# Maneuvering Target Tracking Algorithm Based on Interacting Multiple Models

Authors: Gannan Yuan; Wei Zhu; Wei Wang; Bo Yin
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Aiming at improving the accuracy and quick response of the filter in nonlinear maneuvering target tracking problems, the Interacting Multiple Models Cubature Information Filter (IMMCIF) is proposed. In IMMCIF, the Cubature Information Filter (CIF) is brought into Interacting Multiple Model (IMM), which can not only improve the accuracy but also enhance the quick response of the filter. CIF is a multisensor nonlinear filtering algorithm; it evaluates the information vector and information matrix rather than state vector and covariance, which can reduce the error of nonlinear filtering algorithm. IMM disposes all the models simultaneously through Markov Chain, which can enhance the quick response of the filter. Finally, the simulation results show that the proposed filter exhibits fast and smooth switching when disposing different maneuver models; it performs better than the IMMCKF and IMMUKF on tracking accuracy.


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## Body

## 1. Introduction

Nowadays, target tracking has broad application in different areas, such as radar tracking, aircraft surveillance, and vehicle navigation [1–4]. In order to improve the tracking precision when the motion of the target is complex, many filtering algorithms have been proposed based on specifically physical model [5–7]. Blom and Bar-Shalom propose the IMM algorithm based on generalized pseudorandom algorithm to depress the error of the single model algorithm [8]. The IMM algorithm processes all the models simultaneously and then switches to different models by checking the weight of each filter [9]. Analysis shows that IMM algorithm is more robust and more exact than the single model algorithm when tracking the maneuvering target [10, 11].In order to enhance the performance of IMM algorithm, many nonlinear filters have been designed. For example, Interacting Multiple Models Extended Kalman Filter (IMMEKF) and Interacting Multiple Models Unscented Kalman Filter (IMMUKF) [4, 10, 12–14] are proposed as IMM-based nonlinear filtering algorithm. The results show that, in maneuvering targets tracking problems, IMMUKF performs better than IMMEKF in stability and precision [11]. It is because Extended Kalman Filter (EKF) carries on the filtering procession by linearizing the state equation, which inevitably induces the linearization error even divergence of filter. However, when Unscented Kalman Filter (UKF) [15] is used in higher order system, the weights of the sigma points will turn into negative which will result in the divergence of the filter. In order to overcome these issues and improve the performance of filter, Cubature Kalman Filter (CKF) is introduced in IMM to track the maneuvering target [14, 16]. And analysis shows that the IMMCKF exhibits better performance than IMMUKF and IMMEKF in enhancing the accuracy and reducing the computational complexity [14]. In this paper, the Interacting Multiple Models Cubature Information Filter (IMMCIF) based on IMM and the CIF [17, 18] is proposed to improve the tracking accuracy and quick response of maneuvering target. In IMMCIF, the state estimation variables including position and velocity of different models are disposed simultaneously through Markov Chain; then the combined state estimate and covariance of each filter are fed back when the cycle completes. The proposed filter evaluates the information vector and information matrix instead of the state vector and covariance; it propagates in the information space which makes the filter easy to initialize and more precise [17]. Therefore, the accuracy of the IMM is improved, and the novel algorithm in this paper is a promising approach for maneuvering target tracking. The numerical simulation results show that the IMMCIF exhibits better than IMMCKF and IMMUKF on accuracy, robustness, and switching response.The remainder of this paper is organized as follows. In Section2, the Cubature Information Filter is briefly reviewed. The whole procession of IMMCIF used in target tracking problem is derived in Section 3. In Section 4, the performance of IMMCIF is better than IMMCKF and IMMUKF, they are compared in a benchmarked target tracking problem, and the results are analyzed. Conclusions are given in Section 5.


## 2. Cubature Information Filter

The CIF is different with conventional filtering algorithms which are based on Kalman filter. It estimates information state vector and information matrix instead of state vector and covariance matrix, from which the measurement update is simpler than CKF.


### 2.1. Time Update

Consider(1)Yk-1∣k-1-1=Sk-1∣k-1Sk-1∣k-1T,where Yk-1∣k-1 is information matrix. Sk-1∣k-1 can be obtained by factorizing the inverse information matrix, which is used to evaluate the cubature points.Then the cubature point can be obtained as follows:(2)Xi,k-1∣k-1=Sk-1∣k-1ξi+x^k-1∣k-1,Xi,k-1∣k-1∗=fXi,k-1∣k-1,uk-1,where ξi is the matrix with a set of unit vector as shown in (3)n10⋮0,…,0⋮01,-10⋮0,…,0⋮0-1.The predicted state and predicted error covariance can be obtained based on CKF:(4)x^k∣k-1=12n∑i=12nXi,k-1∣k-1∗,Pk∣k-1=12n∑i=12nXi,k-1∣k-1∗Xi,k-1∣k-1∗T-x^k∣k-1x^k∣k-1T+Qk-1,where i=1,2,…,2n, n is the dimension of state vector xk.From (4), the predicted information matrix and the predicted information state vector can be derived:(5)Yk∣k-1=Pk∣k-1-1,(6)yk∣k-1=Pk∣k-1-1x^k∣k-1=Yk∣k-1x^k∣k-1=12nYk∣k-1∑i=12nXi,k-1∣k-1∗.


### 2.2. Measurement Update

Consider(7)Yk∣k-1-1=Sk∣k-1Sk∣k-1T,where Sk∣k-1 can be obtained by factorizing the inverse predicted matrix Yk∣k-1.The cubature points are evaluated as(8)χi,k∣k-1=Sk∣k-1ξi+x^k∣k-1.The propagated cubature points are evaluated as(9)Zi,k∣k-1=hχi,k∣k-1.The predicted measurement is estimated:(10)z^k∣k-1=12n∑i=12nZi,k∣k-1.The cross-covariance matrix is estimated:(11)Pxz,k∣k-1=12n∑i=12nχi,k∣k-1Zi,k∣k-1T-x^k∣k-1z^k∣k-1T.Then information state distribution and information matrix can be obtained from (5) and (11):(12)ik=Yk∣k-1Pxz,k∣k-1Rk-1vk+Pxz,k∣k-1TYk∣k-1Tx^k∣k-1,Ik=Yk∣k-1Pxz,k∣k-1Rk-1Pxz,k∣k-1TYk∣k-1T,where vk=z-z^k∣k-1.From (5) and (12), the updated information matrix and updated information can be given, respectively:(13)Yk∣k=Yk∣k-1+Ik,(14)yk∣k=yk∣k-1+ik.Then the estimated state vectorx^k∣k and estimated error covariance Pk∣k can be derived as follows:(15)x^k∣k=Yk∣k-1yk∣k,Pk∣k=Yk∣k-1In,where In is the n dimensional unit matrix.


## 2.1. Time Update

Consider(1)Yk-1∣k-1-1=Sk-1∣k-1Sk-1∣k-1T,where Yk-1∣k-1 is information matrix. Sk-1∣k-1 can be obtained by factorizing the inverse information matrix, which is used to evaluate the cubature points.Then the cubature point can be obtained as follows:(2)Xi,k-1∣k-1=Sk-1∣k-1ξi+x^k-1∣k-1,Xi,k-1∣k-1∗=fXi,k-1∣k-1,uk-1,where ξi is the matrix with a set of unit vector as shown in (3)n10⋮0,…,0⋮01,-10⋮0,…,0⋮0-1.The predicted state and predicted error covariance can be obtained based on CKF:(4)x^k∣k-1=12n∑i=12nXi,k-1∣k-1∗,Pk∣k-1=12n∑i=12nXi,k-1∣k-1∗Xi,k-1∣k-1∗T-x^k∣k-1x^k∣k-1T+Qk-1,where i=1,2,…,2n, n is the dimension of state vector xk.From (4), the predicted information matrix and the predicted information state vector can be derived:(5)Yk∣k-1=Pk∣k-1-1,(6)yk∣k-1=Pk∣k-1-1x^k∣k-1=Yk∣k-1x^k∣k-1=12nYk∣k-1∑i=12nXi,k-1∣k-1∗.


## 2.2. Measurement Update

Consider(7)Yk∣k-1-1=Sk∣k-1Sk∣k-1T,where Sk∣k-1 can be obtained by factorizing the inverse predicted matrix Yk∣k-1.The cubature points are evaluated as(8)χi,k∣k-1=Sk∣k-1ξi+x^k∣k-1.The propagated cubature points are evaluated as(9)Zi,k∣k-1=hχi,k∣k-1.The predicted measurement is estimated:(10)z^k∣k-1=12n∑i=12nZi,k∣k-1.The cross-covariance matrix is estimated:(11)Pxz,k∣k-1=12n∑i=12nχi,k∣k-1Zi,k∣k-1T-x^k∣k-1z^k∣k-1T.Then information state distribution and information matrix can be obtained from (5) and (11):(12)ik=Yk∣k-1Pxz,k∣k-1Rk-1vk+Pxz,k∣k-1TYk∣k-1Tx^k∣k-1,Ik=Yk∣k-1Pxz,k∣k-1Rk-1Pxz,k∣k-1TYk∣k-1T,where vk=z-z^k∣k-1.From (5) and (12), the updated information matrix and updated information can be given, respectively:(13)Yk∣k=Yk∣k-1+Ik,(14)yk∣k=yk∣k-1+ik.Then the estimated state vectorx^k∣k and estimated error covariance Pk∣k can be derived as follows:(15)x^k∣k=Yk∣k-1yk∣k,Pk∣k=Yk∣k-1In,where In is the n dimensional unit matrix.


## 3. Interacting Multiple Models Cubature Information Filtering Algorithm

As the information filter can be extended straight to design multisensory fusion algorithm, in this paper, the proposed IMMCIF includes the merits of CIF algorithm and IMM algorithm. The main factor of IMMCIF is that it calculates the state distribution and error covariance matrix by choosing the cubature points with equal weight and then processes the filter update by calculating the information vector and information matrix. The IMMCIF algorithm includes input integration, CIF, model probability update, and output integration. The structure diagram is shown as in Figure1.Figure 1
IMMCIF structure diagram.The detailed filtering processes are as follows.


### 3.1. Input Integration

Consider(16)uk-1∣k-1i/j=pijuk-1iCj,X^k-1∣k-10j=∑X^k-1∣k-1iuk-1∣k-1i/j,Pk-1∣k-10j=∑i=1ruk-1∣k-1i/jPk-1∣k-1i+X^k-1∣k-1i-X^k-1∣k-10jX^k-1∣k-1i-X^k-1∣k-10jT,where Cj=∑i=1rpijuk-1i, uk-1∣k-1i/j is the conditional probability of model i at k-1, uk-1i is the probability of model i at k-1, X^k-1∣k-10j is the initial mean value of model j, Pk-1∣k-10j is the initial error covariance, X^k-1∣k-1i is the estimated value of model i at k-1, and Pk-1∣k-1i is the relative covariance.


### 3.2. Cubature Information Filtering

Let the mixed initial value and the measure value (z) as the input of each filter at k. Then new state vector X^k∣kj, the error covariance Pk∣kj, the predicted measured value zk∣k-1j, and the residual vkj can be obtained from the CIF which have been derived in Section 2.The likelihood valueLkj is(17)Lkj=Nz;zk∣k-1j,vkj=12πVkj·exp⁡-12zk-z^k∣k-1jTVkj-1zk-z^k∣k-1j,where Vkj is the associated covariance of residual vkj.


### 3.3. Model Probability Update

Bayesian hypothesis testing method is used to evaluate the model probability and calculate the residual of each filter. It is known that once the filter model matches with the actual model, the filtering residual is zero mean, and the variancevk is Gaussian White Noise. Therefore from the likelihood function of model matching mi at k, the model probability can be updated from (18)ukj=LkjCj∑j=1nmLkjCj.


### 3.4. Output Integration

The probabilities of model are fused with the estimated value of each filter based on the given weights; then the final output of the IMMCIF is calculated as(19)X^k∣k=∑j=1rXk∣kjukj,Pk∣k=∑j=1rukjPk∣kj+X^k∣kj-X^k∣kX^k∣kj-X^k∣kT.


## 3.1. Input Integration

Consider(16)uk-1∣k-1i/j=pijuk-1iCj,X^k-1∣k-10j=∑X^k-1∣k-1iuk-1∣k-1i/j,Pk-1∣k-10j=∑i=1ruk-1∣k-1i/jPk-1∣k-1i+X^k-1∣k-1i-X^k-1∣k-10jX^k-1∣k-1i-X^k-1∣k-10jT,where Cj=∑i=1rpijuk-1i, uk-1∣k-1i/j is the conditional probability of model i at k-1, uk-1i is the probability of model i at k-1, X^k-1∣k-10j is the initial mean value of model j, Pk-1∣k-10j is the initial error covariance, X^k-1∣k-1i is the estimated value of model i at k-1, and Pk-1∣k-1i is the relative covariance.


## 3.2. Cubature Information Filtering

Let the mixed initial value and the measure value (z) as the input of each filter at k. Then new state vector X^k∣kj, the error covariance Pk∣kj, the predicted measured value zk∣k-1j, and the residual vkj can be obtained from the CIF which have been derived in Section 2.The likelihood valueLkj is(17)Lkj=Nz;zk∣k-1j,vkj=12πVkj·exp⁡-12zk-z^k∣k-1jTVkj-1zk-z^k∣k-1j,where Vkj is the associated covariance of residual vkj.


## 3.3. Model Probability Update

Bayesian hypothesis testing method is used to evaluate the model probability and calculate the residual of each filter. It is known that once the filter model matches with the actual model, the filtering residual is zero mean, and the variancevk is Gaussian White Noise. Therefore from the likelihood function of model matching mi at k, the model probability can be updated from (18)ukj=LkjCj∑j=1nmLkjCj.


## 3.4. Output Integration

The probabilities of model are fused with the estimated value of each filter based on the given weights; then the final output of the IMMCIF is calculated as(19)X^k∣k=∑j=1rXk∣kjukj,Pk∣k=∑j=1rukjPk∣kj+X^k∣kj-X^k∣kX^k∣kj-X^k∣kT.


## 4. Results and Discussion

In this section, we consider a classical target tracking scenario where the target trajectory is shown in Figure2. Let the state vector at time k be Xk=x,x˙,y,y˙T, where x and y are the position variable in X- and Y-bearing and x˙ and y˙ are the velocity variable in X- and Y-bearing.Figure 2
Target trajectory.The CT (coordinated turn) model is(20)F2=1sin⁡ωTω0cos⁡⁡ωT-1ω0cos⁡ωT0-sin⁡ωT01-cos⁡⁡ωTω1sin⁡ωTω0sin⁡ωT0cos⁡⁡ωT,where ω is the turn rate of the target and T is the sampling interval.The right turn rate is defined as-3°, and the left turn rate is defined as 3°.The measure equation of the system is(21)Z=10000010+R,where R is the measurement noise of the system.The initial state isX0=1000m,200m/s,1000m,200m/sT, and the initial associate covariance is P0=diag1000,10,1000,10, the process noise Q~N0,q, with q=10,0;0,10, and the process noise weight matrix is G=T2/2,0;T,0;0,T2/2;0,T. The measurement noise R~N0,r, with r=diag200,0.1. The simulation time is simTime=100s, and the step time T=1s. 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(22)pij=0.90.050.050.10.80.10.050.150.8.The root-mean square error (RMSE) of position and velocity is used to compare the performance of two filtering algorithms. The RMSE defined in state vectorX at k is(23)RMSE=∑n=1n1/k∑k=1kXk,n-X^k,n2n,where Xk,n is the real state vector of n Monte Carlo simulation and X^k,n is the estimated state vector of n Monte Carlo simulation. The simulation results are shown in Figures 2, 3, and 4.Figure 3
RMSEs of (a)X-position and (b) Y-position.
(a)
(b)Figure 4
RMSEs of (a)X-velocity and (b) Y-velocity.
(a)
(b)Figures2, 3, and 4 and Table 1 show the contrast of the tracking accuracy with IMMCIF, IMMCKF, and IMMUKF. It can be seen clearly from the figures that IMMCIF does better in tracking precision than IMMCKF and IMMUKF, while all of them exhibit stable characteristics and there is no error divergence.Table 1
The RMSEs of IMMCIF, IMMCKF, and IMMUKF.
IMMCIF
IMMCKF
IMMUKF
RMSE_X (m)
4.9371
6.6262
9.6817
RMSE_X_V (m/s)
2.8419
3.2512
4.2409
RMSE_Y (m)
5.7319
9.0982
9.1862
RMSE_Y_V (m/s)
4.5933
5.6901
5.8375Figures5(a), 5(b), and 5(c) show the mode probabilities of IMMCIF, IMMCKF, and IMMUKF, which demonstrate that all of them can effectively track the target maneuvering characteristics at preliminary stage. And the mode probabilities correspond well to the maneuvering target, which will change when motion state transforms from CV trajectory to CT trajectory and vice versa. Mode 1 is constant velocity (CV) mode, Mode 2 is right coordinate turn (CT) mode, and Mode 3 is left coordinate turn (CT) mode.Figure 5
Mode probabilities of IMMs.
(a)
IMMCIF
(b)
IMMCKF
(c)
IMMUKFThe comparisons of mode probabilities of IMMCIF, IMMCKF, and IMMUKF are shown in Figures6(a), 6(b), and 6(c). The figures show that the IMMCIF, IMMCKF, and IMMUKF can capture the kinematics of maneuvering in time once the motion state changes when t=20s, t=40s, t=60s, and t=80s. However as time goes by, IMMCKF and IMMUKF cannot track the maneuvering target well while the IMMCIF is more stable.Figure 6
Comparisons of mode probability of IMMs.
(a)
Mode 1
(b)
Mode 2
(c)
Mode 3


## 5. Conclusion

In this paper, the Interacting Multiple Models Cubature Information Filtering (IMMCIF) algorithm is proposed to enhance the precision and quick response of nonlinear maneuvering target tracking problem. This algorithm introduces Cubature Information Filter based on Interacting Multiple Models, which disposes all the models simultaneously through Markov Chain. And it evaluates the information vector and information matrix instead of the state vector and covariance to deal with nonlinear filtering. The simulation results show IMMCIF outperforms IMMCKF and IMMUKF which exhibits potential applications in nonlinear target tracking.



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