Weihua Wu,Jing Jiang,and Yang Wan

1.Air Force Early Warning Academy,Wuhan 430019,China;

2.Unit94627 of the PLA,Wuxi214141,China

Tracking a maneuvering targetin clutter with out-of-sequence measurements for airborne radar

Weihua Wu1,*,Jing Jiang1,and Yang Wan2

1.Air Force Early Warning Academy,Wuhan 430019,China;

2.Unit94627 of the PLA,Wuxi214141,China

There are many proposed optimal or suboptimal algorithms to update out-of-sequence measurement(s)(OoSM(s)) for linear-Gaussian systems,but few algorithms are dedicated to track a maneuvering target in clutter by using OoSMs.In order to address the nonlinear OoSMs obtained by the airborne radar located on a moving platform from a maneuvering target in clutter,an interacting multiple model probabilistic data association (IMMPDA)algorithm with the OoSM is developed.To be practical, the algorithm is based on the Earth-centered Earth-fixed(ECEF) coordinate system where it considers the effect of the platform’s attitude and the curvature of the Earth.The proposed method is validated through the Monte Carlo test compared with the performance ofthe standard IMMPDA algorithm ignoring the OoSM,and the conclusions show that using the OoSM can improve the tracking performance,and the shorter the lag step is,the greaterdegree the performance is improved,but when the lag step is large,the performance is not improved any more by using the OoSM,which can provide some references for engineering application.

out-of-sequence measurement(s)(OoSM(s)),Earthcentered Earth-fixed(ECEF),interacting multiple model(IMM), probabilistic data association(PDA),attitude.

1.Introduction

In centralized multisensortracking systems,there often exist different times delays[1]when the data from the localsensors are transmitted to the processing center via the wireless communication network,which leads to the situations where some measurements from the same target obtained by different sensors will arrive out of sequence. Such out-of-sequence measurements(OoSMs)are a common and practicalproblemfaced in multisensorfusion systems.Recently many algorithms have been proposed to update an OoSMoptimally or suboptimally.

A natural way among these methods for the OoSM processing problem was using the caching filters[2].After the initialwork[3]was done,the problem was brought to attention since an optimal algorithm for the one-steplag(i.e.,the time delay is less than one sampling interval)OoSM problem was proposed in[4],where the problem was first defined,and the algorithm was examined in[5].A suboptimal algorithm for the l-step-lag OoSM problem was proposed in[6],which requires l iterations to update.Another one-step approximate solution for the l-step-lag OoSM problem,avoiding l iterations,was proposed in[7]based on an equivalentmeasurementthathas the same dimension as the state vector.In[8],an alternative set of formulas withoutresorting to explicitly specify the equivalentmeasurementwas derived.The firstoptimal algorithm for the general l-step-lag problem appeared in [9]was utilizing the“fading information”approach.However,the computationalcostis high.An optimalalgorithm for the multiple-step as well as one-step update was presented in[10,11]by using fixed-pointsmoothing based on best linear unbiased estimation(BLUE)fusion.Another smoothing-based algorithm for the multiple-step OoSM was proposed in[12]by using the Rauch-Tung-Streibel (RTS)fixed-intervalsmoother.In[13],the OoSMsolution was obtained by using the so-called accumulated state densities(ASD)approach which provides a unified treatment of filtering and retrodiction by marginalizing the ASD.

Besides the above single-OoSM problem,an optimal centralized update algorithm forthe multi-OoSMproblem was proposed in[14]by stacking the multiple OoSMs in a single measurement vector and performing the batchform update based on the assumption that these multiple OoSMs should arrive in succession.In[15],an optimalsolution called the complete in-sequence information(CISI) approach for the multi-OoSM with the arbitrary arriving order was proposed,which updates the states between the OoSMtime and the currenttime,including the states atthe OoSMtime.

The above optimal or suboptimal solutions are solvedunder the linear dynamics and linear measurementmodels with additive Gaussian noises.However,what one faces in the realworld include the nonlinear measurements,maneuvering targets in clutter and so on.In order to process the nonlinear OoSMs,some particle filter(PF)-based algorithms were proposed[16–20].As is well known,the computationalload ofthese PF-OoSMalgorithms is heavy. To track real(maneuvering)targets,some algorithms for incorporating OoSMs into the state of the art tracker—interacting multiple model(IMM)estimator were presented[13,21–23],butthey could notdealwith uncertainties in the measurementorigin in scenarios involving clutter,while some algorithms have been proposed to exploit probabilistic data association(PDA)to address the OoSM problem in clutter[24,25],and a Gaussian-sum probability hypothesis density(GM-PHD)filter was first proposed to solve the OoSMproblem in clutter[26],butthey could nottrack a maneuvering target.In[27],an algorithm which incorporates PDA into an augmented state IMM (AS-IMM)for maneuvering targettracking in clutter with OoSM was presented.Reference[28]presented the solution to the combined problem of handling sensor biases when their measurements arrive outof sequence.Otherrelative studies include the removalofout-of-sequence measurements from tracks[29–31],and the out-of-sequence track(OoST)problem[32,33]and so on.

In this paper,we focus on tracking a maneuvering target in clutter with the OoSM for multiple airborne radars.All the above mentioned algorithms are based on the simplified two-dimensional(2D)or 3D local coordinate system of fixed station,notto mention involving time-varying attitude of the moving airborne platform.By accounting for influence of curvature of the Earth and its realistic attitude factorofthe moving platform,unlike[27]which uses state augmentation,an IMMPDA algorithm with the nonlinear OoSM(IMMPDAwOoSM)of tracking a maneuvering targetin clutterin the Earth-centered Earth-fixed(ECEF)coordinate system is developed.

2.Formulation of OoSMproblem

For the airborne platform,there are severalcoordinate systems(or frames)involved[34].For ease of references,we summarize the coordinate systems adopted in our work.

The origin of the ECEF coordinate system is located at the center of the Earth,the X axis extends from the origin to the intersection ofthe prime meridian(0°longitude) and the equator(0°latitude),and the Z axis is along the spin axis ofthe Earth,pointing to the north pole,and the Y axis is orthogonalto the X and Z axes with the usualrighthanded rule.The airborne-carried north-east-down(NED) system is associated with the flying vehicle.Its origin is atthe center of gravity of the airborne platform,the X,Y and Z axes pointtoward the geodetic north,eastand downward along the ellipsoid Earth normal,respectively.The body coordinate system is directly defined on the body of an airborne platform.Its origin is also located atthe center of gravity.The X,Y and Z axes pointtoward the forward head,right side and downward to comply with the righthanded rule,respectively,which is called the HRD system forshorthere.

The dynamics ofa moving targetin the 3D ECEF Cartesian system are modeled as

where Xj(k)is the state of the target at time tkfor the model Mj,i.e.,itis a vector consisting of position and velocity fora piecewise constantwhite acceleration(secondorder)model M1

or a vector consisting of position,velocity,and acceleration for a piecewise constant Wiener process acceleration (third-order)model M2

where the superscript T denotes the vector or matrix transpose.

The transition matrices are given by

where Inis an n×n identity matrix,and?denotes the Kroneckerproduct.τk=tk?tk?1is this time interval.

The covariance of the zero-mean white Gaussian process noise sequence is given by

The measurements collected by the sensor s on an airborne platform at time k are defined as zs(k)=where mkis the number of the measurements,and

is a 3D vector consisting of range and azimuth and elevation angles for the 3D sensor such as radar, and the clutter y(k)is modeled as independent and uniformly distributed over the observation volume(or gated volume)V,and the distribution of the number of clutter is Poisson,i.e.,P(m)=e?λV(λV)m/m!,λis the number of false alarms per unit volume.For the target measurement,we have(omitthe time index k)

where xH,yHand zHare the elements ofrefers to the four-quadrantinverse tangentfunction.

The covariance of zero-mean Gaussian measurement noises ws(k)is given by

whereδ(k,k?1)is the Kronecker delta function,and

andσr,σa,σeare the standard deviation(std.)for the radar range,azimuth and elevation measurements noises, respectively.

The problem of tracking a maneuvering target in clutter with the OoSM is as follows.At the time t=tk,the standard IMMPDA[35]tracker hasSubsequently,an OoSM zs(κ)at the earlier time tκarrive.The objective is updatingthe OoSM,namely,to calculateP(k|κ)=P(k|k,κ),fordetails refer to[7].

3.IMMPDA algorithm with OoSM

We have developed the algorithm tracking a maneuvering target with l-step-lag OoSM(about its definition refer to [7])in clutter.

(i)Retrodicted states

We can get the retrodicted state toκfrom k for each filter j(j=1,...,r)

and the retrodicted state covariance

where Fj(κ,k)=is the backward transition matrix.The covariancein(17)is given by

The retrodicted OoSMfor the module j is

The corresponding covariance Sj(κ)of the innovation isis the position components of Pj(κ|k);is the Jacobian matrix of target’s position in the NED frame with respect to target’sposition in the ECEF frame and the platform’s geodetic

coordinate evaluated at target’s retrodicted value ofand platform’s measurements value of?,and submatrixis the Jacobian matrix of target’s position in the NED frame with respect to the platform’s geodetic coordinate,R?is the covariance matrix for the geodetic coordinate vector?.Similarly,

is the Jacobian matrix of target’s position in the HRDframe with respectto platform’s attitude.

is the Jacobian matrix of the nonlinear measurementfunction evaluated atthe target’s retrodicted position in the HRD frame,Rωis the covariance matrix for the attitude angle vectorω.

(iii)OoSMvalidation

A validation gate for the sensor s centered at the retrodicted measurementcan be setup,i.e.,

where the Markov chain transition probability matrix is

where Pr(·)denotes the probability of an event.According to the well-known property of a Markov process,if we denoteΠ(1)=Π(k,k?1),thenΠ(κ,k)=Πl(fā)(1).

The measurements are validated as follows:

where g is the gates threshold for setting up the validation regions, and|·|denotes the matrix determinant.

(iv)Estimation with OoSMin each filter

The update of the state atthe currenttime k in each module of the IMM using the validated OoSMis done as follows.

With the OoSMs zs(κ),the state estimate is updated by

and .In the above,m(κ)is the number of validated OoSMs.The filter gain Gj(k,κ)is obtained by

whereΨj(κ)is the Jacobian matrix of?sevaluated at timeκgiven by

Using the chain derivative rule of composite function according to(7)–(9),we have

The innovation forthe measurement n is

The covariance associated with(27)is

The covariance associated with(26)is

The association probabilitiesare computed according to nonparametric probabilistic data association filter(PDAF)[35]as follows:

with PDas the probability ofdetection of the sensor s;PGis the gate? probability that the target is in the validationis the Gaussian probability

density function with innovationmean zeros and covariance Ssj.

The volume of the validated region is given by

where Vnzis the volume of the unithypersphere of dimension nz,the dimension of the measurement z.For the 1D, 2D and 3D measurements,Vnzis equalto 2,πand 4π/3, respectively.

(v)Updating modelprobabilities

Using the above,the mode probabilities are updated as follows:

where the likelihood function is

and the normalization factor is

(vi)State estimate and covariance combination

Finally,the updated combined estimate is obtained by

and the corresponding covariance is given by

4.Simulation results

Now we consider an example of tracking a highly maneuvering target in clutter.The target starts at geodetic coordinate(121.1°E,40.5°N,8 000 m),ends at (121.1°E,40.3°N,8 000 m),goes through(120.2°E, 40.3°N,8 000 m),(120.5°E,40.3°N,8 000 m),(120.6°E, 40.6°N,8 000 m),(120.9°E,40.6°N,8 000 m)in order.The airborne platform 1 starts at(119.0°E,40.6°N, 8 000 m),ends at(120.5°E,40.6°N,8 000 m);the airborne platform2 starts at(119.0°E,40.2°N,8 000 m),ends at(120.5°E,40.2°N,8 000 m).Both of their speeds are 250 m/s.Fig.1 shows their trajectories.

Fig.1 Two platforms’trajectories,target true trajectory and its estimated trajectory with IMMPDAwOoSM-5

Itis assumed thateach platform is equipped with a radar. The two radars(radar 1 and radar 2)are characterized by the following parameters:σr=100 m,σa=1°,σe=1°. Both radars are assumed to be track-while-scan(TWS) radar,and their sampling time areτ=2 s,and the sampling time of the radar2 is lag 1s longer than thatof radar 1 which is used as the fusion center.The l-step-lag OoSM (OoSM-l)is set in the simulation when the measurements from radar 2 are transmitted to radar 1,as is shown in Fig.2.

Fig.2 Sampling scheme and arriving order under l-step-lag OoSM (OoSM-l)case

The elements of the two platforms’position measurement covariance R?are specified by longitude std. 0.001°,latitude std.0.001°and altitude std.100 m.The elements of the platforms’attitude measurementcovariance Rωare assumed that yaw,pitch and rollstd.are the same as 0.1°.

The two models from Section 2 are used in the IMMPDA tracker.M1and M2have process noise withσ1x= σ1y=σ1z=5 m/s2andσ2x=σ2y=σ2z=20 m/s2, respectively.The assumed model switching probabilities are given byΠ12(1)=0.2 andΠ21(1)=0.1.

The clutter is assumed to be Poisson distributed with expected number ofλ=6.5×10?6/(m·mrad2)for both radars.It is assumed that the probability of detection PD=1 for both radars.The gates threshold is set g=4 leading to a gate probability PG=0.999 7.

We compare IMMPDAwith OoSM-l to IMMPDAwithout the OoSM,the former is called IMMPDAwOoSM-l and the latteris called IMMPDAwoOoSM(which is a standard IMMPDA algorithm in fact)forshort,respectively.

A typical tracking result of the IMMPDAwOoSM-5 is shown in Fig.1.We can visually see that the proposed method can track the maneuvering targetsuccessfully.Because the results of other methods are indistinguishable from that of the IMMPDAwOoSM-5,they are not displayed.

The performance comparison with regard to the root mean-square(RMS)error in position and velocity through 300 Monte Carlo runs is shown in Fig.3,and average numbers of false alarms and targetoriented in the validation region are given in Table 1 which shows the proposed algorithm can work under the circumstance in presence of clutter and mis-association.Fig.3(a)gives the RMSposition errorwhile Fig.3(b)gives the corresponding RMS speed error.

As is seen in Fig.3,the performance of the IMMPDAwOoSM-1 is the best,followed by the IMMPDAwOoSM-3,while the performance of the IMMPDAwOoSM-5 and IMMPDAwoOoSM are similar, which shows that the proposed IMMPDAwOoSM-l algorithm is validated,and using the OoSM can improve the tracking performance,and the shorter the lag step is,the greater degree the performance is improved,but when the lag step is large,the performance is notimproved any more by using the OoSM.

Fig.3 Performance comparison of IMMPDAwoOoSM and IMMPDAwOoSM-l(l=1,3,5)

5.Conclusions

We propose an algorithm which is capable of tracking a highly maneuvering targetwith the nonlinear multiple-lag OoSMin clutter in the ECEF system accounting for timevarying attitude and curvature of Earth.Simulation results show thatthe algorithm is validated,and using the OoSM can improve the tracking performance,and the shorter the lag step is,the greaterdegree the performance is improved,but when the lag step is large,the performance is not improved any more by using the OoSM.Owing to accounting for many realistic problems,the algorithm is expected to be used in engineering practice.Future work will explore multiple maneuvering targets tracking in clutterwith OoSMs in the ECEF system.

Acknowledgment

The first author would like to extend sincere gratitude to the teacher Haiying Du for his examination of English expression of this paper.

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Biographies

Weihua Wuwas born in 1987.He received his B.S.degree in electronic warfare command and engineering in 2009 and M.S.degree in signal and information processing in 2011 respectively, both from Air Force Radar Academy.Now,he is a Ph.D.candidate in the same academy.He had been engaged in the National College Mathematical Modeling and Electronic Design Contest in HubeiProvince and won the second prize.His research interests include passive location and tracking,multi-targettracking,and nonlinear filtering as wellas multi-sensor data fusion applications.

E-mail:weihuawu1987@163.com

Jing Jiangwas born in 1964.He received his M.S. degree in 1996 from National University of Defense Technology and Ph.D.degree in 2006 from Wuhan University,now he is a professor and doctor supervisor in Air Force Early Warning Academy.His research interests include radar data processing,modern information processing,and information fusion. E-mail:jiangj36@sina.com

Yang Wanwas born in 1984.He received his M.S. degree in 2009 and Ph.D.degree in 2013 respectively from Air Force Early Warning Academy,now he is an engineer in Unit 94627 of the PLA.His research interests include radar information processing,tracking before detect.

E-mail:wanyang19850122@163.com

10.1109/JSEE.2015.00083

Manuscript received May 28,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61102168).