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Research Article

A practical adaptive nonlinear tracking algorithm with range rate measurement

International Journal of Distributed Sensor Networks 2018, Vol. 14(5) Ó The Author(s) 2018 DOI: 10.1177/1550147718776863 journals.sagepub.com/home/dsn

Haiyan Yang, Hongqiang Liu, Zhongliang Zhou and An Xu

Abstract It is difficult to answer the problem whether the range rate measurement should be adopted to track a target in a tracking scenario. A practical adaptive nonlinear tracking algorithm with the range rate measurement is proposed, which avoids this problem and achieves good accuracy of target state estimation. First, three popular nonlinear filtering algorithms only with the position measurement are surveyed. Second, three popular nonlinear filtering algorithms with the position and range rate measurements are surveyed. Then, a novel tracking algorithm with range rate measurement is proposed based on the cumulative sum detector and the above two kinds of nonlinear algorithms. The results of simulation experiment demonstrate that the range rate measurement could reduce accuracy of the target state estimation in mismatch tracking scenarios. The results of simulation experiment also verify that the performance of proposed algorithm is better than the current state and the art interacting multiple-model algorithm and can well follow the state estimation output of the measurement equation matching the tracking scenario. Keywords Nonlinear filter, measurement equation, range rate, adaptive filter, target tracking, cumulative sum detector

Date received: 2 December 2017; accepted: 13 April 2018 Handling Editor: Janos Botzheim

Introduction Target tracking algorithm with the range and bearing measurements from the radar is widely applied in practice. Normally, because of the range and bearing measurements in the radar coordinates and the target state in the Cartesian coordinates, the target tracking algorithm is nonlinear. At present, many nonlinear filtering algorithms are applied to solve this problem, such as the extended Kalman filter (EKF) algorithm,1 the unscented Kalman filter (UKF) algorithm,2 the cubature Kalman filter (CKF) algorithm,3 the unbiased converted measurement Kalman filter (UCMKF) algorithm,4 and the decorrelated unbiased converted measurement Kalman filter (DUCMKF) algorithm.5 The EKF first widely used in this tracking problem uses the Taylor expansion of the measurement function to linearly approximate the nonlinear measurement

function. If the nonlinearity of measurement function is strong, the effect of linear approximation is not ideal. The UKF and the CKF can be classified as the generalized Gaussian filtering.6 The algorithm assumes that the state estimation and prediction obey the Gaussian distribution, and the numerical integration method is used to compute the mean and the covariance of the assumed Gaussian density function. Compared with the EKF, the UKF and the CKF have the advantages of not requiring differential operation or calculating the Jacobian matrix, but the disadvantage is that more Air Force Engineering University, Xi’an, P.R. China Corresponding author: Zhongliang Zhou, Air Force Engineering University, Xi’an 710038, P.R. China. Email: [email protected]

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2 computational steps are needed. The UCMKF and the DUCMKF are a class of measurement conversion algorithm whose idea is first to convert the raw radar measurement into Cartesian coordinate system before filtering and then use the linear Kalman filter (KF).7 Lerro and Bar-Shalom4 claim that the performance of the UCMKF is better than that of the EKF in the mixed coordinate system. On the basis of the UCMKF, the DUCMKF replaces the raw measurement from the radar with the state prediction from the filtering to eliminate the correlation between the measurement error covariance estimation and the measurement noise and then the estimation bias is reduced. However, if the state estimation error is large or the model mismatch occurs, the new prediction bias will be introduced due to the use of the state prediction. In addition to range and bearing measurements called position measurement, the Doppler radar can also obtain the range rate measurement. Recently, the range rate measurement is used to estimate the parameters of target motion, because the range rate measurement includes the velocity information. An estimation method of turn rate is proposed by combining the range rate measurement and the coordinate point measurement for a fast curvilinear target.8 In the passive location problem, the high-precision range rate measurement from single satellite using third-order Taylor expansion ensures the accuracy of location.9 Harris and Whitcomb10 report an observation model for range rate measurement with a delayed-state EKF to estimate the state of underwater vehicles in the communication and navigation. The modified approach of probability weight in the interacting multiple-model (IMM) is developed by the normal acceleration statistics derived from the range rate.11 In a multistatic radar, range and range rate measurement from multiple bistatic radars are adopted to determine the targets velocity and location.12 The kinematics and geometry relationships between range rate and target state have been developed in many applications gradually. In the above works, only basic nonlinear filters, such as the EKF, the UCMKF and the CKF, are used for the nonlinear estimation with range rate measurement. The strong nonlinear relationship between the state and the range rate measurement is ignored. Specially, the results of target tracking with the position and range rate measurements obtained by the EKF, the UKF, the CKF, and the UCMKF directly have potential inconsistencies and deviations.13 In recent years, Lei and Han,14 Jiao et al.,15 and Bordonaro et al.16 propose several filtering algorithms with range rate measurement to reduce the stronger nonlinearity of the measurement equation containing the position and range rate measurement. The sequential unscented Kalman filter (SUKF) algorithm, the sequential extended Kalman filter (SEKF) algorithm

International Journal of Distributed Sensor Networks and the converted measurement Kalman filter with range rate (CMKFRR) algorithm are accepted for the better stability and higher accuracy of state estimation than others. The SUKF14 converts the position measurement into Cartesian coordinates and remains the raw range rate in the nonlinear measurement equation. Then, perform the KF for the converted position measurement and perform the UKF for the raw range rate measurement sequentially. The SEKF15 converts the position measurement into Cartesian coordinate system and replaces the raw measurement with the pseudomeasurement that is the product of range and range rate in order to reduce the nonlinearity of the measurement equation. Then, perform the KF for the converted position measurement and perform the second-order EKF for the pseudo-measurement sequentially. The sequential filtering methods with the conversion measurement algorithm have better tracking performance than those of standard UKF and EKF. Bordonaro et al.16 propose the CMKFRR that converts the range, bearing, and range rate measurements to the position and velocity in Cartesian coordinate system. However, the CMKFRR uses the cross-range rate, which needs the prior knowledge to model the expected probability distribution of crossrange rate measurement because the cross-range rate cannot be observed. Although the above SUKF, SEKF, and CMKFRR can effectively reduce the nonlinearity of target tracking with the range rate measurement, it is not feasible to use the range rate measurement in any tracking scenario to improve accuracy of target state estimation. In other words, in some scenarios, the position measurement can be used to achieve the higher accuracy of target state estimation than the position and range rate measurements; in other scenarios, the position and range rate measurement are better. In order to obtain better accuracy of state estimation, it is necessary to distinguish where the position measurement should be used and where the position and range rate measurements should be used. However, this boundary is difficult to determine, because the accuracy of state estimation is influenced not only by the nonlinear measurement equation, also by the true movement of target, measurement noise, filtering algorithm, target motion model, and so on. In order to avoid the decrease in the target tracking performance due to the improper use of range rate measurement, a practical adaptive tracking algorithm with the range rate measurement based on the cumulative sum (CUSUM) detector is proposed in this article to achieve the good accuracy of target state estimation, which is close to the best performance of tracking algorithm with single measurement equation in different scenarios. The content of this article is designed as follows. In section ‘‘Formulation of the problem,’’ the background

Yang et al.

3

of the problem is described and the measurement equation containing the position measurement and the measurement equation containing the position and range rate measurements are described. In section ‘‘Nonlinear tracking algorithms with the position measurement,’’ the popular nonlinear tracking algorithms only with the position measurement, namely, the UKF, the UCMKF, and the DUCMKF, are surveyed. In section ‘‘Nonlinear tracking algorithms with the position and range rate measurement,’’ the popular nonlinear tracking algorithms with the position and range rate measurements, namely, the SUKF, the SEKF, and the CMKFRR, are surveyed and the performance of the algorithms described in section ‘‘Nonlinear tracking algorithms with the position measurement’’ and the algorithms described in this section are compared by the simulation experiment. In section ‘‘A practical adaptive nonlinear tracking algorithm with the range rate measurement,’’ a practical adaptive nonlinear tracking algorithm with the range rate measurement is proposed and its performance is verified. In section ‘‘Conclusion and future works,’’ the conclusions and future works are given.

Formulation of the problem For a target in two-dimensional (2D) plane, a coordinate decoupled dynamic model of the target is described in Cartesian coordinate system Xk + 1 = Fk Xk + G k Wk

ð1Þ

where Xk = ½xk , yk , x_ k , y_ k , s1 3 (n4) 0 is the target motion state at time k; xk , and yk are the target position components; x_ k and y_ k are the target velocity components; s1 3 (n4) is the other components, for example, the target acceleration components and jerk components; Fk 2 Rn 3 n is the state transition matrix; G k is the coefficient matrix of the corresponding dimension; and Wk is the zero mean white Gauss process noise with the covariance matrix Qk . When the position measurement only is adopted to track a target, the measurement vector is denoted as ZPk = ½rk , uk 0 . Assuming the radar is located at the origin of the coordinate system, the measurement equation is described as ZPk = hPk ðXk Þ + VPk

ð2Þ

hPk (Xk )

where is the measurement function only with the position measurement " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # x2k + y2k hPk ðXk Þ = arctanðyk =xk Þ

ð3Þ

where rk is the range measurement; uk is the bearing measurement; VPk is the zero mean white Gauss measurement noise vector with the covariance matrix RP =



s2r 0

0 s2u

 ð4Þ

where sr and su are the standard deviations of range and bearing measurements. When the position and range rate measurement are adopted to track a target, the measurement vector is 0 denoted as ZD k = ½rk , uk , r_ k  . The corresponding measurement equation is denoted as D D ZD k = hk ðXk Þ + Vk

ð5Þ

where hD k (Xk ) is the measurement function with the position and range rate measurements qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 x2k + y2k 6   7 7 6 6 arctan y2k xk 7 D hk ðXk Þ = 6 7 6 xk x_ k + yk y_ k 7 4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 x2k + y2k 2

ð6Þ

where r_ k is the range rate measurement; VD k is the zero mean white Gauss measurement noise vector with the covariance matrix 2

s2r D 4 R = 0 rsr sr_

0 s2u 0

3 rsr sr_ 0 5 s2r_

ð7Þ

where sr_ is the standard deviation of the range rate measurement; r is the correlation coefficient between the range rate measurement and the range measurement.17 It can be seen from formulas (3) and (6) that P hD k (Xk ) has the stronger nonlinearity than hk (Xk ).

Nonlinear tracking algorithms with the position measurement When the position measurement is used to tracking a target, the target state equation and measurement equation are formulas (1) and (2), respectively. For the target tracking problem, the current popular methods are the UKF, the UCMKF, and the DUCMKF.

The UKF algorithm The UKF is an approximate filtering algorithm based on the unscented transformation.18 The filtering distribution is assumed to be a Gaussian distribution in the UKF   p Xk jZP1:k ’ NðXk jmk , Pk Þ

ð8Þ

4

International Journal of Distributed Sensor Networks

where mk and Pk are the state estimation and the state estimation covariance. The UKF is as follow

3.

Prediction. 1. Form the sigma points x(0) k1

= mk1

pffiffiffiffiffiffiffiffiffiffiffiffihpffiffiffiffiffiffiffiffiffiffii = mk1 + n + l Pk1 i pffiffiffiffiffiffiffiffiffiffiffiffihpffiffiffiffiffiffiffiffiffiffii (i + n) xk1 = mk1  n + l Pk1

ð9Þ

 (i)  (i) 0 Wi(c) j^k  mk j^k  mk + RP

2n X

  (i) 0 ^ (i) j^k  mk Wi(c) x  m k k

= Fk1 x(i) k1 ,

i = 0, . . . , 2n

4.

Compute the filter gain Kk , the filtered state mean mk , and the covariance, conditional on the measurement Pk Kk = Ck S1 k

P  mk = m k + Kk Zk  mk 0 Pk = P k  Kk Sk K k

ð10Þ

Compute the predicted mean m k and the predicted covariance P k

m k =

2n X

^ (i) Wi(m) x k

i=0

=

2n X

Wi(c)



^ (i) x k



ð15Þ

i=0

i=0

where l = a2 (n + k)  n; the parameters a and k determine the coverage area of sigma points. 2. Propagate the sigma points through the state transform matrix

m k

 0  ^ (i)  m + Qk1 x k k

ð11Þ

i=0

where the weights Wi(m) and Wi(c) are calculated by l n+l   l + 1  a2 + b W0(c) = n+l l , i = 1, . . . , 2n Wi(m) = Wi(c) = 2ð n + l Þ

W0(m) =

ð12Þ

where b is an additional algorithm parameter that can be used for incorporating prior information. Update. 1. Form the sigma points x(0) = m k k

pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ffi x(i) = m P k i k + n+l k pffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi ffi xk(i + n) = m n + l P k i k 

ð13Þ

i = 1, . . . , n

2.

Sk =

2n X

Ck =

i = 1, . . . , n

P k

(i)

Wi(m) j^k

i=0

i

3.

2n X

mk =

x(i) k1

^ (i) x k

Compute the predicted mean mk , the predicted covariance of the measurement Sk , and the cross-covariance of the state and the measurement Ck

Propagate sigma points through the measurement function  (i) j^k = hPk x(i) , k

i = 0, . . . , 2n

ð14Þ

ð16Þ

The advantage of the UKF over the EKF is that the UKF is not based on a linear approximation at a single point, but uses further points in approximating the nonlinearity. The unscented transform is able to capture the higher order moments caused by the nonlinear transform better than Taylor series–based approximations.19 The UKF can be classified as a special kind of Gaussian filtering, which includes the Gauss–Hermite Kalman filter and the CKF. The UKF performance is close to the Gauss–Hermite Kalman filter and the CKF. Although the mean estimate of the UKF is exact for polynomials up to third order, the covariance computation is only exact for polynomials up to the first order.19

The UCMKF algorithm The conversion measurement Kalman filter (CMKF) algorithm is a common tracking algorithm for the target state in Cartesian coordinate system and the measurement in the polar coordinate system. The CMKF has two bias sources. The first source is the conversion bias. If the bias conversion measurement is adopted in the KF, it will lead to performance degradation. The UCMKF is proposed in Longbin et al.20 to eliminate the bias by multiplicative compensation. The UCMKF with multiplicative compensation is as follows ZuP k =



xuk yuk



"

# r cos ð u Þ l1 k k = u1 , lu rk sinðuk Þ

lu = esu =2 2

ð17Þ

The corresponding conversion measurement covariance matrix is

Yang et al.

5 " RPu

=

RPu11

RPu12

RPu21

RPu22

# ð18Þ

   2 2 1 2 r + s2r ð1 + l0 u cos 2uk Þ RPu11 = l2 u  2 rk cos uk + 2 k    2 1 2 rk + s2r l0 u sin 2uk RPu12 = RPu21 = l2 u  2 rk sinuk cos uk + 2   2 2 RPu22 = l2 u  2 rk sin uk  1 2 + rk2 + s2r ð1  l0 u cos 2uk Þ, l0 u = e2su 2

ð18aÞ

The converted measurement ZuP k and the covariance matrix RPu are combined with the KF to obtain the UCMKF. Although the UCMKF eliminates the conversion bias, the conversion measurement covariance matrix and measurement noise are correlated in formula (18a), which can lead to the second bias source called estimation bias.

The DUCMKF algorithm The DUCMKF based on the UCMKF replaces the current measurement with the one-step state prediction to eliminate the correlation between the conversion measurement covariance matrix and measurement noise and then the estimation bias is reduced. Like the UCMKF, formula (17) is used to perform the unbiased converted measurement. The range and bearing measurement predictions are denoted as rk + 1jk and uk + 1jk at time k + 1, and their standard deviations are denoted as sr, k + 1 and su, k + 1 . The state prediction is denoted as X^k + 1jk = ½xk + 1jk , yk + 1jk , x_ k + 1jk , y_ k + 1jk 0 at time k + 1, and the state prediction covariance is denoted as PPk + 1jk . Then qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2k + 1jk + y2k + 1jk yk + 1jk uk + 1jk = arctan xk + 1jk

rk + 1jk =

ð19Þ

The variables sr, k + 1 and su, k + 1 are calculated by s2r, k + 1

=

s2u, k + 1 =

1 rk2 + 1jk



xk + 1jk



yk + 1jk PPk + 1jk xk + 1jk

1

yk + 1jk rk4 + 1jk

yk + 1jk



xk + 1jk PPk + 1jk yk + 1jk

0

ð20Þ

The decorrelated conversion measurement error covariance matrix is denoted as " RPdu =

RPdu11

RPdu12

RPdu21

RPdu22

ð21aÞ

The DUCMKF can be obtained through combining the converted measurement ZuP and the covariance k matrix RPdu with the KF. Although the DUCMKF avoids the correlation between the measurement error covariance matrix and the measurement noise, the state prediction error is also introduced. When the state predicted error is larger than the measurement error, the performance of the DUCMKF is not better than that of the UCMKF algorithm. This may occur when the measurement error is small or the motion model does not match the true movement of target.

Nonlinear tracking algorithms with the position and range rate measurement When the position and range rate measurements are used to tracking a target, the target state equation and measurement equation are formulas (1) and (5), respectively. For the target tracking problem, the current popular methods are the SEKF, the SUKF, and the CMKFRR.

The SEKF algorithm

0

xk + 1jk

1 2 rk + 1jk + s2r + s2k + 1jk h2 i 2   2 2s 2 1 + cos 2uk + 1jk e2su e u, k + 1jk esu h i   2s2 1  rk2 + 1jk + s2r, k + 1jk 1 + cos 2uk + 1jk e u, k + 1jk 2 1 2 P P rk + 1jk + s2r + s2k + 1jk Rdu12 = Rdu21 = h  2  i 2 2 2s2 sin 2uk + 1jk e2su e u, k + 1jk esu h  i  2s2 1  rk2 + 1jk + s2r, k + 1jk sin 2uk + 1jk e u, k + 1jk 2 1 2 P rk + 1jk + s2r + s2k + 1jk Rdu22 = h2 i 2   2 2s2 1  cos 2uk + 1jk e2su e u, k + 1jk esu h i   2s2 1  rk2 + 1jk + s2r, k + 1jk 1  cos 2uk + 1jk e u, k + 1jk 2 RPdu11 =

# ð21Þ

The SEKF replaces the raw range rate r_ k with the pseudo-measurement rk r_ k . The KF is applied to the converted position measurement, and the second-order EKF is used for the pseudo-measurement sequentially. The steps of the SEKF are as follows.

Conversion measurement. Apply formula (17) to convert the position measurement, and the conversion formula of pseudo-measurement is jk = rk r_ k = x_x + y_y  rsr sr_

ð22Þ

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International Journal of Distributed Sensor Networks

The corresponding covariance matrix of conversion measurement is 2

RPu11

RPu12

6 P 6 RD pu = 4 Ru21

RD pu13

7 7 RD pu23 5

RPu22

RD pu31

3

RD pu32

ð23Þ

RD pu33

 2  D s2u RD pu13 = Rpu31 = s r r_ k + rk rs r sr_ cos uk e  2  D s2u RD pu32 = Rpu32 = s r r_ k + rk rs r sr_ sin uk e   2 2 2 2 2 2 2 RD pu33 = rk s r_ + sr r_ k + 3 1 + r s r s r_ + rk r_ k rs r sr_ ð23aÞ

The decoupled method in Jiao et al.15 is used to decouple the position and the pseudo-measurements.

mP k = Fk1 mk1

2.

The SUKF directly adopts the raw range rate measurement to perform the sequential filter. First, the KF is used for the converted position measurement and then the UKF is used for the raw range rate measurement. The steps of SUKF are as follows.

2

RPu11 6 P D Rru = 4 Ru21 RD ru31

ð24Þ

0

= Fk1 Pk1 F k1

Position measurement updating filtering estimation Hp =



1 0

0 1

0 0 0 0

RD ru32

3 RD ru13 7 RD ru23 5

RD ru33 ð25Þ

0 H = L1 + x_ kp , L2 + y_ kp , xkp , ykp e

d2k = 2Pkp ð1, 3Þ + 2Pkp ð2, 4Þ Ak = Pkp ð1, 1ÞPkp ð3, 3Þ + Pkp ð2, 2ÞPkp ð4, 4Þ + 2Pkp ð1, 2ÞPkp ð3, 4Þ + 2Pkp ð1, 4ÞPkp ð2, 3Þ + Pkp ð1, 3Þ2 + Pkp ð2, 4Þ2 h i D Rjpu = RD pu31 , Rpu32  1  P 1 j 0 j Rpu + Ak Kek = Pkp He0 He Pkp He0 + RD pu33  Rpu Ru

1 mk = mPk + Kek ek  L1 xkp  L2 ykp  xkp x_ kp  ykp y_ kp  d2k 2   Pk = I4  Kek He Pkp

ð27aÞ

= s2r_

In order to do sequential processing, it is necessary to decouple the position measurement and the range rate measurement, and the decoupling method is the same as the SEKF. Sequential filtering. 1. Time update filtering estimation with formula (24). 2. Position measurement updating filtering estimation with formula (25). 3. Range rate measurement updating filtering estimation. Form the sigma points P g (0) k = mk

ð26Þ

and

ð27Þ

RD ru33

D RD ru32 = Rru32 = lu rsr s r_ sin uk

Pseudo-measurement updating filtering estimation

where ½L1 , L2  =  Rjpu (RPu )1 L2 yuk + jk .

RPu12 RPu22

D RD ru13 = Rru31 = lu rsr s r_ cos uk



p p p0 1 p0 Kpk = Pp H Pk H + RPu k H  p uP p P mPk = mP k + Kk Zk  H mk   Pkp = I4  Kpk Hp Pp k

3.

The SUKF algorithm

Conversion measurement. Apply formula (17) to convert the position measurement, and remain the raw range rate measurement, that is, r_ k = r_ k . The corresponding covariance matrix is

Sequential filtering. 1. Time update filtering estimation

PP k

The SEKF adopts the pseudo-measurement and indeed reduces the nonlinearity of the original measurement function. However, if the range is far and the range rate is large, the multiplication of them will give a high gain to the pseudo-measurement noise, which will lead to the decrease in the tracking performance and even the divergence.

ek = L1 xuk +

  pffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffip P g (i) = m + n + l P k k k i qffiffiffiffiffiffi  pffiffiffiffiffiffiffiffiffiffiffiffi (i + n) P gk = mk  n + l Pkp ,

ð28Þ i = 1, . . . , n

i

 k and the covariance Pk Calculate the mean m

Yang et al.

7 2n X

k = m

and takes the KF of information form. The steps of CMKFRR are as follows.

Wi(m) g (i) k

i=0 2n X

Pk =

Wi(c)



g(i) k

k m



g(i) k

k m

0

ð29Þ

Conversion measurement

i=0

2

where the weights Wi(m) and Wi(c) are calculated by formula (12). Substitute the sigma points into the measurement function qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  x2k + y2k h ðXÞ = L1 xk + L2 yk + ðxk x_ k + yk y_ k Þ  1

 D ½L1 , L2  =  Rru RPu , Rru = RD ru31 , Rru32

xuk

3

2

rk

3

6 yu 7 607 2 6 7 6 7 Zuk = 6 ku 7 = esu =2 Dðuk Þ6 7 4 x_ k 5 4 r_ k 5  Dðuk Þ =

y_ uk dð uk Þ 0



0 , dð uk Þ

 dðuk Þ =

c_ k cos uk sin uk

 ðiÞ ek = h g (i) k , i = 0;2n

2

ðiÞ

Wi(m) ek

i=0 2n X

  0 ðiÞ ðiÞ Wi(c) ek  ek ek  ek

ð32Þ

i=0

Pxe k =

2n X

  0 ðiÞ  k ek  ek Wi(c) g (i) k m

i=0

Output the state estimation mk and covariance estimation Pk  ee 1 Kek = Pxe k Pk  k + Kek ðr_ k  ek Þ mk = m Pk = Pk + Ke Pee Ke k

k

3 rk cos uk 6 rk sin uk 7 2 7 Zuk = esu =2 6 4 r_ k cos uk 5 r_ k sin uk

ð31Þ

Compute the predicted measurement mean ek , the measurement predicted covariance Pee k , and the crosscovariance of the state and the measurement Pxe k

Pee k =



where c_ k is the cross-range rate and ck = 0

to predict the measurement

2n X

 sin uk cos uk

ð35Þ

ð30Þ

ek =

ð34Þ

Then, the measurement matrix is H = I4 . The KF of information form. m k = F k1 mk1  Pk = Fk1 Pk1 F0 k1 + Qk1 h  i1 1 1 0 1 Wk = P + H R H H0 Rk k k

u   mk = m k + Wk Z k  m k h  i1 1 0 1 Pk = P + H R H k k

0 1 R1 k = DðaÞRR DðaÞ    1:3 1 0 R1:3, R R1 = R 0 0

k

The CMKFRR algorithm The SEKF and the SUKF are the sequential filtering methods, and their measurement equations still contain nonlinear components. The CMKFRR directly uses the KF for the position and range rate measurement. The algorithm converts the range, bearing, and range rate measurements into Cartesian coordinate system

ð37Þ

where

ð33Þ

Although the mean estimate of the SUKF is the same as that of the SEKF, the covariance computation of the SEKF is exact for polynomials up to the third order. Besides, the SUKF needs more computation steps to deal with the nonlinear measurement function of the range rate.

ð36Þ

ð38Þ ð39Þ

1:3 The matrix R1:3, can be calculated by R

i 2 1 h 1 2 2 2s2u 2s2a + s e x^R + P11 1 + e es a R r 2 i 1 h  1 2 2s2a  x^R + P11 1 + e R 2 12 RR = 0 2  1 1 3 13 2s2u 2s2a ^ ^ 1 + e R13 = + P + rs s e x x es a r r _ R R 2 R R  1 2s2a  x^1R x^3R + P13 1 + e R 2 h 2 i 2 1 1 11 2 2s2u 2s2a ^ R22 = + P + s e x 1  e es u R R r R 2  1   1 2 2s2a  x^R + P11 1  e R 2 R11 R =

8

International Journal of Distributed Sensor Networks

2  1 1 4 2s2u 2s2a 1  e e x^R x^R + P14 es u R 2  1 2s2a  x^1R x^4R + P14 1  e R 2 h i 2   1 3 2 33 2 2s2u 2s2a ^ 1 + e es u R33 = + P + s e x R R r_ R 2  1  3 2 2  x^R + P33 1 + e2sa R 2 i 2 1 h 4 2 2 2 2 1  e2su e2sa esu + x^R + P44 R + s c_ 2  1  4 2 2  x^R + P44 1  e2sa ð39aÞ R 2

R23 R =

0  1 where x^R = DðaÞ0 m k ,.PR = DðaÞ Pk DðaÞ, a = tan  2  1  2   2 x^1R . The parameter sc_ is the mk mk , sa = P22 R

standard deviation of cross range rate. The CMKFRR introduces the unknown cross-range rate c_ k whose standard deviation sc_ should be set with the prior knowledge. The parameter sc_ influences the performance of CMKFRR algorithm. Bordonaro et al.16 claim that the tracking performance of CMKFRR with an appropriate sc_ is better than those of SEKF and SUKF.

Performance of tracking algorithms The simulation experiment is to demonstrate that the performance of the UKF, the UCMKF, the DUCMKF, the SEKF, the SUKF, and the CMKFRR algorithms. Besides, the posterior Crame´r–Rao lower bound (PCRLB)21 for the position measurement called PCRLBR and the PCRLB for the range, bearing, and range rate measurements called PCRLBRR are calculated. Three target tracking scenarios are set up artificially in the simulation experiment. The mean square error (MSE) is used to measure the accuracy of state estimation. The MSE for target position and the MSE for target velocity are defined as, respectively MSEð^x, ^yÞ =

N h i 1X ðx  ^xÞ2 + ðy  ^yÞ2 N i=1

N h   2 i 2  1X MSE ^x_ , ^y_ = x_  ^x_ + y_  ^y_ N i=1

ð40Þ

where N is the Monte Carlo experiment time; x and y are the true position components; ^x_ and ^y_ are the true velocity components; ^x and ^y are the estimated position components; and ^x_ and ^y_ are the estimated position components: Scenario 1: the initial position of target is (6 km, 10 km); the initial magnitude and direction angle of velocity is (50 m=s, p=6 rad); the standard deviation of range measurement is sr = 20m; the standard deviation of bearing measurement is su = 0:18.

Scenario 2: the initial position of target is (20 km, 40 km); the initial magnitude and direction angle of velocity is (50m=s, p=6 rad); the standard deviation of range measurement is sr = 50 m; the standard deviation of bearing measurement is su = 28. Scenario 3: the initial position of target is (6 km, 10 km); the initial magnitude and direction angle of velocity is (200 m=s, p=6 rad); the standard deviation of range measurement is sr = 200 m; the standard deviation of bearing measurement is su = 28. The standard deviation of range rate measurement is sr_ = 0:1 m=s and the correlation coefficient is r =  0:9 in all scenarios. The target follows a nearly constant velocity (CV) track in three scenarios. In order to eliminate the influence of motion model mismatch on tracking performance, all the algorithms adopt the CV motion model in the simulation and the power spectral density of the noise covariance matrix is set to q = 1 W=Hz. The initial position components in tracking algorithm are set as the first converted measurement and the initial velocity components are set to 0 m=s. The initial state estimation covariance matrix is set to P0 = diag(104 , 104 , 50, 50). The parameters of UKF and SUKF are set to l =  1, a = 1, b = 2, and n = 4. For the CMKFRR, the medium standard deviation of cross range rate is set as sc_ = 10m=s. Figures 1–3 show the MSE for position and velocity with 1000 Monte Carlo simulations in the scenarios 1– 3, respectively. The PCRLBRR is lower than the PCRLBR in each tracking scenario. Thus, in theory, introducing the range rate measurement has the potential to improve the accuracy of target state estimation. In Figure 1, the MSEs of the SEKF, the SUKF, and the CMKFRR for position and velocity are obviously better than those of the UKF, the UCMKF, and the DUCMKF. The results demonstrate that the use of range rate measurement indeed improves the accuracy of state estimation in the tracking scenario 1. In Figure 2, after about 70s, the MSEs of all algorithms for position and velocity are interwoven together, but the CMKFRR, which demonstrates that the use of range rate measurement, has little influence on the accuracy of state estimation in the tracking scenario 2. In Figure 3, the MSEs of the SEKF, the SUKF, and the CMKFRR for position and velocity are lower than those of the UKF, the UCMKF, and the DUCMKF during the initial tracking period. However, the MSEs of the SEKF, the SUKF, and the CMKFRR for position are higher than those of the UKF, the UCMKF, and the DUCMKF during the middle and late tracking periods. It demonstrates that the use of range rate measurement is not always beneficial for tracking task.

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Figure 1. MSE for position and velocity with 1000 Monte Carlo simulations in the scenario 1: (a) MSE for position and (b) MSE for velocity.

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Figure 3. MSE for position and velocity with 1000 Monte Carlo simulations in the scenario 3: (a) MSE for position and (b) MSE for velocity.

A practical adaptive nonlinear tracking algorithm with the range rate measurement The CUSUM detector based tracking algorithm with range rate

Figure 2. MSE for position and velocity with 1000 Monte Carlo simulations in the scenario 2: (a) MSE for position and (b) MSE for velocity.

It can be seen from the results of simulation experiment in section ‘‘The CMKFRR algorithm’’ that the range rate measurement cannot improve the accuracy of the tracking algorithm in the all tracking scenarios. In some tracking scenarios, for example, the tracking scenarios of Lei and Han,14 Jiao et al.,15 and Bordonaro et al.16 and the scenario 1 in section ‘‘The CMKFRR algorithm,’’ the use of the range rate measurement indeed improves the state estimation accuracy. However, before performing the tracking algorithm, as the target state is stochastic and the tracking performance is influenced by the true movement of target, measurement noise, filtering algorithm and target motion model, and so on, it is difficult to judge whether the range rate measurement is beneficial to the target tracking task. In essence, this problem can boil down to the problem of whether the measurement model in the target tracking algorithm matches the current tracking scenario. When the measurement model with range rate is matched, the state estimation accuracy of the target can be improved;

10 otherwise, it cannot. The famous IMM algorithm22 is used in the field of tracking to solve the problem of model mismatch. The IMM algorithm models the switching between the models as Markov random process, and realizes the soft switching between the models. The IMM aims at the mismatch between the target motion state model and the real target motion, and can solve the maneuvering target tracking well.23 The IMM may be applied directly to the mismatch problem of the measurement models in this article. However, unlike the target maneuver, the characteristics of the nonmaneuvering tracking scenario have the spatio-temporal stability which the IMM is not suitable.24 Besides, the performance of IMM is affected by the initial model probability and the model transfer probability that need to be set manually based on prior information, while we does not have such a priori information for the measurement model. Except the IMM, the other to solve the model mismatch problem is the decision-based method,25 which mainly executes a single model in target tracking while a detector is performed to detect whether the current model matches the true motion. Once the model mismatch is detected, it will immediately switch the model. This hard switching method is more suitable for the mismatch between the measurement model and the tracking scenario, because the fusion between two types of measurement models with inclusion relationship in this article is not needed during the tracking process. In the decision-based method, a detector should be designed to detect the mismatch of the measurement model. The current-decision-based method includes the measurement-residual (MR)-based chi-square detector algorithm, the generalized likelihood ratio (GLR) test detector algorithm, the marginalized likelihood ratio (MLR) detector algorithm, the cumulative sum–based detector (CUSUM) algorithm.26 The MR, the GLR, and the MLR are a class of batch detection algorithms, and CUSUM is a kind of sequential detection algorithm. Although the CUSUM has been put forward for decades for the change-point detection, it has recently been emphasized in theory and application. Cho and Fryzlewicz27 propose the sparsified binary segmentation (SBS) algorithm which aggregates the CUSUM statistics by adding only those that pass a certain threshold. An adaptive CUSUM is applied to a realtime detection framework for false data injection in smart grid networks.28 The CUSUM change-point detection technology is applied to the stationary evaluation method of univariate time series.29 A nonparameter CUSUM is designed for quickest detection of parameter changes in stochastic regression.30 The CUSUM is more appropriate in the target tracking problem discussed in the article, because the measurement vector of radar comes according to the time

International Journal of Distributed Sensor Networks sequence. Therefore, we use the CUSUM to detect the measurement mismatch in this article. The standard CUSUM detector is used for maneuvering target tracking, which is specifically expressed as (

  ) p Z~k jH1 , Z k1 ,0 , Lk = max Lk1 + log  p Z~ jH0 , Z k1

L0 = 0

k

ð41Þ

Decision rules are as follows: 1. 2.

Accept H1 , if Lk  k, and the maneuver time ^ = minfk : Lk  kg; m Continue detecting k = k + 1, if Lk \k. D

where Zk ¼ (Z1 , . . . , Zk ); k is the detection threshold; Z~k is the measurement prediction residual at time k; p(  j  ) is the conditional probability; H0 is the assumption that the target does not maneuver; and H1 is the assumption that the target maneuvers. According to the standard CUSUM detector, the modified CUSUM detector for the measurement mismatch detection in this article is designed as follows  p 9 8 < = p Z~k jzk1 , Z k1 ,0 , Lk = max Lk1 + log  p : ; p Z~k jzk1 , Z k1

L0 = 0 ð42Þ

Decision rules are as follows: 1. 2.

If Lk  k, then zk = zk1 , zk = zk1 , Lk = L0 , and k = k + 1; If Lk \k, then zk = zk1 , zk = zk1 , and k = k + 1.

p where Z~k is the measurement prediction residual at time k; zk1 is a Boolean variable that zk1 = 1 means the measurement model with position and range rate measurement and zk1 = 0 means the measurement model only with position measurement; and ‘‘–’’ is the NOT operator. Based on the modified CUSUM detector, a practical tracking algorithm with range rate measurement is proposed in this article, which is the CUSUM detector based tracking algorithm with range rate measurement, called CUSUM-RR. The algorithm is described as follows:

Step 1: initialization. Set up m00 , P00 , Q0 , R0 , m10 , P10 , Q1 , R1 , z0 , and k, where the superscript ‘‘1’’ corresponds to the measurement model with position and range rate measurement and

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the superscript ‘‘0’’ corresponds to the measurement model only with position measurement. Step 2: parallel filtering. Perform one of the filter algorithms described in section ‘‘Nonlinear tracking algorithms with the position measurement’’ and one of the filter algorithms described in section ‘‘Nonlinear tracking algorithms with the position and range rate measurement’’ to j obtain the variables fZ~k , Skj , mkj , Pkj g1j = 0 at time k, j where Z~k is the measurement prediction residual, Skj is the residual covariance, mkj is the state estimation, and Pkj is the state estimation error covariance. Step 3: calculation likelihood function.

   p  1=2 1 0  1 p p Z~k jj, Z k1 = 2pSkp  exp  Z~pk Skp Z~k ð43Þ 2

where Skp is the position measurement prediction resip 0 dual covariance. If j = 0, then Z~k = Z~k and Skp = S0k . If j=1  1 0 p 1 Z~k = Z~k ð1Þ, Z~k ð2Þ 

S1 ð1, 1Þ = k1 Sk ð2, 1Þ

S1k ð1, 2Þ S1k ð2, 2Þ

ð44Þ  ð45Þ

Step 4: detection the measurement mismatch. Let  p  p p Z~k jzk1 , Z k1 = p Z~k jj = zk1 , Z k1

ð46Þ

 p  p p Z~ jzk1 , Z k1 = p Z~ jj = zk1 , Z k1

ð47Þ

k

k

Then   ) p Z~kp jzk1 , Z k1 ,0 , Lk = max Lk1 + log  p p Z~ jzk1 , Z k1

   k = log 1  pf pm

ð49Þ

where pf is the false alarm probability and pm is the false dismissal probability.

Performance of CUSUM-RR algorithm

p Assume that Z~k obeys the zero mean Gauss distribution

Skp

For the CUSUM-RR algorithm, two tracking algorithms with two measurement models are performed at the same time, respectively, and the state estimation which has the higher accuracy is outputted. The CUSUM-RR algorithm could adaptively choose the measurement model in the different tracking scenarios, which does not depend on the prior information to choose the measurement model artificially, to avoid the measurement mismatch. The threshold value k is calculated by

(

L0 = 0

k

ð48Þ

If Lk  k, then zk = zk1 , zk = zk1 , Lk = L0 ; if Lk \k, then zk = zk1 , zk = zk1 . Output the tracking result mkj and Pkj according to j = zk . Step 5: time updating. Let k = k + 1, return to the parallel filtering step.

The purpose of this section is to demonstrate the performance of the proposed CUSUM-RR algorithm. The SEKF and UCMKF are selected as the filters of CUSUM-RR according to two measurement equations. In the simulation experiment, the CUSUM-RR is compared with the SEKF, UCMKF, and IMM. The filters of IMM also consist of the SEKF and UCMKF with two measurement models, respectively. The target motion parameters are the same as the tracking scenario 3 in section ‘‘The CMKFRR algorithm.’’ We set three position measurement standard deviations (50 m, 0:58), (100 m, 18), and (200 m, 28). For the CUSUM-RR, pf is set as 0.1%, pm is set as 1% and thus k ’ 2. The outputted initial state estimation is from the filter only with the position measurement, z0 = 0. For the IMM, because of the unknown tracking scenario, the initial model probability distribution is set as m0 = ½0:5, 0:50 , and the transition probability matrix is set as p = ½0:9, 0:1; 0:1, 0:9. The other parameters are the same as in section ‘‘The CMKFRR algorithm.’’ The MSEs of CUSUM-RR, IMM, SEKF, and UCMKF for position with 1000 Monte Carlo simulations are shown in Figures 4(a)–6(a). During the initial tracking period, as the MSE of SEKF is the lower than that of UCMKF, the Boolean variable changes from z0 = 0 to z1 = 1 immediately and keeps zk = 1, that means to output the state estimation from the SEKF. Then, the MSE of CUSUM-RR is the same as the SEKF almost. The MSE of the IMM is located between the MSEs of the SEKF and the UCMKF. During the middle tracking period, as the MSE of UCMKF gradually lower than that of SEKF, the MSEs of CUSUMRR and IMM are adaptively toward the MSE of UCMKF. During the late tracking period, the MSE of CUSUM-RR converges to the MSE of UCMKF that is lower than the MSE of SEKF steadily. The IMM with the state fusion mechanism is inevitably influenced by the random noise and the high MSE of state estimation, and the convergence rate is slower than that of the

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International Journal of Distributed Sensor Networks

Figure 4. The results of position measurement standard deviation (50 m, 0:58): (a) MSE for position with 1000 Monte Carlo simulations, (b) log-likelihood ratio of CUSUM-RR at 500th Monte Carlo simulation, and (c) Lk of CUSUM-RR at 500th Monte Carlo simulation.

CUSUM-RR algorithm. The CUSUM-RR with the CUSUM detector can effectively reduce the influence of random noise by setting up the appropriate threshold. The log-likelihood ratio achieved at 500th Monte Carlo simulation experiment is shown in Figures 4(b)– 6(b). The log-likelihood ratio in Figures 4(b)–6(b) is calculated by  p p Z~k jj = 1, Z k1 log  p p Z~k jj = 0, Z k1

ð50Þ

The parameter Lk in the CUSUM detector at 500th Monte Carlo simulation experiment is shown in Figure 4(c)–6(c). From Figures 4(b)–6(b), during the initial tracking period, the log-likelihood ratio far exceeds the threshold value 2. Thus, in Figures 4(c)–6(c), the value of Lk is almost 0 in the corresponding time period. In the middle tracking period, the outputted state

estimation of CUSUM-RR changes between the SEKF and UCMKF, because the log-likelihood ratio fluctuates on the zero line. This is why the MSE of CUSUMRR does not immediately follow the MSE of the UCMKF when the MSE of the UCMKF is lower than the MSE of the SEKF in the middle tracking period with 1000 Monte Carlo simulation experiment. In general, the proposed CUSUM-RR can effectively solve the problem of measurement model mismatch and provides an application path for the tracking target with range rate measurement in practice. However, the disadvantage of CUSUM-RR is that CUSUM-RR needs about two times the amount of calculation than that of SEKF or UCMKF. The CUSUM-RR also cannot obtain the higher accuracy of state estimation given SEKF and UCMKF because of the hard decision. Although the ideal MSE curves of CUSUM-RR should always follow the lower between the MSE curves of SEKF and UCMKF, due to the interference of random

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Figure 5. The results of position measurement standard deviation (100 m, 18): (a) MSE for position with 1000 Monte Carlo simulations, (b) log-likelihood ratio of CUSUM-RR at 500th Monte Carlo simulation, and (c) Lk of CUSUM-RR at 500th Monte Carlo simulation.

noise, the time of CUSUM mismatch detection is not the same in each Monte Carlo simulation. In other words, the detection delay could not been avoided. The value of threshold k is the key factor for controlling the detection delay. However, this parameter is difficult to give a reasonable priori value.

Conclusion and future works The use of range rate measurement may not improve the target tracking performance, as the special tracking algorithm performance is related to the special tracking scenario. However, since the accuracy of state estimation is affected by many other factors besides the nonlinear measurement equation, it is difficult to establish the application relationship between the tracking scenario and measurement equation. In this article, the nonlinear

filtering algorithms which only use the position measurement and the nonlinear filtering algorithms which use position and range rate measurement are surveyed. The performances of these algorithms are compared by simulation experiments. The experimental results demonstrate that the range rate measurement cannot improve target state estimation accuracy in any tracking scenario. In order to solve this problem, this article proposes a practical tracking algorithm with adaptive ability based on the CUSUM detector, that is, the CUSUM-RR algorithm. The algorithm can achieve good accuracy of state estimation in different tracking scenarios, which avoids the performance degradation due to the misuse of measurement equations. The decision-based algorithm prevents the bad fusion between multiple models and the interference of random noise through setting the appropriate threshold value. The simulation results

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Figure 6. The results of position measurement standard deviation (200 m, 28): (a) MSE for position with 1000 Monte Carlo simulations, (b) log-likelihood ratio of CUSUM-RR at 500th Monte Carlo simulation, and (c) Lk of CUSUM-RR at 500th Monte Carlo simulation.

demonstrate that the CUSUM-RR can better follow the state estimation corresponding to the measurement equation matching the tracking scenario and then has a lower MSE than the IMM. In the future works, the nonlinear tracking algorithm with range rate measurement should be studied more deeply to improve the accuracy of state estimation. The particle filter algorithm may be more suitable for the strong nonlinear measurement equation. If the particle filter is used as a filter of CUSUM-RR, it is necessary to redefine the likelihood ratio to design an effective CUSUM detector. The threshold setting of the CUSUM detector should be further studied to ensure a more rapid detection for the measurement model mismatch. For the maneuvering target tracking problem, the CUSUM-RR proposed in this article cannot be applied directly. The strategy is to add an additional

CUSUM detector to detect the target maneuver, or to use the CUSUM-RR as the sub-filter of the IMM. One of the future research directions is to design a composite CUSUM detector by exploring the relationship between range rate measurement and factors involved in the target tracking. The accuracy of angular velocity and normal acceleration also could be improved by the range rate measurement and the state estimation with the CUSUM-RR. In the following study, it is of importance to develop the method of fusing the range rate measurements from multiple Doppler radars in sensor networks to solve the problems of multiple target tracking and recognition. Author’s Note The owner of the fund is the author Haiyan Yang.

Yang et al. Declaration of conflicting interests The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study is supported by the National Natural Science Foundation of China (61472441).

References 1. Ljung L. Asymptotic behavior of the extended Kalman filter as a parameter estimator for linear systems. IEEE T Automat Contr 1979; 24(1): 36–50. 2. Kandepu R, Foss B and Imsland L. Applying the unscented Kalman filter for nonlinear state estimation. J Pr Contr 2008; 18(7): 753–768. 3. Arasaratnam I and Haykin S. Cubature Kalman filters. IEEE T Automat Contr 2009; 54(6): 1254–1269. 4. Lerro D and Bar-Shalom Y. Tracking with debiased consistent converted measurements versus EKF. IEEE T Aero Elec Sys 1993; 29(3): 1015–1022. 5. Bordonaro SV, Willett P and Bar-Shalom Y. Decorrelated unbiased converted measurement Kalman filter. IEEE T Aero Elec Sys 2014; 50(2): 1431–1444. 6. Ito K and Xiong K. Gaussian filters for nonlinear filtering problems. IEEE T Automat Contr 2000; 45(5): 910–927. 7. Chan YT, Hu AGC and Plant JB. A Kalman filter based tracking scheme with input estimation. IEEE T Aero Elec Sys 1979; 2: 237–244. 8. Frencl VB, do Val JBR, Mendes RS, et al. Turn rate estimation using range rate measurements for fast manoeuvring tracking. IET Radar Sonar Nav 2017; 11(7): 1099–1107. 9. Zhu Y and Zhang S. Passive location based on an accurate Doppler measurement by single satellite. In: Proceedings of the radar conference (RadarConf), Seattle, WA, 8–12 May 2017, pp.1424–1427. New York: IEEE. 10. Harris ZJ and Whitcomb LL. Preliminary feasibility study of cooperative navigation of underwater vehicles with range and range-rate observations. In: Proceedings of the OCEANS’15 MTS/IEEE Washington, Washington, DC, 19–22 October 2015, pp.1–6. New York: IEEE. 11. Liu H, Zhou Z and Yang H. Tracking maneuver target using interacting multiple model-square root cubature Kalman filter based on range rate measurement. Int J Distrib Sens Netw 2017; 13(12): 1–11. 12. Kumar RSR, Ramaiah MV and Kumar JR. Performance comparision of a-b-g filter and kalman filter for CA, NCA target tracking using bistatic range and range rate measurements. In: Proceedings of the 2014 international conference on communications and signal processing (ICCSP), Melmaruvathur, India, 3–5 April 2014, pp.1462–1466. New York: IEEE. 13. Bordonaro SV. Converted measurement trackers for systems with nonlinear measurement functions. PhD Dissertation, University of Connecticut, Mansfield, CT, 2015.

15 14. Lei M and Han C. Sequential nonlinear tracking using UKF and raw range-rate measurements. IEEE T Aero Elec Sys 2007; 43(1): 239–250. 15. Jiao L, Pan Q, Liang Y, et al. A nonlinear tracking algorithm with range-rate measurements based on unbiased measurement conversion. In: Proceedings of the 15th international conference on information fusion, Singapore, 9–12 July 2012, pp.1400–1405. New York: IEEE. 16. Bordonaro S, Willett P and Bar-Shalom Y. Consistent linear tracker with converted range, bearing and range rate measurements. IEEE T Aero Elec Sys 2017; 53: 3135–3149. 17. Bar-Shalom Y. Negative correlation and optimal tracking with Doppler measurements. IEEE T Aero Elec Sys 2001; 37(3): 1117–1120. 18. Julier SJ and Uhlmann JK. A general method for approximating nonlinear transformations of probability distributions. Technical report, Robotics Research Group, Department of Engineering Science, University of Oxford, Oxford, 1 November 1996. 19. Sa¨rkka¨ S. Bayesian filtering and smoothing. Cambridge: Cambridge University Press, 2013, pp.88–92. 20. Longbin M, Xiaoquan S, Yiyu Z, et al. Unbiased converted measurements for tracking. IEEE T Aero Elec Sys 1998; 34(3): 1023–1027. 21. Tichavsky P, Muravchik CH and Nehorai A. Posterior Crame´r-Rao bounds for discrete-time nonlinear filtering. IEEE T Signal Proces 1998; 46(5): 1386–1396. 22. Blom HAP and Bar-Shalom Y. The interacting multiple model algorithm for systems with Markovian switching coefficients. IEEE T Automat Contr 1988; 33(8): 780–783. 23. Sheng H, Zhao W and Wang J. Interacting multiple model tracking algorithm fusing input estimation and best linear unbiased estimation filter. IET Radar Sonar Nav 2016; 11(1): 70–77. 24. Hwang I, Seah CE and Lee S. A study on stability of the interacting multiple model algorithm. IEEE T Automat Contr 2017; 62(2): 901–906. 25. Li XR and Jilkov VP. A survey of maneuvering target tracking—part IV: decision-based methods. Proc SPIE 2002; 4728: 511–534. 26. Ru J, Bashi A and Li XR. Performance comparison of target maneuver onset detection algorithms. Proc SPIE 2004; 5428: 419–428. 27. Cho H and Fryzlewicz P. Multiple-change-point detection for high dimensional time series via sparsified binary segmentation. J Roy Stat Soc B 2015; 77(2): 475–507. 28. Huang Y, Tang J, Cheng Y, et al. Real-time detection of false data injection in smart grid networks: an adaptive CUSUM method and analysis. IEEE Syst J 2016; 10(2): 532–543. 29. Bu¨cher A, Fermanian JD and Kojadinovic I. Combining cumulative sum change-point detection tests for assessing the stationarity of univariate time series. Ithaca, NY: Cornell University Library. 30. Konev V and Vorobeychikov S. Quickest detection of parameter changes in stochastic regression: nonparametric CUSUM. IEEE T Inform Theory 2017; 63(9): 5588–5602.