modeling and filtering for tracking maneuvering targets

7/27/2019 Modeling and Filtering for Tracking Maneuvering Targets

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Al-KhwarizmiEngineering

Journal

Al-Khwarizmi Engineering Journal, Vol. 5, No. 2, PP 1 - 9 (2009)

Modeling and Filtering for Tracking Maneuvering Targets

Sadiq J. Abou-LoukhElectrical Engineering Department / College of Engineering / University of Baghdad

(Received 1 September 2008; accepted 25 May 2009)

Abstract

A new mathematical model describing the motion of manned maneuvering targets is presented. This model is simpleto be implemented and closely represents the motion of maneuvering targets. The target maneuver or acceleration is

correlated in time. Optimal Kalman filter is used as a tracking filter which results in effective tracker that prevents theloss of track or filter divergency that often occurs with conventional tracking filter when the target performs a moderateor heavy maneuver. Computer simulation studies show that the proposed tracker provides sufficient accuracy.

Keywords: Kalman Filter, Modelling and Filtering,Tracking Maneuvering Targets.

1. Introduction

For years lots of effort has been spent on the

development of sophisticated digital filtering

algorithms for tracking maneuvering targets.These algorithms can be classified into classicaland modern. The classical algorithms include least

squares and polynomial filter [1, 2, 3], Wienerfilter [4], and filter [5, 6]. The two-point

extrapolator is considered as a non-recursive filterand can be implemented without any need for astorage device [3]. The function of this filter issimply obtained through the use of the last twodata points. The other simple approach is theWiener filter. It is a constant gain filter which is

equivalent to the steady state gain of the regularKalman filter [7]. Wiener filter does not requirethe calculation of the covariance elements; thusthis filter does not account for the variation andthe statistics in the target maneuver. Furthermore,

this scheme incurred the problem of tracking boththe maneuvering and non maneuvering targets

with the same accuracy, as well as might evenloose the track or diverge.

The filter is another classical trackingscheme extensively utilized in most modesttracking scenarios [5, 6]. It is designed to

minimize the mean square error in the filteredstate under the assumption that the target moves

along straight line trajectory, so it has small

capability to track severely maneuvering targets.

For this reason, various maneuvering detectors areoften attached to facilitate its job against evasive

vehicles.

The modern algorithms involve the use of statespace estimation and adaptive Kalman filtering[8]. Gurfil et. al. [9] suggest an attractivealternative method to the standard Kalman filter to

optimally estimate three dimensional states ofmaneuvering target in two steps: the first is linear

and the second is nonlinear. Another techniquedescribed by Sinha et. al. [10], involves switchingbetween the Kalman-levy filter and the standardKalman filter. The Kalman-levy filter is more

effective in response to large error due to theonset acceleration or deceleration; while theperformance of this filter is worse in the non-maneuvering portion. For this reason the system

switched to the standard Kalman filter.In this paper a simple and accurate target

model is developed. The maneuver equations are

derived for the actual continuous time targetmotion and then expressed in discrete timeaccording to the standard discretization procedureproviding accurate statistical representation of the

true target behavior [11]. The remaining parts ofthis paper are devoted to dynamic equation of

target maneuver, discrete time target equations ofmotion, optimal Kalman tracking filter andcomputer simulation.


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Sadiq J. Abou-Loukh Al -Khwar izmi Engineer ing Journal, Vol. 5, No. 2, PP 1 - 9 (2009)

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2. Dynamic Equations of Target Maneuver

The model stimulated in this section is basedon the fact that without maneuver the target under

consideration, e.g. aircraft, generally follows astraight line constant speed trajectory. Turn,

evasive maneuvers and accelerations due toatmospheric turbulence may be viewed as

perturbations on this flying trajectory. Thecontinuous time target equation of motion may berepresented by [11]:

)(.)(.)( taGtXFtXdt

d (1)

where: a(t) is target acceleration

)(

)()(

tv

tr

tX

and

1

0G

The acceleration term a(t) is assumed to bewhite Gaussian noise. The normality assumptionof the noise is one of the necessary conditions forapplying the theory of optimum Kalman filter [7].

However, the whiteness here seems to be

inappropriate justification for the real-world auto-commanded vehicles. For such vehicles, the targetacceleration and hence the target maneuver arecorrelated in time: namely, if the target isaccelerated at time t, it is likely to be acceleratedat time (t + ) for sufficiently small . Forexample, a lazy turn will often give rise tocorrelated acceleration inputs for up to one

minute; evasive maneuvers will provide correlatedacceleration inputs for periods between ten to

thirty seconds and atmospheric turbulence mayprovide correlated acceleration inputs for one to

two seconds. A typical representative model ofthe correlation function c() associated with thetarget acceleration is assumed to be:

0)(.)()( //2 betataEc bm (2)

where,2

m is the variance of the target

acceleration and b is the reciprocal of themaneuver (acceleration) time constant.

For example:b 1/60 for a lazy turn, b 1/20

for an evasive maneuver and b 1 foratmospheric turbulence.

Now, taking the Laplace transform of bothsides of Eq. (2) and by partitioning the result, one

can get

(3)

where : { . } is the Laplace transform operator,

and

The term H(s) is the transfer function of thephysical shaping filter for a(t), and W(s) is the

transform of the white noise w(t) that drives a(t).

The resulting equation of the shaping filter in timedomain is

)()(.)( twtabta (4)

For which )(Wc is the correlation function of

the input white noise which satisfies

)(..2)( 2 tbc mW (5)

This secondary system is blended with the

pervious two state per coordinate target modelEq.(1) to obtain the overall augmented targetmodel which is driven by a white noise w(t) asfollows :

)(.)(.)( twGtXFtXdt

d (6)

where :

)(

)(

)(

)(

ta

tv

tr

tX

w(t) is a zero mean white Gaussian noisedriving function with covariance equal

to2

2 mb,

b

F

00

100

010

and

1

0

0

G

00

10F

)(.)(.)(

)()(

2)()(

2

sWsHsH

bsbs

bcsC m

target range at time t

target velocity at time t

22)(

)(

1)(

mbsW

bssH


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3

3. Discrete Time Target Equations of

Motion

The discrete form of the target model canreadily be found by discretizing the continuous

form of the target equation of motion described in

Eq.(6) by simply using the standard discretizationprocedure explained in [11] . This is done by

integrating Eq.(6) over the interval (t , t+T) to get

dwGetXeTtX

Tt

t

TtFTF)(.)(.)(

))((

(7)

Rewriting Eq.(7) in appropriate form, then

)()(.),1()1( kukXkkkX (8)

where)(

1212),(

Fe

dwGeku

Tk

kT

TkF.)(..)(

)1(

)1(

(9)

and t = kT

It can be easily verified that the state transitionmatrix ),1( kk is

(10)

And when bT is small, this matrix can be reducedto the Newtonian matrix

100

10

2/1 2

T

TT

(11)

Furthermore, the input vector of the maneuver

excitation noise u(k) given in the target modelEq.(8) is not equivalent to the sampled version of

the continuous time white noise w(t) as it is seenin Eq.(9). After substituting F and G in Eq.(9), theinput noise vector is determined as follows :

w()d

(12)

Since w(t) is a zero mean white Gaussian noise,then u(k) is a discrete time white Gaussian

sequence with zero mean and covariance matrix

Q(k):

)(.)()(.)(0)(

jkkQjukuE

kuE

T

(13)

where: (.) is Kronecker delta symbol, and

(14)

After substituting the matrices G , TG , andT

in Eq. (14), the covariance matrix is simplified to:

333231

232221

131211

2.2

qqq

qqqqqq

bm

(15)

where:

Tk

kT

TT

m dTkGbGTkkQ

)1(

2).,)1((2.),)1(()(

Tk

kT

m d

nnnnn

nnnnn

nnnnn

bkQ

)1(

2

32313

32

2

212

3121

2

1

2

)()()()()(

)()()()()(

)()()()()(

.2)(

T

kk

00

10

1

),1( bTe

b]1[

1

bTebTb

112

bTe),1( kk

Tkb

Tkbb

Tkb

Tkbb

ku

Tk

kT

)1(exp

])1([exp1[

1

)1(exp

)1(1[1

)(

2

)1(

222

412

2233

2

511

22212

1

.423

221

2

1

TbbTebTeeb

q

ebTTbTb

bTeb

q

bTbTbT

bTbT

]

exp[-b{(k+1)T-}]]

Tkb

Tkbb

)1(exp

])1([exp1[1

exp[-b{(k+1)T-}]--

Tk

kT

dw

n

n

n)1(

3

2

1

)(.

)(

)(

)(


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4

When bT is sufficiently small then

TTT

TTT

TTT

bkQLimit mbT

26

238

6820

2)(23

234

345

2

0

Reflecting the fact that for sufficiently short

time periods the physical target moves atessentially constant velocity. For a fixed samplingperiod T,

as b

Furthermore, to be able to apply the theory ofoptimal Kalman filtering, an output equation is

needed to supply the desired information aboutthe system. Along each independent coordinate

(range, elevation or azimuth angle) beinganalyzed and processed, an observation or outputmodel should be defined. This model describesthe tracking sensor or measuring channel which issimply modeled as a sampled version of the

observation disturbed by an additive whiteGaussian noise corrupting the measured

information. Again the range channel isconsidered here as follows:

)()()( knkrky rr (16)

where: yr(k) is the measured range,

r(k) is the exact range,nr(k) is the additive white Gaussiannoise uncorrelated with u(k) and have the

following statistics

0)( knE r )(.)(.)( 2 ikinknE rrr

.&0)(.)( ikallforiuknE r And

2

r is the variance of the observation

channel noise.Rewriting Eq.(16) in terms of the target state,

)()(.)( knkXHky rr (17)

where: 001H is the observation matrix.

The target and observation model for elevation

angle )(k and azimuth angle (k) can be easily

derived using exactly the same manipulations thatare used to derive the range model. However, thefinal form of these models are given here and as

follows :

Elevation Angle

Target model :

)()(.),1()1( kukXkkkX eee

Observation model : )()(.)( knkXHky eee

Prior statistics: 0)( kuE e

kiallforinkuE

ikinknE

ikkQiukuE

knE

ee

eee

e

T

ee

e

&0)(.)(

)(.)(.)(

)(.)()(.)(

0)(

2

Matrices:

100

10

2/1

),1(

2

T

TT

kk ,

,

and

Azimuth Angle

Target model :

)()(.),1()1( kukXkkkX aaa

Observation model : )()(.)( knkXHky aaa

Prior statistics: 0)( kuE a

200

000

000

)(

m

bkQLimit

2332

1331

1221

2

33

2

223

2

322

2

313

12

1

.21

2

1

23.42

1

.212

1

qq

qq

qq

eb

q

ee

b

q

bTeeb

q

ebTeb

q

bT

bTbT

bTbT

bTbT

001H

Te kkkkX )()()()(


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5

kiallforinkuE

ikinknE

ikkQiukuE

knE

aa

aaa

a

T

aa

a

&0)(.)(

)(.)(.)(

)(.)()(.)(

0)(

2

Matrices :

100

10

2/1

),1(

2

T

TT

kk ,

001H ,

and Ta kkkkX )()()()(

where:

)(,)(,)( kkk are elevation angle, first

and second derivative of elevation angle

respectively,

)(,)(,)( kkk are azimuth angle, first

and second derivative of azimuth anglerespectively,

22 , ae are the error (variance of observation

channel noise) in the measured elevation

and the azimuth angles respectively,

)(,)( kyky ae are the measured elevation and

the azimuth angles respectively.

It is clear that the developed models for range,

elevation and azimuth channels are decoupled,because there is no cross-coupling or dependencebetween any two associated items of any channel.Thus, these coordinates can be processed andestimated via implementing three independenttracking filters.

4. Optimal Kalman Tracking Filter

The aforementioned target and observationmodels for range, elevation and azimuth

coordinates have similar aspects and they aresuitably to confirm the requirements of

implementation Kalman filtering algorithm. It isrecommended here to define Kalman trackingfilter for one channel only (the range channel),while the others are exactly the same. The

following equations summarize the recursiveKalman tracking filter for the range

coordinate [7]:Target model:

)()(.),1()1( kukXkkkX

Observation model : )()(.)( knkXHky rr

Filtered estimate:

Predicted estimate: )/(.)/1( kkXkkX

Kalman gain:

12)/1(..)/1()1( r

TTHkkPHHkkPkK

Covariance matrix of predicted error:

)(.)/(.)/1( kQkkPkkPT

Covariance matrix of filtered error:

)/1(.)1()1/1( kkPHkKIkkP

Estimate of maneuvering target range

coordinate by these Kalman filter recursive

equations require an initial estimates of )0/0(X

and )0/0(P to be inspired. The initialization is

based on the first two observations as follows:

)0()0/0( ryr

Tyyvrr )0()1()0/0(

0)0/0( a

TavrX )0/0()0/0()0/0()0/0(

where: yr(0) and yr(1) are, respectively, the firstand second received sensor measurements. The

corresponding covariance matrix of the filterederror estimated is defined as:

2

11 )0/0( rP

TPP r /)0/0()0/0(2

2112 22

22 /2)0/0( TP r

5. Computer Simulations

Computer simulation studies are used to verify,

compare and evaluate the performance of the

developed model. The tracking filter is exercisedunder different flight environments. Tracking

.)1()/(.)1/1( kKkkXkkX )1(kyr )/(.. kkXH

0)0/0()0/0( 3332 pp

)0/0()0/0()0/0( 233113 PPP


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performance is evaluated by tracking an accuratefigure given by:

21

2

. )/()(1

)(

N

i

est iirirN

r

where it is interpreted as the Root-Mean-Square(RMS) of the range estimate error of N estimated

points on the tracked trajectory.However, computer simulation requires two

additional subroutines. The first is used togenerate a wide class of maneuvering targettrajectories, from lightly to heavily maneuveredtargets and with three different turns ( 90 , 180,270 ). The second subroutine generates a white

Gaussian noise with different strengthsrepresenting the additive observation channel

corruptions (.)rn .

For different cases are examined as follows:

Case One:

Different target aviations are simulated and

unified to the datum of mr 150 , sT 1.0 ,

initial velocity smv /500)0( . The targetperforms three independent turns of 90, 180, and

270 for each single flight and with differentaccelerations: 1, 10, 20, 30, 40, and 50 m/s

2.

These trajectories are generated and sampled at aninterval T=0.1 s. Observations are formed usingwhite Gaussian noise generator with the specified

standard deviation ( mr 150 ). For each run,

1200 observations (N=1200) are constructed.These data are then filtered by standard Kalmanfilter based on the developed model assuming a

moderate value for 2/2 smmm and for alltrajectories. The tracking accuracy is computed(MSE) using 1200 estimated points. The resultsare listed in table (1).

Table 1,

Range Tracking Accuracy for Different Target

Maneuvers and Turns.

Targetacceleration

( m/s2

)

Range tracking accuracy

mrest )(.

90 turn 180 turn 270 turn

110

2030

4050

38.6738.89

40.5346.65

49.8455.26

38.7340.62

43.4048.17

53.3158.07

39.3240.93

44.7648.52

54.1658.78

Case Two:

For the purpose of evaluating the tracking

accuracy of the proposed tracker, the trackingperformance of the proposed tracker is compared

with the performance of other tracking filters such

as filter [6] and Wiener filter [4] underdifferent flight environments. It is assumed that

1200,1.0,150 NsTmr and 90 turn.Computer results are shown in table (2).

Table 2,

Range Tracking Accuracy of Different Filters.

Targetacceleration

( m/s2

)

Tracking accuracy

mrest )(. Proposed

filter

filter

Wiener

filter1

10

2030

4050

38.6738.89

40.5346.65

49.8455.26

30.7136.13

72.62divergent

divergentdivergent

28.2634.88

78.04divergent

divergentdivergent

It is clearly seen from these results that Wienerfilter and filter are suitable only for trackingnon-maneuvering or slowly fluctuating targets.

Case Three:

All parameters in target and observation

models can be specified with sufficient accuracybefore processing the trajectory of enemymaneuvering target except for the variance of thetarget acceleration or maneuver since this

parameter describes the target behavior or statisticof target maneuverability during its flight.

Actually, the target usually behaves inundetermined aspects unknown to the tracking

filter. This fact leads to incorrect choice ofm

and hence degradation in filter tracking accuracy.

The effect of uncertainty in m on the range

tracking accuracy is investigated by simulatingvarious trajectories with the following parameters:

msT r 150,1.0 , 90 turn and a = 1, 10, 20,30, 40, and 50 m/s2 and processing each trajectory

using three different values of m as :

. The resultsare listed in table (3).

The symbol*

in each row of table (3) denotes

the highest tracking accuracy achieved for theconsidered target maneuver or acceleration. It is

222/5/2,/5.0 smandsmsm


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7

well evident from these results that for lightlymaneuvered target ( a 10 m/s2 )

2/5.0 smm is more suitable while for

heavily maneuvered targets ( a 30 m/s2 )2

/5 smm is more suitable than 5.0m or

2/2 sm .

Table 3,

Range Tracking Accuracy for Different Standard

Deviation of Target Acceleration.

Target

acceler-

ation

( m/s2

)

Range tracking accuracy

mrest )(. 2/5.0 smm

2/2 smm 2/5 smm

110

203040

50

24.13*25.30*

49.0678.82106.55

129.28

38.6738.89

40.53*46.6549.84

55.26

63.2254.87

48.4542.13*44.28*

51.76*

Case Four:

Although2

/5.0 smm and2

/5 smm

provide high tracking accuracy for processing

trajectories of lightly and heavily maneuvered

targets respectively; however, these values are notthe proper or optimum m . In this run, optimal

value of m , that yields the highest tracking

accuracy, is searched for the simulationtrajectories of case three. These attributes areshown in table (4).

Table 4,

Range Tracking Accuracy at Optimum StandardDeviation of Target Acceleration for Different

Target Maneuvers.

Target

acceleration

( m/s2

)

Range

tracking

accuracy

mrest )(.

Optimum

)/( 2smm

1

10203040

50

19.37

23.6828.7629.5328.13

30.49

0.02

0.83.25.96.7

8.3

These results show that optimal variance of targetacceleration varies in wide extents and have great

influence on the tracking performance of thefilter.

6. Conclusion

Using a simple target model that accountsstatistically for the magnitude and duration oftarget maneuver has shown how a Kalman filtercan be constructed to track maneuvering targets.

The important features of the presented targetmodel are : firstly, it is simple to be implemented,

secondly, it is able to describe wide class ofmaneuvering target trajectories from lightly toheavily maneuvered targets, and thirdly, It isderived in a decoupled form for the range,elevation and azimuth angles. Thus, these

coordinates can be processed and estimated viaimplementing three independent tracking filters.

This advantage facilitates the tracker activity intwo ways. First, the computational efforts aregreatly reduced since the overall systemdimension is reduced from 9 x 9 to three separate

models of 3 x 3 dimensional subsystems for eachcoordinate. Secondly, the system reliability isfurther enhanced when applied for on-line tacticalcombat conditions.

The tracking performance of the proposedKalman filter has been analyzed and tested byusing different computer simulation studies. It is

shown that using the proposed filter, the error insensor range measurement is reduced from

)150(150 mm r to (4060) meters dependingon the target maneuverability as shown in

table (1).The tracking performance of the filter is also

compared with the tracking filter [5, 6] andWiener filter [4] under various flightenvironments. These two filters exhibit higher

tracking accuracy than the suggested Kalmanfilter. In case of applying these two filters in realworld, the filter may lose the track or divergewhen the target performs moderate or heavymaneuver (a30 m/s

2table (2)) while the target

presented here will never diverge.The main problem addressed by computer

simulation studies is the degradation in tracking

performance due to uncertainty in modelparameters especially the variance of target

acceleration2

m . A comparison between table (4)

and table (3) illustrates how the tracking accuracyis significantly improved when m is properly


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selected to fit the target maneuver. In all computer

simulation studies, it is assumed that m is

constant for the whole trajectory; however, thisassumption is not always correct. Thus, the

demand for on-line adaptation of m is greatly

highlighted to enhance the filter performance infront of any sudden changes encountered duringtarget flight trajectory.

7. References

[1] Eli Brooker, " Tracking and Kalman FilteringMade Easy ", John Wiley & Sons, Inc., 2001.

[2] Lennert Ljung, " System Identification Theoryfor the User ", Prentice Hall, Inc., 1999.

[3] C. Schell, S.P. Linder and J.R. Zeidler,

"Tracking Highly Maneuverable Target withUnknown Behaviour", Proceedings of theIEEE, Vol. 92, No.3, pp. 558-574, March2004.

[4] X.R. Li and V.P Jilkov, "Survey ofManeuverable Target Tracking-Part I:

Dynamic Models", IEEE Trans. on Aerospaceand Electronic systems, Vol. AES-39, Issue4,pp. 1333-1364, Oct. 2003.

[5] Jae-Chem Yoo and Young-Soo Kim, "Alpha-Beta Tracking Index (--) Filter", Signal

Processing, Elsevier, Vol.83, Issue 1, pp. 169-180, Jan. 2003.

[6] T.L. Ogle and W.D. Blair, "Fixed-Lag -Filter for Target Trajectory Smoothing, IEEETrans. on Aerospace and Electronic Systems,Vol. AES-40, Issue 4, pp. 1417-1421 Oct.

2004.[7] Dan Simon, "Optimal State Estimation:

Kalman H-infinity and NonlinearApproaches", John Wiley & Sons, Inc., 2006.

[8] Y. Yu and Q. Cheng, "Particle Filters forManeuvering Target Tracking Problems",Signal Processing, Elsevier, Vol. 86, Issue 1,

pp. 195-203, Jan. 2006.[9] P. Gurfil and N.J. Kasdin, "Two Step Optimal

Estimator for Three Dimensional TargetTracking", IEEE Trans. on Aerospace and

Electronic Systems, Vol. AES-41, Issue 3, pp.

780-793, July 2005.[10] A. Sinha, T.K. Barajan and Y. Bar-Shalom,

"Application of the Kalman-Levy Filter forTracking Maneuvering Targets", IEEETrans. on Aerospace and ElectronicSystems, Vol. AES-43, Issue 3, pp. 1097-

1107, July 2007.[11] C.K. Chui, "Kalman Filtering: with Real-time

Applications", Second Edition, New York,Springer-Verlag, 1999.

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