Bayesian Estimation and Tracking

ANTON J. HAUG, PhD, is member of the technical staff at the Applied Physics Laboratory at The Johns Hopkins University, where he develops advanced target tracking methods in support of the Air and Missile Defense Department. Throughout his career, Dr. Haug has worked across diverse areas such as target tracking; signal and array processing and processor design; active and passive radar and sonar design; digital communications and coding theory; and time- frequency analysis.

• Preface xvAcknowledgments xviiList of Figures XixList of Tables xxvPART I PRELIMINARIES1 Introduction 31.1 Bayesian Inference 41.2 Bayesian Hierarchy of Estimation Methods 51.3 Scope of This Text 61.3.1 Objective 61.3.2 Chapter Overview and Prerequisites 61.4 Modeling and Simulation with MATLAB® 8References 92 Preliminary Mathematical Concepts 112.1 A Very Brief Overview of Matrix Linear Algebra 112.1.1 Vector and Matrix Conventions and Notation 112.1.2 Sums and Products 122.1.3 Matrix Inversion 132.1.4 Block Matrix Inversion 142.1.5 Matrix Square Root 152.2 Vector Point Generators 162.3 Approximating Nonlinear Multidimensional Functions with Multidimensional Arguments 192.3.1 Approximating Scalar Nonlinear Functions 192.3.2 Approximating Multidimensional Nonlinear Functions 232.4 Overview of Multivariate Statistics 292.4.1 General Definitions 292.4.2 The Gaussian Density 32References 403 General Concepts of Bayesian Estimation 423.1 Bayesian Estimation 433.2 Point Estimators 433.3 Introduction to Recursive Bayesian Filtering of Probability Density Functions 463.4 Introduction to Recursive Bayesian Estimation of the State Mean and Covariance 493.4.1 State Vector Prediction 503.4.2 State Vector Update 513.5 Discussion of General Estimation Methods 55References 554 Case Studies: Preliminary Discussions 564.1 The Overall Simulation/Estimation/Evaluation Process 574.2 A Scenario Simulator for Tracking a Constant Velocity Target Through a DIFAR Buoy Field 584.2.1 Ship Dynamics Model 584.2.2 Multiple Buoy Observation Model 594.2.3 Scenario Specifics 594.3 DIFAR Buoy Signal Processing 624.4 The DIFAR Likelihood Function 67References 69PART II THE GAUSSIAN ASSUMPTION: A FAMILY OF KALMAN FILTER ESTIMATORS5 The Gaussian Noise Case: Multidimensional Integration of Gaussian-Weighted Distributions 735.1 Summary of Important Results From Chapter 3 745.2 Derivation of the Kalman Filter Correction (Update) Equations Revisited 765.3 The General Bayesian Point Prediction Integrals for Gaussian Densities 785.3.1 Refining the Process Through an Affine Transformation 805.3.2 General Methodology for Solving Gaussian-Weighted Integrals 82References 856 The Linear Class of Kalman Filters 866.1 Linear Dynamic Models 866.2 Linear Observation Models 876.3 The Linear Kalman Filter 886.4 Application of the LKF to DIFAR Buoy Bearing Estimation 88References 927 The Analytical Linearization Class of Kalman Filters: The Extended Kalman Filter 937.1 One-Dimensional Consideration 937.1.1 One-Dimensional State Prediction 947.1.2 One-Dimensional State Estimation Error Variance Prediction 957.1.3 One-Dimensional Observation Prediction Equations 967.1.4 Transformation of One-Dimensional Prediction Equations 967.1.5 The One-Dimensional Linearized EKF Process 987.2 Multidimensional Consideration 987.2.1 The State Prediction Equation 997.2.2 The State Covariance Prediction Equation 1007.2.3 Observation Prediction Equations 1027.2.4 Transformation of Multidimensional Prediction Equations 1037.2.5 The Linearized Multidimensional Extended Kalman Filter Process 1057.2.6 Second-Order Extended Kalman Filter 1057.3 An Alternate Derivation of the Multidimensional Covariance Prediction Equations 1077.4 Application of the EKF to the DIFAR Ship Tracking Case Study 1087.4.1 The Ship Motion Dynamics Model 1087.4.2 The DIFAR Buoy Field Observation Model 1097.4.3 Initialization for All Filters of the Kalman Filter Class 1117.4.4 Choosing a Value for the Acceleration Noise 1127.4.5 The EKF Tracking Filter Results 112References 1148 The Sigma Point Class: The Finite Difference Kalman Filter 1158.1 One-Dimensional Finite Difference Kalman Filter 1168.1.1 One-Dimensional Finite Difference State Prediction 1168.1.2 One-Dimensional Finite Difference State Variance Prediction 1178.1.3 One-Dimensional Finite Difference Observation Prediction Equations 1188.1.4 The One-Dimensional Finite Difference Kalman Filter Process 1188.1.5 Simplified One-Dimensional Finite Difference Prediction Equations 1188.2 Multidimensional Finite Difference Kalman Filters 1208.2.1 Multidimensional Finite Difference State Prediction 1208.2.2 Multidimensional Finite Difference State Covariance Prediction 1238.2.3 Multidimensional Finite Difference Observation Prediction Equations 1248.2.4 The Multidimensional Finite Difference Kalman Filter Process 1258.3 An Alternate Derivation of the Multidimensional Finite Difference Covariance Prediction Equations 125References 1279 The Sigma Point Class: The Unscented Kalman Filter 1289.1 Introduction to Monomial Cubature Integration Rules 1289.2 The Unscented Kalman Filter 1309.2.1 Background 1309.2.2 The UKF Developed 1319.2.3 The UKF State Vector Prediction Equation 1349.2.4 The UKF State Vector Covariance Prediction Equation 1349.2.5 The UKF Observation Prediction Equations 1359.2.6 The Unscented Kalman Filter Process 1359.2.7 An Alternate Version of the Unscented Kalman Filter 1359.3 Application of the UKF to the DIFAR Ship Tracking Case Study 137References 13810 The Sigma Point Class: The Spherical Simplex Kalman Filter 14010.1 One-Dimensional Spherical Simplex Sigma Points 14110.2 Two-Dimensional Spherical Simplex Sigma Points 14210.3 Higher Dimensional Spherical Simplex Sigma Points 14410.4 The Spherical Simplex Kalman Filter 14410.5 The Spherical Simplex Kalman Filter Process 14510.6 Application of the SSKF to the DIFAR Ship Tracking Case Study 146Reference 14711 The Sigma Point Class: The Gauss–Hermite Kalman Filter 14811.1 One-Dimensional Gauss–Hermite Quadrature 14911.2 One-Dimensional Gauss–Hermite Kalman Filter 15311.3 Multidimensional Gauss–Hermite Kalman Filter 15511.4 Sparse Grid Approximation for High Dimension/High Polynomial Order 16011.5 Application of the GHKF to the DIFAR Ship Tracking Case Study 163References 16312 The Monte Carlo Kalman Filter 16412.1 The Monte Carlo Kalman Filter 167Reference 16713 Summary of Gaussian Kalman Filters 16813.1 Analytical Kalman Filters 16813.2 Sigma Point Kalman Filters 17013.3 A More Practical Approach to Utilizing the Family of Kalman Filters 174References 17514 Performance Measures for the Family of Kalman Filters 17614.1 Error Ellipses 17614.1.1 The Canonical Ellipse 17714.1.2 Determining the Eigenvalues of P 17814.1.3 Determining the Error Ellipse Rotation Angle 17914.1.4 Determination of the Containment Area 18014.1.5 Parametric Plotting of Error Ellipse 18114.1.6 Error Ellipse Example 18214.2 Root Mean Squared Errors 18214.3 Divergent Tracks 18314.4 Cramer–Rao Lower Bound 18414.4.1 The One-Dimensional Case 18414.4.2 The Multidimensional Case 18614.4.3 A Recursive Approach to the CRLB 18614.4.4 The Cramer–Rao Lower Bound for Gaussian Additive Noise 19014.4.5 The Gaussian Cramer–Rao Lower Bound with Zero Process Noise 19114.4.6 The Gaussian Cramer–Rao Lower Bound with Linear Models 19114.5 Performance of Kalman Class DIFAR Track Estimators 192References 198PART III MONTE CARLO METHODS15 Introduction to Monte Carlo Methods 20115.1 Approximating a Density From a Set of Monte Carlo Samples 20215.1.1 Generating Samples from a Two-Dimensional Gaussian Mixture Density 20215.1.2 Approximating a Density by Its Multidimensional Histogram 20215.1.3 Kernel Density Approximation 20415.2 General Concepts Importance Sampling 21015.3 Summary 215References 21616 Sequential Importance Sampling Particle Filters 21816.1 General Concept of Sequential Importance Sampling 21816.2 Resampling and Regularization (Move) for SIS Particle Filters 22216.2.1 The Inverse Transform Method 22216.2.2 SIS Particle Filter with Resampling 22616.2.3 Regularization 22716.3 The Bootstrap Particle Filter 23016.3.1 Application of the BPF to DIFAR Buoy Tracking 23116.4 The Optimal SIS Particle Filter 23316.4.1 Gaussian Optimal SIS Particle Filter 23516.4.2 Locally Linearized Gaussian Optimal SIS Particle Filter 23616.5 The SIS Auxiliary Particle Filter 23816.5.1 Application of the APF to DIFAR Buoy Tracking 24216.6 Approximations to the SIS Auxiliary Particle Filter 24316.6.1 The Extended Kalman Particle Filter 24316.6.2 The Unscented Particle Filter 24316.7 Reducing the Computational Load Through Rao-Blackwellization 245References 24517 The Generalized Monte Carlo Particle Filter 24717.1 The Gaussian Particle Filter 24817.2 The Combination Particle Filter 25017.2.1 Application of the CPF–UKF to DIFAR Buoy Tracking 25217.3 Performance Comparison of All DIFAR Tracking Filters 253References 255PART IV ADDITIONAL CASE STUDIES18 A Spherical Constant Velocity Model for Target Tracking in Three Dimensions 25918.1 Tracking a Target in Cartesian Coordinates 26118.1.1 Object Dynamic Motion Model 26218.1.2 Sensor Data Model 26318.1.3 GaussianTracking Algorithms for a Cartesian StateVector 26418.2 Tracking a Target in Spherical Coordinates 26518.2.1 State Vector Position and Velocity Components in Spherical Coordinates 26618.2.2 Spherical State Vector Dynamic Equation 26718.2.3 Observation Equations with a Spherical State Vector 27018.2.4 GaussianTracking Algorithms for a Spherical StateVector 27018.3 Implementation of Cartesian and Spherical Tracking Filters 27318.3.1 Setting Values for q 27318.3.2 Simulating Radar Observation Data 27418.3.3 Filter Initialization 27618.4 Performance Comparison for Various Estimation Methods 27818.4.1 Characteristics of the Trajectories Used for Performance Analysis 27818.4.2 Filter Performance Comparisons 28218.5 Some Observations and Future Considerations 293APPENDIX 18.A Three-Dimensional Constant Turn Rate Kinematics 29418.A.1 General Velocity Components for Constant Turn Rate Motion 29418.A.2 General Position Components for Constant Turn Rate Motion 29718.A.3 Combined Trajectory Transition Equation 29918.A.4 Turn Rate Setting Based on a Desired Turn Acceleration 299APPENDIX 18.B Three-Dimensional Coordinate Transformations 30118.B.1 Cartesian-to-Spherical Transformation 30218.B.2 Spherical-to-Cartesian Transformation 305References 30619 Tracking a Falling Rigid Body Using Photogrammetry 30819.1 Introduction 30819.2 The Process (Dynamic) Model for Rigid Body Motion 31119.2.1 Dynamic Transition of the Translational Motion of a Rigid Body 31119.2.2 Dynamic Transition of the Rotational Motion of a Rigid Body 31319.2.3 Combined Dynamic Process Model 31619.2.4 The Dynamic Process Noise Models 31719.3 Components of the Observation Model 31819.4 Estimation Methods 32119.4.1 A Nonlinear Least Squares Estimation Method 32119.4.2 An Unscented Kalman Filter Method 32319.4.3 Estimation Using the Unscented Combination Particle Filter 32519.4.4 Initializing the Estimator 32619.5 The Generation of Synthetic Data 32819.5.1 Synthetic Rigid Body Feature Points 32819.5.2 Synthetic Trajectory 32819.5.3 Synthetic Cameras 33319.5.4 Synthetic Measurements 33319.6 Performance Comparison Analysis 33419.6.1 Filter Performance Comparison Methodology 33519.6.2 Filter Comparison Results 33819.6.3 Conclusions and Future Considerations 341APPENDIX 19.A Quaternions Axis-Angle Vectors and Rotations 34219.A.1 Conversions Between Rotation Representations 34219.A.2 Representation of Orientation and Rotation 34319.A.3 Point Rotations and Frame Rotations 344References 34520 Sensor Fusion Using Photogrammetric and Inertial Measurements 34620.1 Introduction 34620.2 The Process (Dynamic) Model for Rigid Body Motion 34720.3 The Sensor Fusion Observational Model 34820.3.1 The Inertial Measurement Unit Component of the Observation Model 34820.3.2 The Photogrammetric Component of the Observation Model 35020.3.3 The Combined Sensor Fusion Observation Model 35120.4 The Generation of Synthetic Data 35220.4.1 Synthetic Trajectory 35220.4.2 Synthetic Cameras 35220.4.3 Synthetic Measurements 35220.5 Estimation Methods 35420.5.1 Initial Value Problem Solver for IMU Data 35420.6 Performance Comparison Analysis 35720.6.1 Filter Performance Comparison Methodology 35920.6.2 Filter Comparison Results 36020.7 Conclusions 36120.8 Future Work 362References 364Index 367