Integrated Tracking, Classification, and Sensor Management E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1133

Veröffentlichungsjahr: 2012

Beschreibung

A unique guide to the state of the art of tracking, classification, and sensor management This book addresses the tremendous progress made over the last few decades in algorithm development and mathematical analysis for filtering, multi-target multi-sensor tracking, sensor management and control, and target classification. It provides for the first time an integrated treatment of these advanced topics, complete with careful mathematical formulation, clear description of the theory, and real-world applications. Written by experts in the field, Integrated Tracking, Classification, and Sensor Management provides readers with easy access to key Bayesian modeling and filtering methods, multi-target tracking approaches, target classification procedures, and large scale sensor management problem-solving techniques. Features include: * An accessible coverage of random finite set based multi-target filtering algorithms such as the Probability Hypothesis Density filters and multi-Bernoulli filters with focus on problem solving * A succinct overview of the track-oriented MHT that comprehensively collates all significant developments in filtering and tracking * A state-of-the-art algorithm for hybrid Bayesian network (BN) inference that is efficient and scalable for complex classification models * New structural results in stochastic sensor scheduling and algorithms for dynamic sensor scheduling and management * Coverage of the posterior Cramer-Rao lower bound (PCRLB) for target tracking and sensor management * Insight into cutting-edge military and civilian applications, including intelligence, surveillance, and reconnaissance (ISR) With its emphasis on the latest research results, Integrated Tracking, Classification, and Sensor Management is an invaluable guide for researchers and practitioners in statistical signal processing, radar systems, operations research, and control theory.

Leseprobe

Contents

Cover

Series Page

Title Page

Copyright

Preface

Acknowledgments

References

Contributors

Part I: Filtering

Chapter 1: Angle-Only Filtering in Three Dimensions

1.1 Introduction

1.2 Statement of Problem

1.3 Tracker and Sensor Coordinate Frames

1.4 Coordinate Systems for Target and Ownship States

1.5 Dynamic Models

1.6 Measurement Models

1.7 Filter Initialization

1.8 Extended Kalman Filters

1.9 Unscented Kalman Filters

1.10 Particle Filters

1.11 Numerical Simulations and Results

1.12 Conclusions

Appendix 1A Derivations for Stochastic Differential Equations in MSC

Appendix 1B Transformations between Relative Cartesian Coordinates and MSC

Appendix 1C Filter Initialization for Relative Cartesian Coordinates and MSC

Acknowledgments

References

Chapter 2: Particle Filtering Combined with Interval Methods for Tracking Applications

2.1 Introduction

2.2 Related Works

2.3 Interval Analysis

2.4 Bayesian Filtering

2.5 Box Particle Filtering

2.6 Box Particle Filtering Derived from the Bayesian Inference Using a Mixture of Uniform Probability Density Functions

2.7 Box-PF Illustration over a Target Tracking Example

2.8 Application for a Vehicle Dynamic Localization Problem

2.9 Conclusions

Acknowledgments

References

Chapter 3: Bayesian Multiple Target Filtering Using Random Finite Sets

3.1 Introduction

3.2 Overview of the Random Finite Set Approach to Multitarget Filtering

3.3 Random Finite Sets

3.4 Multiple Target Filtering and Estimation

3.5 Multitarget Miss Distances

3.6 The Probability Hypothesis Density (PHD) Filter

3.7 The Cardinalized PHD Filter

3.8 Numerical Examples

3.9 MeMBer Filter

Acknowledgments

References

Chapter 4: The Continuous Time Roots of the Interacting Multiple Model Filter

4.1 Introduction

4.2 Hidden Markov Model filter

4.3 System with Markovian Coefficients

4.4 Markov Jump Linear System

4.5 Continuous-Discrete Filtering

4.6 Concluding Remarks

Appendix 4A Differentiation Rule for Discontinuous Semimartingales

Appendix 4B Derivation of Differential for

References

Part II: Multitarget Multisensor Tracking

Chapter 5: Multitarget Tracking Using Multiple Hypothesis Tracking

5.1 Introduction

5.2 Tracking Algorithms

5.3 Track Filtering

5.4 MHT Algorithms

5.5 Hybrid-State Derivations of MHT Equations

5.6 The Target-Death Problem

5.7 Examples for MHT

5.8 Summary

Acknowledgments

References

Chapter 6: Tracking and Data Fusion for Ground Surveillance

6.1 Introduction to Ground Surveillance

6.2 GMTI Sensor Model

6.3 Bayesian Approach to Ground Moving Target Tracking

6.4 Exploitation of Road Network Data

6.5 Convoy Track Maintenance using Random Matrices

6.6 Convoy Tracking with the Cardinalized Probability Hypothesis Density Filter

References

Chapter 7: Performance Bounds for Target Tracking: Computationally Efficient Formulations and Associated Applications

7.1 Introduction

7.2 Bayesian Performance Bounds

7.3 PCRLB Formulations in Cluttered Environments

7.4 An Approximate PCRLB for Maneuering Target Tracking

7.5 A General Framework for the Deployment of Stationary Sensors

7.6 UAV Trajectory Planning

7.7 Summary and Conclusions

Acknowledgments

References

Chapter 8: Track-Before-Detect Techniques

8.1 Introduction

8.2 Models

8.3 Baum Welch Algorithm

8.4 Dynamic Programming: Viterbi Algorithm

8.5 Particle Filter

8.6 ML-PDA

8.7 H-PMHT

8.8 Performance Analysis

8.9 Applications: Radar and IRST Fusion

8.10 Future Directions

References

Chapter 9: Advances in Data Fusion Architectures

9.1 Introduction

9.2 Dense-Target Scenarios

9.3 Multiscale Sensor Scenarios

9.4 Tracking in Large Sensor Networks

9.5 Multiscale Objects

9.6 Measurement Aggregation

9.7 Conclusions

References

Chapter 10: Intent Inference and Detection of Anomalous Trajectories: A Metalevel Tracking Approach

10.1 Introduction

10.2 Anomalous Trajectory Classification Framework

10.3 Trajectory Modeling and Inference Using Stochastic Context-Free Grammars

10.4 Trajectory Modeling and Inference using Reciprocal Processes (RP)

10.5 Example 1: Metalevel Tracking for GMTI Radar

10.6 Example 2: Data Fusion in a Multicamera Network

10.7 Conclusion

Acknowledgment

References

Part III: Sensor Management and Control

Chapter 11: Radar Resource Management for Target Tracking—A Stochastic Control Approach

11.1 Introduction

11.2 Problem Formulation

11.3 Structural Results and Lattice Programming for Micromanagement

11.4 Radar Scheduling for Maneuvering Targets Modeled as Jump Markov Linear System

11.5 Summary

References

Chapter 12: Sensor Management for Large-Scale Multisensor-Multitarget Tracking

12.1 Introduction

12.2 Target Tracking Architectures

12.3 Posterior Cramér–Rao Lower Bound

12.4 Sensor Array Management for Centralized Tracking

12.5 Sensor Array Management for Distributed Tracking

12.6 Sensor Array Management for Decentralized Tracking

12.7 Conclusions

Appendix 12A Local Search

Appendix 12B Genetic Algorithm

Appendix 12C Ant Colony Optimization

References

Part IV: Estimation and Classification

Chapter 13: Efficient Inference in General Hybrid Bayesian Networks for Classification

13.1 Introduction

13.2 Message Passing: Representation and Propagation

13.3 Network Partition and Message Integration for Hybrid Model

13.4 Hybrid Message Passing Algorithm for Classification

13.5 Numerical Experiments

13.6 Concluding Remarks

References

Chapter 14: Evaluating Multisensor Classification Performance with Bayesian Networks

14.1 Introduction

14.2 Single-Sensor Model

14.3 Multisensor Fusion Systems—Design and Performance Evaluation

14.4 Summary and Continuing Questions

Appendix 14A Developing a Sensor's Local Confusion Matrix

Appendix 14B Solving for the Off-Diagonal Elements of the Global Classification Matrix

Appendix 14C A Graph-Theoretic Representation of the Recursive Approach for Estimating the Diagonal Elements of the GCM

Appendix 14D Designing Monte Carlo Simulations of the GCM

Appendix 14E Proof of Approximation 1

References

Chapter 15: Detection and Estimation of Radiological Sources

15.1 Introduction

15.2 Estimation of Point Sources

15.3 Estimation of Distributed Sources

15.4 Searching for Point Sources

15.5 Conclusions

Acknowledgments

References

Part V: Decision Fusion and Decision Support

Chapter 16: Distributed Detection and Decision Fusion with Applications to Wireless Sensor Networks

16.1 Introduction

16.2 Elements of Detection Theory

16.3 Distributed Detection with Multiple Sensors

16.4 Distributed Detection in Wireless Sensor Networks

16.5 Copula-Based Fusion of Correlated Decisions

16.6 Conclusion

Acknowledgments

Appendix 16A Performance Analysis of a Network with Nonidentical Sensors via Approximations

References

Chapter 17: Evidential Networks for Decision Support in Surveillance Systems

17.1 Introduction

17.2 Valuation Algebras

17.3 Local Computation in a VA

17.4 Theory of Evidence as a Valuation Algebra

17.5 Examples of Decision Support Systems

Appendix 17A Construction of a BJT

Appendix 17B Inward Propagation

References

Index

IEEE Press445 Hoes LanePiscataway, NJ 08854

IEEE Press Editorial BoardJohn B. Anderson, Editor in Chief

Kenneth Moore, Director of IEEE Book and Information Services (BIS)

Technical ReviewersSamuel S. Blackman, RaytheonProfessor Rob Evans, University of Melbourne, AustraliaRamanarayanan Viswanathan, Southern Illinois University Carbondale

Cover Illustration: Courtesy of Ba-Ngu Vo

Cover Design: John Wiley & Sons, Inc.

Copyright © 2013 by The Institute of Electrical and Electronics Engineers, Inc.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.

Published simultaneously in Canada.

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ISBN: 978-0-470-63905-4

Preface

This book has been a long time in the making, starting with a series of conversations in 2007 during the Colloquium on Information Fusion in Xi'an China, followed by many discussions at various conferences as well as phone calls from half way around the globe. These conversations were centered on the ever-growing interest in tracking and sensor management in the wider community and the accessibility of the state-of-the-art techniques to graduate students, researchers, and engineers.

The research on multitarget tracking and sensor management was driven by aerospace and space applications such as radar, sonar, guidance, navigation, air traffic control, and space exploration in the 1960s. Since then, these research areas have flourished into other diverse disciplines such as image processing, oceanography, autonomous vehicles and robotics, remote sensing, biomedical research, and space debris tracking. Recent efficient multitarget tracking techniques and advances in sensing and computing technology have opened up prospective applications in areas such as driving safety and traffic monitoring, homeland security, and surveillance of public facilities.

While interest in this discipline is rapidly growing with many exciting advances during the last decade, comprehensive and accessible account of significant developments in the field are few and far between. The focus of our book is on expository writing, clear description of theoretical developments, and real-world applications in these areas. The chapters of the book are divided into five groups under the headings: Filtering, Multitarget Multisensor Tracking, Sensor Management and Control, Estimation and Classification, and Decision Fusion and Decision Support. Each chapter is solicited from internationally renowned experts in their respective areas. By providing concise and detailed descriptions, such as pseudo codes for algorithms, we endeavor to facilitate the implementations of the state-of-the-art algorithms, thereby making a wealth of approaches and techniques accessible to a wider audience.

Chapter 1 develops three classes of filtering algorithms for the angle-only filtering problem in 3D using bearing and elevation measurements. The dynamic models used by these filtering algorithms are the nearly constant velocity model for the relative Cartesian state vector, exact discrete-time dynamic model for modified spherical coordinates (MSC), and exact continuous-time dynamic model for MSC. The extended Kalman filter (EKF), unscented Kalman filter (UKF), and particle filter (PF) are developed for each class, of which the UKF and PF based on the exact continuous-time dynamic model for MSC represent new algorithms. Finally, a comparative evaluation of their accuracy and computational complexity is presented using Monte Carlo simulations.

Chapter 2 presents a recently introduced approach called box particle filtering which emerged from the synergy between sequential Monte Carlo (SMC) methods and interval analysis. A theoretical derivation of the box particle filter is given based on mixtures of uniform probability density functions with box supports. Experiments with both simulated and real data show the advantages of the box particle filter over the conventional particle filter for certain classes of problems.

Chapter 3 presents an accessible account of developments in the random finite set approach to the multitarget tracking problem. This chapter is classified under the filtering part of the book because fundamentally, the random finite set approach poses the multitarget tracking problem as a Bayesian filtering problem (in the space of finite subsets or simple finite point patterns). In this chapter, we discuss the notion of a mathematically consistent error metric for multitarget tracking and present arguments for the finite set representation of the multitarget state. We also detail random finite set-based algorithms such as the probability hypothesis density (PHD), Cardinalized PHD (CPHD), and Multitarget Multi-Bernoulli filters.

The interacting multiple model (IMM) filter is a well-established and widely used algorithm at present for maneuvering target tracking. Currently, almost all IMM filtering algorithms used are discrete-time filtering algorithms. However, it is rather unknown that the original IMM filter was developed in a purely continuous-time setting, which subsequently led to the development of the discrete-time IMM filter. Chapter 4 presents in detail the mathematical development of exact continuous-time nonlinear filtering for jump Markov systems, including the continuous-time IMM filter as well as continuous-discrete-time IMM and particle filters.

The track-oriented multiple hypothesis tracking (MHT) for multisensor multi-target tracking is regarded as one of the most advanced tracking algorithms at present, relative to which other tracking algorithms are compared. Chapter 5 presents a hybrid-state derivation of the track-oriented MHT equations that is closely related to the original treatment by Kurien [1] with some minor modifications. The target death problem inherent in PHD filtering is also addressed and it is shown that it does not arise in the track-oriented MHT. A number of illustrative examples are considered to demonstrate the merits of MHT. In order to make the chapter self-contained, a comprehensive review of the state-of-the-art filtering and tracking algorithms are summarized in the beginning of the chapter, with extensive references.

Chapter 6 describes several strategies to improve airborne ground surveillance by enhanced tracking performance. The following topics are considered: specific sensor modeling, improved data association using signal strength measurements, exploitation of digital road maps, and detection and tracking of target groups. The proposed algorithms are shown to enhance track precision and track continuity over conventional techniques.

Chapter 7 presents a review of recent developments in the calculation of mean square error tracker performance bounds, together with examples that demonstrate how such bounds can be used as a basis for performing online sensor management. The review concentrates on the posterior Cramér–Rao lower bound (PCRLB), and describes computationally efficient formulations of the PCRLB that take account of real-world complexity. Two applications, concerned with the deployment of passive sonobuoys, and UAV trajectory planning, demonstrate that the PCRLB provides an efficient mechanism for performing sensor management in order to accurately track an evasive target.

Chapter 8 presents a review of the track-before-detect (TBD) problem, namely tracking when the measurement is an intensity map. It describes the different methods that have been applied to this problem and compares their performance on a simple scenario. A case study fusing data from an infra-red camera and microwave radar illustrate the advantages that can be gained through the improved sensitivity offered by the track-before-detect algorithm.

While centralized detection and estimation are known to outperform distributed approaches, the same is not always true when one is confronted with measurement origin uncertainty. Indeed, all known approaches to multitarget tracking are suboptimal. Thus, judicious multistage processing may outperform single-stage processing. In a sense, we are choosing between (suboptimal) distributed and (suboptimal) centralized processing. Chapter 9 identifies a number of scenarios where multistage fusion architectures lead to promising results.

Chapter 10 presents an overview of meta-level tracking algorithms for inferring target intent. Such meta-level trackers are fully compatible with existing target tracking algorithms and form the sensor–human interface. To capture the complex spatial trajectories of targets, stochastic context free grammars are used. Then Bayesian signal processing algorithms are used to estimate the target trajectory.

Chapter 11 presents an overview of stochastic control methods for radar resource management. Radar resource management is intrinsically a partially observed stochastic control problem since decisions need to be made based on the estimates provided by a tracker. Such problems are typically intractable unless the underlying structure is exploited. The chapter shows how supermodularity and lattice programming methods can be used to characterize the structure of the optimal radar scheduling policy.

Chapter 12 addresses the problem of multisensor resource management with application to multitarget tracking. Specifically, sensor selection, sensor placement, and performance evaluation are considered in detail. A particular contribution of this chapter is the derivation of the Posterior Cramér–Rao Lower Bound (PCRLB) to quantify the achievable estimation accuracy in multitarget tracking problem, which is used as the key metric for sensor management.

Chapter 13 on efficient inference in general hybrid Bayesian networks for classification introduces a probabilistic inference framework for hybrid Bayesian networks, in which both discrete and continuous variables are present and their functional relationship can be nonlinear. This type of model is very common in classification applications where discrete random variables representing entity types or situational hypotheses are to be assessed given noisy observations represented by mixed discrete and continuous variables.

Chapter 14 presents a new analytical approach for quantifying the long-run performance of a multisensor classification system modeled by a Bayesian network. The methodology has been applied to fusion performance evaluation of practical tracking and classification systems involving multiple sensor types. It illustrates the use of off-line evaluation to estimate marginal performance gains and sensor mode selection using measures and metrics derived herein.

Chapter 15 considers the problem of detecting, estimating, and searching for point and distributed sources of radiation. A Bayesian approach is adopted with the posterior density approximated using the notion of progressive correction combined with either Monte Carlo approximation or linearization.

In Chapter 16, important problems of distributed detection and decision fusion for a multisensor system are discussed. With known local sensors' performance indices, the design for optimal decision fusion rule at the fusion center and the optimal local decision rules at sensors are presented in both parallel and serial networks under either the Bayesian or Neyman–Pearson criterion. When local sensors are nonidentical and their performance indices are unknown, the counting rule is proposed and its exact as well as approximated performance are analyzed. For the challenging problem of distributed detection with correlated observations, a decision fusion framework using copula theory is described, which is shown particularly useful for non-Gaussian distributed and nonlinearly dependent sensor observations.

Chapter 17 presents the development of an automatic knowledge-based information fusion system to support the decision making process in a reliable, timely, and consistent manner even in conditions of uncertainty. This is obtained by using the framework of valuation algebra for knowledge representation and reasoning under uncertainty together with the algorithms for performing local computations in valuation algebra. These algorithms are then specialized to the theory of belief functions. Two practical examples are discussed: decision support systems for target identification and threat assessment.

Acknowledgments

We are indebted to Dr. Sankar Basu of National Science Foundation who first suggested the idea of writing a book for Wiley/IEEE to Mahendra Mallick. Dr. Basu emphasized that the book should pay special attention to solving practical problems of interest with sound algorithms and examples.

The editors would like to thank their respective universities for the provision of the facilities for completing this book, namely the University of British Columbia and Curtin University. In the preparation of this book, the third editor, Professor Vo is supported in part by the Australian Research Council under the discovery grant DP0878158.

The contents of the book have greatly benefitted from interactions with numerous researchers from diverse fields. We express our sincere thanks to the late Jean-Pierre Le Cadre (IRISA/CNRS, France), Samuel S. Blackman (Raytheon Systems Company, USA), Barbara La Scala (National Australia Bank, Australia), Yvo Boers (THALES Nederland), and David Salmond (QinetiQ, UK).

Finally, we would like to acknowledge our families for their support and patience during the writing, correcting, and editing of this book.

References

1. T. Kurien, Issues in the design of practical multitarget tracking algorithms, in: Y. Bar-Shalom (Ed.), Multitarget-Multisensor Tracking: Advanced Applications, Artech House, Norwood, MA, USA, 1990, Chapter 3.

Contributors

Fahed Abdallah, HEUDIASYC, UMR CNRS 6599, Université de Technologie de Compi‘egne, France

Sanjeev Arulampalam, Submarine Combat Systems, Maritime Operations Division, Defence Science & Technology Organisation, Edinburgh, South Australia, Australia

Alessio Benavoli, Istituto “Dalle Molle” di Studi sull'Intelligenza Artificiale (IDSIA), Manno (Lugano), Switzerland

Henk Blom, National Aerospace Laboratory NLR, Amsterdam, The Netherlands

Craig Carthel, Compunetix, Inc., Monroeville, PA, USA

Kuo-Chu Chang, Systems Engineering and Operations Research, Volgenau School of Engineering, George Mason University, Fairfax, VA, USA

Qi Cheng, School of Electrical & Computer Engineering, Oklahoma State University, Stillwater, OK, USA

Daniel Clark, Department of Electrical, Electronic and Computing Engineering, Heriot-Watt University, Riccarton, Edinburgh, UK

Stefano Coraluppi, Compunetix, Inc., Monroeville, PA, USA

Samuel Davey, Intelligence Surveillance & Reconnaissance Division, Defence Science & Technology Organization, Edinburgh, South Australia, Australia

Michael Feldmann, Department Sensor Data and Information Fusion, Fraunhofer FKIE Wachtberg, Germany

Amadou Gning, Department of Computer Science, University College London, London, UK

Neil Gordon, Intelligence Surveillance & Reconnaissance Division, Defence Science & Technology Organisation, Edinburgh, South Australia, Australia

Marcel Hernandez, Hernandez Technical Solutions Ltd., Malvern, UK

Thia Kirubarajan, Electrical and Computer Engineering Department, Communications Research Laboratory, McMaster University, Hamilton, Ontario, Canada

Wolfgang Koch, Department Sensor Data and Information Fusion, Fraunhofer FKIE, Wachtberg, Germany

Vikram Krishnamurthy, Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, Canada

Mahendra Mallick, Propagation Research Associates, Inc., Marietta, GA, USA

Michael Mertens, Department Sensor Data and Information Fusion, Fraunhofer FKIE Wachtberg, Germany

Lyudmila Mihaylova, School of Computing and Communications, InfoLab21, Lancaster University, Lancaster, UK

Mark Morelande, Department of Electrical and Electronic Engineering, The University of Melbourne, Parkville, Victoria, Australia

Ruixin Niu, Department of Electrical & Computer Engineering, Virginia Commonwealth University, Richmond, VA, USA

Branko Ristic, Intelligence Surveillance & Reconnaissance Division, Defence Science & Technology Organization, Fishermans Bend, Victoria, Australia

Mark Rutten, Intelligence Surveillance & Reconnaissance Division, Defence Science & Technology Organization, Edinburgh, South Australia, Australia

Eswar Sivaraman, United Airlines, Enterprise Optimization, Chicago, IL, USA

Wei Sun, SEOR and C4I Center, George Mason University, Fairfax, VA, USA

Ashok Sundaresan, GE Global Research, Niskayuna, NY, USA

Ratnasingham Tharmarasa, Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada

Martin Ulmke, Department Sensor Data and Information Fusion, Fraunhofer FKIE, Wachtberg, Germany

Pramod Varshney, Department of Electrical Engineering & Computer Science, Syracuse University, Syracuse, NY, USA

Ba-Ngu Vo, Department of Electrical and Computer Engineering, Curtin University, WA, Australia

Ba-Tuong Vo, Department of Electrical and Computer Engineering, Curtin University, WA, Australia

Yanjun Yan, ARCON Corporation, Waltham, MA, USA

Part I

Filtering

Chapter 1

Angle-Only Filtering in Three Dimensions

Mahendra Mallick, Mark Morelande, Lyudmila Mihaylova, Sanjeev Arulampalam, and Yanjun Yan

1.1 Introduction

The angle-only filtering problem in 3D using bearing and elevation measurements is an important problem in many passive tracking applications. For example, it arises in passive ranging using an infrared search and track (IRST) sensor [1, 2], passive sonar, passive radar in the presence of jamming, and satellite to satellite passive tracking [3, 4]. It is the counterpart of the bearings-only filtering problem in 2D. For the 3D case, the objective is to estimate the three-dimensional state of a target, such as position and velocity, using noisy measurements of bearing and elevation from a single maneuvering platform. A great deal of research has been carried out for the bearings-only filtering problem in 2D—see for example, [5–9] and the references therein. However, the number of publications for the angle-only filtering problem in 3D is relatively small [3, 4, 10–18].

Research in angle-only filtering in 3D began by extending the methods developed for the counterpart problem in 2D. For the 2D bearings-only filtering problem, it is well known that, for a target moving with uniform motion, target range cannot be observed without an ownship (sensor) maneuver [19]. Though the prior distribution of the initial state aids in improving observability, its contribution degrades with time. In addition, the accuracy of the state estimate is highly dependent on the nature of the maneuver and the particular target–observer geometry. Early recursive algorithms for this problem were based on the extended Kalman filter (EKF) [20–22] using Cartesian coordinates [23]. Researchers noted that the performance of these algorithms was poor due to premature collapse of the covariance matrix. This led to the formulation of the modified polar coordinates (MPC) [5, 24, 25], in which improved performance was demonstrated.

The state vector in MPC consists of bearing, bearing-rate, range-rate divided by range, and the inverse of range [5, 9, 24]. The important difference between the MPC and the Cartesian coordinates is that in MPC, the first three elements of the state are observable even before an ownship maneuver. By decoupling the observable and unobservable components of the state vector, this approach was demonstrated to prevent ill-conditioning of the covariance matrix which led to better filter performance [5, 24, 25]. The continuous-time dynamic model for the MPC is nonlinear and is represented by four continuous-time stochastic differential equations (SDEs). The key difficulty of using MPC is that the commonly applied nearly constant velocity model (NCVM) for nonmaneuvering targets is highly nonlinear in MPC. In fact, there has been some confusion in the literature as to how to convert the widely used NCVM from Cartesian coordinates to MPC. In the original work on bearings-only filtering in MPC [24, 25], these equations are numerically integrated to obtain the predicted state and covariance at the discrete measurement times. Subsequently, Aidala and Hammel [5] noted that exact, closed-form discrete-time stochastic difference equations in MPC can be obtained by using the nonlinear transformations between MPC and Cartesian coordinates. They proposed an EKF in these coordinates and claimed superior performance relative to its Cartesian counterpart.

Angle-only filtering in 3D is beset by the same observability issues that arise in the 2D case [19, 26]. As such, most of the research in the 3D angle-only filtering problem has focused on developing algorithms in the modified spherical coordinates (MSC) [17] —the 3D equivalent of MPC. The components of MSC are elevation, elevation-rate, bearing, bearing-rate times cosine of elevation, the inverse of range, and range-rate divided by range. As with MPC in 2D filtering, the main problem when using MSC in 3D filtering is the nonlinear dynamic model which arises when a target moves with the NCVM in Cartesian coordinates. Again, a number of ways of transforming the NCVM in Cartesian coordinates to MSC have been proposed.