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