Tracking Filters for Radar Systems Wing Ip Tam A thesis submitted in conformity with the requirernents for the degree of Master of Applied Science Graduate Department of Electrical and Cornputer Engineering University of Toronto @Copyright by Wing Ip Tarn 1997 National Library Bibliothèque nationale 1*1 ofCanada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON KIA ON4 Ottawa ON KI A ON4 Canada Canada The author has granted a non- L'auteur a accordé une licence non exclusive licence dowing the exclusive permettant ii la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seli reproduire, prêter, distribuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la fome de rnicrofiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thése ni des extraits substantiels may be printed or othemke de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. Tracking Filters for Radar Systems by Wig Ip Tam Master of Applied Science, 1997 Depart ment of Elect rical and Computer Engineering, University of Toront O Abstract In this paper we discuss the problem of target tracking in Cartesian coordinates with polar measurements and propose two efncient tracking algorithms. The first dgorithm uses the multidimensional Gauss-Hermite quadrature to evaluate the op- timal estimate as a weighted sum of functional dues. To reduce the computational requirements of this quadrature technique we have suggested several ways to reduce the dimension, the number and the order of the quadratures required for a given accu- racy. The second tracking dgorithm is based on the Gaussian sum filter. To alleviate the computationd buden associated with the Gaussian sum fdter, we have found two efficient and systematic ways to approximates a non-Gaussian and measurement- dependent bction by a weighted surn of Gaussian density function and we have derived the formula for updating the paxameters involved in the bank of Kahan- type flters and we also have proposed new techniques to control the number of terms in the Gaussian mixture at each iteration. Simulation results show that these two proposed methods are more accurate than the classical method, such as the extended Khanfilter (EKF). Acknowledgements 1am especidy gratefd for the extensive and generous assistance of my superior, Pro- fessor Dimitrios Hatzinakos, who spent time with me each week to guide me through the thesis and made numerous helpfid suggestions to me. 1wish to thank' Kostas Plataniotis for his thorough reviews and perceptive com- ments which greatly improve the quality of my thesis work. I also wish to thank Professor Pasupat hy and Professor Kwong for their consultations. 1 thank my friends William Ma, Erwin Pang, Jacky Yan, Richard Lee, Wilson Wong and Ryan Lee and his farnily for all the fun time. Lastly, much credit must be given to the members of my family, who have sup- ported me and have borne with me during this period. Contents 1 Introduction 1 2 Problem Definition and Literature Survey 4 2.1 Basic Mode1 for Radar Systems ..................... 4 2.1.1 Target Dynamics ......................... 5 2.1.2 Radar Measurernent ....................... 7 2.2 The Bayesian Approach ......................... 8 2.3 Sub-Optimal Nonlinear Filters for Radar Tracking ........... 12 3 An Efficient Radar Tkacking Algorithm Using Multidimensionai Gauss- Hermite Quadratures 18 3.1 Introduction ................................ 18 3.2 Gauss.HermiteQuadrature ........................ 19 3.3 Basic Principles of the Proposed Filter ................. 21 3.4 Complexity Analysis ........................... 24 3.5 tete ........................... 25 3.6 Simulation Results ............................ 26 4 Adaptive Gaussian Sum Algorithm for Radar 'Ikacking 29 4.1 Introduction ................................ 29 4.2 Theoretical Foundations ......................... 30 4.3 Basic Principles of the Proposed Filter ................. 30 4.3.1 Approximation of Densities ......... ,. ........ 31 4.3.2 BaskofKalmanFilters . 40 4.3.3 Growing Memory Problem . 41 4.4 Filter Structure . , , . 45 4.5 Simulation Results . 46 5 Conclusion 54 A Discretization of the Continuous-Time Equation of Motion B Track Initialization C Detail Derivations of the Curve-fitting Approach D Detail Denvation of the 'Randormation Approach E Reproducing Property of Gaussian Densities F Bank of Kalman Fiiters List of Figures 2.1 Flow Diagram for the Bayesian Estimator ............... 3.1 With the factorization the order of the quadrature is reduced for a given accuracy............................... 3.2 Flow Diagram of the Proposed Filter .................. 3.3 Cornparison of the position errors ..................... 3.4 Cornparison of the velocity errors ..................... 4.1 Plots for the me-fitting approadi ................... 4.2 Basic Principles of the Proposed Gaussian mixture Approximation Method .................................. 4.3 Fundamentals of the proposed Gaussian sum approximation method . 4.4 A . The function p(z, lx-) can be obtained through a nonlinear transfor- mation from the density p(w,) . B . Similady. some initial parameters fiorn the statistics of the noise W. can be transformed into a Gaussian sum approximation for the Çiction p(z&) .............. 4.5 Plots of the original function p(z, lx-) and the approximation ..... 4.6 Mixture before Reduction (50 components) ................ 4.7 Mixture after Reduction (10 components) ................. 4.8 Flow Diagram of the Proposed Gaussian Sum Filter .......... 4.9 Cornparison of the position errors for measurement noise level 1 ...... 4.10 Cornparison of the velocity errors for measurement noise level 1 ...... 4.11 Comparison of the position mors for measurement noise level 2 ...... 4.12 Cornparison of the vdouty enors for measurement noise Ievel 2 ..... 4.13 Superposition of 100 realizations of experiment based on Extended Kalman Filter .................................... 52 4.14 Superposition of 100 realizations of experiment based on Extended Kalman Fe ................................... 52 4.15 Cornparison of the position errors for 3-D tracking scenario ........ 53 4.16 Cornparison of the velocity errors for 3-D tracking scenario ...... 53 C.1 Weighted Gaussian Flnction ....................... 61 C.2 Aftine Transformation of the eIlipsoidal base .............. 62 List of Tables 4.1 Comparison of the accuracy and the efficiency of the proposed Gaussian sum approximation met hod and the classical method. Note that the classical method is based on the Marquardt algonthm and the number of Gaussian terms N is fixed to be 5 for both methods. 36 4.2 Comparison of the accuracy and the efficiency of the proposed Gaussian surn approximation method and the classical method. Note that the classical method is based on the Marquardt algorithm and the number of Gaussian terms N is fixed to be 5 for both methods. 40 Chapter 1 Introduction For nearly three decades the target tradcing-trajectory estimation problem has been a fruitfd applications area for state estimation. It has found wide applications in both military and cornmerciai areas [l,21 such as inertial navigation, guidance & control; global positioning system (GPS);differential global positioning system (DGPS);wide area augmentation system (WAAS); inertial navigation system (INS) ; missiles guid- ance system; satellite orbit determination; maritime surveillance; air trac control; fieeway tr&c system; fie control system; automobile navigation system; fleet man- agement; underwater target tracking system. Many problems have been solved, yet new and diversified applications still challenge engineers. This paper addresses the problem of target tracking in Cartesian coordinates with polar measurements. In tracking applications the target motion is usually best mod- eled in a simple fashion using Cartesian coordinates. Unfortunately, in most radar systems the target position measurements are provided in polar coordinates (range and azimuth) with respect to the sensor location. Tracking in Cartesian coordinates using polar measurements is a problern of nonlinear estimation. A ~~O~OUStreat- ment of the nonlinear estimation problem requises the use of stochastic integrals and stochastic differential equations [3]. In this paper, we dladapt the formal manipu- lation of the white noise process and omit the ~~O~OUSderivations using Ito calnilus. The nonlineu estimation problem is very challenging because the distribution of the state is generally non-Gaussian. Without the Gaussian property, efficient computa- tion of the conditional mean is very difficult because the optimal (conditional mean) nonlinear estimator cannot be realized with a finite-dimensional implementation; con- sequently all practicd nonlinear filters are suboptimal. These suboptimal nonlinear mten can be divided into two categories in general: the most popular technique uses a Taylor-series expmsion to approximate the nonlinear system model, e.g. the extended Kalman filter (EKF);the other approximates the conditional probability density function in such a fashion that makes the computation of the conditional mean