Asynchronous Anti-Bias Track-to-Track Association Algorithm Based on Nearest Neighbor Interval Average Distance for Multi-Sensor Tracking Systems

Abstract

Due to sensor characteristics, geographical environment, electromagnetic interference, electromagnetic silence, information countermeasures, and other reasons, there may be significant system errors in sensors in multi-sensor tracking systems, resulting in poor track-to-track association (TTTA) effect of the system. In order to solve the problem of TTTA under large system errors, this paper proposes an asynchronous anti-bias TTTA algorithm that utilizes the average distance between the nearest neighbor intervals between tracks. This algorithm proposes a systematic error interval processing method to track coordinates, and then defines the nearest neighbor interval average distance between interval coordinate datasets and interval coordinate points, and then uses grey theory to calculate the correlation degree between tracks. Finally, the Jonker–Volgenant algorithm is combined to use the canonical allocation method for TTTA judgment. The algorithm requires less prior information and does not require error registration. The simulation results show that the algorithm can ensure a high average correct association rate (over 98%) of asynchronous unequal rate tracks under large system errors, and achieve stable association, with good association and anti-bias performance. Compared with other algorithms, the algorithm maintains good performance for different target numbers and processing cycles, and has good superiority and robustness.

1. Introduction

Multi-sensor tracking systems [1,2] have become a hot research field in recent years because they can expand the information acquisition range, improve the accuracy and reliability of reconnaissance systems, and improve the stability of target track and information [3,4]. The distributed multi-sensor tracking system has become the preferred solution for research in this field due to its advantages such as low communication bandwidth requirements, low computational complexity, and high survivability [5,6], and how to judge tracks from different sensors as the same target is one of the core problems to be solved, that is, TTTA [7,8,9,10,11]. Typical tracks include the flight path of aerial targets, the navigation path of naval vessels, etc. At present, TTTA technology is widely used in engineering practice, such as Marine Vessel Automatic Identification System (AIS) [12], Automatic Dependent Surveillance-broadcast System (ADS-B) [13], and battlefield air information fusion. Especially in modern battlefields, which have expanded to the five-dimensional space of land, sea, air, space, and electromagnetic, TTTA judgment directly affects the final decision of commanders. Due to the influence of sensor characteristics, geographical environment, electromagnetic interference, electromagnetic silence, information countermeasures and other factors, the systematic errors of the sensor will be changed during use [14]. The existence of systematic errors increases the difficulty of TTTA, which makes TTTA and error registration need to be carried out simultaneously [15].

Singer and Kanyuck [16] developed the first TTTA algorithm in 1970 to estimate two tracks from two different sensors based on the use of gates. The work in [17] extends the traditional nearest neighborhood method and K-nearest neighborhood method, and makes amendments and improvements in association criteria, quality design, and ambiguity processing, etc. The work in [18] has suggested a new TTTA algorithm based on the well-known iterative closest point (ICP) and the global nearest neighbor (GNN). These algorithms do not take into account the systematic errors of the sensors. In order to overcome the influence of systematic errors, some scholars have proposed effective methods to solve the problem of TTTA and error estimation. A target tracking method based on sensor selection and a possible variable bias registration method are proposed in reference [19]. Reference [20] aim to solve the spatial registration and track association problem in the case where incomplete measurements are provided by different sensors, this paper proposes a residual bias estimation registration (RBER) method based on maximum likelihood and the sequential m-best track association algorithm based on the new target density (SMBTANTD). A joint approach for solving the problem of TTTA and sensor bias estimation is designed by Zhu and Wang [21]. By the work in [22], an algorithm for processing spatiotemporal bias and state estimation of asynchronous multi-sensor systems is proposed to obtain the enhanced state vector and establish the enhanced state model. Reference [23] uses the obtained radar track and corresponding automatic identification system (AIS) track data to calculate the measurement errors of distance, azimuth, and Doppler velocity, and a Gaussian distribution model is derived through probability distribution fitting. The work in [24], according to the statistical characteristics of Gaussian random vectors, led to the design for an anti-bias TTTA technique for aircraft platforms. Tian et al. stated a TTTA algorithm based on the reference topology (RET) feature [25]. The method proposed in the literature [26] bypasses the translation and distance deviation of sensors. Reference [27] developed a spatiotemporal method for TTTA, but it requires a large amount of data collection as a prerequisite. Reference [28] proposed a track segment association method based on a bidirectional Holt–Winters prediction and fuzzy analysis [29], which can effectively solve the track association problem where the target label attributes change before and after track breakage. But these algorithms all require time-domain registration [30] to synchronize the tracks, which can lead to error propagation and accumulation during the time-domain registration process, affecting the effectiveness of TTTA.

In order to avoid situations such as time-domain registration and bias estimation, some scholars directly establish a correlation matrix for tracks for TTTA. In the literature, Ref. [31] introduces grey theory [32] into the algorithm and proposes a new TTTA algorithm. In the work of [33], the original track points are replaced by the track sequence after the interval data and real data mixed sequence transformation, and then a TTTA algorithm based on this sequence’s similarity degree is proposed by using the maximum association criterion. However, the new bias introduced by this transformation will affect the correct TTTA. In the work of [34], segmented sequence division rules are defined for TTTA, but a certain amount of data are required as the premise, so the effect of TTTA is greatly affected by the length of sampling time. At the same time, these algorithms do not consider the case of large systematic errors.

Aiming at the problem of asynchronous unequal rate TTTA under large systematic errors, this paper defines a new distance measurement standard between tracks. First, the track coordinates are processed into systematic error intervals, and then the nearest neighbor interval average distance between the interval coordinate datasets and the interval coordinate point is used to measure the distance between a single coordinate point and the track coordinate dataset. Then, grey correlation theory is used to obtain the track correlation degree between asynchronous tracks and establish the correlation matrix. Finally, the Jonker–Volgenant algorithm and classical allocation method are combined to realize TTTA. This work aims to achieve track correlation under large systematic errors, and verify the effectiveness and superiority of the algorithm by analyzing the average correct association rate, the number of false associations, and the maximum false association rate under different simulation environments.

The purpose of this article is to design and implement effective asynchronous unequal rate TTTA for multi-sensor tracking systems with large systematic errors. Its main contributions are as follows:

• By processing the systematic errors interval of track coordinates and defining the nearest neighbor interval average distance between interval coordinate datasets and interval coordinate points, the correlation degree between the tracks is established, and an asynchronous anti-bias TTTA algorithm based on the nearest neighbor interval average distance is proposed. This algorithm does not require time domain alignment and error registration, effectively avoiding the introduction of new estimation errors and achieving TTTA under large systematic errors.
• The average correct association rate of the tracks of the algorithm under different cycle ratios, different delay startup times, and different noise distribution forms is analyzed, demonstrating its anti-interference and effectiveness.
• By analyzing the TTTA effect of the algorithm under different systematic errors and comparing the algorithms, the good anti-bias performance of the algorithm was verified.
• The average correct association rate, the number of false associations, and the maximum false association rate of various algorithms are compared under the simulation conditions of different target numbers and processing cycles, which proves that the proposed algorithm has strong robustness and superiority.

The main innovation of this paper is to derive and propose an asynchronous anti-bias TTTA algorithm based on the nearest neighbor interval average distance. The structure of the article is as follows. Section 2 defines the nearest neighbor interval average distance and the degree of correlation between interval coordinate datasets, and the asynchronous anti-bias TTTA algorithm is derived in detail. Section 3 carries out simulation experiments and analyzes the results according to performance evaluation indicators. Finally, Section 4 provides conclusions.

2. Materials and Methods

2.1. Mathematical Formula Definition

To facilitate the direct correlation of different tracks with large systematic errors without time domain registration in the future, and to establish correlation degree standards between tracks, two new mathematical definitions are given below.

Definition 1.

The nearest neighbor interval average distance between interval coordinate datasets and interval coordinate points.

Setting the non-empty interval coordinate dataset as: ${\tilde{H}}^i=\left\{{\tilde{h}}^i\left(1\right),{\tilde{h}}^i\left(2\right),\cdots ,{\tilde{h}}^i\left(l\right)\right\}$, $i=1,2,\cdots ,m$, in which the interval coordinate points ${\tilde{h}}^i\left(q\right)=\left([x_{{}^f}^i\left(q\right),x_{{}^u}^i\left(q\right)],[y_{{}^f}^i\left(q\right),y_{{}^u}^i\left(q\right)],[z_{{}^f}^i\left(q\right),z_{{}^u}^i\left(q\right)]\right)$,$q=1,2,\cdots l$, represent the range of values for each component. If the interval average distance between any interval coordinate point ${\tilde{h}}^j\left(p\right)=\left([x_{{}^f}^j\left(p\right),x_{{}^u}^j\left(p\right)],[y_{{}^f}^j\left(p\right),y_{{}^u}^j\left(p\right)],[z_{{}^f}^j\left(p\right),z_{{}^u}^j\left(p\right)]\right)\notin {\tilde{H}}^i$, $j=1,2,\cdots ,n$,$p=1,2,\cdots k$ and interval coordinate point ${\tilde{h}}^i\left(q\right)$ is defined as:

$$\begin{array}{c}{\tilde{d}}_q=(\sqrt{{\left(x_f^i\left(q\right)-x_f^j\left(p\right)\right)}^2+{\left(y_f^i\left(q\right)-y_f^j\left(p\right)\right)}^2+{\left(z_f^i\left(q\right)-z_f^j\left(p\right)\right)}^2}+\\ \hspace{1em}\hspace{1em}\sqrt{{\left(x_u^i\left(q\right)-x_u^j\left(p\right)\right)}^2+{\left(y_u^i\left(q\right)-y_u^j\left(p\right)\right)}^2+{\left(z_u^i\left(q\right)-z_u^j\left(p\right)\right)}^2})÷2\end{array}$$

(1)

Then called

$${\tilde{d}}_{N-ij}=\tilde{d}\left({\tilde{H}}^i,{\tilde{h}}^j\left(p\right)\right)=\min \left({\tilde{d}}_1,{\tilde{d}}_2,\cdots ,{\tilde{d}}_l\right)$$

(2)

is the nearest neighbor interval average distance between interval coordinate dataset ${\tilde{H}}^i$ and interval coordinate point ${\tilde{h}}^j\left(p\right)$.

Definition 2.

Correlation degree between unequal interval coordinate datasets.

Assuming that there are m interval coordinate datasets array $\tilde{H}=\left\{{\tilde{H}}^1,{\tilde{H}}^2,\cdots ,{\tilde{H}}^i,\cdots ,{\tilde{H}}^m\right\}$ and any other interval coordinate dataset ${\tilde{H}}^j=\left\{{\tilde{h}}^j\left(1\right),{\tilde{h}}^j\left(2\right),\cdots ,{\tilde{h}}^j\left(k\right)\right\}\notin \tilde{H}$, where the interval coordinate points ${\tilde{h}}^j\left(p\right)=\left([x_{{}^f}^j\left(p\right),x_{{}^u}^j\left(p\right)],[y_{{}^f}^j\left(p\right),y_{{}^u}^j\left(p\right)],[z_{{}^f}^j\left(p\right),z_{{}^u}^j\left(p\right)]\right)$,$p=1,2,\cdots k$. Then the correlation degree between interval coordinate datasets is defined as:

$$\phi \left({\tilde{H}}^i,{\tilde{H}}^j\right)=\frac{1}{k}{\displaystyle \sum _{p=1}^k\eta \left({\tilde{H}}^i,{\tilde{h}}^j\left(p\right)\right)}.$$

(3)

In the formula:

$$\eta \left({\tilde{H}}^i,{\tilde{h}}^j\left(p\right)\right)=\frac{\underset{i}{\min }\underset{p}{\min }{\tilde{d}}_{ij}\left(p\right)+\delta \underset{i}{\max }\underset{p}{\max }{\tilde{d}}_{ij}\left(p\right)}{{\tilde{d}}_{ij}\left(p\right)+\delta \underset{i}{\max }\underset{p}{\max }{\tilde{d}}_{ij}\left(p\right)}.$$

(4)

Indicates the grey correlation coefficient of interval coordinate dataset ${\tilde{H}}^j$ and ${\tilde{H}}^i$ [35]. The real number represents the resolution coefficient, which is generally 0.5, and ${\tilde{d}}_{ij}\left(p\right)={\tilde{d}}_{N-ij}\left(p\right)=\tilde{d}\left(H^i,h^j\left(p\right)\right)$.

2.2. Mathematical Model Establishment

This article takes a three-dimensional situation as an example, assuming that there is a target i in the air, with the sensor v position as the coordinate origin, and the true state vector of the target at a certain time is $X_v^i=\left(r,\theta ,\beta \right)$, where $r$ is the oblique distance, $\theta$ is the azimuth angle and $\beta$ is the pitch angle. The target state estimation vector observed by sensor v is ${\widehat{X}}_v^i=\left(\widehat{r},\widehat{\theta },\widehat{\beta }\right)$, and the maximum systematic errors measured by sensor v are $\Delta r_m,\Delta {\theta }_m,\Delta {\beta }_m$. Due to the uncertainty of the deviation direction of the measurement systematic errors of the sensor, the observation state vector of the sensor can be subjected to bilateral interval processing to ensure that the true state vector of the target is within the processed range.

The observed value and true value of the sensor meet:

$$\{\begin{array}{c}\widehat{r}=r+\Delta r+n_r\\ \widehat{\theta }=\theta +\Delta \theta +n_{\theta }\\ \widehat{\beta }=\beta +\Delta \beta +n_{\beta }\end{array}$$

(5)

where, $n_r,n_{\theta },n_{\beta }$ is the random error of Gaussian white noise, which can be eliminated by filtering. This article mainly considers the systematic errors.

As shown in Figure 1, the possible distribution area of the true position extrapolated from the observation value ${\widehat{X}}_v^i=\left(\widehat{r},\widehat{\theta },\widehat{\beta }\right)$ is G, and G is jointly determined by intervals $\left(\widehat{r}-\left|\Delta r_m\right|,\widehat{r}+\left|\Delta r_m\right|\right)$, $\left(\widehat{\theta }-\left|\Delta {\theta }_m\right|,\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)$ and $\left(\widehat{\beta }-\left|\Delta {\beta }_m\right|,\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)$. This process is called bilateral intervalization.

2.3. Asynchronous Anti-Bias TTTA Algorithm Based on the Nearest Neighbor Interval Average Distance

This article takes a multi-sensor tracking system composed of two sensors and a fusion center as an example to derive a detailed asynchronous anti-bias TTTA algorithm based on the nearest neighbor interval average distance. The flowchart is shown in Figure 2.

The two sensors v and w search and track multiple batches of targets within the detection range, and the output track serial number sets are $T_v=\left\{1,2,\cdots ,m\right\}$ and $T_w=\left\{1,2,\cdots ,n\right\}$, respectively. In addition, due to the differences in sensor detection range, startup time, and search frequency, the number of tracks in sets $T_v$ and $T_w$ is not equal, that is, $m\ne n$. Then the correlation degree formula ${\phi }_{ij}=\phi \left(H_v^i,H_w^j\right)$ between different tracks of two sensors can be expressed using the correlation degree formula between interval coordinate datasets as

$${\tilde{\phi }}_{ij}=\phi \left({\tilde{H}}_v^i,{\tilde{H}}_w^j\right).$$

(6)

In the formula: $H_v^i,H_w^j$ represent the true state vector dataset of the ith track of sensor v and the jth track of sensor w, respectively; ${\tilde{H}}_v^i,{\tilde{H}}_w^j$ represent the datasets of the state estimation vectors of sensor v for the ith track of the captured target and sensor w for the jth track of the captured target after interval processing.

Firstly, the target observation state vectors are intervalized to obtain an interval coordinate datasets array ${\tilde{H}}_v=\left\{{\tilde{H}}_v^1,{\tilde{H}}_v^2,\cdots ,{\tilde{H}}_v^i,\cdots ,{\tilde{H}}_v^m\right\}$, ${\tilde{H}}_w=\left\{{\tilde{H}}_w^1,{\tilde{H}}_w^2,\cdots ,{\tilde{H}}_w^i,\cdots ,{\tilde{H}}_w^m\right\}$. Subsequently, the interval coordinate datasets are transformed to the Cartesian coordinate system of the fusion center through coordinates, and the datasets of the sensor with low sampling rate are selected as the reference datasets, so as to calculate the nearest neighbor interval average distance between the jth reference dataset and all comparison datasets according to Definition 1; according to Definition 2, the correlation degree ${\phi }_j$ between the jth reference dataset and all the comparison datasets is obtained. By analogy, the correlation degree between other reference datasets and all the comparison datasets is calculated, and the correlation degree matrix ${\phi }_{ij}$ of all the datasets of the two sensors is established. ${\phi }_{ij}$ reflects the degree of correlation between tracks. The higher the value, the greater the probability that two tracks are the same target. Finally, the corresponding objective function is established according to the correlation degree matrix, and the Jonker–Volgenant algorithm and the classical allocation method are combined to traverse the global track assignment to obtain the optimal solution.

The following is a detailed derivation of the algorithm process using two three-coordinate sensors as an example, and analogical calculations can be performed for multiple sensors.

2.3.1. Data Reception

Assuming that the two sensors are asynchronous and have different scanning cycles, T is the data processing cycle of the fusion center. Taking the c processing cycle $\left[\left(c-1\right)T,T\right]$ as an example, the number of tracks reported by sensors v and w are m and n, respectively, then the reported track coordinate datasets $A_v\left(c\right)$ and $A_w\left(c\right)$ can be expressed as

$$A_v\left(c\right)=\left\{A_v^1\left(c\right),A_v^2\left(c\right),\cdots ,A_v^i\left(c\right),\cdots ,A_v^m\left(c\right)\right\},$$

(7)

$$A_w\left(c\right)=\left\{A_w^1\left(c\right),A_w^2\left(c\right),\cdots ,A_w^j\left(c\right),\cdots ,A_w^n\left(c\right)\right\}.$$

(8)

In the formula, $A_v^i\left(c\right),A_w^j\left(c\right)$, respectively, represent the ith and jth track data of sensors v and w within the c processing cycle. Considering that the sensor network system has leakage points due to transmission channel damage in practice, the number of track points reported by the sensor may not be equal. If the ith track of sensors v contains $n_v^i$ track points, the coordinate dataset of the ith track is

$${\widehat{A}}_v^i\left(c\right)=\left\{{\widehat{X}}_v^i\left(t_v^1\right),{\widehat{X}}_v^i\left(t_v^2\right),\cdots ,{\widehat{X}}_v^i\left(t_v^{n_v^i}\right)\right\}$$

(9)

In the formula, ${\widehat{X}}_v^i\left(t_v^q\right)$ represents the polar coordinate state estimate of the qth track point of the ith track reported by sensors v for the cth processing cycle, $q=1,2,\cdots n_v^i$; $t_v^q$ represents the time stamp of the corresponding track point.

Similarly, the coordinate dataset of the jth track of sensors w containing $n_w^j$ track points can be expressed as

$${\widehat{A}}_w^j\left(c\right)=\left\{{\widehat{X}}_w^j\left(t_w^1\right),{\widehat{X}}_w^j\left(t_w^2\right),\cdots ,{\widehat{X}}_w^j\left(t_w^{n_w^j}\right)\right\}.$$

(10)

The track points $\widehat{X}$ in the track dataset reported by the sensor include oblique range $r$, azimuth angle $\theta$, and pitch angle $\beta$.

2.3.2. Systematic Errors Intervalization

As shown in Figure 1, sensor v is located at the origin. Due to the uncertainty of the systematic error direction, the observation values are processed using a bilateral interval method. Therefore, the shadow space G is jointly determined by $\left(\widehat{r}-\left|\Delta r_m\right|,\widehat{r}+\left|\Delta r_m\right|\right)$,$\left(\widehat{\theta }-\left|\Delta {\theta }_m\right|,\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)$, and $\left(\widehat{\beta }-\left|\Delta {\beta }_m\right|,\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)$, and the spatial vertex is denoted as $C_1,C_2,C_3,C_4,C_5,C_6,C_7,C_8$. To convert vertex polar coordinates to Cartesian coordinate system with sensor v as the origin

$$\left[\begin{array}{c}X\\ Y\end{array}\right]=R·\Phi ·{\rm B},Z=R^1·{{\rm B}}^1$$

(11)

In the formula:

$$\begin{array}{c}X=\left[\begin{array}{cccccccc}{x}_{C_1}& x_{C_2}& x_{C_3}& x_{C_4}& x_{C_5}& x_{C_6}& x_{C_7}& x_{C_8}\end{array}\right]\\ Y=\left[\begin{array}{cccccccc}{y}_{C_1}& y_{C_2}& y_{C_3}& y_{C_4}& y_{C_5}& y_{C_6}& y_{C_7}& y_{C_8}\end{array}\right]\\ Z=\left[\begin{array}{cccccccc}{z}_{C_1}& z_{C_2}& z_{C_3}& z_{C_4}& z_{C_5}& z_{C_6}& z_{C_7}& z_{C_8}\end{array}\right]\\ R=\left[\begin{array}{cccccccc}\widehat{r}+\left|\Delta r_m\right|\hfill & & \widehat{r}-\left|\Delta r_m\right|& & \widehat{r}+\left|\Delta r_m\right|& & \widehat{r}-\left|\Delta r_m\right|& \\ & \widehat{r}+\left|\Delta r_m\right|& & \widehat{r}-\left|\Delta r_m\right|& & \widehat{r}+\left|\Delta r_m\right|& & \widehat{r}-\left|\Delta r_m\right|\end{array}\right]\\ \Phi =\left[\begin{array}{cccccccc}\sin \left(\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)& \sin \left(\widehat{\theta }-\left|\Delta {\theta }_m\right|\right)& 0& 0& 0& 0& 0& 0\\ \cos \left(\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)& \cos \left(\widehat{\theta }-\left|\Delta {\theta }_m\right|\right)& 0& 0& 0& 0& 0& 0\\ 0& 0& \sin \left(\widehat{\theta }-\left|\Delta {\theta }_m\right|\right)& \sin \left(\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)& 0& 0& 0& 0\\ 0& 0& \cos \left(\widehat{\theta }-\left|\Delta {\theta }_m\right|\right)& \cos \left(\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)& 0& 0& 0& 0\\ 0& 0& 0& 0& \sin \left(\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)& \sin \left(\widehat{\theta }-\left|\Delta {\theta }_m\right|\right)& 0& 0\\ 0& 0& 0& 0& \cos \left(\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)& \cos \left(\widehat{\theta }-\left|\Delta {\theta }_m\right|\right)& 0& 0\\ 0& 0& 0& 0& 0& 0& \sin \left(\widehat{\theta }-\left|\Delta {\theta }_m\right|\right)& \sin \left(\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)\\ 0& 0& 0& 0& 0& 0& \cos \left(\widehat{\theta }-\left|\Delta {\theta }_m\right|\right)& \cos \left(\widehat{\theta }+\left|\Delta {\theta }_m\right|\right)\end{array}\right]\\ {\rm B}=\left[\begin{array}{cccccccc}\cos \left(\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)& 0& 0& 0& 0& 0& 0& 0\\ 0& \cos \left(\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)& 0& 0& 0& 0& 0& 0\\ 0& 0& \cos \left(\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)& 0& 0& 0& 0& 0\\ 0& 0& 0& \cos \left(\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)& 0& 0& 0& 0\\ 0& 0& 0& 0& \cos \left(\widehat{\beta }-\left|\Delta {\beta }_m\right|\right)& 0& 0& 0\\ 0& 0& 0& 0& 0& \cos \left(\widehat{\beta }-\left|\Delta {\beta }_m\right|\right)& 0& 0\\ 0& 0& 0& 0& 0& 0& \cos \left(\widehat{\beta }-\left|\Delta {\beta }_m\right|\right)& 0\\ 0& 0& 0& 0& 0& 0& 0& \cos \left(\widehat{\beta }-\left|\Delta {\beta }_m\right|\right)\end{array}\right]\\ R^1=\left[\begin{array}{cccccccc}\widehat{r}+\left|\Delta r_m\right|& \widehat{r}+\left|\Delta r_m\right|& \widehat{r}-\left|\Delta r_m\right|& \widehat{r}-\left|\Delta r_m\right|& \widehat{r}+\left|\Delta r_m\right|& \widehat{r}+\left|\Delta r_m\right|& \widehat{r}-\left|\Delta r_m\right|& \widehat{r}-\left|\Delta r_m\right|\end{array}\right]\\ {{\rm B}}^1=\left[\begin{array}{cccccccc}\sin \left(\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)& \sin \left(\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)& \sin \left(\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)& \sin \left(\widehat{\beta }+\left|\Delta {\beta }_m\right|\right)& \sin \left(\widehat{\beta }-\left|\Delta {\beta }_m\right|\right)& \sin \left(\widehat{\beta }-\left|\Delta {\beta }_m\right|\right)& \sin \left(\widehat{\beta }-\left|\Delta {\beta }_m\right|\right)& \sin \left(\widehat{\beta }-\left|\Delta {\beta }_m\right|\right)\end{array}\right]\end{array}$$

The interval coordinates after the systematic errors are intervalized can be expressed as:

$$\{\begin{array}{c}{\tilde{h}}_v=\left(\left[x_f,x_u\right],\left[y_f,y_u\right],\left[z_f,z_u\right]\right)\\ x_f=\min \left\{x_{C_1},x_{C_2},x_{C_3},x_{C_4},x_{C_5},x_{C_6},x_{C_7},x_{C_8}\right\}\hfill \\ y_f=\min \left\{y_{C_1},y_{C_2},y_{C_3},y_{C_4},y_{C_5},y_{C_6},y_{C_7},y_{C_8}\right\}\hfill \\ z_f=\min \left\{z_{C_1},z_{C_2},z_{C_3},z_{C_4},z_{C_5},z_{C_6},z_{C_7},z_{C_8}\right\}\hfill \\ x_u=\max \left\{x_{C_1},x_{C_2},x_{C_3},x_{C_4},x_{C_5},x_{C_6},x_{C_7},x_{C_8}\right\}\hfill \\ y_u=\max \left\{y_{C_1},y_{C_2},y_{C_3},y_{C_4},y_{C_5},y_{C_6},y_{C_7},y_{C_8}\right\}\hfill \\ z_u=\max \left\{z_{C_1},z_{C_2},z_{C_3},z_{C_4},z_{C_5},z_{C_6},z_{C_7},z_{C_8}\right\}\hfill \end{array}$$

(12)

Using Formula (12), the ith track coordinate dataset of sensor v in the cth processing cycle can be converted into a track interval coordinate dataset

$${\tilde{S}}_v^i\left(c\right)=\left\{{\tilde{s}}_v^i\left(1\right),{\tilde{s}}_v^i\left(2\right),\cdots ,{\tilde{s}}_v^i\left(n_v^i\right)\right\}$$

(13)

In the formula: ${\tilde{s}}_v^i\left(t_v^q\right)=\left([x_{{}^f}^i\left(t_v^q\right),x_{{}^u}^i\left(t_v^q\right)],[y_{{}^f}^i\left(t_v^q\right),y_{{}^u}^i\left(t_v^q\right)],[z_{{}^f}^i\left(t_v^q\right),z_{{}^u}^i\left(t_v^q\right)]\right)$ represents the interval coordinate vector of the qth track point after the ith track reported by sensor v has undergone interval processing, $q=1,2,\cdots n_v^i$, and $t_v^q$ represent the time stamp of the corresponding track point.

Similarly, the jth track coordinate dataset of sensor w in the cth processing cycle is converted into a track interval coordinate dataset

$${\tilde{S}}_w^j\left(c\right)=\left\{{\tilde{s}}_w^j\left(1\right),{\tilde{s}}_w^j\left(2\right),\cdots ,{\tilde{s}}_w^j\left(n_w^j\right)\right\}$$

(14)

2.3.3. Coordinate Transformation

The correlation of tracks in a multi-sensor tracking system needs to be carried out in a unified coordinate system. Due to the fact that stations in the actual system are not at the same point and are far away, and considering the influence of the Earth’s curvature, the sensor and fusion center are not in the same plane, as shown in Figure 3.

If we simply establish their respective rectangular coordinate systems for coordinate transformation, coordinate transformation error will be introduced to affect the TTTA. Therefore, it is necessary to take the FC station center rectangular coordinate system as the unified coordinate system for data processing under the Earth-Centered Earth-Fixed (ECEF) coordinate system, and the interval-processed track data are converted to the Cartesian coordinate system of the interval coordinate through geographical coordinate transformation [36]. Subsequently, the ith track interval coordinate dataset of sensor v in the cth processing cycle can be obtained:

$${\tilde{H}}_v^i\left(c\right)=\left\{{\tilde{h}}_v^i\left(1\right),{\tilde{h}}_v^i\left(2\right),\cdots ,{\tilde{h}}_v^i\left(n_v^i\right)\right\}$$

(15)

In the formula: ${\tilde{h}}_v^i\left(t_v^q\right)=\left([x_{{}^f}^i\left(t_v^q\right),x_{{}^u}^i\left(t_v^q\right)],[y_{{}^f}^i\left(t_v^q\right),y_{{}^u}^i\left(t_v^q\right)],[z_{{}^f}^i\left(t_v^q\right),z_{{}^u}^i\left(t_v^q\right)]\right)$ the interval coordinate of the qth track point of the ith track from sensor v in the same coordinate system for the cth processing cycle, $q=1,2,\cdots ,n_v^i$

Similarly, the the jth track interval coordinate dataset of sensor w in the cth processing cycle can be obtained:

$${\tilde{H}}_w^j\left(c\right)=\left\{{\tilde{h}}_w^j\left(1\right),{\tilde{h}}_w^j\left(2\right),\cdots ,{\tilde{h}}_w^j\left(n_w^j\right)\right\}$$

(16)

Finally, the interval coordinate datasets arrays for sensor v and sensor w reporting tracks can be obtained, which are

$${\tilde{H}}_v\left(c\right)=\left\{{\tilde{H}}_v^1\left(c\right),{\tilde{H}}_v^2\left(c\right),\cdots ,{\tilde{H}}_v^i\left(c\right),\cdots ,{\tilde{H}}_v^m\left(c\right)\right\}$$

(17)

$${\tilde{H}}_w\left(c\right)=\left\{{\tilde{H}}_w^1\left(c\right),{\tilde{H}}_w^2\left(c\right),\cdots ,{\tilde{H}}_w^i\left(c\right),\cdots ,{\tilde{H}}_w^m\left(c\right)\right\}$$

(18)

2.3.4. Reference Datasets Selection

Because the sensors are asynchronous and the sampling rate is not equal, selecting the suitable reference datasets can reduce the introduction of errors in calculating the nearest neighbor interval average distance which can represent the real distance between two tracks. As shown in Figure 4, assuming two tracks come from the same target of different sensors, the sampling rate of sensor w is smaller than that of sensor v, and the startup time is inconsistent. Due to the existence of errors, the tracks do not overlap.

From the figure, it can be seen that the track points of sensors v and w are distributed in a disordered manner. Within time t₁, when selecting the jth track as the reference dataset, there are only two nearest neighbor interval average distances obtained from $j_1,j_2$; When selecting the ith track as the reference dataset, there are four nearest neighbor interval average distances (calculated by $i_1,i_2,i_3,i_4$). Obviously, the nearest neighbor interval average distance obtained from $j_1,j_2$ is closer to the true distance between the two tracks, and the information is more representative; moreover, due to the deviation between the nearest neighbor interval average distance and the true distance, and the fact that the number of the nearest neighbor interval average distance obtained from the reference dataset of the ith track is significantly greater than that of the jth track, the calculation error will increase. Therefore, before calculating the track correlation degree, selecting the track dataset from sensors with a lower sampling rate as the reference dataset is necessary.

2.3.5. Derivation of Track Correlation Degree Matrix

Assuming that the sampling rate of sensor w is low, the interval coordinate datasets of its tracks are selected as the reference datasets. According to the definition of the nearest neighbor interval average distance between the interval coordinate dataset and the interval coordinate point in Definition 1, calculate the nearest neighbor interval average distance matrix between the interval coordinate datasets of all comparison tracks and the jth reference dataset in the c processing cycle as

$${\tilde{\Psi }}_j\left(c\right)={\left[{\tilde{d}}_{i,p}\right]}_{m\times n_w^j}=\left[\begin{array}{cccc}{\tilde{d}}_{1,1}& {\tilde{d}}_{1,2}& \cdots & {\tilde{d}}_{1,n_w^j}\\ {\tilde{d}}_{2,1}& {\tilde{d}}_{2,2}& \cdots & {\tilde{d}}_{2,n_w^j}\\ ⋮& ⋮& & ⋮\\ {\tilde{d}}_{m,1}& {\tilde{d}}_{m,2}& \cdots & {\tilde{d}}_{m,n_w^j}\end{array}\right]$$

(19)

In the formula:

$${\tilde{d}}_{i,p}=\tilde{d}\left({\tilde{H}}_v^i\left(c\right),{\tilde{h}}_w^j\left(p\right)\right),i=1,2,\cdots ,m,{\tilde{h}}_w^j\left(p\right)\in {\tilde{H}}_w^j\left(c\right)$$

Subsequently, according to Definition 2, the column vector of the correlation degree between the jth track of sensor w and all tracks of sensor v can be obtained as

$${\phi }_j\left(c\right)={\left[{\tilde{\phi }}_{ij}\left(c\right)\right]}_{m\times 1}={\left[\phi \left({\tilde{H}}_v^i\left(c\right),{\tilde{H}}_w^j\left(c\right)\right)\right]}_{m\times 1}$$

(20)

In the formula:

$$\begin{gathered} \phi \left({\tilde{H}}_v^i\left(c\right),{\tilde{H}}_w^j\left(c\right)\right)=\frac{1}{n}{\displaystyle \sum _{p=1}^n\eta \left({\tilde{H}}_v^i\left(c\right),{\tilde{h}}_w^j\left(p\right)\right)} \\ \hspace{1em}\hspace{1em}\eta \left({\tilde{H}}_v^i\left(c\right),{\tilde{h}}_w^j\left(p\right)\right)=\frac{\underset{i}{\min }\underset{p}{\min }{\tilde{d}}_{i,p}+\delta \underset{i}{\max }\underset{p}{\max }{\tilde{d}}_{i,p}}{{\tilde{d}}_{i,p}+\delta \underset{i}{\max }\underset{p}{\max }{\tilde{d}}_{i,p}} \end{gathered}$$

By analogy, the correlation degree column vectors of all tracks of sensor w and all tracks of sensor v are obtained, and finally the correlation degree matrix of tracks is

$${\phi }_{ij}\left(c\right)={\left[{\phi }_1\left(c\right),{\phi }_2\left(c\right),\cdots ,{\phi }_n\left(c\right)\right]}_{m\times n}$$

(21)

2.3.6. Asynchronous TTTA Judgment

The $m\times n$ dimensional asynchronous track correlation degree matrix ${\phi }_{ij}\left(c\right)$ of Formula (21) is obtained from the previous article, and $m\ne n$. Considering the problem of tracks missing due to transmission channel damage and completing all TTTA, expand ${\phi }_{ij}\left(c\right)$ into a square matrix, take ${\rm M}=\max \left(m,n\right)$, and use 0 to complete ${\phi }_{ij}\left(c\right)$ into an M-order square matrix ${\mu }_{ij}\left(c\right)$, and the supplemented elements are considered as virtual targets. Then, the Jonker–Volgenant algorithm and the classical allocation method are combined to perform the correlation judgment on the tracks.

Setting:

$${\delta }_{ij}=\{\begin{array}{l}1\begin{array}{ccc}& & \mathrm{Indicates}\mathrm{that}\mathrm{the}\mathrm{i}\mathrm{th}\mathrm{track}\mathrm{and}\mathrm{the}\mathrm{j}\mathrm{th}\mathrm{track}\mathrm{are}\mathrm{the}\mathrm{same}\mathrm{target}\end{array}\hfill \\ 0\begin{array}{ccc}& & \mathrm{Indicates}\mathrm{that}\mathrm{the}\mathrm{i}\mathrm{th}\mathrm{track}\mathrm{and}\mathrm{the}\mathrm{j}\mathrm{th}\mathrm{track}\mathrm{are}\mathrm{not}\mathrm{the}\mathrm{same}\mathrm{target}\end{array}\hfill \end{array}.$$

(22)

The objective function is marked as:

$$L\left(c\right)={\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^M{\delta }_{ij}\cdot }}{\mu }_{ij}\left(c\right).$$

(23)

Since the greater the correlation degree between tracks means the greater the probability of tracks being the same target, the objective function in (23) can be converted into the following two-dimensional allocation problem:

$$\{\begin{array}{c}\underset{{\delta }_{ij}}{\min }{\displaystyle \sum _{i=1}^M{\displaystyle \sum _{j=1}^M{\delta }_{ij}\cdot \left(1-{\mu }_{ij}\left(c\right)\right)}}\\ {\displaystyle \sum _{i=1}^M{\delta }_{ij}=1,\forall j\in \left\{1,2,\cdots ,M\right\}}\\ {\displaystyle \sum _{j=1}^M{\delta }_{ij}=1,\forall i\in \left\{1,2,\cdots ,M\right\}}\end{array}.$$

(24)

Then the Jonker–Volgenant algorithm is used to minimize the value of the objective function, and the correlation degree matrix between tracks can be solved. Among them, the Jonker–Volgenant algorithm is much faster than the famous Hungarian algorithm. The complexity of this algorithm is lower, and its computational complexity is O(N³). The specific steps will not be repeated here. Finally, the TTTA judgment can be obtained.

3. Experiments and Performance Analysis

3.1. Simulation Environment and Evaluation Index

The simulation environment is the Windows 10 64-bit operating system and Matlab R2021a software platform. The main parameters of the computer used in the simulation are as follows: the processor is Intel Core i5-11400 H, the main frequency is 2.70 GHz, and the memory is 16.0 GB.

Assuming that there are two three-dimensional coordinate sensors in a multi-sensor tracking system that search and track targets in a common area, and the sensors have no omissions or errors in the target search: the targets move uniformly in a straight line within the area, with the initial direction and velocity uniformly distributed within the range of [0, 2π] and [100 m/s, 300 m/s]. The sensor parameter settings are shown in

Table 1. The sensor parameter settings.

Parameter Settings	Sampling Cycle	Geographical Coordinates	Oblique Distance Random Error	Azimuth Angle Random Error	Pitch Angle Random Error
v	6 s	(0°, 0°, 0 m)	50 m	0.001 rad	0.001 rad
w	4 s	(0°, 0.2°, 0 m)	50 m	0.001 rad	0.001 rad

.

In order to ensure the reliability and superiority of the experimental results, each group of data was subjected to $W=100$ Monte Carlo experiments. The number of false associations $F_{\max }\left(c\right)$, the maximum false association rate $N_{\max }\left(c\right)$, and the average correct association rate $Ez\left(c\right)$ are used as the evaluation indexes of the TTTA effect. $F_{\max }\left(c\right)$ is defined as the number of experiments with track false association in $W$ times Monte Carlo experiments. $N_{\max }\left(c\right)$ is defined as the maximum false association rate when the false association occurs in $W$ times Monte Carlo experiments. $Ez\left(c\right)$ is defined as follow:

$$Ez\left(c\right)=\frac{{\displaystyle \sum _{n=1}^W{Z}_n\left(c\right)}}{WL}\times 100\%.$$

(25)

In the formula: $Z_n\left(c\right)$ is the number of tracks correctly associated by the FC in the nth experiment within the c processing cycle. L is the total number of target tracks in the common observation area.

3.2. Algorithm Performance Analysis

To demonstrate the effectiveness, superiority, and anti-bias performance of the algorithm designed in this article (it is called NN-IAD), we changed the simulation scenario conditions and conducted simulation comparative experiments with SD-IRS [33] and KNN-I [37] algorithms.

3.2.1. Algorithm Effectiveness Analysis

The parameter settings for simulation scenario 1 are shown in

Table 2. The parameter settings for simulation scenario 1.

The Processing Cycle of the FC	The Number of Targets	Maximum Oblique Distance Random Error	Maximum Azimuth Angle Random Error	Maximum Pitch Angle Random Error
50 s	20	1000 m	0.01 rad	0.02 rad

.

Assuming that the sensors have the same maximum systematic errors, the sampling cycle of sensor w is 4 s, and k represents the ratio of the sampling period of sensor v to the sampling period of sensor w.

$$k=\frac{T_w}{T_v}.$$

(26)

Table 3. Time consuming of algorithms with different values of k.

Sampling Period Ratio	k = 1	k = 1.5	k = 2	k = 2.5	k = 3
time consumed	0.07023	0.05654	0.05030	0.04888	0.04235

lists the comparison of calculation time under different cycle ratios. It can be seen that as the ratio increases from 1 to 3, the time consumption decreases significantly. This is because the increase of the ratio is equivalent to the decrease of the number of sensor v track points that need to be processed, thus reducing the amount of calculation.

Table 4. The change of the false association times $F_{\max }\left(c\right)$ and the average correct association rate $Ez\left(c\right)$ with different k values and startup time difference.

Startup Time Difference (s)	The Evaluation Indexes	Sampling Cycle Ratio
		k = 1	k = 1.5	k = 2	k = 2.5	k = 3
1	$F_{\max }\left(c\right)$	1	0	1	0	2
	$Ez\left(c\right)$	99.9%	100%	99.9%	100%	99.8%
1.5	$F_{\max }\left(c\right)$	2	2	1	0	2
	$Ez\left(c\right)$	99.8%	99.8%	99.9%	100%	99.8%
2	$F_{\max }\left(c\right)$	0	0	0	3	3
	$Ez\left(c\right)$	100%	100%	100%	99.7%	99.7%
2.5	$F_{\max }\left(c\right)$	3	1	2	1	3
	$Ez\left(c\right)$	99.7%	99.9%	99.8%	99.9%	99.7%

lists the false association times $F_{\max }\left(c\right)$ and the average correct association rate $Ez\left(c\right)$ under different cycle ratios and startup times. It can be seen that the average correct association rate is 99.7~100%, and the maximum track false association occurs only 3 times. The high association rate indicates that the startup time and radar sampling rate have no special impact on the correct association rate of the algorithm. This is because different sampling rates only affect the number of track points. However, the center of gravity of the algorithm in this paper is to calculate the nearest neighbor interval average distance between each point of all comparison tracks and the reference track points, and there is no requirement for the track length. In essence, the time difference of sensor startup also makes the time of track points different. However, the algorithm in this paper does not require time parameters, so it has no significant impact on the final settlement result. The effectiveness of the algorithm is proved.

Table 5. Average correct association rate $Ez\left(c\right)$ of different noise distributions.

Different Noise Distribution Forms	Gaussian Distribution	Rayleigh Distribution	Exponential Distribution	Uniform Distribution
$Ez\left(c\right)$	99.8%	99.7%	99.8%	100%

shows the average correct association rate of the algorithm in different noise distribution forms. It can be seen from the data in the table that the algorithm in this paper can maintain a high average correct association rate under the four noise distribution forms of Gaussian, Rayleigh, Exponential, and Uniform, and is less affected by the noise distribution forms, which proves that the algorithm has strong anti-noise performance.

3.2.2. Systematic Errors Analysis

Assuming that sensors v and w are turned on simultaneously, the parameter settings for simulation scenario 2 are shown in

Table 6. The parameter settings for simulation scenario 2.

The Processing Cycle of the FC	The Number of Targets	Maximum Oblique Distance Random Error	Maximum Azimuth Angle Random Error	Maximum Pitch Angle Random Error
30 s	20	500 m~2000 m	0.005 rad~0.02 rad	0.005 rad~0.02 rad

. In the experiment, simulation analysis was conducted by fixing two systematic error parameters of two sensors and changing one systematic error parameter.

Figure 5 shows the average correct association rate of tracks by changing different systematic error parameters. From the three graphs, it can be seen that the changes in systematic errors have a small impact on the average correct association rate of the algorithm, all of which remain above 98%. The algorithm has been proven to be effective and has strong anti-bias performance.

3.2.3. Comparison of Algorithm Performance

In this section, the superiority of this algorithm compared to other comparative algorithms is analyzed from three aspects: different maximum systematic error parameters, processing cycles, and target numbers.

Simulation scenario 3: Assuming that the maximum systematic errors of sensor v are 1000 m, 0.01 rad, and 0.01 rad, respectively. The sensor w takes different maximum systematic errors and compares the algorithm’s anti-bias performance. The parameter settings are shown in

Table 7. The parameter settings for simulation scenario 3.

The Processing Cycle of the FC	The Number of Targets	Maximum Oblique Distance Random Error	Maximum Azimuth Angle Random Error	Maximum Pitch Angle Random Error
30 s	20	100~1500 m	0.005~0.015 rad	0.005~0.015 rad

, and the simulation results are shown in

Table 8. Algorithm performance under different maximum systematic errors.

Maximum Systematic Errors			NN-IAD			SD-IRS
$\Delta r_m$	$\Delta {\theta }_m$	$\Delta {\beta }_m$	$Ez\left(c\right)$	$F_{\max }\left(c\right)$	$N_{\max }\left(c\right)$	$Ez\left(c\right)$	$F_{\max }\left(c\right)$	$N_{\max }\left(c\right)$
100	0.01	0.01	100.0%	0	0	90.6%	91	30%
500	0.01	0.01	99.9%	1	10%	75.4%	100	45%
1000	0.01	0.01	99.8%	2	10%	57.8%	100	50%
1500	0.01	0.01	99.9%	1	10%	53.4%	100	55%
1000	0.005	0.01	100.0%	0	0	53.6%	100	55%
1000	0.015	0.01	99.9%	1	10%	63.6%	100	50%
1500	0.01	0.005	99.9%	1	10%	55.6%	100	50%
1500	0.01	0.015	100.0%	0	0	55.5%	100	55%
500	0.005	0.005	99.9%	1	10%	74.1%	100	45%
500	0.015	0.015	99.6%	4	10%	75.9%	100	45%
1000	0.005	0.005	100.0%	0	0	63.7%	100	55%
1000	0.015	0.015	100.0%	0	0	56.6%	100	55%
1500	0.005	0.005	99.9%	1	10%	53.5%	100	55%
1500	0.015	0.015	99.8%	2	10%	59.4%	100	55%

.

From Table 8, it can be seen that whether it is the maximum distance error or the maximum angle (azimuth, pitch angle) error, as the maximum systematic errors increase, the performance of the SD-IRS algorithm sharply decreases. The average correct association rate decreases from 90.55% in the case of small system errors to only about 55%, and the maximum false association rate reaches 55%, indicating poor correlation effect. This is because the SD-IRS algorithm further introduces errors during the process of virtual and real changes, without compensating for systematic errors, and the selection of threshold directly affects the correlation effect. The algorithm in this article considers systematic errors and selects the nearest neighbor interval average distance as the standard for measuring the distance between tracks, which is closer to the actual distance and has good anti-bias performance. Therefore, the algorithm is less affected by the maximum systematic errors, proving its superiority.

Simulation scenario 4: Assuming that the maximum systematic errors of sensors v and w are the same, change the target numbers and fusion center processing cycles to compare the superiority of the algorithm. The parameter settings are shown in

Table 9. The parameter settings for simulation scenario 4.

The Processing Cycle of the FC	The Number of Targets	Maximum Oblique Distance Random Error	Maximum Azimuth Angle Random Error	Maximum Pitch Angle Random Error
20:2:50 s	10:1:50	1000 m	0.01 rad	0.02 rad

.

Figure 6 shows the comparison of the false association numbers $F_{\max }\left(c\right)$, the maximum false association rate $N_{\max }\left(c\right)$, and the average correct association rate $Ez\left(c\right)$ of the three algorithms under different target numbers and processing cycles. From the figure, it can be seen that although the SD-IRS algorithm does not perform time-domain registration and directly establishes a correlation degree matrix for association, transforming the real number sequence into a mixed sequence of interval and real numbers can reduce the accuracy of track calculation. At the same time, due to not considering the impact of large systematic errors, it affects the average correct association rate of TTTA, which decreases with the increase of target numbers. From the number of false associations, it can be seen that the algorithm has poor association performance and cannot complete a complete correct association. The maximum false association rate remains around 50%. Although the KNN-I algorithm considers the existence of systematic errors and has a high average correct association rate of over 98%, the measurement criteria between tracks in the algorithm is greatly influenced by the distribution of targets. As the number of targets increases, the average correct association rate of the algorithm decreases significantly overall, with a minimum of only 95.5%. Even from the perspective of maximum false association rate, a complete false association with jitter reaching 100% occurs at uncertain moments. The NN-IAD algorithm in this article not only considers the impact of systematic errors, but also processes the systematic errors through intervalization. At the same time, the interval average distance is used as the distance measurement between tracks, which is closer to the actual nearest neighbor distance compared to the KNN-I algorithm and does not accumulate and propagate time-domain registration error; the algorithm maintains an average correct association rate of over 99% for different target numbers and processing cycles. Whether it is the number of the false association or the maximum false association rate, the algorithm proposed in this paper remains at a relatively low level compared to other algorithms, and the fluctuation is relatively small, ensuring accurate TTTA in different situations. The algorithm has strong superiority and robustness.

3.2.4. Algorithm Complexity Analysis

On the premise of the same experimental environment, the complexity of the algorithm is analyzed by analyzing the time consumption of the algorithm under different numbers of targets. The parameter settings for simulation scenario 5 are shown in

Table 10. The parameter settings for simulation scenario 5.

The Processing Cycle of the FC	The Number of Targets	Maximum Oblique Distance Random Error	Maximum Azimuth Angle Random Error	Maximum Pitch Angle Random Error
30 s	10:2:24	1000 m	0.01 rad	0.02 rad

.

Figure 7 shows a comparison of the time consumption of three algorithms under different target numbers. It can be seen that as the number increases, the algorithm’s time consumption gradually increases. Compared to the other two algorithms, the algorithm proposed in this paper takes more time because the track points are subjected to systematic error interval processing while also considering the influence of Earth curvature, which increases the amount of computation, and is in line with practical operations. This indicates that the algorithm proposed in this paper sacrifices the amount of computation to improve the accuracy of TTTA and has a certain degree of complexity.

4. Conclusions

In order to solve the problem of asynchronous unequal rate track association under large systematic errors, the systematic error interval processing was performed on the track data, and the nearest neighbor interval average distance was defined. An asynchronous anti-bias TTTA algorithm based on the nearest neighbor interval average distance was proposed. This algorithm does not require time-domain registration and error registration, and directly correlates asynchronous unequal rate tracks without being affected by processing cycles and noise distribution forms, achieving high-precision TTTA under large systematic errors; this algorithm maintains low false association numbers, low maximum false association rate, and high correct association rate in different situations, with good anti-bias performance, superiority, and robustness.