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An Improved Interacting Multiple Model Filtering Algorithm Based on the Cubature Kalman Filter for Maneuvering Target Tracking Wei Zhu, Wei Wang * and Gannan Yuan College of Automation, Harbin Engineering University, No. 145 Nantong Street, Harbin 150001, China; [email protected] (W.Z.); [email protected] (G.Y.) * Correspondence: [email protected]; Tel.: +86-451-8256-8488 Academic Editor: Xue-Bo Jin Received: 9 April 2016; Accepted: 26 May 2016; Published: 1 June 2016

Abstract: In order to improve the tracking accuracy, model estimation accuracy and quick response of multiple model maneuvering target tracking, the interacting multiple models five degree cubature Kalman filter (IMM5CKF) is proposed in this paper. In the proposed algorithm, the interacting multiple models (IMM) algorithm processes all the models through a Markov Chain to simultaneously enhance the model tracking accuracy of target tracking. Then a five degree cubature Kalman filter (5CKF) evaluates the surface integral by a higher but deterministic odd ordered spherical cubature rule to improve the tracking accuracy and the model switch sensitivity of the IMM algorithm. Finally, the simulation results demonstrate that the proposed algorithm exhibits quick and smooth switching when disposing different maneuver models, and it also performs better than the interacting multiple models cubature Kalman filter (IMMCKF), interacting multiple models unscented Kalman filter (IMMUKF), 5CKF and the optimal mode transition matrix IMM (OMTM-IMM). Keywords: target tracking; cubature Kalman filter; unscented Kalman filter; interacting multiple models

1. Introduction Bayes filtering algorithms have been broadly used in target tracking systems [1–4], while a large number of Gaussian approximation filters and Monte Carlo filters have been introduced to solve target tracking problems [5]. Although the particle filter (PF) can deal with non-linear and non-Gaussian systems, the computational complexity always makes its use prohibitive [6]. Gaussian approximation filtering algorithms are more efficient. Among the Gaussian approximation filters, the extended Kalman filter (EKF) has been widely used in nonlinear systems [7,8]. It uses first order Taylor series expansion, which can induce deviations when the systems have higher order and complex non-linear character. In order to reduce the system linearization errors, the unscented Kalman filter (UKF) was introduced to deal with nonlinear systems and it outperforms EKF [9]. Recently, Arasaratnam and Haykin presented the cubature Kalman filter (CKF) based on the spherical-radial cubature rule [10,11]. The CKF has a rigid mathematical proof that is different from the UKF, and both UKF and CKF can approximate the model of the system using specially chosen points. It has been proved that when the dimension of the system is three, the CKF has the same performance as the UKF [12]. Blom and Shalom have proposed the interacting multiple model (IMM) algorithm based on a generalized pseudo-random algorithm to decrease the error of single model algorithm, which will improve the quick response and accuracy of target tracking [13]. The IMM algorithm processes all the models simultaneously and changes different models by checking their weights. It has been proved that the IMM algorithm performs better than any single model algorithm in complex tracking problems [14].

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Many filters have been integrated with the IMM algorithm to enhance the accuracy and quick response of nonlinear target tracking [14–16]. The performance of interacting multiple models unscented Kalman filter (IMMUKF) is compared with the interacting multiple models extended Kalman filter (IMMEKF), and the results show that IMMUKF performs better than IMMEKF in bearings-only maneuvering tracking problems [15]. However, when the dimension of the system is more than three, the weights of UKF are negative which will cause the divergence of the filter [12,17]. Then the CKF is introduced in IMM to overcome the issue, and the new algorithm can reduce the computational complexity and improve the accuracy of the filter [17,18]. Lee, Motai and Choi have proposed the multichannel interacting multiple model estimator (MC-IMME) to improve the overall performance of the traditional particle filter, ensemble KF and IMME [19]. The multiple delta quaternion extended Kalman filter is proposed in [20] for head orientation prediction. The proposed multiple model delta quaternion (DQ) (MMDQ) filters integrate constant velocity (CV) and constant acceleration (CA) DQ filters in an IMME framework, and the experimental results show that the new filter performs better than DQ-EKF albeit with increased computation. In [21], the authors proposed a sensor fusion algorithm which introduces dynamic noise covariance matrix into interacting multiple models. The proposed filter is more accurate than the Kalman filter when there are abrupt changes in the path of the vehicle. In order to improve the accuracy of the traditional IMM algorithm, the optimal mode transition matrix IMM (OMTM-IMM) algorithm was proposed in [22]. The OMTM-IMM utilizes the linear minimum variance theory to minimize the error of the initial state and the simulation results show that it outperforms the traditional IMM when the sojourn times of the system are not known. In this paper, the interacting multiple models five degree cubature Kalman filter (IMM5CKF) based on a five degree cubature Kalman filter and IMM algorithm is proposed to improve the tracking accuracy, model estimation accuracy and quick response of target tracking algorithms. The negative weights of 5CKF go to 0 when the system dimensions go to 8, so 5CKF is more stable than UKF [23]. The simulation results show that the IMM5CKF exhibits better accuracy and switching sensitivity performance than IMMCKF, IMMUKF, 5CKF and OMTM-IMM. The remainder of the paper is organized as follows: the high degree of cubature Kalman filter is analyzed in Section 2. In Section 3, IMM5CKF is derived. The performance of the target tracking algorithms are compared in a benchmarked target tracking problem in Section 4. Conclusions are given in Section 5. 2. Five Degree Cubature Kalman Filter The five degree cubature Kalman filter is proposed to improve the accuracy of the traditional Cubature Kalman Filter [23]. It chooses deterministic odd points to transfer the nonlinear functions to calculate the posterior mean and covariance of the system. Supposing state variables x “ N px, Pq, where x is mathematical expectation of x, P is the covariance of x. The five degree Cubature Kalman Filter includes two steps, time update and measurement update. 2.1. Time Update (1)

Factorize The Cholesky decomposition of Pk´1|k´1 is calculated as:

(2)

Pk´1|k´1 “ Sk´1|k´1 SkT´1|k´1

(1)

X0i,k´1|k´1 “ xˆ k´1|k´1 a X1i,k´1|k´1 “ pn ` 2qSk´1|k´1 ei ` xˆ k´1|k´1 a X2i,k´1|k´1 “ ´ pn ` 2qSk´1|k´1 ei ` xˆ k´1|k´1

(2)

Evaluate the cubature points:

(3) (4)

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a pn ` 2qSk´1|k´1 s` i ` xˆ k´1|k´1 a X4i,k´1|k´1 “ ´ pn ` 2qSk´1|k´1 s` i ` xˆ k´1|k´1 a X5i,k´1|k´1 “ pn ` 2qSk´1|k´1 s´ i ` xˆ k´1|k´1 a X6i,k´1|k´1 “ ´ pn ` 2qSk´1|k´1 s´ i ` xˆ k´1|k´1 + #c ˘ 1` ` e ` el : j ă l, j, l “ 1, 2, ..., n si “ 2 j + #c ˘ 1` ´ si “ e ´ el : j ă l, j, l “ 1, 2, ..., n 2 j X3i,k´1|k´1 “

w0 “ w1 “ w2 “

(3)

2 n`2

4´n

pn ` 2q2

(6) (7) (8) (9)

(10) (11)

, i “ 1, 2, ¨ ¨ ¨ n

(12)

, i “ 1, 2, ..., npn ´ 1q{2

(13)

2pn ` 2q2 2

(5)

where w0 , w1 and w2 is the weights of the cubature points, ei is the unit vector. Evaluate the propagated cubature points The sample points are obtained by propagating the cubature points through the state equation as:

(4)

´ ¯ ˚ X0i,k “ f X 0i,k ´ 1 | k ´ 1 | k ´1

(14)

´ ¯ ˚ X1i,k |k´1 “ f X1i,k´1|k´1 ´ ¯ ˚ X2i,k “ f X 2i,k ´ 1 | k ´ 1 | k ´1 ´ ¯ ˚ X3i,k “ f X 3i,k´1|k´1 | k ´1 ´ ¯ ˚ X4i,k |k´1 “ f X4i,k´1|k´1 ´ ¯ ˚ X5i,k “ f X 5i,k´1|k´1 | k ´1 ´ ¯ ˚ X6i,k |k´1 “ f X6i,k´1|k´1

(15) (16) (17) (18) (19) (20)

Estimate the predicted points State prediction xˆ k|k´1 is then calculated by the weighted combination of sample points as: ˚ ˚ ˚ xˆ k|k´1 “ w0 X0i,k |k´1 ` w1 X1 ` w2 X2

X1˚ “

n ÿ

˚ ˚ pX1i,k |k´1 `X2i,k|k´1 q

(21) (22)

i “1

X2˚ “

npnÿ ´1q{2

˚ ˚ ˚ ˚ pX3i,k |k´1 `X4i,k|k´1 ` X5i,k|k´1 ` X6i,k|k´1 q

(23)

i “1

(5)

Estimate the predicted error covariance: ˚ ˚T Pk|k´1 “ w0 Xoi,k |k´1 Xoi,k|k´1 ` P1 ` P2

(24)

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P1 “ w1

n ´ ÿ

˚ ˚T ˚ ˚T X1i,k |k´1 X1i,k|k´1 ` X2i,k|k´1 X2i,k|k´1

¯ (25)

i “1

P2 “ w2

n p nř ´1q{2 ´ i “1

˚ ˚T ˚ ˚T ˚ ˚T ˚ ˚T X3i,k |k´1 X3i,k|k´1 ` X4i,k|k´1 X4i,k|k´1 ` X5i,k|k´1 X5i,k|k´1 ` X6i,k|k´1 X6i,k|k´1

¯

(26)

2.2. Measurement Update (1)

(2)

(3)

Factorize: Pk|k´1 “ Sk|k´1 SkT|k´1

(27)

X0i,k|k´1 “ xˆ k|k´1 a X1i,k|k´1 “ pn ` 2qSk|k´1 ei ` xˆ k|k´1 a X2i,k|k´1 “ ´ pn ` 2qSk|k´1 ei ` xˆ k|k´1 a X3i,k|k´1 “ pn ` 2qSk|k´1 s` i ` xˆ k|k´1 a X4i,k|k´1 “ ´ pn ` 2qSk|k´1 s` i ` xˆ k|k´1 a X5i,k|k´1 “ pn ` 2qSk|k´1 s´ i ` xˆ k|k´1 a X6i,k|k´1 “ ´ pn ` 2qSk|k´1 s´ i ` xˆ k|k´1

(28)

Evaluate the cubature points:

(29) (30) (31) (32) (33) (34)

Evaluate the propagated cubature points

The sample points are obtained by propagating the cubature points through the observation equation: ´ ¯ Z0i,k|k´1 “ h X0i,k|k´1 (35) ´ ¯ Z1i,k|k´1 “ h X1i,k|k´1 (36) ´ ¯ Z2i,k|k´1 “ h X2i,k|k´1 (37) ´ ¯ Z3i,k|k´1 “ h X3i,k|k´1 (38) ´ ¯ Z4i,k|k´1 “ h X4i,k|k´1 (39) ´ ¯ Z5i,k|k´1 “ h X5i,k|k´1 (40) ´ ¯ Z6i,k|k´1 “ h X6i,k|k´1 (41) (4)

Estimate the predicted measurement: zˆk|k´1 “ w.0 Z0i,k|k´1 ` w1 Z1 ` w2 Z2 Z1 “

n ´ ÿ

(42)

¯ Z1i,k|k´1 ` Z2i,k|k´1

(43)

i“1

Z2 “

npnÿ ´1q{2 ´ i “1

¯ Z3i,k|k´1 ` Z4i,k|k´1 ` Z5i,k|k´1 ` Z6i,k|k´1

(44)

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(5)

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Estimate the innovation covariance matrix: T T Pzz,k|k´1 “ w0 Z0i,k|k´1 Z0i,k |k´1 ` Pzz1 ` Pzz2 ´ zˆ k|k´1 zˆ k|k´1 ` Rk

Pzz1 “ w1

n ´ ÿ

T T Z1i,k|k´1 Z1i,k |k´1 ` Z2i,k|k´1 Z2i,k|k´1

(45)

¯ (46)

i“1

Pzz2 “ w2

npnř ´1q2 ´ i “1

(6)

T T T T Z3i,k|k´1 Z3i,k ` Z4i,k|k´1 Z4i,k ` Z5i,k|k´1 Z5i,k ` Z6i,k|k´1 Z6i,k | k ´1 | k ´1 | k ´1 | k ´1

¯

(47)

Estimate the cross-covariance matrix: T T Pxz,k|k´1 “ w0 X0i,k|k´1 Z0i,k |k´1 ` Pxz1 ` Pxz2 ´ xˆ k|k´1 zˆ k|k´1

Pxz1 “ w1

n ´ ÿ

T T X1i,k|k´1 Z1i,k |k´1 ` X2i,k|k´1 Z2i,k|k´1

(48)

¯ (49)

i “1

Pxz2 “ w2

n p nř ´1q{2 ´ i “1

(7) (8)

(9)

T T T T X3i,k|k´1 Z3i,k ` X4i,k|k´1 Z4i,k ` X5i,k|k´1 Z5i,k ` X6i,k|k´1 Z6i,k | k ´1 |k´1 | k ´1 | k ´1

¯

(50)

Estimate the Kalman gain: ´1 Wk “ Pxz,k|k´1 Pzz,k | k ´1

(51)

´ ¯ xˆ k|k “ xˆ k|k´1 ` Wk zk ´ zˆk|k´1

(52)

Estimate the updated state:

Estimate the corresponding error covariance: Pk|k “ Pk|k´1 ´ Wk Pzz,k|k´1 WkT

(53)

3. IMM High Degree Cubature Kalman Filter In the paper, the proposed IMM5CKF includes the merits of the 5CKF algorithm and IMM algorithm. The main character of IMM5CKF is that it calculates the state distribution and error covariance matrix by choosing an odd number of special cubature points with equal weights, and the negative weights go to 0 when the dimension of the system goes to 8. This means that it is more stable than UKF. The IMM-5CKF algorithm includes input integration, five degree cubature Kalman filter, model probability update and output integration. The structure diagram is shown in Figure 1. The filtering processes are shown in the following subsection. 3.1. Input Integration i{ j

uk´1|k´1 “ 0j Xˆ k´1|k´1 “

0j Pk´1|k´1

r ÿ

“

i{ j uk´1|k´1

" Pki ´1|k´1

”

ÿ

pij uik´1

(54)

Cj

i{ j Xˆ ki ´1|k´1 uk´1|k´1

0j ` Xˆ ki ´1|k´1 ´ Xˆ k´1|k´1

ı”

(55) 0j Xˆ ki ´1|k´1 ´ Xˆ k´1|k´1

ıT *

(56)

i“1

where Cj “

r ř i “1

i{ j

pij uik´1 , uk´1|k´1 is the conditional probability of model i at time k ´ 1, uik´1 is the

0j 0j probability of model i at time k ´ 1, Xˆ k´1|k´1 is the initial mean value of model j, Pk´1|k´1 is the

Wk  Pxz , k |k 1 Pzz,1k |k 1

(51)

xˆk |k  xˆk |k 1  Wk  zk  zˆk | k 1 

(52)

(8) Estimate the updated state: Sensors 2016, 16, 805

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(9) Estimate the corresponding error covariance:

WkT model i at time k ´ 1, Pi (53)is the initial error covariance, Xˆ ki ´1|k´1 is the Pestimated value k | k  Pk | k 1  W k Pzz , k | k 1of k´1|k´1 relative covariance.

Xˆ k11|k 1

Pk11|k 1

Pkr1|k 1

Xˆ kr1|k 1

...

Input Integrate

Xˆ k011|k 1

Pk011|k 1

...

Xˆ k0r1|k 1 Pk0r1|k 1

...

5CKF

Z k 5CKF

L1k

L rk

...

 k |k Mode Probability Update

k|k

Output Integrate

Xˆ k1 | k

Pk1| k

Xˆ kr | k

...

Pk r| k

Pk | k

Figure 1. IMM-5CKF structure diagram. Figure 1. IMM-5CKF structure diagram.

3.2. Five Degree Cubature Kalman Filtering The mixed initial value and measure value (z) are the input of each filter at time k. Then the new j j j j state vector Xˆ k|k , error covariance Pk|k , predicted measure value zk|k´1 and residual vk can be got from the 5CKF. j The likelihood value Lk is: j Lk

´ “N

j j z; zk|k´1 , vk

1

¯

ı˙ ıT ´ ¯´1 ” 1” j j j zk ´ zˆk|k´1 ¨ exp ´ zk ´ zˆk|k´1 Vk j 2 ˆ

“b 2πVk

j

(57)

j

where Vk is the associated covariance of residual vk . 3.3. Model Probability Update It has been known that if the filter model matches with the actual model, the filter residual is zero and the variance v pkq is Gaussian White Noise. Then the model probability can be updated by Equation (58): j L Cj j uk “ nm k (58) ř j Lk Cj j “1

3.4. Output Integration The probabilities of the model are integrated with the estimated value of each filter based on the given weights. The output of IMM-5CKF can be calculated as: Xˆ k|k “

r ÿ

j

j

Xk |k u k

(59)

" ” ı” ıT * j j j j uk Pk|k ` Xˆ k|k ´ Xˆ k|k Xˆ k|k ´ Xˆ k|k

(60)

j “1

Pk|k “

r ÿ j “1

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4. Results and Discussion In this section, the IMM-5CKF is compared with IMMCKF, IMMUKF, 5CKF and OMTM-IMM in . . T a benchmark target tracking scenario. The state variable at time k is Xk “ rx, x, y, ys , where x and y . . are the position variables, x and y are the velocity variables. The coordinated turn model is: » sinpωT q pcospωT q´1q fi 1 0 ω ω — 0 cos pωTq 0 ´sin pωTq ffi ffi — (61) F2 “ — ffi sinpωT q fl – 0 p1´cospωTqq 1 ω

0

ω

sin pωTq

0

cos pωTq

where ω is the turn rate and T is the sampling interval. The right turn rate is defined as ´3˝ , and the left turn rate is 3˝ . The measurement equation of the system is: « Z“

1 0

0 0

0 1

0 0

ff `R

(62)

where R is the measurement noise of the system. The initial state X0 “ r1000 m, 200 m/s, 1000 m, 200 m/ssT , initial associate covariance is P0 “ diag pr1000, noise Q „ N p0, qq, q “ r10, 0; 0, 10s, process noise weight ” 10, 1000, 10sq, process ı 2

2

matrix is G “ T2 , 0; T, 0; 0, T2 ; 0, T . The measure noise R „ N p0, rq, with r “ diag pr20, 0.1sq. The simulation time simTime “ 100s, the step time T “ 1 s. The target turns right during 20 s „ 40 s, turns left during 60 s „ 80 s, and maintains uniform motion during the other time. The model transition probability is: »

fi 0.9 0.05 0.05 — ffi pij “ – 0.1 0.8 0.1 fl 0.05 0.15 0.8

(63)

The root-mean square error (RMSE) of position and velocity are used to contrast the performance of the filtering algorithms. The RMSE defined in state vector X at k is: g f n f ř f e RMSE “

n“1

1 k

k ř

2

pXk,n ´ Xˆ k,n q

k “1

n

(64)

Figure 2 shows the target trajectory after 100 Monte Carlo simulations, from which it can be found that all the algorithms could track the trajectory of the target. Figures 3 and 4 show that the estimated RMSEs in position and velocity of IMM5CKF, IMMCKF, IMMUKF, 5CKF and OMTM-IMM respectively. From Figures 3 and 4, it can be found that all the algorithms exhibit stable characteristics, and there are no error divergence during the simulation time. In addition, the results show that the RMSEs of IMM5CKF are less than those of the other algorithms and the performance is more stable. In Figures 3 and 4, the RMSEs of 5CKF is the largest, which means a single model algorithm cannot adapt to changeable target tracking problems. In [21], authors had proved that OMTM-IMM performs better than traditional IMM algorithm. In Figures 3 and 4, it can be seen that the accuracy of OMTM-IMM is better than 5CKF, but worse than that of the other algorithms which are based on improved nonlinear filters.

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is more stable. In Figures 3 and 4, the RMSEs of 5CKF is the largest, which means a single model is more stable. In Figures 3 and 4, the RMSEs of 5CKF is the largest, which means a single model algorithm cannot adapt to changeable target tracking problems. In [21], authors had proved that algorithm cannot adapt to changeable target tracking problems. In [21], authors had proved that OMTM-IMM performs better than traditional IMM algorithm. In Figures 3 and 4, it can be seen that OMTM-IMM performs better than traditional IMM algorithm. In Figures 3 and 4, it can be seen that the accuracy of OMTM-IMM is better than 5CKF, but worse than that of the other algorithms which Sensors 2016, 16, 805 8 of 12 the accuracy of OMTM-IMM is better than 5CKF, but worse than that of the other algorithms which are based on improved nonlinear filters. are based on improved nonlinear filters. Trajectory Trajectory

10000 10000

y/m y/m

9000 9000 8000 8000

y/m y/m

7000 7000

6220 6220 6200 6200 6180 6180 6160 6160

8000 8020 8040 8060 8000 8020 8040 8060

6000 6000 5000 5000 4000 4000 3000 3000 2000 2000 1000 1000 0 0

t=1 t=1

True True

0.5 0.5

1 1

IMM-5CKF IMM-5CKF

x/m x/m

IMM-UKF IMM-UKF

1.5 1.5

2 2

IMM-CKF IMM-CKF

2.5 2.5 4 x 10 4 x 10 OMTM-IMM OMTM-IMM

5CKF 5CKF

X-RMSE/m X-RMSE/m

Figure Figure 2. 2. Target Target Trajectory. Trajectory. Figure 2. Target Trajectory. 30 30 25 25 20 20 15 15 10 10 5 5 0 00 0

(a) (a)

10 10

20 20

30 30

40 40

10 10

20 20

30 30

40 40

Y-RMSE/m Y-RMSE/m

25 25

50

60 60

70 70

80 80

90 90

100 100

50

60 60

70 70

80 80

90 90

100 100

50 t/s t/s (b) (b) (b) (b)

20 20 15 15 10 10 5 5 0 00 0

IMMUKF IMMUKF

IMMCKF IMMCKF

50 t/s t/s

IMM5CKF IMM5CKF

5CKF 5CKF

OMTM-IMM OMTM-IMM

Figure 3. RMSEs of (a) X-position and (b) Y-position. Y-position. Figure 3. RMSEs of (a) X-position and (b) Y-position.

The RMSEs of the IMM5CKF, IMMCKF, IMMUKF, 5CKF and OMTM-IMM are shown in Table 1. The data shows that the tracking accuracy of IMM-5CKF is better than that of the other algorithms with increasing computational load.

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6

XV-RMSE/m/s XV-RMSE/m/s

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4

(a)

6 2 4 0

0 2

15 0

YV-RMSE/m/s YV-RMSE/m/s

9 of 12 9 of 12

10

20

30

40

50

60

70

80

90

100

60

70

80

90

100

60

70

80

90

100

t/s (b)

0

10

20

30

40

50

t/s (b)

10 15 5 10 0 0 5

10

20

30

50

t/s IMMUKF

0

40

0

IMMCKF

10

20

30

IMM5CKF 40

50

5CKF 60

OMTM-IMM

70

80

Figure 4. RMSEs of (a) X-velocity and (b) Y-velocity. t/s IMMUKF

IMMCKF

IMM5CKF

5CKF

90

100

OMTM-IMM

The RMSEs of the IMM5CKF, IMMCKF, IMMUKF, 5CKF and OMTM-IMM are shown in Table 1. The data shows that the tracking is better than that of the other algorithms Figure 4.accuracy (b) Y-velocity. RMSEs ofof(a)IMM-5CKF X-velocity and Y-velocity. with increasing computational load. The RMSEs of theTable IMM5CKF, IMMCKF, 5CKF and OMTM-IMM 1. The RMSEs of theIMMUKF, different target tracking algorithms. are shown in Table 1. Table 1. The RMSEs of the target algorithms. The data shows that the tracking accuracy of different IMM-5CKF is tracking better than that of the other algorithms RMSE IMM5CKF IMMCKF IMMUKF 5CKF OMTM-IMM with increasing computational load. RMSE IMM5CKF IMMCKF IMMUKF 5CKF OMTM-IMM RMSE_X 2.6675 2.4847 2.5392 27.4975 RMSE_X (m)(m) 2.6675 2.4847 2.5392 27.4975 Table 1. The RMSEs of the different target tracking algorithms. RMSE_X_V (m/s) 1.1245 1.8306 1.8930 5.7001 RMSE_X_V (m/s) 1.1245 1.8306 1.8930 5.7001 RMSE_Y (m) 2.5255 2.8534 3.0362 21.7947 RMSE_Y (m) 2.5255 2.8534 3.0362 21.7947 RMSE (m/s) IMM5CKF IMMCKF IMMUKF 12.2331 5CKF RMSE_Y_V 1.4972 2.9201 2.8488 RMSE_Y_V (m/s) 1.4972 2.9201 2.8488 12.2331 RMSE_X 2.6675 2.4847 2.5392 27.4975 Time(m) (s) 14.9726 7.2549 7.3785 5.3101 Time (s)(m/s) 14.9726 7.2549 7.3785 5.3101 RMSE_X_V 1.1245 1.8306 1.8930 5.7001

5.6211 5.6211 3.2510 3.2510 6.0674 6.0674 OMTM-IMM 4.9938 4.9938 5.6211 6.0314 6.0314 3.2510 RMSE_Y (m) 2.5255 2.8534 3.0362 21.7947 6.0674 Figures 5–7 show probabilities of IMMCKF, model1.4972 probabilities2.9201 of IMM5CKF, IMM5CKF, IMMCKF,IMMUKF IMMUKFand andOMTM-IMM. OMTM-IMM. RMSE_Y_V (m/s)the model 2.8488 12.2331 4.9938 Time (s) 14.9726 7.2549 7.3785 5.3101 6.0314 1

0.9 model probabilities of IMM5CKF, IMMCKF, IMMUKF and OMTM-IMM. Figures 5–7 show the 0.8 1

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Figures Figures 5–7 5–7 demonstrate demonstrate that that IMM5CKF, IMM5CKF, IMMCKF, IMMCKF, IMMUKF IMMUKF and and OMTM-IMM OMTM-IMM can can effectively effectively Figures 5–7 demonstrate that IMM5CKF, IMMCKF, IMMUKF and OMTM-IMM can effectively track the model characteristics of a maneuvering target. It is also found that the IMM-5CKF track the model characteristics of a maneuvering target. It is also found that the IMM-5CKF can can track the the model characteristics of a maneuvering target. It the is also foundstate that the IMM-5CKF cantcapture  20 capture kinematics of maneuvering in time once motion changes at time capture the kinematics of maneuvering in time once the motion state changes at time t  20 ss ,, the kinematics of maneuvering in time once the motion state changes at time t “ 20 s, t “ 40 s, t “ 60 s tt  ss ,, tt  60 ss and tt  .. The results show that IMM5CKF has advantage  40 40  80 80 ssresults The simulation simulation show has thatan IMM5CKF has an an obvious obvious and t“ 80 s.60 The and simulation show thatresults IMM5CKF obvious advantage overadvantage the other over the other algorithms in target tracking problems. over the other algorithms in target tracking problems. algorithms in target tracking problems. 5. Conclusions 5. Conclusions Conclusions In this this paper, IMM5CKF is to the accuracy, model In is proposed to enhance the tracking accuracy, model estimation accuracy thispaper, paper,IMM5CKF IMM5CKF is proposed proposed to enhance enhance the tracking tracking accuracy, model estimation estimation accuracy and response sensitivity of nonlinear maneuvering target tracking problems. The algorithm and response of nonlinear maneuvering target tracking problems. algorithm accuracy and sensitivity response sensitivity of nonlinear maneuvering target tracking The problems. Theintroduces algorithm five Kalman into interacting multiple which aintroduces five degree aacubature Kalmancubature filter into interacting multiple which simultaneously disposes of introduces five degree degree cubature Kalman filter filter intomodels interacting multiple models models which simultaneously disposes of all the models through a Markov Chain. A classical target tracking all the models through a Markov Chain. A classical target trackingChain. problem utilized target to demonstrate simultaneously disposes of all the models through a Markov A isclassical tracking problem is to demonstrate the can improve the response that the IMM5CKF improvethat the quick response sensitivity of target tracking algorithm, and problem is utilized utilizedcan toindeed demonstrate that the IMM5CKF IMM5CKF can indeed indeed improve the quick quick response sensitivity of tracking it accurate IMMUKF, it exhibits more accurate thanalgorithm, IMMCKF, and IMMUKF, CKFmore and OMTM-IMM. In our future research,CKF the sensitivity of target target tracking algorithm, and it exhibits exhibits more accurate than than IMMCKF, IMMCKF, IMMUKF, CKF and OMTM-IMM. In our future research, the study may focus on multisensor navigation and and OMTM-IMM. In our future research, the study may focus on multisensor navigation and positioning positioning systems. systems. The The proposed proposed algorithm algorithm should should be be suitable suitable for for the the complex complex real real environments environments according to the analysis. according to the analysis.

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study may focus on multisensor navigation and positioning systems. The proposed algorithm should be suitable for the complex real environments according to the analysis. Acknowledgments: The authors would like to thank all the reviewers for helping improve the clarity of the presentation of this paper. This work is supported by the National Natural Science Foundation (61571148), China Postdoctoral Science Foundation Grant (2014M550182), Heilongjiang Postdoctoral Special Fund (LBH-TZ0410) and Innovation of Science and Technology Talents in Harbin (2013RFXXJ016). Author Contributions: Wei Zhu and Wei Wang conceived, designed and performed the experiments; Wei Zhu and Wei Wang analyzed the data; Gannan Yuan contributed the analysis tools; Wei Zhu wrote the paper. Conflicts of Interest: The authors declare no conflict of interest.

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