2D rigid-body target modelling for tracking andidentification with GMTI/HRR measurements

S. Wu, L. Hong and J.R. Layne

Abstract: Joint ground moving-target tracking identification is a crucial task in a modern combatoperation. Due to the entirely different environment, ground moving-target tracking is quitedifferent from airborne target tracking. A major difference lies in target modelling. In airbornetarget tracking, a target is usually treated as a point, while for ground target tracking, a target isconsidered a rigid body. Two approaches for 2D rigid-body target modelling are proposed.Equipped with ground moving-target indicator and high-resolution range sensors, the newapproaches effectively explore the concepts of local and global motions of a rigid body. Thekinematics of a global motion is described by a constant acceleration model, and a local motionis modelled by the pivoting centre and pseudocentre approaches. The proposed models areimplemented by the extended Kalman filter with and without a probabilistic data association filter.The simulation results show that the proposed approaches not only correctly track a rigid-bodytarget in a complicated scenario but also simultaneously report its structural information.

1 Introduction

While target tracking has been a major task in an air-to-aircombat engagement, simultaneously tracking and identify-ing ground moving targets is a much needed capability inmodern air-to-ground combat operations. The importantdifference between air-to-air and air-to-ground operations isthe target modelling where in air-to-air tracking, a target istreated as a point and a target is dealt with as a rigid body inair-to-ground tracking and identification (ID). Because ofthe rigid-body assumption, a ground moving target carriesboth kinematics and ID information. How to effectivelyexploit kinematics and ID information of a ground movingtarget is an active research area. This paper focuses on rigid-body target modelling with application to simultaneoustracking and ID.

Recent progress in sensor devices, such as syntheticaperture radar (SAR) and high-resolution range (HRR)profiling radar, has made simultaneously tracking andidentifying ground moving targets feasible. Traditionalpoint-target tracking algorithms establish target trajectoriesand other kinematic information, such as velocities andaccelerations, from sequences of noisy measurements in thepresence of false alarms, countermeasures, missed detectionand manoeuvers. On the other hand, target recognition isused to identify stationary targets based on spatial domainfeatures. Therefore a successful joint tracking and identifi-cation algorithm should simultaneously exploit bothtrajectory (temporal) and spatial information.

Although various mathematical models have beendeveloped for tracking a point target in the past threedecades (see, e.g. a recent review [1]), there is littleliterature dealing with the simultaneous tracking and IDproblem. Stuff [2] initiated an attempt on applying invariantconstraints to tracking and identifying moving targets.Jacobs and O’Sullivan [3, 4] introduced a joint tracking andrecognition technique using an HRR model and a like-lihood-based approach. Miller et al. [5] proposed a methodfor generating the conditional mean estimation of targetpositions, orientation and types in recognition, and thetracking of an unknown number of targets by using themeasurements from both narrowband sensor array (forproviding azimuth and elevation angles) and optical or radarimagers (for obtaining the target type and orientation). Thejump-diffusion technique was adopted by Sworder andBoyd [6] in their tracking and recognition work. Blasch andHong [7, 8] developed a belief function based algorithm forjoint target tracking and ID using ground moving targetindicator (GMTI) and HRR measurements.

More recently, Gu and Hong proposed an algorithm totrack 2D rigid-body targets with invariant constraints [9].In [9], the motion of the target centre is estimated by usingan interval turn model [10], and the spatial features at eachinstant are updated by cartesian measurements with rigid-body constraints. Hong, Wu and Layne [11] proposed aninvariant-based interacting multiple-template algorithm forsimultaneous tracking and ID using GMTI=HRR. However,in [11] the target’s pivoting centre is assumed fixed, thoughunknown. (The pivoting centre is the point about which thetarget’s rotational motion is made. More details inSection 2.) In reality, the pivoting centre is moving in anarbitrary manner which makes the simultaneous trackingand ID problem very challenging.

This paper investigates two approaches of modelling thisgeneral case of moving rigid-body targets. One approachtries to estimate the target’s actual moving pivoting centrebased on a set of constraints, and another assumes a fixedpseudocentre (the pseudocentre may not be the actualpivoting centre) and puts the pseudocentre at an arbitrarystructure feature point. (The nature of structure feature

q IEE, 2004

IEE Proceedings online no. 20040694

doi: 10.1049/ip-cta:20040694

S. Wu and L. Hong are with the Department of Electrical Engineering,Wright State University, Dayton, OH 45435, USA

J.R. Layne is with the Target Recognition Branch SNAT, Air ForceResearch Laboratory, Wright-Patterson AFB, OH 45433, USA

Paper first received 2nd September 2003 and in revised form 23rd April2004

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 429

points depends on the type of sensors. For a radar sensor thestructure points could be electromagnetic scatteringcentres.) It is shown that it is not necessary to estimate thetarget true moving pivoting centre; instead, a pseudocentredescribes equally well the kinematic behaviour of a rigidmoving target. Both modelling approaches are implementedwith the extended Kalman filter (EKF) and the EKF with aprobabilistic data association (PDA) filter to simultaneouslytrack and provide ID information of ground moving targetsusing GMTI and HRR measurements.

2 Problem setup

2.1 Sensor target geometry

To simplify the development we assume targets are 2Dtargets, there is only one target in a surveillance region,and there are no false alarms. The method presented in thispaper can be extended for multiple targets in a clutteredenvironment. The approaches developed can be extended to3D rigid-body targets. Also, for the scenario where thesensing distance is much greater than the target size, a 3Dtarget can be well approximated by a 2D target.

Figure 1a shows a sample 2D rigid target with prominentfeature points (PFPs) marked by circles and the pivotingpoint by a square. Figure 1b presents a sensor targetgeometry where both GMTI and HRR sensors are located atthe origin and the target is moving along its trajectorymeanwhile rotating about its pivoting point. The GMTIsensor measures entire target’s kinematic information, suchas range, range-rate and azimuth angle, at a coarseresolution (yards), while the HRR sensor provides rangemeasurements ðRisÞ of PFPs at a much higher resolution(feet). With an assumption that the distance from the sensorsto the target is much greater than the target dimension size,the range rays are in parallel to each other, as shown inFig. 1b. Moreover, we denote the distance from the sensors

to the pivoting centre ðPcÞ by Rc and the azimuth angle ofthe centre range ray by yc: The distance between thepseudocentre and its ith HRR reflection PFP is described byri; and yi is defined as the angle from the local referencedirection to ri:

2.2 Definition of global and local motion

It is very important that the concept of global and localmotion is clearly defined. Figure 2 shows the motion patternof a rigid body. A rigid-body motion is composed of twokinds of motion: one is the translational motion of thepivoting centre which is defined as global motion, and theother is the rotational motion relative to the pivoting centrewhich is defined as local motion. Global motion is the sameas the point-target motion widely used by the trackingcommunity and a local motion is a new concept that we areintroducing. Here are some comments on global and localmotions: a global motion reflects the entire object motionand carries no ID information, while a local motion isencoded with object structural information i.e. how a groupof PFPs are moving together; and a local motion modulatesa global motion.

It can be proved that given composite motionmeasurements, the decomposition of local and globalmotions is not unique. Figure 3a shows true global andlocal motions, forming the composite motion of a rigidbody moving from tk in thin dashed line to tkþ1 in boldsolid line. The true global motion is a translation of thepivoting centre Tk and the true local motion is the rotationangle yk about the pivoting point. Notice that thepivoting centre is not fixed on the target; it changes itslocation from tk to tkþ1: So this is a general rigid bodymotion case. The same composite rigid body motion isshown in Fig. 3b. But this time the global and localmotions are different: T 0

k and y0k: The global motion isthe translation of one vertex, called the pseudocentre, and

Fig. 1 Rigid body target

a Prominent features and pivoting centreb Sensor target geometry

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004430

the local motion is made around this pseudocentre. Thetrue pivoting point is not needed. This nonuniqueness ofglobal and location motion decomposition actually givesus freedom in rigid-body target modelling.

3 Rigid-body target modelling

Two modelling approaches for a 2D rigid-body target arepresented. One is the pivoting centre based where the truetarget pivoting centre is estimated, and the other ispseudocentre based where an arbitrary pseudocentre is used.

3.1 Pivoting centre based modelling

As shown in Fig. 4, the rigid-body target motion can bedescribed by a sequence of actions. I: a translation of theentire target, II: a pivoting centre motion and III: a rotationaround the new pivoting centre. Further, the global motionis a combination of the translation and the pivoting centremotion, while the local motion is determined by the pivotingcentre motion and the rotation around the pivoting centre.Based on this phenomenon, in the pivoting centre basedapproach, we try to mimic the three-step motions by theglobal and local motions.

Fig. 3 Two equivalent local and global motion decompositions

a Around pivoting centreb Around pseudo centre

Fig. 4 Pivoting-centre-based modelling

Fig. 2 Local and global motion

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 431

3.1.1 Definition of state vector: We use xgk to

describe the global motion state which can be expressed as

xgk ¼ ½xc; yc; _xxc; _yyc; €xxc; €yyc�Tk ð1Þ

where ðxc; ycÞ is the target pivoting centre co-ordinate in thereference co-ordinate system. The state vector of the localmotion is defined as

xlk ¼ ½r1; r2; . . . ; rM ; y1; y2; . . . ; yM ; _yy; €yy�Tk ð2Þ

where M is the number of scattering centres, ri ði ¼1; . . . ;MÞ and yi ði ¼ 1; . . . ;MÞ are illustrated in Fig. 4.Quantities _yy and €yy express the angular velocity andacceleration (Because of the rigidity assumption, we have_yyi ¼ _yyj ¼ _yy and €yyi ¼ €yyj ¼ €yy:). Therefore the whole statevector is defined by

xk ¼ xgk

� �T; xl

k

� �Th iT

ð3Þ

3.1.2 System equation: The global motion of thetarget can be described by a constant-acceleration (CA)model (other maneuvering models, such as Singer model,can also be used for the global motion), i.e.

xckþ1¼ f1ðxkÞ ¼ xck

þ T _xxckþ T2

2€xxck

þ v1k

yckþ1¼ f2ðxkÞ ¼ yck

þ T _yyckþ T2

2€yyck

þ v2k

_xxckþ1¼ f3ðxkÞ ¼ _xxck

þ T €xxckþ v3k

_yyckþ1¼ f4ðxkÞ ¼ _yyck

þ T €yyckþ v4k

€xxckþ1¼ f5ðxkÞ ¼ €xxck

þ v5k

€yyckþ1¼ f6ðxkÞ ¼ €yyck

þ v6kð4Þ

where T is the sampling interval, and vik; i ¼ 1; . . . ; 6 is

white noise with a zero-mean normal distribution. Figure 4shows a rigid-body target moving from time tk to time tkþ1;where the target has four PFPs ðP1; P2; P3; P4Þ and thepivoting centre moves from Pck

to Pckþ1: Suppose the

pivoting centre local motion at time tkþ1 in the targetattached co-ordinate system can be approximated by itsvalue at tk; then radii at tkþ1 can be determined by

rikþ1¼ f6þiðxkÞ¼ ½ð�r1k

cosy1kþ rr1k�1

cosyy1k�1þ rik

cosyikÞ2þ

ð�r1ksiny1k

þ rr1k�1sinyy1k�1

þ riksinyik

Þ2�1=2 þ v6þi;

for i ¼ 1; 2; . . . ;M ð5Þ

where the estimates from tk�1; rr1k�1and yy1k�1

; are used.From the geometric relationship in Fig. 4, we can get

yyikþ1¼ tan�1 �r1k

cosy1kþ rr1k�1

cosyy1k�1þ rik

sinyik

�r1ksiny1k

þ rr1k�1sinyy1k�1

þ rikcosyik

;

for i ¼ 1; . . . ;M ð6Þ

Based on (6), considering the rigid-body constraint andusing the CA model to describe the angle motion, we have

yikþ1¼ f6þMþiðxkÞ

¼ yyikþ1þ T _yyk þ

T2

2€yyk þ v6þMþi; for i ¼ 1; . . . ;M ð7Þ

_yykþ1 ¼ f2Mþ7ðxkÞ ¼ _yyk þ T €yyk þ v2Mþ7 ð8Þ

€yykþ1 ¼ f2Mþ8ðxkÞ ¼ €yyk þ v2Mþ8 ð9ÞEquations (5) to (9) constitute a nonlinear local motiondynamic system. Besides, the global and local dynamicsystems are tightly coupled. In the following implemen-tation, an extended Kalman filter is used which adopts afirst-order linear approximation with jacobians. Since thesystem equations are too complex to derive their jacobiansby hand, we use the symbolic tools of Matlab.

3.1.3 Measurement equations: Two sensors,HRR and GMTI, are used in this approach, each of whichmeasures different information of the target. The GMTImode measures the whole target’s global motion (range,range-rate and azimuth angle) with a coarse resolutionwhere

zRg

z _RRg

zyg

264

375

k

¼h1

h2

h3

264

375

k

¼Rc

_RRc

yc

264

375

k

þw1

w2

w3

264

375

k

ð10Þ

where

Rck¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

c þ y2c

q; _RRck

¼ xc _xxc þ yc _yyc

Rck

; yck¼ tan�1 yc

xc

�

ð11Þand

w1

w2

w3

264

375

k

N 0;

s2Rc

0 0

0 s2_RRc

0

0 0 s2yc

2664

3775

k

0BB@

1CCA ð12Þ

is the GMTI sensing uncertainty which is relatively large.The HRR sensor measures composite motions of PFPs in theform of ranges

zR1

zR2

..

.

zRM

26664

37775

k

¼

h4

h5

..

.

hMþ3

26664

37775

k

¼

Rc þ r1 cosðy1 � ycÞRc þ r2 cosðy2 � ycÞ

..

.

Rc þ rM cosðyM � ycÞ

26664

37775

k

þ

w4

w5

..

.

wMþ3

26664

37775

k

ð13Þ

Since M PFPs are attached to the same rigid-body target themeasurement uncertainty for the ith point should becorrelated to that of the jth point. Therefore we definecommon range uncertainty, as well as individual rangeuncertainty, to reflect the rigidity constraint. The covarianceof (13) is then given by

COVfwi wig¼s2Rþs2

ri; COVfwi wjg¼s2

R;

4� i; j� Mþ3 ð14Þ

where sR is common uncertainty and sriis individual

uncertainty. The smaller the value of sR is, the less therigidity constraint is enforced.

3.2 Pseudocentre based modelling

In the psudocentre modelling approach, the location of thepseudocentre could be arbitrary. Let’s assume that thepseudocente is fixed at one of the PFPs, say the first featurepoint P1: As shown in Fig. 5, ri ði ¼ 2; . . . ;MÞ is the

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004432

distance from the ith PFP to the pseudocentre and yi ði ¼2; . . . ;MÞ is the angle between the positive reference x axisand the vector from the pseudocentre to the ith PFP. Using

the rigid-body constraint, i.e. _rri ¼ 0; _yyi ¼ _yyj ¼ _yy; and €yyi ¼€yyj ¼ €yy; the local state vector can be written as

xlk ¼ xT

r ; xTy

� �T ð15Þ

where xr ¼ ½r2; . . . ; rM�T and xy ¼ ½y2; . . . ; yM ; _yy; €yy�T : Withthe rigidity constraint, the model for xr is given by

xrkþ1¼ IM�1xrk

þ vrkð16Þ

where IM�1 is an ðM � 1Þ-dimensional identity matrix andvrk

is a zero-mean gaussian white noise. Assuming constantacceleration kinematics for angles, xy satisfies the followingmodel

xykþ1¼ Fyk

xykþ vyk

ð17Þ

where

Fyk¼

IM�1 TIM�1;1T2

2IM�1;1

O1;M�1 1 T

O1;M�1 0 1

24

35 ð18Þ

where O1;M�1 is a 1-by-ðM� 1Þ zeros matrix, IM�1;1 is anðM � 1Þ-by-1 matrix of ones, and T is the sampling period;vyk

is also assumed as zero-mean gaussian white. Putting theglobal and local motion equations together, we have adynamic system model for the pseudocentre based approach

xkþ1 ¼ Fkxk þ vk ð19Þ

where

xk ¼ ½xc; yc; _xxc; _yyc; €xxc; €yyc; r2; . . . ; rM ; y2; . . . ; yM; _yy; €yy�T ð20Þ

Fk ¼Fgk

O6;M�1 O6;Mþ1

OM�1;6 IM�1 OM�1;Mþ1

OMþ1;6 OMþ1;M�1 Fyk

24

35 ð21Þ

Fgk¼

I2 TI2

T2

2I2

O2 I2 TI2

O2 O2 I2

264

375 ð22Þ

and vk ¼ vTgk; vT

rk; vT

yk

h iT

which is a 2M þ 6 dimensional

vector of zero-mean gaussian white noise. Notice that theglobal motion in this approach is the motion ofthe pseudocentre P1 and local motions are made w.r.t. thepseudocentre.

The measurement equations for the pseudocentre basedapproach are similar to those of the pivoting centre basedapproach, except that the GMTI sensor measures the globalmotion of the pseudocentre (with a coarse resolution)instead.

A comment on the two approaches developed so far isnow in order. Trying to estimate true global and localmotions of a rigid-body target, the system model of thepivoting centre approach is highly nonlinear; while thesystem model of the pseudocentre approach is linear andsimple, based on the premise that different global and localmotions describe a same target rigid-body motion.

4 Target structural information extraction

A rigid-body moving target carries both kinematic and IDinformation. Since the global motion describes the kin-ematic behaviour of the entire target by a single point’sdynamics (either pivoting point or pseudopoint), no IDinformation can be extracted from the global motion state.On the other hand, the local motion state exhibits both thetarget’s structural information (therefore the ID informationof the target) and local kinematics. This Section discussesthe extraction of target structural information only and theextraction and application of target local kinematics can befound in [12].

The ID information is conveyed by the target structuralinformation. For a rigid-body target, the structural infor-mation is coded in the distribution of PFPs, i.e. the distancesbetween any two PFPs. Once we have estimated the localstate vector by either the pivoting centre approach or by thepseudocentre approach, one can calculate the distancesbetween any two PFPs. In the following we present suchcalculations for the pivoting point approach (similarcalculations can be applied to the pseudocentre approach).Assume we have estimated xrk

¼ ½r1; r2; . . . ; rM�Tk and xyk¼

½y1; y2; . . . ; yM ; _yy; €yy�Tk : From Fig. 4 we calculate the distancebetween points Pi and Pj by

PiPj ¼ fðrikÞ2 þ ðrjk

Þ2 � 2rikrjkcosðyjk

� yikÞg1=2 ð23Þ

where i ¼ 1; . . . ;M; and j ¼ i þ 1 for i<M; j ¼ 1 fori ¼ M: The set of distances fPiPjg carries the target IDinformation, though there is some redundant information inthe set. With these distances one can develop a method toidentify the object by using techniques such as geometrichashing or bayesian classifier. An interacting multiple-template approach using the structural information for targetID can be found in [11].

5 Simulations

As shown in Fig. 6a, a 2D rigid-body target with four PFPsis moving in a circle anticlockwise, where an enlargedversion of the target is shown in Fig. 6b. The centre of thecircle at (10,000 ft, 10,000 ft) in a reference co-ordinatesystem and the radius of the circle is R0 ¼ 500 ft: Whiletravelling at a global angular velocity o0 ¼ 0:03 rad=s the

Fig. 5 Pseudocentre-based modelling

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 433

target is also rotating with respect to its pivoting centre at avelocity O ¼ 0:11 rad=s. At the same time, the pivotingcentre is moving in a circular manner with an angularvelocity o ¼ 0:06 rad=s, and radius R ¼ 3 (Fig. 6b). Theco-ordinates of the four feature points in the target centrebased cartesian system are (25, 10), ð25; �10Þ;ð�19; �10Þ; ð�19; 10Þ: In addition, we assume that theHRR and GMTI sensors are located at (0, 0) and thesampling period T ¼ 0:02 s:

For the scenario shown in Fig. 6, our data generationprogram provides measurement data for both the pivotingcentre and pseudocetre approaches. The noise intensities ofHRR and GMTI measurements are fixed as sR ¼ 1 ft; sri

¼0:1 ft; sRc

¼ 10 ft; s _RRc¼ 5 ft=s and syc

¼ 0:01 rad: On thesystem modelling side, we choose the noise intensities asqc ¼ 0:5; qy ¼ 0:01; qr ¼ 0:01 for both modellingapproaches. As an example, sample HRR and GMTImeasurements for the pivoting centre approach are shownin Fig. 7.

Assuming the correspondences between PFPs and rangemeasurements are known, the standard extended Kalmanfiltering algorithm is used to estimate the state vector for theboth approaches to test the correctness of the modellingmethods. Figures 8 and 9 show the estimation results of thepivoting centre and the pseudocentre approaches respect-ively, where solid curves are the trues and dashed curves arethe estimated. Since the pivoting centre and pseudocentreapproaches define different state vectors, it is difficult todirectly compare them. However, because they bothreconstruct the target structural information, e.g. (23), weplot the reconstructed distances, as well as the trues, betweenPFPs for both approaches in the left column of Fig. 12.

When we do not have knowledge of the correspondencesbetween PFPs and range measurements, the probabilisticdata association (PDA) algorithm is needed. In the PDAalgorithm the permutation technique is used to solve thecorrespondence problem. In this way there are a total of M!possible measurement arrangements at each instant. Amongthe M! possible arrangements only one set reflects the realtrue-measurement pairing and the others are considered tobe false alarms. To handle multiple measurements withuncertainty, the EKF with probabilistic data association(EKF-PDA) algorithm [13] is used. The tracking results ofthe pivoting centre and pseudocentre approaches with anEKF-PDA algorithm are shown in Figs. 10 and 11,respectively. A comparison of reconstructed structuralinformation with both modelling approaches using anEKF-PDA algorithm is given in the right column ofFig. 12. To numerically compare the two modellingapproaches, the averaged RMS errors of 50 Monte Carloruns for the cases of with and without permutations arepresented in Table 1. As expected, without knowledge ofcorrespondences, the results are not as good as the ones withthe knowledge of correspondences.

Fig. 7 HRR ðzRi; i ¼ 1; . . . ; 4Þ and GMTI ðzRc

; z _RRc; and zyc

Þ measurements for pivoting centre approach

Fig. 6 Simulation scenario with global motion

a 2D rigid body target with four PFPs moves in circle anticlockwiseb Enlarged version of target

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004434

Fig. 8 Average (with 50 Monte Carlo runs) estimated and true state vectors of pivoting centre modelling approach with knowledge ofcorrespondences

Fig. 9 Average (with 50 Monte Carlo runs) estimated and true state vectors of pseudocentre modelling approach with knowledgeof correspondences

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 435

Fig. 10 Average (with 50 Monte Carlo runs) estimated and true state vectors of pivoting centre modelling approach without knowledge ofcorrespondences

Fig. 11 Average (with 50 Monte Carlo runs) estimated and true state vectors of pseudocentre modelling approach without knowledge ofcorrespondences

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004436

By examining the simulation Figures and Table 1, thefollowing remarks can be made. Both the pivoting centreand pseudocentre approaches not only estimate decentglobal and local motion information, but also providecorrect structural information. And the pseudocentreapproach in general outperforms the pivoting centreapproach owing to its linear and simple system modelling.

6 Conclusions

We have developed two modelling approaches for a rigid-body moving target: one based on pivoting centre estimationand the other assumes an arbitrary pseudocentre. Whenimplemented with the EKF and the EKF-PDA, it turns outthat the pseudocentre approach has a better performance

Fig. 12 Average (with 50 Monte Carlo runs) reconstructed target structural information

a With knowledge of correspondencesb Without knowledge of correspondences

Table 1: Averaged RMS errors of state vectors over 50 Monte Carlo runs

Approach

Pivoting centre ‘know’

correspondences

Pseudocentre ‘know’

correspondences

Pivoting centre ‘don’t

know’ correspondences

Pseudocentre ‘don’t know’

correspondences

xc 13.46 13.67 15.84 16.02

yc 13.45 13.66 15.83 16.07

_xxc 7.21 7.36 7.20 5.96

_yyc 7.21 7.36 7.20 5.92

€xxc 2.14 2.43 1.79 1.16

€yyc 2.14 2.43 1.79 1.12

r1 0.4616 0.0187 0.5184 0.0687

r2 0.4119 0.0207 0.5261 0.0697

r3 0.4510 0.0124 0.5332 0.0504

r4 0.4245 0.0000 0.5302 0.0000

�1 0.2632 0.0957 0.2892 0.1517

�2 0.3081 0.1009 0.3278 0.1239

�3 0.3539 0.1187 0.3685 0.1497

�4 0.2622 0.0000 0.2699 0.0000

_�� 0.0090 0.0151 0.0090 0.0056

€�� 0.0248 0.0295 0.0192 0.0086

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004 437

because of its linear and simple system model. The modelsdeveloped in this paper play an important role in trackingand identifying rigid-body moving targets.

7 Acknowledgment

The authors would like to thank the anonymous reviewersfor suggesting references [1] and [5], and their helpfulcomments.

8 References

1 Li, X.R., and Jilkov, V.P.: ‘Survey of maneuvering target tracking. PartI: Dynamic models’, IEEE Trans. Aerosp. Electron. Syst., 2003, 39, (4),pp. 1333–1364

2 Stuff, M.A.: ‘Three-dimensional analysis of moving target radarsignals: methods and implications for ATR and feature-aided tracking’,Proc. SPIE Int. Soc. Opt. Eng., Orlando, FL, 1999, 3721, pp. 485–496

3 Jacobs, S.P., and O’Sullivan, J.A.: ‘High-resolution radar model forjoint tracking and recognition’. Proc. IEEE National Radar Conf., May1997, pp. 99–104

4 O’Sullivan, J.A., Jacobs, S.P., Miller, M.I., and Snyder, D.L.:‘A likelihood-based approach to joint target tracking and identification’.

Proc. 27th Asilomar Conf. on Signals, Systems, and Computers,November 1993, vol. 1, pp. 290–294

5 Miller, M.I., Srivastava, A., and Grenander, U.: ‘Conditional-meanestimation via jump-diffusion processes in multiple-target tracking/recognition’, IEEE Trans. Signal Process., 1995, 43, (11),pp. 2678–2690

6 Sworder, D.D., and Boyd, J.E.: ‘Jump-diffusion processes intracking=recognition’, IEEE Trans. Signal Process., 1998, 46, (1),pp. 235–239

7 Blasch, E., and Hong, L.: ‘Sensor fusion cognition using belief filteringfor tracking and identification’. Proc. SPIE Int. Soc. Opt. Eng., 1999,3719, pp. 250–259

8 Blasch, E., and Hong, L.: ‘Simultaneous feature-based identificationand track fusion’. Proc. of IEEE Int. Conf. on Decision and Control,Tampa, FL, December 1998, pp. 239–244

9 Gu, B., and Hong, L.: ‘Tracking 2-D rigid targets with invariantconstraint’, Inf. Sci., 2001, 138, pp. 79–97

10 Cong, S., and Hong, L.: ‘Interval models for target tracking algorithms’,Math. Comput. Model., 2001, 34, pp. 593–602

11 Hong, L., Wu, S., and Layne, J.: ‘Invariant-based probabilistic targettracking and identification with GMTI/HRR measurements’, submittedto IEE Proc., Radar, Sonar Nav.

12 Hong, L., Cui, N., Pronobis, M., and Scott, S.: ‘Local motion feature-aided ground moving-target tracking with GMTI and HRR measure-ments’, IEEE Trans. Autom. Control (to be published)

13 Bar-Shalom, Y., and Fortman, T.E.: ‘Tracking and data association’(Academic Press, San Diego, CA, 1988)

IEE Proc.-Control Theory Appl., Vol. 151, No. 4, July 2004438