Tracking Filters for Radar Systems

Wing Ip Tam

A thesis submitted in conformity with the requirernents

for the degree of Master of Applied Science

Graduate Department of Electrical and Cornputer Engineering

University of Toronto

@Copyright by Wing Ip Tarn 1997

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Tracking Filters for Radar Systems

by

Wig Ip Tam

Master of Applied Science, 1997

Depart ment of Elec t rical and Computer Engineering, University of Toront O

Abstract

In this paper we discuss the problem of target tracking in Cartesian coordinates

with polar measurements and propose two efncient tracking algorithms. The first

dgorithm uses the multidimensional Gauss-Hermite quadrature to evaluate the op-

timal estimate as a weighted s u m of functional dues. To reduce the computational

requirements of this quadrature technique we have suggested several ways to reduce

the dimension, the number and the order of the quadratures required for a given accu-

racy. The second tracking dgorithm is based on the Gaussian s u m filter. To alleviate

the computationd buden associated with the Gaussian sum fdter, we have found

two efficient and systematic ways to approximates a non-Gaussian and measurement-

dependent b c t i o n by a weighted surn of Gaussian density function and we have

derived the formula for updating the paxameters involved in the bank of Kahan-

type flters and we also have proposed new techniques to control the number of terms

in the Gaussian mixture at each iteration. Simulation results show that these two

proposed methods are more accurate than the classical method, such as the extended

K h a n filter (EKF) .

Acknowledgements

1 am especidy gratefd for the extensive and generous assistance of my superior, Pro-

fessor Dimitrios Hatzinakos, who spent time with me each week to guide me through

the thesis and made numerous helpfid suggestions to me.

1 wish to thank' Kostas Plataniotis for his thorough reviews and perceptive com-

ments which greatly improve the quality of my thesis work. I also wish to thank

Professor Pasupat hy and Professor Kwong for their consultations.

1 thank my friends William Ma, Erwin Pang, Jacky Yan, Richard Lee, Wilson

Wong and Ryan Lee and his farnily for all the fun time.

Lastly, much credit must be given to the members of my family, who have sup-

ported me and have borne with me during this period.

Contents

1 Introduction 1

2 Problem Definition and Literature Survey 4

2.1 Basic Mode1 for Radar Systems . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Target Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.2 Radar Measurernent . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 The Bayesian Approach . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Sub-Optimal Nonlinear Filters for Radar Tracking . . . . . . . . . . . 12

3 An Efficient Radar Tkacking Algorithm Using Multidimensionai Gauss-

Hermite Quadratures 18

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.2 Gauss.HermiteQuadrature . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Basic Principles of the Proposed Filter . . . . . . . . . . . . . . . . . 21

3.4 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 t e t e . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.6 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Adaptive Gaussian Sum Algorithm for Radar 'Ikacking 29

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Theoretical Foundations . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3 Basic Principles of the Proposed Filter . . . . . . . . . . . . . . . . . 30

4.3.1 Approximation of Densities . . . . . . . . . ,. . . . . . . . . 31

4.3.2 BaskofKalmanFilters . . . . . . . . . . . . . . . . . . . . . 40

4.3.3 Growing Memory Problem . . . . . . . . . . . . . . . . . . . . 41

4.4 Filter Structure . . . . . . . . . . . . . , , . . . . . . . . . . . . . . . 45

4.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Conclusion 54

A Discretization of the Continuous-Time Equation of Motion

B Track Initialization

C Detail Derivations of the Curve-fitting Approach

D Detail Denvation of the 'Randormation Approach

E Reproducing Property of Gaussian Densities

F Bank of Kalman Fiiters

List of Figures

2.1 Flow Diagram for the Bayesian Estimator . . . . . . . . . . . . . . .

3.1 With the factorization the order of the quadrature is reduced for a

given accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Flow Diagram of the Proposed Filter . . . . . . . . . . . . . . . . . . 3.3 Cornparison of the position errors . . . . . . . . . . . . . . . . . . . . . 3.4 Cornparison of the velocity errors . . . . . . . . . . . . . . . . . . . . .

4.1 Plots for the me-fitting approadi . . . . . . . . . . . . . . . . . . . 4.2 Basic Principles of the Proposed Gaussian mixture Approximation

Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Fundamentals of the proposed Gaussian sum approximation method . 4.4 A . The function p ( z , lx-) can be obtained through a nonlinear transfor-

mation from the density p ( w , ) . B . Similady. some initial parameters

fiorn the statistics of the noise W. can be transformed into a Gaussian

sum approximation for the Çiction p(z&) . . . . . . . . . . . . . . 4.5 Plots of the original function p ( z , lx-) and the approximation . . . . . 4.6 Mixture before Reduction (50 components) . . . . . . . . . . . . . . . . 4.7 Mixture after Reduction (10 components) . . . . . . . . . . . . . . . . . 4.8 Flow Diagram of the Proposed Gaussian Sum Filter . . . . . . . . . . 4.9 Cornparison of the position errors for measurement noise level 1 . . . . . . 4.10 Cornparison of the velocity errors for measurement noise level 1 . . . . . . 4.11 Comparison of the position mors for measurement noise level 2 . . . . . . 4.12 Cornparison of the vdouty enors for measurement noise Ievel 2 . . . . .

4.13 Superposition of 100 realizations of experiment based on Extended Kalman

Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.14 Superposition of 100 realizations of experiment based on Extended Kalman

F e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.15 Cornparison of the position errors for 3-D tracking scenario . . . . . . . . 53

4.16 Cornparison of the velocity errors for 3-D tracking scenario . . . . . . 53

C.1 Weighted Gaussian Flnction . . . . . . . . . . . . . . . . . . . . . . . 61

C.2 Aftine Transformation of the eIlipsoidal base . . . . . . . . . . . . . . 62

List of Tables

4.1 Comparison of the accuracy and the efficiency of the proposed Gaussian sum

approximation met hod and the classical method. Note that the classical

method is based on the Marquardt algonthm and the number of Gaussian

terms N is fixed to be 5 for both methods. . . . . . . . . . . . . . . . . . 36

4.2 Comparison of the accuracy and the efficiency of the proposed Gaussian

surn approximation method and the classical method. Note that the

classical method is based on the Marquardt algorithm and the number

of Gaussian terms N is fixed to be 5 for both methods. . . . . . . . . 40

Chapter 1

Introduction

For nearly three decades the target tradcing-trajectory estimation problem has been a

fruitfd applications area for state estimation. It has found wide applications in both

military and cornmerciai areas [l, 21 such as inertial navigation, guidance & control;

global positioning system (GPS); differential global positioning system (DGPS); wide

area augmentation system (WAAS); inertial navigation system (INS) ; missiles guid-

ance system; satellite orbit determination; maritime surveillance; air t r a c control;

fieeway tr&c system; f i e control system; automobile navigation system; fleet man-

agement; underwater target tracking system. Many problems have been solved, yet

new and diversified applications still challenge engineers.

This paper addresses the problem of target tracking in Cartesian coordinates with

polar measurements. In tracking applications the target motion is usually best mod-

eled in a simple fashion using Cartesian coordinates. Unfortunately, in most radar

systems the target position measurements are provided in polar coordinates (range

and azimuth) with respect to the sensor location. Tracking in Cartesian coordinates

using polar measurements is a problern of nonlinear estimation. A ~ ~ O ~ O U S treat-

ment of the nonlinear estimation problem requises the use of stochastic integrals and

stochastic differential equations [3]. In this paper, we d l adapt the formal manipu-

lation of the white noise process and

The nonlineu estimation problem is

omit the ~ ~ O ~ O U S derivations using Ito calnilus.

very challenging because the distribution of the

state is generally non-Gaussian. Without the Gaussian property, efficient computa-

tion of the conditional mean is very difficult because the optimal (conditional mean)

nonlinear estimator cannot be realized with a finite-dimensional implementation; con-

sequently all practicd nonlinear filters are suboptimal. These suboptimal nonlinear

mten can be divided into two categories in general: the most popular technique

uses a Taylor-series expmsion to approximate the nonlinear system model, e.g. the

extended Kalman filter (EKF); the other approximates the conditional probability

density function in such a fashion that makes the computation of the conditional

mean efficient, e.g. the Gaussian sum filter. Both approaches yield computationally

feasible algorithms; however, each has shortcoming that must be addressed. The f is t

approach is efficient but not accurate and often resdts in filter divergence whereas

the second approach yields very accurate result but the complexity is so high that

precludes practical real-time applications.

To improve the performance of the existing approaches, we propose two subopti-

mal tracking algorithms which are believed to be more accurate and efficient than the

most widely used Taylor series methods, e.g. the EKF. The fîrst method uses the mul-

tidimensional Gauss-Hermite quadrature to evaluate the optimal (conditional mean)

estimate of the target states directly fiom the Bayesian equations. This quadrature

technique replaces the integrals in the Bayesian equations with a weighted sum of sam-

ples of the integrand and this approximation can be very accurate if the integrand

is smooth. However, this method has a major drawback, namely its computational

complexity because of the dimension of the problem and the number and the order

of the quadratures involved the number of samples required are usually too large to

be handled in seal-tirne. In this paper we have suggested several ways to reduce the

computational requirements of this quadrature technique by reducing the dimension,

the number and the order of the quadratures required for a given accuracy. The

second tracking algorithm is based on the Gaussian sum filter. The Gaussian sum

approximation method can approximate any probability density function as closely as

desired [4]. Moreover, the optimal estimate (conditional mean) can be computed in a

simple manner as a weighted sum of the estimates from a bank of Kaknan-type filters

operating in pardel. However, this method has a very high complexity since the

number of Gaussian terms required in the approximation increases exponentially in

time. To deviate the computational burden associated with the Gaussian sum filter,

we have found two efficient and systematic ways to approximates a non-Gaussian and

measurement-dependent function by a weighted sum of Gaussian density fimction

and we have derived the formula for updating the parameters involved in the bank of

Kalman-type filters and we also have suggested several ways to control the number of

terms in the Gaussian mixture a t each iteration. Simulation results show that these

two proposed methods are more accurate than the classical method, such as the EKF.

The rest of the paper is organized as follows: ki chapter 2, mathematical models

for target tracking applications and the existing tracking algorithms are discussed.

In chap ter 3, the tracking algorithm based on the multidimensional Gauss-Henni te

quadrature is introduced. Motivation and implementation issues for this technique

are discussed and simulation results are presented also in this chapter. In chapter

4, the tracking algorithm based on the Gaussian sum filter is discussed. Modified

Gaussian sum approximation method is analyzed and several mixture reduction al-

gorithms are presented in this chap ter. Simulation results of this technique and the

pedorrnance of this proposed tracking filter is described. Finally, chapter 5 sumrna-

rizes our conclusions aad future works.

Chapter 2

Problem Definition and Literature

Survey

2.1 Basic Mode1 for Radar Systems

The tracking problem is a state estimation problem, i.e. assuming the state of a

target evolves in continuous time according to the equation

The state vector x( t ) usually contains target position, velocity and sometimes accel-

eration as state variables and its corresponding discrete measurement vector is given

by

where v(t) and w, are system and measurement noise processes respectively and

they are assumed to be zero mean white noise processes. The covariance of the sys-

tem noise, Q, is selected to compensate for modelling errors (disaepancies between

the mode1 and the actual process). The cowiance of the measurement noise pro-

cess sequence, &, should also be chosen to represent a l l possible excursions such as

measurement biases, fdse measurements, etc.

2.1.1 Target D ynamics

Constant Velocity (CV) Model For some targets such as airplanes, a constant

velocity (CV) model is sdicient, i.e. the state vector contains six variables

where (x(t), y(t), z ( t ) ) are coordinates of a Cartesian coordinates system used to

describe the target dynamics. The state equations axe

where vi(t) is the system noise terms used to characterîze modelling errors.

Constant Accelerating (CA) Model If the target being tracked is maneuvering

(accelerating) , a constant accelerating (CA) model is sometimes used, i.e..

The state equations can be written accordingly, i.e.,

xi ( t ) = xi+ i ( t )

( t ) = zif2(t) i = 1 7 4 y 7 1 (2.6)

5i+2 ( t ) = vi ( f )

These two state equations are also referred to as the first- and second-order polyno-

mi al dynamics respect ively.

Target with Sudden Maneuvers Targets with sudden maneuvers can be mod-

elled as systems with abrupt changes [5]. We can modify equation (2.1) to become

two sets of equations, one representing the premaneuver dynamics and the other

incorporating the maneuver feature

where x,(t) is the vector representing maneuvering force and satisfies

fort < t ,

where t , is the tirne the maneuver begins. Tm(-) is the maneuvering dynamics, and

v,(t) is the system noise for the maneuvering dynamics fm(*). For targets with sud-

den maneuvers, tm is unknown, f, (*) and vrn(t) may be unknown or partially known.

The target maneuver x,(t) is correlated in time; namely, if a target is accelerating

at time t, it is likely to be accelerating at time t + r for sufficiently s m d T. For

example, a lazy tuni will often give rise to correlated acceleration inputs for up to

one minute, evasive maneuvers will provide correlated acceleration inputs for periods

between ten and thirty seconds, and atrnospheric turbulence may provide correlated

acceleration inputs for one or two seconds. A typical representative mode1 often used

in airplane tracking is

where a is the correlation constant and va@) is a noise process. Methods for selecting

values for a and statistics for v&) can be found in [6].

Discrete Time Equations of Motion The above target equations of motion must

be fust discretized [7] into appropriate discrete t h e equations in order for the appli-

cation of suitable digital filters [6]. In this paper, we assume the motion of the target

(such as aircrafts , and missiles) generally foIIows a straight-line const ant-velocity (CV)

trajectory. Turns, evasive maneuvers, and acceleration due to turbulence of the sur-

rounding environment may be modeled as perturbations upon the constant-velocity

trajectory. Assume the target is moving in a two-dimensional plane, i.e. the state

vector x, contains four variables

T X n = [%, V s , n , Y n i vg ,n ]

lThe discretization of continuous-time system is presented in Appendix A

where (x, , yn) are the target positions and ( v ~ , ~ , v,,) are the target velocities at tirne

step n. The discrete time target equations of motions are given as

where the target accelerations a , , and a , , are modeled as stationary Gaussian dis-

tributed random processes with zero mean and variance 0:. It is assumed that the

accelerations in x and y directions are mutudy statistically independent. AT is the

radar scan interval.

In matrix notation

where vn is the noise process sequence [ ~ , , a , , ] ~ with zero mean and covariance

matrix Qn (which equals 4). F is c d e d the transition matrix and G the noise gain

matrix.

2.1.2 Radar Measurement

When a radas is used to measure the position of a target moving in the two-dimensional

space, the measurements are reported in range and bearing which are in polar CO-

ordinates. The polar coordinate measurernent of the target position related to the

Cartesian coordinate target state is denoted as

where r, and 8, are the measured range and bearing of the moving target and w,, and

wd,n are the measurernent noise of the range and the bearing; they are assumed to be

zero mean white Gaussian processes with variances and c r i respectively. Moreover,

the range and the beating of the measurement are assumed to be uncorrelated.

In rnatrix notation:

where z, is the vector of polar coordinates memuement [r, 9,jT. h( -) is the Cartesian-

tepolar coordinate transformation. w, is the observation noise process [w,, w,] T

which is assumeci to be zerwnean white Gaussian noise process with c d a n c e ma- * l

The basic problem is to estimate the m e n t state x, based on the sequence of

observations Zn = {%)hl, the statisria of the noise input and the memuernent

and system modek. Generally, one choose the optimal estimate by extremizing some

performance critaion such as the mean-square enor. Regardless of the perf'ce

criterion, given the a podniori density fiinction p(scJZn) any type of esrimate can

be detennioed. Th-, the estimation problem can be approached as the problem of

determining the a posteriori density. This approach is generaiiy refmed to as the

Bayesian approach.

2.2 The Bayesian Approach

In this papa, we dl choose the optima estimate based on the m;n;miim mean

squared ~ O T ( b M E ) criterion. In this case, the optimal estimate G,,, at t h e n

is jus the meaa of the srate b i t y hct ion c o n d i t i d on the meastuement k o r y

Similady, because the measurement z,, depends on the state Xn and a white observa-

tion noise sequence w,,

Utilizing the above property the a posteriori density p(xn 12") can be determined

recursively by the foIlowing Bayesian equations in two stages: prediction and u p

date [9].

estatistic of v, stafistic of w, etarget mode1 measurement mode1

Figure 2.1: Flow Diagram for the Bayesiaa Estirnator

Prediction: Suppose that the conditional density p ( ~ n - ~ [Zn-') at t h e n - 1 is known.

Then using the state equation(2.12) it is possible to obtain the conditional

density of the state p(x, (Zn-') at time n

This equation is cded Chapmaa-Kolmogorov equation.

The probabilistic model of the state evolution, p ( ~ lx,+l) can be defined by the

state equation and the known statistics of vn-l

where 6(*) is the Dirac delta function. The delta function is used because if x,-~

and vn-1 are known, then x,, can be obtained deterministically. Since the noise

sequence vn is assumed to be an zero-mean white Gaussian random process

with covariance matrix Q, and the system dynamics f (*) is linear; therefore the

density p ( ~ , k - ~ ) is also white Gaussian of the form

Update: The conditional density p(xn 12") can be written using Bayes' formula

where the normalizing constant is given by

The conditional density of zn given x,, p(znlxn) is defined by the measurement

mode1 (2.14) and the known statistics of wn

The observation noise sequence wn is white Gaussian and the measurement

equation is nonlùiear; This density becomes

The initial conditional density p(xol Z O ) is given by

where p(xo) is usudy assumed to be white Gaussian 2.

This optimal (conditional mean) nonlinear filter cannot be realized with a finite-

dimensional implementation. It is because the distribution of the state is generally

non-Gaussian; thus, the integration indicated in equations (2.18) and (2.23) is gener-

ally impossible to accomplish in closed form except when the measurement equation

is linear (i.e. the measurement function h(x,,) is linear) and the statistics of the

initial state and the noise sequences are rtll white Gaussian. In that case, the equa-

tions (2.18) and (2.22) can be evaluated in closed form and the conditional density

p(xn(Zn) is reduced to a Gaussian density function. The propagation of its mean and

covariance matrix is collectively known as the K h a n filter equations.

The Kalman Filter Assume the measurement equation is linear

where H, is the measurement matrix and the noise process wn is modeled as a zero-

mean white Gaussian random process with covariance matrix &.

The linear-Gaussian assumption of the mode1 leads to the presenmtion of the

Gaussian property of the conditional density p(x, 1 Zn). Thus, they are completely

characterized by their means and their covariance matrices, which can be ob tained

recursively in two stages: prediction and update [IO].

Updat e

'The track is initialized fiorn thefirst two rneasurements using the algorithm presented in A p

pendix B

where

is called the Kalman gain.

Most of the sub-optimal nonlineu filter are based on the linear Kalman filter

equations by transforrning the nonlineu measurement equation into a linear equation

with white additive Gaussian noise, i.e. forcing the requirements of the K h a n filter

equations satisfied. The next section presents several sub-optimal filters used in target

tracking in Cartesias coordinates with noisy polar measurements.

2.3 Sub-Opt imal Nonlinear Filters for Radar Track-

ing

The most widely used tracking filter is the extended K h a n filter (EKF) [IO, 81 which

employs the fist-order Taylor series approximation to adapt the linear Kalrnan filter

to the nonlinear system described by equations (2.12) and (2.14). Since the state is

in Cartesian coordinates and the measurements are in polar coordinates. Therefore,

there is a nonlinear measurement function h(&) and linearization of the measurement

equation is required for the state and covariasce update. The prediction stage of the

EKF proceeds as in equations (2.28) and (2.29) but the update stage is described as

follows:

where

is the

is the

;iy on

K h a n gain and

Jacobian of the nonlineu function h(xn). Its accuracy however depends heav-

the stability of the Jacobian matrix. In practice, due to the modelling error

the Jacobian matrix is often numerically unstable resulting in filter divergence. This

Taylor series expansion method can be extended further to include higher order t ems

in the expansion for the nonlinear function h(xn) to seek better approximation, such

as the second-order filter [9]. It is however questionable whether higher-order approx-

imations would improve performance in cases where extended K h a n filter (EKF)

diverges [9]. The crucial defect in the approximations is that replace global proper-

ties of a function by its local properties (its derivatives). Clearly, if the true state

lies outside the region in which the nonlinear system is accurately represented by its

Taylor series, not only will the first-order approximation be inadequate but so will

any higher-order approximation.

There exists similar approaches to the EKF which utilizes mixed coordinate filters,

i.e. the state is in Cartesian coordinates and the measurements are in polar coordi-

nate. They a.U share the same filter structure; they are only different in the way their

filter gains Kn are computed. One technique is the iterated extended K h a n fil-

ter [IO]. It improves the accuracy of the extended Kalman filter (EKF) by repeatedly

updating the estimate XnIn, and the kalman gain Kn based on first-order Taylor series

expansion about the most recent estimate. Denote the ilh estimate of x,ln by IinlnVi,

i = 0 , 1 , . . . , N and XnlnVo = îCnln-i* The foIlowing equations surnmaxize the iterated

extended K h a n filter (KF) measurement update:

Similar to the extended K h a n filter (EKF). Its accuracy also depends heavily on

the stability of the Jacobian matrix. If the Jacobian matrix is numerically unstable,

it may result in slowed or even non-convergence of the estimates. A new nonlinear

iterated filter based on the Juiier et al's discrete approximation and the Levenberg-

Marquaxdt algorithm [Il] has been shown to be more accurate and robust than the

EKF and the IKF. There is also a "quasi-extended" Kaknan filter [12], which shows

improvements when tracking maneuvering trtrgets at close range. There is another

technique, modified gain extended Kalman filter (MGEKF) [13], has been shown to

gumantee stability and exponential convergence for a special class of nonlinearities.

Another technique uses the statistical lineaxization as opposed to the Taylor se-

ries expansion [IO] to approximate the nonlinear function h(xJ such that the mean

square enor between the linear approximation and the nonlinear function is mini-

mized. The computational requirements of this method is greater than that of the

extended Kalman filter; however, the performance advantages oEered by this method

rnay make the addi tional comput ation wor t hwhile. There exists similar techniques

which are based on various series expansions, such as Edgeworth and Gram-Charlier

expansion. In some cases, because of the computational constraints, it may be im-

practical to compute the filter gain in real time. Under such conditions one must use

either a set of precomputed filter gains or a constant gain filter. One popular constant

gain filter is a - ,O - 7 filter (or a - ,B filter when using a CV model) [14, 151.

The extended Kalman filter (EKF) uses the polar coordinates measurement . There

is an alternative approach which transforms the polar coordinates measurements to

the Cartesian coordinate systems. Then the conventional Kalrnan filter is applied.

This approach is generally called the converted measurement Kalman filter (CMKF).

With the conver ted measurement Kalman filter [16], the polar coordinate measure-

ment zP, is first converted to the Cartesian coordinate measurements zC, using the

inverse trassformation h-'(z~) The original noise process w, acting on the converted

measurement zn no longer behaves rigorody as an additive t e m , but in some corn-

plicated fashion. However, at le& when the covariance of the noise wn is s m d , the

new Cartesian coordinate measurement equation can be written as follows:

where D = 1 1 O 0 0 1

and wn is

=Dx,+W, (2.41)

approximated as a white Gaussian noise process

4

on the convched measurement 2: with zero mean and covariance matrix M n

Mn = E [ W ~ + ; ~ ] = Jb-1 (Pnln-i)&Jb-l T (2.42)

where

As a result, the new measurement equation in Cartesian coordinates becomes linear

and the noise process is Gaussian, the standard Kalrnan flter can be applied to the

problem with the measurement equation defined in Equation (2.41) and the statistics

of the measurement noise given as in Equation (2.42). This method however is an

acceptable approximation only for moderate cross-range errors (cross-range error is

defined as the product of the range and the bearing angle). An improved method

using debiased converted measurements [17] which accounts for the sensor inaccu-

racies over all practical geometries and acmacies and provides consistent estimates

(i.e. compatible with the filter calcukted covariances) for all practical situations.

There is another technique [2] which preprocesses the transfonned measurements to

reduce measurement uncertainty and increase state estimation acniracy by explicitly

employing the knowledge available about taiget motion and trajectory.

So far we have discussed various algorithms based on first-order approximation

techniques. For the rest of the section , we will present several well-known nonlinear

algorithms based on global approximation techniques. These nonlinear algorithms are

no t cornmonly used for real-the applications, because the computation time and data

storage requirements are too excessive. One global approach is to approxirnate the

densities directly in such a manner that makes the integrations involved in Bayesian

equations (2.22) and (2.18) as trackable as possible. The simplest method is the point-

mass method [18], where the densities are approximated by point masses located on

a rectangular grid. As a result, the integrations in the Bayesian equations can be

accomplished numerically as a discrete nonLinear convolution which requires O ( p )

operations to cornpute, where N is the number of grid points. An improved method

approximates the densities with a piecewise constant b c t i o n [19] and this approxi-

mation is slightly more sophisticated than the point mass method, but it converts the

Bayesian equations to a discrete linear convolution which requires only O(N log N)

operations. There is another technique which uses a Gaussian-sum approximation

for the density [4]. In this case, the Bayesian equations become a bank of K h a n

filter equations. However, the number of components in the approximating sum grows

exponentiaily with time. More sophisticated interpolation schemes such as different

spline approximations [20, 21, 22, 231 which require fewer grid points for a given

accuracy are also studied. There exists alternative technique, called the bootstrap

filter [24], which represents and recursively propagates the required density function

as a set of random samples, rather than a function over state space. As the number

of samples becomes very large, they effectively provide an exact, equivalent, repre-

sentation of the required density function. However, the number of sampies required

to give a satisfactory representation of the densities for the filter operation is very

difficult to determine. Another global approach is to approximate the integrations

involved in the Bayesian equations directly. All such methods amount to replacing

the integral with a weighted sum of samples of the integrand based on some particular

quadrature formula [25, 26, 271. The basic issue again is trading off computational

complexity against the numbes of grid points required to achieve a desired accuracy.

Fuzzy logic and neural network have been considered extensively for t ~ g e t track-

ing recently [28, 291. An fuzzy-Khan estimator [30] which uses a fuzzy system to

estimate the current t aqe t maneuvering acceleration and a KaIman filter to estimate

the target position is studied. The resulting estimator would keep the features of the

Kalman filter but with improved accuracy if good estimates of target maneuvering

acceleration can be produced. Similarly, neurofuzzy estimators [31] are used to ini-

tialize the Kahan Bter and extended K h a n filter. It is shown that flters which

have been initialized with neurofuzzy estimates converge faster and give irnproved

performance. Fuzzy logic is also introduced into a conventional a - P constant gain

filter [32] in an attempt to improve its tracking ability for maneuvering targets.

Chapter 3

An Efficient Radar Tracking

Algorit hm Using Multidimensional

Gauss-Hermite Quadratures

3.1 Introduction

In this chapter, a new tradcing algorithm based on multidimensional Gauss-Hermite

quadrature is presented. This quadrature technique evaluates the optimal (condi-

tional mean) estimate by replacing the integrals involved in the Bayesian equations

with a weighted sum of samples of the integrand and this approximation can be very

accurate if the integrand is smooth. However, this technique requires excessive corn-

putational time and data storage, because of the dimension, the number and the

order of the quadrature involved the number of samples required are usually too large

to be handled in red-the. In this chapter, we suggest several ways to reduce the

computational requirements of this quadrature technique by reducing the dimension,

the number and the order of the quadratures required for a given acniracy.

3.2 Gauss-Hermite Quadrature

Based upon the orthogonality property of the Hermite polynomials, one can develop

the Gauss-Hermite quadrature (331.

where Bi are the weights and tii are the roots of the Hermite polynomial of degree k

and they are chosen so that the sum of the k appropriately weighted functional values

yields the integral exactly when f (u) is a polynomial of degree k or less.

The above one-dimensional quadrature formula can be easily extended to the

more general multi-dimensional quadrature formula [34]. Consider the evaluation of

the following multidimensional integral

where r and P are n-dimensional vectors and P is a n x n nonsingular matrix. If

P is a positive diagonal matrix, then the multidimensional quadrature formulae by

Stroud [34] may be applied, which is obtained by successively applying the one di-

mensional Gauss-Hermite formula. If P is not diagonal, an affine transformation of

variables can make P diagonal. One approach is to h d the eigenvector decomposition

of the matrix P:

where D is a diagonal matrix of eigenvalues and V is a full matrk whose col-

are the corresponding eigenvectors. The expression

achîeves the diagonalkation. A simples approach is to fmd the square-root of P, W

using the Cholesky algorithm (351 such that

and the expression

also achieves diagonalization. Then the multidimensional integral in equation (3.2)

can be evaluated numerically by applying the one dimensional Gauss-Hermite quadra-

ture formula to one variable at a time and it becomes

where

where ri, ,.-- ,iN are the N-dirnen~iona.1 grid points and Bi, ,... ,,, are the corresponding

weights. Bi,(m = 1,. . . , N ) and uim(rn = 1,. . . , N) are the weights and the grid

point for the one dimensional Gauss-Hermite quadrature fkom equation (3.1). The

grid r is c d e d a floating grid and it is centered at the m e n t mean 2. The square-root

approach has the advantage over the eigenvector approach in that the computation of

the square root of a matrix is faster than computation of the eigenvectors of a matrix.

The quadrature formula in equation (3.7) involves the summation of kN weighted

functional values, where N is the dimension of the quadrature and k is the order of

the quadrature. For hi& dimensional problems it is desirable to keep the order of

quadrature s m d in order to reduce the complexity of the quadrature. However, the

result will not be acnirate if the function f(e) cannot be weU approxîmated by a low

order algebraic expression. In this case, one may factor f (-) into two components: one

is nearly algebraic and the other being Gaussian. The Gaussian component c m then

be combined with the original Gaussian weighting function to obtain a new Gaussian

weighting function.

3.3 Basic Principles of the Proposed Filter

The central issue here is to compute the optimal (conditional mean) estimate based on

the Bayesian equations accurately and efficiently using the multidimensional Gauss-

Hermite quadrature technique. To reduce the number of quadratures required, ana-

lytic results are applied to the prediction stage ( t h e update), and the application of

quadrature techniques is restricted o d y to the update stage (measurement update),

because the equation of target motion (2.12) is linear and the measurement equation

(2.14) is nonlinear. Assume at time step n the mean &-lln-l and the covariance

mat& Pn-lln-i of the conditional density p(~,_~lZ"-') is known. Then we approx-

imate the distribution of the state prediction p(xnlZn-l) as a Gaussian distribution

with mean f and covariance matrix Pnln-l.

This algonthm rests on the assumption that the state prediction density is Gaussian

at each step. This assumption is closely satisfied in many situations of practical in-

terest, in particular if the state noise vn is Gaussian [36].

Then the conditional mean (optimal estimate) kIn and the conditional covariance

Pnln can be determined fiom the Bayesian equations as follows:

The above integrations generally cannot be accomplished in dosed form because the

densities of the state are non-Gaussian, To make the dculations of the conditional

mean and the conditional covariance possible and &cient, we replace the integrak

involved in the Bayesian equations with a weîghted sum of samples of the integrand

based on the Gauss-Hermite quadrature formula. This requires the evaluation of the

integration of the form

as the Gaussian weighting function, but the remaining expression

is not algebraic. To determine the d u e of the integral in equation (3.15) accurately

with &(xJ as the Gaussian weighting function wiU require a high-order quadrature

formula. However, if the expression F2(xn) is factored into two expressions: one is

Gaussian, F:(x~) ; the other is nearly algebraic within the desired region, Fi(xn)

and a new weighting function determined by Fl(xn) F!(x.) is used, then the same

accuracy can be obtained with a lower order integration formula (see Figure 3.1).

To determine F;(x,) and F;(xn), we use the linear approximation to the function

h(xn )

where the matrices H and Ho can be determined either by first-order Taylor series

expansion or the statistical linearization. The error of this linear approximation is

then given by

Figure 3 L: With the factorization the order of the quadrature is reduced for a given

accuracy. For example, in A, if F2(x) is the weighting function, the number of grid

points required to cover the integrand Fi(x) completely will be 7; however, if the

integrand Fi (x) is factorized and the new weighting function is F2(x) x F: (x), then

the number of grid points required is reduced to 4 as shown in B.

The expression Fz(xn) can be rewritten as

where

As a result, the new weighting function determined by F ( x ~ ) = F~(x,) Fi@,)

becomes

where

These equations are exactly the linearized Kalman filter equâtions and the constant

term C2 is not necessary to evaluate as it wiU be canceled later.

Applying the multidimensiond Gauss-Hermite quadrature formula to the integral

in Equation (3.15) with F(xn) in Equation (3.20) as the new weighting function and

A(xn) F;(xn) in Equation (3 .l9) as the new integrand, the integral in Equation (3 -15)

becomes

where

w,w,T = Pnln (3.25)

&,il ,... ,iN = Wnuil,... , i ~ f %ln (3.26) T

Uil ,... ,ir = [uil 1 t uiN] (3.27)

Bil, ..., iN = IwnlBil **BiN (3.28)

where Wn is the square root of PnIn from the Cholesky algorithm; ~, , i~ , . . . , i , is a

N-dimensional grid point which is constructed from N one-dimensional grid points

uij(j = 1, . . . , N) and Bi,,... ,i, is the corresponding weight which is just the product

of N ondimensional weights Bij(j = 1,. . . , N). Utilizing the results from Equation

(3.24), the conditional mean (optimal estimate) and the conditional covariance

3.4 Complexity Analysis

This tracking algorithm basicdy consists of two parts: the first part is based on

the linearized K h a n filter to obtain a rough estimate of the conditional mean and

the conditional covariance; the second part is based on the multidimensiond Gauss-

Hermite quadrature formula to compensate for any error introduced from the linear

approximation.

The evaluation of the initial grid points ui,,...,iN and its corresponding weights

Bi,,... ,i, in equation (3.27) and (3.28) requires the most computations; they are de-

termined from a set of N one-dimensional grid points uij and weights Bij. Fortunately,

these initial N-dimensional grid points and the weights can be computed off-line. As-

sume the dimension of the quadrature is N and the order of the quadrature is k. The

computationd requirements at each iteration are as follows:

1. the calculation of the square root W, of PnIn from the Cholesky algorithm

(3.25) requires 0 ( N 2 ) operations;

2. the affine transformation (3.26) requires Na scalar multiplications and N2 scdar

additions for each grid points ui, ,... , iN.

3. the caldation of the conditional mean 1î,(, and the conditional covariance Pni,

requires kN evaluations of Fi(-), (fl + N + 1) kN scalar multiplications and

(N2 + N + 1) (k - l)N scalar additions.

The whole algorithm requires excessive amount of additions and multiplications and

data storage.

3.5 Filter Structure

The block diagram of the proposed filter is presented in Figure 3.2. It consists of the

foIlowing t hree stages :

Stage O: Previous estimates:

Assume at step n the previous mean and covariance estimates X, and PnI, aze

known.

Figure 3.2: Flow Diagram of the Proposed Filter

Stage 1: Linearized Kalman filter:

The new mean f and the new covariance Pnln are first estimated by the

linearized Kalman iîlter equations.

Stage 3: Error Compensation:

To compensate for the errors introduced in the linear approximation in Stage 2

multidimensional Gauss-Hermite quadrature is used to evaluate the conditional

mean Xn and the conditional covariance PnI,

3.6 Simulation Results

To compare the performance of our proposed filter with that of currently popular

approximate filters a two-dimensional target tracking application described by the

system equation (2.12) and the measurement equation (2.14) with the following pa-

rameters is simtdated.

The following algorithms are used in the simulation: the extended Kalman Nter

(EKF), the converted measurement Kalman filter (CMKF) and proposed quadrature-

based fiiter. AU these filters require an initial filtered estimate hlo and an initial error

covariance Polo. They are estimated fiom the first two measurements using the al-

gorithm presented in Appendix B. The dimensions and the order of the quadrature

we used is 4 and 16 respectively, i.e. the proposed filter has a computational com-

plexity of order 16~. The results presented in Figures 3.3 and 3.4 are based on 100

measurements averaged over 500 independent realizations of the experiment with the

sampling interval of one second and the initial state xo is chosen randomly for each

realizations of the experiment.

The proposed filter is compared with the weU-known classical filters, e.g. the EKF

and the converted measurement Kalman filter (CMKF). The position errors and the

velocity errors for each filters are shown in Figs. 3.3 and 3.4 where the error is defined

as the root mean square of the difference between the actual value and the estimated

value. Our proposed method converges faster and yields results of smaller error than

the EKF and the converted measurement Kalman filter (CMKF) does whereas the

EKF on average diverges due to the instability of the Jacobian matrix. Although this

quadrature-based filter does yield more accurate estimates, the computations and the

data storage it required are still too excessive. To reduce this computational burden,

we must investigate some other means to represent the densities such that the optimal

estimate can be computed in a simple and efficient fashion.

.... ................:.........:......... i... h .

Figure 3.3: Cornparison of the position errors

Figure 3.4: Cornparison of the vdoaty errors

6000

5000

4000 n E E Y

I I I I I I I 1 I

I ......... ...... ..... ...... i ..-.....: r . ........; ..-.. ..: ..-

! i ! I ....-..* ...... ........ 4 ....-..-; .--...-..; ;- -=. - ! i

L

.................. ........ ........ ......... .........

S O - P

................ L.........,........-.....*..~........-........I............................,

...... ........ ...... L I *..: ........-

*-.--.:--.-.--.-:--.--..-.:*.----.-* i

O 10 20 30 40 50 60 70 80 90 100 "lime

Chapter 4

Adapt ive

for Radar

Gaussian

Tracking

4.1 Introduction

Sum Algorit hm

R e c a that it is generaily impossible to accomplish the integration involved in the

Bayesian equations in closed form when the measurement equation is nonlinear. This

problem leads to either the numerical evaluation of the conditional densities or the

investigation of density approximation for which the required integration can be ac-

cornplished in a straightforward manner. Zn this chapter, the Gaussian sum approx-

imation method is used to approximate the conditional densities, such as p(xnlZn)

such that the integration indicated in equations (2.18) and (2.22) can be accomplished

in closed form, i.e. the optimal estimate can be computed analyticdy. The Gaus-

sian sum approximation method can approximate any probability density hinction

as closely as desired. Moreover, it is compnsed of a bank of Kalman-type &ers

operating in pardel with each individual estimate possessing its own corresponding

weighting term; hence, the optimal estimate can be computed efficiently as a weighted

sum of the estimates from each Kalman-type filters.

4.2 Theoretical Foundations

Lemma: Given any probability density function p(x), it is possible to approximate

it as closely as desired in the Hilbert space &(Rn) by a linear combination of Gaussian

densi ties.

N

P A ( X ) = C d ( x -mi, Bi)

where

and the weighting factors ai have the constraints:

- m i ) T ~ y ' ( ~ - mi)] (4.2)

and mi is a mean vector and Bi is a positive definite covariance matrix. For a proof

of this result, see Sorenson and Aspach [4].

These properties insure t hat the approximation is always positive and integrates

to unity. This approximation is thus a true probability density function in its own

right and when this representation is used the Bayesian recursion relations can be

solved exactly in closed form.

4.3 Basic Principles of the Proposed Filter

To evaluate the state prediction density lZndL) efticiently fiom equation(2. la), we

will assume the conditional density p(xna1 12"-') to be Gaussian with mean îCn-lln-l

and conriance matrix Pn-lln-L- Based on this assumption, the equation(2.18) can

be evaluated in closed form and the resulting state prediction density p(xn is

a Gaussian density function with mean and covariance mat& Pn!n-l.

Even though the density ~ ( 2 ~ 1 % ) is a Gaussian density function with mean h(x,) and

covariance matrk &, the integration involved in equation (2.22) cannot be accom-

plished in closed form, because the integration involved is done with respect to x,, not

zn. The density p(z. lx,) when expressed in tenns of x, is a non-Gaussian function.

The word function is used instead of density function because the density p(znIxn)

when expressed in terms of x,,, denoted as p(zn lx,), has all the properties of a density

b c t i o n , except it doesn't have an area of unity, i.e. jTw p(zn IxJdx, f 1. Since the

function p(z , IL) is non-Gaussian, the integration involved in equation (2.22) cannot

be accomplished analytically. The Gaussian sum approximation method is used to

approximate this function such that the integration involved in the Bayesian equa-

tions can be accomplished in a simple manner, and the optimal state estimate hr,nln can be computed analytically and efficiently as a weighted sum of the estimates from

a bank of Khan-type filters.

4.3.1 Approximation of Densities

Since the function p(z,lx-) has all the properties of a density function, except it

doesnyt have an area of unity, the Gaussian sum approximation may be modified such

that it can be used to approximate this b c t i o n with a weighted sum of Gaussian

density function, where the sum of weights is equal to the area of the function, instead

of unity.

N w k e Ckl an,i = p(zn lx,)& and D = [: : : :]. It isbonvcruerydiB-

cult to choose the %bestn parameters an,i, mi and Bi efficiently because the function

p(znl&) depends on the current d u e of the measurement 2,.

Classical Optimization Approach

This approach selects the parameters Bn,i, mi and an,i so that the Lk n o m be-

tween the actud function p(zn i s ) and the Gaussian sum approximation p c s ( ~ ) is

As the number of terms N increases and as the conriance Bns decreases to zero, the

n o m between the actual density and the approximation must mnish. However,

for finite N and nonzero comriance matrix, it is reasonable to attempt to minimize

the Lk n o m such as this. In doing this, the stochastic estimation problem becomes

a deterministic optimization problem.

To actually carry out this optimization procedure, we m u t obtain a set of equally

spaced samples fiom the function p(zn lxn) and fiom the samples we determine the

"bestn sum of Gawians [37]. Thus, the problem becomes to choose the parameters

m,,i and an,,- such that

where {xnj : j = 1,. . . , K ) is the set of uniformly spaced points and E is the pre-

scribed accuracy. Note that in this approximation, the number of Gaussians used, N,

&O has to be determined, and it is desirable to fmd the smallest N possible.

Assume the dimension of problem is m and the L2 n o m is used; then a Gaussian

te- determined by parameters B.,i, mnti and kVi has (m + l ) (m + 2)/2 unknowns.

For a known number of Gawian terms N, this optimization problem is equivalent to

that of solving a system of N(m + 1) (m + 2)/2 nonlinear equations by Ieast-squares

method using K » N(m + l)(m + 2)/2 data points. To solve a system of nonlinear

equations, we can use the Marquardt algorithm (381. In this algorithm, starting fkom

an initial set of parameters, the parameters are iteratively modified until a local mini-

mum is reached in the sum-of-squared enor between the data and the approximating

This approach is not very suitable for the radar tracking applications, because

the function p(z&) we want to approximate depends on the current value of the

measurement 2,. Thus, the optimization procedure indicated in equation (4.7) must

be accomplished on-line for each new measurement, which is impossible because the

dimensions of the problem and the number of parameters involved are too large to be

handled in real-time. Consequently we propose two efficient and yet accurate methods

to approximate the function p(zn IL).

Curve-fitting Approach

Our method utilizes the special symmetric geornetry of the function p(znIx-) to select

the parameters an,i, m,,; and Bric to obtain the near-optimal Gaussian sum approxi-

mation for the function p(z,lx-). The function p(z,,(xJ has a heavy-tailed structure

and its projection onto the x,,-plane has a crescent shape. If we can fit this crescent

shape with some ellipses geometricaily, then we can represent the function p(znl&)

as a sum of Gaussian mixture obtained fiom those ellipses, because the projection of

a Gaussian function onto the x.-space is an ellipse (see Figure 4.1).

Figure 4.1: (a)Plots of the fundion p(zn 1s) and (b) its projection

Figure 4.2: Basic Principles of the Proposed Gaussian mixture Approximation

Method

Step O: Defme the contour of the projection as shown in Figure 4.lb. Eere we as-

sume the region outside three standard derivations of the density p(zn lx,) has

negligible probability; i.e.

This contour when plotted in terms of z, is an ellipse, but when plotted in terms

of xn is the shape shown in Figure 4.lb. Define Pt. 1 be the measurement 2,;

Line 1 be the h e joining Pt. 1 and the origin; Line 2 be the line perpendicdar

to Line 1 and passing through Pt. 1.

Step 1: Denote the four interceptions of Contour C, Line 1 and Line 2 (see Fig- ure 4.2a) as Pt. la-d. Then the fist Gaussian term is defined to have a

ellipsoidal base bounded by Pt. la-d and a height equal to the value of the

function p ( z , 1%) evaluated at Pt. 1. Or the first Gaussian term can be defined

by the foliowing weight al, mean mi and covariance matrix BI: ':

where

G r = p(z, (Dx,, = mi)

kl = 9/(9 + 2 1n(2rrlR,J1/*ii~))

01, = \Pt. 1-Pt. la(/S

Q L ~ = (Pt. 1 -Pt. lb1/3

Step 2: Define Arc A be the arc r = rn; Line 3 be the line joining Pt. l a and the

origin; Line 4 be the line perpendicular to Line 3 and passing through Pt. 2,

where Pt. 2 is defined as the interception of Line 3 and Arc A. Repeat Step

1 with Line 1 replaced by Line 3, Line 2 by Line 4 and Pt. 1 by Pt. 2 and

we can obtain the second Gaussian term (see Figure 4.2b). The rest Gaussian

terms are obtained in a similar manner. This algonthm is terminated either

when the number of terms exceeds a certain prescribed value or when Pt. 2 is

close to the tips of Contour C (see Figure 4.2d).

Numerical Exam~les

We compare the acniracy and the efficiency of the proposed approximation method

with the classical optimization method in approximating the function ~ ( z , lx,) with

a sum of Gaussian terms. We assume the noise w, is a white Gaussian process

sequence with zero mean and covariance matrix R, = 1; i ] . ~ e h t h e r u s u m e

lThe detail derivation of this curve-fitting approach is praented in Appendi C

the measurement z, is equal to 1 rn = lokm 1. The cornparison ir basecl on t h e e O,, = n/4rad

Heren t scenarios: in Case 1 or = 10m and os = 0.lrad; in Case 2 or = 20m and

cg = =.?rad; in Case 3 or = 50m and 00 = 0.25rad. The simulation results are

Case 2

% error

Case 1

Case 2

Classical method

1.21

Case 3

processing time (s)

Case 1

Proposed method

4.66

Table 4.1: Cornparison of the accuracy and the efficiency of the proposed Gaussian sum

approximation method and the classical method. Note that the classicai method is based

on the Marquardt algonthm and the number of Gaussian terms N is fixed to be 5 for both

methods.

1.76

3.5641e3

Case 3 I 3.4165e3

sumrnarized in Table 4.1. They show that the proposed Gaussian sum approximation

method is very efficient but yet yields accurate results comparable to the classical

optimization method.

4.78

1.59

1.49

Transformation Approach

Recall that the fimction p(zn lx-) can be defined by measurement equation (2.14) and

the known statistics of wn

where 6(-) is the Dirac delta function. The delta function is used because if x,

and w, are known, then x. can be obtained deterministicaily. Thus, the function

p(zn lx-) can be obtained by applying the transformation w, = 8, - h(x,) to the

density function p,(w,) (see Figures 4.4A and 4.3A). Utilizing this observation, we

select some initial parameters innIi and Bn,i from the known statistics of the

noise w,, and transform these parameters from the wn-space to the xn-space based

on the transformation w, = z, - h(x,,) and fmally collect them as a Gaussian sum

approximation for the function p(z&) (see Figures 4.4B and 4.3B). This approach is

more efficient than the classical approach because most of the computations is involved

in the initialization which can be done off-line; the only computation required at each

i teration is for updating the initial parameters.

Figure 4.3: Fundament als of the proposed Gaussian sum approximation met hod

Step O: For initialization, select the parameters mnIi and B , , ~ for a prescribed

d u e of N such that the following sum-of-squared error is minimized.

where {wnj : j = 1,. . . , K) is the set of uniformly spaced points distributed

through the region containing non-negligible probability and E is the prescribed

accuracy.

Figure 4.4: A. The function p(zn lx,) can be obtained through a nonlinear trans-

formation from the density p(w,). B. Similady, some initial parameters fiom the

statistics of the noise wn can be transformed into a Gaussian s u m approximation for

the function p(z, IL)

Step 1: For each new measurement z,, update the new parameters anl i , m,,i and B n 1 i

such that

where

m n , i = Zn - m n , i

2The detail derivation of this transformation approach is presented in Appendix D

Numerical Examples

We compare the accuracy and the efficiency of the proposed approximation method

with the classical curve-fitting method in approximating the fimction p(z , lx,) with

a s u of Gaussian terms. We assume the noise w, is a white Gaussian process se- c

quence with zero mean and covariance matrix R, = 1 0 1 where ur = lûm anci

r i

00 = O.Irad. We further assume the current measurement z, = 1 '" 1 is known

where r. = lkm and 8, = rf4rad.

(a) Original function p(z.l&) (b) Gaussian Sum Approximation

Figure 4.5: Plots of the original function p ( z , 1%) and the approximation

The simulation results are summarized in Figure 4.5 and Table 4.2. They show

that the proposed Gaussian sum approximation method is very efficient but yet yields

accurate results comparable to the classical optimization method.

4.3.2 Bank of Kalman Filters

% error

processing time (s)

Either using the curve-fitting approach or using the transformation approach, we wiU

end up with a Gaussian sum approximation for the density ~ ( z ~ I x ~ ) If we substitute

this approximation into equation (2.22) and complete the squares 3, the a posteriori

density p(xn 1 Zn) becomes

where

Table 4.2: Comparison of the accuracy and the efficiency of the proposed Gaussian

sum approximation method and the classical method. Note that the classical method

is based on the Marquardt algorithm and the number of Gaussian tenns N is fixed

to be 5 for both methods.

Classical method

1.21

3.5641e3

3The detail derivation is presented in Appendix E

Proposed method

2.23

0.17

4.3.3 Growing Memory Problem

Inherent in the Gaussian sum algorithm is a serious memory growing problem that

causes the number of tems in the Gaussian sum ta increase exponentidy at each

iteration. h this section we will present two approaches to alleviate this prob

Equivalent Gaussian Approach

For simple applications the Gaussian sum approximation of the density p(xn

em.

Zn) is

collapsed into one equivdent Gaussian term [39]. Consequently, the total num ber

of Gaussian terms is fixed to be N . The mean and the covariance matrix of the

equivalent Gaussian term, denoted XnIn and Pnl, respectively, are given as follows:

This method is simple asd elegant and it usually yields very accurate estimates, be-

cause if the a posteriori density p(xn1Zn) can be approximated very closely by a sum

of Gaussian density functions, then its mean can be estimated very accurately as a

weighted sum of the means of those Gaussian density functions. Furthemore, to

control the number of terms in the approximation we collapse the Gaussian mixture

into one equivdent Gaussian term at the end of each iteration. However, by doing

so we actually collapse the non-Gaussian nature of the a posterion' density p(x,lZn)

into a Gaussian density function, which of course can senously degrade the perfor-

mance of the filter. Hence, it w u be desirable to reduce the number of terms in the

approximation but at the same time retain the general structure of the distribution

of the a posteriori density ~ ( x ~ I Z " ) . This approach is called the mixture reduction

approach.

Mixture Reduction Approach

For complex applications the number of Gaussian terms in the approximation is re-

duced to some specified limit M for M < N. As a result, the total number of Gaussian

terms is fixed to be N - M 4. This approach results in another Gaussian sum approx-

imation with lesser terms without modifying the structure of the distribution beyond

some acceptable limit and it should preserve the mean and the covariance rnatrix of

the original Gaussian sum. Suppose that the nurnber of onginal Gaussian sum ap-

proximation is reduced by rnerging several terms together. If c is the set of Gaussian

terms to be merged, then in order to preserve the mean and cotariance matrix of the

original Gaussian sum, the weight the mean inIn, and the covariance matrix

PnI, , should be chosen [40] as follows:

Ideally the partition of the Gaussian terms for merging should be such that some

cost function is minimized. However, to reduce the Gaussian sum fkom N terms to

M tems this could involve the evaluation of the criterion for every possible partition

to identify the minimum. Such a procedure would be fax too time-consuming; so a

suboptimal approach has been adapted, in which a pair of Gaussian terms are merged

at every iteration of the algorithm to minimize some chosen criterion.

The Joining Algorithm

This algorithm uses the Mahalanobis distance [40] as the cost function

' Y 3 j Gj = - (4 - 4)T(~i + Pj)-'(Xi - 'Yi + 7j

where 3, xi and Pi are the weight, the mean and the covariance mat& of the i th

Gaussian term; rj, Xi and Pi are the weight, the mean and the covariance matrice of

the j th Gaussian tenn. At each iteration, the two Gaussian terms which is closest in

the sense of the Mahalanobis distance are combined to fonn a new Gaussian dehed

'When this mixture reduction approach is used, the Gaussian sum fiiter equations have to be

modifieci. The modification is presented in Appendix F

by equations (4.30- 4.32). To implement this algonthm a symmetric matnx contain-

ing the distance between every pair of Gaussian tenns in the onginai Gaussian sum

is evaluated. This rnerging should proceed until the minimum distance exceeds some

specified threshold. At each iteration, the smdest element of the matrix is found and

the corresponding pair of components are merged. This algorithm has a complexity

of order N2 where N is the number of components in the original Gaussian sum. It

can fiuther simplified by merging the tems with the smallest and the second s m d -

est weights together at each iteration until the number of remaining terms has been

reduced below the specified value M. This method works because the te= with the

smdest and the second smdest weights cary the l e s t important information; thus,

merging them will not m o w the original mixture very much. This simplified joining

algorithm has a complexity of order N.

The Clustering Algorit hm

This algorithm is based on the K-means clustering algorithm [41] which partitions

the original Gaussian mixture into M clusters such that the Mahalanobis distance

between the Gaussian tenns and each cluster center is minimized. Each cluster is

then approximated by a single Gaussian defmed by equations (4.30-4.32).

Modified K-means Clustering Algonthm

0 Choose randomly M of the Gaussian terms as the centers of the cluster cl,. . . , CM.

0 Vlhile there are no changes in the position of the cluster centers.

- For each cluster

* Use the cluster centers to classify the Gaussian terms into chsters by

evaluating a N x M matrix D with the entry dG dehed as the Ma-

halanobis distance between the ith Gaussian term and the j th cluster

center cj. To classify the ith Gaussian term into clusters for instance,

we must fmd the mallest element on the ith row of the m a t e D and

the column that element belongs to corresponds to the cluster center

that Gaussian term should go.

* Update the new cluster centers by collapsing each cluster into one

single Gaussian using equations (4.30-4.32).

- end For

O end While

One drawback of this algorithm is that the results produced depend on the initial

values for the clus ter cent ers, and it fiequent ly happens t hat subop timal partitions

are found.

Numerical Examples

To demonstrate this clustering algorithm we generate a Gaussian mixture with 50

randomly chosen components and then apply the algorithm to reduce its number of

components to 10. Figure 4.6 shows the onginal Gaussiaa mixture and Figure 4.7

shows the Gaussian mixture after applying the dustering algorithm.

Figure 4.6: Mixture before Reduction Figure 4.7: Mixture after Reduction (10

(50 components) components)

4.4 Filter Structure

The blodc diagram of the proposed Gaussian sum filter is presented in Figure 4.8. It

consists of four stages:

Gaussian Sum Approximation

Figure 4.8: Flow Diagram of the Proposed Gaussian Sum Filter

Stage O: Previous estimate

Assume at time n the Gaussian mixture approximation of the conditional den-

s i s P(X,-~ (Zn-') is known.

Stage 1: Gaussian Sum Approximation

The density p ( z , lx,) is approximated systematically by a weighted sum of Gaus-

sian terms either by the curvefitting approach or the transformation approach.

Stage 2: Bank of Kalman Filters

The Gaussian mixtures fiom Stage O and Stage 1 are passed to a bank of N x

M K h a n filters which evaluate the new Gaussian 'mixture for the density

P(X~ Izn)-

Stage 4: Mixture Reduction

To control the number of Gaussian te=, the density p(xJZn) is either col-

lapsed into one equident Gaussian term or reduced to M Gaussian terms by

the joining algonthm or the clustering algorithm.

4.5 Simulation Results

To compare the performance of our proposed filter with that of currently popular

approximate filters two different scenarios are considered.

Scenario 1:

A two-dimensional target tracking application described by the system equation (2.12)

and the measurement equation (2.14) with the following parameters and two different

noise levels is simulated.

The following algorithms are used in the simulation: the extended Kahan filter

(EKF), the converted measurement K h a n filter (CMKF) and the proposed adap

tive Gaussian sum filter (AGSF) which uses the Transformation approach in the

Gaussian sum approximation and the Equivalent Gaussian approach to control the

number of terms in the approximations. All these filters require an initial Qtered

estimate hlo and an initial error covariance Polo. They are estimated fiom the fmt

two measurements using the algorithm presented in Appendix B. The number of the

Gaussian terms we used in the approximation is 9 and they are collapsed into one

equivalent Gaussian at the end of each iteration. The result s presented in Figures 4.9-

4.12 are based on 100 measurements averaged over 500 independent realizations of the

experiment with the sampling interval of one second and two different measurement

noise levels and the initial state xo is chosen randomly for each realizations of the

exp eriment .

The proposed filter is compared with the well-known classical filters, e.g. the ex-

tended Kalman filter (EKF) and the converted measurement K h a n fiter (CMKF).

The position errors and the velocity errors for each iîlters for each measurement noise

levels are shown in Figs. 4.9, 4.10, 4.11 and 4.12 where the error is defmed as the

root mean square of the difference between the actual value and the estimated value.

Our proposed method converges faster and yields estimates of smder error than the

EKF aod the converted measurement K h a n filter (CMKF) does in a.LI cases. Note

that the extended Kalman filter (EKF) diverges in only some of the cases due to the

instability of the Jacobian matrix as shown in figure 4.13 and 4.14.

Scenario 2:

A threedimensional target tracking application described by the system equation

(2.12) and the measurement equation (2.14) with the foilowing puameters is simu-

lated.

xn+~ =

The following algorithms are used in the simulation: the extended Kaknan filter

(EKF), the converted measurement ICalman flter (CMKF) and the proposed adaptive

Gaussian surn filter (AGSF) which uses the Transformation approach in the Gaussian

sum approximation and the Mixture Reduction approach to control the number of

terms in the approximations. All these filters require an initial filtered estimate

and an initial error covariance Polo. They are estimated fiom the first two rneasure-

ments using the algorithm presented in Appendix B. The number of the Gaussian

terms we used in the approximation is 49 and they are reduced to 25 equivalent

Gaussian terms at the end of each iteration. The results presented in Figures 4.15

and 4.16 are based on 100 measurements averaged over 10 independent realizations

of the experiment with the sampling interval of one second and the initial state xo is

chosen randomly for each realizations of the experiment.

The proposed filter is compared with the well-known classicd filters, e.g. the ex-

tended Kalman filter (EKF) and the converted measurement Kalman filter (CMKF).

The position errors and the velocity errors for each filters for each measurement

noise levels are shown in Figs. 4.15 and 4.16 where the error is defined as the root

mean square of the difference between the actual value and the estimated value. Our

proposed met hod clearly outperforms the others by converging faster and yielding

estimates of smder enor. Note that both the extended K h a n filter (EKF) and the

converted measurement K h a n filter (CMKF) diverges due to the highly nonlinear

nature of the problem whereas our proposed adaptive Gaussian sum filter (AGSF)

converges because it is able to retain the non-Gaussian nature of the distribution of

the a posten'on' density p(x, [Zn) at each iteration.

Comparison of Position Emrs for Noise Level I lo3 .......... i.........!.........!.........:.........!, .........! ......... 1.. . ......: .........! ........ .......*..,.........,.........,..

...... ........................................ .... .. "..'".'.. ..... ......

......... .; ........ .; ........ .; ........ .;. ...... ..:... ...... ......

.... ......... .......... ; ......... ; :=:&.,. f ...-.-. -.CA- r::

......./.............................,..-..*.....*.*..**.............* C . . . . . . . . . . . . . . . , . . , . . . . . . . . .

h

E - i Y

1.1 ; p

........ ......... ......... ........ ......... ......... ........ ......... i ......... i i i i i i i .:.. ;. ........ ;...*.....;.........:..........:..........:.........c.........;.................. ........ ;.* .......; .......: .................... .........; .........; .................. . -. ..:. ..................................... --. ....... "......*.. ... ..................................... . . . ... ......... ..... ..... *..;. :.. . .;. .. ......; ........ .. a..... -..; .., ...... ,.. ..,.... .............. . -

J .- O 10 20 30 40 50 60 70 80 90 100

Tirne (s)

Figure 4.9: Comparison of the position errors for rneasurement noise

Ievel 1 Corn parison of Velocity Enors for Noise Level I

......... ....... ........ ........ ......... .......-- ........ .......... .........

......... ......... ....*...

,::::::::: .. -*.... ::::::::: i :::::::::. ::::::::: t j: :::::::::; :::;II ............................................................................................. .- ......?...............................'..,................<.........i.........i.............~...... S.. i..... ........... ;-+.......; ....... : . . . : . . . - - . * . ;* ........ ;.-,,.,.... ,....... . .

..-.....-. ;.-.... ... ........................................................................................... - - - - - * * -

I O - ' ~ 1 1 1 1 1 1 I 1 1

10 20 30 40 50 60 70 80 90 100 Time (s)

Figure 4.10: Comparison of the velocity errors for measurement noise

level 1 50

O TTme (s)

J I O 20 30 40 50 60 70 80 90 100

Comparison of Position Errors for Noise Level I lo3 .......... ! .........! .........! ........ 2 .........: ......... 2 .........; ......... 4 ......... ........

Figure 4.11: Comparison of the position errors for measurement noise

n

f Y

L .

level 2 Comparison of Vetocity Errors for Noise Level II

10' t. ........ 1. - -...... &.-... .... i ...-.. -.+ .-... ..-.r ........ 4 ......... 4 ......... 4 ........ 6 ......... 4 *............*.....,.....*.........~...,..ri...-.-......*...5.i-.-..........-. ......... ..*.......--.*.~.-...*.---.*.*.~....*-.-.~..........................*....~.--..*-.....~~~. I........-.........C.............................. ................................................... .......... ;*...... .. ;.........---..-.-;-........;..-.-....;..........~.........;.-.....*.~*..*....~.~*....*.;.......*.~*.......*:*.*...~~., ........ .... .......... P......... ;.... .....; .......-.: 1.-.*.a ....: ...-......: .......*.; .....*--.; .*...;.*

..............................-.........-........-........-............................-..........

...-.--... %...-• .-.- a - . ~ - - - . - ~ ~ . * . . . . . * . ~ - ~ . - - - . . . . . - . - - - . . . . . . ' . . - . ~ . - - . . : . ~ . . . . . * . d ~ ~ . . - . . . . ~ ~ - ~ * . * . - . . ....~..........~.....~...~..~.~..~.......~..~.....~.~~~....... a. .. ......,.........,.................... 1.**... ............. ..........-.....;........:.............................. ---.. -....-----.--...-.*~.*.-...-5.-.-----.~--~.~~.~ A ........-.........a....................

1 O-' I I f l f t I 1 I

O 10 20 30 40 50 60 70 80 90 100 Tirne (s)

...................................................................................................... _. ..........................._........-........ ...................................... .......... i . . ....:= ~ ~ , ~ . ~ . ~ ~ ~ ~ . - . - . . . ~ ~ ~ : ~ : . ~ : . ~ ~ . ~ . ~ ~ ~ : - : . - : . - . ' . - ' . ~ ~ - ~ . ~ . ~ ~ ~ . - . - ~ - ~ ~ ~ . - - / .......... i-'> ........................................................................................ /-:

. . . . . "'../..*'," ....... i .........,............. . . f i

1'. .....:......... ; ................. .-i .. . ..... I ..** 1 ......... ......... ..... ......... ......... .......... ......... . . . . .K.: ; ; : : - . + : " . ' . . . . 'i.. . .* .

m . .

Figure 4.12: Comparison of the velocity errors for measurement noise

.......*.. l . . . ......S.......... %.........'.....................'...... ...................................

1 O' 1 1 I I I I I 1 I

level 2

Figure 4.13: Superposition of 100 realizations of experiment based on

Extended Kdman Filter I I I r a l I I I i

Figure 4.14: Superposition of 100 reaiizations of experiment based on

Extended K h a n Filter

Figure 4.15: Comparison of the position errors for 3-D tracking scenario 16000 ; I I I I L I I 1 . .

* . -. . . . . .

Figure 4.16: Cornparison of the veiouty errors for 3-D tracking scenario

Chapter 5

Conclusion

In this paper we have discussed the problem of target tracking in Cartesian coordi-

nates with polar measurements. Two suboptimal tracking algorithms which is be-

lieved to be more accurate and efficient than the most widely used Taylor series

methods, e.g. the EKF are introduced. The first method uses the multidimensional

Gauss-Hermite quadrature to evaluate the optimal (conditional mean) est imate of

the target st ates directly frorn the Bayesian equations. This quadrature technique

replaces the integrds in the Bayesian equations with a weighted sum of samples of

the integrand and this approximation can be very accurate if the integrand is smooth.

However, this method has a major drawback, namely its computational complexity

because of the dimension of the problem and the number and the order of the quadra-

tures the number of samples involved are usually too large to be handled in red-time.

In this paper we have suggested several ways to reduce the computational require-

ments of this quadrature technique by reducing the dimension, the number and the

order of the quadratures required for a given accuracy. To reduce the number of

quadrature required, analytic results are applied to the prediction stage and the ap-

plication of quadrature techniques is restricted to the update stage because only the

measurement equation is nonlinear. To reduce the order of quadrature, the integand

is factored into two components: one is Gaussian and the other is nearly algebraic

within the desired region. In this case, the Gaussian component can be combined the

original Gaussian weighting function to obtain a new Gaussian weighting function.

Simulation results show this quadrature approach is more accurate than the classicd

methods, such as the EKF and the converted measurement K h a n filter (CMKF),

but it still has a high computational complexity.

The second tradcing dgonthm is based on the Gaussian sum filter. The Gaussian

sum approximation method yields very accurate result because it c m approximate

any probability density k c t i o n as closely as desired; moreover, the optimal estimate

(conditional mean) can be computed in a simple rnanner as a weighted sum of the

estimates from a bank of Kalman-type filters operating in pardel. However, this

method has a very high complexity since the number of Gaussian terms required in

the approximation increases exponentialiy in tirne. To alleviate the computational

burden associated with the Gaussian sum filter, we have found two efficient and sys-

tematic ways to approximates a non-Gaussian and measurement-dependent function

by a weighted sum of Gaussian density function. The f i s t approach utilizes the spe-

cial symmetric geometry of the function to select the number and the parameters

of Gaussians in the mixture. The second approach selects some initial parameters

fiom the known statistics of the noise and transform these parameters based on the

measurement equation to obtain the Gaussian sum approximation. We have derived

the formula for updating the weights and the parameters involved in the bank of

Kalman-type filters and we also have suggested several ways to control the number

of terms in the Gaussian mixture at each iteration. The fvst approach is to collapse

the mixture into one equivalent Gaussian and the second approach is to reduce the

number of Gaussians to some specified Lunit using the proposed joining algorithm or

the k-means clustering algorithm. Simulation resul ts show t hat this Gaussian sum

filter is more accurate than the classical methods, such as the EKJ? and the converted

measurement Kalman filter (CMKF), and yet as efficient as they are.

Future Works

First of all, there are still some problems rernains unsolved and requires more atten-

tions:

What is the optimal number of Gaussian terms we should use in the approxi-

ma tion?

What is the number of t e m s we should keep in the mixture reduction process?

So far these nurnbers are pre-determined and therefore it will be desirable to have

certain means to determine them analyticaily. However in the long run we should

investigate

r A new adaptive and intelligent Gaussian sum approximation method for a gen-

eral non-Gaussian and non-st ationary distribution;

A new adaptive and intelligent mixture reduction method which reduces the

number of Gaussian terms but retains the general structure of the distribution.

We should also apply this filter to other applications such as the underwater target

tracking [42].

Appendix A

Discretization of the

Continuous-Time Equation of

Motion

Assume in a single physical dimension, the target equations of motions may be rep

resented by

~ ( t ) = Fx(t) + Gv(t) (A. 1)

where x(t) = [ x ( t ) , i.(t), I(t)jT and v(t) is a white noise driving function with variance

The sensors have a constant scan interval sampling target position every AT seconds,

i.e. the measurements are taken in discrete time. The appropriate discrete t h e target

equations of motions are given by

it follows that

It can be verified that

where @(AT,a) is the target state transition matrix and un is the inhomogenous

driving input. Since

x(t + AT) = eFATx( t ) + Lttl e F ( A T - ' ) ~ v ( r ) d r

When aAT is s m d , @(AT, a) reduces to the Newtonian matrix

The noise sequence un satisfies

Since v(t) is white noise, E[u~u,+~] = O for i # O so that un is a disaete t h e white

noise sequence.

Appendix B

Track Initializat ion

Assume the initial density p(xo) is white Gaussian. It mean jro and its comiance

matrix PO can be estimated from the f i s t two measurements z-L and 20.

R e c d that the polar measurement z i at time n is related to the Cartesian target

state as follows:

This polar coordinate measurement z{ is first converted to the Cartesian coordinate

measurements z i using the inverse transformation h-'(z;)

The noise W, = [CI,, 6,,IT on the converted measurement zk can be assumed to

be a zer-mean white Gaussian noise process with covariance matrix Mn defbed by

equation (2.42).

Then the initial mean ko is defined as

where

E[xo] = E[&,o - &,O] = &,O

E[v=,o] = E[(G,o - &,-*)/AT] = (&,O - &,-,)/AT

Similarly, the initial covariance matrix Po is defined as

where

and the matrices M-1 and Mo are evaluated fiom equation (2.42) based on the first

two measwements 2-1 and zo.

Appendix C

Detail Derivations of the

Curve-fitting Approach

The objective here is to determine a weighted Gaussian denisty function such as it

has the ellipsoidal base defined by Pt. la-d as shown in figure C.1 and the weights

equal to the value of the function p ( z , I L ) at Pt. 1, where Pt. 1 is just the mean of

the firs t Gaussian term.

Rld Rlb

Figure C.1: Weighted Gaussian Function

The ellipsoidal base can be determined by rotating the ellipse defined by the

lengths 01, and c 1 b in Figure C.2A through the angle di; as a result the equivalent

covariance matrix can be detemiined by applying an rotation f i e tradormation to

the covariance rnatrix ( 1 a ci 1) Mmows:

where

Figure C.2: A 5 e Transformation of the ellipsoidal base

The value of kl and the weight ai axe determined as follows: At Pt. 1,

At three standard derivations level,

Substitue equation (C.6) into equation (C.5), the value of kl becomes

kl = 9

9 + 2 q2ral 1% 1112)

Appendix D

Detail Derivation of the

Transformation Approach

The function p(zn 13,) c m be rewritten as

where finti = Z, - mn,i and and Ên,i are the initial parameters determined

from the known statistics of the noise w, from equation (4.17) and

is a non-Gaussian function of xn.

Denote E(x.) as the exponential component O

1

f the h c t i o n h

Let consider the Taylor series expansion of the h c t i o n Fi(xn) about m,:

where Jn(uinti) and HeFi(mns) are the Jacobiaa and the Hessian of the function

Fi(xn) respectively.

where Jh(Zn) and Heh(mnti) are the Jacobian and the Hessian of the function h(x,)

respectively. If mn,i is chosen such that mnc = h-'(ni,,+)' , the constant term

F(mn,,-) becomes zero and the first order term JF,(mns) aiso becomes zero; only the

second order term and the higher order terms remain. Thus, the function N(ninqi - h(x,), Ëni) can be approximated as a Gaussian function by taking ody the second

order term and ignoring the remaining higher order terms.

where

Finally, substitute equation (D.7) into equation (D.l), the density p(znlxn) becomes

l Here, we assume the function is invertible; however, if the inverse does not exist, then we m u t

choose mn,i to be the most iikely solution given m,,i = h(mn,i)

Appendix E

Reproducing Property of Gaussian

Densities

First we make some definitions:

be an r x 1 matrix (column vector),

be an r x r symmetric, positive definite matrix,

be an s x 1 matrix (column vector),

be an s x s symmetric, positive defmite matrix,

be an r x s matrix,

be an s x 1 random matrix (column vector).

Also d&e N,(x - a, A) be a r-dimensional Gaussian density function with mean a

and covariance matrix A.

Theorem Let variables a, A, r, b, B, s, c, C and D be dehed as above. Then

Proof The proof is straightforward, though somewhat tedious, and involves com-

pleting the square and applying the matrix inversioo lemma.

the definition of c and C from equations (E.3) and (E.1), the above equation

becornes

- - 1 1 .-f [(x-C) (x-C) - C ~ C - ~ C + R * A - ~ ~ + b T ~ - ' b]

(2?r)r/21A1112 (2~)8/~1B11/2

- - 1 ,-f [ (x-c)~c-~ (x-c)] 1 1 e-~[aT~-l~+bT~-'b-e~C-~

(2a)~ /~I Cil/2 IA1l/21B1112

Knowing the relations that

Ic~'/~ - - 1 IA11/21B11/2 (DBDT + A)l12 (E-7)

a T ~ - ' a + b T ~ - ' b - = (Db - a ) T ( ~ + DBD=)-'(D~ - a) (E.8)

the above equation can simplified as follows:

Note that this theorem can be used to shift the dependence on x from a pair of

Gaussians to a single Gaussian. From the theorem the product of two Gaussians is

Gaussian whose normalking cons tant is the reciprocal of ano ther Gaussian indepen-

dent of x.

Appendix F

Bank of Kalman Filters

When the mixture reduction dgorithm is used instead of the equivdent Gaussian

dgorithm, the resulting conditional density p(xnlZn) will be expressed as a weighted

s u m of M Gaussian terms

Using this resdt the state prediction density p(xn+l (Zn) can be evaluated andytically

fkom equation (2.18).

where

Utilizing the proposed Gaussian sum approximation, we can approximate the density

p(zn+* ( x ~ + ~ ) as the following weighted sum of Gaussian terms

Substitute this result into equation (2.22) the conditional density p(xn+ (z"*') be-

cornes

where foo tnotesize

Finally, the optimal estimate k+lln+l at time step n + 1 becomes

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