models and algorithms for tracking target with coordinated

Research ArticleModels and Algorithms for Tracking Target withCoordinated Turn Motion

Xianghui Yuan Feng Lian and Chongzhao Han

Ministry of Education Key Laboratory for Intelligent Networks and Network Security (MOE KLINNS)School of Electronics and Information Engineering Xirsquoan Jiaotong University Xirsquoan 710049 China

Correspondence should be addressed to Feng Lian lianfeng1981mailxjtueducn

Received 23 July 2013 Accepted 15 December 2013 Published 12 January 2014

Academic Editor Shuli Sun

Copyright copy 2014 Xianghui Yuan et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Tracking target with coordinated turn (CT) motion is highly dependent on the models and algorithms First the widely usedmodels are compared in this papermdashcoordinated turn (CT)modelwith known turn rate augmented coordinated turn (ACT)modelwith Cartesian velocity ACT model with polar velocity CT model using a kinematic constraint and maneuver centered circularmotion model Then in the single model tracking framework the tracking algorithms for the last four models are compared andthe suggestions on the choice of models for different practical target tracking problems are given Finally in the multiple models(MM) framework the algorithm based on expectation maximization (EM) algorithm is derived including both the batch formand the recursive form Compared with the widely used interacting multiple model (IMM) algorithm the EM algorithm shows itseffectiveness

1 Introduction

Theproblem of tracking a single target with coordinated turn(CT) motion is considered The motion of a civil aircraft canusually be modeled as moving by constant speed in straightlines and circle segments The former is known as constantvelocity (CV)model and the latter is coordinated turnmodelIn tracking applications only the position part of the state canbe measured by the sensor and the turn rate 120596 is often un-known So the measurement data can be seen as the incom-plete data This is a resource-constrained problem for track-ing target with coordinated turn motion

CTmodel is highly dependent on the choice of state com-ponents [1] The turn rate 120596 can be augmented in the CTmodel called ACT model There are two types of ACT mod-els ACTmodel with Cartesian velocity and ACTmodel withpolar velocity The state vectors are [119909 119910 119910 120596]

1015840 and[119909 119910 V 120601 120596]1015840 respectively The two are both nonlinear mod-els and have been compared in [2 3] based on EKF ForunscentedKalman filter (UKF) is a very efficient tool for non-linear estimation [4 5] here the two models are comparedbased on UKF

When the target with CT motion has a constant speed itsatisfies a kinematic constraint119881sdot119860 = 0 where119881 is the target

velocity vector and 119860 is the target acceleration vector If thedynamic model incorporates the constraint directly it willbecome a highly nonlinear one To avoid this nonlinearity thekinematic constraint was incorporated into a pseudomea-surement model [6ndash8]

A maneuver-centered model is introduced in [9] Thestate components are [119903 120579 120596]1015840Themodelrsquos state equation hasa linear form but its measurement equation is pseudolinearbecause the noise covariance is actually state dependent [10]The center of the turn should be accurately determinedwhich is inherently a nonlinear problem

Target dynamic models and tracking algorithms haveintimate ties [1] In the single model tracking framework thetracking algorithms are interpreted and compared

The interactingmultiplemodel (IMM) approach has beengenerally considered to be the mainstream approach tomaneuvering target tracking It utilizes a bank of 119873 Kalmanfilters each designed tomodel a differentmaneuver [11] IMMalgorithm is a suboptimal algorithm based on the minimummean square error (MMSE) criterionUnder theMMSE crite-rion to get the optimal estimation of the target state the com-putational load grows exponentially when the measurementsare increasing In recent years tracking target based onmaxi-mum a posteriori (MAP) criterion has received a lot of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 649276 10 pageshttpdxdoiorg1011552014649276

2 Mathematical Problems in Engineering

interest [12ndash17] Expectation maximization (EM) algorithmis the state estimation approach based on MAP criterion Using EMalgorithm the computational load grows linearly dur-ing per iteration and the optimal estimation based on MAPcriterion can be achieved finally

The existing EM algorithm to track maneuvering targetcan be classified into two categories one formulates themaneuver as the unknown input [12ndash14] and the other for-mulates themaneuver as the systemrsquos process noise [15] Aim-ing at the problem to track a target with CTmaneuver an EMalgorithm is presented The maneuver is formulated by theturn rate First the turn rate sequence is estimated using theEM algorithm Then with the estimated turn rate sequencethe target state sequence is estimated accurately

The rest of this paper is organized as follows Section 2presents all the CTmodelsrsquo state equations and measurementequationsThe tracking algorithms based on single model areinterpreted in Section 3 the simulations are also presented InSection 4 the batch and recursive EM algorithms are derivedand compared with the IMM algorithm in simulationSection 5 provides the paperrsquos conclusions

2 Dynamic Models for CT Motion

A maneuvering target can be modeled by

119883119896+1

= 119891119896(119883119896) + 119908119896

119911119896= ℎ119896(119883119896) + 119890119896

(1)

where 119883119896and 119911

119896are target state and observation respec-

tively at discrete time 119905119896 119908119896and 119890

119896are process noise and

measurement noise sequences respectively 119891119896and ℎ

119896are

vector-valued functions

21 CT Model with Known Turn Rate The coordinated turnmotion can be described by the following equation

119883119896+1

= 119865 (120596119896)119883119896+ 119908119896 (2)

The measurement equation is

119911119896= 119867CT119883119896 + 119890119896 (3)

The components of state are119883 = [119909 119910 119910]1015840 120596119896stands

for the turn rate in time 119896Where

119865 (120596119896) =

[[[[[[[[[[[[

[

1

sin (120596119896119879)

120596119896

0 minus

1 minus cos (120596119896119879)

120596119896

0 cos (120596119896119879) 0 minus sin (120596

119896119879)

0

1 minus cos (120596119896119879)

120596119896

1

sin (120596119896119879)

120596119896

0 sin (120596119896119879) 0 cos (120596

119896119879)

]]]]]]]]]]]]

]

119908 = [119908119909119908119910]

1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876CT120575119896119897

(4)

Assume only position could be measured where

119867CT = [1 0 0 0

0 0 1 0]

119864 [119890119896] = 0 119864 [119890

1198961198901015840

119897] = 119877120575

119896119897

(5)

This model assumes that the turn rate is known or couldbe estimated When the range rate measurements are avail-able the turn rate could be estimated by using range ratemea-surements [18 19] The tracking performance will be deteri-orated when the assumed turn rate is far away from the trueoneThismodel is usually used as one of themodels in amul-tiple models framework

22 ACT Model with Cartesian Velocity In this model thestate vector is chosen to be 119883 = [119909 119910 119910 120596]

1015840 the state spaceequation can be written as

119883119896+1

= 119891ACT1 (119883119896) + 119866ACT1119908119896 (6)

where

119891ACT1 (119883) =

[[[[[[[[[[[[[[

[

119909 +

120596

sin (120596119879) minus119910

120596

(1 minus cos (120596119879))

119910 +

120596

(1 minus cos (120596119879)) +119910

120596

(sin (120596119879))

cos (120596119879) minus 119910 sin (120596119879)

sin (120596119879) + 119910 cos (120596119879)

120596

]]]]]]]]]]]]]]

]

(7)

119866ACT1 =[[[[

[

1198792

2

0 119879 0 0

0

1198792

2

0 119879 0

0 0 0 0 1

]]]]

]

1015840

(8)

119908 = [119908119909119908119910119908120596]

1015840

(9)

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876ACT1120575119896119897 (10)

Assume only position could be measured the measure-ment equation can be written as

119911119896= 119867ACT1119883119896 + 119890119896 (11)

where

119867ACT1 = [1 0 0 0 0

0 1 0 0 0] (12)

119864 [119890119896] = 0 119864 [119890

1198961198901015840

119897] = 119877120575

119896119897 (13)

23 ACT Model with Polar Velocity This modelrsquos state vectoris119883 = [119909 119910 V 120601 120596]1015840 and the dynamic state equation is givenby

119883119896+1

= 119891ACT2 (119883119896) + 119866ACT2119908119896 (14)

Mathematical Problems in Engineering 3

where

119891ACT2 (119883) =

[[[[[[[[[[[

[

119909 + (

2V120596

) sin(1205961198792

) cos(120601 + 1205961198792

)

119910 + (

2V120596

) sin(1205961198792

) sin(120601 + 1205961198792

)

V

120601 + 120596119879

120596

]]]]]]]]]]]

]

119866ACT2 =[

[

0 0 1198792

0 0

0 0 0

1198792

2

1198792

]

]

1015840

119908 = [119908V 119908120596]1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876ACT2120575119896119897

(15)

However themeasurement equation is the same as (11) to (13)

24 Kinematic Constraint Model For a constant speed targetthe acceleration vector is orthogonal to the velocity vector

119862 (119883) = 119881 sdot 119860 = 0 (16)where 119881 is the target velocity vector and119860 is the target accel-eration vector

This kinematic constraint can be used as a pseudom-easurement The state vector is chosen to be 119883 =

[119909 119910 119910 119910]1015840 So the dynamic model is the constant

acceleration (CA) model given by119883119896+1

= 119865119883119896+ 119866119908119896 (17)

where

119865 =

[[[[[[[[[[[

[

1 119879

1198792

2

0 0 0

0 1 119879 0 0 0

0 0 1 0 0 0

0 0 0 1 119879

1198792

2

0 0 0 0 1 119879

0 0 0 0 0 1

]]]]]]]]]]]

]

119866 =

[[[

[

1198792

2

119879 1 0 0 0

0 0 0

1198792

2

119879 1

]]]

]

1015840

119908 = [119908119909119908119910]

1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876CA120575119896119897

(18)

The measurement equation is given by119911119896= 119867119909119896+ 119890119896 (19)

where

119867 = [

1 0 0 0 0 0

0 0 0 1 0 0]

119864 [119890119896] = 0 119864 [119890

1198961198901015840

119897] = 119877120575

119896119897

(20)

The pseudomeasurement is

119881119896|119896

119878119896|119896

sdot 119860119896+ 120583119896= 0 (21)

where 119881119896|119896= [119896|119896

119910119896|119896]

1015840 and 119860119896= [119896

119910119896]1015840

119878119896|119896

is the filtered speed at time 119896

119878119896|119896= radic2

119896|119896+ 1199102

119896|119896 (22)

120583119896sim 119873 (0 119877

120583

119896) (23)

119877120583

119896= 1199031(120575)119896+ 1199030 0 le 120575 lt 1 (24)

where 1199031is chosen to be large for initialization and 119903

0is chosen

for steady-state conditions

25 Maneuver-Centered CT Model This modelrsquos state vectoris given by119883 = [119903 120579 120596]

1015840 The process state space equation is

119883119896+1

= Φ119883119896+ Γ119908119896 (25)

where

Φ =[

[

1 0 0

0 1 119879

0 0 1

]

]

Γ =[[

[

1 0

0

119879

2

0 1

]]

]

119908 = [119908119903119908120596]1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876

119898120575119896119897

(26)

Assume the center of the CTmotion is (119909119888 119910119888)The trans-

formation between Cartesian coordinates and maneuver-centered coordinates is given by

119903 = radic(119909 minus 119909119888)2

+ (119910 minus 119910119888)2

120579 = tanminus1 (119910 minus 119910119888

119909 minus 119909119888

)

(27)

So the measurement equation is given by

119911119896= 119867119898119883119896+ 119890119896 (28)

where

119867119898= [

1 0 0

0 1 0]

119864 [1198901198961198901015840

119897] = 119877119898= 1198691199031205791198771198691015840

119903120579

(29)

119869119903120579is the Jacobian matrix based on (27) which leads to

119869119903120579=[

[

cos 120579 sin 120579

minus sin 120579119903

cos 120579119903

]

]

(30)


3 Tracking Algorithms ina Single Model Framework

31 UKF Filter with ACT Models If the turn rate is aug-mented to the state vector it will become a nonlinear pro-blem The extended Kalman filter (EKF) has been used totrack this kind of motion Since unscented Kalman filter(UKF) is very suitable for nonlinear estimation [4 5] here theUKF algorithm is introduced

(i) Calculate the Weights of Sigma Points

119882119898

0=

120582

(119899 + 120582)

119882119888

0

=

120582

(119899 + 120582)

+ (1 minus 1205722+ 120573)119882

119898

119894

= 119882119888

119894=

05

(119899 + 120582)

119894 = 1 2 2119899

(31)

where 119899 is the dimension of the state vector 120582 =

1205722(119899 + 120581) minus 119899 is a scaling parameter 120572 determines

the sigma points around 119909 and is usually set to a smallpositive value (eg 1119890 minus 3) 120581 is a secondary scalingparameter which is usually set to 0 and 120573 = 2

is optimal forGauss distributionsWhere the (radic(119899 + 120582)119875119909)119894is

the 119894th row of the matrix square root

(ii) Calculate the Sigma Points

1205850

119896minus1|119896minus1= 119883119896minus1|119896minus1

120585(119894)


+ (radic(119899 + 120582) 119875119909)

119894

119894 = 1 2 119899

120585(119894)


minus (radic(119899 + 120582) 119875119909)

119894

119894 = 119899 + 1 119899 + 2 2119899

(32)

(iii) Time Update

120585(119894)

119896= 119891119896(120585(119894)

119896minus1|119896minus1) 119894 = 0 1 2119899119883

119896|119896minus1

=

2119899

sum

119894=0

119882119898

119894120585(119894)

119896119875119896|119896minus1

=

2119899

sum

119894=0

119882119888

119894(120585(119894)

119896minus 119883119896|119896minus1

) (120585(119894)

119896minus 119883119896|119896minus1

)

1015840

+ 119866119876119896minus11198661015840

(33)

(iv) Measurement Update Because we assume the measure-ment equation is linear the following is just the same as thetraditional Kalman filter

119896|119896minus1

= 119867119883119896|119896minus1

119878119896

= 119867119875119896|119896minus1

1198671015840+ 119877119896119870119896

= 119875119896|119896minus1

1198671015840119878minus1

119896119883119896|119896

= 119883119896|119896minus1

+ 119870119896(119911119896minus 119911119896|119896minus1

) 119875119896|119896

= 119875119896|119896minus1

minus 1198701198961198781198961198701015840

119896

(34)

For the cases where the measurement equation is also non-linear the measurement update can be referred to [10] fordetails

32 Kinematic Constraint Tracking Filter The Kalman filter-ing equations for processing this kinematic constraint as apseudomeasurement are given below where the filtered stateestimate and error covariance after the constraint have beenapplied are denoted by119883119862

119896|119896and 119875119862

119896|119896 respectively [8]

(i) Time Update

119883119896|119896minus1

= 119865119883119862

119896minus1|119896minus1

119875119896|119896minus1

= 119865119875119862

119896minus1|119896minus11198651015840+ 119866119876119896minus11198661015840

(35)

(ii) Measurement Update The measurement update is thesame as (34)

(iii) Constraint Update

119870119862

119896= 119875119896|119896119862119879

119896[119862119896119875119896|1198961198621015840

119896+ 119877120583

119896]

minus1

119883119862

119896|119896

= [119868 minus 119870119862

119896119862119896]119883119896|119896119875119862

119896|119896

= [119868 minus 119870119862

119896119862119896] 119875119896|119896

(36)

where

119862119896=

1

119878119896|119896

[0 0 119909119896|119896

0 0 119910119896|119896] (37)

33 Maneuver-Centered Tracking Filter

(i) EstimatingCenter ofManeuverThe center of themaneuvershould be estimated from the measurements It can be esti-mated through least square method which requires an iter-ative search procedure The following simple geometricallyoriented procedure of estimating the center was proposed in[9] The main idea is as follows if two points are on a circlethen the perpendicular bisector of the chord between thosepoints will pass through the center of the circleThe slope (119898)and 119910 intercept (119887) of the perpendicular bisector is given by

119898 =

(1199091minus 1199092)

(1199102minus 1199101)

119887 =

(1199101+ 1199102)

2

minus 119898

(1199091+ 1199092)

2

(38)


where (1199091 1199101) and (119909

2 1199102) are the coordinates of the two

points The center can be given by

119909119888=

(1198871minus 1198872)

(1198982minus 1198981)

119910119888=

(11989811198872minus 11989821198871)

(1198981minus 1198982)

(39)

(ii) Maneuver Detection In the absence of a maneuver thetarget is assumed to be traveling in a straight line andmodeledby a constant velocity (CV)motion (CVmodel is very simpleand commonly used which will not be listed here)When themaneuver is detected the filter switches to the maneuver-centerCTmodelWhile the endof amaneuver is detected thefilter will then switch back to CV model

Here a fading memory average of the innovations is usedto detect if a maneuver occurs The equation is given by

119906119896= 120588119906119896minus1

+ 119889119896

(40)

with

119889119896= ]1015840119896119878minus1

119896]119896 (41)

where 0 lt 120588 lt 1 ]119896is the innovation vector and 119878

119896is its cov-

ariance matrix119906119896will have a chi-squared distribution with degrees

119899119906= 119899119911

1 + 120588

1 minus 120588

(42)

where 119899119911is the dimension of the measurement vector When

119906119896exceeds a threshold (eg 95 or 99 confidence interval)

then a maneuver onset is declared The end time of a maneu-verwill be determined in a similar fashionTheprocedure canbe referred to [9] for details

34 Simulation Results

(i)The ScenarioThe scenario simulated here is very similar tothat described in [20] It includes few rectilinear stages andfew CT maneuvers Four consecutive 180∘ turns with rates120596 = 187minus28 56minus468 are simulated respectively for scans[56 150] [182 245] [285 314] and [343 379] The targettrajectory can be seen in Figure 1

The initial target position and velocities are 1198830= 60 km

1198840= 40 km

0= minus172 km and

0= 246 km It is assumed

that the sensor measures Cartesian coordinates 119883 and 119884 di-rectly It is also assumed that120590

119883= 120590119884= 100mand the sample

rate 119879 = 1

(ii) Algorithmsrsquo Parameters UKF controlled ACT modelrsquosparameter

120572 = 10minus3 120573 = 2 120581 = 0

119876ACT1 = diag 1 1 10minus4

119876ACT2 = diag 1 10minus4

(43)

1 15 2 25 3 35 4 45 5 55 62

25

3

35

4

45

5

55

6

k = 0

times104

times104

Y(m

)

X (m)

Figure 1 The test trajectory

Kinematic constraint modelrsquos parameter

119876CA = diag 1 1

120575 = 092 1199030= 1 119903

1= 200

(44)

Maneuver centered modelrsquos parameter

119876119898= diag 106 10minus4

120588 = 08

(45)

(iii) Results The four models are listed as follows

Method 1 ACT model with Cartesian velocityMethod 2 ACT model with polar velocityMethod 3 kinematic constraint modelMethod 4 maneuver-centered CT model

Rootmean squared errors (RMSE) are used here for com-parison The RME position errors are defined as follows

RMSPE (119896) = radic 1

119872

119872

sum

119894=1

[(119909119894

119896minus 119909119894

119896)2

+ (119910119894

119896minus 119910119894

119896)2

] (46)

where119872 = 200 are the Monte-Carlo simulation runs 119909119894119896and

119910119894

119896stand for the true position while 119909119894

119896|119896and 119910119894

119896are the posi-

tion estimatesThe RMS position errors of all but the first ten are shown

in Figure 2Table 1 summarizes the average RMS of the position

errorsTable 2 summarizes the relative computational complex-

ity normalized to method 4It can be seen from the figure and tables thatmethod 2 has

the best performance and its computational load is roughly


0 50 100 150 200 250 300 350 40050

100

150

200

250

300

350

Time (s)

RMS

posit

ion

erro

rs (m

)

Method 4

Method 3

Method 2

Method 1

Figure 2 RMS position errors of the four methods

Table 1 Average RMS of position errors

Method Average RMS of position errors (m)1 94262 81573 109514 18346

Table 2 Relative computational load

Method Relative computational load1 7262 7073 1424 1

the same as method 1 So we can conclude that ACT modelwith polar velocity is better than ACT model with Cartesianvelocity Method 4 has the least computational load but itsperformance is poor Method 3 is slightly more complexthan method 4 but can decrease the error greatly So if thecomputational load is of great concern kinematic constraintmodel is a good choice

4 The Expectation Maximization (EM)Algorithm for Tracking CT Motion Target

In this part the model in Section 21 is usedThe turn rate 120596

119896can be described by a Markov chain [21

22] and has 119903 possible values

120596119896isin 119872119903= 120596 (1) 120596 (2) 120596 (119903) (47)

Assume the initial probability 120591119894and the one-step transi-

tion matrix are known as follows

120591119894= 119901 (120596

0= 120596 (119894)) 119894 = 1 2 119903

120587119894119895= 119901 (120596

119896+1= 120596 (119895) | 120596

119896= 120596 (119894)) 119894 119895 = 1 2 119903

(48)

The measurement sequence is defined by 1198851119873

=

1199111 1199112 119911

119873 state sequence is 119883

1119873= 1198831 1198832 119883

119873

and maneuver sequence isΩ1119873

= 1205961 1205962 120596

119873

41 Batch EMAlgorithm Assume themeasurement sequenceis known this algorithm focuses on finding the best maneu-ver sequence based onMAP criterionThere is one best man-euver sequenceΩ(B)

1119873in 119903119873 possible sequences that makes the

conditional probability density function be the maximumWhenΩ(B)

1119873is achieved the state sequence119883

1119873can be estim-

ated accuratelyAccording to EM algorithm 119885

1119873is considered to be the

incomplete data1198831119873

to be the ldquolostrdquo data andΩ1119873

to be thedata that needs to be estimated EM algorithm carries out thefollowing two steps iteratively

(1) Expectation Step (E step)

119869 (Ω1119873 Ω(119895)

1119873)

= 119864X1119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

(49)

where 119869(Ω1119873 Ω(119895)

1119873) is defined as the cost functionΩ(119895)

1119873is the

maneuver sequence estimation after 119895 times iteration

(2) Maximization step (M step)

Ω(119895+1)

1119873= argmax

Ω1119873

119869 (Ω1119873 Ω(119895)

1119873) (50)

If the initial value is given the above E step and M step arecarried out repeatedly until convergence

(i) E step The union probability density function can be de-composed as follows

119901 (1198831119873 1198851119873 Ω1119873)

=

119873

prod

119896=1

119901 (119911119896| 119883119896) times

119873

prod

119896=1

119901 (119883119896| 119883119896minus1 120596119896minus1) times 119901 (119883

0)

times

119873

prod

119894=1

119901 (120596119894120596119894minus1) times 119901 (120596

0)

(51)

119901(119883119896| 119883119896minus1 120596119896minus1) and 119901(120596

119894| 120596119894minus1) rely on the maneuver

sequenceΩ1119873

The state equation is Gaussian distribution

119901 (119883119896| 119883119896minus1 120596119896minus1) = 119873 119883

119896minus 119865 (120596

119896minus1)119883119896minus1 119876119896 (52)

where 119873120583 Σ is the Gaussian probability density functionwith mean 120583 and covariance Σ


From the above analysis

119869 (Ω1119873 Ω(119895)

1119873)

= 1198641198831119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

=

119873

sum

119896=1

ln119901 (120596119896| 120596119896minus1) minus

1

2

(119883119896|119873

minus 119865(120596119896minus1)119883119896minus1|119873

)

1015840

times 119876minus1

119896(119883119896|119873

minus 119865 (120596119896minus1)119883119896minus1|119873

)

(53)

where

119883119896|119873

= 119864 [119883119896| 1198851119873 Ω(119895)

1119873] (54)

Those terms which are independent of Ω1119873

are omittedhere

In the E step if Ω(119895)1119873

is given the cost function can beachieved using Kalman smoothing algorithm

(ii) M step In themaximization step a newΩ1119873

is chosen fora higher conditional probabilityThen a better parameter esti-mation is achieved compared to the former iterationThe fol-lowing Viterbi algorithm can solve this problem perfectly

Viterbi algorithm is a recursive algorithm looking for thebest path As shown in Figure 3 the path connects the adjac-ent points with theweights to be the logarithm function of thelikelihood named cost The pathrsquos total cost is the sum of itseach pointrsquos cost The best path has the maximum cost Thedetailed method to find the best path can be found in [12]

(iii) Calculating Algorithm

(1) Initialization the initial maneuver sequenceΩ(1)1119873

andthreshold 120576 should be given

(2) Iteration for each circle (119895 = 1 2 ) carry outthe following steps (1) E step according to (53)calculate the cost between the adjacent point (2) Mstep according to Viterbi algorithm find a bettermaneuver sequence

(3) Stop if Ω(119895+1)1119873

minusΩ(119895)

1119873 le 120576 then stop the iterationThe

best maneuver sequence is Ω(B)1119873

= Ω(119895+1)

1119873 then the

state estimation sequence is calculated according toΩ(B)1119873

42 Recursive EM Algorithm In target tracking applicationsthe targetrsquos state always needs online estimation So a recur-sive EM algorithm is needed for calculating 120596

119896

(i) Recursive Equation Under the MAP criterion

Ω(B)1119896= argmax

Ω1119896

119901 (Ω1119896| 1198851119896) (55)

k = 1 k = 2 k = 3 k = 4

1205961

1205962

1205963

1205964 1205964 1205964 1205964

1205963 1205963 1205963

1205962 1205962 1205962

12059611205961 1205961

Figure 3 Viterbi algorithm for path following

where 119901(Ω1119896| 1198851119896) can be calculated online

119901 (Ω1119896| 1198851119896) = 119901 (Ω

1119896| 119911119896 1198851119896minus1

)

=

119901 (119911119896| Ω119896 1198851119896minus1

) 119901 (Ω119896| 1198851119896minus1

)

119901 (119911119896| Ω1119896minus1

)

= 119901 (119911119896| Ω1119896 1198851119896minus1

) 119901 (120596119896| Ω1119896minus1

)

times 119901 (Ω1119896minus1

| 1198851119896minus1

) (119901 (119911119896| 1198851119896minus1

))minus1

(56)

Because Ω1119896

is Markov chain

119901 (120596119896| Ω1119896minus1

) = 119901 (120596119896| 120596119896minus1) (57)

The possible maneuver sequence grows exponentially as thetime grows For the computation to be feasibility it is assumedthat

119901 (119911119896| Ω1119896 1198851119896minus1

) asymp 119901 (119911119896| 120596119896 1198851119896minus1

)

= 119873 (120592119896 S119896)

(58)

where 120592119896is the Kalman filterrsquos innovation and S

119896is the covari-

ance of the innovationThe cost function is defined as

119869 (120596119896(119894)) = ln119901 (Ω

1119896 120596119896(119894) 1198851119896) 119894 = 1 2 119903

(59)

which stands for the cost to model 119894 until time 119896From (57) to (59)

119869 (120596119896(119895)) = 119869 (120596

119896minus1(119894)) + ln120587

119894119895

minus

1

2

1205921015840

119896(119894 119895) Sminus1

119896(119894 119895) 120592

119896(119894 119895) 119894 119895 = 1 2 119903

(60)

where 120592119896(119894 119895) stands for the innovation when model 119894 is

chosen in time 119896minus1 andmodel 119895 is chosen in time 119896 S119896(119894 119895) is

the corresponding covariance


Because of using the assumption (58) the iteration algo-rithm is not the optimal algorithm under MAP criterion buta suboptimal one

(ii) Calculating Algorithm Only one-step iteration is listedhere

(1) E Step Calculation Using (10) calculate each costfrom time 119896 minus 1 to 119896 1199032 costs are needed

(2) M Step Calculation According to Viterbi algorithmfind out themaximum cost 119869max(120596119896(119894)) related to eachmodel 119869max(120596119896(119894)) is the initial value to be the nextiteration

(3) Filtering According to the path which reaches eachmodel calculate each modelrsquos state estimation 119883

119896(119894)

and covariance 119875119896|119896(119894) 119894 = 1 2 119903

(4) The Final Results From 119869max(120596119896(119894)) 119894 = 1 2 119903choose the maximum one as the final filtering result

119895 = argmax119894

119869max (120596119896(119894))119903

119894=1 (61)

119883(B)119896= 119883119896(119895) 119875

(B)119896|119896

= 119875119896|119896(119895) (62)


(i) Simulation Scenario Target initial state is 1198830

=

[60000m minus172ms 40000m 246ms]1015840 The sample rate119879 = 1 s The covariance of process noise

119876 = [

119876119909

0

0 119876119910

] 119876119909= 119876119910=

[[[[[

[

1198794

3

1198793

2

1198793

2

1198792

]]]]]

]

(63)

Assume only position can bemeasured themeasurementequation is the following

119911119896= [

1 0 0 0

0 0 1 0]119883119896+ V119896 (64)

The covariance of measurement noise 119877 = 2500I where Iis the 2 times 2 unit matrix

The simulation lasts for 300 s Targetrsquos true turn rate is

120596119896=

0 0 le 119896 lt 103

0033 rads 104 le 119896 lt 198

0 198 le 119896 lt 300

(65)

Figure 4 gives the targetrsquos true trajectoryAssume targetrsquos maximum centripetal acceleration is

30ms2 Under the speed 300ms the corresponding turnrate is 01 rads Seven models are used for this simulationFrom -01 to 01 the sevenmodels are distributed evenlyTheirvalues are minus01 minus0067 minus0033 0 0033 0067 and 01 Theinitial probability matrix is

120591 = [

1

7

1

7

1

7

1

7

1

7

1

7

1

7

] (66)

20 25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

x (km)

y(k

m)

Figure 4 Target trajectory

The model transition matrix is

120587119894119895=

07 119894 = 119895

005 119894 = 119895

119894 119895 = 1 2 7

(67)

(ii) Simulation Results and Analysis Batch EM algorithmrecursive algorithm and IMMalgorithmare compared in thisscenario Rootmean squared errors (RMSE) are used here forcomparisonThe RME position errors are defined as (46) andvelocity error are defined as follows

RMSVE (119896) = radic 1

119872

119872

sum

119894=1

[(119894

119896minus 119909

119894

119896)

2

+ ( 119910119894

119896minus 119910

119894

119896)

2

] (68)

where119872 = 200 are Monte-Carlo simulation runs and 119894119896 119910119894

119896

and 119909119894119896 119910119894119896stand for the true and estimated velocity at time 119896

in the 119894th simulation runs respectivelyFigures 5 and 6 show the position and velocity perfor-

mance comparison It can be concluded that the batch EMalgorithm has much less tracking errors compared to IMMalgorithm During maneuver onset time and terminationtime the IMM algorithm is better than recursive EM algo-rithm But on stable period the recursive EM algorithmperforms better

5 Conclusions

Aiming at the CTmotion target tracking several models andalgorithms are introduced and simulated in this paper

In single model framework four CT models have beencompared for tracking applications ACT model with Carte-sian velocity ACT model with polar velocity kinematic con-straint model and maneuver-centered model The Monte-Carlo simulations show that the ACT model with polar


0 50 100 150 200 250 30010

20

30

40

50

60

70

Batch EMRecursive EMIMM

t (s)

Ep

(m)

Figure 5 Position performance comparison

0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)


Figure 6 Velocity performance comparison

velocity has the best tracking performance but the compu-tational load is a bit heavier The kinematic constraint modelhas a moderate tracking performance but its computationalload decreases greatly compared with UKF controlled ACTmodel So if the computational load is of a great concernthe kinematic constraint model is suggested If the trackingperformance is very important and the computational load isnot a problem the ACTmodel with polar velocity is suitable

In multiple models framework EM algorithm is used fortracking CT motion target First a batch EM algorithm isderived The turn rate is acted as the maneuver sequence andestimated based on the MAP criterion Under the E step thecost function is calculated using the Kalman smoothing algo-rithm Under the M step Viterbi algorithm is used for pathfollowing to find out the pathwithmaximumcost Simulationresults show that the Batch EM algorithm has better trackingperformance than IMM algorithm Through modification of

the cost function a recursive EM algorithm is presentedThealgorithm can track the target online Compared with theIMM algorithm on the stable period the recursive EM algo-rithm has better tracking performance

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research work was supported by the National KeyFundamental Research amp Development Programs (973) ofChina (2013CB329405) Foundation for Innovative ResearchGroups of the National Natural Science Foundation of China(61221063) Natural Science Foundation of China (6120322161004087) PhD ProgramsFoundation ofMinistry of Educa-tion of China (20100201120036) China Postdoctoral ScienceFoundation (2011M501442 20100481338) and FundamentalResearch Funds for the Central University

References

[1] X R Li and V P Jilkov ldquoSurvey of maneuvering Target Track-ingmdashpart I dynamic modelsrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 39 no 4 pp 1333ndash1364 2003

[2] F Gustafsson and A J Isaksson ldquoBest choice of coordinate sys-tem for tracking coordinated turnsrdquo in Proceedings of the 35thIEEE Conference on Decision and Control pp 3145ndash3150 KobeJapan December 1996

[3] A J Isaksson and F Gustaffsson ldquoComparison of some Kalmanfilter based on methods for maneuver tracking and detectionrdquoin Proceedings of the 34th IEEEConference onDecision and Con-trol pp 1525ndash1530 New Orleans La USA December 1995

[4] S Julier J Uhlmann and H F Durrant-Whyte ldquoA newmethodfor the nonlinear transformation of means and covariances infilters and estimatorsrdquo IEEE Transactions on Automatic Controlvol 45 no 3 pp 477ndash482 2000

[5] S J Julier and J K Uhlmann ldquoUnscented filtering and nonlin-ear estimationrdquo Proceedings of the IEEE vol 92 no 3 pp 401ndash422 2004

[6] M Tahk and J L Speyer ldquoTarget Tracking problems subject tokinematic constraintsrdquo IEEE Transactions on Automatic Con-trol vol 35 no 3 pp 324ndash326 1990

[7] A T Alouani andW D Blair ldquoUse of a kinematic constraint intracking constant speed maneuvering targetsrdquo in Proceedings ofthe 30th IEEE Conference on Decision and Control pp 1055ndash1058 Brighton UK December 1991

[8] A T Alouani andW D Blair ldquoUse of a kinematic constraint intracking constant speed maneuvering targetsrdquo IEEE Transac-tions on Automatic Control vol 38 no 7 pp 1107ndash1111 1993

[9] J A Roecker and C D McGillem ldquoTarget Tracking in maneu-ver-centered coordinatesrdquo IEEE Transactions on Aerospace andElectronic Systems vol 25 no 6 pp 836ndash843 1989

[10] X R Li and V P J ILkov ldquoA Survey of maneuvering TargetTrackingmdashpart III measurementmodelsrdquo inConference on Sig-nal and Data Processing of Small Targets vol 4437 of Proceedingof SPIE pp 423ndash446 San Diego Calif USA 2001


[11] Y Bar-Shalom K C Chang and H A P Blom ldquoTracking amaneuvering target using input estimation versus the interact-ing multiple model algorithmrdquo IEEE Transactions on Aerospaceand Electronic Systems vol 25 no 2 pp 296ndash300 1989

[12] A Averbuch S Itzikowitz and T Kapon ldquoRadar Target Track-ingmdashviterbi versus IMMrdquo IEEE Transactions on Aerospace andElectronic Systems vol 27 no 3 pp 550ndash563 1991

[13] G W Pulford and B F La Scala ldquoMAP estimation of targetmanoeuvre sequence with the expectation-maximization algo-rithmrdquo IEEE Transactions on Aerospace and Electronic Systemsvol 38 no 2 pp 367ndash377 2002

[14] A Logothetis V Krishnamurthy and J Holst ldquoA Bayesian EMalgorithm for optimal tracking of a maneuvering target in clut-terrdquo Signal Processing vol 82 no 3 pp 473ndash490 2002

[15] P Willett Y Ruan and R Streit ldquoPMHT problems and somesolutionsrdquo IEEE Transactions on Aerospace and Electronic Sys-tems vol 38 no 3 pp 738ndash754 2002

[16] Z Li S Chen H Leung and E Bosse ldquoJoint data associationregistration and fusion using EM-KFrdquo IEEE Transactions onAerospace and Electronic Systems vol 46 no 2 pp 496ndash5072010

[17] D L Huang H Leung and E Boose ldquoA pseudo-measurementapproach to simultaneous registration and track fusionrdquo IEEETransactions on Aerospace and Electronic Systems vol 48 no 3pp 2315ndash2331 2012

[18] X Yuan CHan Z Duan andM Lei ldquoAdaptive turn rate estim-ation using range rate measurementsrdquo IEEE Transactions onAerospace and Electronic Systems vol 42 no 4 pp 1532ndash15402006

[19] V B Frencl and J B R do Val ldquoTracking with range rate mea-surements turn rate estimation and particle filteringrdquo in Pro-ceedings of the IEEE Radar Conference pp 287ndash292 Atlanta GaUSA May 2012

[20] E Semerdjiev L Mihaylova and X R Li ldquoVariable- and fixed-structure augmented IMM algorithms using coordinated turnmodelrdquo in Proceedings of the 3rd International Conference onInformation Fusion pp 10ndash13 Paris France July 2000

[21] Y Bar-shalom X R Li and T Kirubarajan Estimation withApplications to Tracking andNavigationTheory Algorithms andSoftware John Wiley amp Sons New York NY USA 2001

[22] J P Iielferty ldquoImproved tracking ofmaneuvering targets the useof turn-rate distributions for acceleration modelingrdquo IEEETransactions on Aerospace and Electronic Systems vol 32 no 4pp 1355ndash1361 1996

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


interest [12ndash17] Expectation maximization (EM) algorithmis the state estimation approach based on MAP criterion Using EMalgorithm the computational load grows linearly dur-ing per iteration and the optimal estimation based on MAPcriterion can be achieved finally

The existing EM algorithm to track maneuvering targetcan be classified into two categories one formulates themaneuver as the unknown input [12ndash14] and the other for-mulates themaneuver as the systemrsquos process noise [15] Aim-ing at the problem to track a target with CTmaneuver an EMalgorithm is presented The maneuver is formulated by theturn rate First the turn rate sequence is estimated using theEM algorithm Then with the estimated turn rate sequencethe target state sequence is estimated accurately

The rest of this paper is organized as follows Section 2presents all the CTmodelsrsquo state equations and measurementequationsThe tracking algorithms based on single model areinterpreted in Section 3 the simulations are also presented InSection 4 the batch and recursive EM algorithms are derivedand compared with the IMM algorithm in simulationSection 5 provides the paperrsquos conclusions

2 Dynamic Models for CT Motion

A maneuvering target can be modeled by

119883119896+1

= 119891119896(119883119896) + 119908119896

119911119896= ℎ119896(119883119896) + 119890119896

(1)

where 119883119896and 119911

119896are target state and observation respec-

tively at discrete time 119905119896 119908119896and 119890

119896are process noise and

measurement noise sequences respectively 119891119896and ℎ

119896are

vector-valued functions

21 CT Model with Known Turn Rate The coordinated turnmotion can be described by the following equation

119883119896+1

= 119865 (120596119896)119883119896+ 119908119896 (2)

The measurement equation is

119911119896= 119867CT119883119896 + 119890119896 (3)

The components of state are119883 = [119909 119910 119910]1015840 120596119896stands

for the turn rate in time 119896Where

119865 (120596119896) =

[[[[[[[[[[[[

[

1

sin (120596119896119879)

120596119896

0 minus

1 minus cos (120596119896119879)

120596119896

0 cos (120596119896119879) 0 minus sin (120596

119896119879)

0

1 minus cos (120596119896119879)

120596119896

1

sin (120596119896119879)

120596119896

0 sin (120596119896119879) 0 cos (120596

119896119879)

]]]]]]]]]]]]

]

119908 = [119908119909119908119910]

1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876CT120575119896119897

(4)

Assume only position could be measured where

119867CT = [1 0 0 0

0 0 1 0]

119864 [119890119896] = 0 119864 [119890

1198961198901015840

119897] = 119877120575

119896119897

(5)

This model assumes that the turn rate is known or couldbe estimated When the range rate measurements are avail-able the turn rate could be estimated by using range ratemea-surements [18 19] The tracking performance will be deteri-orated when the assumed turn rate is far away from the trueoneThismodel is usually used as one of themodels in amul-tiple models framework

22 ACT Model with Cartesian Velocity In this model thestate vector is chosen to be 119883 = [119909 119910 119910 120596]

1015840 the state spaceequation can be written as

119883119896+1

= 119891ACT1 (119883119896) + 119866ACT1119908119896 (6)

where

119891ACT1 (119883) =

[[[[[[[[[[[[[[

[

119909 +

120596

sin (120596119879) minus119910

120596

(1 minus cos (120596119879))

119910 +

120596

(1 minus cos (120596119879)) +119910

120596

(sin (120596119879))

cos (120596119879) minus 119910 sin (120596119879)

sin (120596119879) + 119910 cos (120596119879)

120596

]]]]]]]]]]]]]]

]

(7)

119866ACT1 =[[[[

[

1198792

2

0 119879 0 0

0

1198792

2

0 119879 0

0 0 0 0 1

]]]]

]

1015840

(8)

119908 = [119908119909119908119910119908120596]

1015840

(9)

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876ACT1120575119896119897 (10)

Assume only position could be measured the measure-ment equation can be written as

119911119896= 119867ACT1119883119896 + 119890119896 (11)

where

119867ACT1 = [1 0 0 0 0

0 1 0 0 0] (12)

119864 [119890119896] = 0 119864 [119890

1198961198901015840

119897] = 119877120575

119896119897 (13)

23 ACT Model with Polar Velocity This modelrsquos state vectoris119883 = [119909 119910 V 120601 120596]1015840 and the dynamic state equation is givenby

119883119896+1

= 119891ACT2 (119883119896) + 119866ACT2119908119896 (14)


where

119891ACT2 (119883) =

[[[[[[[[[[[

[

119909 + (

2V120596

) sin(1205961198792

) cos(120601 + 1205961198792

)

119910 + (

2V120596

) sin(1205961198792

) sin(120601 + 1205961198792

)

V

120601 + 120596119879

120596

]]]]]]]]]]]

]

119866ACT2 =[

[

0 0 1198792

0 0

0 0 0

1198792

2

1198792

]

]

1015840

119908 = [119908V 119908120596]1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876ACT2120575119896119897

(15)







= 119865119883119896+ 119866119908119896 (17)

where

119865 =

[[[[[[[[[[[

[

1 119879

1198792

2

0 0 0

0 1 119879 0 0 0

0 0 1 0 0 0

0 0 0 1 119879

1198792

2

0 0 0 0 1 119879

0 0 0 0 0 1

]]]]]]]]]]]

]

119866 =

[[[

[

1198792

2

119879 1 0 0 0

0 0 0

1198792

2

119879 1

]]]

]

1015840

119908 = [119908119909119908119910]

1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876CA120575119896119897

(18)


where

119867 = [

1 0 0 0 0 0

0 0 0 1 0 0]

119864 [119890119896] = 0 119864 [119890

1198961198901015840

119897] = 119877120575

119896119897

(20)


119881119896|119896

119878119896|119896

sdot 119860119896+ 120583119896= 0 (21)

where 119881119896|119896= [119896|119896

119910119896|119896]

1015840 and 119860119896= [119896

119910119896]1015840

119878119896|119896


119878119896|119896= radic2

119896|119896+ 1199102

119896|119896 (22)

120583119896sim 119873 (0 119877

120583

119896) (23)

119877120583

119896= 1199031(120575)119896+ 1199030 0 le 120575 lt 1 (24)


0is chosen




119883119896+1

= Φ119883119896+ Γ119908119896 (25)

where

Φ =[

[

1 0 0

0 1 119879

0 0 1

]

]

Γ =[[

[

1 0

0

119879

2

0 1

]]

]

119908 = [119908119903119908120596]1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876

119898120575119896119897

(26)



119903 = radic(119909 minus 119909119888)2

+ (119910 minus 119910119888)2

120579 = tanminus1 (119910 minus 119910119888

119909 minus 119909119888

)

(27)


119911119896= 119867119898119883119896+ 119890119896 (28)

where

119867119898= [

1 0 0

0 1 0]

119864 [1198901198961198901015840

119897] = 119877119898= 1198691199031205791198771198691015840

119903120579

(29)


119869119903120579=[

[

cos 120579 sin 120579

minus sin 120579119903

cos 120579119903

]

]

(30)





119882119898

0=

120582

(119899 + 120582)

119882119888

0

=

120582

(119899 + 120582)

+ (1 minus 1205722+ 120573)119882

119898

119894

= 119882119888

119894=

05

(119899 + 120582)

119894 = 1 2 2119899

(31)







1205850


120585(119894)


+ (radic(119899 + 120582) 119875119909)

119894

119894 = 1 2 119899

120585(119894)


minus (radic(119899 + 120582) 119875119909)

119894

119894 = 119899 + 1 119899 + 2 2119899

(32)

(iii) Time Update

120585(119894)

119896= 119891119896(120585(119894)

119896minus1|119896minus1) 119894 = 0 1 2119899119883

119896|119896minus1

=

2119899

sum

119894=0

119882119898

119894120585(119894)

119896119875119896|119896minus1

=

2119899

sum

119894=0

119882119888

119894(120585(119894)

119896minus 119883119896|119896minus1

) (120585(119894)

119896minus 119883119896|119896minus1

)

1015840

+ 119866119876119896minus11198661015840

(33)


119896|119896minus1

= 119867119883119896|119896minus1

119878119896

= 119867119875119896|119896minus1

1198671015840+ 119877119896119870119896

= 119875119896|119896minus1

1198671015840119878minus1

119896119883119896|119896

= 119883119896|119896minus1

+ 119870119896(119911119896minus 119911119896|119896minus1

) 119875119896|119896

= 119875119896|119896minus1

minus 1198701198961198781198961198701015840

119896

(34)



119896|119896and 119875119862


(i) Time Update

119883119896|119896minus1

= 119865119883119862


119875119896|119896minus1

= 119865119875119862


(35)



119870119862

119896= 119875119896|119896119862119879

119896[119862119896119875119896|1198961198621015840

119896+ 119877120583

119896]

minus1

119883119862

119896|119896

= [119868 minus 119870119862

119896119862119896]119883119896|119896119875119862

119896|119896

= [119868 minus 119870119862

119896119862119896] 119875119896|119896

(36)

where

119862119896=

1

119878119896|119896

[0 0 119909119896|119896

0 0 119910119896|119896] (37)



119898 =

(1199091minus 1199092)

(1199102minus 1199101)

119887 =

(1199101+ 1199102)

2

minus 119898

(1199091+ 1199092)

2

(38)


where (1199091 1199101) and (119909



119909119888=

(1198871minus 1198872)

(1198982minus 1198981)

119910119888=

(11989811198872minus 11989821198871)

(1198981minus 1198982)

(39)



119906119896= 120588119906119896minus1

+ 119889119896

(40)

with

119889119896= ]1015840119896119878minus1

119896]119896 (41)


119896is its cov-


119899119906= 119899119911

1 + 120588

1 minus 120588

(42)







1198840= 40 km

0= minus172 km and



119883= 120590119884= 100mand the sample

rate 119879 = 1


120572 = 10minus3 120573 = 2 120581 = 0



(43)

1 15 2 25 3 35 4 45 5 55 62

25

3

35

4

45

5

55

6

k = 0

times104

times104

Y(m

)

X (m)



119876CA = diag 1 1

120575 = 092 1199030= 1 119903

1= 200

(44)


119876119898= diag 106 10minus4

120588 = 08

(45)





119872

119872

sum

119894=1

[(119909119894

119896minus 119909119894

119896)2

+ (119910119894

119896minus 119910119894

119896)2

] (46)


119910119894


119896|119896and 119910119894

119896are the posi-







0 50 100 150 200 250 300 350 40050

100

150

200

250

300

350

Time (s)

RMS

posit

ion

erro

rs (m

)

Method 4

Method 3

Method 2

Method 1











120596119896isin 119872119903= 120596 (1) 120596 (2) 120596 (119903) (47)



120591119894= 119901 (120596

0= 120596 (119894)) 119894 = 1 2 119903

120587119894119895= 119901 (120596

119896+1= 120596 (119895) | 120596

119896= 120596 (119894)) 119894 119895 = 1 2 119903

(48)


=

1199111 1199112 119911


1119873= 1198831 1198832 119883

119873


= 1205961 1205962 120596

119873












119869 (Ω1119873 Ω(119895)

1119873)

= 119864X1119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

(49)

where 119869(Ω1119873 Ω(119895)


1119873is the



Ω(119895+1)

1119873= argmax

Ω1119873

119869 (Ω1119873 Ω(119895)

1119873) (50)



119901 (1198831119873 1198851119873 Ω1119873)

=

119873

prod

119896=1

119901 (119911119896| 119883119896) times

119873

prod

119896=1


0)

times

119873

prod

119894=1

119901 (120596119894120596119894minus1) times 119901 (120596

0)

(51)

119901(119883119896| 119883119896minus1 120596119896minus1) and 119901(120596


sequenceΩ1119873


119901 (119883119896| 119883119896minus1 120596119896minus1) = 119873 119883

119896minus 119865 (120596

119896minus1)119883119896minus1 119876119896 (52)




119869 (Ω1119873 Ω(119895)

1119873)

= 1198641198831119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

=

119873

sum

119896=1

ln119901 (120596119896| 120596119896minus1) minus

1

2

(119883119896|119873


)

1015840

times 119876minus1

119896(119883119896|119873


)

(53)

where

119883119896|119873

= 119864 [119883119896| 1198851119873 Ω(119895)

1119873] (54)


are omittedhere

In the E step if Ω(119895)1119873









(3) Stop if Ω(119895+1)1119873

minusΩ(119895)



= Ω(119895+1)

1119873 then the



119896



Ω1119896

119901 (Ω1119896| 1198851119896) (55)

k = 1 k = 2 k = 3 k = 4

1205961

1205962

1205963

1205964 1205964 1205964 1205964

1205963 1205963 1205963

1205962 1205962 1205962

12059611205961 1205961



119901 (Ω1119896| 1198851119896) = 119901 (Ω

1119896| 119911119896 1198851119896minus1

)

=

119901 (119911119896| Ω119896 1198851119896minus1

) 119901 (Ω119896| 1198851119896minus1

)

119901 (119911119896| Ω1119896minus1

)

= 119901 (119911119896| Ω1119896 1198851119896minus1

) 119901 (120596119896| Ω1119896minus1

)

times 119901 (Ω1119896minus1

| 1198851119896minus1

) (119901 (119911119896| 1198851119896minus1

))minus1

(56)

Because Ω1119896

is Markov chain

119901 (120596119896| Ω1119896minus1

) = 119901 (120596119896| 120596119896minus1) (57)


119901 (119911119896| Ω1119896 1198851119896minus1

) asymp 119901 (119911119896| 120596119896 1198851119896minus1

)

= 119873 (120592119896 S119896)

(58)




119869 (120596119896(119894)) = ln119901 (Ω

1119896 120596119896(119894) 1198851119896) 119894 = 1 2 119903

(59)


119869 (120596119896(119895)) = 119869 (120596

119896minus1(119894)) + ln120587

119894119895

minus

1

2

1205921015840

119896(119894 119895) Sminus1

119896(119894 119895) 120592

119896(119894 119895) 119894 119895 = 1 2 119903

(60)










119896(119894)

and covariance 119875119896|119896(119894) 119894 = 1 2 119903


119895 = argmax119894

119869max (120596119896(119894))119903

119894=1 (61)

119883(B)119896= 119883119896(119895) 119875

(B)119896|119896

= 119875119896|119896(119895) (62)



=


119876 = [

119876119909

0

0 119876119910

] 119876119909= 119876119910=

[[[[[

[

1198794

3

1198793

2

1198793

2

1198792

]]]]]

]

(63)


119911119896= [

1 0 0 0

0 0 1 0]119883119896+ V119896 (64)



120596119896=

0 0 le 119896 lt 103

0033 rads 104 le 119896 lt 198

0 198 le 119896 lt 300

(65)



120591 = [

1

7

1

7

1

7

1

7

1

7

1

7

1

7

] (66)

20 25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

x (km)

y(k

m)



120587119894119895=

07 119894 = 119895

005 119894 = 119895

119894 119895 = 1 2 7

(67)



119872

119872

sum

119894=1

[(119894

119896minus 119909

119894

119896)

2

+ ( 119910119894

119896minus 119910

119894

119896)

2

] (68)


119896




5 Conclusions




0 50 100 150 200 250 30010

20

30

40

50

60

70


t (s)

Ep

(m)


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)








Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

119891ACT2 (119883) =

[[[[[[[[[[[

[

119909 + (

2V120596

) sin(1205961198792

) cos(120601 + 1205961198792

)

119910 + (

2V120596

) sin(1205961198792

) sin(120601 + 1205961198792

)

V

120601 + 120596119879

120596

]]]]]]]]]]]

]

119866ACT2 =[

[

0 0 1198792

0 0

0 0 0

1198792

2

1198792

]

]

1015840

119908 = [119908V 119908120596]1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876ACT2120575119896119897

(15)







= 119865119883119896+ 119866119908119896 (17)

where

119865 =

[[[[[[[[[[[

[

1 119879

1198792

2

0 0 0

0 1 119879 0 0 0

0 0 1 0 0 0

0 0 0 1 119879

1198792

2

0 0 0 0 1 119879

0 0 0 0 0 1

]]]]]]]]]]]

]

119866 =

[[[

[

1198792

2

119879 1 0 0 0

0 0 0

1198792

2

119879 1

]]]

]

1015840

119908 = [119908119909119908119910]

1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876CA120575119896119897

(18)


where

119867 = [

1 0 0 0 0 0

0 0 0 1 0 0]

119864 [119890119896] = 0 119864 [119890

1198961198901015840

119897] = 119877120575

119896119897

(20)


119881119896|119896

119878119896|119896

sdot 119860119896+ 120583119896= 0 (21)

where 119881119896|119896= [119896|119896

119910119896|119896]

1015840 and 119860119896= [119896

119910119896]1015840

119878119896|119896


119878119896|119896= radic2

119896|119896+ 1199102

119896|119896 (22)

120583119896sim 119873 (0 119877

120583

119896) (23)

119877120583

119896= 1199031(120575)119896+ 1199030 0 le 120575 lt 1 (24)


0is chosen




119883119896+1

= Φ119883119896+ Γ119908119896 (25)

where

Φ =[

[

1 0 0

0 1 119879

0 0 1

]

]

Γ =[[

[

1 0

0

119879

2

0 1

]]

]

119908 = [119908119903119908120596]1015840

119864 [119908119896] = 0 119864 [119908

1198961199081015840

119897] = 119876

119898120575119896119897

(26)



119903 = radic(119909 minus 119909119888)2

+ (119910 minus 119910119888)2

120579 = tanminus1 (119910 minus 119910119888

119909 minus 119909119888

)

(27)


119911119896= 119867119898119883119896+ 119890119896 (28)

where

119867119898= [

1 0 0

0 1 0]

119864 [1198901198961198901015840

119897] = 119877119898= 1198691199031205791198771198691015840

119903120579

(29)


119869119903120579=[

[

cos 120579 sin 120579

minus sin 120579119903

cos 120579119903

]

]

(30)





119882119898

0=

120582

(119899 + 120582)

119882119888

0

=

120582

(119899 + 120582)

+ (1 minus 1205722+ 120573)119882

119898

119894

= 119882119888

119894=

05

(119899 + 120582)

119894 = 1 2 2119899

(31)







1205850


120585(119894)


+ (radic(119899 + 120582) 119875119909)

119894

119894 = 1 2 119899

120585(119894)


minus (radic(119899 + 120582) 119875119909)

119894

119894 = 119899 + 1 119899 + 2 2119899

(32)

(iii) Time Update

120585(119894)

119896= 119891119896(120585(119894)

119896minus1|119896minus1) 119894 = 0 1 2119899119883

119896|119896minus1

=

2119899

sum

119894=0

119882119898

119894120585(119894)

119896119875119896|119896minus1

=

2119899

sum

119894=0

119882119888

119894(120585(119894)

119896minus 119883119896|119896minus1

) (120585(119894)

119896minus 119883119896|119896minus1

)

1015840

+ 119866119876119896minus11198661015840

(33)


119896|119896minus1

= 119867119883119896|119896minus1

119878119896

= 119867119875119896|119896minus1

1198671015840+ 119877119896119870119896

= 119875119896|119896minus1

1198671015840119878minus1

119896119883119896|119896

= 119883119896|119896minus1

+ 119870119896(119911119896minus 119911119896|119896minus1

) 119875119896|119896

= 119875119896|119896minus1

minus 1198701198961198781198961198701015840

119896

(34)



119896|119896and 119875119862


(i) Time Update

119883119896|119896minus1

= 119865119883119862


119875119896|119896minus1

= 119865119875119862


(35)



119870119862

119896= 119875119896|119896119862119879

119896[119862119896119875119896|1198961198621015840

119896+ 119877120583

119896]

minus1

119883119862

119896|119896

= [119868 minus 119870119862

119896119862119896]119883119896|119896119875119862

119896|119896

= [119868 minus 119870119862

119896119862119896] 119875119896|119896

(36)

where

119862119896=

1

119878119896|119896

[0 0 119909119896|119896

0 0 119910119896|119896] (37)



119898 =

(1199091minus 1199092)

(1199102minus 1199101)

119887 =

(1199101+ 1199102)

2

minus 119898

(1199091+ 1199092)

2

(38)


where (1199091 1199101) and (119909



119909119888=

(1198871minus 1198872)

(1198982minus 1198981)

119910119888=

(11989811198872minus 11989821198871)

(1198981minus 1198982)

(39)



119906119896= 120588119906119896minus1

+ 119889119896

(40)

with

119889119896= ]1015840119896119878minus1

119896]119896 (41)


119896is its cov-


119899119906= 119899119911

1 + 120588

1 minus 120588

(42)







1198840= 40 km

0= minus172 km and



119883= 120590119884= 100mand the sample

rate 119879 = 1


120572 = 10minus3 120573 = 2 120581 = 0



(43)

1 15 2 25 3 35 4 45 5 55 62

25

3

35

4

45

5

55

6

k = 0

times104

times104

Y(m

)

X (m)



119876CA = diag 1 1

120575 = 092 1199030= 1 119903

1= 200

(44)


119876119898= diag 106 10minus4

120588 = 08

(45)





119872

119872

sum

119894=1

[(119909119894

119896minus 119909119894

119896)2

+ (119910119894

119896minus 119910119894

119896)2

] (46)


119910119894


119896|119896and 119910119894

119896are the posi-







0 50 100 150 200 250 300 350 40050

100

150

200

250

300

350

Time (s)

RMS

posit

ion

erro

rs (m

)

Method 4

Method 3

Method 2

Method 1











120596119896isin 119872119903= 120596 (1) 120596 (2) 120596 (119903) (47)



120591119894= 119901 (120596

0= 120596 (119894)) 119894 = 1 2 119903

120587119894119895= 119901 (120596

119896+1= 120596 (119895) | 120596

119896= 120596 (119894)) 119894 119895 = 1 2 119903

(48)


=

1199111 1199112 119911


1119873= 1198831 1198832 119883

119873


= 1205961 1205962 120596

119873












119869 (Ω1119873 Ω(119895)

1119873)

= 119864X1119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

(49)

where 119869(Ω1119873 Ω(119895)


1119873is the



Ω(119895+1)

1119873= argmax

Ω1119873

119869 (Ω1119873 Ω(119895)

1119873) (50)



119901 (1198831119873 1198851119873 Ω1119873)

=

119873

prod

119896=1

119901 (119911119896| 119883119896) times

119873

prod

119896=1


0)

times

119873

prod

119894=1

119901 (120596119894120596119894minus1) times 119901 (120596

0)

(51)

119901(119883119896| 119883119896minus1 120596119896minus1) and 119901(120596


sequenceΩ1119873


119901 (119883119896| 119883119896minus1 120596119896minus1) = 119873 119883

119896minus 119865 (120596

119896minus1)119883119896minus1 119876119896 (52)




119869 (Ω1119873 Ω(119895)

1119873)

= 1198641198831119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

=

119873

sum

119896=1

ln119901 (120596119896| 120596119896minus1) minus

1

2

(119883119896|119873


)

1015840

times 119876minus1

119896(119883119896|119873


)

(53)

where

119883119896|119873

= 119864 [119883119896| 1198851119873 Ω(119895)

1119873] (54)


are omittedhere

In the E step if Ω(119895)1119873









(3) Stop if Ω(119895+1)1119873

minusΩ(119895)



= Ω(119895+1)

1119873 then the



119896



Ω1119896

119901 (Ω1119896| 1198851119896) (55)

k = 1 k = 2 k = 3 k = 4

1205961

1205962

1205963

1205964 1205964 1205964 1205964

1205963 1205963 1205963

1205962 1205962 1205962

12059611205961 1205961



119901 (Ω1119896| 1198851119896) = 119901 (Ω

1119896| 119911119896 1198851119896minus1

)

=

119901 (119911119896| Ω119896 1198851119896minus1

) 119901 (Ω119896| 1198851119896minus1

)

119901 (119911119896| Ω1119896minus1

)

= 119901 (119911119896| Ω1119896 1198851119896minus1

) 119901 (120596119896| Ω1119896minus1

)

times 119901 (Ω1119896minus1

| 1198851119896minus1

) (119901 (119911119896| 1198851119896minus1

))minus1

(56)

Because Ω1119896

is Markov chain

119901 (120596119896| Ω1119896minus1

) = 119901 (120596119896| 120596119896minus1) (57)


119901 (119911119896| Ω1119896 1198851119896minus1

) asymp 119901 (119911119896| 120596119896 1198851119896minus1

)

= 119873 (120592119896 S119896)

(58)




119869 (120596119896(119894)) = ln119901 (Ω

1119896 120596119896(119894) 1198851119896) 119894 = 1 2 119903

(59)


119869 (120596119896(119895)) = 119869 (120596

119896minus1(119894)) + ln120587

119894119895

minus

1

2

1205921015840

119896(119894 119895) Sminus1

119896(119894 119895) 120592

119896(119894 119895) 119894 119895 = 1 2 119903

(60)










119896(119894)

and covariance 119875119896|119896(119894) 119894 = 1 2 119903


119895 = argmax119894

119869max (120596119896(119894))119903

119894=1 (61)

119883(B)119896= 119883119896(119895) 119875

(B)119896|119896

= 119875119896|119896(119895) (62)



=


119876 = [

119876119909

0

0 119876119910

] 119876119909= 119876119910=

[[[[[

[

1198794

3

1198793

2

1198793

2

1198792

]]]]]

]

(63)


119911119896= [

1 0 0 0

0 0 1 0]119883119896+ V119896 (64)



120596119896=

0 0 le 119896 lt 103

0033 rads 104 le 119896 lt 198

0 198 le 119896 lt 300

(65)



120591 = [

1

7

1

7

1

7

1

7

1

7

1

7

1

7

] (66)

20 25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

x (km)

y(k

m)



120587119894119895=

07 119894 = 119895

005 119894 = 119895

119894 119895 = 1 2 7

(67)



119872

119872

sum

119894=1

[(119894

119896minus 119909

119894

119896)

2

+ ( 119910119894

119896minus 119910

119894

119896)

2

] (68)


119896




5 Conclusions




0 50 100 150 200 250 30010

20

30

40

50

60

70


t (s)

Ep

(m)


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)








Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119882119898

0=

120582

(119899 + 120582)

119882119888

0

=

120582

(119899 + 120582)

+ (1 minus 1205722+ 120573)119882

119898

119894

= 119882119888

119894=

05

(119899 + 120582)

119894 = 1 2 2119899

(31)







1205850


120585(119894)


+ (radic(119899 + 120582) 119875119909)

119894

119894 = 1 2 119899

120585(119894)


minus (radic(119899 + 120582) 119875119909)

119894

119894 = 119899 + 1 119899 + 2 2119899

(32)

(iii) Time Update

120585(119894)

119896= 119891119896(120585(119894)

119896minus1|119896minus1) 119894 = 0 1 2119899119883

119896|119896minus1

=

2119899

sum

119894=0

119882119898

119894120585(119894)

119896119875119896|119896minus1

=

2119899

sum

119894=0

119882119888

119894(120585(119894)

119896minus 119883119896|119896minus1

) (120585(119894)

119896minus 119883119896|119896minus1

)

1015840

+ 119866119876119896minus11198661015840

(33)


119896|119896minus1

= 119867119883119896|119896minus1

119878119896

= 119867119875119896|119896minus1

1198671015840+ 119877119896119870119896

= 119875119896|119896minus1

1198671015840119878minus1

119896119883119896|119896

= 119883119896|119896minus1

+ 119870119896(119911119896minus 119911119896|119896minus1

) 119875119896|119896

= 119875119896|119896minus1

minus 1198701198961198781198961198701015840

119896

(34)



119896|119896and 119875119862


(i) Time Update

119883119896|119896minus1

= 119865119883119862


119875119896|119896minus1

= 119865119875119862


(35)



119870119862

119896= 119875119896|119896119862119879

119896[119862119896119875119896|1198961198621015840

119896+ 119877120583

119896]

minus1

119883119862

119896|119896

= [119868 minus 119870119862

119896119862119896]119883119896|119896119875119862

119896|119896

= [119868 minus 119870119862

119896119862119896] 119875119896|119896

(36)

where

119862119896=

1

119878119896|119896

[0 0 119909119896|119896

0 0 119910119896|119896] (37)



119898 =

(1199091minus 1199092)

(1199102minus 1199101)

119887 =

(1199101+ 1199102)

2

minus 119898

(1199091+ 1199092)

2

(38)


where (1199091 1199101) and (119909



119909119888=

(1198871minus 1198872)

(1198982minus 1198981)

119910119888=

(11989811198872minus 11989821198871)

(1198981minus 1198982)

(39)



119906119896= 120588119906119896minus1

+ 119889119896

(40)

with

119889119896= ]1015840119896119878minus1

119896]119896 (41)


119896is its cov-


119899119906= 119899119911

1 + 120588

1 minus 120588

(42)







1198840= 40 km

0= minus172 km and



119883= 120590119884= 100mand the sample

rate 119879 = 1


120572 = 10minus3 120573 = 2 120581 = 0



(43)

1 15 2 25 3 35 4 45 5 55 62

25

3

35

4

45

5

55

6

k = 0

times104

times104

Y(m

)

X (m)



119876CA = diag 1 1

120575 = 092 1199030= 1 119903

1= 200

(44)


119876119898= diag 106 10minus4

120588 = 08

(45)





119872

119872

sum

119894=1

[(119909119894

119896minus 119909119894

119896)2

+ (119910119894

119896minus 119910119894

119896)2

] (46)


119910119894


119896|119896and 119910119894

119896are the posi-







0 50 100 150 200 250 300 350 40050

100

150

200

250

300

350

Time (s)

RMS

posit

ion

erro

rs (m

)

Method 4

Method 3

Method 2

Method 1











120596119896isin 119872119903= 120596 (1) 120596 (2) 120596 (119903) (47)



120591119894= 119901 (120596

0= 120596 (119894)) 119894 = 1 2 119903

120587119894119895= 119901 (120596

119896+1= 120596 (119895) | 120596

119896= 120596 (119894)) 119894 119895 = 1 2 119903

(48)


=

1199111 1199112 119911


1119873= 1198831 1198832 119883

119873


= 1205961 1205962 120596

119873












119869 (Ω1119873 Ω(119895)

1119873)

= 119864X1119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

(49)

where 119869(Ω1119873 Ω(119895)


1119873is the



Ω(119895+1)

1119873= argmax

Ω1119873

119869 (Ω1119873 Ω(119895)

1119873) (50)



119901 (1198831119873 1198851119873 Ω1119873)

=

119873

prod

119896=1

119901 (119911119896| 119883119896) times

119873

prod

119896=1


0)

times

119873

prod

119894=1

119901 (120596119894120596119894minus1) times 119901 (120596

0)

(51)

119901(119883119896| 119883119896minus1 120596119896minus1) and 119901(120596


sequenceΩ1119873


119901 (119883119896| 119883119896minus1 120596119896minus1) = 119873 119883

119896minus 119865 (120596

119896minus1)119883119896minus1 119876119896 (52)




119869 (Ω1119873 Ω(119895)

1119873)

= 1198641198831119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

=

119873

sum

119896=1

ln119901 (120596119896| 120596119896minus1) minus

1

2

(119883119896|119873


)

1015840

times 119876minus1

119896(119883119896|119873


)

(53)

where

119883119896|119873

= 119864 [119883119896| 1198851119873 Ω(119895)

1119873] (54)


are omittedhere

In the E step if Ω(119895)1119873









(3) Stop if Ω(119895+1)1119873

minusΩ(119895)



= Ω(119895+1)

1119873 then the



119896



Ω1119896

119901 (Ω1119896| 1198851119896) (55)

k = 1 k = 2 k = 3 k = 4

1205961

1205962

1205963

1205964 1205964 1205964 1205964

1205963 1205963 1205963

1205962 1205962 1205962

12059611205961 1205961



119901 (Ω1119896| 1198851119896) = 119901 (Ω

1119896| 119911119896 1198851119896minus1

)

=

119901 (119911119896| Ω119896 1198851119896minus1

) 119901 (Ω119896| 1198851119896minus1

)

119901 (119911119896| Ω1119896minus1

)

= 119901 (119911119896| Ω1119896 1198851119896minus1

) 119901 (120596119896| Ω1119896minus1

)

times 119901 (Ω1119896minus1

| 1198851119896minus1

) (119901 (119911119896| 1198851119896minus1

))minus1

(56)

Because Ω1119896

is Markov chain

119901 (120596119896| Ω1119896minus1

) = 119901 (120596119896| 120596119896minus1) (57)


119901 (119911119896| Ω1119896 1198851119896minus1

) asymp 119901 (119911119896| 120596119896 1198851119896minus1

)

= 119873 (120592119896 S119896)

(58)




119869 (120596119896(119894)) = ln119901 (Ω

1119896 120596119896(119894) 1198851119896) 119894 = 1 2 119903

(59)


119869 (120596119896(119895)) = 119869 (120596

119896minus1(119894)) + ln120587

119894119895

minus

1

2

1205921015840

119896(119894 119895) Sminus1

119896(119894 119895) 120592

119896(119894 119895) 119894 119895 = 1 2 119903

(60)










119896(119894)

and covariance 119875119896|119896(119894) 119894 = 1 2 119903


119895 = argmax119894

119869max (120596119896(119894))119903

119894=1 (61)

119883(B)119896= 119883119896(119895) 119875

(B)119896|119896

= 119875119896|119896(119895) (62)



=


119876 = [

119876119909

0

0 119876119910

] 119876119909= 119876119910=

[[[[[

[

1198794

3

1198793

2

1198793

2

1198792

]]]]]

]

(63)


119911119896= [

1 0 0 0

0 0 1 0]119883119896+ V119896 (64)



120596119896=

0 0 le 119896 lt 103

0033 rads 104 le 119896 lt 198

0 198 le 119896 lt 300

(65)



120591 = [

1

7

1

7

1

7

1

7

1

7

1

7

1

7

] (66)

20 25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

x (km)

y(k

m)



120587119894119895=

07 119894 = 119895

005 119894 = 119895

119894 119895 = 1 2 7

(67)



119872

119872

sum

119894=1

[(119894

119896minus 119909

119894

119896)

2

+ ( 119910119894

119896minus 119910

119894

119896)

2

] (68)


119896




5 Conclusions




0 50 100 150 200 250 30010

20

30

40

50

60

70


t (s)

Ep

(m)


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)








Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where (1199091 1199101) and (119909



119909119888=

(1198871minus 1198872)

(1198982minus 1198981)

119910119888=

(11989811198872minus 11989821198871)

(1198981minus 1198982)

(39)



119906119896= 120588119906119896minus1

+ 119889119896

(40)

with

119889119896= ]1015840119896119878minus1

119896]119896 (41)


119896is its cov-


119899119906= 119899119911

1 + 120588

1 minus 120588

(42)







1198840= 40 km

0= minus172 km and



119883= 120590119884= 100mand the sample

rate 119879 = 1


120572 = 10minus3 120573 = 2 120581 = 0



(43)

1 15 2 25 3 35 4 45 5 55 62

25

3

35

4

45

5

55

6

k = 0

times104

times104

Y(m

)

X (m)



119876CA = diag 1 1

120575 = 092 1199030= 1 119903

1= 200

(44)


119876119898= diag 106 10minus4

120588 = 08

(45)





119872

119872

sum

119894=1

[(119909119894

119896minus 119909119894

119896)2

+ (119910119894

119896minus 119910119894

119896)2

] (46)


119910119894


119896|119896and 119910119894

119896are the posi-







0 50 100 150 200 250 300 350 40050

100

150

200

250

300

350

Time (s)

RMS

posit

ion

erro

rs (m

)

Method 4

Method 3

Method 2

Method 1











120596119896isin 119872119903= 120596 (1) 120596 (2) 120596 (119903) (47)



120591119894= 119901 (120596

0= 120596 (119894)) 119894 = 1 2 119903

120587119894119895= 119901 (120596

119896+1= 120596 (119895) | 120596

119896= 120596 (119894)) 119894 119895 = 1 2 119903

(48)


=

1199111 1199112 119911


1119873= 1198831 1198832 119883

119873


= 1205961 1205962 120596

119873












119869 (Ω1119873 Ω(119895)

1119873)

= 119864X1119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

(49)

where 119869(Ω1119873 Ω(119895)


1119873is the



Ω(119895+1)

1119873= argmax

Ω1119873

119869 (Ω1119873 Ω(119895)

1119873) (50)



119901 (1198831119873 1198851119873 Ω1119873)

=

119873

prod

119896=1

119901 (119911119896| 119883119896) times

119873

prod

119896=1


0)

times

119873

prod

119894=1

119901 (120596119894120596119894minus1) times 119901 (120596

0)

(51)

119901(119883119896| 119883119896minus1 120596119896minus1) and 119901(120596


sequenceΩ1119873


119901 (119883119896| 119883119896minus1 120596119896minus1) = 119873 119883

119896minus 119865 (120596

119896minus1)119883119896minus1 119876119896 (52)




119869 (Ω1119873 Ω(119895)

1119873)

= 1198641198831119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

=

119873

sum

119896=1

ln119901 (120596119896| 120596119896minus1) minus

1

2

(119883119896|119873


)

1015840

times 119876minus1

119896(119883119896|119873


)

(53)

where

119883119896|119873

= 119864 [119883119896| 1198851119873 Ω(119895)

1119873] (54)


are omittedhere

In the E step if Ω(119895)1119873









(3) Stop if Ω(119895+1)1119873

minusΩ(119895)



= Ω(119895+1)

1119873 then the



119896



Ω1119896

119901 (Ω1119896| 1198851119896) (55)

k = 1 k = 2 k = 3 k = 4

1205961

1205962

1205963

1205964 1205964 1205964 1205964

1205963 1205963 1205963

1205962 1205962 1205962

12059611205961 1205961



119901 (Ω1119896| 1198851119896) = 119901 (Ω

1119896| 119911119896 1198851119896minus1

)

=

119901 (119911119896| Ω119896 1198851119896minus1

) 119901 (Ω119896| 1198851119896minus1

)

119901 (119911119896| Ω1119896minus1

)

= 119901 (119911119896| Ω1119896 1198851119896minus1

) 119901 (120596119896| Ω1119896minus1

)

times 119901 (Ω1119896minus1

| 1198851119896minus1

) (119901 (119911119896| 1198851119896minus1

))minus1

(56)

Because Ω1119896

is Markov chain

119901 (120596119896| Ω1119896minus1

) = 119901 (120596119896| 120596119896minus1) (57)


119901 (119911119896| Ω1119896 1198851119896minus1

) asymp 119901 (119911119896| 120596119896 1198851119896minus1

)

= 119873 (120592119896 S119896)

(58)




119869 (120596119896(119894)) = ln119901 (Ω

1119896 120596119896(119894) 1198851119896) 119894 = 1 2 119903

(59)


119869 (120596119896(119895)) = 119869 (120596

119896minus1(119894)) + ln120587

119894119895

minus

1

2

1205921015840

119896(119894 119895) Sminus1

119896(119894 119895) 120592

119896(119894 119895) 119894 119895 = 1 2 119903

(60)










119896(119894)

and covariance 119875119896|119896(119894) 119894 = 1 2 119903


119895 = argmax119894

119869max (120596119896(119894))119903

119894=1 (61)

119883(B)119896= 119883119896(119895) 119875

(B)119896|119896

= 119875119896|119896(119895) (62)



=


119876 = [

119876119909

0

0 119876119910

] 119876119909= 119876119910=

[[[[[

[

1198794

3

1198793

2

1198793

2

1198792

]]]]]

]

(63)


119911119896= [

1 0 0 0

0 0 1 0]119883119896+ V119896 (64)



120596119896=

0 0 le 119896 lt 103

0033 rads 104 le 119896 lt 198

0 198 le 119896 lt 300

(65)



120591 = [

1

7

1

7

1

7

1

7

1

7

1

7

1

7

] (66)

20 25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

x (km)

y(k

m)



120587119894119895=

07 119894 = 119895

005 119894 = 119895

119894 119895 = 1 2 7

(67)



119872

119872

sum

119894=1

[(119894

119896minus 119909

119894

119896)

2

+ ( 119910119894

119896minus 119910

119894

119896)

2

] (68)


119896




5 Conclusions




0 50 100 150 200 250 30010

20

30

40

50

60

70


t (s)

Ep

(m)


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)








Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 300 350 40050

100

150

200

250

300

350

Time (s)

RMS

posit

ion

erro

rs (m

)

Method 4

Method 3

Method 2

Method 1











120596119896isin 119872119903= 120596 (1) 120596 (2) 120596 (119903) (47)



120591119894= 119901 (120596

0= 120596 (119894)) 119894 = 1 2 119903

120587119894119895= 119901 (120596

119896+1= 120596 (119895) | 120596

119896= 120596 (119894)) 119894 119895 = 1 2 119903

(48)


=

1199111 1199112 119911


1119873= 1198831 1198832 119883

119873


= 1205961 1205962 120596

119873












119869 (Ω1119873 Ω(119895)

1119873)

= 119864X1119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

(49)

where 119869(Ω1119873 Ω(119895)


1119873is the



Ω(119895+1)

1119873= argmax

Ω1119873

119869 (Ω1119873 Ω(119895)

1119873) (50)



119901 (1198831119873 1198851119873 Ω1119873)

=

119873

prod

119896=1

119901 (119911119896| 119883119896) times

119873

prod

119896=1


0)

times

119873

prod

119894=1

119901 (120596119894120596119894minus1) times 119901 (120596

0)

(51)

119901(119883119896| 119883119896minus1 120596119896minus1) and 119901(120596


sequenceΩ1119873


119901 (119883119896| 119883119896minus1 120596119896minus1) = 119873 119883

119896minus 119865 (120596

119896minus1)119883119896minus1 119876119896 (52)




119869 (Ω1119873 Ω(119895)

1119873)

= 1198641198831119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

=

119873

sum

119896=1

ln119901 (120596119896| 120596119896minus1) minus

1

2

(119883119896|119873


)

1015840

times 119876minus1

119896(119883119896|119873


)

(53)

where

119883119896|119873

= 119864 [119883119896| 1198851119873 Ω(119895)

1119873] (54)


are omittedhere

In the E step if Ω(119895)1119873









(3) Stop if Ω(119895+1)1119873

minusΩ(119895)



= Ω(119895+1)

1119873 then the



119896



Ω1119896

119901 (Ω1119896| 1198851119896) (55)

k = 1 k = 2 k = 3 k = 4

1205961

1205962

1205963

1205964 1205964 1205964 1205964

1205963 1205963 1205963

1205962 1205962 1205962

12059611205961 1205961



119901 (Ω1119896| 1198851119896) = 119901 (Ω

1119896| 119911119896 1198851119896minus1

)

=

119901 (119911119896| Ω119896 1198851119896minus1

) 119901 (Ω119896| 1198851119896minus1

)

119901 (119911119896| Ω1119896minus1

)

= 119901 (119911119896| Ω1119896 1198851119896minus1

) 119901 (120596119896| Ω1119896minus1

)

times 119901 (Ω1119896minus1

| 1198851119896minus1

) (119901 (119911119896| 1198851119896minus1

))minus1

(56)

Because Ω1119896

is Markov chain

119901 (120596119896| Ω1119896minus1

) = 119901 (120596119896| 120596119896minus1) (57)


119901 (119911119896| Ω1119896 1198851119896minus1

) asymp 119901 (119911119896| 120596119896 1198851119896minus1

)

= 119873 (120592119896 S119896)

(58)




119869 (120596119896(119894)) = ln119901 (Ω

1119896 120596119896(119894) 1198851119896) 119894 = 1 2 119903

(59)


119869 (120596119896(119895)) = 119869 (120596

119896minus1(119894)) + ln120587

119894119895

minus

1

2

1205921015840

119896(119894 119895) Sminus1

119896(119894 119895) 120592

119896(119894 119895) 119894 119895 = 1 2 119903

(60)










119896(119894)

and covariance 119875119896|119896(119894) 119894 = 1 2 119903


119895 = argmax119894

119869max (120596119896(119894))119903

119894=1 (61)

119883(B)119896= 119883119896(119895) 119875

(B)119896|119896

= 119875119896|119896(119895) (62)



=


119876 = [

119876119909

0

0 119876119910

] 119876119909= 119876119910=

[[[[[

[

1198794

3

1198793

2

1198793

2

1198792

]]]]]

]

(63)


119911119896= [

1 0 0 0

0 0 1 0]119883119896+ V119896 (64)



120596119896=

0 0 le 119896 lt 103

0033 rads 104 le 119896 lt 198

0 198 le 119896 lt 300

(65)



120591 = [

1

7

1

7

1

7

1

7

1

7

1

7

1

7

] (66)

20 25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

x (km)

y(k

m)



120587119894119895=

07 119894 = 119895

005 119894 = 119895

119894 119895 = 1 2 7

(67)



119872

119872

sum

119894=1

[(119894

119896minus 119909

119894

119896)

2

+ ( 119910119894

119896minus 119910

119894

119896)

2

] (68)


119896




5 Conclusions




0 50 100 150 200 250 30010

20

30

40

50

60

70


t (s)

Ep

(m)


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)








Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119869 (Ω1119873 Ω(119895)

1119873)

= 1198641198831119873

ln119901 (1198831119873 1198851119873 Ω1119873) | 1198851119873 Ω(119895)

1119873

=

119873

sum

119896=1

ln119901 (120596119896| 120596119896minus1) minus

1

2

(119883119896|119873


)

1015840

times 119876minus1

119896(119883119896|119873


)

(53)

where

119883119896|119873

= 119864 [119883119896| 1198851119873 Ω(119895)

1119873] (54)


are omittedhere

In the E step if Ω(119895)1119873









(3) Stop if Ω(119895+1)1119873

minusΩ(119895)



= Ω(119895+1)

1119873 then the



119896



Ω1119896

119901 (Ω1119896| 1198851119896) (55)

k = 1 k = 2 k = 3 k = 4

1205961

1205962

1205963

1205964 1205964 1205964 1205964

1205963 1205963 1205963

1205962 1205962 1205962

12059611205961 1205961



119901 (Ω1119896| 1198851119896) = 119901 (Ω

1119896| 119911119896 1198851119896minus1

)

=

119901 (119911119896| Ω119896 1198851119896minus1

) 119901 (Ω119896| 1198851119896minus1

)

119901 (119911119896| Ω1119896minus1

)

= 119901 (119911119896| Ω1119896 1198851119896minus1

) 119901 (120596119896| Ω1119896minus1

)

times 119901 (Ω1119896minus1

| 1198851119896minus1

) (119901 (119911119896| 1198851119896minus1

))minus1

(56)

Because Ω1119896

is Markov chain

119901 (120596119896| Ω1119896minus1

) = 119901 (120596119896| 120596119896minus1) (57)


119901 (119911119896| Ω1119896 1198851119896minus1

) asymp 119901 (119911119896| 120596119896 1198851119896minus1

)

= 119873 (120592119896 S119896)

(58)




119869 (120596119896(119894)) = ln119901 (Ω

1119896 120596119896(119894) 1198851119896) 119894 = 1 2 119903

(59)


119869 (120596119896(119895)) = 119869 (120596

119896minus1(119894)) + ln120587

119894119895

minus

1

2

1205921015840

119896(119894 119895) Sminus1

119896(119894 119895) 120592

119896(119894 119895) 119894 119895 = 1 2 119903

(60)










119896(119894)

and covariance 119875119896|119896(119894) 119894 = 1 2 119903


119895 = argmax119894

119869max (120596119896(119894))119903

119894=1 (61)

119883(B)119896= 119883119896(119895) 119875

(B)119896|119896

= 119875119896|119896(119895) (62)



=


119876 = [

119876119909

0

0 119876119910

] 119876119909= 119876119910=

[[[[[

[

1198794

3

1198793

2

1198793

2

1198792

]]]]]

]

(63)


119911119896= [

1 0 0 0

0 0 1 0]119883119896+ V119896 (64)



120596119896=

0 0 le 119896 lt 103

0033 rads 104 le 119896 lt 198

0 198 le 119896 lt 300

(65)



120591 = [

1

7

1

7

1

7

1

7

1

7

1

7

1

7

] (66)

20 25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

x (km)

y(k

m)



120587119894119895=

07 119894 = 119895

005 119894 = 119895

119894 119895 = 1 2 7

(67)



119872

119872

sum

119894=1

[(119894

119896minus 119909

119894

119896)

2

+ ( 119910119894

119896minus 119910

119894

119896)

2

] (68)


119896




5 Conclusions




0 50 100 150 200 250 30010

20

30

40

50

60

70


t (s)

Ep

(m)


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)








Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119896(119894)

and covariance 119875119896|119896(119894) 119894 = 1 2 119903


119895 = argmax119894

119869max (120596119896(119894))119903

119894=1 (61)

119883(B)119896= 119883119896(119895) 119875

(B)119896|119896

= 119875119896|119896(119895) (62)



=


119876 = [

119876119909

0

0 119876119910

] 119876119909= 119876119910=

[[[[[

[

1198794

3

1198793

2

1198793

2

1198792

]]]]]

]

(63)


119911119896= [

1 0 0 0

0 0 1 0]119883119896+ V119896 (64)



120596119896=

0 0 le 119896 lt 103

0033 rads 104 le 119896 lt 198

0 198 le 119896 lt 300

(65)



120591 = [

1

7

1

7

1

7

1

7

1

7

1

7

1

7

] (66)

20 25 30 35 40 45 50 55 6025

30

35

40

45

50

55

60

65

x (km)

y(k

m)



120587119894119895=

07 119894 = 119895

005 119894 = 119895

119894 119895 = 1 2 7

(67)



119872

119872

sum

119894=1

[(119894

119896minus 119909

119894

119896)

2

+ ( 119910119894

119896minus 119910

119894

119896)

2

] (68)


119896




5 Conclusions




0 50 100 150 200 250 30010

20

30

40

50

60

70


t (s)

Ep

(m)


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)








Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 250 30010

20

30

40

50

60

70


t (s)

Ep

(m)


0 50 100 150 200 250 3000

10

20

30

40

50

60

70

t (s)

E

(m)








Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









models and algorithms for tracking target with coordinated

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