Maneuvering target tracking algorithm based on CDKF in observation bootstrapping strategy①

Hu Zhentao (胡振濤), Zhang Jin, Fu Chunling, Li Xian

(*Instituteof Image Processing and Pattern Recognition, Henan University, Kaifeng 475004, P.R.China) (**School of Physics and Electronics, Henan University, Kaifeng 475004, P.R.China)

Maneuvering target tracking algorithm based on CDKF in observation bootstrapping strategy①

Hu Zhentao (胡振濤)*, Zhang Jin*, Fu Chunling②**, Li Xian*

(*Instituteof Image Processing and Pattern Recognition, Henan University, Kaifeng 475004, P.R.China) (**School of Physics and Electronics, Henan University, Kaifeng 475004, P.R.China)

The selection and optimization of model filters affect the precision of motion pattern identification and state estimation in maneuvering target tracking directly. Aiming at improving performance of model filters, a novel maneuvering target tracking algorithm based on central difference Kalman filter in observation bootstrapping strategy is proposed. The framework of interactive multiple model (IMM) is used to realize identification of motion pattern, and a central difference Kalman filter(CDKF) is selected as the model filter of IMM. Considering the advantage of multi-sensor fusion method in improving the stability and reliability of observation information, the hardware cost of the observation system for multiple sensors is adopted, meanwhile, according to the data assimilation technique in Ensemble Kalman filter(EnKF), a bootstrapping observation set is constructed by integrating the latest observation and the prior information of observation noise. On that basis, these bootstrapping observations are reasonably used to optimize the filtering performance of CDKF by means of weight fusion way. The object of new algorithm is to improve the tracking precision of observed target by the multi-sensor fusion method without increasing the number of physical sensors. The theoretical analysis and experimental results show the feasibility and efficiency of the proposed algorithm.

maneuvering target tracking, interacting multiple model (IMM), central difference Kalman filter (CDKF), bootstrapping observation

0 Introduction

The key of target tracking is to estimate its motion state by using the priori pattern information of target motion and the latest observation. The availability and reliability of an algorithm depend on two aspects including the matching level between motion model and real motion pattern, and the performance of model filter. According to the pattern and strength of target motion, it is usually divided into non-maneuvering target tracking and maneuvering target tracking[1]. When an observed target moves in the non-maneuvering pattern, it can be described by single model. At this point, it is not related to the model matching problem, and the precision of state estimation mainly relies on the performance of the used filter. When the observed target moves in maneuvering pattern, the structure of multiple models needs to be generally adopted because of the uncertainty of the motion model. For such problem, a group of models usually needs to be designed to describe the different motion behavior. In the model set, each model matches a specific behavior pattern, and the estimation results of more than one parallel filters are organically integrated to constitute state estimation. According to the differences of model switching principle, the structure of multiple models is divided into static multiple models estimation[2]and dynamic multiple models estimation[3]. The hard decision mechanism of binary decision is adopted in the static multiple models estimation, and target motion model is identified by the accumulative result of estimation error. Its weakness is that the threshold value of model decision relies heavily on expert knowledge. Besides, the accumulative process of estimation error results in the delay of model switching time. The typical realization of dynamic multiple models estimation is IMM[4]. A kind of soft decision mechanism of model selection is used in IMM, which adopts the balance strategy between the precision of model identification and the precision of state estimation. So it avoids the dependence for expert knowledge. At present, IMM is considered as the mainstream approach to solve the maneuvering target tracking problem.

When state model and observation model are linear or weak nonlinear, in order to obtain better performance in the process of model identification and state estimation, Kalman filter (KF)[5]or extended Kalman filter(EKF)[6]are used as a model filter in IMM. However, when they are strong nonlinear, KF and EKF are no longer applicable. Considering that it is much easier to approximate probability density distribution of nonlinear function than nonlinear function itself, meanwhile, accompanied by rapid development of computer performance, the filter design according to sampling strategy becomes the most active research hotspot in nonlinear estimation[7,8]. Recently, some domestic and foreign scholars put forward a series of solutions for the design and optimization of nonlinear filters. Those realizing principles can roughly be divided into deterministic sampling and random sampling. The typical method of deterministic sampling is unscented Kalman filter (UKF)[9]. Its basic idea is that a set of carefully chosen sigma points are used to deliver the statistic characteristics of random variables by UT transform, and then the mean and the covariance can be estimated by the weighted statistical linear regression way. Its advantages are that UKF is insensitive to system nonlinear degree, meanwhile, it avoids the calculation process of Jacobian matrix appearing in EKF. However, the filtering precision of UKF is limited by parameter selection of sigma point and weight, and the non-positive definite problem of estimation error covariance appears easily in filtering iteration. Similar to the implementation of UKF, there are some solutions such as Gauss-Hermite filter(GHF)[10]adopting the numerical integration principle of Gaussian-Hermite, cubature Kalman filtering(CKF)[11]adopting the third-order volume integral principle and so on. The typical methods of random sampling are particle filter(PF)[12]and Ensemble Kalman filter (EnKF)[13], and their common disadvantages are that the filtering precision and computation complexity are limited by the dimension of estimated state and the number of samples. Considering the parallel filtering way used in IMM, many PFs or EnKFs need to be run at the same time. Therefore, the calculation amount will be increased sharply along with the number of target motion models, and real-time is damaged. Aiming at solving the problem, combining with the Stirling interpolation principle, the central difference Kalman filter(CDKF) gives a novel realizing structure of deterministic sampling[14], and it will deals with the contradiction between the estimation precision of nonlinear state and the computational complexity. According to above analysis, through the dynamic combination of IMM and CDKF in observation bootstrap strategy, a novel maneuvering target tracking algorithm is designed in the paper, and the feasibility and efficiency of the algorithm are verified by emulation experiment.

1 The central difference Kalman filter in observation bootstrapping strategy

1.1 Central difference Kalman filter

CDKF is considered as a classic nonlinear filter based on the Stirling interpolation principle. In realization of CDKF, sigma points are sampled according to state prior distribution of observed system, and its posterior distribution is expressed by sigma points using linear regression transformation[15]. LetLdenotethestatedimensionoftheobservedsystem,thenumberofsigmapointsis2L+1.Inordertomakesigmapointshavethesamemeanvalue,varianceandhigher-ordercenterdistancewithrealstate,sigmapointsandtheircorrespondingweightareexpressedas

(1)

(2)

1) Initialization

(3)

(4)

2) Time update

(5)

(6)

(7)

3) Observation update

(8)

(9)

(10)

(11)

Kk=Pxz(Pzz)-1

(12)

(13)

Pk|k=Pk|k-1-KkPzz(Kk)Τ

(14)

1.2 CDKF in observation bootstrapping strategy

n=1,2,…,N(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

2 Maneuvering target tracking algorithm based on central difference Kalman filter in observation bootstrapping strategy

2.1 Interacting multiple model

The key of IMM is that multiple models working in parallel are respectively used to match different modes of maneuvering target. Among models, they are transferred according to probability matrix. Based on cutting and merging the hypothesis of each model, the estimation of multiple parallel filters is synthesized. IMM overcomes the influence of error caused by the mismatch between motion state and model when single model is used to describe the estimated system. Considering the following multi-model system with model switching characteristics

xk=f(xk-1,γk, uk-1)

(25)

zk=h(xk, vk)

(26)

γk～p(γk|γk-1)

(27)

(28)

(29)

2.2 Interacting multiple model based on cubature Kalman filter with observation iterated update

Considering that CDKF-OBS has high estimation precision, CDKF-OBS is selected as the model filter in IMM. The objective is to improve the overall performance of IMM by promoting the state estimation of each pattern. On the basis of that, the IMM algorithm based on CDKF in observation bootstrapping strategy (IMMCDKF- OBS) is proposed. In order to facilitate understanding the concrete implementation of IMMCDKF-OBS, the form of pseudo code is given.

Initialization：＾xi0|0=x0，Pi0|0=P0，πij=π0，μi0=μ01)Inputinteraction μik-1，μijk-1|k-1，xik-1|k-1andPik-1|k-1arecalculatedby μik-1=∑Ji=1πijμjk-1 μijk-1|k-1=πijμjk-1/μik-1 xjk-1|k-1=∑Ji=1＾xik-1|k-1μijk-1|k-1 Pjk-1|k-1=∑Ji=1[Pik-1|k-1+(＾xik-1|k-1-xjk-1|k-1) (＾xik-1|k-1-xjk-1|k-1)Τ]μijk-1|k-1 μjk-1denotesthemodelprobabilityofmodeljattimek-1，andπijdenotesthetransitionprobabilityfrommodelitomodelj．2)ModelfilteringTakingxik-1|k-1andPik-1|k-1as＾xk-1|k-1andPk-1|k-1inEq．(1)，calculate＾xk|k-1andPk|k-1canbecalculatedinac?cordancewithEq．(2)toEq．(7)．Combiningwiththeboot?strappingobservation，＾xn，ik|kandPn，ik|konthebasisofΘnk，modelicanbesolvedbyEq．(8)toEq．(10)andEq．(18)toEq．(21)．Then，accordingtoEq．(22)toEq．(24)，＾xik|kandPik|kcanbeobtainedforeachmodel．3)Modelprobabilityupdatinglikisfirstlycalculatedbyfollowingequations．ln，ik=|(2π)Pn，izz|-12 exp{-[(Θn，ik-＾zik|k-1)Τ(Pn，izz)-1(Θn，ik-＾zik|k-1)]/2} lik=∑Nn=1n，ikln，ikAndthenmodelprobabilityμikafterupdatingisexpressedas μik=μik-1lik/∑Jj=1(μjk-1ljk)4)OutputinteractionCombiningwithEq．(28)andEq．(29)，＾xk|kandPk|kcanbecalculated．5)Letk=k+1，returntostep1)．

3 Simulation result and analysis

To verify the feasibility and availability of the proposed algorithm, the simulation scenario is set as the maneuvering target tracking by using the observations of two-coordinate radar. The sampling intervalτis1sandthesamplingstepsare35.ThenumberofMonteCarlosimulationis100.TheexperimentplatformadoptsPC,Pentium4 (CPU), 3.26GHzdominantfrequency, 2Gmemory,Windows7,andtheprogramminglanguageisMatlab2012b.Themodeoftargetmotioninradarscanningareaisasfollows.Theestimatedtargetmoveinuniformcircularmodeinthefirst10samplingperiods,anditsturningangularvelocityis+0.3rad/s.Inthesamplingperiodsfrom10to25andfrom11to35,itsturningangularvelocitiesare-0.15rad/sand+0.3rad/s,respectively,where“+”and“-”denotethattheestimatedtargetmovethedirectionofanticlockwiseandclockwise,respectively.Combiningwiththedynamiccharacteristicsofmaneuveringtargetmotionandthephysicalcharacteristicofradarsensors,thesystemstateequationandtheobservationequationofestimatedtargetareasfollows.

Results from Fig.1 to Fig.5 show the model matching probability of five algorithms. In total, it is easy to see the model matching probability of IMMCDKF and IMMCDKF-OBS are superior to IMMUKF, IMMEKF and IMMCKF, furthermore, IMMCDKF-OBS is better than IMMCDKF. The reason is that the pros and cons of model filter selection directly effect the reliability of model identification in IMM. Because of introducing observation bootstrapping strategy in IMMCDKF-OBS, the performance of CDKF-OBS is superior to CDKF. When the feature is introduced into IMM, it reflects the improvement of real-time, precision and stability of models identification. Fig.6 and Fig.7 show the RMSE comparison of five algorithms. It is clear that the RMSE of IMMCDKF-OBS is less than other four algorithms, that is, the precision of IMMCDKF-OBS is the highest. From the figures one can also know that RMSE of IMMCDKF-OBS always keeps at low level and relatively stabilized. Table 1 quantitatively gives the mean of RMSE and the average time over 100 independent runs. It can be clearly found that the data of mean of RMSE describing algorithm filtering precision verifies the above analyzed results. The time cost is used to assess the computational complexity of these algorithms. The above results are conducive to reasonable selection of filters in practical engineering applications. It can be seen that the run time of IMMEKF is minimum, but its precision is also the lowest. The run time of IMMCDKF-OBS is slightly increased relative to IMMCDKF. However, its precision is certainly superior to the other algorithms. According to the above results shown in this paper, the five types of maneuvering target tracking algorithms provide guidance significance in the practical engineering application. The significance of above results gives the reasonable selection direction of model filter in the maneuvering target tracking system.

Fig.1 IMMUKF

Fig.2 IMMEKF

Fig.3 IMMCKF

Fig.4 IMMCDKF

Fig.5 IMMCDKF-OBS

Fig.6 Horizontal direction

Fig.7 Vertical direction

AlgorithmHorizontaldirection(km)Verticaldirection(km)Timecost(s)IMMUKF0．06480．11650．0096IMMEKF0．06150．11180．0025IMMCKF0．06050．11230．0138IMMCDKF0．04330．07210．0096IMMCDKF?OBS0．02890．03790．0352

4 Conclusions

The method of interactive multiple model solves the model matching problem by sacrificing filtering precision, a maneuvering target tracking algorithm based on CDKF in observation bootstrapping strategy is proposed. In process of IMMCDKF-OBS, through the dynamic connection among observation bootstrapping strategy, central difference Kalman filter and interacting multiple model, the valid identification and estimation of the mode and state for maneuvering target tracking are realized. Compared with the existing processing method, the advantages of the new algorithm are as follows: Firstly, based on the method of interacting multiple model, the problem of state estimation in multi-model system is solved in the process of IMMCDKF-OBS. Secondly, observation bootstrapping strategy is used to simulate observation information of multiple sensors and the information will be extracted and utilized by weight fusion strategy. New algorithm improves filtering precision on condition that hardware cost of the system is of no growth.

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10.3772/j.issn.1006-6748.2017.02.005

①Supported by the Postdoctoral Science Foundation of China (No. 2014M551999) and the Open Foundation of Key Laboratory of Spectral Imaging Technology of the Chinese Academy of Sciences (No. LSIT201711D).

②To whom correspondence should be addressed. E-mail: fuchunling@henu.edu.cn

on May 26， 2016

o, born in 1979. He received his Ph.D degrees in Control Science and Engineering from Northwestern Polytechnical University in 2010. He also received his B.S. and M.S. degrees from Henan University in 2003 and 2006 respectively. Now, he is an assistant professor of college of computer and information engineering, Henan University. His research interests include complex system modeling and estimation, target tracking and particle filter, etc.