Feedback Particle Filter-based Multiple Target Tracking usingBearing-only Measurements

Adam Tilton, Tao Yang, Huibing Yin and Prashant G. Mehta

Abstract—This paper describes the joint probabilistic dataassociation-feedback particle filter (JPDA-FPF) introduced in ourearlier paper [1]. The JPDA-FPF is based on the feedback parti-cle filter concept (see [2],[3]). A remarkable feature of the JPDA-FPF algorithm is its innovation error-based feedback structure,even with data association uncertainty in the general nonlinearcase. The classical Kalman filter-based joint probabilistic dataassociation filter (JPDAF) is shown to be a special case of theJPDA-FPF.

A multiple target tracking application is presented: In theapplication, bearing only measurements with multiple sensorsare used to track targets in the presence of data associationuncertainty. It is shown that the algorithm is successfully able totrack targets with significant uncertainty in initial estimate, andeven in the presence of certain “track coalescence” scenarios.

I. INTRODUCTION

Filtering with data association uncertainty is important toa number of applications, including, air and missile defensesystems, air traffic surveillance, weather surveillance, groundmapping, geophysical surveys, remote sensing, autonomousnavigation and robotics [4], [5], [6]. In each of these appli-cations, there exists data association uncertainty in the sensethat one can not assign individual measurements to individualtargets in an apriori manner.

Given the large number of applications, algorithms forfiltering problems with data association uncertainty have beenextensively studied in the past; cf., [4], [7] and referencestherein. One important and widely used algorithm is the jointprobabilistic data association filter (JPDAF) [4]. In this algo-rithm, a Kalman filter is used to solve the filtering problem.For the data association problem, the central object of interestis the computation (or approximation) of the measurement-to-target association probability. The association probability isapproximated using the Bayes’ formula, and integrated withthe Kalman filter to obtain the JPDAF algorithm.

The development of particle filters has naturally led to in-vestigation of data association algorithms based on importancesampling techniques. This remains an active area of research;cf., [8], [9], [10], [11] and references therein. One earlycontribution is the multitarget particle filter (MPFT) in [12].In this paper and related studies (e.g., [13], [14]), a particlefilter is used to solve the filtering problem. The associationprobabilities are obtained via the use of Markov Chain Monte-Carlo (MCMC) techniques.

Financial support from NSF grants EECS-0925534 and the AFOSR grantFA9550-09-1-0190 is gratefully acknowledged.

A. Tilton, T. Yang, H. Yin, and P. G. Mehta are with the Coordinated Sci-ence Laboratory and the Department of Mechanical Science and Engineeringat the University of Illinois at Urbana-Champaign (UIUC) {atilton2;taoyang1;yin3; mehtapg}@illinois.edu

In a recent work [1], we introduced a novel particle filteralgorithm for solution of the joint filtering-data associationproblem. The proposed algorithm is referred to as joint proba-bilistic data association-feedback particle filter (JPDA-FPF).The JPDA-FPF algorithm is based on the feedback particlefilter concept (see [2],[3]). A feedback particle filter is acontrolled system to approximate the solution of the nonlinearfiltering problem. The filter has a feedback structure similarto the Kalman filter: At each time t, the control is obtainedby using a proportional gain feedback with respect to acertain modified form of the innovation error. The filter designamounts to design of the proportional gain – the solution isgiven by the Kalman gain in the linear Gaussian case. Figure 1depicts a comparison of the Kalman filter and the feedbackparticle filer.

It was shown in [1] that the joint probabilistic dataassociation-feedback particle filter (JPDA-FPF) represents ageneralization of the Kalman filter-based joint probabilisticdata association (JPDAF). One remarkable conclusion is thatthe JPDA-FPF retains the innovation error-based feedbackstructure even for the nonlinear problem. The innovation error-based feedback structure is expected to be useful because ofthe coupled nature of the filtering and the data associationproblems.

The aim of this paper is to describe applications of theJPDA-FPF algorithm. Beyond [1], the contributions of thispaper are as follows:

• Notation. The algorithm is described for an ODE model forthe signal and the observation process, as opposed to the SDEformalism employed in [1]. The formalism is better suited forimplementation and application of the JPDA-FPF algorithm.

• JPDA-FPF algorithms. The basic PDA-FPF algorithmpresented in [1] is now generalized to the multi-target problemwith data association uncertainty. Notation for associations inthe general case is introduced, the resulting feedback particlefilter algorithm described and compared with the linear case(see Table I).

• Multiple Target Tracking Application. A multiple targettracking application is presented: In this application, bearingonly measurements with multiple sensors are used to tracktargets in the presence of data association uncertainty. It isshown that the algorithm is successfully able to track targetswith significant uncertainty in initial estimate, and even in thepresence of certain “track coalescence” problem scenarios.

The outline of the remainder of this paper is as follows.The JPDA-FPF algorithm is first described for single target inthe presence of clutter, in Sec. II. The multiple target case is

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discussed in Sec. III. The application results are described inSec. IV.

II. FEEDBACK PARTICLE FILTER WITH DATAASSOCIATION UNCERTAINTY

In this section, we describe the basic probabilistic dataassociation-feedback particle filter (PDA-FPF) for tracking asingle target in the presence of multiple measurements. Thefilter for multiple targets is obtained as a straightforwardextension, and described in Sec. III.

A. Problem statement, Assumptions and NotationThe following notation is adopted:(i) At time t, the target state is denoted by Xt ∈ Rd .(ii) At time t, the observation vector Y t :=(Y 1

t ,Y2

t , . . . ,Ym

t )T , where m is assumed fixed andY m

t ∈ Rs.(iii) At time t, the association random variable is denotedas At ∈ {0,1, . . . ,m}. It is used to associate one mea-surement to the target: At = m signifies that the mth-measurement Y m

t is ‘associated’ with the target, and At = 0means that the target is not detected at time t.

The following models are assumed for the state and obser-vation processes:

(i) The state Xt evolves according to a nonlinear ODE:

Xt = a(Xt)+σBBt , (1)

where a(·) are C1 function, and {Bt} is assumed to be astandard d−dim white noise with σB ∈ R.

(ii) In [1], the association random process At is modeledas a jump Markov process in continuous-time. For thediscussion here, we simply assume a uniform prior on At(This would correspond to the mixing rate of the Markovprocess being very large).

(iii) At and Xt are assumed to be mutually independent.(iv) At time t, the observation model is given by,

Y mt = 1[At=m]h(Xt)+W m

t , (2)

for m ∈ {1, . . . ,m}, where {W mt } are mutually indepen-

dent white noise processes. The covariance matrix ofthe observation noise is denoted as R, assumed to be apositive definite symmetric matrix. We define:

1[At=m] :=

{1 when At = m

0 otherwise.

The problem is to obtain the posterior distribution of Xtgiven the history of observations (filtration) Y t := σ(Y τ : τ ≤t). The posterior is denoted by p, so that for any measurableset A⊂ Rd , ∫

X∈Ap(x, t)dx = P{Xt ∈ A|Z t}

The methodology comprises of the following two parts:(i) Evaluation of association probability, and(ii) Integration of association probability in the feedbackparticle filter framework.

B. Association Probability for a Single Target

The association probability is defined as the probability ofthe association [At = m] conditioned on Z t :

βmt , Pr([At = m]|Z t), m = 0,1, ...,m. (3)

Since the events are mutually exclusive and exhaustive,∑mm=0 β m

t = 1.In the following, we integrate association probability with

the feedback particle filter, which is used to approximateevolution of the posterior. Next, an algorithm to approximatethe association probability is discussed.

Separate algorithms for data association and posterior aremotivated in part by the classical JPDA filtering literature [15],[4], [7]. A separate treatment is also useful while consideringmultiple target tracking problems. For such problems, one canextend algorithms for data association in a straightforwardmanner, while the algorithm for posterior remains as before.Additional details appear in Sec III.

C. Feedback Particle Filter

Following the feedback particle filter methodology, themodel for the particle filter is given by,

X it = a(X i

t )+σBBit +U i

t , (4)

where X it ∈ Rd is the state for the ith particle at time t, U i

t isits control input, and {Bi

t} are mutually independent standardwhite noise processes, i ∈ {1,2, . . . ,N}. We assume the initialconditions {X i

0}Ni=1 are i.i.d., independent of {Bi

t}, and drawnfrom the initial distribution p∗(x,0) of X0. Both {Bi

t} and {X i0}

are also assumed to be independent of Xt ,Yt .There are two types of conditional distributions of interest

in our analysis:1) p(x, t): Defines the conditional dist. of X i

t given Zt .2) p∗(x, t): Defines the conditional dist. of Xt given Zt .

The control input U it are said to be optimal if p ≡ p∗. That

is, given p∗(·,0) = p(·,0), our goal is to choose U it in the

feedback particle filter so that the evolution equations of theseconditional distributions coincide.

It is shown in [1] that the optimally controlled dynamics ofthe ith particle have the following form,

X it = a(X i

t )+σBBit +

m

∑m=1

βmt K(X i

t , t)Ii,mt (5)

where Ii,mt is a modified form of the innovation process,

Ii,mt := Y m

t − [β m

t

2h(X i

t )+(1− β mt

2)ht ], (6)

where ht := E[h(Xt)|Z t ] =∫

h(x)p(x, t)dx. In a numericalimplementation, we approximate h≈ 1

N ∑Ni=1 h(X i

t ) =: h(N).The gain function K is the solution of a certain Euler-

Lagrange boundary value problem (E-L BVP):

∇ · (pK) =−(h− h)T R−1 p. (7)

The filter (5)-(7) is referred to as the feedback particlefilter. It is shown in [1] that the feedback particle filter is

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Fig. 1. Innovation error-based feedback structure for (a) Kalman filter and (b) nonlinear feedback particle filter.

consistent with the nonlinear filter, given consistent initializa-tions p( · ,0) = p∗( · ,0). Consequently, if the initial conditions{X i

0}Ni=1 are drawn from the initial distribution p∗( · ,0) of

X0, then, as N→ ∞, the empirical distribution of the particlesystem approximates the posterior distribution p∗( · , t) for eacht.

The filter requires solution of the E-L BVP (7) for eachvalue of time t. For the numerical results presented in thispaper, we use the following approximation of this solution

K≈ 1N

N

∑i=1

X it

(h(X i

t )− h(N)t

)TR−1. (8)

The justification for the approximation appears in [16].

D. Filter for Association Probability

In [1], a Wonham filter is described for evolution of theassociation probability. In the limit of large transition rate, thefollowing heuristic is valid:

βmt =

L(Y mt )

∑mj=1 L(Y j

t ), m = 1,2, ...,m, (9)

where

L(Y mt ) =

1

(2π)d2 |R| 12

∫Rd

exp[−‖Y m

t −h(x)‖2R]

p(x, t)dx,

(10)

where the weighted norm with respect to R is defined as:

‖V‖R :=(

12

V T R−1V) 1

2(11)

The formula (10) is the nonlinear counterpart of the formulaused to obtain association probability in the classical PDAF(see Section 6.4 in [15]). As in the PDAF, the formula followsfrom application of the Bayes’ rule. The derivation appears inAppendix D in [1].

For the numerical results presented in this paper, we use thefollowing approximation of (10):

L(Y mt )≈ 1

(2π)d2 |R| 12

N

∑i=1

exp[−||Y mt −h(X i

t )||2R].

E. Comparison of PDA-FPF and PDAF

In this section, we compare the PDA-FPF to PDAF with theaid of a linear example. Consider the following linear model:

Xt = A Xt + Bt , (12a)Yt = H Xt +Wt , (12b)

where A is a d × d matrix and H is an m× d matrix. Weassume the initial distribution p∗(x,0) is Gaussian with meanµ0 and covariance matrix Σ0.

The following lemma provides the solution of the gainfunction K(x, t) in the linear case.

Lemma 1: Consider the linear observation (12b). Supposep(x, t) = 1

(2π)d2 |Σt |

12

exp[− 1

2 (x−µt)T ΣT

t (x−µt)]

is assumed to

be Gaussian with mean µt and variance Σt . Then the solutionof the E-L BVP (7) is given by:

K(x, t) = ΣtHT R−1 (13)

The formula (13) is verified by direct substitution in (7)where the distribution p is Gaussian.

Using the formula for the gain function, the linear PDA-FPFis given by,

X it = A X i

t +σBBit

+ΣtHT R−1m

∑m=1

βmt

[Y m

t −H(β m

t

2X i

t +(1− β mt

2)µt)

]. (14)

The following theorem states that p = p∗ in this case. Thatis, the conditional distributions of Xt and X i

t coincide. Theproof is omitted.

Theorem 2: Consider the single target tracking problemwith a linear model defined by the state-observation equations(12a,12b). The PDA-FPF is given by (14). In this case theposterior distributions of Xt and X i

t coincide, whose conditionalmean and covariance are given by the following,

µt = Aµt +ΣtHT R−1m

∑m=1

βmt (Y m

t −Hµt) (15)

Σt = AΣt +ΣtAT + I−ΣtHT R−1HΣt

m

∑m=1

(β mt )2. (16)

In practice {µt ,Σt} in (14)- (16) are approximated as samplemeans and sample covariances from the ensemble {X i

t }Ni=1.

µt ≈ µ(N)t :=

1N

N

∑i=1

X it ,

Σt ≈ Σ(N)t :=

1N−1

N

∑i=1

(X it −µ

(N)t )2.

(17)

The data association probability β mt is also evaluated by

using particles:

βmt ∝

1N

1

(2π)d2 |R| 12

N

∑i=1

exp[−‖Y m

t −HX it ‖2

R].

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TABLE I

COMPARISON OF THE NONLINEAR PDA-FPF ALGORITHM WITH THE LINEAR PDAF ALGORITHM

PDAF PDA-FPF

State model Xt = AXt +σBBt Xt = a(Xt)+σBBt

Observation model Yt = HXt +Wt Yt = h(Xt)+Wt

Assoc. Prob. β mt := P([At = m]|Z t)

Prediction Yt = Hµt Y i,mt = [

β mt2 h(X i

t )+(1− β mt2 )ht ]

Innovation error Imt = Y m

t − Yt Ii,mt = Y m

t − Y i,mt

Feedback control Umt = KgIm

t U i,mt = K(X i

t , t)Ii,mt

Gain Kalman Gain Sol. of a BVP (7)

Control input ∑mm=1 β m

t Umt ∑

mm=1 β m

t U i,mt

Formula for β Mt ∝ exp

[−‖Y m

t − Yt‖2R]

∝1N ∑

Ni=1 exp

[−‖Y m

t −h(X it )‖2

R]

Filter equation µt = Aµt +Kg ∑m β mt Um

t X it = a(X i

t )+σBBit +K(X i

t , t)∑m β mt U i,m

t

F. Comparison of PDA-FPF and PDAF

Table I provides a comparison of the PDA-FPF and thePDAF algorithms. The main point to note is that the feedbackparticle filter has an innovation error-based structure: In effect,the ith particle makes a prediction Y i,m

t as a weighted-averageof h(X i

t ) and ht . This is then used to compute an innovationerror Ii,m

t . The Bayes’ update step involves gain feedback ofthe innovation error.

III. MULTIPLE TARGET TRACKING USING FEEDBACKPARTICLE FILTER

In this section, we extend the PDA-FPF to the multipletarget tracking problem. The resulting filter is referred toas the joint probabilistic data association feedback particlefilter (JPDA-FPF). For notational ease, we assume that allmeasurements originate from targets (That is, there is noclutter).

A. Problem Statement, Assumptions and Notation

The following notation is adopted:(i) There are n distinct targets. The set of targets is denotedby an index set N = {1,2, . . . , n}. The set of permutationsof N is denoted by P(N ). The cardinality of the setP(N ), |P(N )| = n!. A typical elements of P(N )is denoted as a = (a1,a2, . . . ,an), where an ∈N for alln ∈N .

(ii) At time t, the state of the nth target is denoted as Xnt ∈

Rd for n ∈N .(iii) At time t, there is exactly one measurement per targetfor a total of n measurements. There are no missingmeasurements and no measurements because of clutter.The observation vector Y t := (Y 1

t ,Y2

t , . . . ,Yn

t )T , where thenth entry, Y n

t ∈ Rs, originates from one of the targets inN .

(iv) At time t, the association random variable is denotedas At ∈ P(N ). It is used to associate targets withthe measurements: At = at = (a1

t ,a2t , . . . ,a

nt ) ∈ P(N )

signifies that the measurement Y 1t originates from target

a1t , Y 2

t originates from target a2t , . . . , Y n

t originates fromtarget ant .

The following models are assumed for the three stochasticprocesses.

(i) The dynamics of the nth target evolves according to thenonlinear ODE:

Xnt = an(Xn

t )+σnBBn

t , (18)

where {Bnt ∈ Rd}nn=1 are mutually independent white

noise processes.(ii) The observation model is given by:Y 1

t...

Y nt

= Ψ(At)

h(X1t )

...h(X n

t )

+W 1

t...

W nt

, (19)

where {W nt } are mutually independent white noise pro-

cesses with covariance matrix R whose dimension is s×s.Furthermore, {W n

t } are mutually independent with {Bnt }

and Ψ(At) is the permutation matrix for association At .Example 1: Suppose n= 2. The permutation matrices are

Ψ({1,2}) =[

Is 00 Is

], Ψ({2,1}) =

[0 IsIs 0

], (20)

where Is is an identity matrix with dimension s.(iii) The model for At is the same as the model assumedin discussions of PDA-FPF (see Sec. II-A)

(iv) At and X t are assumed to be mutually independent.The problem is to design n feedback particle filters, where

the nth filter is intended to estimate the posterior distribution ofthe nth target given the history of all unassociated observations(filtration) Y t := σ (Y s : s≤ t). The posterior distribution isdenoted as pn(x, t).

As before, the algorithm comprises of the following twoparts:

(i) Evaluation of the measurement-to-target associationprobability, and

(ii) Integration of association probability with the feedbackparticle filter.

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B. JPDA-FPF

The association probability is defined as the conditionalprobability of association [ar

t = n]:

βn,rt := Pr

([ar

t = n]∣∣Z t ,Y t

). (21)

In words, βn,rt is the conditional probability that the rth

measurement, Y rt , originates from the nth target.

Using these probabilities, and denoting the state of the ith

particle for the nth target at time t as Xn,it , i ∈ {1,2, . . . ,N},

the JPDA-FPF algorithm for the nth target is as follows:

Xn,it = an(Xn,i

t )+σiBBn

t +n

∑r=1

βn,rt Kn(Xn,i

t , t)In,i,rt , (22)

where In,i,rt is a modified form of the innovation process,

In,i,rt := Y r

t −[

βn,rt

2h(Xn,i

t )+(1− βn,rt

2)hn

t

](23)

with hnt := E[h(Xn

t )|Z t ], and Kn is control gain obtained fromsolving a certain EL-BVP:

∇ · (pnKn) =−(h− hn

t )T R−1 pn, (24)

where pn(x, t) denotes the posterior distribution. In practice itis approximated by a constant-valued formula:

Kn =1N

N

∑i=1

Xn,it

(h(Xn,i

t )− hn(N)t

)TR−1, (25)

where hn(N)t = 1

N ∑Ni=1 h(Xn,i

t ).Finally, the association probabilities are calculated easily

using Bayes’ rule as in Sec. II-D. In particular,

βn,rt = ∑

{b∈P(N ):br=n}γ

bt , (26)

where

γbt ∝

n

∏k=1

Pr(

Y kt

∣∣∣ [akt = bk]

),

=1

(2π)d2 |R| 12

n

∏k=1

(∫Rd

exp[−‖Y k

t −h(x)‖2R

]pbk(x, t)dx

)

≈ 1

(2π)d2 |R| 12

n

∏k=1

(1N

N

∑i=1

exp[−‖Y k

t −h(Xbk,it )‖2

R

]).

As the number of targets increases, there is an exponentialgrowth in the number of associations. In practice, one mayalso wish to consider approaches to reduce filter complexity,e.g., by assigning gating regions for the measurements; cf.,Sec. 4.2.3 in [15]. This is the subject of future investigation.

C. Multi-sensor Case

Now suppose there are m sensors that provide indepen-dent measurements of n targets. The filter for this case isa straightforward generalization of the single-sensor filterdescribed in the preceding section: The notation specific tosensor m ∈ {1,2, . . . ,m} is tagged with sensor index m.

In particular, we denote the measurements for the mth sensoras Y m

t = (Y 1;mt , . . . ,Y n;m

t )T . The association probabilities arenow denoted as

βn,r;mt = Pr

([ar;m

t = n]∣∣Z t ,Y

mt)

(27)

for m ∈ {1,2, . . . ,m} and n,r ∈ N . It is the conditionalprobability that the rth measurement from the mth sensororiginates from the nth target.

Using the association probabilities, the filter for the nth

target is

Xn,it = an(Xn,i

t )+σiBBn,i

t +m

∑m=1

n

∑r=1

βn,r;mt Kn;m(Xn,i

t , t)In,i,r;mt ,

where formulae for innovation error and the gain function aresimilar to (23)- (25).

The association probabilities are obtained, separately foreach sensor, using formulae similar to (26).

IV. NUMERICS

A. Single target tracking problem

Consider a target tracking problem with two bearing-onlysensors [17]. A single target moves in a two-dimensional(2d) plane according to the standard white-noise accelerationmodel:

Xt = AXt +ΓBt , (28)

where X :=(X1,V1,X2,V2)T ∈R4, (X1,X2) denotes the position

and (V1,V2) denotes the velocity. The matrices,

A =

0 1 0 00 0 0 00 0 0 10 0 0 0

, Γ = σB

0 01 00 00 1

,and Bt is a standard 2d white noise.

The observation model is given by,

Yt = h(Xt)+σWWt , (29)

where Wt is a standard 2d white noise, h = (h1,h2)T and

h j(x1,v1,x2,v2) = arctan

(x2− x(sen j)

2

x1− x(sen j)1

), j = 1,2,

where (x(sen j)1 ,x(sen j)

2 ) denote the position of sensor j.Figure 2 depicts a sample path obtained for a typical

numerical experiment. The sensor and target locations aredepicted together with an estimate (conditional mean) that isapproximated using a feedback particle filter. Since there isonly single target, there is no data association uncertainty. Thebackground depicts the ensemble of observations that weremade over the simulation run. Each point in the ensemble isobtained by using the process of triangulation based on two(noisy) angle measurements.

The simulation parameters are: The initial position of thetarget is depicted, the initial velocity was chosen as (0.2,−5)and σB = 0.1; The two sensor positions are depicted andσW = 0.017; The particle filter comprised of N = 200 particles

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Meas.

Tgt.Ptcle. Mean

Sensor

Tgt. Init.

Ptcle. Init

30

40

20

10

0

0 10 20 30-30 -20 -10-10

Fig. 2. Simulation results: Comparison of the true target trajectory with theestimate obtained using FPF.

Fig. 3. Simulation results for two target example.

whose initial position is chosen from a Gaussian distributionwhose mean is depicted. The gain function is obtained usingthe constant gain approximation in (25). The simulation resultsshow that the filter can adequately track the target.

B. Two targets

Consider a tracking problem with two targets as depictedin Fig. 3. The targets move in the 2d plane according to thestandard white-noise model (28).

There are two bearing-only sensors also depicted in thefigure. At time t, each sensor obtains two angle only mea-surements according to model (29).

There is data association uncertainty in this case, in thesense that one cannot assign measurements to individualtargets in an apriori manner. In this particular example, errorsin data association can lead to appearance of a “ghost” target(see Fig. 4).

In the simulation study, tracks are initialized at the locationof the “ghost” target (black circles in Fig. 3). Particles are ini-tialized by drawing from a Gaussian distribution whose meanis set to be the “ghost” target location. As depicted in Fig. 4,this position is identified by using the process of triangulationbased on initial positions of the two targets. Figure 3 depictsthe estimate (mean of particles) obtained using the JPDA-FPF algorithm. At each time t, four association probabilitiesare approximated for each sensor: β

1,1;1t , β

1,2;1t are the two

Fig. 4. Illustration of “ghost” target in the two-sensor two-target case: Theghost appears because of incorrect data association.

association probabilities for target 1, and β2,1;1t , β

2,2;1t are the

probabilities for target 2 (see (27)).The simulation parameters are: The two targets start at po-

sition (−20,50) and (20,50), respectively. The initial velocityvectors are V 1

0 =V 20 = (0.0,−5.0). σB = 0.1, σW = 0.017 and

the number of particles N = 200.

C. Track coalescence exampleTrack coalescence is a common problem in multiple target

tracking applications. Track coalescence can occur when twoclosely spaced targets move with approximately the samevelocity over a time period [6], [18]. With standard implemen-tations of JPDAF and SIR particle filter algorithms, the targettracks tends to coalesce even after the targets have movedapart [18]. In the following example, we describe simulationresults for JPDA-FPF for a model problem scenario relatedto [19].

As in the preceding example, there are two targets in the2d plane, modeled by (28). Initially, the two targets are atposition (−40,600) and (40,600), respectively. They movetowards each other with initial velocity vector (4.5,−5.0) and(−4.5,−5.0), respectively. When they are close to each other,they move vertically together for some time (velocity alongx is set to 0). Finally, they move away from each other withvelocity vectors (−4.5,−5.0) and (4.5,−5.0), respectively.

Two sensors obtain two bearing-only measurements, butwith data association uncertainty.

Figure 5(a) depicts the results of a single simulation: Thetracks are initialized at the actual target location. The estimatesare obtained using the JPDA-FPF with N = 200 particles.Figure 5(b) depicts the association probability during thesimulation run.

As shown in the figure, the tracks coalesce when the twotargets are close. However, the filter tracks the targets oncethey move away. That is, the track coalescence problem issuccessfully avoided.

D. Three targetsIn this numerical experiment, the objective is to track

three targets with two bearing-only sensors. The initial target

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Fig. 5. (a) Simulation results for the track coalescence example; (b) Plot of data association probability.

Fig. 6. Simulation results for the three targets example.

positions are (−40,300), (0,300) and (40,300), respectively.The initial velocity vectors are V 1 =V 2 =V 3 = (0,−5).

Figure 6 depicts the results of a simulation study. With onlytwo sensors, the filter performance degrades as the number oftargets increases beyond 3.

ACKNOWLEDGEMENT

We are grateful to Dr. Samuel Blackman for suggesting theapplication problem described in this paper.

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