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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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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. 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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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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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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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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5 2 1-9(200 9)
9
/ /
: . . . ((Kalman filter (tracking)
. .