Pure Azimuth Target Location Method Based on Mobile Nodes

Abstract

The method for bearing-only target localization based on mobile nodes disclosed by the present invention comprises the following steps: establishing a noise model; constructing a linear least squares estimator; constructing an auxiliary variable estimator; constructing a generalized pseudo-linear equation set; introducing an auxiliary variable matrix to obtain the target position. The method for bearing-only target localization based on mobile nodes of the present invention is based on a distance-related noise model. By improving the least squares estimation method, a closed-form solution can be obtained, avoiding complex iterative calculations. At the same time, prior information of measurement noise is not required, avoiding an increase in system complexity. Moreover, by constructing a generalized pseudo-linear equation and introducing an auxiliary variable matrix, the present invention can effectively improve the accuracy and stability of target localization.

Description

Technical Field

[0001] The present invention belongs to the technical field of target positioning, and particularly relates to a bearing-only target positioning method based on mobile nodes. Background Art

[0002] Target positioning methods based on angle-of-arrival (AOA) measurement information have been widely applied in civil and military fields such as wireless sensor networks, aerospace electronic systems, and automatic control, and thus have attracted a large number of scholars to conduct research on them for decades. AOA measurement information is usually obtained by sensors listening to the signals emitted by the target. On a two-dimensional plane, the target position can be estimated using the AOA information measured by a single mobile sensor. The mobile sensor can be an unmanned aerial vehicle or an unmanned underwater vehicle.

[0003] From the perspective of estimation theory, AOA positioning is a non-linear estimation problem, and its main challenge comes from the highly non-linear relationship between the AOA measurement value and the true target position. The current mainstream methods include grid search method, maximum likelihood estimation method, closed-form pseudo-linear estimation method, etc. The grid search method obtains the best fit of the measurement by searching all possible target positions, which is time-consuming in calculation. The maximum likelihood estimation method has asymptotic unbiasedness and effectiveness, but usually requires a search method (such as the steepest descent algorithm, Gauss-Newton iteration algorithm) to obtain the estimated value, with high computational complexity. In addition, when the initial value is set unreasonably, the estimation performance of the maximum likelihood estimation method will be unstable.

[0004] The closed-form pseudo-linear estimation method usually constructs a linear least squares estimator and can obtain a closed-form solution, so the amount of calculation is small. However, due to the correlation between the measurement matrix and the pseudo-linear noise vector, when the measurement noise is large, the estimation accuracy will seriously decline. To improve the estimation accuracy, scholars have proposed many improvement methods based on the linear least squares estimator, such as bias compensation method, weighted auxiliary variable estimation method, weighted least squares estimation method, etc. These improvement methods require the statistical information (such as variance) of the measurement noise as prior information to calculate the bias compensation or weighted matrix. However, in actual engineering, the prior information usually needs to be estimated. On the one hand, estimating the prior information will increase the complexity of the system; on the other hand, when the estimation error of the prior information is large, the bias compensation or weighted matrix will also contain large noise, which has an adverse impact on the target positioning accuracy and is difficult to meet the positioning requirements. Summary of the Invention

[0005] The purpose of the present invention is to provide a bearing-only target positioning method based on mobile nodes, which can not only obtain a closed-form solution with low computational complexity, but also effectively improve the positioning accuracy and has good stability.

[0006] The technical solution adopted by the present invention is: a pure bearing target positioning method based on a mobile node, comprising the following steps:

[0007] Step 1: Establish an AOA measurement value model based on the nonlinear function of the real AOA value of the target signal and the AOA measurement noise, where the AOA measurement noise is an additive Gaussian white noise with zero mean and variance related to the target-node distance;

[0008] Step 2: Substitute the AOA measurement value obtained in step 1 into the nonlinear function of the true AOA value in step 1 to obtain a pseudo-linear equation, and solve it by the least squares method to obtain the estimated position of the target;

[0009] Step 3: Use the target estimated position obtained in step 2 to calculate the AOA estimate, establish an auxiliary variable matrix, and use the auxiliary variable estimation method to obtain a new target estimated position;

[0010] Step 4, calculate the second-order expression of the pseudo-linear equation in step 2 and expand it into a vector form, and then combine the vector form with the pseudo-linear equation in step 2 to obtain a generalized pseudo-linear equation system;

[0011] Step 5: Use the Taylor series expansion formula to rewrite the generalized pseudo-linear equations obtained in step 4 into new pseudo-linear equations. Based on the new target estimated position obtained in step 3, a new auxiliary variable matrix is ​​established and introduced to obtain the target positioning estimate.

[0012] The present invention is also characterized in that:

[0013] Step 1 is as follows: At time k∈{0,1,2,...,N-1}, the actual AOA value θ of the target signal k for:

[0014]

[0015] In formula (1), r k =[x k ,y k ] T is the position of the mobile node at time k, p = [p x ,p y ] T is the target position, and the function f(.,.) represents θ k Non-linear relationships with targets and node positions;

[0016] The AOA measurement value is obtained by observing the target through the mobile node for:

[0017]

[0018] In formula (2), n kis the AOA measurement noise, which is additive white Gaussian noise with zero mean and variance i.e., The variance is related to the signal-to-noise ratio, and the signal-to-noise ratio is related to the distance between the target and the node. Therefore, the variance is modeled as a function related to the distance:

[0019]

[0020] In Equation (3), d k represents the distance from the target to the node at time k, α is the path loss exponent, and d 0 is the reference distance. Set d 0 = 1, represents the variance of the AOA measurement noise at d 0 .

[0021] Step 2 is specifically: Substitute the AOA measurement value into Equation (1) to obtain the following pseudo-linear equation:

[0022] F p = b + η (4)

[0023] In Equation (4), F is the measurement matrix, which is represented by the following Equation (5a); b is the measurement vector, which is represented by the following Equation (5b); and η is the noise vector, which is represented by the following Equation (5c):

[0024]

[0025]

[0026] η = [η 0 , η 1 ,..., η N-1 T (5c)

[0027] In Equations (5a) and (5b), u k is represented by the following Equation (5d):

[0028]

[0029] The target estimated position is obtained by solving with the least squares method:

[0030]

[0031] In Equation (6), the superscript T represents the transpose operation of the matrix.

[0032] The specific steps of Step 3 are:

[0033] ​Step 3.1: Estimated target position obtained in step 2 Substitute into formula (1) to calculate the AOA estimate

[0034] Step 3.2: Based on the results of step 3.1 Create auxiliary variable matrix H:

[0035]

[0036] Step 3.3: Use H T Replace the measurement matrix F in formula (6) T , and get the new estimated target position:

[0037]

[0038] Step 4 is specifically as follows: Calculate the pseudo linear equation, i.e., the second-order expression of formula (4):

[0039] bb T =(Fp-η)(Fp-η) T =Fpp T F T +Γ (9)

[0040] In formula (9), Γ = ηη T -Fpη T -ηp T F T Represents the noise term, and writes formula (9) in vector form:

[0041] Vech(bb T )=Vech(Fpp T F T )+Vech(Γ) (10)

[0042] In formula (10), Vech(Fpp T F T ) is a vector of size N(N+1) / 2×1, and its elements are:

[0043]

[0044]

[0045] Combining formula (4) with formula (10), we get a set of generalized pseudo-linear equations:

[0046]

[0047] In formula (13), A(p) represents the quadratic function of p, and ε is the noise term.

[0048] Step 5 is specifically as follows: By means of the first-order Taylor series expansion, formula (13) is linearly expanded at to obtain the following result:

[0049]

[0050] In formula (14), is obtained by replacing p in A(p) with and is given by the following formula:

[0051]

[0052]

[0053]

[0054] After algebraic operations, formula (14) becomes a new pseudo-linear equation:

[0055]

[0056] In formula (18), G is the measurement matrix, is the data vector, and -ε is the pseudo-linear noise vector;

[0057] Using the new target estimated position in formula (7) and substituting it into formula (1) to calculate the new AOA estimated value and constructing a new auxiliary variable matrix from Multiplying both sides of formula (18) by D T , and replacing G on the right side of the equation with D and ignoring the noise term, we get:

[0058]

[0059] Solving it gives the target location estimation value:

[0060]

[0061] The beneficial effects of the present invention are as follows: The pure azimuth target location method based on mobile nodes of the present invention, based on the distance-related noise model, can obtain a closed-form solution by improving the least squares estimation method, avoiding the complex calculations of iteration; at the same time, it does not require the prior information of measurement noise, avoiding increasing the complexity of the system. Moreover, the present invention can effectively improve the accuracy and stability of target location by constructing a generalized pseudo-linear equation and introducing an auxiliary variable matrix. Description of the Drawings

[0062] Figure 1 is the target location scenario of mobile nodes based on AOA in the present invention;

[0063] Figure 2 is the simulation positioning scenario;

[0064] Figure 3 is when 0.04° ≤ σ θ ≤ 0.22°, the deviation norm relationship curves of the maximum likelihood estimation, least squares method, instrumental variables method, and the method of the present invention;

[0065] Figure 4 is when 0.04° ≤ σ θ ≤ 0.22°, the mean square error relationship curves of the maximum likelihood estimation, least squares method, instrumental variables method, and the method of the present invention. Detailed implementation manner

[0066] The present invention will be described in detail below in conjunction with the accompanying drawings and specific implementation manners.

[0067] The present invention provides a bearing-only target positioning method based on mobile nodes, and the specific steps are as follows:

[0068] Step 1, establish a noise model;

[0069] In the AOA positioning problem, the target position is unknown and needs to be estimated. As Figure 1 shown, at time k ∈ {0, 1, 2,..., N - 1}, the true AOA value θ k of the target signal is:

[0070]

[0071] where r k = [x k , y k T is the position of the mobile node at time k, p = [p x , p y T is the target position, θ k is the direction angle between the stationary target p and the node r k at time k, and the function f(.,.) represents the non-linear relationship between θ k and the target and node positions. In actual engineering, the true AOA value is unknown. By observing the target through the mobile node, the measured AOA value is:

[0072]

[0073] where n k is the AOA measurement noise, modeled as independent Gaussian white noise with zero mean and variance , that is In the present invention, the prior knowledge set is unknown.

[0074] The variance of the AOA noise is related to the signal-to-noise ratio, and the signal-to-noise ratio is related to the distance between the target and the node. Therefore, is modeled as a distance-related function:

[0075]

[0076] where d k represents the distance from the target to the node at time k, α is the path loss exponent, and d 0 is the reference distance (in the present invention, d 0 is set to 1), represents the variance of the AOA measurement noise at d 0 .

[0077] Step 2, construct a linear least squares estimator;

[0078] Using the established AOA target location model, the target location of the linear least squares estimate is deduced. Substituting the AOA measurement value into formula (1) and after rearrangement, the following pseudo-linear equation can be obtained:

[0079] F p = b + η (4)

[0080] where

[0081]

[0082]

[0083] η = [η 0 , η 1 ,..., η N-1 T (5c)

[0084]

[0085] Using the least squares estimation method, the target estimated location can be obtained:

[0086]

[0087] The least squares solution result is called the least squares estimator, where b is the measurement vector and the superscript T represents the transpose operation of the matrix.

[0088] Step 3, construct an auxiliary variable estimator;

[0089] Due to the correlation between the measurement matrix F and the noise vector η, the least squares estimator is biased. To reduce the bias, an auxiliary variable matrix H is established and used to replace the measurement matrix F T . Therefore, the pseudo-linear equation F T F p = F T b is modified to the new normal equation H T F p = H T b, and thus the auxiliary variable estimator is obtained. The estimated result of the target position is as follows T where the auxiliary variable matrix H is constructed from the AOA estimation values

[0090]

[0091] , that is: where the AOA estimation values

[0092]

[0093] are calculated by substituting the in formula (6) into formula (1), that is Step 4, construct the generalized pseudo-linear equations;

[0094] Although the accuracy of the auxiliary variable estimator is better than that of the least squares estimator, it still has a certain bias when the measurement noise is large. To further improve the positioning performance, the present invention constructs a set of generalized pseudo-linear equations. First, calculate the second-order expression of the pseudo-linear equation (4):

[0095] bb

[0096] bb T =(Fp - η)(Fp - η) T =Fpp T F T +Γ (9)

[0097] where Γ = ηη T -Fpη T -ηp T F T represents the noise term. Since both bb T and Fpp T F T are symmetric matrices, only the diagonal elements and the lower diagonal elements of the matrix in (9) are considered. Therefore, equation (9) can be written in vector form

[0098] Vech(bb T ) = Vech(Fpp T F T ) + Vech(Γ) (10)

[0099] Among them, Vech(Fpp T F T ) is a vector of size N(N + 1) / 2×1, and the elements inside are

[0100]

[0101]

[0102] Combining formula (4) and formula (10), a set of generalized pseudo-linear equations is obtained

[0103]

[0104] where A(p) represents a quadratic function of p, and ε is the noise term.

[0105] Step 5, introduce an auxiliary variable matrix to obtain the target position.

[0106] With the help of the first-order Taylor series expansion, formula (13) is linearly expanded at The result is as follows:[[]]

[0107]

[0108] Among them, is obtained by replacing p in A(p) with and is given by the following formula:[[]]

[0109]

[0110]

[0111]

[0112] After algebraic operations, formula (14) becomes

[0113]

[0114] The above equation can be regarded as a new pseudo-linear equation, where G is the measurement matrix, is the data vector, and -ε is the pseudo-linear noise vector.

[0115] Using in formula (7) and substituting it into formula (1) to calculate the new AOA estimate value And from this estimated value construct a new auxiliary variable matrix Multiply both sides of formula (18) by D T , and replace G in the right side of the equation with D, ignoring the noise term, to obtain

[0116]

[0117] Therefore, a new closed - form solution is obtained.

[0118]

[0119] The result calculated using formula (20) is the target location estimate proposed by the present invention.

[0120] Performance Analysis

[0121] Using the Monte Carlo simulation statistical method, the positioning performance of the proposed algorithm of the present invention is compared with the least - squares algorithm, the instrumental variable algorithm, and the maximum - likelihood estimation algorithm. Among them, the number of iterations of the maximum - likelihood estimation is specified as K = 10. The estimation error and the mean - square error are used as the accuracy measurement criteria for performance comparison. The estimation error and the mean - square error are calculated by the following formulas respectively:

[0122]

[0123]

[0124] Among them, represents the estimated value of p in the i - th Monte Carlo trial, and M = 2000 represents the number of Monte Carlo trials. In the simulation scenario, it is assumed that the target is located at [25, 300] T , and the mobile node moves along the p y = 0.8p x + 10 linear path. When 5 ≤ p x ≤ 35 and k = 0, 1,..., 49, the simulated positioning scenario is as Figure 2 shown, where the triangles represent the trajectories of the mobile node movements, and the circles represent the positions of the stationary targets.

[0125] When σ θ increases from 0.04° to 0.22° with a step size of 0.03°, the bias - norm results of the four estimators are as Figure 3 shown. Among the four estimators, the least - squares method shows the largest error, indicating that it has a serious bias problem. Although the instrumental variable method reduces the bias compared with the least - squares method, it still has a larger bias than the method of the present invention. For σ θ ≤ 0.19°, the bias of the maximum - likelihood estimation is less than that of the least - squares method, but for larger AOA noise (e.g., σ θ> 0.22°), the error of the maximum likelihood estimation will increase rapidly. This observation indicates that the maximum likelihood estimation method has a divergence problem. Since the method of the present invention utilizes the constructed generalized pseudo-linear equations, can utilize more information, and borrows the advantages of the auxiliary variable method, the method of the present invention is superior to other estimators.

[0126] Figure 4 The mean square error performance of the maximum likelihood estimation, least squares method, auxiliary variable method and the method of the present invention was compared for different σ θ values. When σ θ ≤ 0.19°, the least squares method is inferior to the other three estimators. Compared with the least squares method, the maximum likelihood estimation and the auxiliary variable method have smaller mean square errors. However, when σ θ is greater than 0.19°, the mean square error of the auxiliary variable method becomes very large. This observation indicates that for larger AOA noises, the mean square error performance of the auxiliary variable method becomes unstable. In addition, when σ θ > 0.19°, the problem of divergence of the estimation result will occur in the mean square error of the maximum likelihood estimation. Among the four estimation methods, the method of the present invention exhibits superior target localization performance.

Claims

1. Bearing-only target positioning method based on mobile nodes, It is characterized in that The following steps are involved: Step 1: Establish an AOA measurement value model based on the non-linear function of the true AOA value of the target signal and the AOA measurement noise. The AOA measurement noise is additive Gaussian white noise with zero mean and variance related to the target-node distance; at time , the true AOA value of the target signal is: （1） In formula (1), is the position of the mobile node at time k , is the target position, and the function represents the non-linear relationship between Step 2: Substitute the AOA measurement value obtained in Step 1 into the true AOA value non-linear function in Step 1, i.e., formula (1), to obtain the following pseudo-linear equation: （4） In Equation (4), is the measurement matrix; b is the measurement vector; is the noise vector; The estimated target position is obtained by solving the least squares method: （6） In Equation (6), the superscript T represents the transpose operation of a matrix; Step 3: Use the target estimated position obtained in step 2 to calculate the AOA estimate, establish an auxiliary variable matrix, and use the auxiliary variable estimation method to obtain a new target estimated position; the specific steps are: Step 3.

1. Substitute the target estimated position obtained in Step 2 into Formula (1) to calculate the AOA estimated value ; Step 3.

2. Based on the result obtained in Step 3.1 Establish an auxiliary variable matrix H : （8） Step 3.3: Use to replace the measurement matrix in formula (6) F T , and obtain a new estimated target position: （7） Step 4: Calculate the second-order expression of the pseudo-linear equation in step 2 and expand it into a vector form, then combine the vector form with the pseudo-linear equation in step 2 to obtain a generalized pseudo-linear equation system; specifically: calculate the second-order expression of the pseudo-linear equation, that is, formula (4): （9） In Equation (9), represents the noise term. Equation (9) can be written in vector form as: （10） In formula (10), is a vector of size ; Combining formula (4) with formula (10), we get a set of generalized pseudo-linear equations: （13） In formula (13), represents a quadratic function of and is the noise term; Step 5. Using the Taylor series expansion formula, rewrite the generalized pseudo-linear equations obtained in Step 4 into new pseudo-linear equations, and establish and introduce a new auxiliary variable matrix based on the new target estimated position obtained in Step 3 to obtain the target positioning estimation value. Specifically: By means of the first-order Taylor series expansion, linearly expand formula (13) at to obtain the following result: （14） In formula (14), is obtained by replacing in with and is given by the following formula: （15） （16） After algebraic operations, formula (14) becomes a new pseudo-linear equation: （18） In Equation (18), G is the measurement matrix, is the data vector, is the pseudo-linear noise vector; Use the new target estimated position in Equation (7) Substitute it into Equation (1) to calculate the new AOA estimated value , and Construct a new auxiliary variable matrix Multiply both sides of Equation (18) by D T , and use D to replace G on the right side of the equation. Ignoring the noise term, we get: （19） Solve to get the target location estimate: （20）。 2. The method for locating a target based on a mobile node according to claim 1, It is characterized in that The specific content of step 1 is: At time , the true AOA value of the target signal is: （1） In formula (1), is the position of the mobile node at time k , is the target position, and the function represents the non-linear relationship between The AOA measurement value obtained by observing the target through the moving nodes is as follows: （2） In Equation (2), is the AOA measurement noise, which is additive white Gaussian noise with zero mean and variance , that is ; Variance is related to the signal-to-noise ratio, and the signal-to-noise ratio is related to the distance between the target and the node. Therefore, the variance is modeled as a function related to the distance: （3） In Equation (3), d k represents the time k when the distance from the target to the node, is the path loss exponent, d 0 is the reference distance, set d 0 = 1, represents d 0 the variance of the AOA measurement noise at 3. The method for locating a target based on a mobile node according to claim 2, It is characterized in that The specific content of step 2 is as follows: Substitute the AOA measurement value into formula (1) to obtain the following pseudo-linear equation: （4） In formula (4), is the measurement matrix, which is represented by the following formula (5a); b is the measurement vector, which is represented by the following formula (5b); is the noise vector, which is represented by the following formula (5c): （5a） （5b） （5c） In formulas (5a) and (5b), u k It is represented by the following formula (5d): （5d） The estimated target position is obtained by solving the least squares method: （6） In Equation (6), the superscript T represents the transpose operation of a matrix.

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