- Email: [email protected]
Data association based on target signal classification information
Journal of Systems Engineering and Electronics Vol. 19, No. 2, 2008, pp.246–251
Data association based on target signal classification information∗ Guo Lei1,2 , Tang Bin1 & Liu Gang2 1. School of Physical Electronic, Univ. of Electronic Science and Technology of China, Chengdu 610054, P. R. China; 2. National Key Lab of Information Integrated Control, Chengdu 610036, P. R. China (Received September 25, 2006)
Abstract: In most of the passive tracking systems, only the target kinematical information is used in the measurement-to-track association, which results in error tracking in a multitarget environment, where the targets are too close to each other. To enhance the tracking accuracy, the target signal classification information (TSCI) should be used to improve the data association. The TSCI is integrated in the data association process using the JPDA (joint probabilistic data association). The use of the TSCI in the data association can improve discrimination by yielding a purer track and preserving continuity. To verify the validity of the application of TSCI, two simulation experiments are done on an air target-tracing problem, that is, one using the TSCI and the other not using the TSCI. The final comparison shows that the use of the TSCI can effectively improve tracking accuracy.
Keywords: passive tracking, joint probabilistic data association, target signal classification information.
1. Introduction Data association is one of the critical research aspects in multitarget tracking. There are some typical algorithms for data association, for example, the joint probabilistic data association (JPDA) algorithm[1−3] , the multiple-hypothesis-tracking (MTH) algorithm [4] , the multiple-frame assignments measurement-to-track algorithm[5] , and so on. In the algorithms mentioned earlier, only the kinematical measurement is used to solve the data association problem and the target signal classification information (TSCI) is seldom used. Such treatment is only effective in the fields of radar and sonar, because the measurement error is smaller than that of the passive radar. However, in the passive radar, the result of the data association is not satisfactory if only the kinematical measurement is used. Compared with the ordinary radars, the passive radar can output TSCI, which is very useful in data association. Therefore, TSCI should be applied in data association to improve the tracking accuracy and purity. A new algorithm is presented to incorporate the TSCI into the JPDA. In the new algorithm, the TSCI
is modeled as a signal feature validation matrix, which reflects the performance of the target signal classifier. Subsequently, after the Schur-Hadamard product of the signal feature validation matrix and the validation matrix of the JPDA is completed, the output will generate the feasible matrix that is used in the JPDA. The simulation experiments show that the new algorithm is very effective for data association of multitargets tracking.
2. The validation matrix of JPDA Data association algorithms decide the track that should be updated by the measurements that have been received. In the JPDA algorithm, one of typical data association algorithms, the association state between the latest measurements and multiple targets in a clutter is initially formulated as the validation matrix. Next, the joint posterior probability of measurements is computed. At the end, the product between the joint posterior probability and the innovation information of the Kalman filter is updated in the state estimator. k Let Z(k) = {zj (k)}m j=1 be the m-dimensional kinematical measurement at time k, and Z k = {Z(i)}ki=1
* This project was supported by the Youth Science and Technology Foundection of University of Electronic Science and Technology of China (JX0622).
Data association based on target signal classification information be the entire set of kinematical measurements up to time k. If the measurement j is in the kinematical validation gate, it can be used to update the target state. On the contrary, it is thought to originate from the clutter and cannot be used in the state estimator. The kinematical validation gate is defined as v(k) = {z : [z − zˆ(k|k − 1)]
·
S −1 (k)[z − zˆ(k|k − 1)] γ} =
{z : v (k)S
−1
matrix helps to derive the association probability between every measurement and every target. βjt = P {θjt |Z k } =
where θ(k) =
m k j=1
(3)
θjt . By using the Bayes’ rule for the
(1) P {θ(k)|Z k } =
However, the valid gates of the different targets overlap each other in a multitarget environment. Therefore, it is difficult to judge whether a measurement originates from one target or another. To solve such a problem in JPDA, the kinematical validation matrix is defined as follows
⎧ ⎨ 1 ωjt (θ) = ⎩ 0
P {θ(k)|Z k }ˆ ωjt (θ(k))
θ
(k)v(k) γ}
j = 1, . . . , m;
feasible matrix, the joint posterior probabilities of all the measurements, up to the present time are
where v is the measurement residual, S is the measurement prediction covariance and γ is the gate threshold.
Ω = [ωjt ],
247
t = 0, 1, . . . , T
1 p[Z(k)|θ(k), Z k−1 ]P {θ(k)} c
(4)
where c is a normalized constant obtained by summing θ, on the right-hand side. The first term on the righthand side of (5) is the conditional probability density function (pdf) of individual measurement. The second term on the right-hand side of (5) is the prior probability of a joint event. Finally, by using the association probability between every measurement and every target, the Kalman filter is updated in the state estimator as follows mk ˆ t (k|k) = ˆ t (k|k)βjt (k) X X (5) j j=0
if θjt occurs
(2)
otherwise
The elements in the matrix Ω correspond to the association of measurement-to-target, where j represents the number of measurements and t represents the number of targets. Note that t = 0 means the fact that there is no target, that is, a measurement is from the clutter. The binary element ωjt indicates whether or not the measurement j lies in the valid gate of the target t. θjt represents the fact that measurement j lies in the valid gate of the target t. On the other hand, to process the measurement from the multitarget merge, the feasible events in the algorithm are defined as follows [2−3] : (1) No more than one measurement originates from each true target. (2) No more than two targets can give the same measurement. Then, the validation matrix Ω is scanned to generˆ , whose element ate one or more feasible matrices, Ω is ω ˆ jt . Computing the same target in every feasible
From the analysis of the JPDA algorithm cited earlier it can be found that there are many effective methods to improve the accuracy of the data association, among which the most valid method is to ensure that the main diagonal elements in the validation matrix are one, and the other elements are zero. This matrix indicates that the association between the measurement and the target is unique.
3. Data association based on target signal classification information 3.1
The TSCI matrix
In the JPDA algorithm, the main-diagonal elements Ωi,i of the validation matrix should be 1.0 and the other elements Ωi,j = 0.0 for i = j under ideal conditions. However, this is not always true when the targets are close to each other. The measurement of a particular target may be associated with more than one multiple track, which unfortunately results in track impurity. If the signal classification validation matrix E is used to improve the measurement-to-track
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Guo Lei, Tang Bin & Liu Gang
association, the validation matrix purity will be significantly improved. However, the signal pdf of each target is usually unknown. If the signal pdf distribution is assumed to be independent of the Gaussian densities[6] , the signal mean and covariance are obtained by the use of the maximum likelihood function method. Usually, the track is divided into three stages, that is, the track initiation, track maintenance, and track end. If the track of each target in the stage of the track initiation can be split[6] , the initiation mean and covariance of each target can be obtained, that is, n 1 t µ ˆt0 = ζ N i=1 i
n ˆt = 1 Σ (ζ t −ˆ µt0 )(ζit −ˆ µt0 ) 0 N i=1 i
t = 1, . . . , T
where ζjt (k) represents the sensor’s signal measurement set about target t at time k. In the stage of the track maintenance, when the sensor receives a new signal measurement, ζj (k), emitted by unknown targets at time k, the distance between the TSCI region of each target and the new signal measurement is first calculated as follows
·
εjt
t = 0, 1, . . . , T
⎧ ⎨ 1 ∀ζ (k) ∈ {V } j t = ⎩ 0 ∀ζj (k) ∈ / {Vt }
εjt
j=1
⎛ ⎞ mk 1 ⎝ µ ˆt = αˆ µt0 + (1 − α) ζj εjt ⎠ ; num validate j=1 ˆt µ ˆt0 = µ ˆt = Σ
1 num validate
mk
(9)
(ζj εjt − µ ˆt )(ζj εjt − µ ˆt ) (10)
j=1
ˆ t are the signal mean and covariance where µ ˆ and Σ of the target t, and α is the weight. The variable num validate denotes the number of the signals lying in the valid gate of target t. 3.2
Date association based on the TSCI matrix
To integrate TSCI into the traditional JPDA, the measurement-to-track association matrix, Λ, is defined as the product of the validation matrix, Ω , given ˆ given in (9). in (2) with the TSCI validation matrix E Therefore, the Λ is Λi,j = Ω Eˆ = [ωjt εjt ]
where κ is the gate threshold of each target signal classification. If Vt (k) κ, it means that ζj (k) lies in the signal valid gate of target t. Otherwise, it does not. Note that the TSCI validation matrix is defined as follows j = 1, . . . , m;
mk
(7)
ˆ0t )−1 (ˆ (Σ µt0 − ζj (k)) < κ}
ˆ = [εjt ], E
num validate =
t
(6)
Vt (k) = {ζj : (ˆ µt0 − ζj (k))
Secondly, the new signal measurements lying in the signal valid gate are used to update the signal mean and covariance of the target. The corresponding equations are listed below.
(8)
where εjt is the element of the TSCI validation matrix ˆ If it is 1, it means that a signal measurement is asE. sociated with a target. In addition, the position of an element in matrix Eˆ denotes the relation between the measurement and target, that is, the row represents the signal measurement and the column represents the target.
(11)
where denotes the schur-hadamard product (term by term). For example, the validation matrix is ⎤ ⎡ 1 1 0 0 1 ⎥ ⎢ ⎢ 0 0 1 1 0 ⎥ ⎥ ⎢ (12) Ωi,j = ⎢ ⎥ ⎢ 1 1 0 1 0 ⎥ ⎦ ⎣ 0 1 0 1 1 From the association matrix mentioned earlier, it can be found that each measurement associates with more than one target. If the TSCI validation matrix is given as ⎡ ⎤ 0 1 0 1 0 ⎢ ⎥ ⎢ 0 0 1 0 0 ⎥ ⎢ ⎥ E=⎢ (13) ⎥ ⎢ 0 0 0 1 0 ⎥ ⎣ ⎦ 0 1 1 0 1
Data association based on target signal classification information The measurement-to-track association matrix Λ is obtained by the product of the validation matrix with the TSCI validation matrix, which is presented below. ⎡ Λi,j
0 1
⎢ ⎢ 0 0 ⎢ =⎢ ⎢ 0 0 ⎣ 0 1
0 0 1 0 0 1 0 0
0
⎤
⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎦ 1
(14)
The comparison between (16) and (18) shows that the TSCI significantly reduces the merge association and thus makes the tracks purer. Therefore, the measurement-to-track association matrix Λ effectively reveals the relation between measurement and target, hence, it is superior to the traditional matrix, owing to the fact that the TSCI of each target is integrated into Λ. 3.3
JPDA algorithm based on the TSCI
In this section, the implementation of the JPDA algorithm based on the TSCI will be summarized. First, in the track initiation stage, the initiation means and covariance of the target signals are computed through (6). Second, in the stage of the track maintenance, there are four main steps involved, which are as follows. Step 1 Use the kinematical validation gate to obtain the validation matrix Ω through (2); Step 2 Use the TSCI gate to generate the TSCI validation matrix E through (8); Step 3 Use the schur-hadamard product of the previous two matrices to derive the measurement-totrack association matrix Λ through (11); Step 4 The feasible matrix is generated; Finally, the procedure that follows is similar to that of the traditional JPDA algorithm.
4. Simulation experiments The target motion model in the Cartesian coordinate system is xk = F xk−1 + Gwk−1 (15) where xk = [xk x˙ k yk y˙ k ]. xk and yk are the xdirectional and y-directional positions of the target in the Cartesian coordinates, respectively. x˙ k and y˙ k are the x-directional and y-directional velocities of the
249
target, respectively. F is the translation matrix, wk−1 is the process noise. Q is process noise standard deviation. ⎡ ⎤ ⎤ ⎡ T 0 1 T 0 0 ⎢ 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 0 1 0 0 ⎥ 1 0 ⎢ ⎥ ⎥ ⎢ ⎥ F =⎢ ⎥, G = ⎢ ⎢ ⎥ ⎢ 0 0 1 T ⎥ ⎢ 0 T ⎥ ⎥ ⎢ ⎢ ⎦ ⎣ 2 ⎥ ⎣ ⎦ 0 0 0 1 0 1 E(wk ) = 0; E(wk wj ) = Qδkj
(16)
In passive tracking, the TDOA (time difference of arrival) of targets is measured. The kinematical part of the measurement vector zk is given as zk = h(xk ) + vk
(17)
where h is the measurement translation function, v is the measurement noise, ⎤ ⎤ ⎡ ⎡ ∆t1 ∆r1 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ; h(xn ) = ⎢ ... ⎥ ; zk = ⎢ ⎥ ⎥ ⎢ ⎢ ⎦ ⎦ ⎣ ⎣ ∆tm ∆rm E(vk ) = 0;
E(vk vj ) = Rδkj
(18)
∆ti is the time difference when the target signals arrive at the master station from the slave station, and ∆ri is the range difference between the master station and slave station. R is the standard deviation of the measurement noise. In the 2-D case, the targets are tracked by three sensors located at (0 km, 50 km), (0 km, 170 km), and (80 km, 80 km), respectively. The time difference measurement standard deviation (SD) is 20 ns. The sample time is T = 1 s and the process noise √ standard deviation is Qii = 0.02. The standard de√ viation of the measurement noise is Rii = 0.15 km. Figure 1 shows the true target trajectories in the simulation scenario. The targets are moving for 400 s at the speed of 250 m/s. Such a scenario is a typical air target tracking problem. It consists of four targets whose signal feature classes are assumed to be 1, 2, 3, and 4, respectively (starting from the left in Fig.1). The target detection probability and the false alarm probability are 1.0 and 0.0, respectively.
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Guo Lei, Tang Bin & Liu Gang
Fig. 1
Target trajectories
The estimation results for the JPDA are based on 100 Monte Carlo runs. Figures 2(a) and 2(b) show the tracks obtained by the JPDA, with and without signal class information, respectively. It can be seen that TSCI significantly improves the tracking performance, and results in cleaner tracks. Figures 3(a) and 3(b) show the root mean square (RMS) position error with and without signal class information, respectively. It can be seen that the use of TSCI yields a purer track and improves RMS position errors.
Fig. 3
RMS position error
Furthermore, actual measurement data are used in the algorithm to verify its validity. In Fig. 4(a), the actual measurement data by two targets flying parallelly are presented. It can be found that the distribution of measurement data is uniform around the actual tracking and there are also some false measurement data. The RMS position error using TSCI and not using TSCI are presented in Figs. 4(b) and (c), respectively. The maximum value of the RMS position error is about 2 km2 in Fig. 4(b), however, in Fig. 4(c) the maximum value of the RMS position error is about 7 km2 . From comparing Fig. 4(b) with (c), it can be seen that the improvement on tracking performance with the use of TSCI is distinctness.
5. Conclusions Fig. 2
Tracking target trajectories
When kinematical information of targets is very similar, there arises ambiguity in passive tracking if kine-
Data association based on target signal classification information
251
merical experiments in this article confirm the benefit of using TSCI data association for improving target tracking, when there is association uncertainty in the kinematical measurements. Acknowledgments The authors are thankful to the reviewers for their comment and advice. At the same time, the authors would like to thank Dr. Xiao Fei for his help.
References [1] Bar-Shalom Y, Li X R. Multitarget-multisensor tracking: principle and techniques.
Storrs, CT: YBS Publishing,
1995: 205–211. [2] Bar-Shalom Y, Fortmann T E. Tracking and data association. Boston: Academic Press, 1988: 138–145. [3] Kuo C C, Bar-shalom Y. Join probabilistic data association for multitarget tracking with possibly unresolved measurement and maneuvers. IEEE Tran. on AC, 1984, 29(7): 585–594. [4] Reid D B. An algorithm for tracking multiple targets. IEEE Trans. on AC, 1979, 24: 843–854. [5] Deb S, Yeddanapudi M, Pattipati K R, et al. A generalized S-dimensional assignment for multisensor-multitarget state estimation. IEEE Trans. on AES, 1997, 33(2): 523–528. [6] He You, Xiu Jianjuan, et al. Radar data processing with application, Beijing, Publishing House of Electronics Industry, 2006: 120–125.
Guo Lei was born in 1971. He is a lecturer in University of Electronic Science and Technology of China (UESTC). He is a Ph. D. candidate. His research interests include multisensor data fusion, passive tracking, sensors management, etc. E-mail: leiguo@uestc. edu.cn
Fig. 4
RMS position error comparison by using actual measure data
matical information is used alone. To improve the association results, target classification information has been integrated into the JPDA algorithm, which is one of the typical data association algorithms. The nu-
Tang Bin was bron in 1964. He is a professor in UESTC. His research interests include radar signal, electron counter reconnaissance. Liu Gang was born in 1973. He is a doctor and his current research interest in electron counter reconnaissance.
Data association based on target signal classification information∗ Guo Lei1,2 , Tang Bin1 & Liu Gang2 1. School of Physical Electronic, Univ. of Electronic Science and Technology of China, Chengdu 610054, P. R. China; 2. National Key Lab of Information Integrated Control, Chengdu 610036, P. R. China (Received September 25, 2006)
Abstract: In most of the passive tracking systems, only the target kinematical information is used in the measurement-to-track association, which results in error tracking in a multitarget environment, where the targets are too close to each other. To enhance the tracking accuracy, the target signal classification information (TSCI) should be used to improve the data association. The TSCI is integrated in the data association process using the JPDA (joint probabilistic data association). The use of the TSCI in the data association can improve discrimination by yielding a purer track and preserving continuity. To verify the validity of the application of TSCI, two simulation experiments are done on an air target-tracing problem, that is, one using the TSCI and the other not using the TSCI. The final comparison shows that the use of the TSCI can effectively improve tracking accuracy.
Keywords: passive tracking, joint probabilistic data association, target signal classification information.
1. Introduction Data association is one of the critical research aspects in multitarget tracking. There are some typical algorithms for data association, for example, the joint probabilistic data association (JPDA) algorithm[1−3] , the multiple-hypothesis-tracking (MTH) algorithm [4] , the multiple-frame assignments measurement-to-track algorithm[5] , and so on. In the algorithms mentioned earlier, only the kinematical measurement is used to solve the data association problem and the target signal classification information (TSCI) is seldom used. Such treatment is only effective in the fields of radar and sonar, because the measurement error is smaller than that of the passive radar. However, in the passive radar, the result of the data association is not satisfactory if only the kinematical measurement is used. Compared with the ordinary radars, the passive radar can output TSCI, which is very useful in data association. Therefore, TSCI should be applied in data association to improve the tracking accuracy and purity. A new algorithm is presented to incorporate the TSCI into the JPDA. In the new algorithm, the TSCI
is modeled as a signal feature validation matrix, which reflects the performance of the target signal classifier. Subsequently, after the Schur-Hadamard product of the signal feature validation matrix and the validation matrix of the JPDA is completed, the output will generate the feasible matrix that is used in the JPDA. The simulation experiments show that the new algorithm is very effective for data association of multitargets tracking.
2. The validation matrix of JPDA Data association algorithms decide the track that should be updated by the measurements that have been received. In the JPDA algorithm, one of typical data association algorithms, the association state between the latest measurements and multiple targets in a clutter is initially formulated as the validation matrix. Next, the joint posterior probability of measurements is computed. At the end, the product between the joint posterior probability and the innovation information of the Kalman filter is updated in the state estimator. k Let Z(k) = {zj (k)}m j=1 be the m-dimensional kinematical measurement at time k, and Z k = {Z(i)}ki=1
* This project was supported by the Youth Science and Technology Foundection of University of Electronic Science and Technology of China (JX0622).
Data association based on target signal classification information be the entire set of kinematical measurements up to time k. If the measurement j is in the kinematical validation gate, it can be used to update the target state. On the contrary, it is thought to originate from the clutter and cannot be used in the state estimator. The kinematical validation gate is defined as v(k) = {z : [z − zˆ(k|k − 1)]
·
S −1 (k)[z − zˆ(k|k − 1)] γ} =
{z : v (k)S
−1
matrix helps to derive the association probability between every measurement and every target. βjt = P {θjt |Z k } =
where θ(k) =
m k j=1
(3)
θjt . By using the Bayes’ rule for the
(1) P {θ(k)|Z k } =
However, the valid gates of the different targets overlap each other in a multitarget environment. Therefore, it is difficult to judge whether a measurement originates from one target or another. To solve such a problem in JPDA, the kinematical validation matrix is defined as follows
⎧ ⎨ 1 ωjt (θ) = ⎩ 0
P {θ(k)|Z k }ˆ ωjt (θ(k))
θ
(k)v(k) γ}
j = 1, . . . , m;
feasible matrix, the joint posterior probabilities of all the measurements, up to the present time are
where v is the measurement residual, S is the measurement prediction covariance and γ is the gate threshold.
Ω = [ωjt ],
247
t = 0, 1, . . . , T
1 p[Z(k)|θ(k), Z k−1 ]P {θ(k)} c
(4)
where c is a normalized constant obtained by summing θ, on the right-hand side. The first term on the righthand side of (5) is the conditional probability density function (pdf) of individual measurement. The second term on the right-hand side of (5) is the prior probability of a joint event. Finally, by using the association probability between every measurement and every target, the Kalman filter is updated in the state estimator as follows mk ˆ t (k|k) = ˆ t (k|k)βjt (k) X X (5) j j=0
if θjt occurs
(2)
otherwise
The elements in the matrix Ω correspond to the association of measurement-to-target, where j represents the number of measurements and t represents the number of targets. Note that t = 0 means the fact that there is no target, that is, a measurement is from the clutter. The binary element ωjt indicates whether or not the measurement j lies in the valid gate of the target t. θjt represents the fact that measurement j lies in the valid gate of the target t. On the other hand, to process the measurement from the multitarget merge, the feasible events in the algorithm are defined as follows [2−3] : (1) No more than one measurement originates from each true target. (2) No more than two targets can give the same measurement. Then, the validation matrix Ω is scanned to generˆ , whose element ate one or more feasible matrices, Ω is ω ˆ jt . Computing the same target in every feasible
From the analysis of the JPDA algorithm cited earlier it can be found that there are many effective methods to improve the accuracy of the data association, among which the most valid method is to ensure that the main diagonal elements in the validation matrix are one, and the other elements are zero. This matrix indicates that the association between the measurement and the target is unique.
3. Data association based on target signal classification information 3.1
The TSCI matrix
In the JPDA algorithm, the main-diagonal elements Ωi,i of the validation matrix should be 1.0 and the other elements Ωi,j = 0.0 for i = j under ideal conditions. However, this is not always true when the targets are close to each other. The measurement of a particular target may be associated with more than one multiple track, which unfortunately results in track impurity. If the signal classification validation matrix E is used to improve the measurement-to-track
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Guo Lei, Tang Bin & Liu Gang
association, the validation matrix purity will be significantly improved. However, the signal pdf of each target is usually unknown. If the signal pdf distribution is assumed to be independent of the Gaussian densities[6] , the signal mean and covariance are obtained by the use of the maximum likelihood function method. Usually, the track is divided into three stages, that is, the track initiation, track maintenance, and track end. If the track of each target in the stage of the track initiation can be split[6] , the initiation mean and covariance of each target can be obtained, that is, n 1 t µ ˆt0 = ζ N i=1 i
n ˆt = 1 Σ (ζ t −ˆ µt0 )(ζit −ˆ µt0 ) 0 N i=1 i
t = 1, . . . , T
where ζjt (k) represents the sensor’s signal measurement set about target t at time k. In the stage of the track maintenance, when the sensor receives a new signal measurement, ζj (k), emitted by unknown targets at time k, the distance between the TSCI region of each target and the new signal measurement is first calculated as follows
·
εjt
t = 0, 1, . . . , T
⎧ ⎨ 1 ∀ζ (k) ∈ {V } j t = ⎩ 0 ∀ζj (k) ∈ / {Vt }
εjt
j=1
⎛ ⎞ mk 1 ⎝ µ ˆt = αˆ µt0 + (1 − α) ζj εjt ⎠ ; num validate j=1 ˆt µ ˆt0 = µ ˆt = Σ
1 num validate
mk
(9)
(ζj εjt − µ ˆt )(ζj εjt − µ ˆt ) (10)
j=1
ˆ t are the signal mean and covariance where µ ˆ and Σ of the target t, and α is the weight. The variable num validate denotes the number of the signals lying in the valid gate of target t. 3.2
Date association based on the TSCI matrix
To integrate TSCI into the traditional JPDA, the measurement-to-track association matrix, Λ, is defined as the product of the validation matrix, Ω , given ˆ given in (9). in (2) with the TSCI validation matrix E Therefore, the Λ is Λi,j = Ω Eˆ = [ωjt εjt ]
where κ is the gate threshold of each target signal classification. If Vt (k) κ, it means that ζj (k) lies in the signal valid gate of target t. Otherwise, it does not. Note that the TSCI validation matrix is defined as follows j = 1, . . . , m;
mk
(7)
ˆ0t )−1 (ˆ (Σ µt0 − ζj (k)) < κ}
ˆ = [εjt ], E
num validate =
t
(6)
Vt (k) = {ζj : (ˆ µt0 − ζj (k))
Secondly, the new signal measurements lying in the signal valid gate are used to update the signal mean and covariance of the target. The corresponding equations are listed below.
(8)
where εjt is the element of the TSCI validation matrix ˆ If it is 1, it means that a signal measurement is asE. sociated with a target. In addition, the position of an element in matrix Eˆ denotes the relation between the measurement and target, that is, the row represents the signal measurement and the column represents the target.
(11)
where denotes the schur-hadamard product (term by term). For example, the validation matrix is ⎤ ⎡ 1 1 0 0 1 ⎥ ⎢ ⎢ 0 0 1 1 0 ⎥ ⎥ ⎢ (12) Ωi,j = ⎢ ⎥ ⎢ 1 1 0 1 0 ⎥ ⎦ ⎣ 0 1 0 1 1 From the association matrix mentioned earlier, it can be found that each measurement associates with more than one target. If the TSCI validation matrix is given as ⎡ ⎤ 0 1 0 1 0 ⎢ ⎥ ⎢ 0 0 1 0 0 ⎥ ⎢ ⎥ E=⎢ (13) ⎥ ⎢ 0 0 0 1 0 ⎥ ⎣ ⎦ 0 1 1 0 1
Data association based on target signal classification information The measurement-to-track association matrix Λ is obtained by the product of the validation matrix with the TSCI validation matrix, which is presented below. ⎡ Λi,j
0 1
⎢ ⎢ 0 0 ⎢ =⎢ ⎢ 0 0 ⎣ 0 1
0 0 1 0 0 1 0 0
0
⎤
⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎦ 1
(14)
The comparison between (16) and (18) shows that the TSCI significantly reduces the merge association and thus makes the tracks purer. Therefore, the measurement-to-track association matrix Λ effectively reveals the relation between measurement and target, hence, it is superior to the traditional matrix, owing to the fact that the TSCI of each target is integrated into Λ. 3.3
JPDA algorithm based on the TSCI
In this section, the implementation of the JPDA algorithm based on the TSCI will be summarized. First, in the track initiation stage, the initiation means and covariance of the target signals are computed through (6). Second, in the stage of the track maintenance, there are four main steps involved, which are as follows. Step 1 Use the kinematical validation gate to obtain the validation matrix Ω through (2); Step 2 Use the TSCI gate to generate the TSCI validation matrix E through (8); Step 3 Use the schur-hadamard product of the previous two matrices to derive the measurement-totrack association matrix Λ through (11); Step 4 The feasible matrix is generated; Finally, the procedure that follows is similar to that of the traditional JPDA algorithm.
4. Simulation experiments The target motion model in the Cartesian coordinate system is xk = F xk−1 + Gwk−1 (15) where xk = [xk x˙ k yk y˙ k ]. xk and yk are the xdirectional and y-directional positions of the target in the Cartesian coordinates, respectively. x˙ k and y˙ k are the x-directional and y-directional velocities of the
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target, respectively. F is the translation matrix, wk−1 is the process noise. Q is process noise standard deviation. ⎡ ⎤ ⎤ ⎡ T 0 1 T 0 0 ⎢ 2 ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ 0 1 0 0 ⎥ 1 0 ⎢ ⎥ ⎥ ⎢ ⎥ F =⎢ ⎥, G = ⎢ ⎢ ⎥ ⎢ 0 0 1 T ⎥ ⎢ 0 T ⎥ ⎥ ⎢ ⎢ ⎦ ⎣ 2 ⎥ ⎣ ⎦ 0 0 0 1 0 1 E(wk ) = 0; E(wk wj ) = Qδkj
(16)
In passive tracking, the TDOA (time difference of arrival) of targets is measured. The kinematical part of the measurement vector zk is given as zk = h(xk ) + vk
(17)
where h is the measurement translation function, v is the measurement noise, ⎤ ⎤ ⎡ ⎡ ∆t1 ∆r1 ⎥ ⎥ ⎢ ⎢ ⎥ ⎢ . ⎥ ⎢ .. ⎥ ; h(xn ) = ⎢ ... ⎥ ; zk = ⎢ ⎥ ⎥ ⎢ ⎢ ⎦ ⎦ ⎣ ⎣ ∆tm ∆rm E(vk ) = 0;
E(vk vj ) = Rδkj
(18)
∆ti is the time difference when the target signals arrive at the master station from the slave station, and ∆ri is the range difference between the master station and slave station. R is the standard deviation of the measurement noise. In the 2-D case, the targets are tracked by three sensors located at (0 km, 50 km), (0 km, 170 km), and (80 km, 80 km), respectively. The time difference measurement standard deviation (SD) is 20 ns. The sample time is T = 1 s and the process noise √ standard deviation is Qii = 0.02. The standard de√ viation of the measurement noise is Rii = 0.15 km. Figure 1 shows the true target trajectories in the simulation scenario. The targets are moving for 400 s at the speed of 250 m/s. Such a scenario is a typical air target tracking problem. It consists of four targets whose signal feature classes are assumed to be 1, 2, 3, and 4, respectively (starting from the left in Fig.1). The target detection probability and the false alarm probability are 1.0 and 0.0, respectively.
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Guo Lei, Tang Bin & Liu Gang
Fig. 1
Target trajectories
The estimation results for the JPDA are based on 100 Monte Carlo runs. Figures 2(a) and 2(b) show the tracks obtained by the JPDA, with and without signal class information, respectively. It can be seen that TSCI significantly improves the tracking performance, and results in cleaner tracks. Figures 3(a) and 3(b) show the root mean square (RMS) position error with and without signal class information, respectively. It can be seen that the use of TSCI yields a purer track and improves RMS position errors.
Fig. 3
RMS position error
Furthermore, actual measurement data are used in the algorithm to verify its validity. In Fig. 4(a), the actual measurement data by two targets flying parallelly are presented. It can be found that the distribution of measurement data is uniform around the actual tracking and there are also some false measurement data. The RMS position error using TSCI and not using TSCI are presented in Figs. 4(b) and (c), respectively. The maximum value of the RMS position error is about 2 km2 in Fig. 4(b), however, in Fig. 4(c) the maximum value of the RMS position error is about 7 km2 . From comparing Fig. 4(b) with (c), it can be seen that the improvement on tracking performance with the use of TSCI is distinctness.
5. Conclusions Fig. 2
Tracking target trajectories
When kinematical information of targets is very similar, there arises ambiguity in passive tracking if kine-
Data association based on target signal classification information
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merical experiments in this article confirm the benefit of using TSCI data association for improving target tracking, when there is association uncertainty in the kinematical measurements. Acknowledgments The authors are thankful to the reviewers for their comment and advice. At the same time, the authors would like to thank Dr. Xiao Fei for his help.
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Storrs, CT: YBS Publishing,
1995: 205–211. [2] Bar-Shalom Y, Fortmann T E. Tracking and data association. Boston: Academic Press, 1988: 138–145. [3] Kuo C C, Bar-shalom Y. Join probabilistic data association for multitarget tracking with possibly unresolved measurement and maneuvers. IEEE Tran. on AC, 1984, 29(7): 585–594. [4] Reid D B. An algorithm for tracking multiple targets. IEEE Trans. on AC, 1979, 24: 843–854. [5] Deb S, Yeddanapudi M, Pattipati K R, et al. A generalized S-dimensional assignment for multisensor-multitarget state estimation. IEEE Trans. on AES, 1997, 33(2): 523–528. [6] He You, Xiu Jianjuan, et al. Radar data processing with application, Beijing, Publishing House of Electronics Industry, 2006: 120–125.
Guo Lei was born in 1971. He is a lecturer in University of Electronic Science and Technology of China (UESTC). He is a Ph. D. candidate. His research interests include multisensor data fusion, passive tracking, sensors management, etc. E-mail: leiguo@uestc. edu.cn
Fig. 4
RMS position error comparison by using actual measure data
matical information is used alone. To improve the association results, target classification information has been integrated into the JPDA algorithm, which is one of the typical data association algorithms. The nu-
Tang Bin was bron in 1964. He is a professor in UESTC. His research interests include radar signal, electron counter reconnaissance. Liu Gang was born in 1973. He is a doctor and his current research interest in electron counter reconnaissance.