation Model 479
12.4.3.1 State Transition Matrices 479
12.4.3.2 Dynamic Simulation 480
12.4.4 Sample Results 480
12.4.4.1 Stand-Alone GNSS Performance 480
12.4.4.2 INS-Only Performance 482
12.4.4.3 Integrated GNSS/INS Performance 484
12.5 Summary 485
Problems 486
References 487
AppendixA Software 489
A.1 Software Sources 489
A.2 Software for Chapter 2 490
A.3 Software for Chapter 3 490
A.4 Software for Chapter 4 490
A.5 Software for Chapter 7 491
A.6 Software for Chapter 10 491
A.7 Software for Chapter 11 492
A.8 Software for Chapter 12 493
A.9 Software for Appendix B 494
A.10 Software for Appendix C 494
A.11 GPS Almanac/Ephemeris Data Sources 495
Appendix B Coordinate Systems and Transformations 497
B.1 CoordinateTransformationMatrices 497
B.1.1 Notation 497
B.1.2 Definitions 498
B.1.3 UnitCoordinateVectors 498
B.1.4 Direction Cosines 499
B.1.5 CompositionofCoordinateTransformations 500
B.2 Inertial Reference Directions 500
B.2.1 Earth’s Polar Axis and the Equatorial Plane 500
B.2.2 ^e Ecliptic and the Vernal Equinox 500
B.2.3 Earth-Centered Inertial (ECI) Coordinates 501
B.3 Application-dependent Coordinate Systems 501
B.3.1 Cartesian and Polar Coordinates 501
B.3.2 Celestial Coordinates 502
B.3.3 Satellite Orbit Coordinates 503
B.3.4 Earth-Centered Inertial (ECI) Coordinates 504
B.3.5 Earth-Centered, Earth-Fixed (ECEF) Coordinates 505
B.3.5.1 Longitude in ECEF Coordinates 505
B.3.5.2 Latitudes in ECEF Coordinates 505
B.3.5.3 Latitude on an Ellipsoidal Earth 506
B.3.5.4 Parametric Latitude 506
B.3.5.5 Geodetic Latitude 507
B.3.5.6 WGS84 Reference Geoid Parameters 510
B.3.5.7 Geocentric Latitude 510
B.3.5.8 Geocentric Radius 512
B.3.6 Ellipsoidal Radius of Curvature 512
B.3.7 Local Tangent Plane (LTP) Coordinates 513
B.3.7.1 Alpha Wander Coordinates 513
B.3.7.2 ENU/NED Coordinates 514
B.3.7.3 ENU/ECEF Coordinates 514
B.3.7.4 NED/ECEF Coordinates 515
B.3.8 Roll-Pitch-Yaw (RPY) Coordinates 516
B.3.9 Vehicle Attitude Euler Angles 516
B.3.9.1 RPY/ENU Coordinates 517
B.3.10 GPS Coordinates 518
B.4 Coordinate Transformation Models 520
B.4.1 Euler Angles 521
B.4.2 Rotation Vectors 522
B.4.2.1 RotationVectortoMatrix 523
B.4.2.2 MatrixtoRotationVector 524
B.4.2.3 Special Cases for sin(0) « 0 526
B.4.2.4 MATLAB® Implementations 527
B.4.2.5 Time Derivatives of Rotation Vectors 527
B.4.2.6 Time Derivatives of Matrix Expressions 533
B.4.2.7 Partial Derivatives with Respect to Rotation Vectors 536
B.4.3 Direction Cosines Matrix 538
B.4.3.1 Rotating Coordinates 538
B.4.4 Quaternions 542
B.4.4.1 QuaternionMatrices 542
B.4.4.2 Addition and Multiplication 543
B.4.4.3 Conjugation 544
B.4.4.4 Representing Rotations 545
B.5 Newtonian Mechanics in Rotating Coordinates 547
B.5.1 Rotating Coordinates 547
B.5.2 TimeDerivativesofMatrixProducts 548
B.5.3 Solving for Centrifugal and Coriolis Accelerations 548
Appendix C PDF Ambiguity Errors in Nonlinear Kalman Filtering 551
C.1 Objective 551
C.2 Methodology 552
C.2.1 Computing Expected Values 552
C.2.2 RepresentativeSampleofPDFs 553
C.2.3 ParametricClassofNonlinearTransformationsUsed 556
C.2.4 Ambiguity Errors in Nonlinearly Transformed Means and Variances 558
C.3 Results 558
C.3.1 Nonlinearly Transformed Means 558
C.3.2 Nonlinearly Transformed Variances 559
C.4 Mitigating Application-specific Ambiguity Errors 563
References 564
Index 565
