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Global Navigation Satellite Systems, Inertial Navigation, and Integration

by ; ;
Edition:
3rd
ISBN13:

9781118447000

ISBN10:
111844700X
Format:
Hardcover
Pub. Date:
2/19/2013
Publisher(s):
Wiley-Interscience
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Summary

The Global Positioning System (GPS) may be used in conjunction with Inertial Navigation Systems (INS) for tracking and navigation and may be applied in the civilian and military sectors to track devices, to locate people and objects, in the air, on the ground, or at sea. The book is intended for readers who need to combine Global Navigation Satellite Systems (GNSS), Inertial Navigation Systems (INS), and Kalman filters. With a focus on providing readers with solutions to real-world problems, the book offers numerous detailed examples and practice problems, as well as software that demonstrates Kalman filter algorithms with GNSS and INS data sets. The book is accompanied by a Solutions Manual for instructors.

Author Biography

MOHINDER S. GREWAL, PhD, PE, is Professor of Electrical Engineering in the College of Engineering and Computer Science at California State University, Fullerton.

ANGUS P. ANDREWS, PhD, was senior scientist (now retired) at the Rockwell Science Center in Thousand Oaks, California.

CHRIS G. BARTONE, PhD, PE, is Professor of Electrical Engineering in the Russ College of Engineering and Technology, School of Electrical Engineering and Computer Science at Ohio University, Athens, Ohio.

Table of Contents

Preface xxvii

Acknowledgments xxxi

Acronyms and Abbreviations xxxiii

1 Introduction, 1

1.1 Navigation, 1

1.1.1 Navigation-Related Technologies, 1

1.1.2 Navigation Modes, 2

1.2 GNSS Overview, 4

1.2.1 GPS, 4

1.2.1.1 GPS Orbits, 4

1.2.1.2 GPS Signals, 4

1.2.1.3 Selective Availability (SA), 5

1.2.1.4 Modernization of GPS, 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 Next Generation GLONASS, 7

1.2.3 Galileo, 7

1.2.3.1 Galileo Navigation Services, 7

1.2.3.2 Galileo Signal Characteristics, 8

1.2.3.3 Updates, 9

1.2.4 Compass (BeiDou-2), 10

1.2.4.1 Compass Satellites, 10

1.2.4.2 Frequency, 10

1.3 Inertial Navigation Overview, 10

1.3.1 Theoretical Foundations, 10

1.3.2 Inertial Sensor Technology, 11

1.3.2.1 Sensor Requirements, 12

1.3.2.2 Motivation, 13

1.3.2.3 Inertial Sensors Prior to Newton, 13

1.3.2.4 Early Momentum Wheel Gyroscopes

(MWGs), 14

1.3.2.5 German Inertial Technology: 1930s–1945, 15

1.3.2.6 Charles Stark Draper (1901–1987), “The Father

of Inertial Navigation”, 19

1.3.2.7 Aerospace Inertial Technology, 20

1.3.2.8 Developments Since the Cold War, 30

1.4 GNSS/INS Integration Overview, 30

1.4.1 The Role of Kalman Filtering, 30

1.4.2 Implementation, 31

1.4.3 Applications, 31

1.4.3.1 Military Applications, 31

1.4.3.2 Civilian and Commercial Applications, 31

Problems, 32

References, 32

2 Fundamentals of Satellite Navigation Systems, 35

2.1 Navigation Systems Considered, 35

2.1.1 Systems Other than GNSS, 35

2.1.2 Comparison Criteria, 36

2.2 Satellite Navigation, 36

2.2.1 Satellite Orbits, 36

2.2.2 Navigation Solution (Two-Dimensional Example), 36

2.2.2.1 Symmetric Solution Using Two Transmitters

on Land, 36

2.2.2.2 Navigation Solution Procedure, 40

2.2.3 Satellite Selection and

Dilution of Precision (DOP), 41

2.2.4 Example Calculation of DOPS, 45

2.2.4.1 Four Satellites, 45

2.3 Time and GPS, 46

2.3.1 Coordinated Universal Time (UTC) Generation, 46

2.3.2 GPS System Time, 46

2.3.3 Receiver Computation of UTC, 47

2.4 Example: User Position Calculations with No Errors, 48

2.4.1 User Position Calculations, 48

2.4.1.1 Position Calculations, 48

2.4.2 User Velocity Calculations, 50

Problems, 51

References, 53

3 Fundamentals of Inertial Navigation, 54

3.1 Chapter Focus, 54

3.2 Basic Terminology, 55

3.3 Inertial Sensor Error Models, 59

3.3.1 Zero-Mean Random Errors, 60

3.3.1.1 White Sensor Noise, 60

3.3.1.2 Exponentially Correlated Noise, 60

3.3.1.3 Random Walk Sensor Errors, 60

3.3.1.4 Harmonic Noise, 61

3.3.1.5 “1/f” Noise, 61

3.3.2 Fixed-Pattern Errors, 61

3.3.3 Sensor Error Stability, 62

3.4 Sensor Calibration and Compensation, 63

3.4.1 Sensor Biases, Scale Factors, and Misalignments, 63

3.4.1.1 Compensation Model Parameters, 63

3.4.1.2 Calibrating Sensor Biases, Scale Factors, and Misalignments, 64

3.4.2 Other Calibration Parameters, 65

3.4.2.1 Nonlinearities, 65

3.4.2.2 Sensitivities to Other

Measurable Conditions, 65

3.4.2.3 Other Accelerometer Models, 66

3.4.3 Calibration Parameter Instabilities, 66

3.4.3.1 Calibration Parameter Changes

between Turn-Ons, 67

3.4.3.2 Calibration Parameter Drift, 67

3.4.4 Auxilliary Sensors before GNSS, 67

3.4.4.1 Attitude Sensors, 67

3.4.4.2 Altitude Sensors, 68

3.4.5 Sensor Performance Ranges, 68

3.5 Earth Models, 68

3.5.1 Terrestrial Navigation Coordinates, 69

3.5.2 Earth Rotation, 70

3.5.3 Gravity Models, 70

3.5.3.1 GNSS Gravity Models, 71

3.5.3.2 INS Gravity Models, 71

3.5.3.3 Longitude and Latitude Rates, 73

3.6 Hardware Implementations, 77

3.6.1 Gimbaled Implementations, 78

3.6.2 Floated Implementation, 80

3.6.3 Carouseling and Indexing, 81

3.6.3.1 Alpha Wander and Carouseling, 81

3.6.3.2 Indexing, 81

3.6.4 Strapdown Systems, 82

3.6.5 Strapdown Carouseling and Indexing, 82

3.7 Software Implementations, 83

3.7.1 Example in One Dimension, 83

3.7.2 Initialization in Nine Dimensions, 84

3.7.2.1 Navigation Initialization, 84

3.7.2.2 INS Alignment Methods, 84

3.7.2.3 Gyrocompass Alignment, 85

3.7.3 Gimbal Attitude Implementations, 87

3.7.3.1 Accelerometer Recalibration, 87

3.7.3.2 Vehicle Attitude Determination, 87

3.7.3.3 ISA Attitude Control, 88

3.7.4 Gimbaled Navigation Implementation, 89

3.7.5 Strapdown Attitude Implementations, 90

3.7.5.1 Strapdown Attitude Problems, 90

3.7.5.2 Coning Motion, 90

3.7.5.3 Rotation Vector Implementation, 93

3.7.5.4 Quaternion Implementation, 95

3.7.5.5 Direction Cosines Implementation, 96

3.7.5.6 MATLAB® Implementations, 97

3.7.6 Strapdown Navigation Implementation, 97

3.7.7 Navigation Computer and Software Requirements, 99

3.7.7.1 Physical and Operational Requirements, 100

3.7.7.2 Operating Systems, 100

3.7.7.3 Interface Requirements, 100

3.7.7.4 Software Development, 100

3.8 INS Performance Standards, 101

3.8.1 Free Inertial Operation, 101

3.8.2 INS Performance Metrics, 101

3.8.3 Performance Standards, 102

3.9 Testing and Evaluation, 102

3.9.1 Laboratory Testing, 102

3.9.2 Field Testing, 103

3.10 Summary, 103

Problems, 104

References, 106

4 GNSS Signal Structure, Characteristics, and Information Utilization, 108

4.1 Legacy GPS Signal Components, Purposes, and Properties, 109

4.1.1 Mathematical Signal Models for the Legacy GPS Signals, 109

4.1.2 Navigation Data Format, 112

4.1.2.1 Z-Count, 114

4.1.2.2 GPS Week Number (WN), 115

4.1.2.3 Information by Subframe, 116

4.1.3 GPS Satellite Position Calculations, 117

4.1.3.1 Ephemeris Data Reference Time Step and Transit Time Correction, 119

4.1.3.2 True, Eccentric, and Mean Anomaly, 119

4.1.3.3 Kepler’s Equation for the Eccentric Anomaly, 120

4.1.3.4 Satellite Time Corrections, 121

4.1.4 C/A-Code and Its Properties, 122

4.1.4.1 Temporal Structure, 124

4.1.4.2 Autocorrelation Function, 124

4.1.4.3 Power Spectrum, 125

4.1.4.4 Despreading of the Signal Spectrum, 126

4.1.4.5 Role of Despreading in Interference Suppression, 127

4.1.4.6 Cross-Correlation Function, 128

4.1.5 P(Y)-Code and Its Properties, 129

4.1.5.1 P-Code Characteristics, 129

4.1.5.2 Y-Code, 130

4.1.6 L1 and L2 Carriers, 130

4.1.6.1 Dual-Frequency Operation, 130

4.1.7 Transmitted Power Levels, 131

4.1.8 Free Space and Other Loss Factors, 131

4.1.9 Received Signal Power, 132

4.2 Modernization of GPS, 132

4.2.1 Areas to Benefi t from Modernization, 133

4.2.2 Elements of the Modernized GPS, 134

4.2.3 L2 Civil Signal (L2C), 135

4.2.4 L5 Signal, 136

4.2.5 M-Code, 138

4.2.6 L1C Signal, 139

4.2.7 GPS Satellite Blocks, 140

4.2.8 GPS III, 141

4.3 GLONASS Signal Structure and Characteristics, 141

4.3.1 Frequency Division Multiple Access (FDMA)

Signals, 142

4.3.1.1 Carrier Components, 142

4.3.1.2 Spreading Codes and Modulation, 142

4.3.1.3 Navigation Data Format, 142

4.3.1.4 Satellite Families, 143

4.3.2 CDMA Modernization, 143

4.4 Galileo, 144

4.4.1 Constellation and Levels of Services, 144

4.4.2 Navigation Data and Signals, 144

4.5 Compass/BD, 146

4.6 QZSS, 146

Problems, 148

References, 150

5 GNSS Antenna Design and Analysis, 152

5.1 Applications, 152

5.2 GNSS Antenna Performance Characteristics, 152

5.2.1 Size and Cost, 153

5.2.2 Frequency and Bandwidth Coverage, 153

5.2.3 Radiation Pattern Characteristics, 155

5.2.4 Antenna Polarization and Axial Ratio, 156

5.2.5 Directivity, Effi ciency, and Gain of a GNSS Antenna, 159

5.2.6 Antenna Impedance, Standing Wave Ratio, and Return Loss, 160

5.2.7 Antenna Bandwidth, 161

5.2.8 Antenna Noise Figure, 163

5.3 Computational Electromagnetic Models (CEMs) for GNSS Antenna Design, 164

5.4 GNSS Antenna Technologies, 166

5.4.1 Dipole-Based GNSS Antennas, 166

5.4.2 GNSS Patch Antennas, 166

5.4.2.1 Edge-Fed, LP, Single-Frequency GNSS Patch Antenna, 168

5.4.2.2 Probe-Fed, LP, Single-Frequency GNSS Patch Antenna, 170

5.4.2.3 Dual Probe-Fed, RHCP, Single-Frequency GNSS Patch Antenna, 171

5.4.2.4 Single Probe-Fed, RCHP, Single-Frequency GNSS Patch Antenna, 172

5.4.2.5 Dual Probe-Fed, RHCP, Multifrequency GNSS Patch Antenna, 175

5.4.3 Survey-Grade/Reference GNSS Antennas, 176

5.4.3.1 Choke Ring-Based GNSS Antennas, 176

5.4.3.2 Advanced Planner-Based GNSS Antennas, 177

5.5 Principles of Adaptable Phased-Array Antennas, 180

5.5.1 Digital Beamforming Adaptive Antenna Array Formulations, 182

5.5.2 STAP, 185

5.5.3 SFAP, 185

5.5.4 Confi gurations of Adaptable Phased-Array Antennas, 185

5.5.5 Relative Merits of Adaptable Phased-Array Antennas, 186

5.6 Application Calibration/Compensation Considerations, 187

Problems, 189

References, 190

6 GNSS Receiver Design and Analysis, 193

6.1 Receiver Design Choices, 193

6.1.1 Global Navigation Satellite System (GNSS) Application to be Supported, 193

6.1.2 Single or Multifrequency Support, 194

6.1.2.1 Dual-Frequency Ionosphere Correction, 194

6.1.2.2 Improved Carrier Phase Ambiguity Resolution in High-Accuracy Differential Positioning, 194

6.1.3 Number of Channels, 195

6.1.4 Code Selections, 195

6.1.5 Differential Capability, 196

6.1.5.1 Corrections Formats, 197

6.1.6 Aiding Inputs, 198

6.2 Receiver Architecture, 199

6.2.1 Radio Frequency (RF) Front End, 199

6.2.2 Frequency Down-Conversion and IF Amplification, 201

6.2.2.1 SNR, 202

6.2.3 Analog-to-Digital Conversion and

Automatic Gain Control, 203

6.2.4 Baseband Signal Processing, 204

6.3 Signal Acquisition and Tracking, 204

6.3.1 Hypothesize about the User Location, 205

6.3.2 Hypothesize about Which GNSS Satellites Are

Visible, 205

6.3.3 Signal Doppler Estimation, 206

6.3.4 Search for Signal in Frequency and Code Phase, 206

6.3.4.1 Sequential Searching in Code Delay, 208

6.3.4.2 Sequential Searching in Frequency, 209

6.3.4.3 Frequency Search Strategy, 209

6.3.4.4 Parallel and Hybrid Search Methods, 210

6.3.5 Signal Detection and Confi rmation, 210

6.3.5.1 Detection Confi rmation, 211

6.3.5.2 Coordination of Frequency Tuning and Code Chipping Rate, 213

6.3.6 Code Tracking Loop, 213

6.3.6.1 Code Loop Bandwidth Considerations, 217

6.3.6.2 Coherent versus Noncoherent Code Tracking, 217

6.3.7 Carrier Phase Tracking Loops, 218

6.3.7.1 PLL Capture Range, 221

6.3.7.2 PLL Order, 221

6.3.7.3 Use of Frequency-Lock Loops (FLLs) for Carrier Capture, 221

6.3.8 Bit Synchronization, 222

6.3.9 Data Bit Demodulation, 222

6.4 Extraction of Information for User Solution, 223

6.4.1 Signal Transmission Time Information, 223

6.4.2 Ephemeris Data for Satellite Position

and Velocity, 224

6.4.3 Pseudorange Measurements Formulation Using Code Phase, 224

6.4.3.1 Pseudorange Positioning Equations, 226

6.4.4 Measurements Using Carrier Phase, 226

6.4.5 Carrier Doppler Measurement, 228

6.4.6 Integrated Doppler Measurements, 229

6.5 Theoretical Considerations in Pseudorange, Carrier Phase, and

Frequency Estimations, 231

6.5.1 Theoretical Error Bounds for Code Phase Measurement, 232

6.5.2 Theoretical Error Bounds for Carrier Phase Measurements, 233

6.5.3 Theoretical Error Bounds for Frequency Measurement, 234

6.6 High-Sensitivity A-GPS Systems, 235

6.6.1 How Assisting Data Improves Receiver Performance, 236

6.6.1.1 Reduction of Frequency Uncertainty, 236

6.6.1.2 Determination of Accurate Time, 237

6.6.1.3 Transmission of Satellite Ephemeris Data, 238

6.6.1.4 Provision of Approximate Client Location, 238

6.6.1.5 Transmission of the Demodulated Navigation Bit Stream, 239

6.6.1.6 Server-Provided Location, 240

6.6.2 Factors Affecting High-Sensitivity Receivers, 240

6.6.2.1 Antenna and Low-Noise RF Design, 240

6.6.2.2 Degradation due to Signal Phase Variations, 240

6.6.2.3 Signal Processing Losses, 241

6.6.2.4 Multipath Fading, 241

6.6.2.5 Susceptibility to Interference and Strong Signals, 241

6.6.2.6 The Problem of Time Synchronization, 242

6.6.2.7 Diffi culties in Reliable Sensitivity Assessment, 242

6.7 Software-Defi ned Radio (SDR) Approach, 242

6.8 Pseudolite Considerations, 243

Problems, 244

References, 246

7 GNSS Data Errors, 250

7.1 Data Errors, 250

7.2 Ionospheric Propagation Errors, 251

7.2.1 Ionospheric Delay Model, 252

7.2.2 GNSS SBAS Ionospheric Algorithms, 254

7.2.2.1 L1L2 Receiver and Satellite Bias and Ionospheric Delay Estimations for GPS, 256

7.2.2.2 Kalman Filter, 259

7.2.2.3 Selection of Q and R, 261

7.2.2.4 Calculation of Ionospheric Delay Using Pseudoranges, 262

7.3 Tropospheric Propagation Errors, 263

7.4 The Multipath Problem, 264

7.4.1 How Multipath Causes Ranging Errors, 264

7.5 Methods of Multipath Mitigation, 266

7.5.1 Spatial Processing Techniques, 267

7.5.1.1 Antenna Location Strategy, 267

7.5.1.2 Ground Plane Antennas, 267

7.5.1.3 Directive Antenna Arrays, 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 Modifi ed Correlator Reference Waveforms, 271

7.5.3 Multipath Mitigation Technology (MMT) Technology, 272

7.5.3.1 Description, 272

7.5.3.2 Maximum-Likelihood (ML) Multipath Estimation, 272

7.5.3.3 The Two-Path ML Estimator (MLE), 273

7.5.3.4 Asymptotic Properties of ML Estimators, 274

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, 275

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 Properties of the Compressed Signal, 279

7.5.3.12 The Compression Theorem, 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, 283

7.6 Theoretical Limits for Multipath Mitigation, 283

7.6.1 Estimation-Theoretic Methods, 283

7.6.1.1 Optimality Criteria, 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 SA Errors, 288

7.11 Error Budgets, 288

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, 299

8.3 GEO with L1L5 Signals, 299

8.3.1 GEO Uplink Subsystem Type 1 (GUST) Control Loop Overview, 302

8.3.1.1 Ionospheric Kalman Filters, 303

8.3.1.2 Range Kalman Filter, 303

8.3.1.3 Code Control Function, 304

8.3.1.4 Frequency Control Function, 304

8.3.1.5 L1L5 Bias Estimation Function, 305

8.3.1.6 L1L5 Bias Estimation Function, 305

8.3.1.7 Carrier Frequency Stability, 306

8.4 GUS Clock Steering Algorithm, 307

8.4.1 Receiver Clock Error Determination, 308

8.4.2 Clock Steering Control Law, 310

8.5 GEO Orbit Determination (OD), 310

8.5.1 OD Covariance Analysis, 312

8.6 Ground-Based Augmentation System (GBAS), 316

8.6.1 Local-Area Augmentation System (LAAS), 316

8.6.2 Joint Precision Approach and Landing

System (JPALS), 317

8.6.3 Enhanced Long-Range Navigation (eLoran), 318

8.7 Measurement/Relative-Based DGNSS, 319

8.7.1 Code Differential Measurements, 319

8.7.1.1 Single-Difference Observations, 320

8.7.1.2 Double-Difference Observations, 320

8.7.2 Carrier Phase Differential Measurements, 321

8.7.2.1 Single-Difference Observations, 321

8.7.2.2 Double-Difference Observations, 321

8.7.2.3 Triple-Difference Observations, 322

8.7.2.4 Combinations of L1 and L2 Carrier Phase Observations, 322

8.7.3 Positioning Using Double-Difference Measurements, 322

8.7.3.1 Code-Based Positioning, 322

8.7.3.2 Carrier Phase-Based Positioning, 322

8.7.3.3 Real-Time Processing versus

Postprocessing, 323

8.8 GNSS Precise Point Positioning Services and Products, 323

8.8.1 The International GNSS Service (IGS), 323

8.8.2 Continuously Operating Reference Stations (CORSs), 324

8.8.3 GPS Inferred Positioning System (GIPSY) and Orbit Analysis Simulation Software (OASIS), 324

8.8.4 Australia’s Online GPS Processing System (AUPOS), 325

8.8.5 Scripps Coordinate Update Tool (SCOUT), 325

8.8.6 The Online Positioning User Service (OPUS), 325

Problems, 325

References, 326

9 GNSS and GEO Signal Integrity, 328

9.1 Introduction, 328

9.1.1 Range Comparison Method, 329

9.1.2 Least-Squares Method, 330

9.1.3 Parity Method, 331

9.2 SBAS and GBAS Integrity Design, 332

9.2.1 SBAS Error Sources and Integrity Threats, 333

9.2.2 GNSS-Associated Errors, 334

9.2.2.1 GNSS Clock Error, 334

9.2.2.2 GNSS Ephemeris Error, 335

9.2.2.3 GNSS Code and Carrier Incoherence, 335

9.2.2.4 GNSS Signal Distortion, 335

9.2.2.5 GNSS L1L2 Bias, 336

9.2.2.6 Environment Errors: Ionosphere, 336

9.2.2.7 Environment Errors: Troposphere, 336

9.2.3 GEO-Associated Errors, 336

9.2.3.1 GEO Code and Carrier Incoherence, 336

9.2.3.2 GEO-Associated Environment Errors: Ionosphere, 337

9.2.3.3 GEO-Associated Environment Errors: Troposphere, 337

9.2.4 Receiver and Measurement Processing Errors, 337

9.2.4.1 Receiver Measurement Error, 337

9.2.4.2 Intercard Bias, 337

9.2.4.3 Multipath, 338

9.2.4.4 L1L2 Bias, 338

9.2.4.5 Receiver Clock Error, 338

9.2.4.6 Measurement Processing Unpack/Pack Corruption, 338

9.2.5 Estimation Errors, 338

9.2.5.1 Reference Time Offset Estimation Error, 338

9.2.5.2 Clock Estimation Error, 339

9.2.5.3 Ephemeris Correction Error, 339

9.2.5.4 L1L2 Wide-Area Reference Equipment (WRE) and GPS Satellite Bias

Estimation Error, 339

9.2.6 Integrity-Bound Associated Errors, 339

9.2.6.1 Ionospheric Modeling Errors, 339

9.2.6.2 Fringe Area Ephemeris Error, 340

9.2.6.3 Small-Sigma Errors, 340

Problems, 325

References, 326

9 GNSS and GEO Signal Integrity, 328

9.1 Introduction, 328

9.1.1 Range Comparison Method, 329

9.1.2 Least-Squares Method, 330

9.1.3 Parity Method, 331

9.2 SBAS and GBAS Integrity Design, 332

9.2.1 SBAS Error Sources and Integrity Threats, 333

9.2.2 GNSS-Associated Errors, 334

9.2.2.1 GNSS Clock Error, 334

9.2.2.2 GNSS Ephemeris Error, 335

9.2.2.3 GNSS Code and Carrier Incoherence, 335

9.2.2.4 GNSS Signal Distortion, 335

9.2.2.5 GNSS L1L2 Bias, 336

9.2.2.6 Environment Errors: Ionosphere, 336

9.2.2.7 Environment Errors: Troposphere, 336

9.2.3 GEO-Associated Errors, 336

9.2.3.1 GEO Code and Carrier Incoherence, 336

9.2.3.2 GEO-Associated Environment Errors: Ionosphere, 337

9.2.3.3 GEO-Associated Environment Errors: Troposphere, 337

9.2.4 Receiver and Measurement Processing Errors, 337

9.2.4.1 Receiver Measurement Error, 337

9.2.4.2 Intercard Bias, 337

9.2.4.3 Multipath, 338

9.2.4.4 L1L2 Bias, 338

9.2.4.5 Receiver Clock Error, 338

9.2.4.6 Measurement Processing Unpack/Pack Corruption, 338

9.2.5 Estimation Errors, 338

9.2.5.1 Reference Time Offset Estimation Error, 338

9.2.5.2 Clock Estimation Error, 339

9.2.5.3 Ephemeris Correction Error, 339

9.2.5.4 L1L2 Wide-Area Reference Equipment (WRE) and GPS Satellite Bias

Estimation Error, 339

9.2.6 Integrity-Bound Associated Errors, 339

9.2.6.1 Ionospheric Modeling Errors, 339

9.2.6.2 Fringe Area Ephemeris Error, 340

9.2.6.3 Small-Sigma Errors, 340

9.2.6.4 Missed Message: Old But Active Data (OBAD), 340

9.2.6.5 Time to Alarm (TTA) Exceeded, 340

9.2.7 GEO Uplink Errors, 340

9.2.7.1 GEO Uplink System Fails to Receive SBAS Message, 340

9.2.8 Mitigation of Integrity Threats, 340

9.2.8.1 Mitigation of GNSS Associated Errors, 341

9.2.8.2 Mitigation of GEO-Associated Errors, 343

9.2.8.3 Mitigation of Receiver and Measurement Processing Errors, 343

9.2.8.4 Mitigation of Estimation Errors, 344

9.2.8.5 Mitigation of Integrity-Bound-Associated Errors, 345

9.3 SBAS Example, 346

9.4 Summary, 347

9.5 Future: GIC, 348

Problem, 348

References, 348

10 Kalman Filtering, 350

10.1 Introduction, 350

10.1.1 What Is a Kalman Filter?, 351

10.1.2 How Does It Work?, 352

10.1.2.1 Prediction and Correction, 353

10.1.3 How Is It Used?, 353

10.2 Kalman Filter Correction Update, 354

10.2.1 Deriving the Kalman Gain, 354

10.2.1.1 Approaches to Deriving the Kalman Gain, 355

10.2.1.2 Gaussian Probability Density Functions, 355

10.2.1.3 Properties of Likelihood Functions, 356

10.2.1.4 Solving for Combined Information Matrix, 358

10.2.1.5 Solving for Combined Argmax, 359

10.2.1.6 Noisy Measurement Likelihoods, 360

10.2.1.7 Gaussian Maximum-Likelihood Estimate (MLE), 362

10.2.1.8 Estimate Correction, 364

10.2.1.9 Kalman Gain Matrix for MLE, 364

10.2.2 Estimate Correction Using the Kalman Gain, 364

10.2.3 Covariance Correction for Using Measurements, 365

10.3 Kalman Filter Prediction Update, 365

10.3.1 Stochastic Systems in Continuous Time, 365

10.3.1.1 White-Noise Processes, 365

10.3.1.2 Stochastic Differential Equations, 365

10.3.1.3 Systems of First-Order Linear Differential Equations, 367

10.3.1.4 Representation in Terms of Vectors and Matrices, 368

10.3.1.5 Eigenvalues of Dynamic

Coeffi cient Matrices, 369

10.3.1.6 Matrix Exponential Function, 371

10.3.1.7 Forward Solution, 371

10.3.1.8 Time-Invariant Systems, 371

10.3.2 Stochastic Systems in Discrete Time, 372

10.3.2.1 Zero-Mean White Gaussian Noise Sequences, 372

10.3.2.2 Gaussian Linear Stochastic Processes in Discrete Time, 372

10.3.3 State Space Models for Discrete Time, 373

10.3.4 Dynamic Disturbance Noise Distribution Matrices, 374

10.3.5 Predictor Equations, 374

10.4 Summary of Kalman Filter Equations, 375

10.4.1 Essential Equations, 375

10.4.2 Common Terminology, 375

10.4.3 Data Flow Diagrams, 376

10.5 Accommodating Time-Correlated Noise, 377

10.5.1 Correlated Noise Models, 378

10.5.1.1 Autocovariance Functions, 378

10.5.1.2 Random Walks, 378

10.5.1.3 Exponentially Correlated Noise, 379

10.5.1.4 Harmonic Noise, 379

10.5.1.5 Selective Availability (SA), 379

10.5.1.6 Slow Variables, 380

10.5.2 Empirical Modeling of Sensor Noise, 380

10.5.2.1 Spectral Characterization, 381

10.5.2.2 Shaping Filters, 381

10.5.3 State Vector Augmentation, 382

10.5.3.1 Correlated Dynamic Disturbance Noise, 382

10.5.3.2 Correlated Sensor Noise, 383

10.5.3.3 Correlated Noise in Continuous Time, 383

10.6 Nonlinear and Adaptive Implementations, 384

10.6.1 Assessing Linear Approximation Errors, 384

10.6.1.1 Statistical Measures of Acceptability, 384

10.6.1.2 Sampling for Acceptability Testing, 385

10.6.2 Nonlinear Dynamics, 390

10.6.2.1 Nonlinear Dynamics with Control, 390

10.6.2.2 Propagating Estimates, 390

10.6.2.3 Propagating Covariances, 390

10.6.3 Nonlinear Sensors, 391

10.6.3.1 Predicted Sensor Outputs, 391

10.6.3.2 Calculating Kalman Gains, 391

10.6.4 Linearized Kalman Filter, 391

10.6.5 Extended Kalman Filtering (EFK), 392

10.6.6 Adaptive Kalman Filtering, 393

10.7 Kalman–Bucy Filter, 395

10.7.1 Implementation Equations, 395

10.7.2 Kalman–Bucy Filter Parameters, 396

10.8 Host Vehicle Tracking Filters for GNSS, 397

10.8.1 Vehicle Tracking Filters, 397

10.8.2 Dynamic Dilution of Information, 397

10.8.2.1 Effect on Position Uncertainty, 398

10.8.3 Specialized Host Vehicle Tracking Filters, 399

10.8.3.1 Unknown Constant Tracking Model, 401

10.8.3.2 Damped Harmonic Resonator, 401

10.8.3.3 Type 2 Tracking Model, 402

10.8.3.4 DAMP1 Tracking Model: Velocity Damping, 403

10.8.3.5 DAMP2 Tracking Model: Velocity and

Acceleration Damping, 403

10.8.3.6 DAMP3 Tracking Model: Position, Velocity, and Acceleration Damping, 405

10.8.3.7 Tracking Models for Highly Constrained Trajectories, 408

10.8.3.8 Filters for Spacecraft, 409

10.8.3.9 Other Specialized Vehicle Filter Models, 409

10.8.3.10 Filters for Different Host Vehicle Types, 409

10.8.3.11 Parameters for Vehicle Dynamics, 409

10.8.3.12 Empirical Modeling of Vehicle Dynamics, 409

10.8.4 Vehicle Tracking Filter Comparison, 411

10.8.4.1 Simulated Trajectory, 411

10.8.4.2 Results, 412

10.8.4.3 Model Dimension versus Model Constraints, 412

10.8.4.4 Role of Model Fidelity, 413

10.9 Alternative Implementations, 413

10.9.1 Schmidt–Kalman Suboptimal Filtering, 413

10.9.1.1 State Vector Partitioning, 414

10.9.1.2 Implementation Equations, 414

10.9.1.3 Simulated Performance in GNSS Position Estimation, 415

10.9.2 Serial Measurement Processing, 416

10.9.2.1 Measurement Decorrelation, 416

10.9.2.2 Serial Processing of Decorrelated Measurements, 417

10.9.3 Improving Numerical Stability, 417

10.9.3.1 Effects of Finite Precision, 417

10.9.3.2 Alternative Implementations, 418

10.9.3.3 Conditioning and Scaling Considerations, 419

10.9.4 Kalman Filter Monitoring, 421

10.9.4.1 Rejecting Anomalous Sensor Data, 421

10.9.4.2 Monitoring Filter Health, 423

10.10 Summary, 425

Problems, 426

References, 428

11 Inertial Navigation Error Analysis, 430

11.1 Chapter Focus, 430

11.2 Errors in the Navigation Solution, 432

11.2.1 The Nine Core INS Error Variables, 432

11.2.2 Coordinates Used for INS Error Analysis, 432

11.2.3 Model Variables and Parameters, 432

11.2.3.1 INS Orientation Variables and Errors, 433

11.2.4 Dynamic Coupling Mechanisms, 439

11.2.4.1 Dynamic Coupling, 439

11.3 Navigation Error Dynamics, 442

11.3.1 Error Dynamics due to Velocity Integration, 442

11.3.2 Error Dynamics due to Gravity Calculations, 443

11.3.2.1 INS Gravity Modeling, 443

11.3.2.2 Navigation Error Model for

Gravity Calculations, 444

11.3.3 Error Dynamics due to Coriolis Acceleration, 445

11.3.4 Error Dynamics due to Centrifugal Acceleration, 446

11.3.5 Error Dynamics due to Earthrate Leveling, 447

11.3.6 Error Dynamics due to Velocity Leveling, 448

11.3.7 Error Dynamics due to Acceleration and Misalignments, 449

11.3.8 Composite Model from All Effects, 450

11.3.9 Vertical Navigation Instability, 452

11.3.9.1 Altimeter Aiding, 454

11.3.10 Schuler Oscillations, 457

11.3.11 Core Model Validation and Tuning, 459

11.3.11.1 Horizontal Inertial Navigation Model, 459

11.4 Inertial Sensor Noise, 459

11.4.1 CEP Rate versus Sensor Noise, 461

11.5 Sensor Compensation Errors, 461

11.5.1 Sensor Compensation Error Models, 462

11.5.1.1 Exponentially Correlated Parameter Drift Model, 463

11.5.1.2 Dynamic Coupling into Navigation Errors, 465

11.5.1.3 Augmented Dynamic Coeffi cient Matrix, 465

11.6 Software Sources, 467

11.7 Summary, 468

Problems, 470

References, 471

12 GNSS/INS Integration, 472

12.1 Chapter Focus, 472

12.1.1 Objective, 472

12.1.2 Order of Presentation, 473

12.2 GNSS/INS Integration Overview, 473

12.2.1 Historical Background, 473

12.2.2 The Loose/Tight Ranking, 475

12.2.2.1 Loosely Coupled Implementations, 476

12.2.2.2 More Tightly Coupled Implementations, 476

12.2.2.3 Ultratightly Coupled Integration, 477

12.2.2.4 Limitations, 477

12.2.3 Unifi ed Navigation Model, 477

12.3 Unifi ed Model for GNSS/INS Integration, 479

12.3.1 GNSS Error Models, 479

12.3.1.1 Receiver Clock Error Model, 479

12.3.1.2 Atmospheric Propagation Delay Model, 480

12.3.1.3 Pseudorange Measurement Noise, 481

12.3.2 INS Error Models, 481

12.3.2.1 Navigation Error Model, 481

12.3.2.2 Sensor Compensation Errors, 481

12.3.3 GNSS/INS Error Model, 482

12.3.3.1 State Variables, 482

12.3.3.2 Numbers of State Variables, 482

12.3.3.3 Dynamic Coeffi cient Matrix, 483

12.3.3.4 Process Noise Covariance, 484

12.3.3.5 Measurement Sensitivities, 484

12.4 Performance Analysis, 485

12.4.1 Dynamic Simulation Model, 485

12.4.1.1 State Transition Matrices (STMs), 485

12.4.1.2 Dynamic Simulation, 486

12.4.2 Results, 486

12.4.2.1 Stand-Alone GNSS Performance, 486

12.4.2.2 Stand-Alone INS Performance, 488

12.4.2.3 Integrated GNSS/INS Performance, 488

12.5 Other Integration Issues, 490

12.5.1 Antenna/ISA Offset Correction, 490

12.5.2 Infl uence of Trajectories on Performance, 491

12.6 Summary, 492

Problem, 493

References, 494

Appendix A Software, 495

A.1 Software Sources, 495

A.2 Software for Chapter 3, 496

A.3 Software for Chapter 4, 496

A.4 Software for Chapter 7, 496

A.5 Software for Chapter 10, 497

A.6 Software for Chapter 11, 498

A.7 Software for Chapter 12, 498

A.8 Almanac/Ephemeris Data Sources, 499

Appendix B Coordinate Systems and Transformations, 500

B.1 Coordinate Transformation Matrices, 500

B.1.1 Notation, 500

B.1.2 Definitions, 501

B.1.3 Unit Coordinate Vectors, 501

B.1.4 Direction Cosines, 502

B.1.5 Composition of Coordinate Transformations, 503

B.2 Inertial Reference Directions, 503

B.3 Application-Dependent Coordinate Systems, 504

B.3.1 Cartesian and Polar Coordinates, 504

B.3.2 Celestial Coordinates, 505

B.3.3 Satellite Orbit Coordinates, 505

B.3.4 ECI Coordinates, 507

B.3.5 Earth-Centered, Earth-Fixed (ECEF) Coordinates, 508

B.3.5.1 Longitudes in ECEF Coordinates, 508

B.3.5.2 Latitudes in ECEF Coordinates, 508

B.3.5.3 Latitude on an Ellipsoidal Earth, 509

B.3.5.4 Parametric Latitude, 509

B.3.5.5 Geodetic Latitude, 510

B.3.5.6 WGS84 Reference Geoid Parameters, 513

B.3.5.7 Geocentric Latitude, 513

B.3.5.8 Geocentric Radius, 514

B.3.6 Ellipsoidal Radius of Curvature, 515

B.3.7 Local Tangent Plane (LTP) Coordinates, 515

B.3.7.1 Alpha Wander Coordinates, 516

B.3.7.2 ENU/NED Coordinates, 516

B.3.7.3 ENU/ECEF Coordinates, 516

B.3.7.4 NED/ECEF Coordinates, 517

B.3.8 RPY Coordinates, 518

B.3.9 Vehicle Attitude Euler Angles, 518

B.3.9.1 RPY/ENU Coordinates, 519

B.3.10 GNSS Navigation Coordinates, 521

B.4 Coordinate Transformation Models, 523

B.4.1 Euler Angles, 523

B.4.2 Rotation Vectors, 524

B.4.2.1 Rotation Vector to Matrix, 525

B.4.2.2 Matrix to Rotation Vector, 527

B.4.2.3 Special Cases for sin(θ) ≈ 0, 528

B.4.2.4 Time Derivatives of Rotation Vectors, 529

B.4.2.5 Time Derivatives of Matrix Expressions, 534

B.4.2.6 Partial Derivatives with Respect to Rotation Vectors, 537

B.4.3 Direction Cosine Matrix, 539

B.4.3.1 Rotating Coordinates, 540

B.4.4 Quaternions, 543

B.4.4.1 Quaternion Matrices, 543

B.4.4.2 Addition and Multiplication, 544

B.4.4.3 Conjugation, 545

B.4.4.4 Representing Rotations, 545

B.5 Newtonian Mechanics in Rotating Coordinates, 548

B.5.1 Rotating Coordinates, 548

B.5.2 Time Derivatives of Matrix Products, 549

B.5.3 Solving for Centrifugal and Coriolis Accelerations, 549

Index 551



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