Simulation Based Methods Target Tracking · can be used. The sequential Monte Carlo method, or...

Linkoping Studies in Science and TechnologyThesis No. 930

Simulation Based Methods

forTarget Tracking

Rickard Karlsson

REGLERTEKNIK

AUTOMATIC CONTROL

LINKÖPING

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Division of Automatic Control & Communication systemsDepartment of Electrical Engineering

Linkopings universitet, SE–581 83 Linkoping, SwedenWWW: http://www.control.isy.liu.se

Email: [email protected]

Linkoping 2002

Simulation Based Methods for Target Tracking

c© 2002 Rickard Karlsson

Department of Electrical Engineering,Linkopings universitet,SE–581 83 Linkoping,

Sweden.

ISBN 91-7373-267-2ISSN 0280-7971

LiU-TEK-LIC-2002:03

Printed by UniTryck, Linkoping, Sweden 2002

To Karin

Abstract

In this thesis we study a Bayesian estimation formulation of the target trackingproblem. Traditionally, linear or linearized models are used, where the uncer-tainty in the sensor and motion models is typically modeled by Gaussian densities.Hence, classical sub-optimal Bayesian methods based on linearized Kalman filterscan be used. The sequential Monte Carlo method, or particle filter, provides anapproximative solution to the non-linear and non-Gaussian estimation problem.The particle filter approximates the optimal solution, hence it can outperform theKalman filter in many cases, given sufficient computational resources. A surveyover relevant tracking literature is presented including aspects as estimation, dataassociation, sensor fusion and target modeling. In various target tracking relatedestimation and data association applications, we extend or modify particle filteringalgorithms.

The passive ranging application when only angle information is available isdiscussed for several problems. In an air-to-sea application it is shown how toincorporate terrain induced constraints using a terrain database. The algorithmis also successfully evaluated on experimental sonar data acquired from a torpedosystem.

In a multi-target data association application a simulation based approach fordata association is proposed and compared to classical algorithms for an air-to-airtracking application. Moreover, the number of particles needed in the particle filteris adapted using a control structure to reduce the computational complexity.

i

Acknowledgments

I would like to thank my supervisor, Professor Fredrik Gustafsson for guidance,inspiring discussions and support during my research project. I would also like tothank Professor Lennart Ljung for giving me the opportunity to join the control& communication group and for creating a creative, stimulating and professionalatmosphere. All the members and staff at the control & communication groupare greatfully acknowledged. Moreover, Ulla Salaneck deserves extra gratitude foradministrative help and cheerful attitude.

During the project many people have been of great help and assistance. Dr.Niclas Bergman introduced me to the particle filtering method and provided valu-able guidance in the area. Many people have contributed with suggestions, ideasand comments for this thesis. Professor Fredrik Gustafsson, Dr. Fredrik Gun-narsson, Dr. Mikael Norrlof, Dr. Sunil Kukreja, Lic. Torbjorn Crona, Lic. FredrikTjarnstrom and Jonas Jansson read and commented the thesis or various partsthereof. I am greatful for all your comments, remarks and suggestions that signifi-cantly improved the quality of my thesis.

I would like to thank Bjorn Gabrielsson who helped me get in touch with SaabBofors Underwater Systems and especially Per-Ola Svensson and Elias Franssonfor providing sonar data from a torpedo system.

The work was supported by the competence center Information Systems forIndustrial Control and Supervision (ISIS), which is gratefully acknowledged. Iwould also like to thank Saab Bofors Dynamics for giving me the opportunity tocombine work with part time PhD studies.

Finally, I would like to thank my parents and sister for support during theseyears and my wife Karin for love and encouragement.

Linkoping, January, 2002

Rickard Karlsson

iii

iv Contents

Contents

1 Introduction 1

1.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Estimation – Theory and Methods 7

2.1 Estimation Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 The Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Discrete-Time Bayesian Estimation . . . . . . . . . . . . . . . 112.3 Kalman Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 The Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Steady State Filters . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 The Information Filter . . . . . . . . . . . . . . . . . . . . . . 142.3.4 The Extended Kalman Filter . . . . . . . . . . . . . . . . . . 15

2.4 Multiple Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Interactive Multiple Models . . . . . . . . . . . . . . . . . . . 182.4.2 Adaptive Forgetting through Multiple Models . . . . . . . . . 232.4.3 Range Parameterized Extended Kalman Filters . . . . . . . . 242.4.4 Bayesian Estimation using Gaussian Sums . . . . . . . . . . . 262.4.5 Change Detection using Multiple Models . . . . . . . . . . . 30

v

vi Contents

3 Numerical Methods for Estimation and Filtering 31

3.1 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 The Point Mass Filter . . . . . . . . . . . . . . . . . . . . . . 32

3.1.2 Stochastic Riemann Approximation . . . . . . . . . . . . . . 32

3.2 Off-line Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . 34

3.2.1 Accept-Reject Methods . . . . . . . . . . . . . . . . . . . . . 34

3.2.2 Importance Sampling . . . . . . . . . . . . . . . . . . . . . . 35

3.2.3 Markov Chain Monte Carlo Methods . . . . . . . . . . . . . . 37

3.2.4 Metropolis–Hastings . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.5 Gibbs Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2.6 Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3 Sequential Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . 41

3.3.1 The Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.2 The Auxiliary Particle Filter . . . . . . . . . . . . . . . . . . 49

3.3.3 Particle Filtering using Gaussian Sums . . . . . . . . . . . . . 51

3.3.4 Depletion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Target Tracking and Data Association 55

4.1 Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.1.1 The Infrared Sensor . . . . . . . . . . . . . . . . . . . . . . . 56

4.1.2 The Radar Sensor . . . . . . . . . . . . . . . . . . . . . . . . 56

4.1.3 Pseudo-Measurements for Kinematic Constraints . . . . . . . 57

4.2 Tracking Models and Coordinate Systems . . . . . . . . . . . . . . . 58

4.2.1 A Second Order Tracking Model . . . . . . . . . . . . . . . . 58

4.2.2 The Singer Acceleration Model . . . . . . . . . . . . . . . . . 58

4.2.3 The Coordinated Turn Model . . . . . . . . . . . . . . . . . . 59

4.2.4 The Nearly Coordinated Turn Model . . . . . . . . . . . . . . 60

4.2.5 Modified Spherical Coordinates . . . . . . . . . . . . . . . . . 61

4.2.6 Process Noise Models . . . . . . . . . . . . . . . . . . . . . . 64

4.3 Track Initiation and Termination . . . . . . . . . . . . . . . . . . . . 65

4.4 Sensor Fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.4.1 Measurement to Track Fusion . . . . . . . . . . . . . . . . . . 68

4.4.2 Track to Track Fusion . . . . . . . . . . . . . . . . . . . . . . 69

4.5 Data Association . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.5.1 Measurement Validation and Gating . . . . . . . . . . . . . . 70

4.5.2 Nearest Neighbor Association . . . . . . . . . . . . . . . . . . 72

4.5.3 Probabilistic Data Association . . . . . . . . . . . . . . . . . 72

4.5.4 Joint Probabilistic Data Association . . . . . . . . . . . . . . 74

4.5.5 Multiple Hypothesis Tracking . . . . . . . . . . . . . . . . . . 75

4.5.6 Association using Monte Carlo Techniques . . . . . . . . . . . 78

Contents vii

5 Particle Filter Applications 79

5.1 Maneuvering Target Tracking for ATC . . . . . . . . . . . . . . . . . 805.2 Passive Ranging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.2.1 Air-to-Air Passive Ranging . . . . . . . . . . . . . . . . . . . 875.2.2 Terrain Induced Constraints for Passive Tracking . . . . . . . 955.2.3 Passive Tracking for a Torpedo System . . . . . . . . . . . . . 99

5.3 Multi Target Tracking and Data Association . . . . . . . . . . . . . . 1035.3.1 The SIR/MCJPDA–Algorithm . . . . . . . . . . . . . . . . . 1045.3.2 SIR/MCJPDA–Simulation Study . . . . . . . . . . . . . . . . 1065.3.3 Particle Number Controller . . . . . . . . . . . . . . . . . . . 109

6 Conclusions 113

Abbreviations ix

Abbreviations

APF Auxiliary Particle FilterAFMM Adaptive Forgetting through Multiple ModelsAR Auto-RegressiveATC Air Traffic ControlCFAR Constant False Alarm RateCDF Cumulative Density FunctionCUSUM Cumulative SumEKF Extended Kalman FilterFLIR Forward Looking InfraredEM Expectation-MaximizationFOR Field of RegardFOV Field of ViewGMA Geometric Moving AverageGNN Global Nearest NeighborGPB Generalized Pseudo-BayesianGPS Global Position SystemGSP Gaussian Sum Particlei.i.d. Independent Identically DistributedIMM Interacting Multiple ModelINS Inertial Navigation SystemsIR InfraredIRST Infrared Search and TrackIS Importance SamplingJDL Joint Directors of LaboratoriesJPDA Joint Probabilistic Data AssociationKF Kalman FilterLMS Least Mean SquareLOS Line-of-sightLS Least SquareMAP Maximum a PosterioriMCMC Markov Chain Monte CarloMHT Multiple Hypotheses TrackingMISE Mean Integrated Square ErrorML Maximum LikelihoodMLE Maximum Likelihood EstimateMMSE Minimum Mean Square ErrorMSC Modified Spherical CoordinatesNN Nearest NeighborNNSF Nearest Neighbor Standard FilterPDA Probabilistic Data AssociationPDF Probability Density FunctionPMF Point Mass Filter

x Notations

PMHT Probabilistic Multi Hypothesis TrackingRADAR Radio Detection and RangingRCS Radar Cross SectionRLS Recursive Least SquaresRMSE Root Mean Square ErrorRPEKF Range Parameterized Extended Kalman FilterRPF Regularized Particle FilterRWR Radar Warning ReceiverSAR Synthetic Aperture RadarSIR Sampling Importance ResamplingSIS Sequential Importance SamplingSNF Strongest Neighbor FilterSPRT Sequential Probability Ratio TestTERNAV Terrain NavigationTTG Time-to-GoTWS Track While Scan

Notations

A Linearized state matrixA′ The transpose of matrix A

CAI Transformation matrix (from inertial to antenna system)

CR Coefficient of variationet Measurement noiseε Residual vector, innovationsf(·) State equation transition mapping (discrete-time)

f(·) State equation transition mapping (continuous-time)ϕ Azimuth angleΦ Transition matrixg(·) Arbitrarily used functionh(·) Measurement relationH Linearized measurement relationI Unity matrixI Used to denote the value of an integralKt Kalman gain matrix at time t

K Kernel functionl(·) Likelihood functionλ Birth/death Markov parameterM Number of measurementsMt IMM modelN Number of particles, samples or modelsNF Number of RPEKF filtersN(µ, P ) Gaussian distribution with mean value µ and covariance P

N(x; µ, P ) Gaussian densityη Position coordinate

Notations xi

O Null matrixPD Probability of detectionPG Gating probabilityPt State error covariance matrix at time t

Pr· Probabilityp(·) Probability density functionp(xt|Yt) Posterior densitypet Measurement noise probability densitypvt Process noise probability densitypx0 Initial state probability densityR Measurement noise covariance matrixSt Kalman filter innovation matrix at time t

σ Standard deviationQ Process noise covariance matrixq(·) Proposal densityT Sample periodθ Elevation angleΘ Gaussian sum particle filter weightsU Uniform distributionvt Process noisext State vector at time t

x(i)t Sample or multiple-model selection of state vector

ξ Position coordinateyt Measurement at time t

Yt The cumulative set of measurements up to and including time t

ζ Position coordinate

ω(i)t The i :th importance weights at time t

Ω Turn rate

xii Notations

1

Introduction

In this thesis we will describe some estimation methods applied to target trackingapplications. Target tracking systems rely heavily upon statistical state estimationtheory. Two main methods are commonly used, the maximum likelihood (ML)method and a Bayesian approach. Depending on the application, different mod-els can be used to describe a dynamical system. Often linear state dynamics isa sufficiently good model description. However, non-linear models are sometimesused, which complicates the solution. In many cases, the measurement relation isnon-linear. Often non-linearities in the state equation or the measurement rela-tion can be handled by linearizations around the estimated state. The unknowndynamics or sensor errors are described by a white Gaussian stochastic process.Many linearized techniques rely upon these assumptions. For a general non-linearsystem with arbitrary but known noise distribution, it is easy to formulate theoptimal equations for a Bayesian approach. These equations are not, in general,analytically solvable. Therefore, it is necessary to study approximate solutions.Several computer intensive statistical methods exist. Some of them are based ondeterministic numerical integration while others rely on stochastic simulation basedapproaches. The main objective here is to investigate and analyze these simula-tion based methods and compare them to classical methods based on linearizedtechniques.

For target tracking applications, modern systems are capable of handling multi-ple targets. This leads to data association issues. As a result, standard techniques

1

2 Introduction

for traditionally implemented systems have been developed. When using simula-tion based methods, new association methods must be introduced or the problemmust be formulated in such a manner that classical methods can be used. In Fig-ure 1.1, a tracking application is presented, where both air to air and air to seaapplications are highlighted. Commonly used sensors are radar and infrared (IR)


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Figure 1.1: Target tracking – different platforms.

sensors, or a combination thereof. Traditionally the main tracking sensor has beenthe radar sensor. When using only passive sensors such as the IR sensor, someapplications rely upon passive ranging; that is, range and velocity estimation usingangle-only measurements. Range estimation is possible by maneuvering the ownplatform so the range will be observable in the estimation problem.

Combining sensors of different accuracy or that observe different features mayenhance tracking performance. This leads to the need for reliable sensor data fusionmethods. Different sensor fusion methods can be used to achieve this goal. Thesensor fusion problem and its solution is often within the usual Bayesian trackingapproach.

For tracking applications there are many different scenarios and setups. Thetracking system can be located on a moving vehicle, such as an aircraft, missile orsubmarine, or located on the ground for surveillance purposes. We address severaltracking problems and study some applications. A radar tracking application forair traffic control (ATC) originally developed in Karlsson and Bergman (2000b,a) ispresented. The passive ranging application from Karlsson and Gustafsson (2001b)is also included. The multiple target tracking problem involving different dataassociation techniques is studied, and some methods described in Karlsson andGustafsson (2001a) are discussed.

Model based signal processing for target tracking relies upon models of the sen-sors involved, the target maneuvering model and the own platform. If accurate

1.1 Problem Formulation 3

models are available these can be incorporated in the estimation problem and ac-curate tracking can be performed. Often these models are non-linear. Uncertaintymodels used in classical tracking applications often assume a Gaussian distribution.If the distribution is non-Gaussian this may decrease tracking performance, if nothandled by the estimation method. To enhance performance for tracking systemsthe choice of coordinate system can be crucial for some applications.

1.1 Problem Formulation

Target tracking applications have been an active research area for many years.There exist several good target tracking books; for instance Bar-Shalom and Fort-mann (1988); Bar-Shalom and Li (1993); Blackman (1986); Blackman and Popoli(1999), where both sensors, target models and estimation theory are thoroughlydescribed. The commonly studied estimation technique relies upon the Kalmanfilter or for non-linear systems the extended Kalman filter. During recent yearsthe growth in computational power has made computer intensive statistical meth-ods feasible. The main breakthrough came with the seminal paper of Gordonet al. (1993). Both Markov Chain Monte Carlo methods and sequential MonteCarlo methods (particle filters) are now implemented in several applications. InBergman (1999) both tracking and navigation applications are described.

The purpose of this thesis is to focus on the target tracking application for singleand multiple targets, and describe related problems such as sensor data fusion, dataassociation, sensor- and target modeling. The classical techniques and the recentMonte Carlo related methods are discussed and compared.

1.2 Outline

The objective is to give a survey of the target tracking area and to investigateseveral simulation based methods. Topics that are covered are estimation, sensorfusion, data association and modeling. The overall goal is to investigate severaltarget tracking applications when simulation based methods are used and com-pare these computational intensive methods to classical estimation and associationmethods.

In this section a brief overview of the thesis is given. In Chapter 1 an introduc-tion to target tracking and data association is given. A short overview of the fieldincluding basic problems and techniques are presented.

Chapter 2 describes common methods in estimation theory. Mainly, the maxi-mum likelihood method and the Bayesian method are discussed. Bayesian frame-work for dynamical systems is highlighted and optimal Bayesian estimation is pre-sented. Several sub-optimal approaches based on Kalman filter theory and multiplemodels are presented as feasible estimation methods.

In Chapter 3 several numerical methods for the estimation and filtering problemare described. Numerical integration based on Riemann sums and on/off-line Monte

4 Introduction

Carlo methods are discussed. Different solutions to recursive estimation problemsare presented as variations to the particle filter method.

Chapter 4 covers target tracking and data association problems. Topics suchas sensor data fusion, track initiation, coordinate systems and sensor modelingare presented. Different data association methods are discussed thoroughly and aMonte Carlo association for particle filters is proposed.

In Chapter 5 applications and extensions of theory, applied to different trackingproblems are illustrated. Both pure simulations and tests on experimental data areperformed.

Lastly, Chapter 6 provides conclusive remarks as well a summary of the workand discusses future research.

1.3 Contributions

Here we outline the significant contributions presented in the thesis:

• Extension and implementation of a multiple model auxiliary particle filter,using hard constraints for an air traffic control (ATC) application, presentedin Section 5.1.

• Passive ranging is investigated using particle filters, and compared with theRPEKF multiple filter, using a Cartesian and MSC representation, in Sec-tion 5.2.

• Passive ranging using terrain induced constraints in Section 5.2.2.

• Application of particle filtering to experimental data acquired from a torpedosystem in Section 5.2.3.

• The Monte Carlo JPDA algorithm presented in Section 4.5.6 together witha novel controller for particles in Section 5.3.3.

• The information filter based initiation and interpretation presented in Sec-tion 4.3.

Parts of the material have been published previously. The auxiliary particle filterfor maneuvering targets was presented in

Karlsson, R. and Bergman, N. (2000a). Auxiliary particle filters fortracking a maneuvering target. In Proc. of the 39th IEEE Conferenceon Decision and Control, pages 3891–3895, Sydney, Australia.

and an intermediate version appeared in

Karlsson, R. and Bergman, N. (2000b). Maneuvering target trackingusing auxiliary particle filters. In Reglermote 2000, pages 278–283.

The target tracking application is put in a general framework in

1.3 Contributions 5

Gustafsson, F., Gunnarsson, F., Bergman, N., Forssell, U., Jansson, J.,Karlsson, R., and Nordlund, P.-J. (2002). Particle filters for positioning,navigation and tracking. IEEE Transactions on Signal Processing. (Feb,2002).

The passive ranging was presented in

Karlsson, R. and Gustafsson, F. (2001b). Range estimation using angle-only target tracking with particle filters. In Proc. of the AmericanControl Conference, volume 5, pages 3743–3748, Arlington, Virginia,USA

The Monte Carlo JPDA association method and the particle filter controller ap-peared in

Karlsson, R. and Gustafsson, F. (2001a). Monte Carlo data associationfor multiple target tracking. In IEE International Seminar on TargetTracking: Algorithms and Applications, Enschede, The Netherlands

6 Introduction

2

Estimation – Theory and

Methods

Dynamical models of the underlying system are central in target tracking applica-tions. The main trade-off is between model complexity, inaccuracies in the systemmodel and errors in the observations. The following discrete-time state space de-scription is often plausible

xt+1 = f(xt, vt)

yt = h(xt, et),

where the state vector xt ∈ Rn consists of, for instance, position and velocity

components at time t. The observation yt ∈ Rm is often a non-linear mapping of the

current state. Both the system model and the measurement relation are inaccurate,due to modeling and/or sensor errors. This is described by the stochastic processesvt and et. The main objective in this chapter is to estimate and predict the state,xt, using the observations up to and including time t, Yt = yit

i=1.

In Section 2.1 several estimation paradigms are discussed. The least squares, themaximum likelihood and maximum a posteriori estimation methods are presented.The Bayesian approach is discussed in Section 2.2. In Section 2.3.1 Kalman filtertheory is presented. Approximative methods are discussed when non-linearities arepresent, for instance the extended Kalman filter. Estimation techniques related tomultiple models are discussed in Section 2.4.

7

8 Estimation – Theory and Methods

2.1 Estimation Paradigms

In this thesis we will focus on state estimation for dynamical systems and study thediscrete-time formulation. Estimation techniques are often categorized as Bayesianor non-Bayesian. The statistical techniques we implement rely on both the non-Bayesian maximum likelihood (ML) method and ordinary Bayesian methods. BothML estimation and maximum a posteriori (MAP) estimators are considered. Forsome special cases, the techniques yield the same estimates, even if the approachesare fundamentally different. The Bayesian approach assumes that the unknownparameter has an initial or prior distribution.

A common technique for state estimation is the least square (LS) method (Bar-Shalom and Fortmann, 1988). The idea is to minimize the mean square erroraccording to

xLS = arg minx

t∑

i=1

(yi − h(x))2. (2.1)

For random parameters the concept is extended by minimizing the expected valuegiven the observations Yt = y1, . . . , yt

xMMSEt = arg min

xE(x− x)2|Yt, (2.2)

which we denote the minimum mean square error (MMSE). It can be shown thatthe solution is given by the conditional mean

xMMSEt = Ex|Yt =

∫

xp(x|Yt)dx. (2.3)

In Bar-Shalom and Fortmann (1988) this is shown if we assume that the conditionalexpected value is differentiable. Differentiation gives

∂

∂xE(x− x)2|Yt = 2(x− Ex|Yt) = 0.

The maximum likelihood (ML) method is a statistical method where a so-calledlikelihood function is constructed. The estimate is chosen to maximize the likeli-hood criterion. If we assume that the state vector or parameter vector is given byx ∈ R

n, the construction of the likelihood function is based on the observations upto present time, Yt. In the general case, we construct the likelihood function bycombining the likelihoods for different times assuming independence

L =t

Πi=1

l(yi|x). (2.4)

The point estimate of the parameter vector, or state vector, is given by the argu-ment that maximizes the likelihood function. The theory relies on the fact that,asymptotically, the ML estimate converges almost surely to the true value, un-der fairly general conditions. In Bar-Shalom and Fortmann (1988) the following

2.1 Estimation Paradigms 9

definitions are used for the ML method and the maximum a posteriori (MAP)method

xML = arg maxx

p(Yt|x) (2.5)

xMAP = arg maxx

p(x|Yt) (2.6)

Example 2.1 ML method for Gaussian distribution.A common example in tracking literature is to present ML and MAP estimation

for a Gaussian noise assumption. If we assume one measurement y and denote theparameter x.

y = x+ e, e ∈ N(e; 0, σ2).

1.The ML method.The likelihood is given by

l(x) = p(y|x) = pe(y − x) =1√2πσ

e−(y−x)2

2σ2 .

The ML estimate is given by maximizing the expression

xML = arg maxx

l(x) = y.

2.The MAP method.Using Bayes’ rule gives

p(x|y) =p(y|x)p(x)

p(y)∝ p(y|x)p(x)

If we assume that the prior p(x) is Gaussian distributed N(x0, σ20), we have

p(x|y) ∝ e−(y−x)2

2σ2 e− (x−x0)2

2σ20 .

Maximization yields the MAP estimate as

xMAP = arg maxx

p(x|y) =x0σ

2 + yσ20

σ2 + σ20

.

3.The LS-method.

Suppose we have several independent measurements from a Gaussian distri-bution

yt = x+ et, et ∈ N(0, σ2), t = 1, 2, . . . , N.


We can form the likelihood function and calculate the maximum likelihoodvalue. It can easily be seen that the ML and LS estimates coincide and that

xML = xLS =1

N

N∑

i=1

yi.

As seen above, the solution is given by the sample mean.

Another interesting aspect of LS and MAP estimation is for the Gaussian case,where it can be shown that

xMMSE = xMAP . (2.7)

For an overview of different LS- and recursive LS-methods we refer to Kailath et al.(2000).

In Bar-Shalom and Fortmann (1988) it is shown that the non-Bayesian estimatebased on ML coincides with the Bayesian MAP approach for the case with a diffuseprior for the Gaussian problem discussed in Example 2.1. A diffuse prior can bedefined as

p(x) =

ε, ifx ∈ [− 12ε ,

12ε ]

0, otherwise. (2.8)

If the Gaussian prior in Example 2.1 has a large variance, the behavior will besimilar to the diffuse prior case. In the limit, when the variance tends to infinity,σ0 → ∞, the same result is obtained.

In Bar-Shalom and Li (1993) the case when the likelihood function can bedecomposed as

L(x) = p(Y|x) = p1(g(Y), x)p2(Y) (2.9)

is considered. Hence, it is then clear that the ML estimates depends only on thefunction g(Y) and not the complete data set Y. This is called the sufficient statistic.

2.2 The Bayesian Approach

The problem of estimating a parameter or the state of a non-linear stochastic sys-tem using noisy measurements as observations has been an active research areafor many years. In Jazwinski (1970) both the non-linear and linear cases are dis-cussed. For the special case, when we have a linear system with additive Gaussiannoise, the Kalman filter, (Anderson and Moore, 1979; Kalman, 1960), yields theanalytical solution to the minimum variance problem.

In the Bayesian theory we use the fact that everything unknown is consideredas a stochastic variable. This leads to a description where we assume some ini-tial or prior distribution. Using the observations we can later revise the estimateby computing the posterior density. The general theory for non-linear filteringwith possible non-Gaussian noise distribution is described thoroughly in Jazwinski(1970); Sorenson and Alspach (1971).

2.2 The Bayesian Approach 11

2.2.1 Discrete-Time Bayesian Estimation

Many recursive estimation problems can be formulated as

xt+1 = f(xt, vt) (2.10)

yt = h(xt, et) (2.11)

It is assumed the noise signals are independent with with probability densities pet

and pvt. Also the initial uncertainty described by the density px0

is assumed in-dependent. Given observations up to time t, Yt = y1, . . . , yt we want to findan optimal estimate xt ∈ R

n from the time update in (2.12) and the measurementupdate in (2.13)

p(xt+1|Yt) =

∫

Rn

p(xt+1|xt)p(xt|Yt) dxt (2.12)

p(xt|Yt) =p(yt|xt)p(xt|Yt−1)

p(yt|Yt−1). (2.13)

These equations can easily be derived using the Markov property, Bayes’ rule andsome standard calculations from probability theory. The measurement updatecomes from

p(xt|Yt) =p(Yt|xt)p(xt)

p(Yt)=p(yt,Yt−1|xt)p(xt)

p(yt,Yt−1)=p(yt|xt,Yt−1)p(Yt−1|xt)p(xt)

p(yt|Yt−1)p(Yt−1)

=p(yt|xt)p(Yt−1|xt)p(xt)

p(yt|Yt−1)p(Yt−1)=p(yt|xt)p(xt|Yt−1)

p(yt|Yt−1). (2.14)

In the first equality, we use Bayes’ rule and in the second Bayes’ rule in combinationwith the definition Yt = Yt−1, yt. Finally, the Markov property and Bayes’ rulegive the result.

The time update equation is given from the following calculations

p(xt+1, xt|Yt) = p(xt+1|xt,Yt)p(xt|Yt) = [Markov] = p(xt+1|xt)p(xt|Yt). (2.15)

Integration of both sides with respect to xt yields,

p(xt+1|Yt) =

∫

Rn

p(xt+1|xt)p(xt|Yt) dxt. (2.16)

For the case when we do not assume additive noise the following theorem fromJazwinski (1970) is applicable.

Theorem 2.1 (Jazwinski, 1970)Let X and Y be two random vectors, with Y = g(X). Suppose that the inverseg−1 exists and both g and g−1 are continuously differentiable. Then

pY (y) = pX(g−1(y))||∂g−1(y)

∂y||, (2.17)

where ||∂g−1(y)∂y || denotes the absolute value of the Jacobian determinant.


Proof (See Jazwinski, 1970) 2

By applying Theorem 2.1, as suggested in Jazwinski (1970) we can calculate thetransition for a more general case as

p(xt+1|xt) = pvt+1(f−1(xt, xt+1))||

∂f−1

∂xt+1||. (2.18)

Using the same approach we calculate the likelihood

p(yt|xt) = pet(h−1(yt, xt))||

∂h−1

∂yt||. (2.19)

Often we use a simplified model assuming additive noise

xt+1 =f(xt) + vt (2.20a)

yt =h(xt) + et. (2.20b)

For the simple additive noise model, we can easily calculate the following relations

p(xt+1|xt) = pvt(xt+1 − f(xt)) (2.21)

p(yt|xt) = pet(yt − h(xt)). (2.22)

2.3 Kalman Filters

In this section, different versions of the Kalman filter are presented. A time-varying Kalman filter for discrete-time is presented in Section 2.3.1. The time-invariant or stationary case is given in Section 2.3.2. For bearings-only applications,the initial covariance in the range direction is very large. The information filterin Section 2.3.3 takes this into account by propagating the inverse covariance inthe filter. For most tracking application, the measurement relation is non-linear,and sometimes the time update, therefore a linearized version can be used. Theextended Kalman filter is presented in Section 2.3.4.

2.3.1 The Kalman Filter

If we assume a linear system with additive Gaussian noise, there exists an analyticalsolution to the Bayesian time and measurement update equations. The solution isgiven by the Kalman filter (KF) equations. Since the system is linear and Gaussian,the update formula will remain Gaussian, and hence since all Gaussian systemscan be described by their first two moments (mean and covariance). The updateequations will consist of mean and covariance update. The original Kalman filter,(Kalman, 1960), was defined in continuous-time, but soon also a discrete versionwas derived. Many books describing different aspects of the Kalman filter exist.Much of the classical theory is described in Anderson and Moore (1979). The book

2.3 Kalman Filters 13

by Kailath et al. (2000) summarizes the estimation field. The theory presentedhere is given in accordance with Gustafsson (2000).

A general time-varying state space model for the Kalman filter is

xt+1 = Atxt +Bu,tut +Bv,tvt (2.23)

yt = Ctxt + et, (2.24)

where the control signal is denoted ut, the process noise vt and the measurementnoise et. All noise realizations are assumed independent with covariance matricesCovvt = Qt and Covet = Rt respectively. The correlated noise problem isdiscussed in Anderson and Moore (1979). The time and measurement updates inthe Kalman filter are given by:

xt+1|t = Atxt|t +Bu,tut (2.25)

Pt+1|t = AtPt|tA′t +Bv,tQtB

′v,t (2.26)

xt|t = xt|t−1 +Ktεt (2.27)

Pt|t = Pt|t−1 −KtCtPt|t−1, (2.28)

where

εt = yt − Ctxt|t−1 (2.29)

St = CtPt|t−1C′t +Rt (2.30)

Kt = Pt|t−1C′tS

−1t . (2.31)

In the equations above A′t denotes the transpose of the system matrix At. If

the initial state x0 and process noise vt and measurement noise et are Gaussianvariables, then given the cumulative set of observations, Yt, we have

xt+1|Yt ∈ N(xt+1|t, Pt+1|t) (2.32)

xt|Yt ∈ N(xt|t, Pt|t) (2.33)

εt ∈ N(0, St) (2.34)

The time index t denotes the samples separated with a sample period of T .

2.3.2 Steady State Filters

The steady state solution to the time-varying Kalman filter in Section 2.3.1 is animportant special case. The stationary Kalman gain, Kt → K and covariancePt → P , t → ∞ are used in the Kalman equations instead of the time-varyingformulas presented in Section 2.3.1. The stationary covariance matrix P is thesolution to the stationary Riccati equation

P = APA′ −APC ′(CPC ′ +R)−1CPA′ +BvQB′v. (2.35)

The covariance matrix from a linear time-invariant system will converge to thesteady-state or stationary value if the system is controllable and observable. If the


absolute values of all eigenvalues of A are strict smaller than one P is also positivedefinite as stated in Anderson and Moore (1979). Note that we the stationarycovariance is a function of the Kalman parameters Q and R.

In classical tracking literature, a common estimation method is the α−β filter orthe α−β−γ filter. The idea is to correct the predicted state with the measurementinnovation εt = yt −Hxt|t−1, using a fixed gain matrix K, yielding

xt|t = xt|t−1 +K(yt −Hxt|t−1). (2.36)

This approach is described in (Bar-Shalom and Fortmann, 1988, p. 89) or (Black-man, 1986, p. 160). Assume that the state consists of position, velocity and accel-

eration in one dimension and that we measure only position, i.e., H =(1 0 0

)′.

The gain matrix is denoted

K =

αβ/Tγ/T 2

, (2.37)

where α, β and γ are dimensionless constant gain values for position, velocity andacceleration respectively. In practice, they are used as design parameters andadjusted to achieve the desired system performance. However if the system iscompared to the stationary Kalman filter, it is possible to relate the coefficients tothe Kalman parameters, Q and R.

2.3.3 The Information Filter

For many practical tracking problems such as bearings-only tracking, the initialuncertainty covariance matrix can be large. As a result the Kalman filter in Sec-tion 2.3.1 may suffer severe numerical problems. A possible solution is to propagatethe inverse covariance rather than the covariance itself. This reduces the numericalproblems allowing the case with arbitrarily large initial uncertainty.

We use the ideas presented in Anderson and Moore (1979); Bar-Shalom and Li(1993) and the Kalman filter introduced in Section 2.3.1. By applying the matrixinversion lemma given in Anderson and Moore (1979); Bar-Shalom and Li (1993),we can reformulate the Kalman filter update by propagating an inverse descriptionof the covariance matrix. This technique is often referred to as the informationfilter.

Theorem 2.2 (The matrix inversion lemma)For valid matrix multiplications and assuming invertability the following equiva-lence holds

(P−1 + C ′R−1C)−1 = P − PC ′(CPC ′ +R)−1

︸︷︷︸

K

CP. (2.38)

Proof (See Anderson and Moore, 1979) 2

2.3 Kalman Filters 15

Taking the inverse of (2.28) and applying Theorem 2.2 yields

P−1t|t = (Pt|t−1 −KtCtPt|t−1)

−1 = P−1t|t−1 + C ′

tR−1t Ct. (2.39)

If we introduce A−1t = AtPt|tA

′t, which is invertible since At is a transition matrix,

we can rewrite the prediction information matrix as

P−1t+1|t = (A−1

t +Bv,tQtB′v,t)

−1. (2.40)

Using Theorem 2.2 again yields

P−1t+1|t = At − AtBv,t(B

′v,tAtBv,t +Q−1

t )−1B′v,tAt, (2.41)

where At = (A′t)

−1P−1t|t A

−1t . In the same way the Kalman gain can be calculated,

yielding

Kt = (P−1t|t−1 + C ′

tR−1t Ct)

−1C ′tR

−1t . (2.42)

It is awkward to find recursions for the estimate. Instead we define

at|t−1 = P−1t|t−1xt|t−1 (2.43a)

at|t = P−1t|t xt|t. (2.43b)

The Kalman gain in (2.42) can be modified using (2.39)

Kt = Pt|tC′tR

−1t . (2.44)

Inserting (2.44) into (2.27) and applying (2.39) together with (2.43) the estimaterecursion becomes

at|t = at|t−1 + C ′tR

−1t yt. (2.45)

The following remarks from Gustafsson (2000) are important:

• C ′tR

−1t Ct is the information for a new measurement.

• Vague prior knowledge can be handled by using P−10 = 0.

2.3.4 The Extended Kalman Filter

Many estimation problems are non-linear, but the noise model is assumed Gaus-sian. The main idea is to linearize the system and apply the Kalman filter. Thisapproach is referred to as the extended Kalman filter (EKF), and several differ-ent implementations exist. Here we apply the discretized-linearization, (Gustafsson,2000), where we first linearize the non-linear continuous-time system and discretizethe system. In Anderson and Moore (1979); Bar-Shalom and Li (1993) the EKF forthe discrete-time is discussed, which will follow directly from the EKF describedbelow.


Consider the following non-linear continuous-time system

x = f(x(t), u(t)) + v(t) (2.46)

yt = h(x(t)) + et, (2.47)

where x ∈ Rn and y ∈ R

m, and the process noise is given by the stochastic processv(t) and the measurement noise by et. The system can be approximated by

x ≈ f(x, u)) + fx(x, u)(x(t) − x) + v(t)

= fx(x, u)x(t) + f(x, u) − fx(x, u)x︸︷︷︸

g(x,u,t)

+v(t), (2.48)

where fx(x) = ∇xf(x). Using Ev(t) = 0 , the system is discretized by solving

d

dτe−fx(x,u)τx(τ) = e−fx(x,u)τ g(x, u, t) (2.49)

e−fx(x,u)(t+T )x(t+ T ) − e−fx(x,u)tx(t) =

∫ t+T

t

e−fx(x,u)τ g(x, u, t)dτ (2.50)

Multiply by efx(x,u)(t+T ) and change integration variable

x(t+ T ) = efx(x,u)Tx(t) +

∫ T

0

efx(x,u)τ g(x, u, t)dτ (2.51)

Note that

efx(x,u)T =

∫ T

0

efx(x,u)τ fx(x, u)dτ + I. (2.52)

Since the true value x is not known we substitute for the estimate x yielding

x(t+ T ) = [

∫ T

0

efx(x,u)τ fx(x, u)dτ + I]x(t)

+

∫ T

0

efx(x,u)T dτ g(x, u, t)

= x(t) +

∫ T

0

efx(x,u)T dτ f(x, u). (2.53)

To summarize we have the following expression for the state prediction

xt+T |t = xt|t +

∫ T

0

efx(xt|t,u)τdτ f(xt|t, u). (2.54)

2.4 Multiple Models 17

It is also possible to derive an expression for the covariance matrix

Pt+T |t = efx(xt|t,u)TPt|tefx(xt|t,u)′T +Qt. (2.55)

To simplify the formulas we use the following notation for the update with sampleperiod T = 1

xt|t = xt|t−1 +Kt(yt − h(xt|t−1)), (2.56)

where

St = HtPt|t−1H′t +Rt (2.57)

Kt = Pt|t−1H′tS

−1t (2.58)

Pt|t = Pt|t−1 −KtStK′t, (2.59)

The linearized measurement matrix is defined as Ht = ∇xh(x)|x=xt|t.

2.4 Multiple Models

Many problems for target tracking applications require a good state estimate, evenif the system changes rapidly. To achieve a fast tracking with small errors, mul-tiple models can be used. Each filter is initialized for specific behavior. Theindividual filters can influence each other in different ways. The estimate is calcu-lated by mixing or switching between the filters. Methods such as the interactivemultiple model (IMM), (Blom and Bar-Shalom, 1988), or the generalized pseudo-Bayesian (GPB) and adaptive forgetting through multiple models (AFMM), (An-dersson, 1985), are examples of estimation techniques using multiple models/filters.By applying a change detector to filtering, the switching can be implemented differ-ently. For passive ranging applications, the range parameterized extended Kalmanfilter (RPEKF) is another example of a multiple model system. The general ideais based on the Gaussian sum approximation, described in Anderson and Moore(1979). Here we discuss the IMM and the RPEKF methods, since they are usedas a comparison to other methods in Chapter 5. Briefly summarizing the multiplemodel approach using a merging technique, we assume N different models/filtersapproximating the density p(x) by mean x(i) and covariance P (i). The merging isthen done according to

p(x) =N∑

i=1

w(i)N(x(i), P (i)) = N(x, P ), (2.60)


where

x =1

w

N∑

i=1

w(i)x(i) (2.61a)

P =1

w

N∑

i=1

w(i)(

P (i) + (x(i) − x)(x(i) − x)′)

, (2.61b)

w =

N∑

i=1

w(i) = 1. (2.61c)

The term∑N

i=1 w(i)(x(i) − x)(x(i) − x)′ is referred to as the spread of the mean.

2.4.1 Interactive Multiple Models

Traditionally, target tracking problems are solved using linearized tracking filters,i.e., extended Kalman filters (EKF) Anderson and Moore (1979). For highly ma-neuvering targets or for low observation rates different maneuvering modes are usedto describe the motion. Therefore, target maneuvers are often described by mul-tiple linearized models. To handle maneuvering targets, multiple models or filterscan be used for the possible maneuvering hypoteses. This could be done withinthe Bayesian framework. However, the main problem is to handle the exponentialgrowth in the number of hypotheses. One important issue for a multiple modelor filter application, is to reduce the number of hypotheses. This can be doneby pruning or merging (mixing). In Blom (1984) a filtering algorithm for lineardiscrete-time filters with Markovian coefficients is given. The suboptimal filter iscalled the interacting multiple model (IMM). State-of-the-art estimation and track-ing literature (Bar-Shalom and Li, 1993, p. 463-464) presents state estimation andprediction performed by switching between models or by mixing. Here we presentthis approach with a slightly different notation. Assume that we use N different

models at time t, denoted M(i)t where the probability for each model is defined as

µ(i)t = PrM(i)

t |Yt. The probability density function at time t is given by thetotal probability theorem using N different models as

p(xt|Yt) =N∑

j=1

p(xt|M(j)t ,Yt)PrM(j)

t |Yt︸︷︷︸

µ(j)t

. (2.62)

Applying Bayes’ rule to the first factor in (2.62) using Yt = yt,Yt−1 gives

p(xt|M(j)t ,Yt) ∝ p(yt|M(j)

t , xt)p(xt|M(j)t ,Yt−1). (2.63)


Applying the total probability theorem to the last factor in (2.63) gives

p(xt|M(j)t ,Yt−1) =

N∑

j=1

p(xt|M(j)t ,M(j)

t−1,Yt−1)PrM(i)t−1|M

(j)t ,Yt−1

︸︷︷︸

µt−1|t−1(i,j)

≈N∑

i=1

p(

xt|M(j)t ,M(i)

t−1, x(l)t−1|t−1, P

(l)t−1|t−1

Nl=1

)

µt−1|t−1(i, j)

=

N∑

i=1

p(xt|M(j)t ,M(i)

t−1, x(i)t−1|t−1, P

(i)t−1|t−1)µt−1|t−1(i, j). (2.64)

The approximation in Equation (2.64) is due to the fact that the models summarizethe history through the estimates and covariances. The mixing probabilities areeasily expressed using Bayes’ rule as

µt−1|t−1(i, j) = PrM(i)t−1|M

(j)t ,Yt−1

∝ PrM(j)t |M(i)

t−1,Yt−1︸︷︷︸

p(i,j)

PrM(i)t−1|Yt−1

︸︷︷︸

µ(i)t−1

, (2.65)

where p(i, j) in practice is used as a design parameter. Condensing (2.64) byapproximating the Gaussian mixture with a single Gaussian gives

p(xt|M(j)t ,Yt−1) =

N∑

j=1

N

(

xt;Ext|M(j)t , x

(i)t−1|t−1, cov(.)

)

µt−1|t−1(i, j)

≈ N

(

xt;N∑

i=1

Ext|M(j)t , x

(i)t−1|t−1µt−1|t−1(i, j), cov(.)

)

= N

(

xt;EN∑

i=1

xt|M(j)t ,

N∑

i=1

x(i)t−1|t−1µt−1|t−1(i, j), cov(.)

)

,

where cov(·) denotes the covariance for each expression.

The IMM method is summarized in Algorithm 2.1, where also all covariancematrices are written out completely. The Kalman filter related topics in the IMMare defined in accordance with the notations in Section 2.3.1.


Algorithm 2.1 (Interactive Multiple Model (IMM), Bar-Shalomand Li, 1993)

1. Calculate the mixing probabilities, using the probability that modeM(i) is in effect at t− 1 given that model M(j) is in effect at time tconditioned upon the measumrents Yt−1.

µt−1|t−1(i, j) = PrM(i)t−1|M

(j)t ,Yt−1

=1

c(j)PrM(j)

t |M(i)t−1,Yt−1

︸︷︷︸

p(i,j)

PrM(i)t−1|Yt−1

︸︷︷︸

µ(i)t−1

,

i, j = 1, . . . , N

c(j) =

N∑

i=1

p(i, j)µ(i)t−1, j = 1, . . . , N

2. Calculate the initial mixing condition for j = 1, . . . , N filters

x(j),0t−1|t−1 =

N∑

i=1

x(i)t−1|t−1µt−1|t−1(i, j)

∆t−1(i, j) = x(i)t−1|t−1 − x

(j),0t−1|t−1

P(j),0t−1|t−1 =

N∑

i=1

µt−1|t−1(i, j)P (i)t−1|t−1 + ∆t−1(i, j)∆

′

t−1(i, j)

3. Likelihood calculation for j = 1, . . . , N

Λ(j)t = N(ε

(j)t ; 0, S

(j)t ), where ε

(j)t = yt − h(x

(j),0t|t−1)

4. Mode probability update for j = 1, . . . , N

µ(j)t = PrM(j)

t |Yt =1

cp(yt|M(j)

t ,Yt−1)PrM(j)t |Yt−1

=1

cΛ

(j)t

N∑

i=1

p(i, j)µ(i)t−1 =

1

cΛ

(j)t c(j)

c =

N∑

j=1

Λ(j)t c(j)

5. Estimate and covariance combination

xt|t =N∑

j=1

x(j)t|t µ

(j)t , Pt|t =

N∑

j=1

µ(j)t P (j)

t|t + [x(j)t|t − xt|t][x

(j)t|t − xt|t]

′


In Example 2.2, a radar tracking application is presented using the IMM methodwith two filters. One filter is used to handle a straight flying path accurately,whereas the other is used to manage maneuvers. Due to the non-linearities in themeasurement equation an EKF is used for the estimation.

Example 2.2 The IMM method for two models.In Figure 2.1, the IMM algorithm is presented graphically for the special case

where only 2 models are used. We consider a radar tracking system where the


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

Estimate

FilterFilter

Interaction/Mixing

Λ(1)t Λ

(2)t

ytyt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t x

(2)t|t , P

(2)t|t

xt|t, Pt|t

µt−1|t−1

µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 2.1: Schematic overview of the IMM algorithm for 2 filters.

distance and angle to the target is measured from a ground based system placed inthe origin. The following Cartesian model with state vector xt = (ξ η ξ η)′, whereξ, η are the Cartesian position coordinates and ξ, η are the velocity components.


The discrete model and measurement relation are given as

xt+1 =

1 0 T 00 1 0 T0 0 1 00 0 0 1

xt +

T 2

2 0

0 T 2

2T 00 T

ut

yt = h(xt) =

(√

ξ2t + η2t

arctan(ηt

ξt)

)

+ et, et ∈ N

((00

)

,

(102 00 0.0012

))

.

The true trajectory is generated as a straight path, followed by a maneuver be-tween t = 20 − 40 and then continuing a straight path. In the IMM filter theprobability to change between the models is 0.05. In Figure 2.2 (a) the scenario,true trajectory, measurements and IMM estimates are given and in Figure 2.2 (b)are the probabilities from the IMM filter (maneuver or non-maneuver).

800 1000 1200 1400 1600 1800 2000 22001000

1100

1200

1300

1400

1500

1600

ξ [m]

η [m

]

True trajectoryIMM−estimateMeasurement


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

(a) Scenario and IMM estimates

0 10 20 30 40 50 600

0.2

0.4

0.6

0.8

1

Time

Pro

babi

lity

No maneuverManeuver


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

(b) IMM probabilites

Figure 2.2: The IMM-2 filter estimates and probabilities.

In Gustafsson (2000) the multiple model approach is investigated further, sum-marizing several similar methods such as IMM and the generalized pseudo-Bayesian(GPB) method. The GPB approach merges the mixture after the measurement up-date, whereas for the IMM method merging is applied after the time update of theweights rather than after the measurement update.


2.4.2 Adaptive Forgetting through Multiple Models

In Andersson (1985) the Adaptive Forgetting through Multiple Models (AFMM)method is proposed. It consists of N filters for tracking, and M models for possiblemaneuvers. The number of filters is kept constant by splitting the most probablefilter and terminate the M − 1 least probable. It is claimed that this method isbest suited for discrete changes in time variable parameters. To avoid terminationof newly initiated filters a minimum life length L of the filters is introduced. Thismethod is adopted to tracking in Bergman (1995), where there are different modelsfor the magnitude of the process noise. The AFMM method is summarized inAlgorithm 2.2.

Algorithm 2.2 (Adaptive Forgetting through Multiple Models,Andersson, 1985)

imin = arg minα(i)t−1|η(i)>L

imax = arg maxα(i)t−1

Q(imin) = Qm

P imin

t−1|t−1 = P imax

t−1|t−1

x(imin)t−1|t−1 = x

(imax)

t−1|t−1

[x(i)t|t , P

(i)t|t ,Λ

(i)t ] = EKF(x

(i)t−1|t−1, P

(i)t−1|t−1, yt, Q

(i)), i = 1, . . . , N

α(i)t = (1 − µ)α

(i)t−1Λ

(i)t , i = 1, . . . , N

α(imin)t =

µ

1 − µα

(imax)t

Q(imin) = Qn

η(i) = η(i+1), i = 1, ..., N

xt|t =

N∑

i=1

α(i)t x

(i)t|t

In the AFMM method N parallel filters are assumed with life length parameter L.The age of the i:th filter is denoted ηi. Initially ηi = 0, ∀i, and Q(i) = Qn, ∀i. Itis assumed that we can handle M different maneuvers. The various maneuveringmodels are represented by different process noise. For the special case implementedhere we have assumed a maneuvering or non-maneuvering target model. Below onlythe process noise is different, where the covariances are given by Qm and Qn forthe maneuvering and non-maneuvering cases

Qt =

Qm,with probability µ

Qn,with probability 1 − µ.


Example 2.3 AFMM: 3 filters and two maneuvering models.The following example from Bergman (1995) is used here to illustrate the AFMMmethod. For each sample the probability, life length and model type are given.In Figure 2.3, at time t = t1 filter number 2 is the most probable. This filter is

-

6

Filter 1

Filter 2

Filter 3

-(0.1;2;n)

-(0.7;2;m)

-(0.2;1;n)

(0.05;1;m)

-(0.75;3;n)

-(0.2;2;n)

-(0.4;2;n)

-(0.35;4;n)

@@

@@R(0.25;1;m)

-(0.7;3;n)

@@

@@R(0.2;1;m)

-(0.1;2;m)

t1 t2 t3 t4

(p;L;type)p = probability

L = life length

type = maneuver, non-maneuver

Figure 2.3: AFMM with 3 parallel filters and 2 models.

then splitted using the two models, maneuvering and non-maneuvering, representedusing m and n respectively. To keep the number of filters constant, the least likelyfilter is removed, that is filter 1. Note that this was possible since the life length offilter 1 was large enough (here greater than 2). At time t2, probability, life lengthand type are given. Filter 2 is still the most probable, therefore it is split. Atthis time the least likely filter (filter 1) can not be removed since its life lengthparameter is below the chosen value 2. Instead we remove filter 3.

2.4.3 Range Parameterized Extended Kalman Filters

Passive ranging or estimation of range and velocity using only a passive sensor,such as the IR sensor, is a difficult problem. The main idea is to maneuver theown platform in such a way that relative range and velocity can be estimated. Inpractice, this means that we have to perform the maneuver in such a way that weout-maneuver the unknown target. We must also assume that a good navigationsystem and reliable sensors with small errors are available, so the own trajectoryand movement can be accurately estimated. There exist several approaches toestimate the range using a single tracking filter. As described in Robinson andYin (1994); Stallard (1987); Simard and Begin (1993) a modified spherical/polarcoordinate system is preferred instead of using a traditional Cartesian system. Theapproach used in this section is a multiple model to estimate the unknown rangeand velocity. We use a special method called the Range Parameterized Extended


Kalman Filter (RPEKF) which consists of a bank of extended Kalman filters, eachtuned to a certain range. The presentation in this section follows the developmentin Karlsson and Gustafsson (2001b). The RPEKF method described in Kronhamn(1998); Arulampalam and Ristic (2000) consists of a bank of extended Kalmanfilters in Cartesian coordinates, initialized to different range assumptions for theangle-only tracking application. In Peach (1995) the filter bank is expressed inmodified polar coordinates.

From Figure 2.4 we define the range gates for i = 1, . . . , NF different rangeassumptions (filters). For a predefined interval (Rmin, Rmax), the filter sub-intervalsare given by

r(i) =Rmin

2(ρi + ρi−1) (2.66)

ρ = (Rmax

Rmin)1/NF . (2.67)

The coefficient of variation CR defines the variance for each sub-interval,

CR =σ(i)

r(i)=

2(ρ− 1)√12(ρ+ 1)

, (2.68)

where r(i) and σ(i) are the range and standard deviation for the different filters.Therefore, the variance for each sub-interval is given as σ(i) = r(i)CR, where CR isdefined in Equation (2.68).


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

RminRmaxRminρ

i−1 Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 2.4: RPEKF range intervals.

The RPEKF uses the likelihood from each EKF, p(yt|i), to recursively updateits probability according to Bayes’ rule

w(i)t = p(i|Yt) ∝ p(yt|i)p(i|Yt−1), (2.69)

where yt is the measurement at time t and Yt = yiti=1 the set of all measurements

up to current time. The prior distribution is assumed uniform, i.e., w(i)t = 1

NF,

i = 1, . . . , NF . However, if other information is available it could be used to enhancethe performance. Under a Gaussian assumption, the likelihood is given from theEKF for the Cartesian case as

l(i)t = p(yt|i) ∝

1√

|S(i)t |

e− 1

2 (yt−h(x(i)

t|t−1))′S

(i)t

−1(yt−h(x

(i)

t|t−1))

(2.70)

S(i)t = H

(i)t P

(i)t|t−1(H

(i)t )′ +Rt, (2.71)


. . .. . .


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1

x(1)

t|t−1 P(1)

t|t−1 yt l(1)t−1

x(1)

t|t P(1)

t|t l(1)t

xt, Pt

Combine

EKF i

x(i)

t|t−1 P(i)

t|t−1 yt l(i)t−1

x(i)

t|t P(i)

t|t l(i)t

xt, Pt

Combine

EKF NF

x(NF )

t|t−1 P(NF )

t|t−1 yt l(NF )t−1

x(NF )

t|t P(NF )

t|t l(NF )t

xt, Pt

Combine

x(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 2.5: The RPEKF structure.

where Rt is the measurement noise covariance matrix and H(i)t = ∇xh(x)|x=x

(i)

t|t−1

.

The measurement update for each filter is given by the Kalman filter. The combinedestimate and covariance can now be expressed using (2.61)

xt|t =

NF∑

i=1

w(i)t x

(i)t|t (2.72)

Pt|t =

NF∑

i=1

w(i)t [P

(i)t|t + (x

(i)t|t − xt|t)(x

(i)t|t − xt|t)

′], (2.73)

where P(i)t|t is the covariance and x

(i)t|t the estimate for the different range filter

i = 1, . . . , NF .In Figure 2.5, the RPEKF idea is summarized. If the filter probability is below

a predefined threshold or if some other criterion, such as if the estimated rangein a filter is outside the (Rmin,Rmax) interval, the filter is removed from furthercalculations.

2.4.4 Bayesian Estimation using Gaussian Sums

The non-linear estimation problem can be stated in a Bayesian framework. Themain difficulty with this approach is that the equations can not, in general, besolved analytically. Approximation is needed to handle the general case. If we applythe EKF, a suboptimal solution is achieved. If this method is used only the first twomoments (mean and covariance) are updated. For a non-linear or non-Gaussiancase, this is not sufficient to describe the probability density completely. Another


approach is to approximate the densities. Several methods have been suggested.In Sorenson and Stubberud (1968) an Edgeworth expansion of the densities isused as an approach. However, this method can lead to densities which do notsatisfy the underlying axioms of probability theory, yielding a non valid density. InSorenson and Alspach (1971) a different approach is used, where the densities areapproximated with Gaussian sums. This method satisfies the probability theoryaxioms and also converges uniformly to any density of practical concern. Thislatter method, which applies recursive Bayesian estimation using Gaussian sums,is described below. The Gaussian mixture equations according to Bar-Shalom andLi (1993) can be defined as

p(x) =

N∑

j=1

w(j)N(x; x(j), P (j)), (2.74)

where the probabilities sum to unity, i.e.,∑N

j=1 w(j) = 1. If we denote the event

that x is Gaussian with mean x(j) and covariance P (j) by H(j), and assume thatall H(j) are mutually exclusive and exhaustive, then

p(x) =

N∑

j=1

p(x|H(j))PrH(j), (2.75)

by the total probability theorem. The mean value can be written as

x =N∑

j=1

w(j)x(j), (2.76)

and the covariance can be decomposed as

P = E(x− x)(x− x)′ =

N∑

j=1

E(x− x)(x− x)′|H(j)w(j)

=N∑

j=1

E(x− x(j) + x(j) − x)(x− x(j) + x(j) − x)′|H(j)w(j)

=N∑

j=1

w(j)E(x− x(j))(x− x(j))′|H(j) +N∑

j=1

w(j)(x(j) − x)(x(j) − x)′, (2.77)

where the last equality is obtained by expanding the expression and applying (2.76).Consider a dynamical system using the following model

xt+1 = f(xt) + g(xt)vt (2.78)

yt = h(xt) + et, (2.79)

where the state vector xt ∈ Rn, process noise vt ∈ N(0, Qt) and measurement noise

et ∈ N(0, Rt) are independent. The main idea with Gaussian sum approximationis to approximate the posterior density, p(xt|Yt), by a sum of Gaussian kernels.We formalize the above discussion in Theorem 2.3.


The measurement noise covariance matrix is denoted Rt and the linearized obser-vation matrix is given by H

(i)t = ∂h(x)

∂x |x=x

(i)

t|t−1

. The coefficients are now given

by

w(i)t =

w(i)t−1N(yt − h(x

(i)t|t−1), S

(i)t )

∑Nj=1 w

(j)t−1N(yt − h(x

(j)t|t−1), S

(j)t )

. (2.90)

From Anderson and Moore (1979) we have the following theorem for justifying theapproach.

Theorem 2.4 (Anderson and Moore, 1979)Assume yt = h(xt) + et with Gaussian noise, et, and probability density

p(xt|Yt−1) =

N∑

i=1

w(i)t−1N(xt − x

(i)t|t−1, P

(i)t|t−1).

Then the Gaussian sum

N∑

i=1

w(i)t N(xt − x

(i)t|t−1, P

(i)t|t−1), (2.91)

approaches p(xt|Yt) uniformly in xt and yt as P(i)t|t−1 → 0, for i = 1, 2, . . . , N . The

coefficients w(i)t are given by (2.90).


The time-update relies on the extended Kalman filter update

x(i)t+1|t = f(x

(i)t|t ) (2.92)

P(i)t+1|t = F

(i)t P

(i)t+1|t(F

(i)t )′ +G(x

(i)t|t )Qt(G(x

(i)t|t ))

′, (2.93)

where F(i)t = ∂f(x)

∂x |x=x

(i)

t|t

. The fact that the EKF update for the filters gives the

correct density is summarized in the following theorem.

Theorem 2.5 (Anderson and Moore, 1979)Using xt+1 = f(xt) + g(xt)vt with Gaussian noise, vt, and probability density

p(xt|Yt) =N∑

i=1

w(i)t N(xt − x

(i)t|t , P

(i)t|t ).

Then the Gaussian sum

N∑

i=1

w(i)t N(xt+1 − x

(i)t+1|t, P

(i)t+1|t), (2.94)

approaches the one-step-ahead a posteriori density, p(xt+1|Yt), uniformly in xt as

P(i)t|t → 0, for i = 1, 2, . . . , N .



In practice, the Gaussian sum approximation is implemented as a bank of parallelextended Kalman filters. The weights are adjusted after each measurement update.If the covariance matrices in the different filters are too large, the Gaussian sumapproximation must be reinitialized. In Kotecha and Djuric (2001) several issues,advantages and drawbacks regarding the GS-filters are discussed.

2.4.5 Change Detection using Multiple Models

In Gustafsson (1992, 2000) several estimation methods using change detection aredescribed. Some methods are closely related to multiple model approaches, de-scribed in previous sections. The main idea in change detection is to use a changedetector, which will indicate changes in the system. These changes can be faults,i.e., indicating a malfunction in the system, or be due to changes in the system,such as a maneuver in the target tracking application. The change detector can beused to adjust or tune an adaptive filter which is used for estimation. In Gustafsson(2000) several change detectors are presented. For instance, exponential forgettingfactor, geometric moving average (GMA) and the cumulative sum (CUSUM).

3

Numerical Methods for

Estimation and Filtering

In Section 2.2, we described the Bayesian approach to state estimation. As men-tioned, only a few special cases can be solved analytically. Here we discuss ap-proximate methods dealing with these optimal estimation equations. Basically,Section 3.1 discusses a deterministic grid-based method to solve the integrals nu-merically. However, for higher state dimension the grid-based approach is notfeasible. The main technique to solve the Bayesian estimation problem for highdimension state space relies on Monte Carlo integration. In Section 3.2 severalclassical off-line applications, for instance Markov Chain Monte Carlo (MCMC)methods are presented. The sequential Monte Carlo method, or particle filter, isintroduced and described in detail in Section 3.3.

3.1 Numerical Integration

Several different approaches exist for solving integrals numerically. Some meth-ods are based on deterministic integration where the integral is approximated bya Riemann sum, others rely on stochastic simulation. Some methods are only ap-plicable to integration on R

1, whereas others are more general. In Section 3.1.1,the point-mass filter for a Bayesian estimation problem is presented. Section 3.1.2discusses a Riemann-sum approach where the integration is based on samples froma stochastic process.

31

32 Numerical Methods for Estimation and Filtering

3.1.1 The Point Mass Filter

In Kramer and Sorenson (1988) the Bayesian approach is investigated for a discrete-time non-linear system. Bayesian time- and measurement updates are solved usingan approximative numerical method, where the density functions are piecewiseconstant on regular regions in the state space. In Bergman (1997) a non-linearand non-Gaussian navigation estimation problem is solved using a deterministicnumerical integration of the Bayesian time- and measurement update equations.This numerical method is sometimes referred to as the point-mass filter (PMF).The navigation system studied is the Saab Bofors Dynamics terrain navigation(TERNAV) system. In Svensson (1999) the PMF is implemented and tested inthe TERNAV system. Performance analysis in terms of Cramer-Rao bounds arepresented in Bergman (1997, 1999).

The PMF grid based approximation for a general integral in Rn is defined as

follows

∫

Rn

g(xt)dxt ≈N∑

i=1

g(x(i)t )δn, (3.1)

where we have assumed a regular grid δ and samples g(x(i)t ). Applying this to a

general Bayesian estimation problem in (2.12)-(2.13), using the model with additivenoise as in (2.10)–(2.11) yields the following approximation

p(x(i)t |Yt) =α−1

t pet(yt − h(x

(i)t ))p(x

(i)t |Yt−1) (3.2)

p(x(i)t+1|Yt) =

N∑

n=1

pvt(x

(i)t+1 − f(x

(n)t ))p(x

(n)t |Yt)δ

n, (3.3)

where the normalization factor αt is given by

αt =

N∑

i=1

pet(yt − h(x

(i)t ))p(x

(i)t |Yt−1)δ

n. (3.4)

For a high dimensional problem this solution method is in practice not feasible.Note that the number of points in the grid approximation increases with time, so areduction method is needed in a practical application. Several such issues and howto use adaptive grid size are discussed in Bergman (1997). Note that the error inthe grid approximation depends on the state dimension.

3.1.2 Stochastic Riemann Approximation

In Robert and Casella (1999) the Riemann-sum approximation of an integral usingrandom samples is discussed. Consider the case where we want to approximate anintegral such as

I =

∫

g(x)p(x)dx, (3.5)

3.1 Numerical Integration 33

where the density p(x) integrates to unity, i.e.,∫p(x)dx = 1. The integral I can

be approximated by the empirical average

I ≈ 1

N

N∑

i=1

g(x(i)), (3.6)

where x(i) are samples from the density. For the one-dimensional case the numericalintegration can be based on random samples according to

I ≈N−1∑

i=0

g(x(i))p(x(i))(x(i+1) − x(i)), (3.7)

where x(0), x(1) . . . , x(n−1) are ordered independent identically distributed (i.i.d)variables. The Riemann sum converges as the number of samples tends to infinity.

As stated in Robert and Casella (1999) a one-dimensional integral over [0, 1],

i.e.,∫ 1

0g(x)dx, with a bounded derivative, ∂g

∂x , on the interval can be approximatedby

∫ 1

0

g(x)dx ≈ g(0)U (0) + g(N)(1 − U (N)) +N−1∑

i=0

g(U (i))(U (i+1) − U (i)), (3.8)

where U (i) are ordered samples from U[0,1]. The variance of the integral is oforder O(N−2), compared to O(N−1) for a pure numerical integration using a grid-approximation. Note however, that this promising result for the stochastic Rie-mann approach is unfortunately not valid for the multi dimensional case. This isreferred to as the curse of dimensionality. We apply this in Example 3.1 to themean value of the Gaussian over the interval [0, 1].

Example 3.1 Riemann-sum approximation.Suppose we want to evaluate the mean value of a one dimensional Gaussian

distribution with mean value 2, and unit variance over the interval [0, 1]. Thisvalue is computed for 100 Monte Carlo simulation, with N samples according toEquation (3.8). The true value of the integral can be computed as

Samples (N) Integral10 0.0650100 0.08151000 0.0836

Table 3.1: Riemann-sum integration.

∫ 1

0

1√2πxe−

(x−2)2

2 dx = . . . = − 1√2π

(e−12 − e−2) + 2(Q(−1) −Q(−2)) ≈ 0.0838,

where Q(x) =∫ x

−∞1√2πe−

x2

2 dx.


3.2 Off-line Monte Carlo Methods

In this section several off-line Monte Carlo methods are described. The main pur-pose is to describe several different ways of generating samples from non-standarddistributions. In order to do so we may require that the distribution is known atleast to a normalization factor. Other methods are specially designed to handlemulti-variable distributions given that the marginalized distributions are simpleto sample from. Several different approaches and examples are given below. Forfurther details we refer to Robert and Casella (1999).

3.2.1 Accept-Reject Methods

There exist many distributions from which it is difficult or impossible to simulatefrom. Sometimes the distribution is known only to a normalization constant. Tosolve these problems we need other ways to compute random variables from thesedistributions. Several methods exist, all based on the accept-reject method.

Theorem 3.1 (Accept-Reject methods, Robert and Casella,1999)Suppose that we know the density p(x) up to a normalization factor, and thatp(x) ≤ Mq(x),∀x, and that we can produce samples X from q(x). Then thefollowing algorithm produces a variable Y distributed according to p(·).

1. Generate X ∼ q, U ∼ U[0,1].

2. Accept Y = X, if U ≤ p(X)Mq(X) .

3. If not accepted return to item 1.

Proof (Robert and Casella,1999)

P (Y ≤ y) = P (X ≤ y|U ≤p(X)

Mq(X)) = [P (A|B) =

P (A, B)

P (B)] =

P (X ≤ y, U ≤ p(X)Mq(X)

)

P (U ≤ p(X)Mq(X)

)

=

∫ y

−∞

∫ p(x)/Mq(x)

0duq(x)dx

∫∞

−∞

∫ p(x)/Mq(x)

0duq(x)dx

=

1M

∫ y

−∞p(x)dx

1M

∫∞

−∞p(x)dx

=

∫ y

−∞

p(x)dx. (3.9)

2

The algorithm provides a generic method how to simulate from a distributionknown up to a normalization factor. Note that in the algorithm the acceptanceprobability is determined by the constant M , and it is exactly 1/M . The mainproblem with algorithms using acceptance-rejection is that there is no upper boundon the number of iterations. For real-time applications other methods must beused, or modifications are needed. The important sampling method presented insection 3.2.2, is one way to deal with this. There are several variants of the accept-rejection technique. In Robert and Casella (1999) some of these extensions tothe original method are described. In Example 3.2, the accept-reject technique isdemonstrated by generating samples from a truncated Gaussian distribution.

3.2 Off-line Monte Carlo Methods 35

Example 3.2 Generate a truncated Gaussian using the Accept-Reject method.Consider the case with

p(x) ≈

1√2πe−

x2

2 , x ∈ [−4, 4]

0, otherwise

By choosing M = 4, the conditions for the Accept-Reject method are fulfilled if weconsider the uniform proposal function q(x) according to

q(x) =

1/8, x ∈ [−4, 4]

0, otherwise

In Figure 3.1 the simulations are presented using N = 10000 samples.

−5 0 50

0.2

0.4

0.6

0.8

x

p(x)

Accept−RejectGaussian pdf

−5 0 50

0.2

0.4

0.6

0.8

x

M q(x)Gaussian pdf


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 3.1: The Accept-Reject method for a truncated Gaussian using a uniformproposal.

3.2.2 Importance Sampling

In Robert and Casella (1999) the importance sampling (IS) method is presented.The problem consists of approximating the expected mean of a function, using theunderlying probability density p(·). The main idea is to use a proposal distributionq(·), from which we can produce samples. When trying to evaluate the mean valuefor an arbitrary function g(·), we use

Eg(X) =

∫

g(x)p(x)dx =

∫

g(x)p(x)

q(x)q(x)dx. (3.10)


Hence, the mean value of g(x) is calculated by computing the mean of g(x) p(x)q(x)

given the density q(x). Approximating the integral and calculating the mean valuegives

Eg(X) ≈ 1

N

N∑

i=1

p(x(i))

q(x(i))︸︷︷︸

w(i)

g(x(i)), (3.11)

where x(1), x(2), . . . , x(N) are samples from q(·) and the importance weights are

defined as w(i) = p(x(i))q(x(i))

.

Example 3.3 Importance sampling for bi-modal distribution.Consider the following distribution

p(x) =1

2N(x; 0, 1) +

1

2N(x; 3, 1/2),

presented in the upper part of Figure 3.2. By calculating the mean value, that is

−10 −5 0 5 100

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

x

p(x)

Bi−modal PDFIS PDF approx


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 3.2: Bi-modal Gaussian distribution used for IS.

using g(x) = x, we get by using q(x) ∈ U(−5, 5) the IS mean estimate, Eg(x) ≈1.4915, when using N = 10000 samples. The true value is simply given byEg(x) = 1.5. In Figure 3.2, the histogram using the importance weights arepresented together with the true PDF.

In an example in (Robert and Casella, 1999, p. 80) it is shown that it mayactually be favorable for some systems to generate samples from a distributionother than that of interest!


3.2.3 Markov Chain Monte Carlo Methods

Monte Carlo integration evaluates the expected mean of an arbitrary function g(X),i.e., Eg(X), by drawing n samples, x(i) : i = 1, 2, . . . , N according to thedistribution p(·) and approximating the population mean of g(X) by the samplemean, that is

Eg(X) ≈ 1

N

N∑

i=1

g(x(i)). (3.12)

If the samples are independent, then the law of large numbers is applicable, hencethe approximation can be made as accurately as desired, by taking sufficiently manysamples. In general it is not possible to draw samples from the density p(·), sinceit could be quite non-standard. The main idea behind Markov Chain Monte Carlo(MCMC) methods is that we can draw samples from another distribution, and byusing a Markov chain, we can generate variables from the chain with a density equalto the desired. For many practical applications it is possible to construct MCMCmethods to approximate the density. Many of these methods are constructed bythe technique described from the original work of Hastings, (Hastings, 1970), whichis a generalization of Metropolis et al. (1953). The presentation here is mainlyaccording to that of Gilks et al. (1996); Robert and Casella (1999).

3.2.4 Metropolis–Hastings

In Hastings (1970) a generalization of the sampling method introduced by Metropo-lis et al. (1953) is given providing an efficient way to handle high dimensional nu-merical problems in statistical analysis. However, the implementation relies on thatsamples can be drawn from a high dimensional distribution. Metropolis-Hastingsalgorithm starts with a target density p(x) and by choosing a conditional den-sity q(y|x). If q(·|x) is easy to simulate from and given up to a normalizationconstant (independent of x) or symmetric (q(y|x) = q(x|y)), then the Metropolis-Hastings algorithm can be implemented in practice, (Robert and Casella, 1999).The Metropolis-Hastings algorithm presented here is according to Bergman (1999)and Robert and Casella (1999).

Algorithm 3.1 (Metropolis–Hastings, Hastings, 1970)

1. Initialize by setting i = 0 and choose x(0) randomly or deterministically.

2. Sample z ∼ q(z|x(i)), u ∼ U(0, 1).

3. Compute the acceptance probability α(x(i), z) = min(

1, p(z)q(x(i)|z)p(x(i))q(z|x(i))

)

.

4. If u ≤ α(x(i), z) then set x(i+1) = z. Otherwise set x(i+1) = x(i).

5. Increase the index i and return to step 2.


The distribution q is referred to as the instrumental or proposal distribution.

Example 3.4 Metropolis-Hastings.The Metropolis-Hastings method is applied to the bi-modal Gaussian mixture fromExample 3.3. We use a similar version of the Metropolis-Hasting algorithm asin Bergman (1999), whereas the proposal yields a random walk, i.e., adding noise tothe current state of the chain and accepting the sample if the likelihood increases.The proposal density used in the simulation was a Gaussian density with zeromean and variance σ2. The number of samples were N = 400. The true densityis given in Figure 3.3 (a), and (b)-(d) represent Metropolis-Hastings simulationsusing different standard deviations in the proposal density. We used (b) σ=0.1, (c)σ=1 and (d) σ=10, to investigate different behavior of the method.

(a) (a) (a)

(b)

(c)

(d)


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 3.3: Metropolis-Hasting simulations.

There has been a lot of work on the Metropolis-Hastings algorithm, resulting inseveral special cases. Here we present some of these. The Independent Metropolis-Hastings algorithm is given in Algorithm 3.2.

Algorithm 3.2 (Independent Metropolis–Hastings, Robert and Casella, 1999)

1. Given x(i).

2. Generate Y ∼ q(y).

3. Let x(i+1) =

Y, with probability min p(Y )q(x(i))p(x(i))q(Y )

, 1x(i), otherwise


The independent Metropolis-Hastings algorithm can be seen as a generalizationof the accept-reject method. Note that even if Y is generated independently, theresulting samples are not independent identically distributed.

In (Robert and Casella, 1999, p. 241) it is pointed out that the independentMetropolis-Hastings algorithm is more efficient than the Accept-Reject algorithm,since on the average more proposed values are accepted.

3.2.5 Gibbs Sampling

In Casella and George (1992) a thorough explanation of the Gibbs sampler tech-nique from Geman and Geman (1984) is given, that is how to generate randomvariables from a (marginal) distribution indirectly, without having to calculate thedensity. Gibbs sampling is based only on elementary properties of Markov chains.Suppose that the density is given by marginalizations as

p(x) =

∫

. . .

∫

p(x, ξ1, ξ2, . . . , ξm)dξ1dξ2 . . . dξm. (3.13)

Rather than to compute or approximate p(x) directly, the Gibbs sampler allowsus effectively to generate samples x(1), . . . , x(N) from p(x) without requiring p(x).The usefulness of the Gibbs sampler increases with state dimension. Assume thatx ∈ R

n with x =(x1, x2, . . . , xn

)′. Denote sample i from component j with

xj,(i).We use the following short hand notation from Bergman (1999) for the state

vector in iteration i excluding component j

x¬j,(i) M= (x1,(i), . . . , xj−1,(i), xj+1,(i−1), . . . , xn,(i−1))′.

The Gibbs sampler is given in Algorithm 3.3.

Algorithm 3.3 (Gibbs sampling, Geman and Geman, 1984)

1. Initialize by setting i = 0 and choose x(0) randomly or deterministically.

2. Cycle through the entries of x and sample from the full conditionals

• x1,(i) ∼ p(x1|x¬1,(i))

• x2,(i) ∼ p(x2|x¬2,(i))...

• xn,(i) ∼ p(xn|x¬n,(i))

3. Output x(i), increase i and return to step 2.


Example 3.5 Gibbs sampler.Consider the interesting family of mixed Gaussian distributions. These are Gaus-sian distributions with a stochastic parameter. Here we consider the case when thevariance is a random variable with an exponential distribution,

X|σ2 ∈ N(0, σ2)

σ2|X ∈ Exp(1)

In Gut (1995) it is shown that X is Laplacian, X ∈ L( 1√2). In Figure 3.4 the

histogram approximating the PDF, using the Gibbs sampler with 10000 samples iscompared to the theoretical PDF.

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

PD

F

Gibbs samplerLaplace L(1/sqrt(2))


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 3.4: Gibbs sampler for a Laplace distribution.

In Robert and Casella (1999) it is mentioned that in fact the Gibbs sampler is aspecial case of the Metropolis-Hastings algorithm, more specifically a combinationof Metropolis-Hastings applied to different components.

3.2.6 Bootstrap

To perform a statistical analysis of a tracking or state estimation scenario, a com-mon technique is to apply Monte Carlos simulations. That means that we areable to re-run the simulation model with a new noise realization or that we cancollect an additional measurement sequence. In many practical applications it isnot possible to collect several measurement sequences. One way to estimate pa-rameters is to apply the bootstrap technique, (Efron, 1979). Here we present thebootstrap algorithm for a time invariant dynamical system as given in Gustafsson(2000).

3.3 Sequential Monte Carlo Methods 41

Algorithm 3.4 (Bootstrap, Efron, 1979)

1. Calculate a point estimate from the data sequence

2. Apply an inverse model (filter) to get a residual sequence

3. Generate a bootstrap noise sequence from the residuals

4. Simulate the system using the bootstrap noise sequence

5. Compute a point estimate

In Example 3.6, we consider the case when the system is described by an auto-regressive (AR) model.

Example 3.6 Bootstrap for an AR(1)-model.In Gustafsson (2000) the bootstrap method is explained via an AR(1)-model.

Consider the following system

yt = −θyt−1 + et ⇒ yt =1

1 + θq−1et, t = 1, . . . , tf .

If we calculate the ML-estimate θ given the observed data, the following residualcan be calculated

et = yt + θyt−1.

From this sequence the bootstrap residual e(i)t can be generated by picking random

samples with replacement. By applying the dynamical model and the currentestimate, new data is given by

y(i)t = −θy(i)

t−1 + e(i)t .

Finally, a new estimate of the parameter vector θ is given from the simulated dataand the scheme can be repeated.

3.3 Sequential Monte Carlo Methods

In target tracking estimation problems the task is to estimate unknown quantitiesfrom noisy observations. Often prior knowledge is available, therefore it is naturalto use a Bayesian approach. For linear systems with a Gaussian noise assumptionit is possible to derive an analytical expression for the estimate. This recursiveexpression is given by the famous Kalman filter. For partially observed linearsystems, the hidden Markov model (HMM) gives an analytical solution. For manypractical tracking problems linear models are not plausible. Sometimes carefullymodeled sensors are not well approximated by a Gaussian distribution.


Sequential Monte Carlo methods, or particle filters, provide general solutions tomany problems where linearizations and Gaussian approximations are intractableor would yield too low performance. Non-Gaussian noise assumptions and incor-poration of constraints on some of the system parameters can also be performed ina natural way by using these simulation based methods. The constraints are dueto limitations in state variables but could be induced by the terrain, such as landavoidance for tracking ships. Monte Carlo techniques have been a growing researcharea lately due to improved computer performance.

In Tanizaki (1997) the non-linear and non-Gaussian problem is analyzed bothusing numerical integration and Monte Carlo integration. Both prediction andsmoothing equations are presented. Several non-linear systems are compared usingdifferent techniques.

The PMF discussed in Bergman (1997); Svensson (1999) is implemented for atwo-dimensional terrain navigation problem. In Ahlstrom and Calais (2000) theTERNAV system is analyzed with a higher order state space model, incorporatingdrift terms in the filter. To avoid implementing the PMF in higher dimension,which is troublesome, a sequential Monte Carlo method based on particle filters isused.

3.3.1 The Particle Filter

Many engineering problems are by nature recursive and require on-line solutions.Therefore, there is a need for accurate state estimation techniques for non-linearand non-Gaussian problems. Monte Carlo techniques have been a growing researcharea lately due to improved computer performance. The seminal paper of Gordonet al. (1993) marks the onset of a rebirth for algorithms based on sequential MonteCarlo simulation techniques for solving the optimal estimation problem. How-ever, similar ideas have been discussed in Handschin and Mayne (1969); Handschin(1970), where the conditional mean and covariance were calculated using impor-tance sampling for recursive Bayesian estimation. In this section the presentationof the sequential Monte Carlo method, or particle filter theory is according toBergman (1999); Doucet et al. (2000a); Pitt and Shephard (1999).

Consider the following non-linear discrete-time system

xt+1 = f(xt, vt)

yt = h(xt, et). (3.14)

Sequential Monte Carlo methods, or particle filters, provide an approximativeBayesian solution to discrete-time recursive problem by updating an approximativedescription of the posterior filtering density. Let xt ∈ R

n denote the state of theobserved system and Yt = yit

i=1 be the set of observations until present time.The Monte Carlo filter approximates the probability density p(xt|Yt) by a large

set of N particles x(i)t N

i=1, where each particle has an assigned relative weight,

w(i)t , such that all weights sum to unity. The location and weight of each particle

reflect the value of the density in the region of the state space. The particle filter


updates the particle location and the corresponding weights recursively with eachnew observation. The non-linear prediction density p(xt|Yt−1) and filtering densityp(xt|Yt) for the Bayesian interference are given by

p(xt|Yt−1) =

∫

Rn

p(xt|xt−1)p(xt−1|Yt−1)dxt−1 (3.15)

p(xt|Yt) =p(yt|xt)p(xt|Yt−1)

p(yt|Yt−1). (3.16)

We refer to Section 2.2.1 for a derivation of the time- and measurement updateequation for the probability densities presented in (3.15) and (3.16).

Often we do not know the normalization factor in Equation (3.16). However,in the methods presented this is not necessary, since we only need to evaluate

p(xt|Yt) ∝ p(yt|xt)p(xt|Yt−1), (3.17)

where the likelihood p(yt|xt) is calculated from Equation (3.14) and using theknown measurement noise density pet

. As stated in Section 2.2.1, an often usedassumption is to use additive noise, so yt = h(xt) + et, yielding

p(yt|xt) = pet(yt − h(xt)). (3.18)

The main idea is to approximate p(xt|Yt−1) with

p(xt|Yt−1) ≈1

N

N∑

i=1

δ(xt − x(i)t ), (3.19)

where δ is the delta-Dirac function. Inserting (3.20) into (3.16) yields

p(xt|Yt−1) ≈N∑

i=1

w(i)t δ(xt − x

(i)t ). (3.20)

The normalized importance weights are defined as

w(i)t =

w(i)t

∑Nj=1 w

(j)t

, i = 1, . . . , N, (3.21)

where w(i)t ∝ p(yt|x(i)

t . If we for each time calculate the weights given by thelikelihood for each sample it is possible to recursively update the weights accordingto

w(i)t = p(yt|x(i)

t )w(i)t−1. (3.22)

This was the original estimation idea. However, this approach leads to divergence,where almost all of the particles have zero weights. By introducing a selection orresampling step as proposed in Gordon et al. (1993) this can be handled. Mainlybecause of the resampling step and the increased computer capacity, there haslately been an increased research activity in the sequential Monte Carlo field. Thisresampling idea from Gordon et al. (1993) is often referred to as Bayesian bootstrapor sampling importance resampling (SIR). The algorithm is given in Algorithm 3.5.


Algorithm 3.5 (Sampling Importance Resampling (SIR), Gordon et al., 1993)

1. Set t = 0 and generate N samples x(i)0 N

i=1 from the initial distribution p(x0).

2. Compute the weights w(i)t = p(yt|x(i)

t ) and normalize, i.e.,

w(i)t = w

(i)t /

∑Nj=1 w

(j)t , i = 1, . . . , N .

3. Generate a new set x(i?)t N

i=1 by resampling with replacement

N times from x(i)t N

i=1, with probability Prx(i?)t = x

(j)t = w

(j)t .

4. Predict (simulate) new particles, i.e., x(i)t+1 = f(x

(i?)t , v

(i)t ), i = 1, . . . , N

using different noise realizations for the particles.

5. Increase t and iterate to step 2.

In Example 3.7, the SIR method is demonstrated for a linear model.

Example 3.7 Probability density from the particle filter.

In Figure 3.5, the initial particle cloud from a Gaussian prior together with theimportance weights are shown. are given. Using a linear transition for the timeupdate, the cloud and the density is spread out according to the figure.

Y−Position

X−Position

Pro

babi

lity

PSfrag replacements

Ax(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 3.5: Probability density using the SIR method.

Sometimes the resampling step is omitted and just imposed when it is needed, toavoid divergence in the filter. This method is referred to as sequential importancesampling (SIS), (Doucet et al., 2000a), and since we do not resample the weights


at each iteration, we need to derive a formula how they are updated recursively.If we recall the IS method in Section 3.2.2, we choose the importance weights in aparticular way. Here, we describe the SIS method in more detail, where the SIRmethod is an important special case.

Consider the following short hand notation x0:t used in Doucet (1998), repre-senting the set of state vectors for different times up to and including t, that is

x0:t = x0, x1, . . . , xt. If the samples x(i)0:t were drawn from the proposal density

q(x0:t|Yt), then the importance weights can be calculated as

w(i)t ∝ p(x

(i)0:t|Yt)

q(x(i)0:t|Yt)

. (3.23)

The complete posteriori density can be rewritten using Bayes’ rule as

p(x0:t|Yt) =p(yt|x0:t,Yt−1)p(x0:t|Yt−1)

p(yt|Yt−1)

=p(yt|x0:t,Yt−1)p(xt|x0:t−1,Yt−1)p(x0:t−1|Yt−1)

p(yt|Yt−1)

=p(yt|xt)p(xt|xt−1)

p(yt|Yt−1)p(x0:t−1|Yt−1) (3.24)

Ignoring the normalization factor we have

p(x0:t|Yt) ∝ p(yt|xt)p(xt|xt−1)p(xt|x0:t−1,Yt−1). (3.25)

Assume that the proposal is chosen in the following form

q(x0:t|Yt) = q(xt|x0:t−1,Yt−1)q(x0:t−1|Yt−1) (3.26)

Inserting (3.25) and (3.26) into Equation (3.23), the weights are recursively updatedas

w(i)t ∝ p(yt|x(i)

t )p(x(i)t |x(i)

t−1)p(x(i)0:t−1|Yt−1)

q(x(i)t |x(i)

0:t−1,Yt)q(x(i)0:t−1|Yt−1)

= w(i)t−1

p(yt|x(i)t )p(x

(i)t |x(i)

t−1)

q(x(i)t |x(i)

0:t−1,Yt). (3.27)

Particularly, the choice of q(x(i)t |x(i)

0:t−1,Yt) = p(x(i)t |x(i)

t−1), gives the following up-date

w(i)t ∝ w

(i)t−1p(yt|x(i)

t ). (3.28)

The SIS method is summarized in Algorithm 3.6. A related method was originallydeveloped in Liu and Chen (1995). Note that the SIR method can be interpretedas SIS when resampling is chosen every time. In that case all weights are set equal.


Algorithm 3.6 (Sequential Importance Sampling (SIS), Doucet et al., 2000a)

1. Set t = 0, generate N samples x(i)0 N

i=1 from the initial distribution p(x0).

Initialize the importance weights w(i)−1 = 1/N , i = 1, . . . , N .

2. Compute the weights w(i)t = w

(i)t−1p(yt|x(i)

t ) and normalize, i.e.,

w(i)t = w

(i)t /

∑Nj=1 w

(j)t , i = 1, . . . , N .

3. If resampling is applied, then generate a new set x(i?)t N

i=1 by

resampling with replacement N times from x(i)t N

i=1, with probability

Prx(i?)t = x

(j)t = w

(j)t ; otherwise let x(i?)

t = x(i)t , i = 1, . . . , N .

4. Predict (simulate) new particles, i.e., x(i)t+1 = f(x

(i?)t , v

(i)t ), i = 1, . . . , N

using different noise realizations for the particles.


Different particle filter methods use a selection or resampling step to avoid diver-gence. As pointed out in Doucet (1998) the variance of the importance weightscan only increase over time. Therefore, if nothing is done to adjust the particlecloud, it will not be possible to avoid divergence or either the empirical densitydoes not reflect the true one. The choice of resampling is often done by usingsome criterion. An often suggested method is to study the effective sample sizeNeff, Bergman (1999); Doucet et al. (2000a); Kong et al. (1994); Liu (1996). Themethod relies on the calculation of how many samples in the particle cloud thatactually contribute to the support of the probability density approximation. Theeffective sample size is defined as

Neff =N

1 + Covq(·|Yt)

(wt(xi

t)) , (3.29)

where q(·|Yt) is the proposal density. It is not possible to evaluate this expression,but an approximation is given by

Neff =1

∑Ni=1(w

(i)t )2

. (3.30)

We can resample if the effective number of samples is less than a predefined thresh-old, i.e., Neff < Nth. Note that, 1 ≤ Neff ≤ N , where the upper bound is attainedwhen all particles have the same weight, and the lower bound when all probabilitymass is at one particle. In Bergman (1999) a suggested threshold is Nth = 2N/3.

In many practical target tracking applications the system presented in (3.14)can be simplified. Standard models assume additive noise of a known distribution(at least known up to a proportional constant). In (3.31) the additive noise model


is recapitulated.

xt+1 = f(xt) + vt

yt = h(xt) + et. (3.31)

This structure simplifies the evaluation of the particle filter. The calculation ofparticle weights is given by (3.32) and the time update is done by adding the noiseaccording to (3.33) for each particle i = 1, . . . , N .

w(i)t = p(yt|x(i)

t ) = pet(yt − h(x

(i)t )) (3.32)

x(i)t+1 = f(x

(i?)t ) + v

(i)t . (3.33)

Estimate and uncertainty region for the particle filter can be calculated as

xMMSEt =

∫

xtp(xt|Yt)dxt ≈N∑

i=1

w(i)t x

(i)t (3.34)

Pt =

∫

(xt − xMMSEt )(xt − xMMSE

t )′p(xt|Yt)dxt

≈N∑

i=1

w(i)t (x

(i)t − xMMSE

t )(x(i)t − xMMSE

t )′. (3.35)

It is possible to use a ML-estimate by simply calculating which particle that hasthe highest weight or probability. However, most common is to use xMMSE as theestimate. As seen from (3.32) the importance weights are sensitive to extremeoutliers. If this is not reflected in the probability density used to calculated theweights there is a risk of ending up with all weights close to zero. To avoid this,these measurements can be discarded.

The computational burden is dependent on the number of particles and on theresampling calculation which is a bottle-neck for parallelization. Fortunately, theresampling step can be efficiently implemented using a classical algorithm for sam-pling N ordered independent identically distributed (i.i.d.) variables, (Ripley, 1988;Doucet, 1998; Bergman, 1999). Assume that we have ordered the i.i.d. variablesu(i)N

i=1 sampled from U(0, 1). Then the following scheme gives the resampling asin O(N) time summarized in Algorithm 3.7.

Algorithm 3.7 (Sampling of ordered U(0, 1) variables, Ripley, p. 96, 1988)

1. Sample u(i) ∼ U(0, 1), for i = 1, . . . , N .

2. Set u(N) =N√u(N).

3. Compute u(i) = u(i+1) i√u(i), for i = N − 1, . . . , 1.

In Bergman (1999) a very compact MATLAB implementation of Algorithm 3.7 isgiven.


1

Particle indexp


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval

∑pi=1 w

(i)t

i − 1 i

w(i)t

N

ξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 3.6: The particle filter resampling method.

In Figure 3.6 the resampling method is presented. The main idea is that parti-cles with small weights are likely to be discarded and particles with large weightsare copied, where the number of copies reflects the probability of the particle. Thiscan be done by iterating over the ordered uniform samples u(j), j = 1, . . . , N andcomparing the cumulative sum of importance weights up to the current index. For

example, in Figure 3.6 particle x(i)t is duplicated.

Another approach is to utilize a deterministic resampling, i.e., to calculate the

number of particles to be copied by usingN (i) = bNω(i)t c, which could be somewhat

faster. Also, instead of using Algorithm 3.7 an efficient sort function can be used.

The resampling step is crucial for reducing the degeneracy problem for particlefilters and provide an accurate approximation of the posterior. Hence, a lot of workhas been done trying to improve it. However, the introduction of the resamplingstep leads to a loss of diversity among the particles. In Doucet et al. (2001) theregularized particle filter (RPF) was proposed as a solution to the problem. A nicesummary is given in Maskell and Gordon (2001). The RPF is identical to the SIRmethod presented except for the resampling step. Instead of drawing samples froma discrete approximation the samples are drawn from a continuous approximationof the density. The RPF uses the following approximation

p(xt|Yt) ≈N∑

i=1

w(i)t Kα(xt − x

(i)t ) (3.36)

Kα(x) =1

αnK(x/α), (3.37)

where K(·) is a kernel function, xt ∈ Rn and α > 0 is the kernel bandwidth. The


kernel density is a symmetric probability density defined by∫

xK(x)dx = 0 (3.38)

∫

||x||2K(x)dx <∞. (3.39)

The kernel is chosen so the mean integrated square error (MISE) between the trueposterior and the corresponding regularized empirical representation is minimized.Optimal choices of kernel function have been proposed for the case when we haveequally weighted samples or if a Gaussian noise with unit variance is used. Forthe general case, it is probably not feasible to obtain an analytical expression, butapproximations could be used.

The main idea behind particle filtering seems rather straight forward and simple.However, the reason why it has been successful is that the empirical density, builtup by the samples, converges to the true density if the number of particles is largeenough. Consider N independent samples drawn from a distribution p(x) anddefine

gN =1

N

N∑

i=1

g(x(i)) (3.40)

as an estimate of the mean of a function g(x), with I = Eg(x) =∫

Rn g(x)p(x)dx.The estimate is asymptotically unbiased and the sum converges almost surely tothe true value

Pr limN→∞

gN = I = 1 (3.41)

by the strong law of large numbers. Moreover, (Geweke, 1989), if

Varg(x) = σ2 =

∫

Rn

(g(x) − I)2p(x) dx =

∫

Rn

g2(x)p(x) dx− I2 <∞, (3.42)

then by the central limit theorem the approximation error converges in distribution,that is

limN→∞

√N(gN − I) ∼ N(0, σ2). (3.43)

The fact that the error is independent of the state dimension makes the sequentialMonte Carlo methods tractable for high dimensional problems, whereas for numer-ical integration methods the error is dependent on the state dimension. For anybounded function gt with norm ||gt|| = sup

x0:t

|gt(x0:t)|, there exist c independent of

N such that

Vargoptt − gpf

t ≤ c||gt||2N

. (3.44)

Further discussions are given in Doucet et al. (2000a) where a uniform convergenceresult is given.


3.3.2 The Auxiliary Particle Filter

There exist several different versions of the basic particle filter. The main tech-niques used are SIR and SIS (see Section 3.3.1). In Pitt and Shephard (1999) theauxiliary particle filter (APF) is proposed as an alternative method. The mainidea is to increase the influence of particles with a large predictive likelihood, ifwe are allowed to wait for the next measurement. This is done by using an extraindex to each particle, so the origin of the particle can be traced back, when thelikelihood at the next time step is evaluated. The effect is that we can re-simulatethose particles that were successful and therefore the particle cloud moves in thedesired direction.

The APF extends the state xt by predicting the state conditional upon particle

k. At time t, the particle set x(i)t N

i=1 and the corresponding weights w(i)t form

the following approximations to the prediction and filter densities for the state ofthe target

p(xt+1|Yt) =

N∑

i=1

p(xt+1|x(i)t )p(x

(i)t |Yt) (3.45)

p(xt+1|Yt+1) ∝ p(yt+1|xt+1)N∑

i=1

p(xt+1|x(i)t )p(x

(i)t |Yt), (3.46)

where Yt is the set of cumulative measurements up to and including time t. Bydefining

p(xt+1, k|Yt+1) ∝ p(yt+1|xt+1)p(xt+1|x(k)t )p(k|Yt), k = 1, . . . , N (3.47)

we can draw from this joint density and then discard the index, to produce a samplefrom the empirical filtering density as required. The index k is referred to as anauxiliary variable. Consider the joint density of particle k at time t and the targetstate at time t+ 1. Bayes’ rule gives

p(xt+1, k|Yt+1) ∝ p(yt+1|xt+1, k)p(xt+1, k|Yt)

= p(yt+1|xt+1)p(xt+1|k,Yt)

= p(yt+1|xt+1)p(xt+1|x(k)t )p(k|Yt)

= p(yt+1|xt+1)p(xt+1|x(k)t )p(k|Yt) (3.48)

Approximating this expression by replacing xt+1 with the expected mean

µ(k)t+1 = E(xt+1|x(k)

t )

in the first factor gives

p(xt+1, k|Yt+1) ≈ c · p(yt+1|µ(k)t+1)p(xt+1|x(k)

t )p(k|Yt). (3.49)


Marginalization over xt+1 yields

p(k|Yt+1) ≈ c · p(yt+1|µ(k)t+1)p(k|Yt), (3.50)

where c is a normalization factor. Sampling from the density (3.48) can now be

performed by resampling with replacement from the set x(i)t N

i=1, where the indexis chosen proportional to (3.50). The resampled candidates are then predicted usingthe system model. In summary, the APF algorithm is given in Algorithm 3.8.

Algorithm 3.8 (Auxiliary Particle Filter (APF), Pitt and Shephard, 1999)

1. Set t = 0, and generate N samples x(i)0 N

i=1 from p(x0), set µ(k)t = x

(k)t ,

w(j)0 = 1/N , k(j) = j, j = 1, . . . , N .

2. Compute µ(k)t+1 = Ext+1|x(k)

t .3. Generate new indices k(j) by sampling N times from

p(k|Yt+1) ∝ w(k)t p(yt+1|µ(k)

t+1) and predict (simulate) the particles, i.e.,

x(j)t+1 = f(x

(k(j))t , vt), j = 1, . . . , N with different noise realizations.

4. Compute the likelihood weights w(j)t =

p(yt+1|x(j)t+1)

p(yt+1|µ(k(j))t+1 )

for j = 1, . . . , N and

normalize, i.e., w(j)t =

w(j)t

∑Mj=1 w

(j)t

.

5. Perform an optional resampling of the set x(i)t+1N

i=1, using the probability

weights. If resampling is chosen then reset w(j)t = 1

N , j =, . . . , N .


3.3.3 Particle Filtering using Gaussian Sums

In Kotecha and Djuric (2001) a particle filtering approach is presented for a Gaus-sian sum model (see Section 2.4.4). As for the Gaussian sum filter in Section 2.4.4the filtering and prediction densities are approximated by finite Gaussian mix-tures. However, the process noise is approximated by a Gaussian mixture andthe sequential update is based on a sampling based method. The Gaussian sumparticle (GSP) method consists of a Gaussian sum part and a part utilizing thesequential importance sampling (SIS) particle filter. Since non-linear models areused the EKF will be used as the filter in the Gaussian sums.

The Gaussian sum (GS) filter assumes that the process noise is Gaussian. Forthe GSP filter this can be relaxed, where the update is done using the particlefiltering technique. Non-Gaussian densities are approximated using Theorem 2.3.


Assume the following model

xt+1 = f(xt) + vt (3.51)

yt = h(xt) + et. (3.52)

The process noise is approximated using GS, as

p(vt) ≈K∑

k=1

α(k)N(vt; µ

(k)t , S

(k)t ). (3.53)

It is assumed that the posterior at time t can be written as

p(xt|Yt) ≈L∑

l=1

ω(l)t N(xt;µ

(l)t , S

(l)t ). (3.54)

Therefore, it is possible to approximate the predicted density

p(xt+1|Yt) =

∫

p(xt+1|xt)p(xt|Yt)dxt

≈∫ K∑

k=1

α(k)N(xt+1; f(xt) + µ

(k)t+1, S

(k)t+1)

L∑

l=1

ω(l)t N(xt;µ

(l)t , S

(l)t )dxt.

(3.55)

By applying the particle filter method, the time update in the GSP algorithm isgiven below, where we used the notation j = l+(k−1)K and J = KL. Initially, we

assume equal weight Θ(j)0 (i) = 1

N for the particles x(j)0 (i), where i = 1, . . . , N

and the GS-models j = 1, . . . , J .

1. Sample x(l)t (i) ∼ N(xt;µ

(l)t , S

(l)t ) for i = 1, . . . , N and l = 1, . . . , L.

2. Obtain samples x(j)t+1(i) ∼ N(xt+1; f(xt = x(j)

t (i)) + µ(k)t+1, S

(k)t+1) for i =

1, . . . , N and j = 1, . . . , J

3. Calculate the weights for the GS-models

ω(j)t+1 =

ω(l)t α(k)

∑Kk=1

∑Ll=1 ω

(l)t α(k)

, j = 1, . . . , J.

4. Calculate mean µ(j)t+1 and covariance S

(j)t+1 from the particle cloud x(j)

t+1(i)

for i = 1, . . . , N and j = 1, . . . , J by taking sample means and covariances.

An approximation of the prediction density is now given by

p(xt+1|Yt) ≈J∑

j=1

ω(j)t+1N(xt+1; µ

(j)t+1, S

(j)t+1) (3.56)


As seen the number of mixands at each time update is increased from L to J = KL.To keep the complexity constant, this will be adjusted by the resampling in themeasurement update step. The density is given by

p(xt+1|Yt+1) ∝ p(yt+1|xt+1)p(xt+1|Yt)

∝ p(yt+1|xt+1)

L∑

j=1

w(j)t+1N(xt+1; µ

(j)t+1, S

(j)t+1). (3.57)

The measurement update is given by

1. Obtain samples x(j)t+1(i) ∼ N(xt+1; µ

(j)t+1, S

(j)t+1) for j = 1, . . . , J , i =

1, . . . , N .

2. Calculate the importance weights for i = 1, . . . , N as

Θ(j)t+1(i) = N

(

yt+1;xt+1 = x(j)t+1(i), S

(j)t+1

)

3. The mean and covariance are estimated for j = 1, . . . , J as

µ(j)t+1 =

∑Ni=1Θ

(l)t+1(i)x(l)

t+1(i)

∑Ni=1Θ

(j)t+1(i)

S(j)t+1 =

∑Ni=1Θ

(j)t+1(i)(x(j)

t+1(i) − µ(j)t+1)(x

(j)t+1(i) − µ

(j)t+1)

′∑N

i=1Θ(j)t+1(i)

.

4. Update the weights as w(j)t+1 =

∑Ni=1Θ

(j)t+1

(i)

∑Lj=1

∑Ni=1Θ

(j)t+1(i)

for j = 1, . . . , J .

5. Normalize the weights w(j)t+1 =

w(j)t+1

∑Ni=1 w

(j)t+1

.

The updated filtering density is now given by

p(xt+1|Yt+1) =J∑

j=1

w(j)t+1N(xt+1;µ

(j)t+1, S

(j)t+1). (3.58)

When applying a resampling step the number of mixands is reduced from J to L.Often a constant number is used and the resampling is performed as described inSection 3.3.1.

3.3.4 Depletion

For many applications using recursive Monte Carlo methods depletion or sampleimpoverishment may occur, i.e. the effective number of samples is reduced. This


means that the particle cloud will not reflect the true density, since only a few ofthe particles will contribute to the approximation of the density. Several differentmethods are proposed in the literature to reduce this problem. By introducing anadditional noise to the samples the depletion problem can be reduced. This tech-nique is called jittering in Fearnhead (1998), but a similar approach was introducedin Gordon et al. (1993) under the name roughening. In Doucet et al. (2000b) thedepletion problem is handled by introducing an additional Markov Chain MonteCarlo (MCMC) step to separate the samples.

4

Target Tracking and Data

Association

In Section 4.1 different sensors and sensor models are discussed. The main trackingsensors are radar and/or IR sensors. Often kinematic properties such as position,velocity and acceleration are estimated, but for more advanced systems, featuresrelated to the signal strength, such as intensity can also be used. The sensor fusionaspects, when multiple sensors are used, are discussed in Section 4.4. In Sections 4.2different tracking models, coordinate systems and process noise descriptions arepresented. An important sub-system for target tracking is the track initializationwhich is discussed in Section 4.3. For multiple target tracking systems the dataassociation is crucial. Different techniques and methods are discussed in Section 4.5.

4.1 Sensors

For target tracking applications many different sensors are available or used. Ifthe platform is moving a good and accurate navigation system must be used toachieve high performance. However, our main focus in this section is on trackingrelated sensors, such as radar and IR sensors. Depending on the application boththe accuracy and the actual shape of the probability distribution of the measure-ment noise may vary. For passive sensors such as IR, the most common measuredsignals are azimuth and elevation angle. For the radar sensor, range and range rate(doppler) are measured. Sometimes also non-kinematic features can be measured,

55

56 Target Tracking and Data Association

such as intensity.

For many tracking systems, both stationary surveillance systems and airborne,the main tracking sensors are radar and IR. A similar tracking sensor is the acousticsonar sensor, used in sub-marines and torpedos. Another important tracking sensoris the laser-radar.

4.1.1 The Infrared Sensor

One of the most common passive sensors for tracking is probably the infrared (IR)sensor. Originally, the term FLIR was used indicating a forward looking infraredsystem, which was a device that produced IR images viewing the temperature asan image. Now complete IR seekers, infrared search and track (IRST) systems areavailable, which often perform detection and tracking of point-targets. These IRdevices are available for different wavelengths, often ranging from about 1−14 [µm].In Blackman and Popoli (1999) the different IR bands are described in detail. Themain kinematic information from an IRST is relative angles to targets (azimuthand elevation). Other non-kinematic features, such as intensity, can also be usedas a measurement signal. The measurement model considered for this sensor is asimple model of the following form

y = h(x) =

(ϕθ

)

+ et, (4.1)

where ϕ is the azimuth angle, θ is the elevation angle. The intensity is an importantsignal in tracking. Under good conditions it can improve tracking performance, butoften it is not reliable and hence not used. A common model for the measurementnoise is to assume an additive white Gaussian noise model.

4.1.2 The Radar Sensor

The radar (radio detection and ranging) sensor is probably the most used sensorfor target tracking applications. A thorough book on radar sensors and radar signalprocessing is Skolnik (1980). A quick tutorial introduction can be found in Kingsleyand Quegan (1992). The radar sensor is mostly used as an active sensor, i.e., itis emitting energy and then detecting the return. Hence, from the time delay andthe speed of light it is possible to calculate the distance to the target. It is possibleto use the sensor in a passive mode, searching for targets emitting radar signals.This is often referred to as a radar warning receiver (RWR).

Depending on the particular application and radar sensor different features aremeasured by the sensor. The most common is to measure angles (azimuth and ele-vation) and range. If a doppler-radar is used, the range rate is available. The rangeand resolution of the measurements are dependent on the used signal processingtechnique and on the physical constraints, mainly the antenna aperture. Differ-ent signal processing techniques used in radar application are constant false alarmrate (CFAR) and synthetic aperture radar (SAR). By using different modulation

4.1 Sensors 57

techniques different features are available. The measurement relation for the radarsensor considered in this thesis is

y = h(x) =

ϕθrr

+ et, (4.2)

where ϕ is the azimuth angle, θ is the elevation, r is the range and r is the rangerate (derived from the doppler shift).

Another important signal is the received power. In radar engineering, the radarequation is often used as an approximation of the radar energy reflected back tothe sensor. In Kingsley and Quegan (1992) the radar equation is expressed as thesignal-to-noise ratio (SNR) according to

SNR =Prec

N=PtranG

2σλ2Ls

(4π)3r4N, (4.3)

where λ is the wavelength, Ptran the transmitted power, Prec the received power, rthe relative range to target, N the average noise power, Ls the loss-factor (Ls < 1),G the antenna gain (assuming transmitted and received gain equal) and σ denotesthe radar cross section (RCS). In Skolnik (1980) the fluctuation of the RCS isdiscussed, where four different models (Swerling cases) are considered. Mainly twodifferent probability densities are used for different statistical assumptions

p(σ) =1

σe−

σσ (4.4)

p(σ) =4σ

σ2e−

2σσ , (4.5)

where σ is a parameter. In Blackman and Popoli (1999) the RCS fluctuations arediscussed and several densities are calculated for both single and multiple signals.Both detection and false alarm densities are considered. In Kingsley and Quegan(1992) theoretical values for the RCS are given for several different geometricaltargets, when the target size is large compared to the wavelength. Often the SNRor received power is not used in tracking filters, since it is not always reliable.

4.1.3 Pseudo-Measurements for Kinematic Constraints

In many tracking applications constraints on the state vector may lead to problemswhen using classical estimation techniques such as the Kalman filter. In Tahkand Speyer (1990) kinematic constraints for tracking applications are discussedin detail. One way to handle constraints is to find a set of state variables so theconstraints can be incorporated in the state equation. This is often difficult and canbe extremely complicated. Instead the constraints can be handled by introducingadditional measurements. The constraint are then incorporated in the existingtracking filter, often an EKF, using these so-called pseudo-measurements.


In Blackman and Popoli (1999) this is discussed further. The constraints areoften expressed as an equation, c(xt) = 0. Hence, by extending the measurementvector by the additional pseudo-measurement form 0 = hc(xt) + ec

t , where ect is a

zero-mean random variable, the constraints are incorporated in the estimation pro-cess. The variance of ec

t controls the relative reliability of the constraint comparedto sensor information.

The pseudo-measurement technique can in some way handle constraints on thestate equation. For the particle filter method described in Section 3.3.1, constraintscan be handled in a non-approximative way. Hard constraints on state variableshave been implemented in for instance Challa and Bergman (2000); Karlsson andBergman (2000a).

4.2 Tracking Models and Coordinate Systems

In this section we describe important target tracking models. The main focus is onthe state dynamics and process noise. For a more detailed presentation of motionmodels we refer to Li and Jilkov (2000). In Gustafsson et al. (2002) relative motionand sensor models are discussed for positioning and tracking applications.

4.2.1 A Second Order Tracking Model

One of the simplest tracking models uses Cartesian coordinates to describe theposition and velocity of a target. In one dimension we use the position (ξ) and thevelocity (ξ) as states in the state vector, thus x(t) = (ξ(t) ξ(t))′. We can use thefollowing continuous-time model

x(t) =

(ξ

ξ

)

=

(0 10 0

)

x(t) +

(01

)

v(t), (4.6)

where we have assumed a perturbation around a constant velocity as a model. Ifwe assume that the noise (acceleration) is constant during the sample period (T ),we can easily calculate a discrete-time model as

xt+1 =

(1 T0 1

)

xt +

(T 2/2T

)

vt. (4.7)

4.2.2 The Singer Acceleration Model

In Singer (1970) a filter based on Kalman techniques is developed for a maneu-vering target. The main idea is to represent the ensemble behavior of differentmaneuvering vehicles. The target maneuver, in this case the acceleration ξ(t) isassumed to be correlated in time. This is due to the assumption that if a target isaccelerating at time t, it is assumed more likely that it will continue acceleratingat time t+ τ , where τ is a small time increment. A model of this is represented bythe auto correlation function

Eξ(t)ξ(t+ τ) = σ2e−α|τ |, (4.8)

4.2 Tracking Models and Coordinate Systems 59

where σ2 is the variance of the target acceleration and α is the reciprocal time con-stant. By performing a spectral decomposition it can be shown that the dynamicsfor the acceleration is given by

d

dtξ(t) = −αξ(t) + v(t), (4.9)

where v(t) is the driving white noise with variance 2ασ2δ(τ). The model is referredto as a first order Gauss-Markov process.

Following the ideas from Singer (1970) and (Blackman and Popoli, 1999, p. 200),we assume that the states are position, velocity and acceleration, summarized inthe state space model

x(t) =

0 1 00 0 10 0 −α

︸︷︷︸

A

x(t) +

001

v(t). (4.10)

Discretizing the system matrix assuming a constant sample time T yields,

Φ = eAT =

1 T 1α2 (e−αT − 1 + αT )

0 1 1α (1 − e−αT )

0 0 e−αT

, (4.11)

Often the sample period is less than the target maneuver time (1/α), therefore thefollowing approximation is often used

Φ ≈

1 T T 2

2

0 1 T (1 − αT2 )

0 0 e−αT

. (4.12)

The limit case is of particular interest, namely

limαT→0

Φ =

1 T T 2

20 1 T0 0 1

, (4.13)

which is the Newtonian matrix for the constant acceleration model. It is possible towrite down analytical expressions for the process noise covariance matrix, (Singer,1970; Blackman and Popoli, 1999).

4.2.3 The Coordinated Turn Model

In Maybeck et al. (1982) a constant-turn model for acceleration is proposed forairborne vehicle motion. Unlike the tracking system presented in Section 4.2.2,using the Gauss-Markov acceleration model, we will here have a non-linear system.This model is known as coordinated turn, and has been discussed in Bar-Shalom



(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

ν

ξ

η

r

ψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 4.1: The definition of heading angle ψ.

and Li (1993); Blackman and Popoli (1999); Gustafsson (2000); Maybeck et al.(1982).

Let xt =(

ξ η ξ η)′

, where ξ and η are the Cartesian position coordinates,

and where the velocity components are denoted ξ and η. If we assume a constant

speed ν =

√

ξ2 + η2 and a constant turn rate Ω = ψ, where (see Figure 4.1) wehave

ξ = ν cos(ψ), η = ν sin(ψ). (4.14)

The acceleration with ν = 0 components are

ξ =d

dtξ = −νΩsin(ψ) = −Ωη (4.15)

η =d

dtη = −νΩcos(ψ) = Ωξ. (4.16)

The state equation can be expressed as

x(t) =d

dt

ξ

ξηη

=

ξ−Ωηη

Ωξ

.

Discretizing the system gives

xt+1 =

1 sin(ΩT )Ω 0 − (1−cos(ΩT ))

Ω0 cos(ΩT ) 0 − sin(ΩT )

0 (1−cos(ΩT ))Ω 1 sin(ΩT )

Ω0 sin(ΩT ) 0 cos(ΩT )

xt +Bvt, (4.17)


4.2.4 The Nearly Coordinated Turn Model

In Blackman and Popoli (1999) an extension to the classical coordinated turn modelpreviously described in Section 4.2.3 is presented, where the turn rate Ω is assumedunknown and part of the estimation process. This model is called the horizontalturn model with velocity states, where in Bar-Shalom and Li (1993) it is referred toas the nearly coordinated turn model. For a two dimensional system it is given as

Xt+1 =

1 sin(ΩT )Ω 0 − (1−cos(ΩT ))

Ω 0 00 cos(ΩT ) 0 − sin(ΩT ) 0 0


Ω 0 00 sin(ΩT ) 0 cos(ΩT ) 0 00 0 0 0 1 00 0 0 0 0 1

Xt +Bvt, (4.18)

where the state vector Xt consists of the Cartesian position and velocity xt =(

ξ η ξ η)′

and the unknown turn rate Ωt, hence Xt = (xt Ωt)′.

4.2.5 Modified Spherical Coordinates

Different systems and techniques are used in tracking applications. Here we con-sider a system defined in relative coordinates, that is the difference between thetarget and the own-ship, when a tracking application is considered. For angle-onlyapplications, it can be shown that the choice of coordinate system is important.Here we discuss one important system, namely the modified spherical coordinate(MSC) system. The MSC is described in Simard and Begin (1993). The followingnotations for the state vector in Cartesian coordinates (x) and MSC (z) are used

x =(

ξ η ζ ξ η ζ)′

(4.19)

z = g(x) =(

1/r ϕ θ r/r Ω θ)′

(4.20)

Ω = ϕ cos θ, (4.21)

where r is the relative range, ϕ the azimuth angle and θ is the elevation angle(from the horizontal plane to the target) according to Figure 4.2. Conversion fromCartesian coordinates to MSC is defined by the transformation z = g(x), with


platform

Tracked object (target)

Tracking


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

N

ξ

η

ζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

r

ϕθ

Tracked object (target)

Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 4.2: The MSC system definition.

components

z1 =1

r=

1√

ξ2 + η2 + ζ2(4.22a)

z2 = ϕ = arctan(η/ξ) (4.22b)

z3 = θ = arctan(−ζ

√

ξ2 + η2) (4.22c)

z4 =r

r=ξξ + ηη + ζζ

ξ2 + η2 + ζ2(4.22d)

z5 = Ω =ηξ − ξη

(ξ2 + η2)cos(arctan(−ζ/

√

ξ2 + η2)) (4.22e)

z6 = θ =−ζ(ξ2 + η2) + ζ(ξξ + ηη)

(ξ2 + η2 + ζ2)√

ξ2 + η2. (4.22f)

To allow tracking not just in the region defined in Figure 4.2 the trigonometricfunctions must be extended to all quadrants.

The continuous-time state equation in MSC with the own acceleration signal inantenna coordinates uA =

(uA

1 uA2 uA

3

)′is given by

z(t) = f(z, uA) + vmsc(t), (4.23)

where vmsc(t) is the process noise. The components in (4.23) are given by (4.24).For derivations of the state equation we refer to Robinson and Yin (1994); Stal-lard (1987). Basically, the MSC state equation is given by differentiation of the


Cartesian system and transformation to MSC. The extension to a nine-state filteris discussed in Robinson and Yin (1994).

z1 =d

dt

1

r= −(

1

r)r

r= −z1z4 (4.24a)

z2 =d

dtϕ =

Ω

cos θ=

z5

cos z3(4.24b)

z3 =d

dtθ = θ = z6 (4.24c)

z4 =d

dt

r

r= θ2 + Ω2 − (

r

r)2 − uA

1

r

= z26 + z2

5 − z24 − z1u

A1 (4.24d)

z5 =d

dtΩ = (−2(

r

r) + θ tan θ)Ω − uA

2

r

= (−2z4 + z6 tan z3)z5 − z1uA2 (4.24e)

z6 =d

dtθ = −2(

r

r)θ − Ω2 tan θ +

uA3

r

= −2z4z6 − z25 tan z3 + z1u

A3 . (4.24f)

The target accelerations are here included in the process noise term, vmsc(t), sincethey are unknown. The system is summarized as z = f(z, uA) + vmsc(t). Thecontrol signal (own acceleration) is converted from inertial to antenna coordinatesusing

uA = CAI u =

cos θ cosϕ cos θ sinϕ − sin θ− sinϕ cosϕ 0

sin θ cosϕ sin θ sinϕ cos θ

u. (4.25)

In Blackman and Popoli (1999), the time-to-go (TTG) estimation for an IRsensor target tracking application is discussed. If we consider a scenario where atarget occupy at most one pixel element in the IR sensor a simple model for thesignal-to-noise-ratio (SNR) is

ySNR = αrβ , (4.26)

where α is a proportional constant. The power exponent β varies according tothe atmosphere, and a typical value is about β ≈ −2.3 (see Blackman and Popoli(1999)). Since we can express the remaining time by using the relative rangeand range rate to the target we defined tTTG = −r/r. By performing a simpledifferentiation we get

tTTG = −rr

= − r

αβrβ−1ySNR= −βy

SNR

ySNR. (4.27)

If we introduce ySNR and its derivative as state variables, then tTTG can be esti-mated. Note that for the MSC system presented in this section, we simply get thetTTG by inverting the system state −z4.


4.2.6 Process Noise Models

In this section, we discuss different ideas on how to define the process noise forthe discrete EKF Q, starting with a continuous model with process noise Q. Weconsider the following continuous and discrete-time tracking models

x(t) = f(x(t)) + v(t), Covv(t) = Q,

xt+T = f(xt) + vt, Covvt = Q.

In (Gustafsson, 2000, p. 322), the discrete state noise covariance is calculated fromthe continuous, for different cases, corresponding to different model assumptions.In the formulas below we use the notation fx = ∇xf(x)|x=x for the functionalmatrix and where f ′

x denotes the corresponding transpose. Basically, five differentalternatives are presented using the following

Qa =

∫ T

0

efxτ Qef ′xτ dτ (4.28)

Qb =1

T

∫ T

0

efxτ dτQ

∫ T

0

ef ′xτ dτ (4.29)

Qc =TefxT Qef ′xT (4.30)

Qd =TQ (4.31)

Qe =TfxQf′x. (4.32)

All expressions are normalized with the sampling time T , so that one and the sameQ can be used for all of the sampling intervals. These methods correspond to moreor less ad hoc assumptions on the process noise for modeling the underlying targetmaneuvers.

a. v(t) is continuous white noise with variance Q.

b. v(t) = vk is a stochastic variable which is constant in each sample interval withvariance Q/T . That is, each maneuver is distributed over the whole sampleinterval.

c. v(t) is a sequence of delta-Dirac pulses, active immediately after a sample istaken. We assume x = f(x) +

∑

k vkδkT−t where vk is discrete white noise

with variance TQ.

d. v(t) is white noise whose total influence during one sample interval is T Q.

e. v(t) is a discrete white noise sequence with variance T Q. We assume that allmaneuvers occur immediately after a sample time, xt+1 = f(xt + vt).

Note that the first two approaches require a linear time invariant model for thestate noise propagation to be exact.

4.3 Track Initiation and Termination 65

4.3 Track Initiation and Termination

The initiation of tracks from measurement data is in many tracking applications adifficult and troublesome task. Many methods could be used, both simple and morecomplex. For the multiple tracking scenario, some data association methods willincorporate track initiation in a natural way whereas other association methodsrelies on some logic. Since the measurement dimension is often lower then thestate dimension, one problem is to initialize the states that could not be computedfrom measurements. Both Bar-Shalom and Fortmann (1988) and Blackman (1986);Blackman and Popoli (1999) discuss several different techniques. Here we describefive initiation methods and briefly discuss how tracks are terminated.

1. Information filter based initiation.

2. The 2/2&m/n cascade logic track formation method.

3. Logic based multi-target track initiator.

4. Sequential probability ratio test confirmation.

5. Bayesian track confirmation.

Information filter based initiation. A state estimate can be given if we allowthe track initiation to be based on consecutive measurements. Consider the follow-ing one-dimensional measurement of position, assuming a Gaussian measurementnoise, where the measurements are given with a sample period T .

yt = ξt + et, et ∼ N(0, R), t = 0, 1, . . . (4.33)

An estimate of the covariance matrix P1|1 can now be calculated using the informa-

tion filter from Section 2.3.3 with no prior knowledge of the covariance, i.e., P−10|−1

is the null matrix (P0|−1 = ∞I). We have

A =

(1 T0 1

)

, B =

(T 2/2T

)

, C =(1 0

)

P−10|−1 =

(0 00 0

)

.

By applying the information filter we have

P−11|1 =

(R R/T

R/T T 2QR + 2R

T 2

)

.

For the general case, the information update is iterated until the inverse covariancegets invertible. Then the traditional Kalman filter can be used.


A special case of this method is given in (Bar-Shalom and Li, 1993, p. 253).The corresponding covariance matrix is given by

P1|1 =

(R R/TR/T 2R/T 2

)

. (4.34)

This method is referred to as the two-point differencing, which guarantees consis-tency of the initialization of the filter, (Bar-Shalom and Fortmann, 1988, p. 82),and corresponds to zero process noise (Q = 0) in the information filter initializationmethod.

The 2/2&m/n cascade logic track formation method. Another track initi-ation method described in Bar-Shalom and Li (1995), under the name a logic basedmulti-target track initiator, uses several measurements in a logic based structure toinitialize the state vector. The 2/2&m/n-method is a cascade based logic for trackformation. In Figure 4.3, this approach is shown for the case m = 2, n = 3, wherewe used the notation D if a measurement is detected and we use it to update orstart a track somewhere in the cascade logic chain. If not detected or if we arenot able to use a measurement we denote it by D. If we are coming back to stateS1, the track was rejected, whereas state S8 represents an accepted track. Thepreliminary track or tentative track is formed in the states S3 − S7. The method


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DD

D

DD D

D

D

D

D

D

D

D

DS1 S2 S3

S4

S5

S6

S7

S8

f(X )

Figure 4.3: The 2/2&m/n track initiation method (m = 2, n = 3).

requires two consecutive measurements, that could be associated to produce aninitial estimate. This is propagated through a Kalman filter, and we require thatof the next three measurements, at least two must be so close that they may beassociated. The method have several different phases, the first where we initiatea track from measurements, the second phase uses a tentative track, which in thefinal step is accepted if all requirements are full-filled.

4.3 Track Initiation and Termination 67

Logic based multi-target track initiator. Yet another method is described in(Bar-Shalom and Fortmann, 1988, p. 252), where a logic-based multi target trackinitiator is proposed. Tracks are started by associating measurements from the firstscan to each candidate from the second scan using an acceptance region (validationregion). The technique is as follows. Assume that we have Mt measurements at

time t (or scan frame) and let y(i),lt be component l of measurement i, where the

measurement vector is assumed yt ∈ Rm. We consider here only the first two

scans, so t = 0, 1. Assuming a priori knowledge of the velocity, we can calculatethe association distance as

dlij = maxy(j),l

1 − y(i),l0 − vl

maxT, 0 + max−y(j),l1 + y

(i),l0 + vl

minT, 0, (4.35)

where T is the time interval between the two initial scans and l = 1, 2, . . . ,m. It ispossible to use different regions in the initiation, for instance conical regions. The

test statistic for the data association is given as Dij = d′ij(R(i)0 +R

(j)1 )−1dij , where

the measurement errors are assumed independent with zero mean and covariances

R(i)1 , R

(j)2 respectively. The measurement is associated if Dij ≤ γ, where γ is a

probability related threshold.

Sequential probability ratio test confirmation. In Blackman (1986), a trackinitiation technique based on sequential analysis of likelihood ratios is discussed.One of the simplest methods is the sequential probability ratio test (SPRT), wherethe following hypotheses are used:

• H(0) = no true target present (observation due to false alarm).

• H(1) = a true target present.

If we assume a scan based system, denoting the number of scans by k and thenumber of detections by m, the likelihood for each hypotheses can be calculated as

l(0) = PmFA(1 − PFA)k−m

l(1) = PmD (1 − PD)k−m,

where the probability of detection and the probability for false alarm are denotedPD and PFA respectively. The SPRT forms the likelihood ratios of the hypothesesand compares this with predefined threshold levels C1 and C2 as:

1. Accept H(0) if, l(1)

l(0)≤ C1.

2. Accept H(1) if, l(1)

l(0)≥ C2.

3. Continue testing if C1 <l(1)

l(0)< C2.


Bayesian track confirmation. In Blackman (1986), a track confirmation methodusing Bayes’ rule is described. Using Bayes’ rule the probability for a true targetgiven the observations Yt can be calculated as

PrHTRUE|Yt =PrYt|HTRUEPrHTRUE

PrYt,

where the probability for the measurement can be expressed as

PrYt = PrYt|HTRUEPrHTRUE + PrYt|HFALSEPrHFALSE.

By using PrHFALSE = 1 − PrHTRUE and denoting the likelihood ratios asL(Yt) = PrYt|HTRUE/PrYt|HFALSE we have

PrHTRUE|Yt =L(Yt)PrHTRUE

L(Yt)PrHTRUE + (1 − PrHTRUE) .

Using Bayes’ rule recursively yields

PrHTRUE|Yt =Lt(Yt)PrHTRUE|Yt−1

Lt(Yt)PrHTRUE|Yt−1 + (1 − PrHTRUE|Yt−1).

For track deletion similar ideas as the SPRT and the Bayesian confirmationmethod can be used. For details we refer to Blackman (1986).

4.4 Sensor Fusion

Sensor fusion, i.e., combining data from several sensors to improve the estimate,can easily be incorporated in the Bayesian approach. Often several sensors of thesame type are used, but different types of sensors could be used. The sensors canbe placed on the same platform or on different platforms. The time update isperformed as before, predicting the density to the next time where we have a newobservation. The measurement update is performed using the model for the sensorresponsible for the observation. In practice some problems must be handled, suchas sensor bias errors. In many applications these can be reduced by calibration orby an off-line bias estimation. In other applications the bias must be recursivelyestimated on-line. Other practical problems may occur for the multiple target case,where correct association is essential for data fusion to be effective. How to handlebias estimation, offsets and trends, colored noise etc, we refer to (Gustafsson, 2000).

Within the data fusion world, effort has been done defining different levels offusion. The Joint Directors of Laboratories (JDL) group has defined and reviseddata fusion models in an attempt to use a common notation. This is described inWaltz and Llinas (1990).

4.5 Data Association 69

4.4.1 Measurement to Track Fusion

One simple data fusion problem that can be put in the Bayesian framework consistsof multiple-sensor measurements and a central tracker. If we assume independentobservations, the tracker consists of sequential updates using the different measure-ment relations. Consider the case with M different sensors, we then have

yt = h(i)(xt, et), i = 1, . . . ,M. (4.36)

The Bayesian estimate is given by solving the time- and measurement updateequations (2.12)–(2.13), using (4.36) sequentially for each received observation. Inmany practical sensor data fusion problems, the measurement relations describedin (4.36) consist of measurements reflecting different parts of the state vector. Forinstance, the radar sensor measures range and angles, so a good measurement willproduce an accurate position, whereas the IR sensor does not measure range.

4.4.2 Track to Track Fusion

Suppose we want to perform sensor data fusion of two independent state estimatesx1 and x2, with covariance matrices P 1 and P 2 respectively. If we assume a linear-Gaussian model, sufficient statistics are mean and covariance, so all informationis available. The minimum variance estimator is given by the fusion formula (seeGustafsson (2000))

x = P((P 1)−1x1 + (P 2)−1x2

)(4.37)

P =((P 1)−1 + (P 2)−1

)−1. (4.38)

It is not difficult to generalize this to the general multi-sensor case, (Gustafsson,2000).

4.5 Data Association

In this section we describe the association problem, that is how we combine mea-surements and state trajectories (tracks). The term clutter is used for returns fromspurious objects or interference, producing false alarms. The observation-to-trackcorrelation or data association is a problem of great importance for multiple targettracking applications. Several methods have been and discussed in estimation andtracking literature, (Bar-Shalom and Fortmann, 1988; Bar-Shalom and Li, 1993;Blackman, 1986; Blackman and Popoli, 1999). In general multi target trackingdeals with state estimation of an unknown number of targets. Some methods arespecial cases which assume that the number of targets is constant or known. Theobservations are considered to originate from targets if detected or from clutter.The clutter is a special model for false alarms, whose statistical properties are dif-ferent from the targets. In many applications only one measurement is assumedfrom each target object, where in other applications several returns are available.This will of course reflect which data association method to use.


Several classical data association methods exist. The simplest is probably thenearest neighbor (NN), which uses only the closest observation to any given stateto perform the measurement update step. The method can be given as a globaloptimization, so the total observation to track statistical distance is minimized.Another multi target tracking association method is the joint probability data as-sociation (JPDA) which is an extension of the probability data association (PDA)algorithm to multi targets. It estimates the states by a sum over all the asso-ciation hypothesis weighted by the probabilities from the likelihood. The mostgeneral method is a time-consuming algorithm called the multi hypothesis tracking(MHT), which calculates every possible update hypothesis. In Reid (1979), severalalgorithms for multiple target tracking are listed and categorized according to theunderlying assumptions. A reference list to the different methods is also given.

Most of these methods rely upon that the mean and covariance is sufficientstatistics for the problem. For linear and Gaussian problems the Kalman filter isthe optimal estimator yielding sufficient information. For non-linear problems theEKF is often used as an approximation. To be able to fully use nonlinear and non-Gaussian estimation methods combined with data association to solve the jointdata association and estimation problem there is a need to develop other methods.In Salmond et al. (1998), this idea is suggested for the particle filter and the problemof maintaining a track on a target in the presence of intermittent spurious objects.In Gordon (1997), a multiple target and multiple sensor estimation and associationproblem is considered using a particle filter based method. In Hue et al. (2000,2001), a method for joint data association and estimation is proposed using aparticle filter and a Gibbs sampler. When a particle filter method is used, eachtrack is considered as samples from a probability density. By estimating covarianceand mean value, traditional methods can be applied for data association. However,it could also be implemented as a part of the Monte Carlo algorithm for multi targetestimation and data association. A Monte Carlo JPDA method was proposed inKarlsson and Gustafsson (2001a), where a comparison with linearized methods wasmade. In Bergman and Doucet (2000), a Markov Chain Monte Carlo (MCMC)technique is used for data association of multiple measurements in an over thehorizon radar application.

4.5.1 Measurement Validation and Gating

In practice, we need a method to select if and which measurement that should beassociated to an existing track when several candidates are available. If we assumethat we already have one or more tracks and we receive new measurements, wecan assume that to be able to associate measurements to tracks they must bein the vicinity of the predicted track. This technique is sometimes called thatthe measurement is within a validation region, or within a gate. Therefore thereduction of possible association candidates is often referred to as gating.

Assume that we have a predicted value of a track xt+1|t and that we can formthe predicted observation, i.e., yt+1|t = h(xt+1|t). This measurement is assumedto have a covariance matrix (innovation covariance) denoted St+1. We make the


assumption that the observation at time t conditioned upon the cumulative mea-surements Yt is Gaussian, that is

p(yt+1|Yt) = N(yt+1; yt+1|t, St+1). (4.39)

In the measurement space, we can define a region within which the observationwould belong with high probability

Vt+1(γ) = yt+1 : εt+1S−1t+1εt+1 ≤ γ, (4.40)

where the innovations are defined as εt+1 = yt+1 − yt+1|t. Under the Gaussianassumption, this expression is χ2-distributed, so the threshold γ can be determinedif we assume that we want some percentage of the measurements to belong tothe region. In Blackman and Popoli (1999), this gating method is referred to asellipsoidal gates, since it could be thought of as an ellipsoid. Another methodwhich is discussed is rectangular gates. In Hall (1992), several other methods arediscussed by introducing different distance measures. For instance, the Euclideanand weighted Euclidean distance, the absolute value or a Minowski norm are used.The χ2-method discussed above is there referred to as the Mahalanobis distance.

The probability that the measurement is within the validation region or gate isdefined as

PG = Pryt+1 ∈ Vt+1(γ). (4.41)

In some applications the region is chosen so PG ≈ 1, and therefore the formulaswhen calculating different hypothesis can be simplified. Note that in many appli-cations observations might not be detected at all. The probability of detection isdefined as

PD = PrThe true measurement is detected. (4.42)

For the multi-target problem, the same technique with validation regions is usedaround each estimated target in the measurement space. The regions are then usedto reduce the number of possible observation-to-track associations. The basic ideais described in Example 4.1.

Example 4.1 Validation regions for multiple targets.For the multi-target application described in Figure 4.4, we have two targets

and several measurements, which are due to clutter or originate from a target.The validation or gating regions are draw around each estimated measurementy = h(x), where x is the predicted state. We see that the measurement y(1) shouldbe associated to track 1, and that y(3) and y(4) should be associated to track2. However, measurement y(2) is within both validation regions. How to solve theconflict, of having several measurements within a gating region, is due to the chosenassociation method. The gating technique is just a way to reduce the number ofpossible combinations.



(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

y

y(3)

y(4)

y(1) y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 4.4: Two tracks in the measurement space (y(i), i = 1, 2) and their valida-tion regions together with measurements (y(j), j = 1, . . . , 4).

4.5.2 Nearest Neighbor Association

One of the simplest data association methods is based on the idea that we asso-ciate the closest measurement and track. This is called the nearest neighbor (NN)method. In Bar-Shalom and Fortmann (1988), this is referred to as the nearest-neighbor standard filter (NNSF). As discussed in Section 4.5.1, several differentnorms or metrics could be used. Here we discuss the following weighted norm.Consider

d2(yt+1) = ε′t+1S−1t+1εt+1, (4.43)

where the residual εt is as defined in Section 4.5.1 and St denotes its covariance.From all measurements within the validation or gating region we chose the onethat maximizes the probability, in other words the combination that minimizes thedistance between observation and track according to the chosen norm. This couldbe done locally, but more common is probably to minimize the overall distancebetween all possible allowed combinations. This is referred to as the all-neighborapproach. In Blackman and Popoli (1999), the term global nearest neighbor (GNN)is used. They also introduce the terms unique-neighbor and all-neighbors. Othersimilar NN approaches exist, for instance in Li (1998) a strongest neighbor filter(SNF) is proposed.

Using the NN method in data association may lead to poor results if spuriousmeasurements occur. This is because the association technique does not accountfor the fact that measurements may originate from other sources than the target.


4.5.3 Probabilistic Data Association

The probabilistic data association filter (PDAF) (Bar-Shalom and Fortmann (1988),p. 163) is a suboptimal Bayesian filter used for the single target application, whenreceiving several measurements, one belonging to the true object and the rest dueto clutter. The sub-optimality is due to the fact that only the latest measurementsare considered. Within the PDAF framework, the probability for detection, theprobability for false alarms or new targets and the association probability are han-dled within the Bayesian framework. We describe the PDA method as given inBar-Shalom and Fortmann (1988), but using notations consistent with the rest ofthe thesis.

The optimal Bayesian approach is another method similar to PDAF. However,the state estimation is done in terms of all combinations of measurements frominitial to current time, whereas the PDAF only uses the latest measurements.

Assume that we have Mt measurements at time t (or scan), denoted by y(i)t , i =

1, . . . ,Mt. We denote this set of measurements at time t by Yt and the set of allcumulative measurement sets up to time t, Yt, i.e.,

Yt = y(i)t Mt

i=1 (4.44)

Yt = Yt, Yt−1, . . .. (4.45)

In the PDAF the estimation is based on the latest set of measurements. The pastis summarized approximately by the following Gaussian assumption

p(xt|Yt−1) = N(xt; xt|t−1, Pt|t−1). (4.46)

Consider the following events

H(0)t = None of the measurement at time t originated from the target (4.47)

H(i)t = y(i)

t is the target-originated measurement, i = 1, . . . ,Mt, (4.48)

with probabilities p(i)t = PrH(i)

t |Yt, i = 0, 1, . . . ,Mt. We assume mutually exclu-

sive and exhaustive events, hence the probabilities p(i)t sum to unity,

∑Mt

i=0 p(i)t = 1.

The estimate is calculated using the total probability theorem

xt|t = Ext|Yt =

Mt∑

i=0

Ext|H(i)t ,YtPrH(i)

t |Yt︸︷︷︸

p(i)t

=

Mt∑

i=0

Ext|H(i)t ,Ytp(i)

t =

Mt∑

i=0

x(i)t|tp

(i)t , (4.49)

where x(i)t|t is the updated state conditioned on the event H(i)

t that y(i)t is the correct

measurement. For the case of no correct measurements, the estimate is a pure


prediction x(0)t|t = xt|t−1. From the Kalman filter we have for i = 1, . . . ,Mt

x(i)t|t = xt|t−1 +Ktε

(i)t (4.50)

ε(i)t = y

(i)t − h(x

(i)t|t−1), (4.51)

where the Kalman gain Kt is the usual one, since by conditioning on H(i)t , there is

no measurement origin uncertainty. Combining the above equations we get

xt|t = xt|t−1 +Ktεt (4.52)

εt =

Mt∑

i=0

p(i)t ε

(i)t . (4.53)

The error covariance matrix is updated as

Pt|t = p(0)t Pt|t−1 + (1 − p

(0)t )P c

t|t + Pt|t, (4.54)

where P ct|t is given by the ordinary measurement update as

P ct|t = [I −KtHt]Pt|t−1 (4.55)

and where Pt|t is given by

Pt|t = Kt[

Mt∑

i=1

p(i)t ε

(i)t (ε

(i)t )′ − εtε

′t]K

′t. (4.56)

For a derivation of the covariance update formula we refer to Bar-Shalom andFortmann (1988).

Below we briefly discuss how the association probabilities p(i)t can be calculated.

By explicitly pointing out that there are Mt measurements in the last measurementset, we get the probabilities

p(i)t = PrH(i)

t |Yt = PrH(i)t |Yt,Mt,Yt−1, i = 0, 1, . . . ,Mt. (4.57)

Applying Bayes’ rule yields

p(i)t ∝ p(Yt|H(i)

t ,Mt,Yt−1)PrH(i)t ,Mt|Yt−1 (4.58)

The probability density for correct measurement is given by

p(y(i)t |H(i)

t ,Mt,Yt) = P−1G N(y

(i)t ;h(xt|t−1), St) = P−1

G N(ε(i)t ; 0, St). (4.59)

Hence, the first factor in (4.58) is

p(Yt|H(i)t ,Mt,Yt−1) =

V 1−Mt

t P−1G N(ε

(i)t ; 0, St), i = 1, . . . ,Mt

V −Mt

t , i = 0,(4.60)


where Vt is the volume of the validation region. The second factor in in (4.58) iscalculated as

PrH(i)t |Mt,Yt−1 = PrH(i)

t ,Mt

=

1

MtPDPG[PDPG + (1 − (PDPG) µF (Mt)

µF (Mt−1) )−1], i = 1, . . . ,Mt

(1 − PDPG) µF (Mt)µF (Mt−1) [PDPG + (1 − (PDPG) µF (Mt)

µF (Mt−1) )−1], i = 0,

(4.61)

where µF (Mt) is the probability mass function for the number of false alarms. Thegate probability (PG) is the probability that the correct measurement is within thevalidation gate and Pd is the detection probability. In Bar-Shalom and Fortmann(1988), two different assumptions are considered for µF (Mt) namely a parametricmodel with a Poisson density with parameter λVt or a non-parametric model usinga diffuse prior.

1. Poisson density. µF (Mt) ∝ e−λVt(λVt)

Mt

Mt! , Mt = 0, 1, . . .

2. Diffuse prior. µF (Mt) ∝ 1M, Mt = 0, 1, . . . ,M − 1, where M is as large as

needed.

4.5.4 Joint Probabilistic Data Association

In Blackman and Popoli (1999), the Joint Probabilistic Data Association (JPDA)is simply defined as being identical to the PDA algorithm, but for the multi-targetcase. The association probabilities are calculated in a similar way, but all hypoth-esis are considered. The state estimation is performed as for the PDA algorithm,only the probability computations (weights) are modified. We describe the JPDAalgorithm via Example 4.2.

Example 4.2 JPDA filter for two targets.Consider the following example from (Blackman and Popoli, 1999, p. 353) pre-

sented in Figure 4.5. We consider two tracks with validation regions, and due toclutter we receive one extra measurement. Due to the validation regions all mea-surements can be associated to track x(1), but only y(2) and y(2) can be associatedto x(2). In Table 4.1 the different hypothesis are calculated according to the JPDAassumption, i.e., all permutations of the measurements are considered when assign-ing the tracks with at most one measurement/track. The notation gij is used forthe Gaussian likelihood (track i, observation j), where

εij = y(j) − h(x(i))

gij =1

(2π)m/2√

|S(i)|e−

12 ε′ij(S

(i))−1εij ,

where i = 1, 2 and y(j) ∈ Rm, j = 1, 2, 3. Note that if gating regions are used to

reduce the complexity, the probability for gating PG should be incorporated in the



(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)x(1)x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 4.5: Validation regions, tracks and measurements for the scenario using theJPDA algorithm.

Hyp. no Track 1 Track 2 False Likelihood1 2 Alarm

1 - - 1,2,3 (1 − PD)2p3FA

2 1 - 2,3 g11PD(1 − PD)p2FA

3 2 - 1,3 g12PD(1 − PD)p2FA

4 3 - 1,2 g13PD(1 − PD)p2FA

5 - 2 1,3 g22PD(1 − PD)p2FA

6 1 2 3 g11g22P2DpFA

7 3 2 1 g13g22P2DpFA

8 - 3 1,2 g23PD(1 − PD)p2FA

9 1 3 2 g11g23P2DpFA

10 2 3 1 g12g23P2DpFA

Table 4.1: The JPDA hypothesis matrix for two tracks and three observations.

equations. Often a value close to unity is used, therefore a good approximation isgiven by the equations already stated above. Also note that a measurement is onlyallowed to update one track within each hypotheses.

4.5.5 Multiple Hypothesis Tracking

One of the most general data association methods is the Multiple Hypothesis Track-ing (MHT) technique. By forming all hypothesis and calculate their probabilities,this computer intensive method can cover all events of interest. Particularly, if theprobability for a measurement-to-track conflict is large, that is if several measure-ments could be associated to the same track, then by calculating all combinationsand assign them appropriate probabilities, the idea is to wait for more observa-tions before a decision is made. In Reid (1979), the MHT algorithm and how toimplement it is described in detail. The implementation is referred to as Reid’salgorithm, which is in principle a hypothesis matrix with pointers or indices de-


scribing how measurements are associated. A simple example of Reid’s hypothesismatrix is given in Example 4.3.

Example 4.3 Hypotheis tree.In Figure 4.6, a hypotheses tree is presented, when an observation based MHT

algorithm is used. The observations are denoted yti(k), i = 1, . . . ,M(k), where

M(k) is the number of measurements in scan k. We assume one existing trackat the beginning of scan k, during which two measurements are received, yt1(k)and yt2(k). These are registered at different times, t1 and t2. The hypothesesconsidered in the example is that the observations could be due to a false alarm(FA), be used as a track continuation, i.e., updating an existing track or be from anew target (NT).


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1) yt2(1) yt1(2)

Scan 1 Scan 2Scan time

FA

FA

FA

TC

TC

TC

NT

NT

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 4.6: The MHT hypothesis tree for a scan based system.

The MHT is based on Bayes’ rule

PrH(l)|Y =PrY|H(l))PrH(l)

PrY , (4.62)

where PrH(l) is the a priori probability that hypothesis H(l) is true. Supposethat the false alarm density is denoted βFA, and that the new target density isβNT . We assume a detection probability PD. So for each new measurement scanor frame, if the prior is substituted by the calculated posterior density, then arecursive version is obtained. We denote the new prior hypoteses by H(l′). InBlackman (1986), the hypotheses probabilities are calculated as


1. False alarm (FA)

PrH(l) =1

cβFA(1 − PD)

Nl′PrH(l′). (4.63)

2. New target (NT)

PrH(l) =1

cβNT (1 − PD)

Nl′PrH(l′). (4.64)

3. Track continuation (TC)

PrH(l) =1

c(1 − PD)

Nl′−1PDgijPrH(l′) (4.65)

εij = y(j) − h(i)(x) (4.66)

d2ij = ε′ijS

−1i εij (4.67)

gij =e−d2

ij/2

(2π)m/2√

|S(i)|. (4.68)

Here c is a normalization constant and the measurement yt ∈ Rm. The innovation

covariance S(i) associated to track i is given from the Kalman filter.The main problem with the MHT method is the exponential growth of hy-

pothesis. Therefore, a lot of work has been done to make the algorithm work inpractice. In many MHT application there is an additional functionality that mergesimilar tracks, in order to decrease the computational burden and remove similar-ities. The typical approach is to merge tracks when they share a common history,often defined as having a fixed number of common observations. In Bar-Shalomand Fortmann (1988), several hypothesis reduction techniques are described. Themost drastic is to use the JPDA technique, combining all the measurements at thecurrent time, considering only the number of known targets with a single hypoth-esis per target. Another method called the N-scan-back method, which combinesall track histories with common observations from k − N to k. This method isdescribed thoroughly in Blackman and Popoli (1999), where it is called the N-scanpruning method. In Bar-Shalom and Fortmann (1988), a reduction method basedon Gaussian mixtures is proposed.

4.5.6 Association using Monte Carlo Techniques

Classical estimation and association methods often rely on the mean and covarianceas sufficient statistics for the problem. The data association and estimation areseparated and do not enter the filter in a natural way. To be able to fully usenonlinear and non-Gaussian estimation methods combined with data associationto solve the joint data association and estimation problem there is a need to developother methods. Even if the multi target application is important it has not beenstudied so much when simulation based methods are used.


In Avitzour (1995), the solution to the assignment problem for data associationis proposed to be within the Bayesian framework by simply incorporate it in theestimation equations. In Salmond et al. (1998), this idea is suggested for theparticle filter, when the problem of maintaining a track on a target in the presenceof intermittent spurious objects. In Gordon (1997), a multiple target and multiplesensor estimation and association problem is solved using the Bayesian bootstrapfilter. Samples are drawn from the overall target probability density. A specialfilter called hybrid bootstrap filter is constructed. In Hue et al. (2000, 2001), themulti target tracking problem is analyzed when recursive Monte Carlo methodsare used. The attempt is to perform a joint data association and estimation. Thesolution to the joint data association and estimation problem relies on the particlefilter method for approximating the posterior density. The particle filter is used inthe estimation step, and the data association is solved using a Gibbs sampler.

In Section 5.3, a joint estimation and association technique based on a particlefilter based JPDA method from Karlsson and Gustafsson (2001a) is presented.

5

Particle Filter Applications

In this chapter several tracking and data association applications are described.Classical estimation and association methods are compared to simulation basedmethods utilizing different particle filters.

In Section 5.1, an air traffic control application is presented for a highly ma-neuvering target, where the auxiliary particle filter from Section 3.3.2 is extendedto handle the maneuvering modes. Also the concept of hard constraints on stateparameters are introduced. The particle filter approach is compared to a multi-ple model method using the IMM filter. In Section 5.2 several passive rangingapplications are discussed. Section 5.2.1 presents a passive ranging air-to-air ap-plication where the aircraft maneuvers to gain range observability since only angleinformation is measured. The particle filter method is compared to a bank of ex-tended Kalman filters, where each filter is tuned to a specific range interval. Theperformance in different coordinate systems are discussed. In Section 5.2.2, thepassive ranging application is investigated further where the focus is on terrain in-duced constraints. In Section 5.2.3, the particle filter for passive ranging is testedon experimental sonar data from a torpedo system. In Section 5.3 a multi targettracking application is presented. Therefore, the assignment or data associationproblem must be handled. Many of the classical data association methods rely onKalman filter estimates of mean and covariance. For non-linear and non-Gaussiansystems, simulation based methods can be used to estimate the full probability den-sity, instead of just the first two moments. A joint tracking and association method

81

82 Particle Filter Applications

is presented based on the particle filter. A comparison with classical methods isgiven. Finally, in Section 5.3.3 a novel control based structure for the number ofsamples is presented.

5.1 Maneuvering Target Tracking for ATC

This section is based on the Air Traffic Control (ATC) application presented inKarlsson and Bergman (2000a); Gustafsson et al. (2002). For ATC tracking, largerotating ground based radar stations are used as the primary tracking sensor asindicated in Figure 5.1. The radar measurements are then used by the trackingsystem to estimate and predict position and velocity. The turn of a civil aircraft isusually done with nearly constant speed and turn rate. If the turn rate is unknownthe non-linear nearly coordinated turn model from Section 4.2.4 can be used. Forhighly maneuvering targets an extension with a model for the turn rate is a feasibleapproach. In this section we perform a simulation study and compare linearizedmultiple models with particle filter methods.

The main idea in this section is to implement the auxiliary particle filter (APF)from Pitt and Shephard (1999), described in Section 3.3.2. using a non-linearATC model based on nearly coordinated turn models, but extending it to allowmaneuvering targets by using a multiple model approach. By using particle filters,non-linearities in the system and measurement models and constraints in systemparameters are incorporated in a natural way. The constraints could be due tonatural limitations in state parameters or due to terrain information. In Karlssonand Bergman (2000a), the ATC model is implemented for a two dimensional case,thus neglecting the influence of the relative height. The used turn rate of themodel coincided with the simulated target trajectory for the different modes ormodels for tutorial reasons. In the survey article, (Gustafsson et al., 2002), theATC model has been extended with a height coordinate and appropriate elevationmeasurements. A more difficult trajectory is generated where the target turn rateis a linear combination of the used modes or models. An extra comparison withthe SIR method is included.


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.1: Tracking of an aircraft using a ground based radar sensor.

5.1 Maneuvering Target Tracking for ATC 83

The main idea with the APF is to increase the influence for particles with alarge predictive likelihood, if we are allowed to wait for the next measurement. Herewe present the idea implemented for a maneuvering target tracking application asproposed in Karlsson and Bergman (2000a), where we have conditioned upon themaneuver sequences. The maneuvers are modeled by a discrete parameter Ω (turnrate) with a Markovian switching structure yielding a mode switching, or jumping,model. In McGinnity and Irwin (1998), a similar idea is used for linear jumpMarkov models, applied to a bootstrap filter.

We introduce a deterministic splitting of each particle into several offsprings,each one representing a different target maneuver. Each offspring is weighted by theMarkov transition probability for the maneuver and its likelihood. In Figure 5.2 (a)an example of three different maneuver assumptions in the deterministic particlesplitting is visualized for the initial particle cloud. The three predicted particleclouds conditioned on the turn rate are clearly distinguished in the figure. Theresampled cloud using the predicted particles is viewed in Figure 5.2 (b).

2020 2040 2060 2080 2100 2120 2140 2160 2180 22001920

1940

1960

1980

2000

2020

2040

2060

2080

Auxiliary particle prediction

ξ [m]

η [m

]

TruePredMeasurement


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

(a) Predicted particle cloud

2020 2040 2060 2080 2100 2120 2140 2160 2180 22001920

1940

1960

1980

2000

2020

2040

2060

2080

After resampling

ξ [m]

η [m

]

TrueResampledMeasurement


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

(b) Resampled APF particle cloud

Figure 5.2: The three different maneuvers for APF.

As described in Section 3.3.2, the APF extends the state xt by predicting the

state conditional upon particle k. At time t, the particle set x(i)t N

i=1 and the

corresponding weights w(i)t approximate the densities in (3.45) and (3.46). The

auxiliary particle filter uses the following definition

p(xt+1, k|Yt+1) ∝ p(yt+1|xt+1)p(xt+1|x(k)t )p(k|Yt), k = 1, . . . , N,

where Yt is the set of cumulative measurements up to time t. From this jointdensity we can draw samples and then discard the index, to produce a sample fromthe empirical filtering density as required. The index k is referred to as an auxiliary


variable. We consider the joint density of particle k at time t and the target stateat time t + 1. The turn rate Ωt+1 indicates that the state is affected during theintegration of the state equation from t to t + 1, i.e., it affects the states at timet+ 1, when applied at time t. Bayes’ rule gives

p(xt+1,Ωt+1, k|Yt+1) ∝ p(yt+1|xt+1,Ωt+1, k)p(xt+1,Ωt+1, k|Yt)

= p(yt+1|xt+1)p(xt+1|Ωt+1, k,Yt)p(Ωt+1, k|Yt)

= p(yt+1|xt+1)p(xt+1|x(k)t ,Ωt+1)p(Ωt+1|k,Yt)p(k|Yt)

= p(yt+1|xt+1)p(xt+1|x(k)t ,Ωt+1)p(Ωt+1|Ω(k)

t )p(k|Yt). (5.1)

Approximating this expression by replacing xt+1 with the expected mean

µ(k)t+1(Ωt+1) = E(xt+1|x(k)

t ,Ωt+1) (5.2)

in the first factor of (5.1) and marginalize over xt+1 gives

p(Ωt+1, k|Yt+1) ≈ c · p(yt+1|µ(k)t+1(Ωt+1))p(Ωt+1|Ω(k)

t )p(k|Yt), (5.3)

where c is a normalization factor. Sampling from the density (5.1) can now be

performed by resampling with replacement from the set x(i)t N

i=1, where the indexis chosen proportional to (5.3). The resampled candidates are then predicted usingthe system model. By implementing three different maneuvers; left turn, straightflying or right turn gives totally 3N particles in the algorithm which slightly in-creases the computational burden. To summarize, the extended APF algorithm isgiven in Algorithm 5.1, where the resampling step is according to Algorithm 3.7.


Algorithm 5.1 (APF for maneuvering target tracking)

1. Set t = 0, and generate N samples x(i)0 N

i=1 from p(x0), set µ(k)t = x

(k)t ,

k(j) = j, w(j)0 = 1/N , j = 1, . . . , N .

2. Compute µ(k)t+1(Ωt+1) = Ext+1|x(k)

t ,Ωt+1 for every Ωt+1 ∈ M(x(k)t ), where

M(x(k)t ) is the set of all feasible (state dependent) maneuvers.

Number of particles after splitting: N .

3. Generate new indices k(j) by sampling N times from

p(k|Yt+1) ∝ w(k)t p(yt+1|µ(k)

t+1)p(Ωt+1|Ω(k)t ) and predict the particles, i.e.,

x(j)t+1 = f(x

(k(j))t , vt), j = 1, . . . , N with different noise realizations.

4. Compute the likelihood weights w(j)t =

p(yt+1|x(j)t+1)

p(yt+1|µ(k(j))t+1 )

for j = 1, . . . , N and

normalize, i.e., w(j)t =

w(j)t

∑Nj=1 w

(j)t

5. Perform an optional resampling of the set x(i)t+1N

i=1, using the probability

weights. If resampling is chosen then reset w(j)t = 1

N , j = 1, . . . , N .


For civil aircraft a common model is to use the nearly coordinated turn system.The discrete two dimensional system is given by

Xt+1 = A(Ωt+1)Xt + [Bv BΩ]vt (5.4)

Xt =(xt Ωt+1

)′, xt =

(

ξ ξ η η)′, (5.5)

where ξ and η are the Cartesian position coordinates and ξ, η are the velocitycomponents. System constraints are incorporated in the model, so that non-feasiblemaneuvers are avoided using the particle filtering technique.

A(Ω) =

1 sin(ΩT )Ω 0 − (1−cos(ΩT ))

Ω 0 00 cos(ΩT ) 0 − sin(ΩT ) 0 0


Ω 0 00 sin(ΩT ) 0 cos(ΩT ) 0 00 0 0 0 1 00 0 0 0 0 1

,


Bv =

T 2

2 0T 0

0 T 2

20 T0 00 0

, BΩ =

000010

.

For military applications or faster maneuvering targets, there is a need to have amodel for the turn rate. Here we impose the following model, assuming a velocity

dependent turn rate, according to Ω = atyp/

√

ξ2 + η2, where atyp is the typicalmaneuvering acceleration which is modeled as a set of three discrete values, havinga Markovian switching structure.

The range, azimuth and elevation radar measurements are modeled as

yt = h(xt) + et =

(√

ξ2 + η2

arctan(ηξ )

)

+ et,

where et is zero mean noise with covariance Rt. Note that for the general case,the measurement equation must be modified so the angle equations are continuous.Independence in time and between the measurement and process noise is assumed.For this model we have neglected the relative height value.

A simulation study using the nearly coordinated turn model is performed. Themaneuvering is done by setting up a flight path in accordance with the used Markovtransition density. Simulations are performed by using the APF method extendedwith the deterministic maneuver splitting. A comparison to a traditional trackingmethod based on an IMM filter consisting of three extended Kalman filters withdifferent turn assumptions is made. In the simulation study the only parameterconstraint considered is a limitation on the target speed to the interval 50 ≤ |v| ≤60 m/s. The simulation step is re-run until a feasible speed is achieved. Thedistribution of the measurement noise is chosen to be Gaussian, with angular anddistance standard deviations of 0.5o and 20 m respectively. The sampling periodis chosen to T = 4 s to emulate a track-while-scan (TWS) behavior. For modernmono-pulse radars the update time and the angle standard deviation may be muchsmaller, but for non-mono pulse system or for air-borne tracking systems muchlarger values should be used.

In Figure 5.3 one realization using Gaussian noise is viewed. The a posterioriprobabilities for each coordinate is presented in Figure 5.4 for the predicted particlesfor one realization. In Figure 5.5 the posterior density after the first update ispresented. In Table 5.1, the position Root Mean Square Error (RMSE) is presented,according to

RMSE =

√√√√ 1

L

L∑

t=1

1

NMC

NMC∑

i=1

(ξit − ξtrue

t )2 + (ηit − ηtrue

t )2, (5.6)

where L = 30 is the simulation path length and ξit, η

it, are the filter position esti-

mates at time t in Monte Carlo run i. The calculations are based on Monte Carlo


2000 2500 3000 3500 40002000

2500

3000

3500

4000

ξ [m]

η [m

]

TrueMeasurementsAuxiliary particlesIMM


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.3: Simulation overview.

1950 2000 2050 2100 2150 2200 22501940

1960

1980

2000

2020

2040

2060Predicted & resampled particles

ξ [m]

η [m

]

Particles (pred)Particles (resampled)MeasurementParticle density

0 0.051900

1950

2000

2050

2100p(η)

η

1900 2000 2100 2200 23000

0.05

0.1

p(ξ)

ξ


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.4: Auxiliary particles and marginal position densities.


19001950200020502100

2100

2150

2200

2250

2300

0

0.01

0.02

0.03

0.04

0.05


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.5: The APF posterior position density.

simulations using NMC = 100 realizations. The APF method extended to the ma-neuvering case using N = 800 particles for each maneuver hypothesis is comparedto the IMM filter. The filtered estimates are compared to the RMSE position esti-mate using only pure measurements. As seen the APF method improves trackingperformance. Both the ability to incorporate constraint information and that thetrue non-linear model can be used without linearization is of importance, as wellas the fact that the APF improves the resampling of important particles for thenext time-scan.

APF IMM-3 MeasurementsRMSE 32.09 37.05 41.5

Table 5.1: RMSE for 100 Monte Carlo simulations.

In Gustafsson et al. (2002), the ATC model has been extended with a heightcoordinate and appropriate elevation measurements. A more difficult trajectoryis generated, with a true turn rate value chosen as an intermediate value of theturn rate used in the multiple model conditioning. In Table 5.2 the APF and SIRmethods extended to the maneuvering case are compared with an IMM filter. Forthe SIR case, two simulations diverged. Depending on the choice of process noise,the slight difference between the IMM and the SIR method may change.

APF SIR IMM-3 MeasurementsRMSE 34.03 40.84 42.20 63.96

Table 5.2: RMSE for 100 Monte Carlo simulations (including height).

5.2 Passive Ranging 89

5.2 Passive Ranging

Target tracking using angle measurements in azimuth and elevation is a commontechnique for both radar, sonar and IR applications. Typically, radar and sonarsensors are used in an active mode, transmitting energy. In this mode range andpossibly range rate are available from the sensor. To avoid the risk of being detectedby a hostile target, one often wants to use the sensors in a passive mode. Hence, onlyangle measurements from target induced energy is available. There are also truepassive sensors, such as the IR sensor, which only detects incoming heat radiation.Therefore only angle information is available. Many airborne systems use severaltracking sensors, particularly for long range tracking systems. It is natural to use apassive sensor to avoid a hostile target to acquire information initially. Therefore,a natural passive ranging application is to use passive tracking when the relativerange is rather high using an IR sensor. Under good transmission conditions, theregion of interest can be above 50 km. In Figure 5.6 an air-to-air passive rangingis presented.


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.6: Air to air passive ranging.

The main idea in passive ranging is to use angle information only to try toestimate the unknown relative range. By using the platform (aircraft, missile,torpedo etc.) and perform maneuvers, it is possible to gain observability in therange direction.

The issue of optimal maneuvers is not discussed here, but can be found inLogothetis et al. (1998, 1997). Often it is difficult to perform maneuvers in anoptimal manner. Therefore, the approach is to perform the maneuvering sequencein a deterministic way, exciting the system sufficiently to gain observability.

Passive ranging applications have been an important research area for severalyears. The classical method is to use a single extended Kalman filter. A com-mon problem is that a single linearized filter may easily diverge. To compensate


for this multiple models and some extra logic are often used techniques. Anotherapproach is to chose a suitable coordinate system, which makes the EKF less sensi-tive to range perturbations. By using sequential Monte Carlo methods, or particlefilters, a single non-linear filter can be used. The unknown range uncertainty iseasier addressed and constraints due to terrain or limitations in the system can beincorporated in a natural way.

For angle-only tracking it is essential that the target maneuvers are capturedby the tracking model. For the long range passive tracking application, it is oftenassumed that the target has not discovered the observer, so it could be assumedthat no severe maneuvers are performed.

5.2.1 Air-to-Air Passive Ranging

In this section, the passive ranging application from Karlsson and Gustafsson(2001b) is presented. Traditionally, angle-only problems are solved using linearizedfilters such as the EKF. The main problem with a single EKF is that it will notestimate the true posterior density, instead it will use estimate of mean and co-variance under the linearization assumption. This causes the filter to diverge inmany cases. To diminish this problem, the choice of different coordinate systemshas been of research interest. In Aidala and Hammel (1983) this is studied and itis investigated how to reduce filter divergence problems by selecting the coordinatesystem. Another approach is to use multiple filters where each filter is parame-terized to a sub-interval. By using a bank of filters, cf. (Peach, 1995), enhancesthe performance by introducing the range parameterized extended Kalman filter(RPEKF), as described in Section 2.4.3.

The passive range estimation is based on a maneuver model of the target.For long range applications, when the target is assumed not to have detected theaircraft a common model used is to assume a straight flying path. For maneuveringtargets, it may be necessary to introduce multiple turn models, which could behandled by using for instance the Interacting Multiple Model (IMM), (Bar-Shalomand Li, 1993), instead of single extended Kalman filters in the filter bank. Anotherpossibility is to use a change detector and adjust the estimate or system whena maneuver is detected, (Holtsberg and Holst, 1991). In Blackman and Popoli(1999), some aspects of the angle-only problem are described in more detail.

In this section we implement the RPEKF for both modified spherical coordi-nates (MSC) and Cartesian coordinates as was proposed in Karlsson and Gustafs-son (2001b) and compare with SIR particle filter based method. The coordinatesystems are presented in more detail in Section 4.2.5.

In the tracking we assume position and velocity coordinates as state variables,and we denote the relative Cartesian state vector x(t), for the difference betweenthe tracked object (target) xtg and the aircraft (tracking platform) xo

x(t) = xtg(t) − xo(t). (5.7)


Both target and aircraft are described by linear state equations

xtg = Axtg(t) +Butg(t) (5.8)

xo = Axo(t) +Buo(t). (5.9)

Hence, in relative coordinates the state equation is

xtg = A (xtg(t) − xo(t))︸︷︷︸

x(t)

+B utg(t)︸︷︷︸

v(t)

−B uo(t)︸︷︷︸

u(t)

. (5.10)

Since the target input signal is unknown, it is considered as process noise in themodel, v(t) = uo(t), hence the following model can be used in Cartesian coordinates

x = f(x, u) = Ax(t) −Bu(t) +Bv(t), (5.11)

where u(t) is the measured aircraft acceleration and v(t) is the process noise, mainlydue to target maneuver. If we assume that the state vector consists of Cartesianposition and velocity components

x =(

ξ η ζ ξ η ζ)′, (5.12)

then the system matrices are given by

A =

(I3x3 TI3x3

O3x3 I3x3

)

, B =

(O3x3

I3x3

)

, (5.13)

where I3x3 and O3x3 are the three-by-three unity and null matrix respectively.For the MSC case we use the notations introduced in Section 4.2.5, where z

denotes the state vector in modified spherical coordinates. The continuous-timestate equation in MSC with the acceleration signal in antenna coordinates is givenby uA =

(uA

1 uA2 u

A3

)′is from Section 4.2.5 given by

z =d

dt

z1z2z3z4z5z6

= f(z, uA) + vmsc =

−z1z4z5

cos z3

z6z26 + z2

5 − z24 − z1u

A1

(−2z4 + z6 tan z3)z5 − z1uA2

−2z4z6 − z25 tan z3 + z1u

A3

. (5.14)

The target accelerations are here included in the process noise term, vmsc, sincethey are unknown. The system is summarized as z = f(z, uA) + vmsc. The inputsignal (aircraft acceleration) is converted from inertial to antenna coordinates using

uA = CAI (z)u =

cos θ cosϕ cos θ sinϕ − sin θ− sinϕ cosϕ 0

sin θ cosϕ sin θ sinϕ cos θ

u, (5.15)


where azimuth (ϕ = z2) and elevation angle (θ = z3) are according to Section 4.2.5.The observation relation consists of azimuth and elevation angle measurements andwe assume an additive noise term

yt = h(xt) =

(ϕθ

)

+ et =

(arctan(ηt/ξt)

arctan(−ζt/√

ξ2t + η2t )

)

+ et. (5.16)

The MSC observation relation yt = h(zt) + et is linear

yt =

(ϕθ

)

+ et =

(0 1 0 0 0 00 0 1 0 0 0

)

zt + et. (5.17)

In Pitt and Shephard (1999), an angle-only application is studied assuming awrapped Cauchy distribution. In our approach, a Gaussian distribution is ap-plied, et ∈ N(0, R). To enhance the performance, measurements of non-kinematicfeatures such as intensity could be incorporated, (Blackman and Popoli, 1999), butin practice it may be troublesome.

One problem with the MSC representation is that it is difficult to initializethe covariance matrices. In Cartesian coordinates it is both easier to interpret andimplement, therefore the covariance is initialized in Cartesian coordinates and thentransformed to MSC. This is done by applying Gauss approximation formula, i.e.,using a Taylor expansion of the coordinate transformation function

z = g(x) ≈ g(x) + gx(x)(x− x), (5.18)

where g(x) is the mapping from Cartesian coordinates to MSC defined in (4.20).The initial state covariance can be expressed as

Pmsc0 = E(z − z)(z − z)′ = E(g(x) − g(x))(g(x) − g(x))′ = Eq (5.18)

≈ gx(x)E(x− x)(x− x)′g′x(x) = gx(x)P0g′x(x), (5.19)

where gx(x) = ∇xg(x)|x=x and P0 is the initial covariance for the Cartesian system.In discrete-time the Cartesian system is given by

xt+1 = f(xt, ut, vt) = Fxt −Gut + vt (5.20)

yt = h(xt) + et, (5.21)

where F = eAT , G =∫ T

0eAτdτB (see Rugh (1996))

The implemented EKF is according to the discretized linearization techniquedescribed in Section 2.3.4, i.e., first linearizing the continuous-time system and thendiscretize. For the MSC system we have the following time update equations forthe estimate and covariance

zt+T |t = zt|t +

∫ T

0

efz(zt|t,uA)τdτ f(zt|t, u

A) (5.22)

Pmsct+T |t = efz(zt|t,u

A)TPmsct|t ef ′

z(zt|t,uA)T +Qmsc. (5.23)


The matrix exponential can be solved using a Taylor expansion. For the Cartesiancase the time update is analytically solvable

xt+T |t = Fxt|t −Gut (5.24)

Pt+T |t = FPt|tF′ +Q. (5.25)

The discretization of the continuous process noise is described in Section 4.2.6.Here we assume a constant signal during the integration, so

Qmsc =1

T

∫ T

0

efzτdτQmsc

∫ T

0

ef ′zτdτ, (5.26)

where Qmsc ≈ gxQg′x and Q = EBv(t)(Bv(t))′. Similar calculation holds for the

Cartesian case.In the implemented angle-only application the filter initialization is performed in

Cartesian coordinates, projecting the assumed range hypothesis to the line-of-sight(LOS) using the measured angles. The velocity in Cartesian coordinates consistsof the known velocity for the tracking platform. The unknown target velocity isaccounted for in the initial uncertainty covariance. The initial state is transformedto the MSC system. This approach allows us to handle the range and range rateinitialization as different hypothesis in the RPEKF approach.

The initial value of the relative state vector assuming no knowledge of the targetvelocity is for each filter

x(i)0 =

r(i) cosϕm cos θm

r(i) sinϕm cos θm

−r(i) sin θm

0 − ξo

0 − ηo

0 − ζo

(5.27)

z(i)0 = g(x

(i)0 ) (5.28)

where ϕm and θm are the measured angle values. The initial state covariancematrix in the Line-of-Sight (LOS) system is in Cartesian coordinates given by

P(i)0LOS

=

(σ(i))2 0 00 (r(i)∆ϕ)2 0 O3x3

0 0 (r(i)∆θ)2

O3x3 (∆v)2I3x3

, (5.29)

where ∆v is the maximal uncertainty in the target velocity, ∆ϕ and ∆θ are theangle measurement noise standard deviation. The initial covariance matrix is cal-culated using the rotation matrix Trot as

P(i)0 = TrotP

(i)0LOS

T ′rot, Trot =

((TA

I )′ O3x3

O3x3 (TAI )′

)

, (5.30)


where

TAI =

cosϕm sinϕm 0− sinϕm cosϕm 0

0 0 1

cos θm 0 − sin θm

0 1 0sin θm 0 cos θm

, (5.31)

evaluated at the initial measurement angles. In MSC the covariance is given by

Pmsc,(i)0 = gx(x)P

(i)0 g′x(x). For other initiation methods, we refer to Section 4.3.

If a single doppler-radar measurement is present the initial range and range rateuncertainty can be set to the measurement error for the radar, thus increasingperformance substantially.

0 0.5 1 1.5 2 2.5 3

x 104

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000Target and aircraft position

ξ [m]

η [m

]

Aircraft (own)Target


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.7: The air-to-air scenario for passive range estimation.

In a simulation study the particle filter method is compared to the linearizedRPEKF approach for both Cartesian and modified spherical coordinates (MSC). Inthe implemented version of the RPEKF method, we used both MSC and Cartesiancoordinates for the measurement- and time update. An alternative is discussed inLarsson (1998), where the measurement update is performed in MSC and the timeupdate in Cartesian coordinates. Thus having a linear time update equation anda linear measurement relation.

We assume that the target is non-maneuvering and that angle observations areavailable with a sample period of T = 1 s. We apply the SIR method to theangle-only tracking problem. The initialization used is similar to the EKF basedapproach, but instead of assuming Gaussian distribution around different working


points, we use a uniform distribution in the range direction. As for the RPEKFmethod we use Cartesian and MSC to utilize the difference. In the Cartesian case,the particle filter implementation is straightforward and in accordance with Sec-tion 3.3.1. When using MSC the continuous-time system is solved by a numericalintegration for the time-update.

The scenario is illustrated in Figure 5.7, where the aircraft (tracking platform)and the target are presented in the horizontal plane. The relative target height is4000 m above the tracking platform. The measurement noise was assumed Gaussianwith angle standard deviation σϕ = σθ = 1 mrad for the IR sensor. The standarddeviation for the initial velocity used to calculate P0 was 200 m/s. The RPEKFshave range interval Rmin = 1 and Rmax = 64 km, with NF =6 filters, yieldingCR = 0.1925 as in Arulampalam and Ristic (2000); Peach (1995). For the particlefilter method, N = 30000 particles are used. In the evaluation Mmc = 50 MonteCarlo simulations were performed, over L = 30 s using the same scenario, but withdifferent measurement noise realizations.

In Figure 5.8, the mean relative range from the Monte Carlo simulations ispresented for the different methods. The performance is evaluated using the root

0 5 10 15 20 25 301.5

2

2.5

3

3.5

4

4.5

5x 10

4 Mean value of relative distance

Time [s]

Mea

n re

lativ

e di

stan

ce [m

]

Cartesian RPEKFMSC RPEKF Cartesian SIR MSC SIR


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.8: Mean range estimation for different methods and coordinate systems.


mean square error for each time, given in Figure 5.9, according to Gustafsson (2000)

RMSE(t) =

√√√√

1

Nmc

Nmc∑

j=1

||xtruet − xt,j ||22, (5.32)

where xt,j denotes the estimate at time t, for Monte Carlo simulation j. Forthe angle-only tracking problem, only position coordinates are used in the RMSEcalculation. The over all performance is analyzed using

0 5 10 15 20 25 300

2000

4000

6000

8000

10000

12000

14000

16000

18000RMSE(t) for different methods

Time [s]

RM

SE

(t)

Cartesian RPEKFMSC RPEKF Cartesian SIR MSC SIR


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.9: Position RMSE(t) for different methods & coordinate systems.

RMSE =

√√√√

1

L

L∑

t=1

Nmc∑

j=1

||xtruet − xt,j ||22. (5.33)

Ignoring transients the position RMSE is presented for the Monte Carlo simulationsin Table 5.3, for t ≥ 20.

To summarize, we have compared the particle filter with the RPEKF method.For the RPEKF method, tracking performance was almost equal using Cartesianor MSC, with NF = 6. The main disadvantage for MSC is the increased compu-tational burden due to the non-linear time update equation. To reduce the RMSEto almost the same level as for the best particle filter, more than NF & 60 filtershad to be used for the Cartesian filter given the scenario. It seems that the MSC


RPEKF SIRCartesian 2669 2044MSC 2685 1233

Table 5.3: Position RMSE ignoring transients.

case can use fewer filters, but to evaluate it, Monte Carlo simulations with differentscenarios must be performed. The particle filter gives better tracking performanceif the initial transient was ignored, for both MSC and Cartesian coordinates, butusing substantially more computational time. Specially for the MSC case, whichis very time consuming. If the number of particles is decreased from N = 30000 to10000, the MSC SIR decreases performance only slightly (RMSE=1274), whereasthe Cartesian case has a lot worse performance than any RPEKF method. Notethat for all simulations, we assumed Gaussian measurement and process noise,making the tracking comparison easier for the Kalman filter. For non-Gaussianassumptions, the particle filter can still be used making it more flexible.


5.2.2 Terrain Induced Constraints for Passive Tracking

In this section we consider an air-to-sea passive ranging application. A passivetracking system based around an IR sensor is used on an aircraft to track a hostileship as illustrated in Figure 5.10. By maneuvering the aircraft estimates of rangeand range rate are available. We use the passive particle filter tracking systemdescribed in Section 5.2.1 for the position estimation. The main objective in this


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.10: Passive ranging for the air-to-sea application.

section is to merge information from a terrain database to discard region uninter-esting to the tracking application. A terrain database over the region of interest isused, which contains terrain type information. Here we only distinguish betweenland and sea, but for other applications the whole classification information fromthe database can be used. Terrain induced tracking constraints are used to improvetracking performance and reduce computational complexity. The tracking filter isimplemented according to the SIR method presented in Algorithm 3.5. In the par-

ticle filter each particle (x(i)t ) represents a possible target location. Hence, applying

a terrain database, particles are discarded if there position is within a restrictedarea (land), whereas particles belonging to accepted regions (sea) are accepted andassigned with a weight according to a likelihood function. Two different approachesare considered, where we use the additive noise system from (2.20).

Terrain constraints via measurement update. A natural approach to intro-duce constraints is by using the importance weights calculated in the measurementupdate. Here we can interpret the database as an extra sensor in a larger sensorfusion context.

w(i)t = pet

(et) = pet(yt − h(x

(i)t )) =

pet(yt − h(x

(i)t )), x

(i)t ∈ Sea area

0, otherwise. (5.34)


Terrain constraints via time update. By introducing the constraints in thestate equation, each particle is accepted if the predicted position after the additivewith additive noise perturbation belongs to a sea region in the database.

pvt(vt) = pvt

(x(i)t+1 − f(x

(i)t )) =

pvt(x

(i)t+1 − f(x

(i)t )), x

(i)t ∈ Sea area

0, otherwise. (5.35)

It is easy to impose other constraints. For example, hard constraints on state-variables such as velocity and acceleration. The fact that sea-targets are close tothe surface can be easily handled. All these constraints are difficult or extremelytroublesome to handle with classical Kalman filter techniques. The positioningapplication in Wijk (2001) uses an indoor mobile robot with a reference map anda sonar sensor.

Time=1

ξ [km]

η [k

m]

1665 1670 1675 1680 1685 1690 1695 1700

6400

6402

6404

6406

6408

6410

6412

6414

6416

6418

Sea targetAircraft


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

(a) SIR estimation of target position (with

constraints).

Time=1

ξ [km]

η [k

m]

1665 1670 1675 1680 1685 1690 1695 1700

6400

6402

6404

6406

6408

6410

6412

6414

6416

6418

Sea targetAircraft


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

(b) SIR estimation of target position

(without constraints).

Figure 5.11: Initial particle cloud for the sea target position in the air-to-sea ap-plication.

In a simulation study we consider the range estimation problem using an IRsensor, which measures azimuth and elevation angles relative to the target. Theparticle filter (SIR) is initialized around the first measurement strobe, where weused an IR sensor with a total angle error of 1 mrad. The range values are drawnfrom a uniform distribution, over the range of interest, and we used N = 5000particles. We consider a track-while-scan (TWS) application, where the sampleperiod is T = 1 s. The aircraft’s velocity was about 250 m/s and a constant heightsinusoidal maneuver was performed to gain observability. The state equation andmeasurement relation are similar to those described in Section 5.2.1, but only theCartesian case is considered. The target model used in the simulations assumes asmall constant velocity. The terrain database has a resolution of 50 m.


In Figure 5.11 the initial particle position estimate is shown together with thetarget, aircraft position and aircraft trajectory. In Figure 5.11 (a) the particlecloud for the SIR method is shown at the initialization using the terrain databaseto discard particles over land areas. In Figure 5.11 (b) the same scenario is pre-sented without the constraints. As a result, many particles and calculation timewas wasted on positions belonging to land areas. The terrain constraints wereincorporated in the calculation of the importance weights as described in (5.34).

In Figure 5.12 the scenario is presented together with the marginal positiondensities in each direction (p(ξ), p(η)) for time t = 1 s, when constraints are used.The marginal distributions are zero over land areas. Otherwise they are almostuniformly distributed, due to the initialization. By performing maneuvers, only

Time=1

1670 1680 1690 1700

6400

6405

6410

6415

Sea targetAircraft

0 0.1 0.2

6400

6405

6410

6415

p(η)

η [k

m]

1670 1680 1690 17000

0.05

0.1

0.15

ξ [km]

p(ξ)


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.12: Target position and marginalized position density using the particlefilter with constraints at t = 1 s.

particles that satisfy the motion and measurement models acquire high enoughprobability. In Figure 5.13 (a), the scenario is presented for time t = 15 s. Anenlargement of the target area is shown in Figure 5.13 (b). As seen in Figure 5.13,the particle cloud is located in the vicinity of the true target.

To summarize, we have shown how to incorporate terrain induced constraintsin an air-to-sea tracking application. This extra information is difficult to utilize ina Kalman filter bank. The terrain information reduces the computational burden,since uninteresting regions are not considered.


Time=15

1670 1680 1690 1700

6400

6405

6410

6415

0 1

6400

6405

6410

6415

p(η)

η [k

m]

1670 1680 1690 17000

0.5

1

ξ [km]

p(ξ)


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

(a) The tracking scenario and particle cloud.

Time=15

ξ [km]

η [k

m]

1677 1678 1679 1680 1681 1682 1683 1684 1685 1686

6400

6401

6402

6403

6404

6405

6406


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

(b) Enlargement of the particle cloud in the target area.

Figure 5.13: Particle filter with constraints at t = 15 s.


5.2.3 Passive Tracking for a Torpedo System

Modern torpedo systems are equipped with an acoustic seeker, which is similar tothe electro-magnetic radar. This sound based sensor is referred to as sonar. Inthe active mode both range and bearing to a target are available. To avoid beingdetected by a hostile target and to reduce the risk of hostile counter-measurements,it is often important to minimize the active mode usage. In the passive mode, theacoustic sensor just listens for target related sounds. Hence in principle only thedirection can be measured. Most sonar based systems do not measure the elevationangle, therefore only bearing information is available. When the torpedo is trackinga sea target using the passive sensor mode the range estimation must be performedby maneuvering the torpedo to gain observability. In this section we focus on atorpedo system and apply the passive tracking techniques from Section 5.2.1, usingboth the RPEKF and particle filter methods. The bearing information is fromexperimental sonar data acquired from a torpedo system provided by Saab BoforsUnderwater Systems.

tA

tB

tC t

D

tEt

F

tG

ξ

η


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

PSfragreplacem

ents

Ax (i)t

+v (i)t

=⇒

u (j+1)u (j)

u (j−1)

Interval

∑

pi=

1 w (i)ti−1

iw (i)t

Nξ

ηζ

Given

trackT1

yt1 (1)

yt2 (1)

yt1 (2)

Scan1

Scan2

Scantim

eFA

TC

NTN

tk1

PF

PF

εt

∆(N

,ε)µ (1)tµ (2)t

1q−1

Resam

plin

gstep

Control structure

|ε|y (1)y (2)y (3)x (1)x (2)y (1)y (2)

yy (3)y (4)y (1)y (2)

νξ

ηr

ψRmin

Rmax

Rm

in ρ i−1

Rm

in ρ ir (i)

rϕ

θ

Trackedobject

(target)

Trackingplatformnxn

yEK

F1

x (1)t|t−1

P (1)t|t−1

ytl (1)t−

1x (1)t|tP (1)t|tl (1)txt , P

t

Com

bineEK

Fi

x (i)t|t−1

P (i)t|t−1

ytl (i)t−

1x (i)t|tP (i)t|tl (i)txt , P

t

Com

bine

EKFNF

x (NF)

t|t−1

P (NF)

t|t−1

yt

l (NF)

t−1x (N

F)

t|tP (NF)

t|tl (NF)

txt , P

t

Com

binex (1)t−1x (2)t−

1

Hypothesesx (1)tx (2)t

EstimateFilter

Interaction/MixingΛ (1)tΛ (2)t

yt

x (1)t−1|t−

1 , P (1)t−1|t−

1

x (2)t−1|t−

1 , P (2)t−1|t−

1

x (1),0t−1|t−

1 , P (1),0t−

1|t−1

x (2),0t−1|t−

1 , P (2),0t−

1|t−1

x (1)t|t , P (1)t|t

x (2)t|t , P (2)t|txt|t , P

t|t

µt−

1|t−1

µt|t , µ

tDDS

1S2S

3S4S

5S6S

7S8

f(X)


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.14: Torpedo trajectory with time indications at way-points and and indi-cation of the ship’s true trajectory.

For range estimation, we use a similar tracking system as described in Sec-tion 5.2.1 but applied to the torpedo system. However, only the Cartesian version


of the tracking model is utilized and the measurement relation uses only bearing(azimuth) information.

The scenario in the position plane is given in Figure 5.14 where torpedo way-points are marked with time indices. The true position of the target (ship) is notknown since no true position information is available. However, it is known thatthe ship follows a rather straight path, with nearly constant velocity as indicatedin the figure. Since the torpedo approaches the target from behind in the finalphase (t > tF ) and impact occurs at t = tG, the probable approximate true targetposition is indicated with a dashed line in Figure 5.14.

Between the initial time tA and first major maneuver at tB a the torpedo fol-lows a relatively straight trajectory. During this time the range estimation is notparticularly good since the small maneuver and the relative geometry to the targetdoes not reveal much range information. At tB the first major maneuver is per-formed to gain observability, followed by maneuvers at tC and tD. During the longinterval tD − tE a straight path is followed, where the main objective is to decreasethe distance to the ship. Finally, at tE and tF maneuvers are performed to gainobservability and to approach the target from behind. The bearing measurementsfrom the sonar are given with a sample period of T s, and the total number of mea-surements is close to 300. The position scale and the actual values of the maneuvertimes are not presented in any plot, since it is confidential information. We apply

ξ

η

p(η)

η

p(ξ)

ξ


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.15: Torpedo trajectory, particle cloud and marginalized densities for targetposition at t = tA.

both the RPEKF and the SIR particle filter to the experimental torpedo data.The filters are initialized in accordance with Section 5.2.1 using the initial bearing


measurement. Here we consider only the Cartesian case and we use N = 15000particles in the SIR method. In Figure 5.15, the output from the particle filter isshown for t = tA. Both the target position and the marginalized target positionprobability densities are given. Initially, no maneuver is made so the range cannot be estimated. The full torpedo position trajectory is shown, where the currenttorpedo position is indicated with a small circle. Here we did not utilize any ex-ternal data about the range uncertainty. In practice the range uncertainty regioncan be reduced by using data from other external sources. After t = tB a sharpmaneuver is initialized, therefore, the range becomes observable and the range un-certainty reduces. In Figure 5.16, the target position estimate from the particlefilter is shown for t ∈ [tB , tC ]. The bearing measurement uncertainty region is alsopresented. Finally, the particle estimate at t & tF is given in Figure 5.17. Since

ξ

η

p(η)

η

p(ξ)

ξ


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.16: Torpedo trajectory, particle cloud and probability of target positionat t ∈ [tB , tC ].

we estimate the relative velocity, the target speed and heading can be calculatedas shown in Figure 5.18 for the SIR and RPEKF method.

The MMSE estimate of target position from the particle filter is presented inFigure 5.19 together with the estimation from the RPEKF method using NF = 6filters. The range uncertainty and how fast it converges depends on how the initialuncertainty region was chosen and on the number of filters and particles. For theSIR method a small jittering noise was added to diminish depletion as discussed inSection 3.3.4.

To summarize, we have applied the RPEKF and the SIR particle filter methodto experimental torpedo data for the bearings-only range estimation case. For the


ξ

η

p(η)

η

p(ξ)

ξ


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.17: Torpedo trajectory, particle cloud and probability of target positionat t & tF .

Targ

et s

peed

Time (sample)

SIRRPEKF

Targ

et h

eadi

ng

Time (sample)


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.18: Target speed and heading estimate.


tA

tB

tC t

D

tE

tF

tG

ξ

η

SIRRPEKFTorpedo


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.19: Torpedo trajectory, MMSE estimation of target position using theSIR and RPEKF methods.

non-maneuvering target case both methods succeeded in range estimation. Since alinear-Gaussian time update model and Gaussian measurement noise is assumed,the only non-linearity was in the bearing measurement relation. The Kalman filterbank and the particle filter have rather similar behavior. However, the particlefilter is more flexible and can easily be adjusted to new data, constraints or noiseassumptions. Multi-path propagation and diffraction problems can be incorporatedin the particle filter framework. Other sea-to-sea applications closely related topassive tracking is positioning using sea-maps, where a combination of techniquesdescribed in this section and those in Section 5.2.2 can be used.

5.3 Multi Target Tracking and Data Association

For multi-target tracking applications, overall tracking performance depends on es-timation and data association methods. Traditionally, state estimation for targettracking is performed using the Kalman filter or linearized versions thereof, suchas the extended Kalman filter (EKF), or multiple models consisting of linearizedfilters. The sufficient statistics for the filter are mean and covariance, so natu-rally the most common data association methods rely on these two moments. InSection 4.5, many different Kalman filter based approaches for data associationare presented. Here we focus on the joint probabilistic data association (JPDA)method, but instead of Kalman filter in the estimation we use the SIR filter de-scribed in Section 3.3.1. This affects the data association step. Therefore, a newmodified algorithm must be introduced. We present the SIR/MCJPDA methodfrom Karlsson and Gustafsson (2001a), where the particle filter data association

5.3 Multi Target Tracking and Data Association 107

based on the JPDA method is considered. The algorithm development is describedin Section 5.3.1 and a simulation study is presented in Section 5.3.2. To reducecomputational complexity, introduced due to the multi target Monte Carlo data as-sociation, a control structure of the number of particles is proposed in Section 5.3.3.

5.3.1 The SIR/MCJPDA–Algorithm

In this section we modify the classical SIR algorithm (Algorithm 3.5) for estimationto handle multiple targets. The proposed association principle is based on a novelMonte Carlo approach for the JPDA algorithm. This SIR/MCJPDA method wasintroduced in Karlsson and Gustafsson (2001a), independent of Schulz et al. (2001),where similar ideas are used for a robotic application. We have assumed time-invariant models for all targets. We use the same Bayesian approach as in Gordon(1997) for the estimation. However, we extend the idea and introduce hypothesiscalculations according to the JPDA method. The resampling is executed over alltarget association hypotheses.

We consider the case of a known number of targets (τ) in a cluttered environ-ment. The tracking system is expressed in relative Cartesian coordinates, wherexj

t denotes the state vector for target j = 1, . . . , τ . We use the following compactrepresentation for the set of all tracks

xt = x1t , x

2t , . . . , x

τt . (5.36)

A special clutter model is used to handle false alarms, x0t (j = 0). Since the SIR

particle filter method is used for the estimation and data association we introducethe following notation for the samples (particles) of the complete state vector

x(i)t Nt

i=1 = x(i),1t , . . . , x

(i),τt Nt

i=1, (5.37)

where Nt denotes the number of particles at time t. The initial target cloud is

denoted x(i),j0 Nt

i=1 for targets (tracks) j = 1, . . . , τ . We consider a TWS system,collecting measurements during each scan. To simplify the notation and reduce thecomputational complexity all measurements within a scan are given the same timeindex. The measurements for each time frame (scan) are denoted yk

t , k = 1, . . . ,Mt.The tracking model in relative coordinates is given as

xjt+1 = Axj

t +Bvjt (5.38)

yjt = h(xj

t ) + ejt , (5.39)

where the process (vjt ) and measurement noise (ej

t ) are considered independentwith distributions pvt

and pet, respectively.


In Table 5.4 an example of a multi target data association using the JPDAmethod is given, considering two tracks and three measurements. The probability ofdetection (PD) and the false alarm probability (PFA) are assumed known constants.The association likelihood (track j, measurement k) is given by pjk = pet

(ykt −

h(xjt )). We consider the case when a measurement is only allowed to update a

track once in a hypothesis.

Hyp. no Track Track False Hyp. probability1 2 Alarm

H1 - - 1,2,3 (1 − PD)2P 3FA

H2 - 1 2,3 (1 − PD)PDp21P2FA

H3 - 2 1,3 (1 − PD)PDp22P2FA

H4 - 3 1,2 (1 − PD)PDp23P2FA

H5 1 - 2,3 (1 − PD)PDp11P2FA

H6 1 2 3 P 2Dp11p22PFA

H7 1 3 2 P 2Dp11p23PFA

H8 2 - 1,3 (1 − PD)PDp12P2FA

H9 2 1 3 P 2Dp12p21PFA

H10 2 3 1 P 2Dp12p23PFA

H11 3 - 1,2 (1 − PD)PDp13P2FA

H12 3 1 2 P 2Dp13p21PFA

H13 3 2 1 P 2Dp13p22PFA

Table 5.4: The hypotheses for two tracks and three measurements.

A general expression for the probability in hypothesis Hn is:

P (Hn) = P τ−Zn

D (1 − PD)ZnPMt−(τ−Zn)FA ln, (5.40)

where Zn is the number of false alarms (FA) and ln is the likelihood part in hy-pothesis n. For the particle filter each particle is associated with a weight

w(i)t =

∑

n

P (H(i)n ). (5.41)

Normalization yields the particle probability w(i)t . The joint particle filtering and

association is summarized in Algorithm 5.2.


Algorithm 5.2 (SIR/MCJPDA)

1. Set t = 0, generate Nt samples from each target j = 1, . . . , τ , i.e.,

x0 = x(i)0 Nt

i=1 = x(i),10 , . . . , x

(i),τ0 Nt

i=1, where x(i),j0 from p(xj

0).

2. For each particle compute the weights for all measurement to track

association w(i)t =

∑

n P (H(i)n ) and normalize for each

measurement, i.e., w(i)t = w

(i)t /

∑Nt

i=1 w(i)t , where P (H

(i)n ) is the

probability for hypothesis n using particle i according to (5.40).

3. Generate a new set x(i?)t Nt

i=1 by resampling with replacement Nt

times from x(i)t Nt

i=1, where Pr(x(i?)t = x

(l)t ) = w

(l)t .

4. Predict (simulate) new particles, i.e., x(i),jt+1 = f(x

(i?),jt , v

(i),jt ),

i = 1, . . . , Nt, using different noise realizations for the particles,for each target j = 1, . . . , τ .


To simplify the algorithm some practical problems are discarded. The mea-surements within a scan are considered given at the same time instances and thenumber of targets (τ) is assumed constant. If the number of targets is unknown orchanging, the algorithm could be modified, using a separate track start hypothe-sis. This can be done within the particle filter framework or by using a linearizedmethod. To allow measurements with different sample times, the prediction stepis modified with an increased computational load as a consequence, i.e., each trackmust be predicted to every measurement time, in the association step. For simplic-ity no gating regions are introduce in the formulas. However, this can be handledas described in Section 4.5.1.

5.3.2 SIR/MCJPDA–Simulation Study

In a simulation study the SIR/MCJPDA-method is tested and compared to classicalmethods. We consider a multi target tracking application in a cluttered environ-ment. The application at hand is an air-to-air scenario, with a missile trackingseveral aircraft. To simplify the simulations we assume that it is always possibleto resolve the targets. The clutter or false alarm model is assumed uniformly dis-tributed in the volume and the number of false alarms for each scan is assumedto be Poisson distributed. Measurements can originate from one object or fromclutter and due to the detection model sometimes no observation is detected at all.

The relative state vector for each track is defined as xjt = xj

t − xmissilet , where

xj is the inertial position and velocity vector for track j and xmissilet the inertial

state vector for the platform (missile). In relative Cartesian coordinates we have

xjt =

(

ξ η ζ ξ η ζ)′, (5.42)


where ξ, η and ζ are the relative Cartesian position coordinates and ξ, η and ζ thevelocity components. We consider a radar sensor measuring relative range andangles, hence the following discrete-time system is used

xjt+1 =

(I3x3 TI3x3

O3x3 I3x3

)

xjt +

(T 2

2 I3x3

TI3x3

)

vjt , (5.43)

yjt = h(xj

t ) =

√

ξ2t + η2t + ζ2

t

arctan(ηt

ξt)

arctan( −ζt√ξ2

t +η2t

)

+ ej

t , j = 1, . . . , τ, (5.44)

where the process noise is assumed Gaussian, vjt ∈ N(0, Q) and the measurement

noise ejt ∈ N(0, R).

The parametric models for false alarms are assumed NFA ∈ Po(λV ), withaverage number of false alarms per unit volume λ and the validation region volumeV . In the simulations ENFA = λV = 0.5 is used. The detection probabilityis assumed PD = 0.9. The number of targets τ = 2 and a sample time of T = 1s. The initial inertial target state vectors xi

0, initial missile state vector xmissile0 ,

measurement noise matrix R, process noise Q and initial state error matrix P0 are

x10 =

6500−10002000−501000

, x20 =

5050−4502000100500

, xmissile0 =

00

3000200−500

,

P0 = diag(1002 1002 1002 502 502 502

),

Q =

102 0 00 102 00 0 102

, R =

502 0 00 0.012 00 0 0.012

.

The implemented EKF is according to the discretized linearization techniqueGustafsson (2000). Initial values for the tracks are drawn from the initial un-certainty region P0 around the true value. We compare the SIR/MCJPDA methodwith an NN data association where the estimation is done by the particle filterand where the covariance matrix needed for the association is calculated for eachtrack using the weights as described in (3.35). A comparison is made to an EKFusing the NN or JPDA association in a similar way. In Figure 5.20, a data as-sociation and estimation using the SIR/MCJPDA filter is presented. To evaluatethe performance a root mean square error (RMSE) analysis is performed over 60simulations. In Table 5.5, the results for the different methods are summarized,using RMSE for the two targets when t ≥ 3, ignoring initial transients. The parti-cle filter used N = 25000 samples. In Figure 5.21, the RMSE values for differenttimes are presented for the methods described in Table 5.5 (track 1). The for bothJPDA based methods no tracks diverged, but for the NN data association there


4000 4500 5000 5500 6000 6500 7000 7500 8000 8500−1500

−1000

−500

0

500

1000

1500

ξ [m]

η [m

]MeasurementEstimateTarget


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.20: SIR/MCJPDA data association and tracking.

Estimation Association RMSE #1 RMSE #2 Divergence

SIR MCJPDA 51.6988 51.3957 -SIR NN 55.8878 55.4883 2EKF JPDA 52.1159 51.5462 -EKF NN 52.6854 54.0163 2

Table 5.5: Association and estimation – RMSE analysis.

were 2 divergent simulations. The reason is that the NN method does not accountfor the clutter problem. Note that the decreased performance in RMSE during theassociation problem for the SIR/NN method is mainly due to one realization, so allthe methods are otherwise quite similar if that realization is discarded. This is dueto the linear time update equation and Gaussian noise assumption and that thelinearization error from the measurement relation is not significant using T = 1.However, it should be emphasized that the particle filter method is more flexible,allowing both non-linear update equation and non-Gaussian noise.


0 5 10 15 2040

50

60

70

80

90

100

110

120

Time [s]

RM

SE

(t)SIR/MCJPDASIR/NNEKF/JPDAEKF/NN


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.21: RMSE(t) for different estimation/association methods.

5.3.3 Particle Number Controller

In this section we consider a novel approach to control the number of particlesneeded in a particle filter based target tracking application. The idea was proposedin Karlsson and Gustafsson (2001a) for the multi-target tracking problem describedin Section 5.3 but could also be used for angle-only tracking, since we initiallyoften use more particles than needed later on. The computational burden for theparticle filter is highly dependent on the number of particles and on the resamplingcalculation. For multiple target tracking applications the computational burdenis heavily increased. Therefore, it is essential to minimize the required number ofparticles in the estimation step. Since it is not known how many particles that areneeded to approximate a density sufficiently well, there have been a lot work tryingto find fundamental limits. For instance, in Boers (1999) an analytical approachcalculating lower bounds for the number of particles needed to approximate theposterior is discussed. In this section we extend the particle filter method byapplying a control structure according to Figure 5.22 in order to control the numberof particles. The number of particles is determined by the controller using the

residual εt = ||µ(1)t − µ

(2)t ||, where µ

(1)t and µ

(2)t are statistical properties from

the particle filters (PFs), using different number of particles. Possible choices arerelevant statistics, such as the mean estimate from the particle filter or utilization


+ −


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt ∆(N, ε)

µ(1)t

µ(2)t

1q−1

Resampling step

Control structure

|ε|

y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.22: Controller of particles.

of the probability density (PDF) or the cumulative density function (CDF). Forinstance the marginal distribution (density for each coordinate) could be used.The original idea was to use two particle filter in parallel, with different numberof particles. By comparing the residual εt, the decision to increase or decrease thenumber of particles could be made. However, by introducing multiple particle filtersthe complexity would increase. Therefore only one filter is used and instead theoutput from the resampling step is compared using different number of particles.

The novel approach to the controller is applied to the SIR/MCJPDA methodfrom Section 5.3.2. The used control structure is a non-linear block consisting of arelay and an integrator.

∆(Nt, εt) =

αinc(Nt), if |εt| > Λ

αdec(Nt), if |εt| ≤ Λ,

In Figure 5.23 the SIR/MCJPDA method using the controller is presented for 20Monte Carlo simulations, where the control parameters are

• Particle number gain: k1 = 12

• Threshold level: Λ = 9.5

• Increment particle gain: αinc(Nt) = 0.2Nt

• Decrement particle gain: αdec(Nt) = −0.1Nt

• µ: Marginal density (compare each coordinated)

For maneuvering targets in a tracking application the controller can reduceor increase the number of particles during the tracking envelope. However, per-formance may now depend on the parameters of the controller. Note that thecontroller is implemented in the resampling step (Algorithm 3.5, step 3). To makethis method efficient, we must have prior knowledge of typical scenarios when thecontrol parameters are chosen, so the number of particles is not decreased too fast.The same idea with a controller of the number of particles could be used for thesingle target in passive ranging, where a large number of particles is needed initially.


0 2 4 6 8 10 12 14 16 18 2045

50

55

60

65

70

75

80

Time [s]

RM

SE

(t)

SIR/MCJPDA: N=25000 (fixed)SIR/MCJPDA: N=5000 (controller)

0 2 4 6 8 10 12 14 16 18 205000

6000

7000

8000

9000

10000

Time [s]

#par

ticle

s


(i)t

+ v(i)t

=⇒

u(j+1)

u(j)

u(j−1)

Interval∑p

i=1 w(i)t

i − 1

i

w(i)t

Nξηζ

Given track

T1

yt1(1)yt2(1)yt1(2)

Scan 1Scan 2

Scan timeFA

TC

NT

Nt

k1

PF

PF

εt∆(N, ε)

µ(1)t

µ(2)t1

q−1

Resampling step


y(1)

y(2)

y(3)

x(1)

x(2)

y(1)

y(2)

yy(3)

y(4)

y(1)

y(2)

νξη

rψ

Rmin

Rmax

Rminρi−1

Rminρi

r(i)

rϕ


Tracking platformnx

ny

EKF 1x

(1)

t|t−1

P(1)

t|t−1

yt

l(1)t−1

x(1)

t|t

P(1)

t|t

l(1)t

xt, Pt

Combine

EKF ix

(i)

t|t−1

P(i)

t|t−1

yt

l(i)t−1

x(i)

t|t

P(i)

t|t

l(i)t

xt, Pt

CombineEKF NF

x(NF )

t|t−1

P(NF )

t|t−1

yt

l(NF )t−1

x(NF )

t|t

P(NF )

t|t

l(NF )t

xt, Pt

Combinex

(1)t−1

x(2)t−1

Hypothesesx

(1)t

x(2)t

EstimateFilter

Interaction/Mixing

Λ(1)t

Λ(2)t

yt

x(1)t−1|t−1, P

(1)t−1|t−1

x(2)t−1|t−1, P

(2)t−1|t−1

x(1),0t−1|t−1, P

(1),0t−1|t−1

x(2),0t−1|t−1, P

(2),0t−1|t−1

x(1)t|t , P

(1)t|t

x(2)t|t , P

(2)t|t


µt|t, µt

DDS1

S2

S3

S4

S5

S6

S7

S8

f(X )

Figure 5.23: The particle number controller using the marginal distribution.

6

Conclusions

In this thesis we have studied several estimation and data association methods fortarget tracking. The main interest has been in sequential Monte Carlo methods,or particle filters, which naturally can increase the tracking performance if non-linear and non-Gaussian models are needed. Classical methods mostly within theBayesian framework have been discussed and used as comparison to the particlefilter methods. In various applications we have compared different particle filterswith classical methods. We have proposed extensions and modifications to existingmethods and algorithms.

We have shown how to implement multiple models in the particle filter to handlemaneuvering targets. A comparison between the classical IMM method and theauxiliary particle filter was made for an ATC application.

The angle-only tracking problem has been discussed in several applications.For an air-to-air implementation the particle filter (SIR) was compared with theRPEKF filter bank. Both Cartesian and modified spherical coordinates were used.For an air-to-sea application, the concept of terrain induced constraints was demon-strated for the particle filter method. A terrain database was used to discard un-interesting position data in the estimation process. The passive range and velocityestimation was also tested on experimental sonar data from a torpedo system. Boththe particle filter and the RPEKF filter bank were successful in the estimation.

For the multi target tracking application we have shown how to incorporate theclassical JPDA data association method within the particle filter framework, and

115

116 Conclusions

also studied an idea how to adapt the number of particles via a control structure.To summarize, the particle filter method is easy to implement for any non-

linear discrete-time system. It can handle non-Gaussian noise and it is easy toimpose constraints to the system. For instance, the non-linear measurement rela-tion or the constraints on the state vector could be defined by a database, so noanalytic relation may exist, which make the classical Kalman filter implementationtroublesome. Both simulation results and tests on experimental data have beenperformed.

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