Maneuvering target tracking in clutter background based on improved interacting multiple-model probabilistic data association algorithm

WU Pan-long, LIU Jia-le, LI Xing-xiu

(1.School of Automation, Nanjing University of Science & Technology, Nanjing 210094, China; 2. School of Science, Nanjing University of Science & Technology, Nanjing 210094, China)

Maneuvering target tracking in clutter background based on improved interacting multiple-model probabilistic data association algorithm

WU Pan-long1, LIU Jia-le, LI Xing-xiu2

(1.School of Automation, Nanjing University of Science & Technology, Nanjing 210094, China; 2. School of Science, Nanjing University of Science & Technology, Nanjing 210094, China)

To improve the performance of tracking a maneuvering target in clutter, an improved interacting multiple model probability data association algorithm (IMMPDA-DCM) is proposed for airborne target tracking. Under the architecture of the proposed algorithm, an interacting multiple model (IMM) is used to deal with the model switching. The debiased converted measurement (DCM) filter is used to compensate the non-linearity in the dynamic system models and then reduce the observation error caused by coordinate transformation. The probability data association (PDA) handles the data association and measurement uncertainties in clutter background. Simulation results show that the proposed algorithm can improve the tracking precision of maneuvering target in clutters, and the position estimation error of IMMPDA-DCM is reduced by 26.38% compared with that of traditional IMMPDA-EKF algorithm.

target tracking; probabilistic data association; clutter background; debiased converted measurement; interacting multiple model

Maneuvering target tracking is a hot topic in the field of target tracking. The key problem of maneuvering target tracking is to establish an accurate target’s movement model and a suitable tracking filtering algorithm. Target’s maneuvering refers to the sudden change in unpredictable circumstances, such as performing some sort of tactic actions, including subduction, acceleration, deceleration, steering and so on.

The main challenge of maneuvering target tracking is the target’s movement uncertainty. Now, the interacting multiple model (IMM) algorithm is widely used for maneuvering target tracking[1-2].

In the system of radar target tacking, the tracking of an airborne target in a cluttered environment might be a challenge due to the several observations for a single airborne target, some tracking measurements do not originate from the airborne target. Therefore, the present study utilizes the probabilistic data association (PDA) filter[3-4]to assign weights to the validated measurements. The PDA filter can extend the tracking capability to a highly cluttered environment. However, the dynamic of target is usually modeled and tracked in the Cartesian coordinates, whereas the measurements are provided in terms of range and angle with respect to the radar sensor location in the polar coordinates. Therefore, the radar target tracking becomes a kind of non-liner estimation problem. One solution to this problem is the extended Kalman filter (EKF) but would results in filter divergence[5-6]. The other solution is debiased converted measurement (DCM) Kalman filter[7].

Combining IMM with appropriate data association algorithm can realize maneuvering target tracking in clutters [8-9], such as ML-PDA, IMMPDA, IMM-MHT and so on. A ML-PDA algorithm has been shown to be robust in a cluttered environment for a constant velocity target, however, it cannot be applied to the situation where targets undergo maneuvers. An adaptive update rate tracking algorithm based on modified IMM-PDA is proposed to avoid tracking loss of maneuvering target tracking in clutters. An interacting multiple model probability data association (IMMPDA) algorithm was proposed to support the navigation and surveillance services of the air traffic management system [10]. In order to gain possible improvement on the tracking performance, the IMMPDA algorithm is combined with the debiased converted measurement (DCM) filter to create an IMMPDA-DCM filter for airborne target tracking in this paper.

1 IMMPDA-DCM algorithm principle

1.1 Radar data debiased converted measurement

In the spherical coordinates, the measured target position (the distance rm, the azimuthmη and the elevation θm) is defined with respect to the true position ( the true distance r, the true azimuth η and the true elevation θ) as[11]

where the errors in distance, azimuth η～ and elevation θ～ are assumed to be independent with zero mean and standard deviation σr,σηand σθrespectively. These polar measurements are converted to the Cartesian coordinate measurements:

From (2), we know that the converted measurements are correlated and nonlinear with respect to the polar measurements (rm,ηmand θm). If the measurement errors of r, η and θ are small and target distance is close, errors statistic approximations obtained in Cartesian coordinates are accurate. These approximations are obtained by taking the first-order terms of a Taylor series expansion for the (2) to approximate the Cartesian coordinate errors as

Note that the approximation transformation errors are unbiased. However, the standard transformations are biased. The average true deviation μcand average true covariance Rcof converted measurement are described as[12]

where

When measurement in the spherical coordinate is converted to be in Cartesian coordinate, the measurement is modified as

1.2 IMMPDA-DCM algorithm

One complete cycle of the proposed IMMPDA-DCM algorithm comprises four major steps: Mixing probabilities calculation, model conditioned filtering in clutter, model probability update and output mixing. Detailed steps of the proposed algorithm is given as follows: 1) Mixing probabilities calculation.

where μj(k - 1) is the conditional probability of the model j at time k-1, Cjis the normalizing constant. μij(k - 1) is the mixing probability, which denotes the probability of system switches from model i to model j at k-1. r is the number of the models. p is the transition probability from model i to model j.

2) Model conditioned filtering in clutter. Computing the DCM based PDA filter for the r models.

(a) Input interaction. Computing the input state and covariance matrices for the r models with

(b) State and covariance prediction.

(c) Validated measurement judgment. During track maintenance, each measurement is validated against the established track by setting up a validation region centered around the predicted measurement, the validation region is

(d) Converted measurement error calculation. Calculating μcand Rcusing (6) and (7).

(e) Probabilistic data association for each validated measurement.

where mkis the number of measurements validated and associated with the track. βi( k ) is the associationprobability of the ith target-originated measurement. β0(k) is the association probability of all measurements are not valid. ～zi( k) is the innovation associated with the mth validated measurement,

PDand PGare the target detection probability and the gate probability, respectively.

(f) State and covariance update. Using the combined innovation to substitute the clutter-free innovation, and calculate the gain matrix, state and covariance updating matrix.

3) Model probability update in clutter.

where Λj( k ) is the likelihood of the jth measurement for jth PDA filter.

Vi( k) is the volume of the ellipse tracking gate,is the dimension of the measurement. cnzis the volume of the unit hypersphere.

4) State Combination.

Finally, all the sub-model conditioned state estimates and covariances are probabilistically combined to find the overall estimate X(k) and its covariance matrix P(k).

2 Target tracking model

The space constellation of BeiDou navigation system The IMM algorithm can estimate the state of a dynamic system with several different models that switch from one to another, and finally get a mixing output. Various nonlinear filtering algorithms can run in the IMM framework. In this paper, the DCM filter is embedded in IMM architecture for maneuvering target tracking, and the CV model and Singer model are selected to build in the IMMDCM algorithm. The CV model is used to describe the basic motion of the target, and the Singer model is used to describe target maneuver. The target dynamic model can be described by

where Xj(k ) is the state of the target at time k for jth model, Fj(k ) is the transition matrix of jth model, the Gj(k ) is the process noise gain matrix. Wj(k ) is the process noise sequences with zero mean and covariance Qj(k ). The state vector are target position, velocity and acceleration at time k,

Where x( k ),y( k ),z( k) denote the position in the x, y, z respectively.denote the velocity in the x, y, z respectively.denote the acceleration in the x, y, z respectively. The state transition matrix and noise gain matrix of CV model are defined as

The state transition matrix and noise gain matrix of Singer model are defined as reference[13-14].

where αx= αy= αz= α, α=0.1 is the reciprocal of the manoeuver time constant. σax,σay, σazare standard deviation of maneuver acceleration in x, y and z direction.

Where a(x,y,z)max(axmax=25, aymax=50, azmax=0) is maximum acceleration of target, pmax= 0.5 is the maximum probability of acceleration or deceleration, p0=0.5 is the probability of without acceleration. The model of the target acceleration probability density is shown in Fig.1[15].

Fig.1 Model of the target acceleration probability density

T is the radar sampling period.w1kis the acceleration in the time step k, w2kis the acceleration increment in the time step k. w1k,w2k,w3kare independent Gaussian white noise.

3 Filter initialization for IMMPDA-DCM

Traditionally, tracking filters are initialized from first few measurements[16]. The initial state X(0) and covariance P(0) can be calculated by two-step extrapolation algorithm using first three received measurements. For singer model, the initialization is based on the first three measurements as follows:

The corresponding covariance is

Where Rij( k )(i, j= 1,2,3) is the ithrow andjthcolumn component of converted measurement covariance matrix calculated from (7) at time k.

In CV model, the target’s acceleration is not considered. To guarantee the received measurements alignment, the first three measurements are also used for the initialization of CV model, however the first measurement is abandoned.

The corresponding covariance is

The IMMPDA-DCM begins the data filtering from the fourth received measurement after the initialization.

4 Simulation and results

Monte Carlo simulation results are presented here to demonstrate the tracking performance of the proposed algorithm, and 100 runs were performed. The scenario of a highly maneuvering airborne target tracking is defined as follows: the sampling rate is T=0.1 s, the target makes five accelerating maneuver with linear segments connecting it. The initial position of the target is (10 000, 6000, 4000) m, and the velocity is (-300, -300, -100) m/s. In the first period of 1-5 s, it flies linearly by constant velocity. From 6-10 s, it makes an accelerating maneuver with (15, 20, 0) m/s2. From 11-15 s, it flies with (5, 5, 0) m/s2. From 16-20 s, it flies with (-5, -20, 0) m/s2. From 21-25 s, it flies with (-5, -20, 0) m/s2. From 26-30 s, it flies with (-10, -10, 0) m/s2. At last it flies linearly from 30-35 s by constant velocity. The relative trajectory of target and aircraft platform is shown in Fig.2.

The noise covariance of the measured distance r, azimuth η and elevation θ are=100 m2,=π2/ 360 000 rad2,= π2/360 000 rad2. γ=16, λ=4e-5, PG= 0.997, PD= 1. The model initial probability is μ= [0.5 0.5], this means that CV and Singer model has the same chance to be selected in the initialization. The variance of the process noise for two models are Q1=0.05625·I3, Q2=1 ·I3. Where, I3is the identity matrix of 3 dimensions. Considering the different process noise level, the transition probability of the system model is chosen as

Fig.2 Trajectory of maneuvering target

Fig.3-Fig.5 are the position comparison in x, y, z directions separately. Fig.6-Fig.8 are the velocity comparison in x, y, z directions separately. The Root Mean Square Error (RMSE) of the target’s position and velocity for the two algorithms are shown in Table.1. Simulation results show that the proposed IMMPDADCM algorithm has the higher tracking precision, which is consistent with the results in Table 1.The position tracking precision of the proposed algorithm decreased by 15.62%, 23.97% and 33.09% compared with the IMMPDA-EKF in x, y, z directions, respectively. The velocity tracking precision of the proposed algorithm decreased by 46.77%, 45.44% and 45.88% compared with the IMMPDA-EKF in x, y, z directions, respectively. The operation time of proposed algorithm is shorter than IMMPDA-EKF.

Tab.1 MSE comparison of two different algorithms

Fig.3 Position error of x direction

Fig.4 Position error of y direction

Fig.5 Position error of z direction

Fig.6 Velocity error of x direction

Fig.7 Velocity error of y direction

Fig.8 Velocity error of z direction

5 Conclusion

In this paper, an interacting multiple model probability data association algorithm based on debiased converted measurement filter (IMMPDA-DCM) is proposed, which is capable of adaptively tracking the maneuvering airborne target. The converted deviation and covariance of DCM filter is estimated using the current measurement. By abandoning the extended Kalman filter framework and using DCM filter in the proposed algorithm, the good tracking precision is achieved with decreasing the computational complexity. The IMMPDADCM algorithm achieves a better tracking performance compared with the IMMPDA-EKF. Monte Carlo simulation results verify the tracking precision and credibility of the proposed algorithm. The proposed algorithm is an effective algorithm for maneuvering target tracking in clutter.

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為了提高雜波條件下的空中機動目標跟蹤精度，提出了一個改進的交互多模型概率數據關聯算法。該算法將交互多模型、去偏轉換測量和概率數據關聯算法相結合，利用交互多模型算法模型集合間不同模型的相互切換來估計跟蹤目標的狀態；利用去偏轉換測量算法對轉換測量誤差進行去偏補償，從而減小觀測數據坐標變換引起的誤差；利用概率數據關聯算法處理數據關聯和測量的不確定性。通過將本文的算法和基于擴展卡爾曼濾波的概率數據關聯算法進行對比分析和驗證，實驗結果表明本文提出的算法可以提高機動目標的跟蹤精度，且跟蹤精度相對基于擴展卡爾曼濾波的概率數據關聯算法減少26.38%的位置誤差。

目標跟蹤；概率數據關聯；雜波；去偏轉換測量；交互多模型

TP24

：A

2015-06-12；

：2015-09-25

國家自然科學基金（61473153,61301217）；江蘇省自然科學基金（BK20131352）；江蘇省“六大人才高峰”項目（2015-XXRJ-006）；高等學校博士學科點基金（20123219120043）；江蘇高校優勢學科建設工程資助項目（PAPD）

吳盤龍（1978—），男，副研究員，博士生導師，從事目標跟蹤和信號處理研究。E-mail: plwu@163.com

1005-6734(2015)06-0755-08

10.13695/j.cnki.12-1222/o3.2015.06.011

基于改進交互多模型概率數據關聯的機動目標跟蹤

吳盤龍1，劉佳樂，李星秀2

（1. 南京理工大學 自動化學院，南京 210094；2. 南京理工大學 理學院，南京 210094）

聯 系 人：李星秀（1981—），女，副教授，從事信號處理技術研究。E-mail: xxlwpl@126.com