analysis of suboptimal filtering algorithms for target tracking

Signal Processing 30 (1993) 221 233 221 Elsevier

Analysis of suboptimal filtering algorithms for target tracking*

M. Farooq, A. Rouhi and S.S. Lim Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, Ontario, Canada K7K 5LO

Received 18 July 1991 Revised 12 June 1992

Abstract. The extended Kalman filter has been applied extensively to the tracking of non-maneuvering targets. Although the filter tracks these targets accurately, its computational burden is generally heavy. In this regard, several suboptimal filtering schemes have been proposed in the literature. In this paper, the authors point out some of the shortcomings of the Baheti filter and present modifications to arrive at a computationally more efficient algorithm with improved performance. The proposed filter is compared with several suboptimal Kalman filtering schemes which require less computational burden than the extended Kalman filter while yielding near optimal performance during both transient and steady state filtering. It is shown that these suboptimal filters are viable alternatives to the extended Kalman filter for target tracking. In addition, an off-line technique for decoupling of the filtering algorithms based upon the correlation coefficient analysis is included in this paper.

Zusammenfassung. Zur Verfolgung von nicht-man6vrierenden Zielen wird sehr h~ufig das erweiterte Kalman-Filter eingesetzt. Die hohe Genauigkeit dieses Verfahrens mul3 jedoch durch einen sehr hohen Rechenaufwand erkauft werden. Daher wurden in der Literatur verschiedene suboptimale Filter-Methoden vorgeschlagen. In der vorliegenden Ver6ffentlichung werden einige Nachteile des Baheti-Filter aufgezeigt undes wird eine modifizierte Form dieses Filters mit verbesserten Eigenschaften pr/isen- tiert, welche auch rechnerisch effizienter ist. Dieses vorgeschlagene Filter wird mit einigen suboptimalen Kalman-Filter- Methoden verglichen, die rechnerisch weniger aufwendig als das erweiterte Kalman-Filter sind, jedoch in ihren Eigenschaften im transienten und stationfiren Betrieb nur unwesentlich vom optimalen Fall abzuweichen. Es wird gezeigt, dab diese suboptimalen Filter-Methoden bei der Zielverfolgung brauchbare Alternativen zum erweiterte Kalman-Filter darstellen. Zusatzlich enth~ilt die Ver6ffentlichung eine auf der Korrelationskoeffizient-Analyse beruhende off-line Methode fiir die Abkupplung der Filteralgorithmen.

R~sum~. Le filtre de Kalman ~tendu a 6t~ abondamment - et avec succ~s utilis6 pour l'interception de cibles non-manoeu- vrantes. Mais la charge de calculs qu'il demande est en g6n6ral ~lev6e. C'est pourquoi diff6rents filtres sous-optimaux ont 6t6 propos+s dans la litt6rature. Dans cet article, les auteurs soulignent certaines limitations du filtre de Baheti et pr+sentent des modifications permettant d'obtenir un algorithme pr6sentant une charge de calcul moins ~lev~e et des performances sup~rieures. Le filtre propos6 est compar6 fi plusieurs variantes sous-optimales de filtres de Kalman, qui demandent une charge de calcul moindre que le filtre de Kalman 6tendu tout en poss6dant des performances quasi-optimales tant pour le filtrage en r6gime permanent que pour le filtrage en r6gime transitoire. I1 est d~montr6 que ces filtres sous-optimaux sont des alternatives viables au filtre de Kalman 6tendu pour des probl6mes d'interception. De plus, cet article comprend une technique off-line qui permet de d6coupler les algorithmes de filtrage et qui est bas6e sur l'analyse du coefficient de corr61ation.

Keywords. Kalman filtering; target tracking.

1. Introduction

Over the past several years, a great deal of research on the application of Kalman filters to

Correspondence to: Professor M. Farooq, Department of Electrical and Computer Engineering, Royal Military College of Canada, Kingston, Ontario, Canada K7K 5LO.

* Portions of this paper were presented at the 28th IEEE Conference on Decision and Control (CDC '89), Tampa, FL, USA, December 1989.

target tracking problems has been reported in the literature. Much effort has been focused on developing computationally efficient Kalman filtering algorithms [1-4, 7, 8, 11-13]. Singer and Sea [11] attempted to decrease the computational load of the Kalman filter by developing an iterative method of computing the estimation error covariance matrix. Daum and Fitzgerald [3] investigated the performance of the Kalman filter

0165-1684/93/$05.00 © 1993 Elsevier Science Publishers B.V. All rights reserved

222 M. Farooq et aL / Suboptimal filtering for target tracking

when it is decoupled in various coordinate systems. Farooq and Bruder [4] compared the computational and storage requirements of the coupled and decoupled Kalman filters and various adaptive Kalman filtering algorithms. Mendel [8] presented the general computational and storage burdens of the discrete Kalman filter, and Gura and Bierman [7] reported the computational and storage requirements of several tracking algorithms, including the standard and stabilized Kalman filters, the sequential least squares filter, and several square root filters. Baheti [1] has proposed an efficient method of computing the Kalman gain based upon the rotation of the filter gain matrix for a target in the radar line-of-sight (referred to as the Baheti filter in this paper). In this paper, the authors point out some of the drawbacks of the Baheti filter and present modifications to arrive at a computationally more efficient algorithm, with improved performance (referred to as the modified Baheti filter). This filter is then compared with other suboptimal filtering algorithms for target tracking. The performance of the modified Baheti filter is illlustrated through numerical simulations. Moreover, the theoretical conditions for decoupling of the filtering algorithms based upon the correlation coefficient analysis is presented in Appendix A. These results, though conservative, can be obtained off-line as opposed to the more costly on-line approach report in [4].

The paper is organized as follows. In the next section, a general model for non-maneuvering targets and their measurements are presented. In Sec- tion 3, several suboptimal tracking filters, which include the extended Kalman filter, a decoupled Kalman filter, a partially decoupled Kalman filter and the Baheti filter, are summarized and their computational load is analyzed. Moreover, the drawbacks of the Baheti filter are discussed and then modifications to yield a computationally more efficient filter with improved performance are presented. The computational requirements of these suboptimal filters are compared quantitatively in Section 4. The computational burden of an extended Kalman filter is also tabulated in order

to provide a benchmark for comparison with the suboptimal techniques. In Section 5 simulation results for two target tracks are presented to demonstrate the relative performance of the tracking algorithms. Concluding remarks are presented in Section 6, and Appendix A contains the analysis on decoupling as a function of noise variances.

2. Target motion model

A general target motion model for a non- maneuvering target in discrete time can be expressed as follows:

x , + , = q~xk+ wk, (1)

where Xk is the state vector of the target, which is composed of position, velocity and acceleration terms in cartesian coordinates, at time tk, • is the linear state transition matrix, and Wk is a zero mean, white noise sequence that represents the pro- cess noise, with E[ W~ W~ ] = Q~ fi~j.

Typically measurements of the target location will be received in spherical coordinates. If the actual measurements are used in filtering then they are given by

Z~=[r~" b~ e'~'lT=hk(Xk)+VSk, (2)

where Z~ is the measurement vector in spherical coordinates, rk, bk and ek represent the range, bearing and elevation measurements, respectively, the superscript m denotes measured quantities, hk(X~) defines the nonlinear relationship between the measurements and the state vector, V~ is a zero mean white noise sequence associated with the measurements and T denotes the transpose of a matrix. The covariance of V~ is given by

cov(V~) zx R~ = diag(0- 2 , 0 -2 , z 0-e)k, (3)

where 0-~, 0-~,, 0-~ are the variances in the range, bearing and elevation, respectively. Since the relationship between the state vector Ark and the measurement vector Z~ is nonlinear in the above formulation, a suboptimal filtering algorithm, such

Signal Processing

M. Farooq et al. / Suboptimal filtering for target tracking 223

as an extended Kalman filter, is generally implemented for target tracking.

An alternative formulation imbeds the nonlinear relationship between the measurements and the state vector in the measurement noise covariance matrix. A linearized observation vector in cartesian coordinates can be approximated by a first order multi-dimensional Taylor series expansion as

Ix: 1 Fr' cos(e' cos(ar ) Z~ = I y~'I = I r : cos(e~')sin(b:)/

Lz~J L r~sin(em) J

[x~ 1 a s c ~ / y ~ / + F~ Vk = HXk + Vk, (4)

/ /

kz~J

where Z~, represents a pseudo-measurement vector that transforms the spherical measurements to cartesian coordinates, x, y, z are elements of the state vector representing the position of a target, the superscript 'a' denotes actual values, H is the observation matrix, V~ represents the additive measurement noise and F~ is the time varying Jacobian matrix defined by

-cos(b~) cos(e~') - r ~ sin(b~) cos(e~) Fk = sin(b~) cos(e~) r~ cos(b~) cos(e~)

sin(e~) 0

- r ~ cos(b~') sin(e~)l

- r ~ sin(b~) sin(e~')/. (5)

r~ cos(e~') J

location and the elements of the noise covariance matrix in spherical coordinates. The specific structure of the state vector, the state transition matrix and the observation matrix for each individual filter will be discussed in the next section.

3. Suboptimal filtering algorithms

In this section we discuss five suboptimal tracking algorithms which include the extended Kalman filter, the decoupled Kalman filter, the partially decoupled Kalman filter, the Baheti filter and the modified Baheti filter. The main drawbacks of the Baheti filter as presented in [1] are highlighted in Section 3.4, while, in Section 3.5, the authors present certain modifications to this filter in order to overcome the original shortcomings as outlined in Section 3.4. This modified filter has a decreased computational burden and yields improved performance.

3.1. Extended Kalman filter (EKF)

The extended Kalman filter yields the linear minimum variance estimate. In this study it plays the role of a benchmark for comparison of the suboptimal filters by establishing the maximum allowable computational threshold. In the filter, we first compute the Kalman gain using

g k ~ T T Pk/k tGk(GkPk/k-lG~ +R~) 1, (7)

Note that the Jacobian is calculated at the measured values of the range, bearing and elevation. The noise covariance matrix in cartesian coordinates is given by

coy{ V~} = CoV{Fk V~}

- - - - ~ R ~ - s T _ _ 2 - FkRkFk - ]~yx 2+21 ~ x y xz 2 0.2 I~yy yz • ( 6 )

It is important to note that the terms cry, i, j = x, y and z, are functions of both the measured target

Ohk Gk(X~/~-I) = ~ xk=2~k , = F[I ' (8)

where Gk is the observation matrix resulting from the Taylor series approximation of ht(Xk) about the predicted estimate, P./. is the error covariance matrix, and R~, is the noise covariance matrix as given in (3). The state estimate and error covariance matrix are then updated as

-~k/k=Xk/k--I + K k ( Z ~ - h k ( X ~ / k - l ) ) , (9)

Pk/k = (I-- KkGk) Pk/k-1, (10) Vol. 30, No. 2. January 1993

2 2 4 M. Farooq et al. / Suboptimal filtering for target tracking

where

£ / . = [~-/-, L., L-, ~-/., ~-/., y./-, 2./., L., L.] ~

is the estimate of the state vector, Z~, is the measurement vector and the bold faced letters denote position, velocity and acceleration. The predicted state estimate and error covariance are given by

)(k+,/k = </>Xk/~, (11)

Pk + 1/k : (I) Pk/k ¢20T+ Qk. (12)

3.2. Decoupled Kalman filter (DECKF)

This version of the Kalman filter uses the pseudo-measurement vector defined by (4). The standard three-dimensional Kalman filter is decoupled into three one-dimensional filters, one for each coordinate in the cartesian space. In the decoupled Kalman filter, the Kalman gain in each coordinate direction is first computed as

i i T i Kk-- Pk/k-1 H (HPk/k_l H v + R~)-t,

i=x , y, z, (13)

where K~ and P!/. are the Kalman gains and error covariance matrices in each coordinate direction, respectively. Note that RT, = Crxx2 , RYk-- O'yy2 , R~= o-~:, and consequently all the Ris are scalars. Furthermore, it should be noted that H = H x= HY=HZ=[1 0 0]. The state estimate and error covariance matrix are then corrected through

^ i - - A i i i ^ i Xk/k--Xk/k-I + Kk ( Z ~ - HXk/k- l ) ,

i = x , y , z , (14)

I - K k H ) P k / k - l , i = x , y , z, (15) P'k/k = ( i i

where

~z=[~,, L L] ~, ~:[y,~ L L] T, ~,~= [~k L L] T.

Finally, the predicted state estimate and error covariance matrix are computed as

^ i __ ~ i Xk+l/k-- ~ lXk /k , i = x , y , Z, (16)

i ~ . _q~lpik/k~T+Qig, i = x , y , z ' Pk+l/k-- (17)

w h e r e (~)l is the state transition matrix associated with each cartesian coordinate axis and is given by

ii s • l = 1 s ]

under the assumption of a constant acceleration. In the above equation Ts is the sampling interval. Here Q~ represents the sub-matrix of Qk that corre- sponds to each filter in x, y, z directions. Obvi- ously, the Kalman filter can be decoupled only when the non-diagonal elements of the measurement noise covariance matrix are assumed negligible. Since this is not always a valid assumption, a criterion for decoupling has been discussed in [4]. This criterion involves the examination of correlation coefficients between two random variables, say x and y, as defined in [9]. The three correlation coefficients between the noises in the i and j direction are defined by

2 2 (T xy (Y xz

c ~ - - - , c ~ - , o- xxO'yy o- xxo- zz

cy:= , (18) (~yy (~zz

where cr 0 and/~i are defined as

o~ = E{(Xi - ~ , ) ( x + - u+)},

u~=E{X~}

for i ¢ j = x, y, z.

In [4], the analysis of the above correlation coefficients for the purpose of decoupling has been carried out through simulations while assuming specific values of the standard deviation of noise in the range (o-r = 1% of the actual range), and in both the bearing and elevation (O" e = O" b = 1 °). However, Appendix A gives theoretical conditions for decou- piing for any value of O'r, o-b and <re. These conditions can be evaluated off-line, while this is not the case for the approach presented in [4] where an on-line procedure was used. In computer simulations, it has been noted [4] that for a correlation

Signal Processing


coefficient less than 0.5, the corresponding off- diagonal elements in the noise covariance matrix can be assumed to be negligible (i.e., if Co.<<.0.5, i, j = x, y, z, i ~j , then o-~ and trj~ can be assumed to be negligible). If this condition is satisfied by the three correlation coefficients simultaneously, then the standard Kalman filter can be replaced by a set of three decoupled filters without a noticeable degradation in performance. Some additional benefits have been noted by Daum and Fitzgerald [3], specifically, in addition to reducing the computational burden, a decoupled filter structure also reduces the effects of nonlinearities and lessens ill- conditioning.

3.3. Partially decoupled Kalman filter

(PDKF) 151

Using the results of Appendix A, it can be shown that for certain conditions of o-~, o- b and o'~ and the bearing and elevation, the filter should not be completely decoupled. For example, in the case of o-~= 1% of the actual range, fib = O'e = 1 °, the correlation coefficient C~y>0.5 for elevation angle exceeding 60 ° . In this case, the off-diagonal terms, Oxy and O-y~, should not be assumed to be negligible especially for the case of low SNR ( < 0 dB). A partially decoupled filter can be used in these cir- cumstances. The partially decoupled filter would maintain coupling in the x - y plane while ignoring in the x-z and y - z planes. Using the measurement model of (4), this algorithm can be implemented on a computer as follows.

The Kalman gain matrices are calculated first.

g x y _ Dx-y Hx y T k - - ~ k / k - - 1 1 1

x (H ~ Y p ~ Y , H ~ S + R{-Y)-', (19a)

K~ = ~ ~r ~ ~ ~T Pk/~-lH (H Pk/k- lH +R~) -1, (19b)

where

o o o o Ool 0 0 1 0 '

Loy -- 2 2 '

and P~./.Y is the error covariance matrix in the x -y

plane. The state estimate vectors and the error covari-

ance matrices are then corrected using the following relations:

f(x-y_C, xy +KT, y~7x -Y - rSxYg~Y ~ (20a) k / k - - A k / k - I ~ L k 1 1 A k / k - I ) ,

A A ^

X~/k =XZ~/k-i + K~ ( Z ~ - H~X~/k_I), (20b)

P x y - t r l g ' x y u x - Y ' ~ D x Y (21a) k / k - - k ~ - l ~ k 1 1 ) l k / k - 1 ,

P~/k = ( I - K~ H z) P~/g_, . (21 b)

Finally, the predicted state estimate and error covariance matrix are given by

^ x - y __ ~ x - y v X y (22a) X k + l / k - - ~ ~ r X k / k ,

z __ g z Xk+l/k-- ~ X~/~, (22b)

x - y - - ~ x _ y D x _ y , r l r ~ X y T ± f i x y (23a) e k + l / k - - ~ l k / k M'" T ~:~.k

P~k+l/k = ~zP~kk~'T + Q~, (23b)

where ~ - Y = d i a g ( ~ l , ~1) and q ~ = ~1.

3.4. Baheti filter (BAHF)

Baheti [1] has proposed an algorithm which per- mits the Kalman gain matrix and the error covariance matrix to be computed in a more efficient manner than that for the standard Kalman filter. The algorithm uses the transformation matrix Tm=diag[Fk, Fk, Fk]. The basic idea of Baheti's paper is that the Kalman gains can initially be computed for a target that is assumed to be in the radar line-of-sight (LOS). These gains can then be multiplied (rotated) by a transformation matrix taking into account the target's actual position with respect to the line-of-sight. The advantage of dealing with a target on the radar line-of-sight is that the gain and the error covariance matrix in each coordinate direction are uncoupled. This per- mits an n-dimensional configuration of the filter to be replaced by n one-dimensional filters. In our case n is either 3 or 2.

Vol. 30, No. 2, January 1993

226 M. Farooq et al. / Suboptirnal filtering for target tracking

Baheti used the following two equations (eqs. (8) and (9) in [1]) to develop a computationally efficient filter:

gk-1 = Pk--1/k-I g[-I [ R i t - l ] -1 , (24)

Pk/, = ( I - Kk-1 gk )Pk/k-J, (25)

where I is a 3 × 3 identity matrix and gk = [Gk 0 0]. For a target on the x-axis where the bearing and elevation angles are zero, the LOS axes coincide with the cartesian reference frame. Hence the Kalman filter equations are decoupled and the gains can be computed for each axis as presented in the following. First, the Kalman filter gain in each direction in radar LOS coordinates is computed by

KOk = T s -I Sk/kgk [Rk] , (26)

state estimate and error covariance matrix are updated and predicted as

)(,/k =X,/k-1 +Kk- , (Z~-hk( f ( , / k - , ) ) , (29)

X~+l/k = ~ X , / , , (30)

Pk+~/k = ~)Pk/k@ T+ Qk. (31)

By establishing (27) and (28) for the filter gain and error covariance matrices in the LOS and cartesian coordinate systems, Baheti obtained three decoupled one-dimensional filters and thereby reduced the computational burden. However, it is important to note that (25) is valid only when K, ~ Kk-~ and hence under any other tracking conditions, for example the transient situations, the filter may diverge. The exact form of (25) (see (4.2.16b) of [6]) is

P~/k = ( I - Kkg, )ek/~-, . (25')

where the superscript 'o' denotes the quantities in the LOS coordinates and Sk/, is the estimation error covariance matrix in the LOS coordinates. Then the filter gain in each direction given by (24) for a target at the range r, bearing b and elevation e in cartesian coordinates is obtained by rotation from the LOS to cartesian coordinates as

Kk = T,,/~k, (27)

where Tm represents the necessary rotation to account for the actual target position. For (26) Baheti utilized the following relationship between the covariance matrices in each direction in the LOS and cartesian coordinates:

Hence, clearly the Baheti filter is not recursive because (24) and (25') are now intercoupled for the same sampling instant k. In this regard, Savelloni [10] corrected the Baheti filter by employing

Kk----- T T Pk/k-I gk [gkPk/*-I g~ + R~] -~, (32)

P k/k = (I-- Kkgk)Pk/k-l( I-- Kkgk) T

+ Kk Rk K~. (33)

in place of (24) for the gain and (25) for the covariance, respectively, in the Baheti filtering scheme. It should be noted that (28) is only an approximation and hence the Baheti filter is suboptimal, which can exhibit divergence in certain situations.

Pk/k ~- TmSk/k T~, (28)

where T~ = diag[F~, F[, F~ ] and Pk/k is the estimation error covariance matrix in cartesian coordinates. The approximation (28) is based on neglecting terms involving the range rate and the angular rates which could have very small influence on the off-diagonal terms of the error covariance matrix in cartesian coordinates for a class of target trajectory under consideration [1]. Finally, the

3.5. Modified Baheti filter (MBAHF)

The authors of this paper have used the relationship established by (28) in a different manner than Baheti. Since (28) establishes the relationship between the estimation error covariance matrices in different coordinate systems, it can be used to develop a filter that switches between the cartesian and LOS coordinates in the following manner to reduce the computational burden.

Signal Processing


First the estimation error covariance matrix is updated in the LOS coordinates;

S~)k = S~)~-1 + HT[R~]-I H. (34)

This matrix is then transformed to cartesian coordinates using (28) and the filter gain is computed using the rotation formula

Kk = "!" vIos~ (35) l m l ~ - k U k ,

where K}, °s is the filter gain in the LOS coordinates and is given by

KlOS_C r_rTtosl-I (36) k - - O k / k 11 I_ l ' tkJ •

Using (12) and (28) into

&+ ~/~ = TT,,~pk+ ~/k[Tm~] ~,

we obtain the predicted covariance matrix in the LOS coordinates as

Sk+l/k = ( T~,I ~ T,,)Sk/k( T~,' ~ Tm) T

+ T~,~Q~[Tm'] T. (37)

Finally the state estimation vector is updated and predicted using

ff~/k =-~k/k--, + Kk(Zk - nXk/k- l ) , (38)

Xk+l/k = ~)(k/~ • (39)

The advantage of the modified Baheti filter is that the band symmetric nature of the error covariance matrix in spherical coordinates is preserved which can be further exploited to reduce the computational load of the tracking algorithm.

responsible for inducing coupling in the resulting Kalman filter assuming Po/-1 is properly initiated.

The relative contribution of the off-diagonal components of R~ can be monitored by calculating the correlation coefficient in the following. It is quite simple to show that inaccurate knowledge of the measurement noise statistics will result in a less accurate filter. The essential question remains as to when this resulting inaccuracy is negligible. A very conservative threshold set in [4] deems the off- diagonal terms negligible if C0.<0.5. This is not to say that the decoupled filter will not function adequately if Cu>0.5. Considering the case of a very high SNR sensor, accurate knowledge of the measurement noise statistics will be almost irrelev- ant and the steady state error will be dominated by the contribution of the state noise covariance. In brief, for a high SNR environment ( > 6 dB) (as is the case in the tracks 1 and 2) the DECKF and MBAHF should provide very similar results. In the low SNR case (<0 dB) the MBAHF will track more accurately during the periods where C~>> 0.5. Note that the correlation coefficient analysis pro- vided in the following allows the decoupling analysis to be performed off-line, as opposed to the more costy on-line approach suggested in [4], but in so doing results in an even more conservative approach to decoupling.

4. Computational burdens

R E M A R K 1. For the problem being considered in this paper the covariances of X0 and Wk are assumed to be block diagonal corresponding to the three cartesian directions. These assumptions in addition to block diagonal structure of q~ render the state dynamics of (1) completely uncoupled in the x, y and z directions, respectively. However, the cartesian pseudo-measurements described in (4) will induce coupling in a Kalman filter used to estimate Ark by virtue of the non-diagonal structure of R~, (eq. (6)) (the covariance of the cartesian pseudo-measurement noise). Due to the block structure of H in (4), the covariance R~ is solely

The relative computational burdens in terms of the time units required per iteration, for the different filters, are compared in this section. An addition was assumed to require 1 time unit while a multiplication requires 1.67 time units and a divi- sion requires 3.33 time units, respectively. The computational requirements are illustrated in Fig. 1 for the two-dimensional case in which the state vector contains position and velocity terms and for the three-dimensional case in which acceleration terms are added to the state vector. From Fig. 1 it is clear that the efficient filters are the decoupled, the partially-decoupled, SaveUoni (SAVF) and the


228 M. Farooq et al. / Suboptimal filtering for target tracking

NO, of Time Units per iteration

5400

4800

4200

3600

3000

2400

1800

1200

600

O

m m m m m

E.K.F.

Two Dimensional Case

Three Dimensional Case Bm

II m m

DEC.K.F. P.D.K.F. BAH.F. SAV.F. M.BAH.F.

Fig. 1. The number of time units required per iteration for the coupled, decoupled and partially-decoupled Kalman filters and for the modified Baheti filters.

modified Baheti filters. Also the modified Baheti filter is more efficient than the partially decoupled, Baheti and Savelloni filters. The decoupled Kalman filter possesses the smallest computational requirement; however, for the given error conditions, it does not track accurately when the elevation angle is greater than the threshold as determined by the conditions in Appendix A (or equivalently when the correlation coefficient is greater than 0.5). For elevation angles that exceed this threshold, a partially-decoupled or fully coupled Kalman filter must be used. If the decoupled filters are to be used, then a simple software routine that switches between the fully decoupled and partially-decoupled Kalman filters as the elevation angle crosses the threshold should be implemented. It should be noted that, for the given observation noise, the decoupling approach previously sug-

gested in [4] required an on-line analysis of the correlation coefficients. However, with the analysis presented in Appendix A, it can now be implemented off-line as a part of a tracking algorithm in order to arrive at the conditions under which com- plete decoupling is not permitted.

5. Simulation results

In this section, the results of computer simulation are presented in order to demonstrate the performance of the five suboptimal filters discussed in Signal Processing

Section 3. In the simulation it was assumed that the measurement noise has <rr = 0.01r, orb = ere = 1 ° and two typical track profiles were used. Track 1 is a straight line trajectory as shown in Fig. 2, where the target travels parallel to the y-axis for 30 sec at a constant speed of 300 m/sec and at a constant altitude of 667 m. Track 2 is more com- plex and is depicted in Fig. 4. This track involves the target making a 90 ° turn during 25 30 sec in the flight, and at the end of the turn the target dives towards the observer. Using the sampling interval Ts = 0.5 sec and a random number generation sub- routine, the filters were implemented in C on an Apollo DN3500 computer system at the Royal Mil- itary College of Canada. To compare the performance of the filters against Track 1 and Track 2, the rms errors of Monte-Carlo runs of each filter are presented in Figs. 3 and 5, respectively.

It is clearly observed from Fig. 3 that, after a transient period of about 10 sec, all the filters

z I Target travels parallel to y-axis for 30 secs.

/ / / ~ at a speed of 300 m/sec, Y

' oO- oo°° / 30s ,'" Os /7500

, ' " 667m i """

x ~ / 17000 m

Fig. 2. Track 1.


800.

700.

600

500

40O

300

200

100

track # I t

• -, - - . . . . E.K.R

I "~ ', BAH.R ~ "" A /~ ~ DEC.K.R

, "V;" . ,

5 10 15 20 25 time in seconds

track #2

E.K.R

. . . . . . DEC.K.F.

64O

560

j - ._c 400

320

I1: 240

~ 160.

5 10 15 20 25 30 35 time in seconds

Fig. 3. x-coordinate RMS error for track number 1. Note: The x-coordinate RMS error is calculated as follows:

I I N -.~j(k/k_l)]Z} '/2 X~s(k) = ~ Z [xAk) / V - - l j = ]

= x coord. RMS error,

where N is the number of runs.

From 0-25 sec t t~get crosses y-axis. From 25-30 secs. turns 90 degrees tovauds observen / ' , From 30-40 secs, dives directly towards obsenter. Constant speed of 250 m/sec throught

Os

500 m

, " 5 0 0 m

• " " " " " ~ 90 degrees

Fig. 4. Track 2.

except for the Baheti filter generate similar estimation errors and the suboptimality does not greatly degrade their performance. As indicated in Section 3.4, the Baheti filter yields acceptable estimates only under the special tracking conditions while other filters produce reasonable estimates in all tracking conditions. For this type of straight line motion (Track 1), the most computationally efficient filter, for example the decoupled Kalman filter, should be implemented for target tracking. We have the same conclusion from the results shown in Fig. 5 for Track 2 as that from Fig. 3 for Track 1. In addition, the Baheti filter in Fig. 5 diverges

Fig. 5. x-coordinate RMS error for track number 2.

following the maneuver while the modified Baheti filter and the other filters manage to maintain fairly good accuracy. The reason for the divergence in the Baheti filter is due to the use of (25) to compute the error covariance matrix in cartesian coordinates. However, this equation is not valid when the target maneuvers or when the target's line-of-sight rate is rapidly changing. This approximation in computing the error covariance matrix directly influences the Kalman gain which, in turn, degrades the overall performance of this filter. On the other hand, the modified Baheti filter employs (34) to compute the Kalman gain hence showing no degradation in performance for all tracking conditions under consideration. As a whole, the simulation results clearly indicate that this technique ( M B A H F ) i s better able to handle target maneuvers and rapid changes in target's line-of- ~xght without involving heavy computational burden.

6. Conclusions

In this paper, the authors have pointed out some of the shortcomings of the Baheti filter, primarily due to the use of Kk-i in computing Pk/k (see (25)), and presented modifications to arrive at a computationally more efficient algorithm, with an improved performance. The proposed filter is compared with several other suboptimal Kalman filtering schemes, which include the extended Kalman


2 3 0 M. Farooq et al. / Subopt imal f i l ter ing f o r target tracking

filter, the decoupled Kalman filter, the partially decoupled Kalman filter and the Baheti filter, for target tracking. The computational requirements of these suboptimal filters are summarized graph- ically for comparison. The computational burden of an extended Kalman filter is also tabulated in order to provide a benchmark for comparison with the other suboptimal techniques. Simulation results for two target tracks are presented to demonstrate the relative performance of the tracking algorithms. The performance of the modified Baheti filter has also been illustrated by numerical simulations. It is clearly shown from the results that the suboptimal techniques involve less computational burden than the extended Kalman filter while yielding reasonably acceptable estimates, and the modified Baheti filter is computationally more efficient than the partially decoupled and Savelloni's filters while maintaining reasonable lock on the target track. These suboptimal filters are viable alternatives to the extended Kalman filter for target tracking without serious degradation of their filter performance.

Appendix A. Correlation coefficient analysis

respectively, and the superscripts 'a' and 'm' denote the actual and the measured values, respectively. The measurements in cartesian coordinates can be generated by resolving the spherical measurements a s

X p m ~ r m C O S b m c o s e m = x a - [ - / ) x ,

Y p m = r m s i n b m c o s e m = y a + L/y, (A2)

Z p m = r m s i n e m = z a -[- u z ,

where the subscript 'pm' denotes pseudo-measurements and Vx, vy and v: are the equivalent noise terms for the pseudo-measurements. The relationship expressing Vx, vy and v~ in terms of v~, Vb and Ve is given by

VC~-FV ~, (A.3)

where the vectors V ~ and V s and the Jacobian matrix F are defined by

Ei l Fvrl V c ~ , V s ~ z) b ,

. LVed

Fcos b m c o s e ~ - r m s in b m c o s e m

F = l sin bm cos e m rm cos bm cos e m

t_ s in e m 0

r m COS b m s in e m ]

- r m s in b m s in e m / .

r m COS e m J

In this appendix, we explain how correlation coefficient analysis can be used to determine when coupled and decoupled (either full or partial) filters are valid, for arbitrary measurement noise statistics. For the sake of brevity, the subscript k to denote the time index is being omitted in the following.

Transformation o f measurement errors f rom

spherical to cartesian coordinates

The actual measurement equations taken in spherical coordinates, is given by

r m = r a + o r , b m = b a + O b , e m = e a + O e ,

(A1)

where vr, Vb and Ve are the additive noise terms for the range, bearing and elevation measurements, Signal Processing

Transformation o f the noise covariance matrix

from spherical to cartesian coordinates

Let the noise covariance matrix in spherical and cartesian coordinates be denoted by R s and R c, respectively. Then R c can be expressed as follows:

R ~ ---F diag(o'~, o "2 , Cre2)r T. (A.4)

For radar target tracking problems O'b and O'e are usually assumed to be stationary, while cr, is assumed to be a fraction (denoted by o-') of the measured range o-,= O"rr m. Replacing O" r by (7'rr m

and defining the vectors a, b, c by

a = [tr'r c o s b m c o s e m - c r h s in b m c o s e m - c r e c o s b m s in era],

b=[tr'rsinbmcose m o - b c o s b m c o s e m - o ' e s i n b m s i n e m ] ,

c = [o" s in e m 0 rye cos era),

(A.5)

M. Farooq et aL / Suboptimal filtering for target tracking

the expression for R ~ is then given by

c Vllall 2 a" b a .c 1 I b .c R ~(rm) 2 b ' a IIbll 2

A Lc" a c. b Ilcll 2

V 2 0.2 0.2 ] 0-XX xy XZ /O.2 2 0.2 VX 0- yy yz ,

1 0.2 ~2 ,x :y 0-2z

(A.6)

where Ilsll is the norm of a vector s and ' - ' denotes a dot product of two vectors.

Analysis o f correlation coefficients

For simplicity, the analysis of the correlation

coefficient will be carried out by squaring the expression given in (18). The tracking area of inter-

est lies between an elevation angle of 0 and 90 ° and between a bearing angle of -180 to 180 °. If the

correlation coefficient Co<<,0.5, for all i, j = x, y, z; i# j , then the corresponding cross-correlation terms o-2 and 2 o'ji can be set to zero without a noticeable loss in tracking accuracy [4]. Thus, a filter with coupling in the i-j plane is not required.

Outside of this range of values, the global maximum value of the correlation coefficient exceeds

0.5. In this case the tracking area can be divided into two regions, namely a region in which C~> 0.5 and a region in which Gj~<0.5. The analysis will

then focus on determining analytical expressions that define the two tracking regions, in terms of the target location and the statistics of the additive measurement noise. Thus, even though the global

maximum value of C0 exceeds 0.5, a filter with coupling in the i-j plane will not be required if the target stays within the tracking region in which

<. o.5.

Using (A.6) into (18) we obtain that

C~2y - (0-2y)22 2 - ( a ' b ) 2 = (a - b) 2. (A.7) 0-xx0-yv Ilall211bll 2 Ilall Ilbll

231

Since a/llall and b/ l lb l l are unit vectors, C2y has its maximum when a and b are parallel. It is clear from (A.5) that a and b are parallel when cos e m= 0 °, i.e. e m= 90 °. Thus the global maximum value of Cxy is 1 for any non-zero values of 0-'r, 0-b and 0-e. Therefore, it is not possible to calculate any range of values of the additive measurement noise for which the global maximum value of C~s will not exceed 0.5. It is however possible to divide the tracking area into two regions by checking whether Cxy<~0.5 or not.

The correlation coefficient Cxy is a function of the target bearing and elevation as well as the statistics of the additive measurement noise. The local maximum values of Cx* occur at elevation angles of 0 and 90 °. Between these angles C*y may attain values less than or equal to 0.5, in which case decoupling in the x-y plane is permitted. The expression for C*y can be derived by evaluating the expression for C~y at the values of bearing at which Cxy attains its maximum value. It can be shown that the bearing values at which Cxv is a maximum are b m= --135 °, - 4 5 °, 45 °, 135 ° at a given elevation angle. The expression for Cx* is given by

O.21 0.2 COS 2 e m 2 e m C* - -vr COS 2e m - +rYeSin 2 xy 2 t 0-~ COS 2 e m + 0-2 c o s 2 e m + 0.2 sin 2 e m"

(A.8)

The elevation angle(s) at which Cx*~ becomes 0.5 is denoted by e* and is expressed as a function of the statistics of the measurement noise:

e * = t a n ~/5C-/3A3B + 4 C ' (A.9)

where A =o'r2', B = 0-e 2 and C = 0 -2 (these definitions will hold throughout the rest of this appendix). Between the elevation angles of 0 and 90 ° e* will take on two values which will be denoted as et and e2. Assume el ~< e2, then 0 ° ~< el ~< e2 ~< 90 °. It can be shown that for e~ ~<e m~<e2, C*y does not exceed 0.5, while for 0~<em<el and e2<em~<90 °, C*y will exceed 0.5. Thus as long as the target remains within the tracking region defined by e, ~<e TM ~<e2 a filter with coupling in the x-y plane is not required.


232 M. Farooq et aL / Suboptimal filtering for target tracking

An analysis o f Cy. and C~., indicates that their

maxima have exactly the same expressions and

hence in the following we present only the case o f

C~:. Using (A.6) we obtain

x~ 0.~x0.~,- Ilall211cll 2 Ilctl (A.10)

Thus it attains its max imum value when a and e

are parallel. A careful examinat ion o f the expres-

sions for a and c reveals that a and c cannot be

made parallel if 0." and 0.e are non-zero. A n alterna-

tive approach to maximizing C~: is to examine the

following term:

( a . c) 2 = (0.'r-- 0.2)2 (COS b m cos e m sin emf.

(A.11)

o f 45 ° and decreases as the elevation angle

approaches 0 or 90 °. The elevation angles at which

Cx* equals 0.5 are given by

e* = t a n - i { [( 1.5A 2 + 1.5B 2 - 4AB) 4- 1.5(A - B)

x x / ( a 2 - 3.3333AB + Bz)]/AB}I/2.

(A.14)

Between the elevation angles o f 0 and 90 ° e* will

take on two values which will be denoted as e3 and

e4. Assume e3 ~< ca. Then 0 ° ~< e3 ~< e4 ~< 90 °. It can

be shown that for e3<em<e4 . Cx* exceeds 0.5, while for 0~<em~<e3 and e4~<em~<90 °, C ' z does not

exceed 0.5. Thus, as long as the target remains out-

side o f the tracking region defined by e3 < e m< e4,

a filter with coupling in the x z plane (similarly for

the filter in the y - z plane) is not required.

In order to maximize (A.11) we select cos b m= 1 and cos em=sin e m, i.e. b m = - 1 8 0 °, 0 °, 180 ° and

e m = 4 5 °. For these values o f b m and e m, CZxz has

max imum

C ~ : ( m a x ) - (0.2, _ 0.2)2 (0.2, _~ 0"2) 2. (A.12)

Thus, (A.12) yields

Cxz(max) > 0.50,

, 1 f o r O <~ 0. r < - - 0. e O r , f 3 0. e < 0. " ,

Cxz(max) < 0.50

for ae ~ ar.~. N/3 0.e.

(A.13)

Thus for ae/x/-3 ~ art ~ ~ 0.e, Cxz cannot exceed

0.5 and the filter that tracks in the x - z plane is

unnecessary. The same is true for Cy~ and the filter

in the y - z plane.

When a'r falls outside the given range it is neces-

sary to determine the tracking region for which

Cx* (which is defined in the same manner as C ' y )

is less than or equal to 0.5. It can be shown that C*: attains its max imum value at an elevation angle Signal Processing

R e f e r e n c e s

[1] R.S. Baheti, "Efficient approximation of Kalman filter for target tracking", IEEE Trans. Aerospace Electron. Sys- tems, Vol. AES-22, No. 1, January 1986, pp. 8 14

[2] F.R. Castella and F.G. Dennebacke, "Analytical results for the x, y Kalman tracking filter", IEEE Trans. Aero- space Electron. Systems, Vol. AES- 10, No. 6, November 1974, pp. 891 895.

[3] F.E. Daum and R.J. Fitzgerald, "Decoupled Kalman filters for phased array radar tracking", IEEE Trans. Auto- mat. Control, Vol. AC-28, No. 3, March 1983, pp. 269-283.

[4] M. Farooq and S. Bruder, "Information type filters for tracking a maneuvering target", IEEE Trans. Aerospace Electron. Systems, Vol. AES-26, No. 3, May 1990, pp. 441 454.

[5] M. Farooq, A. Rouhi and D. Horsman, "Analysis of suboptimal Kalman tracking algorithms", 28 Conf. Decision Control, 13-15 December 1989, pp. 1417-1422.

[6] A. Gelb, ed., Applied Optimal Estimation, The Analytic Sciences Corporation, MIT Press, Cambridge, MA, 1974,

[7] I.A. Gura and A.B. Bierman, "On computational efficiency of linear filtering algorithms", Automatica, Vol. 7, No. 3, May 1971, pp. 299 314.

[8] J.M. Mendel, "Computational requirements for a discrete Kalman filter", IEEE Trans. Automat. Control, Vol. AC- 16, No. 6, December 1971, pp. 748 758.

[9] A. Papoulis, Probability, Random Variables, and Stochastic Processes, McGraw-Hill, New York, 1976.

[10] M.A, Savelloni, M. Eng. Thesis, Rensselaer Polytechnic Institute, Troy, NY, May 1988.


[11] R.A. Singer and R.G. Sea, "Increasing the computational efficiency of discrete Kalman filters", IEEE Trans. Auto- mat. Control. Vol. AC-16, No. 3, June 1971, pp. 254 257.

[12] K.V. Ramachandra, "State estimation of maneuvering targets from noisy radar measurements", lEE Proc., Vol. 135, Pt. F, No. 1, February 1988, pp. 82 84.

[13] K.V. Ramachandra and V.S. Srinivasan, "Steady state results for the X, Y, Z Kalman tracking filter", IEEE Trans. Aerospace Electron. Systems, Vol. AES-13, No. 4, July 1977, pp. 419-423.


analysis of suboptimal filtering algorithms for target tracking

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