Mathematical Problems in Engineering

Volume 2015, Article ID 810613, 7 pages

http://dx.doi.org/10.1155/2015/810613

## Maneuvering Target Tracking Algorithm Based on Interacting Multiple Models

College of Automation, Harbin Engineering University, No. 145, Nantong Street, Harbin 150001, China

Received 2 January 2015; Revised 23 April 2015; Accepted 25 April 2015

Academic Editor: Erik Cuevas

Copyright © 2015 Gannan Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 Section 2, 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

Considerwhere is information matrix. 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:where is the matrix with a set of unit vector as shown in

The predicted state and predicted error covariance can be obtained based on CKF:where , is the dimension of state vector .

From (4), the predicted information matrix and the predicted information state vector can be derived:

##### 2.2. Measurement Update

Considerwhere can be obtained by factorizing the inverse predicted matrix .

The cubature points are evaluated as

The propagated cubature points are evaluated as

The predicted measurement is estimated:

The cross-covariance matrix is estimated:

Then information state distribution and information matrix can be obtained from (5) and (11):where .

From (5) and (12), the updated information matrix and updated information can be given, respectively:

Then the estimated state vector and estimated error covariance can be derived as follows:where is the 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 Figure 1.