A dynamic weighted algorithm for multiple radar fusion in an active protection system

Abstract

The threat detection system based on short-range radars is an essential part of the active protection system (APS) of armored vehicles. The multi-radar data fusion problem is one of the crucial issues in the APS. Firstly, a general algorithm for multi-radar coordinates transformation is given. Then, based on the weighted fusion model and the trajectory characteristics of targets in the APS, a real-time dynamic weighting factor derivation algorithm is proposed. The algorithm is simulated in a dual-radar threat tracking and ballistic prediction scenario. The results prove the correctness and effectiveness of the algorithm.

Keywords

Active protection system radar data fusion weighted fusion algorithm

1. Introduction

The active protection system (APS) is a self-defense system used by armored vehicles to intercept and destroy anti-tank missiles or projectiles.¹ The system consists of the following three parts: the first part is the threat detection system, which detects threats to the vehicle on the battlefield; the second part is the signal processing and decision-making system, which is responsible for processing, analyzing, and making countermeasure decisions; the third part is the confrontation system, which uses a close-range anti-missile defense system to intercept and destroy threats before they hit the vehicle.² The system structure of the APS is shown in Figure 1.

Figure 1.

The structure of the active protection system.

In order to ensure that the intercepting ammunition accurately intersects with the incoming target, it is necessary to obtain information such as the distance, speed, and orientation of the target. Short-range radars installed in different orientations of the vehicle body are usually used to achieve an omnidirectional detection. Therefore, an incoming target in the overlapping coverage area of multiple radars (see Figure 2) may generate several tracks, which need to be fused to a single and more accurate track. In recent years, methods and technologies for the radar data fusion problem have been continuously proposed in various research fields. Xu et al.³ discussed three fusion criteria: one is based on the covariance matrix of estimation error, the second is dependent on the infinite normal number, and the last is related to the element of the state vector. Sarunic and Rutten⁴ described an algorithm for associating and fusing multipath tracks in over-the-horizon radar, which used a model-based approach to incorporate track history in its computation of association probabilities and fused estimate calculations, thus exploiting temporal as well as spatial relationships. Bradaric et al.⁵ used the multistatic ambiguity function to describe the relationships between the system parameters and different performance measures, and then computed the best weighting coefficients for combining the signals from multiple receivers. Shu and Liang⁶ derived multi-input multi-output fusion rules based on the target fluctuation model and compared them against the optimal likelihood ratio method (LR), maximum ratio combiner (MRC), and equal gain combiner (EGC). Soysal and Efe⁷ proposed a new data fusion scheme comprising a covariance intersection algorithm and particle filtering. Charlish et al.⁸ analyzed state of the art algorithms for track-to-track fusion assuming range-Doppler radar measurements, including a central Kalman filter, Naive fusion, and variants of the distributed Kalman filter. Wang et al.⁹ introduced several robust, practical track fusion algorithms, such as track-to-track association, track fusion, and track management.

Figure 2.

A target in the overlapping coverage area.

The weighted average method is a common method in data fusion, which can effectively synthesize multi-sensor information and improve the accuracy of data after fusion.¹⁰ In the field of radar track fusion, there are also many studies based on the weighted fusion algorithm. Li et al.¹¹ calculated the relative radar accuracy by using least-square fitting, and applied radar tracking precision to be the integration of the weighted average method through the fusion of the track-to-track reference. Zhang and Qu¹² derived weighted mean coordinates with the covariance matrix of the sensory, covering the radial distance and the angles. Liu¹³ set up a trajectory membership matrix based on the fuzzy clustering algorithm, calculated ballistic collection trajectory weights in real-time, and fused the trajectory data with corresponding weights. Yang et al.¹⁴ proposed a novel data fusion algorithm to combat the false data injection attack, which introduced data confidence factors into data fusion and adaptively decreased the fusion weights of the injected data.

However, related research in the field of the APS is still relatively rare. Since the target of the APS is high-speed moving ammunitions, their flight duration in the radar detection range is extremely short, so the system is required to achieve rapid real-time fusion calculation of radar data. Besides, the onboard short-range radars used in the APS are susceptible to environmental factors, such as terrain, ground surface characteristics, and target azimuth. It is difficult to accurately estimate the accuracy of the track in traditional weighted averaging methods.

In this paper, a real-time fusion algorithm was proposed for the APS. The spatial alignment of multi-radar is completed by coordinate transformations. Based on the idea of weighted fusion, the data characteristics of target tracks are analyzed, and a weighting factor derivation algorithm is proposed to achieve the dynamic accuracy estimation and real-time multi-radar track fusion. The effectiveness of the algorithm is verified using a recursive least-squares ballistic prediction method in simulation experiments.

2. Coordinate transformations

In order to perform the fusion algorithm to tracks from different radars, the tracks should be transformed into a single coordinate system. Firstly, the tracks are transformed from the spherical coordinate system used in radar detection to the Cartesian coordinate system. Then the coordinate systems of each radar are translated and rotated to the same location and direction.

2.1. Spherical coordinate system of radars

The coordinate system used by radars in the APS of this paper is shown in Figure 3. The azimuth is $0^{°}$ at the center of the radar detection area, with the right-hand side being positive and the left-hand side being negative.

Figure 3.

The coordinate system of radars in the active protection system.

This spherical coordinate system can be transformed into a three-dimensional Cartesian coordinate system according to Equation (1). The $45^{°}$ azimuth direction is the x-axis, the $- 45^{°}$ azimuth direction is the y-axis, and the z-axis is determined by the right-hand rule:

{\begin{matrix} x = r \cos φ \sin (\frac{π}{4} - θ) \\ y = r \cos φ \cos (\frac{π}{4} - θ) \\ z = r \sin φ \end{matrix}

(1)

2.2. Cartesian coordinate system transformation

Considering a point $(x, y, z)$ and translating its coordinate system along the positive direction of the x-axis by a distance a, its coordinate in the new coordinate system $(x', y', z')$ can be given by Equation (2):

[\begin{matrix} x' \\ y' \\ z' \\ 1 \end{matrix}] = T_{x} [\begin{matrix} x \\ y \\ z \\ 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & - a \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \\ 1 \end{matrix}]

(2)

Similarly, the transformation matrices $T_{y}$ , $T_{z}$ of the coordinate system translating along the y-axis and the z-axis (by distances b and c, respectively) can be obtained as shown in Equations (3) and (4):

T_{y} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & - b \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(3)

T_{z} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & - c \\ 0 & 0 & 0 & 1 \end{matrix}]

(4)

Considering a point $(x, y, z)$ and rotating its coordinate system along the positive direction of the x-axis by an angle $α$ , its coordinate in the new coordinate system $(x', y', z')$ can be given by Equation (5):

[\begin{matrix} x' \\ y' \\ z' \\ 1 \end{matrix}] = R_{x} [\begin{matrix} x \\ y \\ z \\ 1 \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & \cos α & \sin α & 0 \\ 0 & - \sin α & \cos α & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \\ 1 \end{matrix}]

(5)

Similarly, the transformation matrices $R_{y}$ , $R_{z}$ of the coordinate system rotating along the y-axis and the z-axis (by angles $β$ and $γ$ , respectively) can be obtained as shown in Equations (6) and (7):

R_{y} = [\begin{matrix} \cos β & 0 & - \sin β & 0 \\ 0 & 1 & 0 & 0 \\ \sin β & 0 & \cos β & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(6)

R_{z} = [\begin{matrix} \cos γ & \sin γ & 0 & 0 \\ - \sin γ & \cos γ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}]

(7)

Then the coordinate of the point $x = {[\begin{matrix} x & y & z & 1 \end{matrix}]}^{T}$ after the translation and rotation of the coordinate system can be given by Equation (8):

x' = R_{x} R_{y} R_{z} T_{x} T_{y} T_{z} x

(8)

3. Multiple radar track fusion

3.1. Architecture of the weighted fusion algorithm

The fusion algorithm used in this paper is also based on a weighted fusion model. Considering a system with K radars to detect one target, the measurement of radar i at time t is $(r_{ti}, θ_{ti}, φ_{ti})$ . If the measurements of each radar are transformed into a unified Cartesian coordinate system as $(x_{ti}, y_{ti}, z_{ti})$ using Equation (1), then the fused coordinates can be derived by the following:

[\begin{matrix} x_{t}^{f} \\ y_{t}^{f} \\ z_{t}^{f} \end{matrix}] = \sum_{i = 1}^{K} w_{ti} [\begin{matrix} x_{ti} \\ y_{ti} \\ z_{ti} \end{matrix}]

(9)

where $w_{ti}$ represents the weighting factors of each radar. In the dynamic weighted algorithm, the weighting factors are given by the estimation of the accuracy of each radar at time t. For every t, the weighting factors must satisfy the following equation:

\sum_{i = 1}^{K} w_{ti} = 1

(10)

The core of the fusion model is the algorithm for weighting factor derivation. The weighting factor derivation algorithm should adequately reflect the accuracy of each radar’s measurement.

3.2. The weighting factor derivation algorithm

Because the working conditions of the APS are complex and variable, and the radars are susceptible to environmental factors, it is not appropriate to use the rated accuracy as the basis for weighting. Instead, the accuracy should be estimated online based on the measured data. Since the real ballistic trajectory cannot be obtained under actual working conditions, it is difficult to calculate the accuracy of the radar data directly. One idea is to take the arithmetic mean of the measurements of each radar as the reference, and then calculate the deviation of each radar.¹⁵ However, the reliability of the reference value thus obtained decreases as the number of radars in the system decreases. Usually, for the onboard radars in the APS, only two radars can discover the same target at the same time. Therefore, taking the arithmetic mean as the reference cannot reflect the difference between the two radars. Considering the characteristics of the target trajectory of the APS, the linearity of the radar track is taken as the weighting indicator in this paper.

The target of the APS is high-speed moving ammunitions. In the radar detection range, the gravity has little influence on them, and their trajectories can be approximated as straight lines. Therefore, the trajectories can be predicted by a linear fit of the radar tracks using the least-squares method.¹⁶ At time t, the difference $δ_{t}$ between the predicted coordinate ${\hat{x}}_{t}$ and the measured coordinate $x_{t}$ represents the coincidence of the measurement with the linear fit at that time. Therefore, its variance $σ^{2} (δ)$ can be used to represent the linearity of the entire track. The specific recursive algorithm is described as follows.

(a) At period N, let the extrapolated coordinate of radar i be ${\hat{x}}_{N}^{i}$ , and the measured coordinate be $x_{N}^{i}$ , then the deviation is given by Equation (11):

δ_{N}^{i} = | {\hat{x}}_{N}^{i} - x_{N}^{i} |

(11)

(b) For each radar, calculate the variance of deviations at all periods recursively. The recursive form of the variance formula is shown as Equation (12):

{\begin{matrix} σ^{2} (δ_{N}^{i}) = \frac{{(δ_{N}^{i} - {\bar{δ}}_{N - 1}^{i})}^{2}}{N} + \frac{N - 2}{N - 1} σ^{2} (δ_{N - 1}^{i}) \\ {\bar{δ}}_{N}^{i} = {\bar{δ}}_{N - 1}^{i} + \frac{δ_{N}^{i} - {\bar{δ}}_{N - 1}^{i}}{N} \end{matrix}

(12)

w_{N}^{i} = \frac{\frac{1}{σ^{2} (δ_{N}^{i})}}{\sum_{j = 1}^{K} \frac{1}{σ^{2} (δ_{N}^{j})}}

(13)

(d) Calculate the fused coordinate $x_{N}^{f} = [x_{N}^{f}, y_{N}^{f}, z_{N}^{f}]$ and distance $d_{N}^{f} = | x_{N}^{f} |$ using Equation (9).

(e) Update the parameters $\hat{θ}$ in the recursive least-squares algorithm using the fused coordinate. Here we take the distance $ψ_{N} = {[d_{N}^{f} 1]}^{T}$ as input, and $x_{N}^{f}$ , $y_{N}^{f}$ , $z_{N}^{f}$ as output. With the fitting of the x coordinate as an example, the recursive algorithm is as follows¹⁷:

{\begin{matrix} L_{N} = P_{N - 1}^{x} ψ_{N} {(1 + ψ_{N}^{T} P_{N - 1}^{x} ψ_{N})}^{- 1} \\ P_{N}^{x} = (I - L_{N} ψ_{N}^{T}) P_{N - 1}^{x} \\ {\hat{θ}}_{N}^{x} = {\hat{θ}}_{N - 1}^{x} + L_{N} (x_{N}^{f} - ψ_{N}^{T} {\hat{θ}}_{N - 1}^{x}) \end{matrix}

(14)

In Equation (14), $P_{N} = (ψ_{N}^{T} ψ_{N})^{- 1}$ is defined for the convenience of the recursive calculation process. $L_{N}$ is the gain factor of the parameter updating step.

In order to start the recursions, the initial values of $\hat{θ}$ and $P$ must be provided. The initial value of $\hat{θ}$ should be given according to a priori information. Since we do not know the direction of the target, a typical initialization will be as follows:

{\hat{θ}}_{0} = [0 0]^{T}

(15)

$P$ should be initialized by a small initial segment of radar data. However, it is impossible for us to know if $ψ_{0}$ is invertible. Therefore, we can simply initialize $P$ with a large value so that its influence on the final result will be significantly small:

P_{0} = 10^{5} I

(16)

(f) At the next period $N + 1$ , extrapolate the coordinate of radar i using Equation (17), then return to step (a):

{\hat{x}}_{N + 1}^{i} = [\begin{matrix} {\hat{x}}_{N + 1}^{i} \\ {\hat{y}}_{N + 1}^{i} \\ {\hat{z}}_{N + 1}^{i} \end{matrix}] = [\begin{matrix} {ψ_{N + 1}^{i}}^{T} {\hat{θ}}_{N}^{x} \\ {ψ_{N + 1}^{i}}^{T} {\hat{θ}}_{N}^{y} \\ {ψ_{N + 1}^{i}}^{T} {\hat{θ}}_{N}^{z} \end{matrix}]

(17)

4. Simulation

In this section, the effectiveness of the proposed algorithm is verified by simulation experiments. The coordinate transformations of different radars are omitted here, and only the weighting process of the algorithm is considered. In the simulation scenario, two radars are used to detect the same incoming target, and an imaginary real trajectory is established. The target is flying at a uniform speed of about 230 m/s. The starting point of the track is about 120 m away from the coordinate origin (i.e., the launch point of the intercepting ammunition). The radar detection period is 3 ms. The measurement errors of both radars satisfy the normal distribution with a mean of $0$ , where the standard deviations of the measurements of radar A and radar B are $0.6$ and $1.2$ , respectively. The simulated target trajectory and the tracks measured by the two radars are shown in Figures 4 and 5.

Figure 4.

The simulated target trajectory and the track from radar A.

Figure 5.

The simulated target trajectory and the track from radar B.

With the algorithm proposed in this paper, recursive fusion and fitting are started from the beginning of the track. In the APS, in order to set aside time for the flight of the intercepting ammunition, the recursive fitting process can only proceed to the cutoff distance $d_{\min}$ , and the position of the target at the interception distance $d_{int}$ is predicted by extrapolation. Here, $d_{\min} = 40 m$ , and the fusion and fitting results are shown in Figure 6. The fitted trajectory is in good agreement with the target trajectory.

Figure 6.

The fitted track after fusion and the simulated track.

In order to compare the fitting quality before and after the dynamic weighted fusion, the Euclidean distance $δ_{int}$ between the predicted intercept point and the actual intercept point on the simulated trajectory is taken as an indicator for evaluating the quality of the fitting. Let $d_{int} = 11.5 m$ , and predict the trajectory using the fused track and the tracks from each radar, respectively. The Monte Carlo simulation experiment was carried out 100 times. The results are shown in Figure 7. The prediction accuracy of the fusion track is generally better than the prediction accuracy of both radars.

Figure 7.

Comparison of the simulation results of the fused track and original radar tracks.

Since the measurement variance of each radar in the simulation environment is known, the optimal fusion weight with the smallest mean square error can be derived as follows¹⁰:

w_{i} = \frac{\frac{1}{σ_{i}^{2}}}{\sum_{j = 1}^{N} \frac{1}{σ_{j}^{2}}}

(18)

In order to verify that the dynamic weighted fusion algorithm (DWF) can effectively estimate the accuracy of each radar, the algorithm was compared with the optimal fusion algorithm based on Equation (18) and the arithmetic mean fusion algorithm (AMF; i.e., the weighting factors of the two radars are both $0.5$ ). The same random seed is used to perform 100 Monte Carlo simulation experiments on the three algorithms. The results are shown in Figure 8. The prediction accuracy of the DWF is close to the optimal fusion algorithm and generally better than that of the arithmetic mean fusion. Particularly when the accuracy of one radar is poor, the DWF can effectively reduce its influence on the fusion result.

Figure 8.

Comparison of the simulation results of the three fusion algorithms.

The mean square errors of the simulation experiments are summarized in Table 1. In the 100 Monte Carlo simulation experiments, the prediction accuracy of the proposed DWF is very close to that of the optimal fusion algorithm, and it is better than that of the single radar fitting and the AMF. The mean square error of the intercept point prediction is $12.5 %$ and $56.3 %$ lower than that of radar A and radar B fitting, respectively, and $20.0 %$ lower than that of the AMF.

Table 1.

Comparison of the fusion algorithm simulation results.

Fusion algorithm	MSE of prediction (m)
Radar A fitting	0.32
Radar B fitting	0.64
Optimal fusion	0.27
AMF	0.35
DWF	0.28

AMF: arithmetic mean fusion algorithm; DWF: dynamic weighted fusion algorithm.

5. Summary

Multi-radar data fusion is one of the crucial issues in the APS of armored vehicles. In this paper, a real-time fusion algorithm was proposed for the APS. Firstly, the coordinate transformation algorithm of multi-radar measurements was given to complete the spatial alignment of radar data. Then, based on the weighted average fusion model, the recursive weighting factor derivation algorithm was proposed according to the extrapolation error variance. The algorithm is simple to implement, fast in calculation, and low in resource consumption, which is compatible with the requirements of the APS. Simulation experiments verified the correctness and effectiveness of the proposed algorithm.

Footnotes

Declaration of conflicting interests

The authors declare that there is no conflict of interest.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Jinli Wang

Author biographies

Jinli Wang is a master’s student at the China North Vehicle Research Institute. He received his bachelor’s degree in engineering from Tsinghua University in Beijing, China. His field of research is the active protection of armored vehicles.

Haiping Song is a PhD student at Beijing Institute of Technology and one of the leaders of the armoured vehicle protection research team at China North Vehicle Research Institute.

Riming Chen is a professor at the China North Vehicle Research Institute. He is one of the leaders of the APS of armored vehicles program.

Yaning Zhang is an associate professor at the China North Vehicle Research Institute. She is a senior member of the APS of armored vehicles program.

References

Vivek

Piramasubramanian

Madhan

, et al. Simulation and performance prediction of an UWB radar for active protection system applications. In: 2016 international conference on communication and signal processing (ICCSP), 6–8 April 2016, pp.0351–0355. Melmaruvathur, India: IEEE.

Song

Xin

Active protection system (APS) theory and design. Beijing, China: Ordnance Industry Press, 2015.

Jin

Zhou

. Several methods of radar data fusion. In: 2002 3rd international symposium on electromagnetic compatibility, 21–24 May 2002, pp.664–667. Beijing, China: IEEE.

Sarunic

Rutten

. Over-the-horizon radar multipath track fusion incorporating track history. In: proceedings of the third international conference on information fusion, 10–13 July 2000, pp.TUC1–13, volume 1. Paris, France: IEEE.

Bradaric

Capraro

Weiner

, et al. Multistatic radar systems signal processing. In: 2006 IEEE conference on radar, 24–27 April 2006, p.8. Verona, NY, USA: IEEE.

Shu

Liang

. Data fusion in a multi-target radar sensor network. In: 2007 IEEE radio and wireless symposium, 9–11 January 2007, pp.129–132. Long Beach, CA, USA: IEEE.

Soysal

Efe

. Data fusion in a multistatic radar network using covariance intersection and particle ﬁltering. In: 14th international conference on information fusion, 5–8 July 2011, pp.1–7. Chicago, IL, USA: IEEE.

Charlish

Govaers

Koch

. Track-to-track fusion schemes for a radar network. In: IET international conference on radar systems (Radar 2012), 22–25 October 2012, pp.1–6. Glasgow, UK: IET.

Wang

Ding

Kong

. Application of multi-sensor track fusion technology in ﬁre control radar network. In: 2013 IEEE 11th international conference on electronic measurement & instruments, 16–19 August 2013, pp.563–567, volume 2. Harbin, China: IEEE.

10.

Zhong

Zhang

Yang

, et al. A weighted fusion algorithm of multi-sensor based on the principle of least squares. Chin J Sci Instrum 2003; 24: 427–430.

11.

Dong

, et al. Model of multi-sensor data fusion and trajectory prediction based on echo state network. In: 2010 international conference on computer, mechatronics, control and electronic engineering, 24–26 August 2010, pp.338–341, volume 1. Changchun, China: IEEE.

12.

Zhang

Fusion estimation of point sets from multiple stations of spherical coordinate instruments utilizing uncertainty estimation based on Monte Carlo. Meas Sci Rev 2012; 12: 40–45.

13.

Liu

Research on the real-time fusion method of target trajectory data. Electron Meas Technol 2012; 6: 66–68.

14.

Yang

Feng

Zhang

, et al. A novel data fusion algorithm to combat false data injection attacks in networked radar systems. IEEE Trans Sign Inform Proc Network 2018; 4: 125–136.

15.

Huang

Zhou

Zhang

, et al. Algorithm of weighting factors dynamic allocation in multi-radar track weighted fusion. J Comput Appl 2008; 9: 80.

16.

Chang

, et al. A new method of target tracking in ultra-short-range radar. In: proceedings of 2013 3rd international conference on computer science and network technology, 12–13 October 2013, pp.995–998. Dalian, China: IEEE.

17.

Liu

Modern identiﬁcation engineering. Beijing, China: National Defence Industry Press, 2006.