Effective joint probabilistic data association using maximum a posteriori estimates of target states

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1 Effective joint probabilistic data association using axiu a posteriori estiates of target states 1 Viji Paul Panakkal, 2 Rajbabu Velurugan 1 Central Research Laboratory, Bharat Electronics Ltd., Bangalore, India 2 Dept. of Electrical Engineering, Indian Institute of Technology Bobay, Mubai, India Abstract In ulti-target tracking MTT, closely oving targets can be tracked accurately with access to the true easureent-to-track M-to-T association configuration, which is not known in practical scenarios. The joint probabilistic data association approach JPDA coputes the association configuration probabilities using the predicted target states. In MTT when targets are close, the target states are obtained by arginalizing the target states coputed over all feasible association configurations. The proposed approach coputes the association configuration probabilities using estiated posterior target states and iproves the estiation accuracy. The posterior target states are obtained as axiu a posteriori MAP estiates. The MAP estiates are obtained using an iterative procedure based on the expectation axiization algorith. Copared to other data association approaches such as Set JPDA and JPDA* the proposed approach provides better estiation accuracy and iproves track identity aintenance for closely oving targets. The identity is aintained in the proposed approach by using the estiated posterior target states for coputation of association configuration probabilities. Monte Carlo siulations verify the advantage of the proposed ethod over other approaches in a siulated, ulti-target, cluttered environent with varying probabilities of detection. I. INTRODUCTION Multi-target tracking MTT proble requires the siultaneous copletion of two tasks: estiating the target state and easureent-to-track association. Estiation is the task of finding the best odel paraeters to describe the observed data. However, the sensor easureents do not identify which target caused the. If data fro one target is istakenly used in the state update of a different target track, then the resulting target state estiate becoes erroneous. The task of assigning correct easureents to correct tracks is the objective of data association process. The difficulty in data association algoriths is illustrated in Fig. 1. In this paper the nuber of targets are assued to be known and fixed. The data association is done using the easureents obtained at the current frae. The tracking syste is provided with a set of easureents, each of which indicates possible presence of a target. However, the tracking syste does not know which easureent corresponds to which target and easureents originated fro clutter. Objective of the data association algorith is to obtain the easureent-to-track association probability distribution so as to iniize the error between estiated target state and the true target state. A. Track identity aintenance In a cluttered ulti-target environent, it is difficult to aintain the identity of target tracks. In such a scenario Fig. 1. Data association in ulti-target tracking: Origin of easureents are unknown. Hence the tracking process does not know which easureent corresponds to which target and which easureents are fro clutter. joint probabilistic data association JPDA technique provides a coparatively better target state estiate [1] than nearest neighbor and probabilistic data association approach. But, JPDA is susceptible to track coalescence when targets ove critically close, as shown in [2]. The exact nearest neighbor PDA approach ENNPDA was proposed in [3] to circuvent track coalescence, but this approach drastically prunes all association hypotheses except the ost likely one. An approach based on selective pruning JPDA*, which effectively weighs the outcoe in favor of the ore likely hypotheses, was proposed in [4], [5]. An approach based on K-best data associations is given in [6] to obtain correct easureent-to-track association. A re-weighting strategy is eployed in [7] to avoid track coalescence by scaling the ost likely association hypotheses. The need to know the location of the targets rather than the identity of the targets has brought out an increasingly popular estiate based on iniu ean optial sub pattern assignent MMOSPA known as Set JPDA SJPDA [8]. SJPDA deals with target state estiation and identity aintenance track labeling as separate probles. Another recent iproveent overlays track identity on targets localized using SJPDA [9].The technique presented in this paper is based on the Iter-JPDA technique developed in [10] for coalescence and swap reduction. This paper explains the working echanis of Iter-JPDA using a static data association exaple and perforance coparison for crossing and aneuvering scenarios. II. JOINT PROBABILISTIC DATA ASSOCIATION USING MAP ESTIMATES Consider the target state transition and easureent odel of the for, X k + 1 = F X k + w k, 1 Z k = HX k + v k, 2 where k is the tie index, X is the target state vector of a single target, Z is the easureent vector, w and v are zero

2 ean Gaussian noise vectors with covariance atrices Q and R respectively. Z k = {z j } :k is the set of easureents obtained at k th scan and k is the nuber of easureents at k th scan. Z k denotes easureents up to tie k. The state transition atrix F and observation atrix H are assued to be known. The initial state is assued Gaussian with ean X 0 0 and covariance P 0 0. Predicted target state at tie k given data up to tie k 1 is denoted as ˆX k k 1 and the corresponding easureent is denoted as Ẑ k k 1. The error in predicted target state is X k k 1 = X k ˆX k k 1, and error in predicted easureent is Z k k 1 = Z k Ẑ k k 1. The error covariance of predicted target state is ] P k k + 1 = E [ X k k 1 X T k k 1, and the residual error covariance ] S k = E [ Z k k 1 Z T k k 1. Target state estiate at tie k given data up to tie k is denoted as X k k. Let A = {A h} N h h=1 be the set of all association events and a i,j A is the association event of j th easureent to i th track. In order to account for the overlap of target state distributions, JPDA fors the joint association events pertaining to current tie k as follows [1], A h = k a j,t, 3 where a j,t {easureent j originated fro target t}, j = 1,..., k ; t = 0, 1,..., T. When targets are located close by as shown in Fig. 2, the validation region of one track ay overlap with another. The overlapping of validation regions can be denoted using a validation atrix M having rows indicating the easureents and coluns indicating the targets. Validation atrix is defined as M = {ω j,t }; with binary eleents ω j,t indicating whether easureent j lies in the validation region of target t. The validation atrix for the scenario shown in Fig. 2 can be obtained as, M = The coluns indicate targets and rows indicate easureents. The colun index zero stands for false target, to which all easureents can be assigned. To select feasible joint events fro the validated easureents, JPDA creates validation atrix for the feasible joint events defined as ˆM = {ˆω j,t }; with binary eleents ˆω j,t. The joint association events obtained using 3 can be split in to two; the feasible events A 1 = {A 1 } N =1 obtained by ˆω j,t A = 1 and non-feasible events A 2 obtained by ˆω j,t A = 0. The feasible and non-feasible joint association hypotheses can be denoted as [1], { 1 A is feasible A A1 ˆω j,t A =. 4 0; A is non-feasible A A 2 The constraints for the feasible joint events are, 1. A easureent can have only one source, i.e, T ˆω j,t A = 1, j = 1 : k 5 t=0 2. No ore than one easureent can originate fro a target, i.e., k ˆω j,t A 1, t = 1 : T. 6 For exaple the validation atrix for the feasible joint event A 1 z 1 and z 4 are fro clutter, z 2 is fro target 1 and z 3 is fro target 2 can be obtained fro M using criteria 1 and 2 as, ˆM = The probability for the joint event A 1 being true is coputed in JPDA as, P { A 1 Z k} = P { A 1 Z k, Z k 1} = 1 c P { Z k A 1, Z k 1} P { A 1 Z k 1}, = 1 } {Z c P k A 1, ˆX k k 1 P {A 1 } 7 The conditioning of A 1 on Z k 1 is considered irrelevant and so the prior for data association P {A 1 } is unifor in JPDA [1]. Therefore the association configuration probabilities coputed using 7 are proportional to the likelihood value. The likelihood function is Gaussian with ean ˆX k k 1 and variance S k, that is, p Z k A 1, ˆX t k k 1, S t k = 8 N Z k ; ˆX k k 1, S k. The target oriented easureents are considered independent and the joint likelihood is evaluated as [1], N Z k ; X tj k k ˆ 1, S tj k = 9 k N z j k ; ˆX tj k k 1, S tj k, where ˆX tj k k 1 is the target to which easureent z j belongs under the given configuration A 1. A. Tracking closely oving targets When targets are located close, as shown in Fig. 2, the observation that belongs to one target ay also belong to soe other target and this causes any association configurations to coexist. In the iniu ean square error MMSE sense,

3 The suation across A 1 is needed because any valid feasible association configuration can exist when targets are close. The prior inforation about the target state X k is independent of A 1 [1], so, P Z k, X k, A 1 P X k Z k = P Z k = P Z k X k, A 1, Z k 1 P X k Z k 1 P A 1 P Z k 13 Fig. 2. Two targets with four easureents. Predicted position of target i T i denoted as ˆX i is shown with dots and easureents Z = {z i } i=1,2,3,4 are shown by squares. Circles around predicted positions are gate area for easureent updation. target state estiates are obtained as the conditional ean [11] by averaging over all association configurations as, X k k = E [ X k Z k] = E [ E [ X k A 1, Z k] A 1, Z k] = E [ X k A 1, Z k] P A 1 Z k 10 A 1 The data association proble in MTT is to copute P A 1 Z k. The following sub-sections bring out the difference between the proposed approach and JPDA. B. Proposed approach The association configuration A 1 is defined by the validation atrix ˆM for feasible joint association events. To evaluate 10 using P A 1 Z k the inforation about rows easureents at k th instant and coluns target state at k th instant of ˆM is needed. In 7 JPDA splits the easureent set as, Z k = { Z k, Z k 1} and uses prior target state obtained fro Z k 1 to coputes P A 1 Z k. Proposed approach uses posterior target state X θ k to copute P A 1 Z k. P A 1 Z k = P A 1 Z k, Z k 1 = P A 1 Z k, X θ k = 1 c P A 1, Z k, X θ k = 1 c P Z k A 1, X θ k P X θ k P A 1, 11 where the quantity X θ i k = α i ˆX i k k 1 is coputed for each target in the proposed approach to axiize the posterior target state distribution P X k Z k. The posterior target state distribution for closely spaced targets can be obtained as, P X k Z k = P X k Z k, Z k 1 = P X k, Z k, Z k 1 P Z k = P Z k, Z k 1, X k, A 1 P Z k 1 = P Z k P Z k X k, A 1, Z k 1 P X k A 1, Z k 1 P A 1 Z k The proposed approach axiizes P X k Z k by axiizing P Z k, X k, A 1. The paraeter X k = X θ k that axiizes p X Z k is coputed iteratively as described in next section. The likelihood function is with ean X θ and variance R k, i.e., p Z k A 1, X θ k = N Z k ; X θ k, R k, k = N z j k ; X θ t j k, R k. 14 The prior distribution of the target states are with ean ˆX k k 1 and variance P k k 1 i.e., p X θ k Z k 1 = 1 X c N θ k ; ˆX k k 1, P k k The distribution of p A 1 is assued to be unifor as in JPDA. If X θ k = ˆX k k 1 and variance is S k in 11 then the proposed approach is sae as JPDA. C. Data association using MAP estiate: An exaple The advantage of the proposed approach, when copared to JPDA, is analyzed here with a scalar nuerical exaple for the static scenario shown in Fig. 2. The predicted position of target tracks T 1 and T 2 are ˆx 1 = 10 and ˆx 2 = 20, respectively. The four easureents are z 1 = 11, z 2 = 13, z 3 = 17 and z 4 = 19. The easureents z 2 and z 3 fall inside the overlapping gate area of tracks T 1 and T 2. The association probabilities obtained with the proposed approach and with JPDA are given in Table I for tracks T 1 and T 2 with p d = 1. The nuerical values of residual error variance S, easureent error variance R and the variance of error in predicted position P are given in Table I. The estiates x i, k k = E [ X i k A 1, Z k] obtained for the ordered easureent set corresponding to association configuration A 1 are also given in Table I. The estiates x θ i k k obtained with proposed approach and x i k k with JPDA are given in Table II along with easureent to track association probabilities p i,j = P A 1 Z k. There are two sets of estiates and association probabilities in Table II with different easureent error covariance R, residual error covariance S and state error covariance P for T 1 and T 2. The first set is coputed with sae paraeters used in Table I. The estiates x θ i k k = x i, k k P A 1 Z k are coputed with the association probabilities P A 1 Z k obtained with the proposed approach. To copute the estiates x i k k obtained

4 TABLE I. ESTIMATES x i, k k AND THE ASSOCIATION PROBABILITY P A 1 Z k ARE TABULATED FOR THE SCENARIO SHOWN IN FIG. 2. THE ORDERED PAIR A 1= {z j1,z j2} INDICATES MEASUREMENT z j1 IS ASSIGNED WITH TRACK T 1 AND z j2 WITH TRACK T 2. P A 1 IS ASSUMED TO BE UNIFORM. Prediction ˆx 1 k k 1 = 10 ˆx 2 k k 1 = 20 A 11= {z 1,z 2 } ] x 1, k k x 2, k k [ Paraeters S=64 R=4 P =60 Proposed approach PA 1 Z k = 1 PZk A c 1,X θ kpx θ k ˆXk k 1P A 1 JPDA PA 1 Z k = 1 c PZk A 1, ˆXk k 1P A 1 A 1 2= {z 1,z 3 } A 1 3= {z 1,z 4 } A 1 4= {z 2,z 3 } A 1 5= {z 2,z 4 } A 1 6= {z 3,z 2 } A 1 7= {z 3,z 4 } TABLE II. PROBABILITY OF ASSOCIATING MEASUREMENTS TO TARGETS p i,j AND THE ESTIMATES WITH THE PROPOSED APPROACH AND WITH JPDA FOR DIFFERENT MEASUREMENT ERROR COVARIANCE R, RESIDUAL ERROR COVARIANCE S AND STATE ERROR COVARIANCE P. Estiate and paraeters R, S and P z 1 = 11 z 2 = 13 z 3 = 17 z 4 = 19 T 1 x θ 1 k k = 11.9 S=64 R=4 P= Proposed approach x θ 1 k k = 11.4 S=16 R=4 P= T 2 x θ 2 k k = 18.1 S=64 R=4 P= x θ 2 k k = 18.6 S=16 R=4 P= T 1 x 1 k k = 12.7 S=64 R=4 P= JPDA x 1 k k = 11.7 S=16 R=4 P= T 2 x 2 k k = 17.3 S=64 R=4 P= x 2 k k = 18.4 S=16 R=4 P= using JPDA the association probabilities coputed using JPDA is used. The proposed approach uses posterior ode x θ i k k to copute P A 1 Z k given in Table I. In the proposed approach the posterior ean obtained using 10 and given in Table II is equal to the posterior ode used for coputing P A 1 Z k. For JPDA posterior ode is not coputed explicitly. The proposed approach coputes ore effective conditional Gaussian posterior having sae ean and ode. The estiates coputed using JPDA are closer to the id point of predicted positions of T 1 and T 2, copared to the estiates obtained using proposed approach. The tendency of estiates to go towards id point and the subsequent coalescence possibility is reduced in the proposed approach. Copared to JPDA the proposed approach increases the probabilities of A 1 =2,3,4,5 as shown in Table I. Siilarly proposed approach reduces the probabilties of A 1 =1,6,7 and thereby reduces coalescence possibility. The second set of estiates and association probabilities with paraeters S=16, R=4, and P=12 provided in Table II is to show that the proposed approach and the JPDA behave siilarly when the error covariance P is sall. With low P the MAP estiates x θ i k k and JPDA estiates x i k k shift closer to the predicted position as shown in Table II. The difference in association probabilities p i,j coputed using proposed approach and JPDA are closer for P = 12 when copared to P = 60. III. ITERATIVE DATA ASSOCIATION ITER-JPDA To copute X θ k in 11 the axiizing paraeter α for each target needs to be obtained. Instead of directly axiizing pθ A1, X θ, Z k the proposed approach axiizes the log of expected value using an EM like algorith. The approach described in this paper obtains X θ k as the arguent axiizing the expectation, E θ [ log pθ A1, X θ k, Z k ]. Copared to the scalar target case described in [10], the description here deals with target state being a vector. Further, the focus here is to copute the axiu a posteriori target state. The expectation of log likelihood can be coputed as [12], [ E-Step: E θ log pθ A1, X θ, Z k ] = log p θ A1, X θ, Z k p θ A1 X θ, Z k. where, 16 log p θ A1, X θ, Z k = 17 log p Z k A 1, X θ p θ X θ A 1 p θ A The coputation of p θ A1 Z k, X θ is done at each iteration with axiizing target states using 11, where X θ = αˆx k k 1 is the target state coputed iteratively. For n targets with Gaussian target state distribution, the function Jα = log p θ A1, X θ, Z k p θ A1 X θ, Z k, 19 can be split in ters of paraeters corresponding to target i. Hence, Jα = Jα i. 20 i=1:n The unknown paraeter α i for each target i is coputed by iniizing the negative log likelihood. M-Step: Jα i = j 2 z j k H ˆX i k k 1 α i R 1 + ˆX i k k 1 α i ˆX i k k 1 2 P 1 β j,i, 21

5 where, β j,i = a i,j A 1 p A 1 Z k, X θ. 22 The axiizing likelihood β j,i is obtained by evaluating the gradient of J with respect to α i, α J = 23 C z j k H ˆX T i k k 1 α i R 1 H ˆX i k k 1 + j ˆX i k k 1 α i ˆX i k k 1 ˆX T P 1 i k k 1 βj,i. and using αi J = 0, where, Nuerator = j α i = Nuerator Denoinator. 24 z j kr 1 H ˆX i k k 1 T + 25 X i k k 1 P 1 ˆX i k k 1 T βj,i Denoinator = H ˆX i k k 1 R 1 H ˆX i k k 1 T + 26 X i k k 1 P 1 ˆX i k k 1 T To obtain β j,i using 22 and 11, beginning with α i = 1, β j,i is coputed as in JPDA. Using α i obtained fro 24, easureent error variance R, and error covariance of predicted target state P, a better P A 1 Z k, X θ p Z k A 1, X θ k p X θ k can be obtained and 22 can be coputed. Here, X θ k = αˆx k k 1 is the paraeter axiizing p X k Z k and p A 1 Z k, X θ p Z k A 1, X θ k p X θ k k N 1 z j k ; α tj ˆX tj k k 1, R n i=1 N 2 α i ˆX i k k 1 ; ˆX i k k 1, P i 27 where N 1 is a Gaussian distribution with ean X θ t j = α tj ˆX tj k k 1 and variance R, N 2 is a Gaussian distribution with ean ˆX tj k k 1 and variance P. By iterating over 22, 24 and 27 the paraeter α i that axiizes E θ can be coputed for each i = 1,..., T. Using α i the target states X θ i for coputing p { A 1 X θ k, Z k } will be obtained. The target state estiates X i k k are obtained by substituting p { A 1 X θ i k, Z k } in 10. A. Description of steps in Iter-JPDA A block diagra representation of the steps involved in JPDA and the proposed approach are shown in Fig. 3. The coputation of MAP estiates of target states X θ in the proposed approach uses the current easureent set along with the predicted target inforation. The predicted target positional inforation is used as initial target state to initialize the EM algorith. The data association process starts at instant k, with easureents obtained at instant k along with predicted target states ˆX i k k 1 for the i th target. The steps involved in Iter- JPDA are as follows, Step 1 : Joint association hypotheses foration The first step in the proposed approach is the foration of all possible valid association hypotheses denoted as A 1. For each hypotheses, likelihood will be evaluated as { } L A 1 = p Z k A 1, αˆx t k k 1, R 28 k = N z j k ; αˆx t k k 1, R. For JPDA and for the first step of iteration process the likelihood is evaluated with α = 1 and with variance S. Step 2 : Updates under A 1 The target state and process error covariance updates will be evaluated for each valid association hypotheses A 1 as [1], X i, k k = ˆX i, k k 1 + K k 29 z j k H ˆX i, k k 1 P i, k k = I KH P i, k k For the case of JPDA the posterior association probabilities used for cobining the updates are coputed as, β A 1 β A 1 = N β =1 A 1 where β A 1 = φ! k! µ F φ V φ p d δa1 1 p d 1 δa1 L A 1, 31 where φ is the nuber of false easureents, p d is the detection probability, µ F is the prior probability ass function of the nuber of false easureents, V is the search volue, and N is the nuber of valid association hypotheses [1]. Step 3 : Coputation of MAP estiates of target states This step distinguishes the proposed approach fro other data association approaches. Proposed approach coputes target state X θ axiizing the posterior target state p X k Z k using expectation axiization algorith. Step 4 : Coputation of posterior Proposed approach coputes{ the posterior for data association } as product of likelihood p Z k A 1, αˆx t k k 1 and prior p X θ Z k. The ter β A 1 in 31 is odified as, β A 1 = φ! k! µ F φ V φ p d δa1 { } 1 p d 1 δa1 p Z k A 1, αˆx tj k k 1 p X θ. 32 Step 5 : Cobining the updates The updates under A 1 are cobined using 10 with weights obtained fro the posterior association probabilities. Step 6 : Prediction

6 a Fig. 3. Block diagra representation for JPDA and the proposed approach are given in a and b, respectively. b The prediction step translates the target state and covariance fro tie instant k 1 to instant k. For linear odels prediction is done using ordinary Kalan filter as, ˆX k k 1 = F X k 1 k 1 P k k 1 = FP k k 1 F T + Q. 33 Steps 1-to-6 are repeated as shown in Fig. 3b, to estiate the target state. IV. SIMULATION Two tracking scenarios are considered for evaluating the perforance of Iter-JPDA. The scenarios are created to check the track aintenance ability when targets cross and also to copare the estiation accuracy when targets are near and when they aneuver. The first scenario shown in Fig. 4 consists of two crossing targets in X-Y plane. Clutter points are uniforly distributed across the rectangle 0 < x < 2000, 50 < y < 150. Measureents are updated with T = 1 s for a duration of 90 s. Two diensional easureents are obtained in Cartesian coordinates fro target state vector X i = [x i ẋ i y i ẏ i ] T with covariance and easureent update atrices, R = σ 2 x 0 0 σ 2 y, H = , where σ x = 10 and σ y = 10. The initial target states are X 1 = [500; 15; 90; 2] T and X 2 = [500; 15; 90; 2] T. The process noise and state transition atrices are, Q Q = q 2 2 0, F = Q 12 2 F , F T 3 T T where Q 1 = T 2, F 2 T 1 =. For scenario 1 siulation has been done with q 0 = 0.03 and to obtain the perforance of Iter-JPDA with higher process error covariance. Tracking perforance is evaluated in ters of estiation accuracy and identity aintenance. To decouple estiation accuracy fro identity aintenance, the target state estiate labels are reordered and obtained as X Ri k k. Here the label R i at instant k is obtained as, arg in { Xi 2 } R i = j X j. 36,2 The positional estiation errors are coputed using X Ri as, e i = X Ri 1 X i X Ri 3 X i 3 2 1/2. The positional root ean square error RMS is coputed for 100 Monte Carlo MC siulations. Track identity aintenance is evaluated in ters of track swap and coalescence percentage obtained during Monte Carlo siulations. Track swap between two tracks i and j is declared if d 2 1,1 + d 2 2,2 < d 2 1,2 + d 2 2,1 37 for 5 consecutive tie instants, where, d 2 i,j = x i x j 2 + ỹ i y j 2 38 for any instant k. The separation between the true trajectories at any instant k is Di,j 2 = x i x j 2 +y i y j 2. A coalescence between tracks i and j is declared in the siulation if d 2 i,j < 1 and D i,j > 10 for 10 consecutive scans. Target state estiation has been done using JPDA, JPDA*, Set-JPDA and Iter-JPDA. JPDA* differs fro JPDA in easureent to track association probability coputation. JPDA* considers a subset of feasible association configurations of JPDA. The sub set is obtained by choosing the ost likely association configuration with respect to each easureent. State estiate and the covariance in Set- JPDA is obtained by allowing perutation of target state as [13], X i k k = p A 1 Z k X ci i, k k 39 P i k k = p A 1 Z k ] ] T P i, k k + [ X ci i, k k X k k [ X ci i, k k X k k 40 where c i is one of the possible perutations of the target state and the optial perutation is the one that iniizes the trace of covariance of 40. In siulations done in this paper covariance given in 40 is coputed for all perutations and the one that iniizes the trace is obtained by sequential evaluation and coparison. Positional RMS error obtained for Fig. 4. Scenario 1: Crossing targets in X-Y plane.

7 crossing scenario using Kalan filter with p d = 1 and no clutter are plotted in Fig. 5. The bup at instant 50 is due to the error in association while crossing. When copared to JPDA* the Iter-JPDA gives better result. The track swap percentages are given in Table III. Fig. 5. Positional RMS error with reordered track labels, p d = 1, no clutter for Scenario 1. To further validate the target crossing scenario, siulations are done with p d = 0.9 and p d = 0.8, along with 10 clutter easureents per frae. The positional RMS errors obtained for the crossing scenario are shown in Fig. 6 and 7 respectively, and the percentage track swap is tabulated in Table III. The sensitivity of JPDA* towards clutter and issed nuber overlay, therefore identity aintenance is copared only with JPDA and JPDA*. JPDA aintains track identity TABLE III. q 0 PERCENTAGE TRACK SWAP FOR 100 M.C RUNS UNDER SCENARIO 1. JP DA [1], [14] JP DA [4], [15] q 0 P 0.03 d = 1, N c = P d = 1, N c = P d = 0.9, N c = P d = 0.8, N c = q 0 P 0.3 d = 0.8, N c = P d = 0.7, N c = Iter JP DA [10] better copared to JPDA* and Iter-JPDA approach reduces the track swap slightly copared to JPDA with q 0 = The identity aintenance in ters of percentage track swap is checked with higher process error covariance by keeping q 0 = 0.3 and the results are shown in last two rows of Table III. With higher process error covariance track swap is considerably reduced with the proposed Iter-JPDA approach copared to JPDA. Track swap percentage for JPDA* is higher copared to JPDA. For p d < 1 the estiation accuracy of JPDA* is better copared to JPDA except near crossing instant as sown in Fig. 6 and Fig. 7. The Scenario 2 consists of two aneuvering targets as shown in Fig. 8. The targets are oving with constant velocity v = 50/s. For first 20 s, targets ove with fixed heading and ake a aneuver of ω = 30 0 and continue to ove parallel for 80 s and ake a aneuver in opposite direction. For Scenario 2 track loss percentage is calculated, because copared to identity aintenance, location of target is a ore significant inforation. The estiation has been done with interacting ultiple odel IMM having constant velocity CV and coordinated turn odel CT. Siulation is done Fig. 6. Positional RMS error with reordered track labels, p d = 0.9, Nuber of clutter easureents per frae N c = 10 for Scenario 1. Fig. 8. Scenario 2: X-Y Trajectory of aneuvering targets T 1 and T 2. Separation distance between targets is denoted as d. Fig. 7. Positional RMS error with reordered track labels, p d = 0.8, N c = 10 clutter easureents per frae for Scenario 1. detections are visible with large bup at crossing instant. Set-JPDA provides better estiation accuracy copared to JPDA* and JPDA. Iter-JPDA estiations are the best aong the considered approaches in ters of estiation accuracy. Siulations with Set-JPDA is done with out aintaining track with σ x = 10, σ y = 10 and q 0 = 30. A track is declared lost if d 2 1,1 + d2 2,2 > 200. The positional RMS errors are plotted in Fig. 9 for a target separation of d = 56. The positional estiates with the proposed approach have better accuracy copared to other approaches. The Set- JPDA siulations are not done with IMM. The large error near k = 100 is due to identity loss during the second aneuver. The track swap, track coalescence and track loss percentages are given in Table IV for d = 56 and d = 26. The proposed approach had no track loss in all the cases, but other two approaches suffer track loss. The vulnerability

8 of detection. One liitation of the proposed iterative approach is its coputational coplexity, due to the recursive coputations of feasible association configuration at each iterations. This can be handled by a factor graph based approach as shown in [17]. Fig. 9. The estiates obtained using IMMJPDA, IMMJPDA*[16] and IMMIter-JPDA for Scenario 2. Siulation is done with d = 56, p d = 1 and with 10 clutter point, Monte Carlo runs where the track is not lost are used for coputing positional RMS error. TABLE IV. PERCENTAGE TRACK SWAP, TRACK COALESCENCE AND TRACK LOSS FOR 100 M.C RUNS FOR SCENARIO 2 WITH DIFFERENT d VALUES USING IMM-JPDA, IMM-JPDA* AND IMM-ITER-JPDA. Separation d JP DA [1],[14] JP DA [4],[15] d=56 with clutter Track loss Swap Coale d=26 with clutter Track loss Swap Coale d=26 with out clutter Track loss Swap Coale Iter JP DA [10] of JPDA to coalescence is seen by decreasing the separation d in Fig. 8. The percentage track loss in JPDA* increases with clutter. The track swap probability for Scenario 2 reains nearly equal for all the approaches. The ability of the proposed approach to give accurate location of the targets is evident fro the reduced track loss probability. The state estiation error obtained fro siulations using Scenario 1 and 2 show that better estiation accuracy can be achieved using the proposed approach, when copared to other approaches in a cluttered ulti-target environent. The advantage of track swap reduction with Iter-JPDA copared to JPDA is evident with higher process noise variance. The percentage track loss with Iter-JPDA is lower when copared to other approaches as shown in Table IV. Track coalescence is avoided with Iter-JPDA when the separation between targets are d = 56 as shown in Table IV. Coalescence percentage of Iter-JPDA is coparable to JPDA* when targets are close. V. CONCLUSION The data association technique developed here effectively coputes the easureent-to-track association probabilities and has better track identity aintenance in closely oving targets. The proposed approach coputes posterior target state estiates and coputes better easureent-to-track association probabilities to reduce the track swap. The posterior target states are obtained as MAP estiates using an EM based approach. Monte Carlo siulations verify the advantage of the proposed ethod over other approaches in a siulated, ulti-target, cluttered environent with varying probabilities ACKNOWLEDGMENT Authors would like to express sincere thanks to Dr. Ajit T Kalghatgi, Director R&D of Bharat Electronics Liited, for the guidance and support to carry out this work. We are also thankful to the Naval Research Board, DRDO, India, for their support and cooperation. REFERENCES [1] Y. Bar-Shalo and T. E. Fortann, Tracking and data association, Acadeic press, inc, [2] R. J. 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