GLOBAL NAVIGATION SATELLITE SYSTEMS, INERTIAL NAVIGATION, AND INTEGRATION 4/E

mation-Theoretic Methods 283
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 Error Budgets 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
8.3 GEO with L1L5 Signals 299
8.3.1 GEO Uplink Subsystem (GUS) Control Loop Overview 302
8.4 GUS Clock Steering Algorithm 307
8.4.1 Receiver Clock Error Determination 309
8.4.2 Clock Steering Control Law 311
8.5 GEO Orbit Determination (OD) 312
8.5.1 OD Covariance Analysis 313
8.6 Ground-Based Augmentation System (GBAS) 318
8.6.1 Local-Area Augmentation System (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 Code Differential Measurements 320
8.7.2 Carrier Phase Differential Measurements 322
8.7.3 Positioning Using Double-Difference Measurements 324
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-Squares Method 332
9.1.3 Parity Method 334
9.2 SBAS and GBAS Integrity Design 334
9.2.1 SBAS Error Sources and Integrity Threats 336
9.2.2 GNSS-Associated Errors 337
9.2.3 GEO-Associated Errors 339
9.2.4 Receiver and Measurement Processing Errors 340
9.2.5 Estimation Errors 341
9.2.6 Integrity-Bound Associated Errors 342
9.2.7 GEO Uplink Errors 343
9.2.8 Mitigation of Integrity Threats 344
9.3 SBAS Example 349
9.4 Summary 351
9.5 Future: GIC 351
Problems 352
References 352
10 Kalman Filtering 355
10.1 Chapter Focus 355
10.2 Frequently Asked Questions 356
10.3 Notation 360
10.3.1 Real Vectors and Matrices 360
10.3.2 Probability Essentials 363
10.3.3 Discrete Time Notation 365
10.4 Kalman Filter Genesis 366
10.4.1 Measurement Update (Corrector) 366
10.4.2 Time Update (Predictor) 373
10.4.3 Basic Kalman Filter Equations 378
10.4.4 The Time-Invariant Case 378
10.4.5 Observability and Stability Issues 378
10.5 Alternative Implementations 380
10.5.1 Implementation Issues 380
10.5.2 Conventional Implementation Improvements 381
10.5.3 James E. Potter (1937–2005) and Square Root Filtering 383
10.5.4 Square Root Matrix Manipulation Methods 384
10.5.5 Alternative Square Root Filter Implementations 386
10.6 Nonlinear Approximations 388
10.6.1 Linear Approximation Errors 389
10.6.2 Adaptive Kalman Filtering 392
10.6.3 Taylor–Maclauren Series Approximations 392
10.6.4 Trajectory Perturbation Modeling 393
10.6.5 Structured Sampling Methods 394
10.7 Diagnostics and Monitoring 397
10.7.1 Covariance Matrix Diagnostics 397
10.7.2 Innovations Monitoring 398
10.8 GNSS-Only Navigation 401
10.8.1 GNSS Dynamic Models 402
10.8.2 GNSS Measurement Models 406
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
11.2.4 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
11.3.3 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 Composite Model from All Effects 438
11.3.9 Vertical Navigation Instability 439
11.3.10 Schuler Oscillations 444
11.3.11 Core Model Validation and Tuning 445
11.4 Inertial Sensor Noise Propagation 447
11.4.1 1∕f Noise 447
11.4.2 White Noise 447
11.4.3 Horizontal CEP Rate Versus Sensor Noise 449
11.5 Sensor Compensation Errors 450
11.5.1 Sensor Compensation Error Models 450
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 New Application Opportunities 462
12.2.1 Integration Advantages 462
12.2.2 Enabling New Capabilities 463
12.2.3 Economic Factors 464
12.3 Integrated Navigation Models 468
12.3.1 Common Navigation Models 468
12.3.2 GNSS Error Models 470
12.3.3 INS Error Models 473
12.3.4 GNSS/INS Error Model 474
12.4 Performance Analysis 476
12.4.1 The Influence of Trajectories 476
12.4.2 Performance Metrics 477
12.4.3 Dynamic Simulation Model 479
12.4.4 Sample Results 480
12.5 Summary 485
Problems 486
References 487
Appendix A 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 Coordinate Transformation Matrices 497
B.1.1 Notation 497
B.1.2 Definitions 498
B.1.3 Unit Coordinate Vectors 498
B.1.4 Direction Cosines 499
B.1.5 Composition of Coordinate Transformations 500
B.2 Inertial Reference Directions 500
B.2.1 Earth’s Polar Axis and the Equatorial Plane 500
B.2.2 The 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.3.6 Ellipsoidal Radius of Curvature 512
B.3.7 Local Tangent Plane (LTP) Coordinates 513
B.3.8 Roll–Pitch–Yaw (RPY) Coordinates 516
B.3.9 Vehicle Attitude Euler Angles 516
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.4.3 Direction Cosines Matrix 538
B.4.4 Quaternions 542
B.5 Newtonian Mechanics in Rotating Coordinates 547
B.5.1 Rotating Coordinates 547
B.5.2 Time Derivatives of Matrix Products 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 Representative Sample of PDFs 553
C.2.3 Parametric Class of Nonlinear Transformations Used 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 

mation-Theoretic Methods 283
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 Error Budgets 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...