Mohinder S. Grewal, PhD, PE, is Professor of Electrical Engineering in the College of Engineering and Computer Science at California State University, Fullerton. He has more than forty years of experience in inertial navigation and control, and his mechanizations are currently used in commercial and military aircraft, surveillance satellites, missile and radar systems, freeway traffic control, and Global Navigation Satellite Systems.

Angus P. Andrews, PhD, is an MIT graduate with a PhD in mathematics from UCLA.  His career in aerospace technology development spans more than 50 years, starting with navigation analysis for the Apollo moon missions, and including a dozen years in the analysis, design, development, and testing of inertial navigation systems.  His discoveries included the orbital navigation method called unknown landmark tracking, alternative solutions for square root filters, and a model for bearing torques of electrostatic gyroscopes.  Since retiring as a senior scientist from the Rockwell Science Center in 2000, he has continued consulting and instructing in sensor error modeling and analysis, and publishing articles and books on these subjects.

Preface to the Fourth Edition ix

Acknowledgements xiii

List of Abbreviations xv

1 Introduction 1

1.1 Chapter Focus 1

1.2 On Kalman Filtering 1

1.3 On Optimal Estimation Methods 6

1.4 Common Notation 28

1.5 Summary 30

Problems 31

References 34

2 Linear Dynamic Systems 37

2.1 Chapter Focus 37

2.2 Deterministic Dynamic System Models 42

2.3 Continuous Linear Systems and their Solutions 47

2.4 Discrete Linear Systems and their Solutions 59

2.5 Observability of Linear Dynamic System Models 61

2.6 Summary 66

Problems 69

References 71

3 Probability and Expectancy 73

3.1 Chapter Focus 73

3.2 Foundations of Probability Theory 74

3.3 Expectancy 79

3.4 Least-Mean-Square Estimate (LMSE) 87

3.5 Transformations of Variates 93

3.6 The Matrix Trace in Statistics 102

3.7 Summary 106

Problems 107

References 110

4 Random Processes 111

4.1 Chapter Focus 111

4.2 Random Variables Processes and Sequences 112

4.3 Statistical Properties 114

4.4 Linear Random Process Models 124

4.5 Shaping Filters (SF) and State Augmentation 131

4.6 Mean and Covariance Propagation 135

4.7 Relationships Between Model Parameters 145

4.8 Orthogonality Principle 153

4.9 Summary 157

Problems 159

References 167

5 Linear Optimal Filters and Predictors 169

5.1 Chapter Focus 169

5.2 Kalman Filter 172

5.3 Kalman–Bucy Filter 197

5.4 Optimal Linear Predictors 200

5.5 Correlated Noise Sources 200

5.6 Relationships Between Kalman and Wiener Filters 201

5.7 Quadratic Loss Functions 202

5.8 Matrix Riccati Differential Equation 204

5.9 Matrix Riccati Equation in Discrete Time 219

5.10 Model Equations for Transformed State Variables 223

5.11 Sample Applications 224

5.12 Summary 228

Problems 232

References 235

6 Optimal Smoothers 239

6.1 Chapter Focus 239

6.2 Fixed-Interval Smoothing 244

6.3 Fixed-Lag Smoothing 256

6.4 Fixed-Point Smoothing 268

6.5 Summary 275

Problems 276

References 278

7 Implementation Methods 281

7.1 Chapter Focus 281

7.2 Computer Roundoff 283

7.3 Effects of Roundoff Errors on Kalman Filters 288

7.4 Factorization Methods for “Square-Root” Filtering 294

7.5 “Square-Root” and UD Filters 318

7.6 SigmaRho Filtering 330

7.7 Other Implementation Methods 346

7.8 Summary 358

Problems 360

References 363

8 Nonlinear Approximations 367

8.1 Chapter Focus 367

8.2 The Affine Kalman Filter 370

8.3 Linear Approximations of Nonlinear Models 372

8.4 Sample-and-Propagate Methods 398

8.5 Unscented Kalman Filters (UKF) 404

8.6 Truly Nonlinear Estimation 417

8.7 Summary 419

Problems 420

References 423

9 Practical Considerations 427

9.1 Chapter Focus 427

9.2 Diagnostic Statistics and Heuristics 428

9.3 Prefiltering and Data Rejection Methods 457

9.4 Stability of Kalman Filters 460

9.5 Suboptimal and Reduced-Order Filters 461

9.6 Schmidt–Kalman Filtering 471

9.7 Memory Throughput and Wordlength Requirements 478

9.8 Ways to Reduce Computational Requirements 486

9.9 Error Budgets and Sensitivity Analysis 491

9.10 Optimizing Measurement Selection Policies 495

9.11 Summary 501

Problems 501

References 502

10 Applications to Navigation 503

10.1 Chapter Focus 503

10.2 Navigation Overview 504

10.3 Global Navigation Satellite Systems (GNSS) 510

10.4 Inertial Navigation Systems (INS) 544

10.5 GNSS/INS Integration 578

10.6 Summary 588

Problems 590

References 591

Appendix A Software 593

A.1 Appendix Focus 593

A.2 Chapter 1 Software 594

A.3 Chapter 2 Software 594

A.4 Chapter 3 Software 595

A.5 Chapter 4 Software 595

A.6 Chapter 5 Software 596

A.7 Chapter 6 Software 596

A.8 Chapter 7 Software 597

A.9 Chapter 8 Software 598

A.10 Chapter 9 Software 599

A.11 Chapter 10 Software 599

A.12 Other Software Sources 601

References 603

Index 605

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