School of Electronics and Information Engineering,Harbin Institute of Technology,Harbin 150001,China

Sequential nonlinear tracking filte without requirement of measurement decorrelation

Gongjian Zhou*,Junhao Xie,Rongqing Xu,and Taifan Quan

School of Electronics and Information Engineering,Harbin Institute of Technology,Harbin 150001,China

Sequential measurement processing is of benefi to both estimation accuracy and computational efficienc.When the noises are correlated across the measurement components, decorrelation based on covariance matrix factorization is required in the previous methods in order to perform sequential updates properly.A new sequential processing method,which carries out the sequential updates directly using the correlated measurement components,is proposed.And a typical sequential processing example is investigated,where the converted position measurements are used to estimate target states by standard Kalman filte ing equations and the converted Doppler measurements are then incorporated into a minimum mean squared error(MMSE) estimator with the updated cross-covariance involved to account for the correlated errors.Numerical simulations demonstrate the superiority of the proposed new sequential processing in terms of better accuracy and consistency than the conventional sequential filte based on measurement decorrelation.

sequential filte,Doppler measurement,measurement decorrelation,minimum mean squared error(MMSE).

1.Introduction

Targettrackingwith commonlyused sensors,such as radar orsonar,is a typical nonlinearproblemwith the targetstate described in the Cartesian coordinates and the measurements reported in the polar or spherical coordinates.The measurements usually include target position(range and one or two angles)as well as Doppler velocity,which is also called radial velocity or range rate.

If only the position measurements are considered,the converted position measurement Kalman filte(CPMKF) [1],which can provide satisfactory performance with the converted measurement errors properly calculated and compensated[1–3],is preferable.The advantage of the CPMKF is the using of nonlinear filter since the original position measurements are converted to a Cartesian frame of reference.When the Doppler measurements are also involvedin state estimation,the using of nonlinear fil tering approaches is inevitable due to the lack of a conversion linear in Cartesian states from the Doppler measurements.In order to make full use of the information contained in the Doppler measurements and take the advantages of the CPMKF simultaneously,the sequential processing methods,which are of benefi to both estimationaccuracyandcomputationalefficien y[4,5],wereproposed in[6–9].The linear component(the convertedpositionmeasurements)andthenonlinearcomponent(theoriginal Doppler or convertedDoppler measurements)are processed sequentiallybythe CPMKF anda nonlinearestimator,the unscented Kalman filte(UKF)[6]or the extended Kalman filte(EKF)[7–9].As introduced in[10,11],the range and range rate measurement errors are statistically correlated for some radar waveforms.To perform proper sequential processing,the measurement decorrelation[12] based on Cholesky factorization was used in[6–9]to produce the uncorrelatedness of the corresponding measurement noises.The main advantage of the sequential processing method is that the approximation of nonlinearity operates based on the current estimated state,rather than the predicted state from the last stage in block processing method.However,the decorrelation is performed according to the measurements during the whole filterin procedure.This may cause loss of estimation accuracy and consistency.

This paper presents a new sequential processing technique,in which the explicit measurement decorrelation is not required[13].The converted position measurements and the converted Doppler measurements(i.e.,the product of the range and range rate measurements[7,8])are used directly in the sequential processing scheme.The correlation is evaluated based on the covariance after the CPMKFinsteadoftheoriginmeasurementcovariance,andis handled implicitly in the minimum mean squared error (MMSE)estimationframework.Thesuperiorityoftheproposed method is illustrated by numerical simulations.

The rest of this paper is organized as follows.Section 2 describes the formulationof the trackingsystem.The measurement conversions and the converted measurement errors are presentedin Section 3.In Section 4,the new tracking filte is presented,including the formulations of the Cartesian state,the covariance update from the CPMKF, and the fina state updatewhich handles the correlationimplicitly.Monte-Carlo simulations are presented in Section 5 followed by conclusions in Section 6.

2.Problem formulation

In Cartesian coordinates,the state of the target is assumed to evolve according to the following equation:

A 2D Doppler radar is assumed to report measurements of targets in polar coordinates,including range,range rate and azimuth.The measurement equation can be expressed as

3.Measurement conversions

As is well known,the position measurements including range and azimuth in polar coordinates can be transformed into linear forms in Cartesian coordinates by

with the associated covariance as

The product of the range and Doppler measurements is used to construct the converted Doppler measurements. The debiased converted Doppler measurements are expressed as

The above is considered as the measurement of the converted Doppler(i.e.,the product of target true range and range rate)

contaminated by zero-mean Gaussian noises with the variance[8]as

The measurement nonlinearity is reduced to a quadratic form by introducing the converted Doppler measurements in(14).This allows the using of Taylor series expansionup to the second order to perform nonlinearity approximation with no truncated error.

Put the converted position and Doppler measurements in one vector,we have

and the associated covariance

Due to the common range measurements in both measurement conversions and the correlation between range andDopplermeasurementerrors,theentirecovariancematrix above is not diagonal.The cross-covariance between the converted position and Doppler measurement errors can also be obtained by the nested conditioning method [8]as

4.Sequential tracking filte

Themeasurementconversionsin the previoussection yield a measurement vector(18)consisting of two components, one of which is linear and the other is nonlinear in Cartesian states.Sequential processing is suitable for this situation.In the conventional sequential processing methods [6–9,12],the covariance matrix factorization is desired to carry out measurement decorrelation before sequential update.

This paper presents a new sequential filterin approach, in which the correlation is not required.The two measurement components are used directly to update the states sequentially.Besides the Cartesian states,the crosscovariance between the Cartesian state errors and the converted Doppler measurement errors is updated in the procedure of converted position measurement processing using a standard Kalman filte.The updated states and crosscovariance are then used to incorporate the converted DopplermeasurementsundertheMMSEestimationframe, which only requires the errors are Gaussian for optimality, regardless of noise dependence.

4.1State update with the linear component

whereHis the measurement matrix and a constant velocity(CV)model is

The measurement noise

is the zero-mean Gaussian noise with known covariance

The time update and measurement update of the target state are implemented as follows:

4.2Cross-covariance update with linear component

To implement the sequential processing properly after the position measurement filtering the cross-covariance between the state errors from the CPMKF,described in the subsection above,and the converted Doppler measurements should be evaluated.

From the formulations of the CPMKF of(1)and(21)–(28),one can rewrite the state estimates at timekas

Then the corresponding estimation error is

MultiplytheabovebytheerroroftheconvertedDoppler measurements,we can get the covariance asHere the assumption that the estimation errors and the process noises at timek?1 are independent of the measurement errors at timekis used and thus the corresponding terms vanish.The cross-covariance between the Cartesian state errors and the converted Doppler measurement errors can be updated by(31)and will be used as the item accounting for the correlation problem.

4.3State update with the nonlinear component

Themeasurementequationofthenonlinearcomponentcan be written as

GiventheupdatedresultsfromtheCPMKF,thestateupdate by the nonlinear component in the measurement vector(18)is done according to the linear MMSE estimation equation

wheredis the nonlinear functionof the convertedDoppler measurements with respect to the Cartesian states.The Jacobian and Hessian ofdare defined respectively,as

The higher-order-terms(HOT)are zero,since the converted Doppler is quadratic in Cartesian states.The prior mean of the measurement is obtained by taking expectation of(34)as

The covariance between the states to be estimated and the measurement can be obtained as

and the covariance of the measurement is

Equation(33)carries out the fina state estimate optimally in the sense of MMSE,as long as the errors are Gaussian,which holds obviously in the assumptions of the problem considered.

The associated covariance is evaluated by

4.4Filter initialization

The basic idea of the two-point differencing initialization method[12]is to estimate the initial position and velocity components of the state using the firs two sets of position measurements and fin the initial covariance by the measurement covariance.

The initial state is given as

and the initial state is

whereTis the sensor scanning interval.

5.Simulation results

This section compares the performances of the new sequential nonlinear tracking filte,which is referred as SEKF-NEW,and the sequential nonlinear filterin approachwithmeasurementdecorrelationbasedonCholesky factorization(SEKF-CF)is presented in[8].The CPMKF [1],which uses only the position measurements,serves hereas thebaseline,whiletheposteriorCramer-Raobound (PCRLB)[16]is used to indicate the theoretically possible performance expected for a given scenario.

A target takes a nearly constant velocity motion starting from position[30 km,30 km]with an initial speed of 10 m/s and an initial heading of 60°.The standard deviation of the process noise is 0.01 m/s2in each coordinate. An active radar is fi ed at the origin of the Cartesian coordinates,measuring target range,azimuth and range rate with an interval ofT=1 s.

Estimation filterin was performed using 100 scans of measurements.Four scenarios are explored and the difference lies in the measurement errors and the correlation coefficienρ,which are shown as follows:

Scenario 1

Scenario 2

Scenario 3

Scenario 4

The root mean squared error(RMSE)and the normalizedestimationerrorsquared(NEES)[12]are usedtoevaluate the accuracy and consistency of the tracking filters respectively.The position RMSE is evaluated according to

and the velocity RMSE is

The metrics are evaluated based on the Monte-Carlo simulations withM=50.The position RMSE,velocity RMSE and NEES are presented in Figs.1–3,4–6 and 7–9 for scenarios 1–3,respectively.

Fig.1 Scenario 1:position RMSE

Fig.2 Scenario 1:velocity RMSE

For Scenario 1,the SEKF-NEW and SEKF-CF have almost the same performance,approaching the PCRLB and consistency.

Fig.3 Scenario 1:consistency test

Fig.4 Scenario 2:position RMSE

Fig.5 Scenario 2:velocity RMSE

As the position measurement errors increase toσr= 0.3 km andσθ=0.3°(Scenario 2),there is obviousdifference between the two sequential filter in NEESs and the proposed SEKF-NEW method has better consistency than the SEKF-CF.The RMSEs of the two sequential filter go up due to the large nonlinear approximation errors in the case of large range and azimuth deviations.

Fig.6 Scenario 2:consistency test

Fig.7 Scenario 3:position RMSE

When the position measurement errors increase to an extremely high level withσr=1.0 km andσθ=1.0°, the position RMSE of the SEKF-CF even exceeds that of the CPMKF,as shown in Fig.7.On the contrary,the proposed SEKF-NEW method can still provide better accuracythanthe CPMKF in Scenario3,andbetter consistency than the SEKF-CF.Evaluating and handling the correlation between measurement components after the CPMKF contributestotheperformanceimprovementofthesequential processing.While in the SEKF-CF,the measurement decorrelation is carried out based on the measurements during the whole filterin procedure.This leads to loses of estimation accuracy and consistency.

Fig.8 Scenario 3:velocity RMSE

Fig.9 Scenario 3:consistency test

The influenc of the correlation coefficien to the tracking performance is examined in Scenario 4,where the simulation setup is the same as Scenario 2 except that is changed to–0.9.The conclusion in Scenario 2 still holds here,the SEKF-NEW provides accuracy similar to and consistency better than the SEKF.

6.Conclusions

A new sequential processing method,without requirement of measurement decorrelationbefore filtering is presented in this paper.The case of tracking in Cartesian coordinates with position and Doppler measurements observed in polar coordinatesis investigated.First,the convertedposition measurements are processed by the CPMKF.The covariance between the state errors of CPMKF and the converted Doppler measurement errors is evaluated based on the fil tering gain and measurement covariance.Then,a fina estimation method was derived,based on the Taylor series expansion and minimum mean squared error estimation, wherethenonlinearityandcorrelationarehandledsimultaneously and implicitly.Simulation results demonstrate theimprovements in accuracy and consistency of using this new sequential processing method.

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Biographies

Gongjian Zhouwas born in 1979.He received his B.S.,M.S.and Ph.D.degrees in information and communication engineering from Harbin Institute of Technology,Harbin,China,in 2000,2002 and 2008,respectively.From April 2011 to May 2012, he was a visiting scholar with the Department of Electrical and Computer Engineering at McMaster University,Hamilton,Ontario,Canada.Currently, he is an associate professor and a supervisor of Ph.D.candidate with the Department of Electronic Engineering at Harbin Institute of Technology.His research interests are estimation,tracking,signal processing, data fusion,detection and radar system.

E-mail:zhougj@hit.edu.cn

Junhao Xiewas born in 1972.He received his B.S.and M.S.degrees in electrical engineering from Harbin Engineering University,Harbin,China,in 1992 and 1995,respectively.In 2001,he received the Ph.D.degree in information and communication engineering from Harbin Institute of Technology,Harbin,China.Currently,he is a professor in the Department of Electronic Engineering,Harbin Institute of Technology.His research interests involve new type radar systems,radar signal processing and radio oceanic remote sensing.

E-mail:xj@hit.edu.cn

Rongqing Xuwas born in 1958.He received his B.S.,M.S.and Ph.D.degrees in information and communication engineering fromHarbin Institute of Technology,Harbin,China,in 1982,1984 and 1990, respectively.Currently he is a professor with the Department of Electronic Engineering at Harbin Institute of Technology.His research interests are radar system,signal processing,clutter cancellation and array signal processing.

E-mail:xurongqing@hit.edu.cn

Taifan Quanwas born in 1949.He received his B.S. degree from the Automation Department,Tsinghua University,Beijing,China,in 1977 and M.S.degree from the Control Engineering Department,Harbin Institute of Technology,Harbin,China,in 1983. He was a visiting professor with the Department of Electrical and Computer Engineering at Osaka University,Osaka,Japan from 1992 to 1993.Currently he is a professor with the Department of Electronic Engineering,Harbin Institute of Technology.His research interests are tracking,neural network,information fusion and signal processing.

E-mail:quantf@hit.edu.cn

10.1109/JSEE.2015.00123

Manuscript received October 28,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61201311,61132005)and the Aerospace Science Foundation of China(20142077010).