# Bayesian Estimation and Tracking: A Practical Guide

## Books A practical approach to estimating and tracking dynamic systems in real-worl applications

Much of the literature on performing estimation for non-Gaussian systems is short on practical methodology, while Gaussian methods often lack a cohesive derivation. Bayesian Estimation and Tracking addresses the gap in the field on both accounts, providing readers with a comprehensive overview of methods for estimating both linear and nonlinear dynamic systems driven by Gaussian and non-Gaussian noices.

Featuring a unified approach to Bayesian estimation and tracking, the book emphasizes the derivation of all tracking algorithms within a Bayesian framework and describes effective numerical methods for evaluating density-weighted integrals, including linear and nonlinear Kalman filters for Gaussian-weighted integrals and particle filters for non-Gaussian cases. The author first emphasizes detailed derivations from first principles of eeach estimation method and goes on to use illustrative and detailed step-by-step instructions for each method that makes coding of the tracking filter simple and easy to understand.

Case studies are employed to showcase applications of the discussed topics. In addition, the book supplies block diagrams for each algorithm, allowing readers to develop their own MATLAB® toolbox of estimation methods.

Bayesian Estimation and Tracking is an excellent book for courses on estimation and tracking methods at the graduate level. The book also serves as a valuable reference for research scientists, mathematicians, and engineers seeking a deeper understanding of the topics.

Preface xv

Acknowledgments xvii

List of Figures Xix

List of Tables xxv

PART I PRELIMINARIES

1 Introduction 3

1.1 Bayesian Inference 4

1.2 Bayesian Hierarchy of Estimation Methods 5

1.3 Scope of This Text 6

1.3.1 Objective 6

1.3.2 Chapter Overview and Prerequisites 6

1.4 Modeling and Simulation with MATLAB® 8

References 9

2 Preliminary Mathematical Concepts 11

2.1 A Very Brief Overview of Matrix Linear Algebra 11

2.1.1 Vector and Matrix Conventions and Notation 11

2.1.2 Sums and Products 12

2.1.3 Matrix Inversion 13

2.1.4 Block Matrix Inversion 14

2.1.5 Matrix Square Root 15

2.2 Vector Point Generators 16

2.3 Approximating Nonlinear Multidimensional Functions with Multidimensional Arguments 19

2.3.1 Approximating Scalar Nonlinear Functions 19

2.3.2 Approximating Multidimensional Nonlinear Functions 23

2.4 Overview of Multivariate Statistics 29

2.4.1 General Definitions 29

2.4.2 The Gaussian Density 32

References 40

3 General Concepts of Bayesian Estimation 42

3.1 Bayesian Estimation 43

3.2 Point Estimators 43

3.3 Introduction to Recursive Bayesian Filtering of Probability Density Functions 46

3.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance 49

3.4.1 State Vector Prediction 50

3.4.2 State Vector Update 51

3.5 Discussion of General Estimation Methods 55

References 55

4 Case Studies: Preliminary Discussions 56

4.1 The Overall Simulation/Estimation/Evaluation Process 57

4.2 A Scenario Simulator for Tracking a Constant Velocity Target Through a DIFAR Buoy Field 58

4.2.1 Ship Dynamics Model 58

4.2.2 Multiple Buoy Observation Model 59

4.2.3 Scenario Specifics 59

4.3 DIFAR Buoy Signal Processing 62

4.4 The DIFAR Likelihood Function 67

References 69

PART II THE GAUSSIAN ASSUMPTION: A FAMILY OF KALMAN FILTER ESTIMATORS

5 The Gaussian Noise Case: Multidimensional Integration of Gaussian-Weighted Distributions 73

5.1 Summary of Important Results From Chapter 3 74

5.2 Derivation of the Kalman Filter Correction (Update) Equations Revisited 76

5.3 The General Bayesian Point Prediction Integrals for Gaussian Densities 78

5.3.1 Refining the Process Through an Affine Transformation 80

5.3.2 General Methodology for Solving Gaussian-Weighted Integrals 82

References 85

6 The Linear Class of Kalman Filters 86

6.1 Linear Dynamic Models 86

6.2 Linear Observation Models 87

6.3 The Linear Kalman Filter 88

6.4 Application of the LKF to DIFAR Buoy Bearing Estimation 88

References 92

7 The Analytical Linearization Class of Kalman Filters: The Extended Kalman Filter 93

7.1 One-Dimensional Consideration 93

7.1.1 One-Dimensional State Prediction 94

7.1.2 One-Dimensional State Estimation Error Variance Prediction 95

7.1.3 One-Dimensional Observation Prediction Equations 96

7.1.4 Transformation of One-Dimensional Prediction Equations 96

7.1.5 The One-Dimensional Linearized EKF Process 98

7.2 Multidimensional Consideration 98

7.2.1 The State Prediction Equation 99

7.2.2 The State Covariance Prediction Equation 100

7.2.3 Observation Prediction Equations 102

7.2.4 Transformation of Multidimensional Prediction Equations 103

7.2.5 The Linearized Multidimensional Extended Kalman Filter Process 105

7.2.6 Second-Order Extended Kalman Filter 105

7.3 An Alternate Derivation of the Multidimensional Covariance Prediction Equations 107

7.4 Application of the EKF to the DIFAR Ship Tracking Case Study 108

7.4.1 The Ship Motion Dynamics Model 108

7.4.2 The DIFAR Buoy Field Observation Model 109

7.4.3 Initialization for All Filters of the Kalman Filter Class 111

7.4.4 Choosing a Value for the Acceleration Noise 112

7.4.5 The EKF Tracking Filter Results 112

References 114

8 The Sigma Point Class: The Finite Difference Kalman Filter 115

8.1 One-Dimensional Finite Difference Kalman Filter 116

8.1.1 One-Dimensional Finite Difference State Prediction 116

8.1.2 One-Dimensional Finite Difference State Variance Prediction 117

8.1.3 One-Dimensional Finite Difference Observation Prediction Equations 118

8.1.4 The One-Dimensional Finite Difference Kalman Filter Process 118

8.1.5 Simplified One-Dimensional Finite Difference Prediction Equations 118

8.2 Multidimensional Finite Difference Kalman Filters 120

8.2.1 Multidimensional Finite Difference State Prediction 120

8.2.2 Multidimensional Finite Difference State Covariance Prediction 123

8.2.3 Multidimensional Finite Difference Observation Prediction Equations 124

8.2.4 The Multidimensional Finite Difference Kalman Filter Process 125

8.3 An Alternate Derivation of the Multidimensional Finite Difference Covariance Prediction Equations 125

References 127

9 The Sigma Point Class: The Unscented Kalman Filter 128

9.1 Introduction to Monomial Cubature Integration Rules 128

9.2 The Unscented Kalman Filter 130

9.2.1 Background 130

9.2.2 The UKF Developed 131

9.2.3 The UKF State Vector Prediction Equation 134

9.2.4 The UKF State Vector Covariance Prediction Equation 134

9.2.5 The UKF Observation Prediction Equations 135

9.2.6 The Unscented Kalman Filter Process 135

9.2.7 An Alternate Version of the Unscented Kalman Filter 135

9.3 Application of the UKF to the DIFAR Ship Tracking Case Study 137

References 138

10 The Sigma Point Class: The Spherical Simplex Kalman Filter 140

10.1 One-Dimensional Spherical Simplex Sigma Points 141

10.2 Two-Dimensional Spherical Simplex Sigma Points 142

10.3 Higher Dimensional Spherical Simplex Sigma Points 144

10.4 The Spherical Simplex Kalman Filter 144

10.5 The Spherical Simplex Kalman Filter Process 145

10.6 Application of the SSKF to the DIFAR Ship Tracking Case Study 146

Reference 147

11 The Sigma Point Class: The Gauss–Hermite Kalman Filter 148

11.2 One-Dimensional Gauss–Hermite Kalman Filter 153

11.3 Multidimensional Gauss–Hermite Kalman Filter 155

11.4 Sparse Grid Approximation for High Dimension/High Polynomial Order 160

11.5 Application of the GHKF to the DIFAR Ship Tracking Case Study 163

References 163

12 The Monte Carlo Kalman Filter 164

12.1 The Monte Carlo Kalman Filter 167

Reference 167

13 Summary of Gaussian Kalman Filters 168

13.1 Analytical Kalman Filters 168

13.2 Sigma Point Kalman Filters 170

13.3 A More Practical Approach to Utilizing the Family of Kalman Filters 174

References 175

14 Performance Measures for the Family of Kalman Filters 176

14.1 Error Ellipses 176

14.1.1 The Canonical Ellipse 177

14.1.2 Determining the Eigenvalues of P 178

14.1.3 Determining the Error Ellipse Rotation Angle 179

14.1.4 Determination of the Containment Area 180

14.1.5 Parametric Plotting of Error Ellipse 181

14.1.6 Error Ellipse Example 182

14.2 Root Mean Squared Errors 182

14.3 Divergent Tracks 183

14.4 Cramer–Rao Lower Bound 184

14.4.1 The One-Dimensional Case 184

14.4.2 The Multidimensional Case 186

14.4.3 A Recursive Approach to the CRLB 186

14.4.4 The Cramer–Rao Lower Bound for Gaussian Additive Noise 190

14.4.5 The Gaussian Cramer–Rao Lower Bound with Zero Process Noise 191

14.4.6 The Gaussian Cramer–Rao Lower Bound with Linear Models 191

14.5 Performance of Kalman Class DIFAR Track Estimators 192

References 198

PART III MONTE CARLO METHODS

15 Introduction to Monte Carlo Methods 201

15.1 Approximating a Density From a Set of Monte Carlo Samples 202

15.1.1 Generating Samples from a Two-Dimensional Gaussian Mixture Density 202

15.1.2 Approximating a Density by Its Multidimensional Histogram 202

15.1.3 Kernel Density Approximation 204

15.2 General Concepts Importance Sampling 210

15.3 Summary 215

References 216

16 Sequential Importance Sampling Particle Filters 218

16.1 General Concept of Sequential Importance Sampling 218

16.2 Resampling and Regularization (Move) for SIS Particle Filters 222

16.2.1 The Inverse Transform Method 222

16.2.2 SIS Particle Filter with Resampling 226

16.2.3 Regularization 227

16.3 The Bootstrap Particle Filter 230

16.3.1 Application of the BPF to DIFAR Buoy Tracking 231

16.4 The Optimal SIS Particle Filter 233

16.4.1 Gaussian Optimal SIS Particle Filter 235

16.4.2 Locally Linearized Gaussian Optimal SIS Particle Filter 236

16.5 The SIS Auxiliary Particle Filter 238

16.5.1 Application of the APF to DIFAR Buoy Tracking 242

16.6 Approximations to the SIS Auxiliary Particle Filter 243

16.6.1 The Extended Kalman Particle Filter 243

16.6.2 The Unscented Particle Filter 243

16.7 Reducing the Computational Load Through Rao-Blackwellization 245

References 245

17 The Generalized Monte Carlo Particle Filter 247

17.1 The Gaussian Particle Filter 248

17.2 The Combination Particle Filter 250

17.2.1 Application of the CPF–UKF to DIFAR Buoy Tracking 252

17.3 Performance Comparison of All DIFAR Tracking Filters 253

References 255

18 A Spherical Constant Velocity Model for Target Tracking in Three Dimensions 259

18.1 Tracking a Target in Cartesian Coordinates 261

18.1.1 Object Dynamic Motion Model 262

18.1.2 Sensor Data Model 263

18.1.3 GaussianTracking Algorithms for a Cartesian StateVector 264

18.2 Tracking a Target in Spherical Coordinates 265

18.2.1 State Vector Position and Velocity Components in Spherical Coordinates 266

18.2.2 Spherical State Vector Dynamic Equation 267

18.2.3 Observation Equations with a Spherical State Vector 270

18.2.4 GaussianTracking Algorithms for a Spherical StateVector 270

18.3 Implementation of Cartesian and Spherical Tracking Filters 273

18.3.1 Setting Values for q 273

18.3.2 Simulating Radar Observation Data 274

18.3.3 Filter Initialization 276

18.4 Performance Comparison for Various Estimation Methods 278

18.4.1 Characteristics of the Trajectories Used for Performance Analysis 278

18.4.2 Filter Performance Comparisons 282

18.5 Some Observations and Future Considerations 293

APPENDIX 18.A Three-Dimensional Constant Turn Rate Kinematics 294

18.A.1 General Velocity Components for Constant Turn Rate Motion 294

18.A.2 General Position Components for Constant Turn Rate Motion 297

18.A.3 Combined Trajectory Transition Equation 299

18.A.4 Turn Rate Setting Based on a Desired Turn Acceleration 299

APPENDIX 18.B Three-Dimensional Coordinate Transformations 301

18.B.1 Cartesian-to-Spherical Transformation 302

18.B.2 Spherical-to-Cartesian Transformation 305

References 306

19 Tracking a Falling Rigid Body Using Photogrammetry 308

19.1 Introduction 308

19.2 The Process (Dynamic) Model for Rigid Body Motion 311

19.2.1 Dynamic Transition of the Translational Motion of a Rigid Body 311

19.2.2 Dynamic Transition of the Rotational Motion of a Rigid Body 313

19.2.3 Combined Dynamic Process Model 316

19.2.4 The Dynamic Process Noise Models 317

19.3 Components of the Observation Model 318

19.4 Estimation Methods 321

19.4.1 A Nonlinear Least Squares Estimation Method 321

19.4.2 An Unscented Kalman Filter Method 323

19.4.3 Estimation Using the Unscented Combination Particle Filter 325

19.4.4 Initializing the Estimator 326

19.5 The Generation of Synthetic Data 328

19.5.1 Synthetic Rigid Body Feature Points 328

19.5.2 Synthetic Trajectory 328

19.5.3 Synthetic Cameras 333

19.5.4 Synthetic Measurements 333

19.6 Performance Comparison Analysis 334

19.6.1 Filter Performance Comparison Methodology 335

19.6.2 Filter Comparison Results 338

19.6.3 Conclusions and Future Considerations 341

APPENDIX 19.A Quaternions Axis-Angle Vectors and Rotations 342

19.A.1 Conversions Between Rotation Representations 342

19.A.2 Representation of Orientation and Rotation 343

19.A.3 Point Rotations and Frame Rotations 344

References 345

20 Sensor Fusion Using Photogrammetric and Inertial Measurements 346

20.1 Introduction 346

20.2 The Process (Dynamic) Model for Rigid Body Motion 347

20.3 The Sensor Fusion Observational Model 348

20.3.1 The Inertial Measurement Unit Component of the Observation Model 348

20.3.2 The Photogrammetric Component of the Observation Model 350

20.3.3 The Combined Sensor Fusion Observation Model 351

20.4 The Generation of Synthetic Data 352

20.4.1 Synthetic Trajectory 352

20.4.2 Synthetic Cameras 352

20.4.3 Synthetic Measurements 352

20.5 Estimation Methods 354

20.5.1 Initial Value Problem Solver for IMU Data 354

20.6 Performance Comparison Analysis 357

20.6.1 Filter Performance Comparison Methodology 359

20.6.2 Filter Comparison Results 360

20.7 Conclusions 361

20.8 Future Work 362

References 364

Index 367

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