Passive bistatic cooperative localization method under azimuth one-dimensional measurement

Abstract

This invention proposes a passive bi-station cooperative positioning method based on one-dimensional azimuth measurement. By constructing an orthogonal space for azimuth observation, a measurement equation can be established between one-dimensional measurement information, time delay information, and the unknown source position. Furthermore, by introducing weighted least squares technology, the near-closed solution can be quickly obtained to determine the target source position. Under Gaussian noise, when the measurement error is small and the deviation is negligible relative to the variance, the mean square error of the positioning can reach the theoretical estimation accuracy. The proposed invention can overcome the target positioning performance limitations caused by the drastic reduction in the number of operational radars due to electronic warfare, and has significant engineering application value.

Description

Technical Field

[0001] This invention belongs to the field of radar technology, and further relates to a cooperative passive target localization technology under the conditions of insufficient observation and fewer than 3 cooperative radars. It can be used for high-precision localization of air targets when the radar can only provide one-dimensional angle measurement and the number of available radar observations is less than 3. Background Technology

[0002] Passive source localization is a fundamental problem in radar, sonar, and communication applications. Various methods have been proposed to address this problem, such as Time of Arrival (TOA), Time Difference of Arrival (TDOA), Angle of Arrival (AOA), and combinations thereof. However, the nonlinear and nonconvex relationship between source location and measurement makes this problem challenging. Maximum Likelihood Estimator (MLE) is asymptotically efficient but requires an initial solution guess to avoid iterative implementation using numerical global search. To ensure global convergence, this guess must be within a small convex neighborhood of the actual solution. To address these challenges, an improved algorithm utilizes TDOAs and their derivatives to develop a closed-form TDOA localization solution that, under certain conditions, reaches the Cramér-Rao bound. Furthermore, by analyzing the necessary optimal conditions of the least-squares cost function of the squared distance difference, a superior localization solution is obtained using the generalized trust region subproblem technique.

[0003] Regarding the use of AOA for passive source localization, most existing methods currently use a one-dimensional AOA, i.e., azimuth angle, for measurement in two-dimensional space. In three-dimensional space, a two-dimensional AOA composed of azimuth and elevation angles is used to estimate the target's three-dimensional positioning information. For two-dimensional AOA estimation, some literature proposes a three-dimensional target motion analysis method based on a partial linear estimator, utilizing azimuth and elevation angles, and exhibiting good positioning performance under large sample conditions. Additionally, some literature proposes a hybrid TDOA and azimuth ranging scheme, which outperforms TDOA alone in accuracy. Based on this, an algorithm that utilizes hybrid azimuth and TDOA measurements and applies geometric constraints to improve performance can be applied to both 2D and 3D scenarios, but requires line search and lacks a closed-form solution.

[0004] In practice, due to the significant differences in the scattering characteristics of targets in different directions (especially targets with special materials and structural designs), multiple radar stations conducting coordinated observations may not all be able to effectively detect the target. In extreme cases, at most only two radar stations may be able to observe it. This problem seems difficult to address with some existing passive localization methods, as they require at least three effective stations to achieve accurate target localization. Therefore, it is meaningful to study the hybrid localization problem using only two stations and missing measurements.

[0005] Under the condition of dual-base cooperative observation with separate transmitting and receiving stations, it is assumed that the two receiving stations and the transmitting station achieve high-precision time-frequency synchronization through navigation discipline. For the above input conditions, this patent mainly studies the high-precision cooperative 3D positioning of the target by combining the TOA information acquired by the two receiving stations and the one-dimensional azimuth measurement information acquired by each station. We propose a simple method to determine the source location by constructing a new relationship between the hybrid measurement and the unknown source location. Theoretical analysis shows that under Gaussian noise, when the measurement error is small and the deviation is negligible relative to the variance, the mean square error (MSE) of the proposed solution can reach the CRB accuracy. Simulations verify the performance of the proposed method. Theoretically, this estimator can be directly extended to more than two stations while maintaining the CRB performance.

[0006] Currently, domestic and international research on dual-station passive positioning mainly focuses on observations such as TDOA positioning, AOA positioning, and TDOA-AOA joint positioning. However, research on joint three-dimensional target positioning methods based on time-frequency synchronization and combined with TOA and dimensional measurement information has not been published. Therefore, there is an urgent need for a three-dimensional target positioning method that can be used with only two stations and one-dimensional angle measurement capability. Summary of the Invention

[0007] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a passive dual-station cooperative positioning method under one-dimensional azimuth measurement, which solves the problem of high-precision three-dimensional positioning of airborne targets under the conditions of a small number of observable radars and missing measurement information in actual engineering.

[0008] The technical solution of this invention is: a passive dual-station cooperative positioning method under one-dimensional azimuth measurement, comprising:

[0009] Establish the two-radar range observation equations for a three-dimensional spatial target;

[0010] Acquire the two radar angle observation variables of a three-dimensional target;

[0011] By constructing orthogonal vectors of radar angle observation vectors, the obtained angle observation variable values ​​are transformed, and the two radar angle observation equations of three-dimensional spatial targets are obtained in the orthogonal domain.

[0012] By combining the obtained range observation equation and the obtained angle observation equation, the two radar observation equations for the three-dimensional target position are calculated.

[0013] Based on the obtained set of observation equations, the estimated position of the space target is obtained using the weighted least squares method.

[0014] The equations for establishing the two-radar range observation of a three-dimensional spatial target include:

[0015] Let the true position of the unknown target in three-dimensional space be u = [x, y, z]. T ∈R 3 The locations of the two observation stations are respectively denoted as and in[] T This is the matrix transpose operation; the distance r from the target to the radar. i Represented as

[0016] r i =||us i || (1)

[0017] Where i = 1, 2 represents the i-th radar;

[0018] In the actual measurement process, the target's distance from the radar is:

[0019]

[0020] In the formula Δn i To define the covariance matrix of a measurement error vector that follows a Gaussian zero-mean distribution, we define its covariance matrix.

[0021] Squaring both sides of equation (1) and substituting equation (2) into the equation, retaining only the first-order error term, we obtain the two radar range observation equations for a three-dimensional spatial target:

[0022]

[0023] The acquisition of the two radar angle observation variables of the three-dimensional spatial target includes:

[0024] The angular observation variable θ relative to the three-dimensional spatial target is measured at each observation station. i for

[0025]

[0026] in Δθ represents the true angle between the radar and the target. i For angle measurement error, the angle measurement error vector form of two-coordinate radar observation in three-dimensional space is Δθ=[Δθ1,Δθ2]. T It follows a Gaussian distribution, and its covariance matrix is ​​defined as E(ΔθΔθ). T )=Q θ .

[0027] The equations for obtaining the two radar angle observations of a three-dimensional spatial target in the orthogonal domain include:

[0028] By combining the azimuth measurement angle with the target and radar station location information, an observation vector in the angle measurement domain is constructed:

[0029] (u(1:2)-s i (1:2))=||u(1:2)-s i (1:2)||·[cosθ i sinθ i ] T (5)

[0030] Construct the direction vector [cosθ] i sinθ i ] T orthogonal vectors [sinθ] i ,-cosθ i Multiply both sides by [sinθ] i ,-cosθ i ] can be obtained

[0031] [sinθ i ,-cosθ i ](u(1:2)-s i (1:2))=0 (6)

[0032] Extending the above equation to three-dimensional vector space, we get the following expression:

[0033] b i T (us i )=0 (7)

[0034] In the formula b i =[sinθ i ,-cosθ i ,0] T ;

[0035] During radar measurement, approximately... but

[0036]

[0037] By substituting equation (8) into equation (7), we obtain the two radar angle observation equations:

[0038]

[0039] In the formula

[0040] The calculation yields two radar observation equations for the three-dimensional target position, including:

[0041] The range and angle observation equations for the first and second radars are calculated. The observation equations for the first and second radars are then combined into a single system, and the range and angle observation equations for the two radars in three-dimensional space are converted into matrix form, as shown in the following equation.

[0042]

[0043] In the formula,

[0044]

[0045] Δφ is the error matrix

[0046] Δφ=[Δn T ,Δθ T ] T (12)

[0047] In the formula, Δn=[Δn1,Δn2] T Δθ=[Δθ1,Δθ2] T ;

[0048] P = bikdiag{[B1,D1]}; (13)

[0049] Where blkdiag{} is a diagonal matrix generating function with matrices as elements, and the expressions for B1 and D1 are as follows:

[0050]

[0051] The expressions for the observation vector h1 and the observation matrix G1 are as follows:

[0052]

[0053]

[0054] The method of obtaining the estimated location of a spatial target using weighted least squares includes:

[0055] The intermediate variables of the observation equation are obtained using the weighted least squares method. The estimated value:

[0056]

[0057] In the formula, the weighting matrix is

[0058] W1=P(bikdiag{Q r Q θ}) -1 P T (18)

[0059] Since the weighted matrix includes target location information, it is replaced with an identity matrix in the first calculation; the initial estimate of the target location is obtained through equation (17), as shown in the following equation:

[0060]

[0061] Based on the first target position estimate, substitute the value into equation (18) to obtain a new weighting matrix, and obtain the precise estimate of the target after iterative processing through equation (17). Repeat the iterative process twice to obtain the position of the converged spatial target.

[0062] The advantages of this invention compared to the prior art are as follows:

[0063] This invention relates to an airborne target localization algorithm under conditions of missing dimensions and a small number of observations, where errors exist. Traditional passive time-of-flight (TOF) localization technology requires at least four receiving stations to achieve high-precision three-dimensional target localization. Introducing two-dimensional angle measurement information can overcome the binary problem, enabling target localization with only three receiving stations. However, in certain application scenarios, radar only possesses one-dimensional high-precision angle measurement capabilities, and it is difficult to guarantee the collaborative operation of multiple receiving stations (≥3) at all times in a battlefield environment. To address the aforementioned problems of missing measurement information and insufficient number of coordinating receiving stations, this invention, based on the engineering foundation of high-precision navigation discipline synchronization on the transceiver platform, proposes a passive collaborative three-dimensional high-precision localization method that combines only the time delay information and one-dimensional azimuth angle measurement information acquired from two receiving stations. By constructing an orthogonal space for azimuth observation, a measurement equation can be established between one-dimensional measurement information, time delay information, and the unknown source position. Secondly, by introducing intermediate variables in the spatial position measurement matrix process, the spatial position information of the target can be directly extracted. Furthermore, to effectively overcome the positioning error problem introduced by system measurement noise, an iterative method based on weighted least squares is introduced to quickly obtain a near-closed solution to determine the target source position. Under Gaussian noise, when the measurement error is small and the deviation is negligible relative to the variance, the mean square error of the positioning can reach the theoretical estimation accuracy. The proposed invention can overcome the target positioning performance challenges caused by the drastic reduction in the number of operational radars due to electronic warfare, and has significant engineering application value. Attached Figure Description

[0064] Figure 1 This is a schematic diagram of dual-station collaborative three-dimensional spatial target observation under one-dimensional azimuth measurement.

[0065] Figure 2 This describes the processing flow of a dual-station collaborative high-precision target positioning method under one-dimensional azimuth measurement.

[0066] Figure 3This is a graph showing the statistical variation of target positioning accuracy with distance measurement error under different height differences between two receiving stations. Detailed Implementation

[0067] The implementation and effects of the present invention will be described in further detail below.

[0068] like Figure 1 , 2 As shown, the application scenario of this invention is as follows: This invention can be applied to the problem of high-precision three-dimensional positioning of airborne targets under conditions of limited observable radar numbers and missing measurement information in practical engineering. By establishing observation vectors under missing-dimensional measurements through orthogonal space, and by introducing intermediate variables and iterative least-squares weighting techniques, high-precision three-dimensional positioning of airborne targets can be achieved under error conditions, improving the stability of the algorithm under noisy measurements. Its main steps are as follows:

[0069] Step 1: Obtain the range observation equations of two radars for a three-dimensional target.

[0070] The true position of the unknown target in three-dimensional space is u = [x, y, z]. T ∈R 3 The locations of the two observation stations are respectively denoted as and (in[] T (This is a matrix transpose operation) The dual-station system has the capability to measure arrival time and azimuth. The distance r of the target reaching the radar can be obtained by multiplying the arrival time measured by the i-th observation station by the propagation speed. i , represented as

[0071] r i =||us i || (1)

[0072] Where i = 1, 2. In actual measurement, the true distance to reach cannot be obtained; only the following measured values ​​are available:

[0073]

[0074] In the formula, Δn i The measurement error vector follows a Gaussian zero-mean distribution, and its covariance matrix is ​​E(Δn). i Δn i T )=Q r .

[0075] Squaring both sides of equation (1) and substituting equation (2) into it, we can obtain the radar range observation equation relative to the target by retaining only the first-order error term:

[0076]

[0077] Step 2: Obtain the two radar angle observation variables of the three-dimensional target.

[0078] The azimuth angle θ relative to the space target measured by each observation station i for

[0079]

[0080] in It is the actual angle between the radar and the target, Δθ i The angle measurement error is represented by the vector form Δθ=[Δθ1,Δθ2]. T Follows a Gaussian distribution, E(ΔθΔθ) T )=Q θ .

[0081] Step 3 transforms the angle measurement values ​​obtained in Step 2 by constructing orthogonal vectors of radar angle observation vectors, and obtains the two radar angle observation equations of the three-dimensional target in the orthogonal domain. The specific process is as follows:

[0082] 1) Combining azimuth measurement angles with target and radar station location information, construct observation vectors in the angle measurement domain:

[0083] (u(1:2)-s i (1:2))=||u(1:2)-s i (1:2)||·[cosθ i sinθ i ] T (5)

[0084] 2) Construct the direction vector [cosθ] i sinθ i ] T orthogonal vectors [sinθ] i ,-cosθ i Multiply both sides of equation (5) by [sinθ] i ,-cosθ i ] can be obtained

[0085] [sinθ i ,-cosθ i ](u(1:2)-s i (1:2))=0 (6)

[0086] 3) To maintain dimensionality with the distance measurement equation, the above equation is extended to a three-dimensional vector space, resulting in the following expression.

[0087] b i T (us i )=0 (7)

[0088] In the formula b i =[sinθ i ,-cosθ i ,0] T .

[0089] In radar measurements, the target's angular error is related to the angular resolution and the target signal-to-noise ratio. Generally, the angular error is small, approximately...

[0090]

[0091] 4) By substituting equation (8) into equation (7), the following equations for the two radar angle observations can be obtained:

[0092]

[0093] In the formula

[0094] Step 4, combined with Step 1 and Step 3, yields two sets of radar observation equations for the three-dimensional target position. The specific process is as follows:

[0095] Expressing equations (3) and (9) in matrix form, we can obtain

[0096]

[0097] In the formula, h1 is the observation vector, and G1 is the observation matrix. Their defining equations are as follows:

[0098]

[0099]

[0100]

[0101] Δφ is the error matrix

[0102]

[0103] In the formula, Δn=[Δn1,Δn2] T Δθ=[Δθ1,Δθ2] T

[0104] P is the error coefficient matrix.

[0105] P = bikdiag{[B1,D1]} (15)

[0106] Where blkdiag{} is a diagonal matrix generating function with matrices as elements, and the expressions for B1 and D1 are shown in the following equations:

[0107]

[0108] Step 5: Based on the observation equations obtained in Step 4 under the measurement error, the position of the space target can be obtained by combining the weighted least squares technique. The specific process is as follows:

[0109] We can obtain the weighted least squares technique.

[0110]

[0111] In the formula, the weighting matrix is

[0112] W1=P(bikdiag{Q r Q θ}) -1 P T (18)

[0113] Since the weighted matrix contains information related to the target location, it can be replaced by an identity matrix. Based on the first coarse estimate of the target location, this value is substituted into (18) to obtain a new weighted matrix, and the fine estimate of the target is obtained through (17). The converged target location result can be obtained by iterating twice. The estimated target location is...

[0114]

[0115] The effects of the present invention will be further illustrated below through simulation experiments.

[0116] Figure 3 (a)-(d) are statistical graphs showing the target position estimation error as a function of the distance measurement variance, given the aerial target position and the spatial positions of the two measuring radars, under the condition of fixed angle measurement error. The target position and the positions of the two coordinate radars in the simulation experiment are shown in Table 1 below:

[0117] Table 1 Spatial Location of Targets and Observation Radars

[0118] x(m) y(m) z(m) Target 100000 100000 10000 Radar 1 500 0 5000 Radar 2 -500 0 4500

[0119] It can be seen that the proposed method achieves aerial target positioning accuracy under different dual-station altitude differences when the distance measurement variance does not exceed 100m. 2 Under these conditions, it maintains high positioning accuracy with minimal fluctuations in positioning error. Even when the distance measurement variance exceeds 100m... 2 Under these circumstances, positioning accuracy gradually deteriorates.

[0120] Furthermore, a comparison of simulation results under different dual-station height difference conditions shows that the target positioning accuracy improves significantly with the increase of the height baseline between the two stations. Under conditions of small range measurement error, the target positioning accuracy improves from the 10km level to 4km under height differences of 500m and 3000m. Since only two radar receiving stations are involved in the positioning, a sufficient height-oriented observation baseline is required to compensate for the loss of height-dimensional information caused by the lack of elevation angle measurements. The proposed method achieves this within a range measurement variance of no more than 100m. 2 Under the condition that the altitude spacing between the two stations reaches 3km, it is possible to achieve kilometer-level three-dimensional high-precision positioning of aerial targets hundreds of kilometers away.

Claims

1. A passive dual-station cooperative positioning method under one-dimensional azimuth measurement, characterized in that, include: Establish the two-radar range observation equations for a three-dimensional spatial target; Acquire the two radar angle observation variables of a three-dimensional target; By constructing orthogonal vectors of radar angle observation vectors, the obtained angle observation variable values ​​are transformed, and the two radar angle observation equations of three-dimensional spatial targets are obtained in the orthogonal domain. By combining the obtained range observation equation and the obtained angle observation equation, the two radar observation equations for the three-dimensional target position are calculated. Based on the obtained set of observation equations, the estimated position of the space target is obtained using the weighted least squares method; The equations for establishing the two-radar range observation of a three-dimensional spatial target include: Let the true position of the unknown target in three-dimensional space be... The locations of the two observation stations are respectively denoted as and ;in[] T For matrix transpose; distance from the target to the radar Represented as （1） in i= 1,2 indicates the first, second, and third. i One radar; In the actual measurement process, the target's distance from the radar is: （2） In the formula To define the covariance matrix of a measurement error vector that follows a Gaussian zero-mean distribution, we define its covariance matrix. ; Squaring both sides of equation (1) and substituting equation (2) into the equation, retaining only the first-order error term, we obtain the two radar range observation equations for a three-dimensional spatial target: (3)； The acquisition of the two radar angle observation variables of the three-dimensional spatial target includes: The angular observation variables relative to the three-dimensional spatial target measured at each observation station for (4) in This represents the actual angle between the radar and the target. For angle measurement error, the vector form of angle measurement error in two-coordinate radar observation in three-dimensional space. It follows a Gaussian distribution, and its covariance matrix is ​​defined. ; The equations for obtaining the two radar angle observations of a three-dimensional spatial target in the orthogonal domain include: By combining the azimuth measurement angle with the target and radar station location information, an observation vector in the angle measurement domain is constructed: （5） Construct direction vector orthogonal vectors Multiply both sides simultaneously achievable （6） Extending the above equation to three-dimensional vector space, we get the following expression: （7） In the formula ; During radar measurement, approximately... ,but （8） By substituting equation (8) into equation (7), we obtain the two radar angle observation equations: （9） In the formula , ; The calculation yields two radar observation equations for the three-dimensional target position, including: The range and angle observation equations for the first and second radars are calculated. The observation equations for the first and second radars are then combined into a single system, and the range and angle observation equations for the two radars in three-dimensional space are converted into matrix form, as shown in the following equation. （10） In the formula, (11)； Error matrix （12）； In the formula, , ; （13）； in blkdiag {} represents a diagonal matrix generation function that uses matrices as elements. and The expressions are as follows: ， (14)。 2. The passive dual-station cooperative positioning method under one-dimensional azimuth measurement according to claim 1, characterized in that, Observation vector and observation matrix The expression is （15） （16）。 3. The passive dual-station cooperative positioning method under one-dimensional azimuth measurement according to claim 1, characterized in that, The method of obtaining the estimated location of a spatial target using weighted least squares includes: The intermediate variables of the observation equation are obtained using the weighted least squares method. The estimated value: (17) In the formula, the weighting matrix is (18) Since the weighted matrix includes target location information, it is replaced with an identity matrix in the first calculation; the initial estimate of the target location is obtained through equation (17), as shown in the following equation: (19) Based on the first target position estimate, substitute the value into equation (18) to obtain a new weighting matrix, and obtain the precise estimate of the target after iterative processing through equation (17). Repeat the iterative process twice to obtain the position of the converged spatial target.

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