LI Lifu and CHENG Ting

School of Information and Communication Engineering,University of Electronic Science and Technology of China,Chengdu 611731,China

Abstract:Phased array radar ’s measurements include two direction cosine and range measurements,which can be obtained in the direction cosine coordinates.State equation of the target is nonlinear with the measurements and in order to solve the nonlinear problem,debiased conversion measurements based target tracking with direction cosine and range measurements in direction cosine coordinates named DCMKFPreDcos is proposed first in this paper,where the predicted information is introduced to calculate the converted measurement errors ’ statistical characteristics to eliminate the correlation between measurement noise and the converted measurement errors covariance.When range rate information can be obtained further,based on the above DCMKF-PreDcos ’ filtering result,the sequential filtering is adopted to process the additional range rate measurement and the DCMKF-PreDcos algorithm with extra range rate information is given.The predicted information is also introduced to calculate the involved statistical characteristics of converted measurements.The effectiveness of the proposed algorithms is shown in simulation results.

Keywords:nonlinear filtering,range rate information,target tracking,direction cosine coordinates,predicted information.

1.Introduction

In practical phased array radar (PAR),target state equation modeled in the Cartesian coordinates is nonlinear with measurements obtained in the spherical or polar coordinates.The extended Kalman filter (EKF) [1,2] is a traditional nonlinear filter that solves the nonlinear problem by utilizing Taylor expansion to linearize the nonlinear measurement equations,but high nonlinearity will lead to poor tracking accuracy of EKF.The unscented Kalman filter (UKF) [3,4] utilizes the sigma points constructed by unscented transformation to approximate the target state’s mean and covariance.The performance of UKF is better than that of EKF while it is more complicated.Particle filter (PF) [5-7] has excellent tracking performance in dealing with the nonlinear problem,but it also has large computational complexity.

Converted measurement Kalman filter is also an effective method to deal with the nonlinear problem.Debiased converted measurement Kalman filter (DCMKF) [8]converts measurements in spherical or polar coordinates to Cartesian coordinates,where the conversion bias is eliminated by subtracting operation.Unbiased converted measurement Kalman filter (UCMKF) was proposed in[9] because the conversion bias is found to be essentially multiplicative.In [10],some comments were made on UCMKF and an improved algorithm is proposed,which recently is utilized in the tracking fusion of asynchornous multi-radar [11].The converted measurement errors’ statistical characteristics in [8-10] are both calculated conditioned on the measurement information,which makes the measurement noise and the converted measurement error covariance correlated.In order to remove the correlation,the predicted information is introduced to calculate the converted measurement errors ’ statistical characteristics and the decorrelated unbiased converted measurement Kalman filter (DUCMKF) [12] is proposed.

Once range rate information can be obtained further,the tracking performance can be improved [13,14],however,the degree of nonlinearity increases.To reduce the degree of nonlinearity while processing the range rate information,sequential filtering [15,16] is introduced.Range rate measurement and position measurements are processed sequentially.The technique of sequential filtering is combined with the idea of the best linear unbised estimation (BLUE) [17] and a totally linear filtering structure was proposed in [18] and [19] respectively.The statically fused converted measurement Kalman filter (SFCMKF) [20] utilizes two separate filters to deal with the range rate and position measurements at the same time,the final result is obtained by statically fusing the outputs of the two separate filters.However,the state equation of the pseudo state is highly related to the target motion mode.Therefore,it is not generally compared with similar algorithms.The recursive BLUE with the range rate was proposed in [21] and the conventional BLUE was introduced into the statically fusion method in [22].CMKF with range rate (CMKFRR) [23] introduces the cross range rate to construct a single linear Kalman filter,but the cross range rate is noninformative and prior information is difficult to obtain.

The measurement information of the above algorithms is given in the spherical or polar coordinates.For direction cosine measurements,the EKF is utilized in the measurement covariance adaptive EKF algorithm (MCAEKF)[24] to process the nonlinear direction cosine measurements,where the noise variance can be adjusted adaptively.By constructing the third direction cosine’s pseudo measurement and utilizing the linear minimum mean square error (LMMSE) criterion,a novel recursive LMMSE filter was proposed in [25].Neither of the above two algorithms consider the range rate information,which can be utilized to further improve the performance of algorithms.In [26],the target tracking problem with the direction cosine measurements and range rate was considered,which also shows the improvement of tracking performance with the range rate.Debiased converted measurement sequential EKF in direction cosine coordinates (DCMSEKFcos) proposed in [27] utilizes a linear filter and a sequential EKF to sequentially filter the debiased converted range rate and position measurements.The SF-CMKF [20] is applied to deal with the measurements of the range rate and position in the direction cosine coordinates in [28].However,in the above two algorithms,the converted meausurement errors’ statistical characteristics are deduced conditioned on the measurement.Based on the above problems,a measurement conversion based sequential filtering method named SEKF-PreDcos was proposed in [29],where a sequential Kalman filtering method with direction cosine measurements is proposed.This paper is an extension of our previous work in [29],where the case with only position measurements and the case with additional range rate measurement are considered comprehensively.The measurement conversion processes in the above two cases are deduced in details.Based on the measurement conversion results,a debiased conversion measurements based target tracking algorithm with direction cosine measurements named DCMKF-PreDcos is proposed.The DCMKF-PreDcos with additional range rate measurement is also given.

The main contributions are listed as follows: first,based on the meausrement in direction cosine coordinates,the measurement conversion is investigated,where the converted meausurement errors ’ statistical characteristics are deduced conditioned on the prediction and a new algorithm called DCMKF-PreDcos is proposed.Second,when the range rate information is available,the DCMKF-PreDcos is extended to this case,where the converted meausurement errors ’ statistical characteristics are deduced conditioned on the prediction.

The remaining parts are organized as follows.The system model is shown in Section 2.The DCMKF conditioned on the predicted information with range rate and position measurements in the direction cosine coordinates is derived in Section 3.Simulation results demonstrate the superiority of our algorithm compared with similar algorithms in Section 4 and the conclusions are given in Section 5.

2.System model

Assuming a PAR is located at the spherical coordinates ’origin.The state equation is constructed in the Cartesian coordinates,which can be modeled as

whereFt-1is the state transition matrix,Gt-1is the state noise input matrix,wt-1is the system state noise,which is white Gaussian and has covarianceQt-1and zero mean.

The measurement equation is formulated as follows:

3.DCMKF conditioned on predicted information with range rate and position measurements in direction cosine coordinates

3.1 DCMKF-PreDcos with direction cosine and range measurements

Firstly,measurement conversion is applied to convert the range and two direction cosines measurements to a pseudo-linear form,which can be written as follows:

The predicted information includes the predicted rangert-and the predicted direction cosines αt-,βt-,γt-,which can be calculated based on the state prediction equation:

The relationship between the predicted information and the true value is as follows:

Therefore,the relationship between the measurements and the predicted information can be obtained:

The debiased converted measurements conditioned on the predicted information can be written as follows:

Similar with the deduction of mean,based on the relationship between the measurements and the predicted information,the elements incan be deduced as follows:

At the last step in the deduction of (32)-(37),the independence between the predicted error and the measurement error attis also utilized.

Pt-is the predicted error covariance,which is calculated as

wherePt-1is the state estimation error covariance.

Based on above measurement conversion,the position Kalman filter can be obtained,which is named as DCMKF conditioned on the predicted information in the direction cosine coordinates (DCMKF-PreDcos).The state update can be expressed as follows:

3.2 DCMKF-PreDcos with additional range rate measurement

With additional range rate information,the degree of nonlinearity increases.The pseudo measurement is constructed to reduce the strong nonlinearity,which can be expressed as

Combining (3)-(5) and (46),the converted measurements are shown as follows:

Apart from the converted position measurement errors’statistical characteristics calculated from (15) to (37),the ones related to the range rate should be deduced too.

The predicted range rate information can be calculated by

Therefore,the debiased converted measurements conditioned on the predicted information can be expressed as follows:

After calculating the errors ’ statistical characteristics,a sequential filter is adopted to process range rate information.The pseudo measurement and the converted position measurements are obviously correlated because the pseudo measurement is constructed by using range rate and range measurements.Therefore,decorrelation is necessary before processing the sequential filtering.

From Cholesky factorization results in [15],one can get (29) and the following formula:

It is obvious that after decorrelation,are uncorrelated.Based on the decorrelation,the position measurements are filtered with the DCMKF-PreDcos proposed in Subsection 3.1 and the pseudo measurement information can be sequentially utilized.The structure of DCMKF-PreDcos with range rate information is given in Fig.1.

Fig.1 Structure of DCMKF-PreDcos with range rate information

According to Fig.1,two direction cosine measurements and range measurement are input into the position filter,where the debiased measurements conversion is obtained by (3) to (5) and (28),the corresponding errors’statistical characteristics are calculated by (31) to (37).The output of the position filter,which is named asandhere,are input together with the range rate and range measurement into the sequential filter,where the pseudo measurement is given in (46) and the related statistical characteristics are given in (55) to (58).The decorrelation between it and the debiased converted position measurements are shown from (61) to (64).Based onafter decorrelation,EKF is utilized as the sequential filter next.The sequential filter’s output as well as the final result are shown as

4.Simulation

Assume a target with the initial position (3 km,8 km,5 km)and the constant velocity (50 m/s,-50 m/s,0 m/s) moves for 100 s.A PAR is located at the spherical coordinates’origin and the range rate,direction cosine range measurements can be obtained.The measurement noises’ standard deviations in six scenes are given in Table 1.The correlation coefficient ρ is set to 0.9.State estimation initialization adopts the three-point initialization method[30].The proposed filter DCMKF-PreDcos is adopted to realize the target tracking,it is compared with the converted position measurement Kalman filter (CPMKF)[28].When the range rate measurement is further obtained,it is compared with DCMSEKFcos in [27].

Table 1 Measurement errors’ standard deviations

The comparison of position root mean square errors(RMSEs) in six scenes are shown from Fig.2 to Fig.7 respectively.Comparing the position RMSEs of DCMKFPreDcos with CPMKF in all scenes,it is shown that the tracking performance of DCMKF-PreDcos is better than that of CPMKF.It indicates the superiority of calculating the statistical characteristics conditioned on the predicted information.

Fig.2 Position RMSEs in Scene 1

When the range rate information can be obtained,DCMKF-PreDcos with range rate measurement and DCMSEKcos show better tracking performance than CPMKF and DCMKF-PreDcos,which indicates the tracking performance will be improved when range rate information is introduced.The tracking performance of DCMKF-PreDcos with range rate measurement and DCMSEKFcos are equivalent in Scene 1 and Scene 2.With the increase of measurement noise,from Fig.4 to Fig.7 one can see that DCMKF-PreDcos with range rate measurement shows better performance than DCMSEKFcos.The reason is that the measurement noise and the converted measurement errors ’ covariance are correlated while the correlation is removed by introducing predicted information in calculating the statistical characteristics in DCMKF-PreDcos,which will help improve the tracking performance while calculation conditioned on the measurement information in DCMSEKFcos does not.

Fig.5 Position RMSEs in Scene 4

Fig.6 Position RMSEs in Scene 5

Fig.7 Position RMSEs in Scene 6

5.Conclusions

For PAR whose measurements are given in direction cosine coordinates,a debiased conversion measurements based target tracking algorithm with direction cosine and range measurements named DCMKF-PreDcos is proposed first.The converted position measurements in the Cartesian coordinates are obtained by carrying out the measurement conversion method,where the predicted information is introduced to calculate the converted measurement errors’ statistical characteristics.With additional range rate measurement,a sequential filtering is utilized in DCMKF-PreDcos and DCMKF-PreDcos with range rate measurement is derived,where the pseudo measurement’s mean and covariance are also deduced conditioned on the predicted information.Simulation results demonstrate that the tracking performance of the proposed algorithm is better than existing algorithms.