AOA convex optimization underwater target fusion localization method considering anchor node drift

Abstract

This invention provides an AOA convex optimization underwater target fusion localization method considering anchor node drift. A Cartesian coordinate system is established as the reference coordinate system in the wireless sensor network. There are N arrays with positional drift and one target with an unknown position. After the target node transmits a signal, the sensor nodes acquire the measurement. The arctangent function is expanded using a Taylor series at the initial point, retaining only the polynomials of degree two and below. Based on the Gaussian assumption of the AOA measurement model, maximum likelihood estimation is used to locate the target. A positive semidefinite relaxation is applied to the maximum likelihood estimation problem. The target coordinates are then obtained by solving the problem. This invention provides better adaptability to node position errors and achieves higher localization accuracy.

Description

Technical Field

[0001] This invention relates to the field of underwater target localization, and in particular to an underwater target localization fusion method. Specifically, it relates to a wireless sensor network target localization method based on AOA (Angle-of-Arrival) measurements when node position errors exist. Background Technology

[0002] Since the beginning of the 21st century, countries around the world have been strengthening research, development, and application of marine science and technology to enhance their international maritime competitiveness. Therefore, the importance of safeguarding maritime rights has become increasingly prominent. Underwater Acoustic Sensor Networks (UASNs), based on acoustic information interaction, have been widely used in recent years.

[0003] Target localization is a crucial function of underwater acoustic sensor networks. Based on the different measurement information acquired by the nodes, target localization methods in wireless sensor networks can be categorized as follows: Time of Arrival (TOA), Time Difference of Arrival (TDOA), Angle-of-Arrival (AOA), Received Signal Strength (RSS), and methods combining multiple measurements. Among these measurements, RSS is highly sensitive to multipath effects and is unsuitable for underwater acoustic environments; TDOA is obtained by subtracting the TOA measurement, which leads to noise superposition and a 3dB performance loss; TOA measurement requires clock synchronization between the target and the node, which is difficult to achieve underwater, especially in non-cooperative target localization. AOA measurement, on the other hand, often yields better localization performance and is more suitable for underwater acoustic environments.

[0004] In recent years, many target localization methods based on AOA (Optical Object Architecture) for sensor networks have been proposed, such as triangulation, maximum likelihood estimation, and least squares estimation. However, triangulation cannot effectively utilize the information obtained from sensors and suffers from significant bias. Maximum likelihood estimation is prone to getting trapped in local optima, and least squares estimation may exhibit large biases due to its linear approximation when the error is substantial. Patent CN117686974A proposes a convex optimization target localization method based on AOA, which shows good performance. However, this method does not consider the impact of node position errors in the sensor network, which are prevalent in underwater acoustic sensor networks.

[0005] In summary, AOA-based target localization methods have better performance than other localization methods. However, existing AOA-based target localization methods still have many imperfections, so it is necessary to conduct in-depth research on AOA-based target localization methods. Summary of the Invention

[0006] To overcome the shortcomings of existing technologies, this invention provides an AOA convex optimization underwater target fusion positioning method considering anchor node drift. This method fully considers the impact of node position errors during modeling and proposes a more accurate maximum likelihood problem, thereby making the positioning method more adaptable to node position errors and achieving the goal of accurately locating the target.

[0007] The technical solution adopted by this invention to solve its technical problem includes the following steps:

[0008] Step 1: Establish a Cartesian coordinate system as the reference coordinate system in the wireless sensor network. There are N arrays with position drift and one target with an unknown position. The position coordinates of the i-th array at the time of deployment are:

[0009] v i =(x i ,y i ) T i = 1, ..., N

[0010] The coordinates of the target to be located are:

[0011] u 0 =(x 0 ,y 0 ) T ,

[0012] Each array passively acquires its AOA (Area of ​​Effect) measurement with the target. Assuming the node position error term follows a Gaussian distribution with zero mean, the actual but unknown position of the i-th array is:

[0013]

[0014] The relationship between the actual location and the deployment location is as follows:

[0015]

[0016] Where s i =(△x) i ,△y i ) T Gaussian white noise with zero mean, s i covariance matrix Φ i =δ i 2 I2, where I2 represents a 2×2 unit diagonal matrix;

[0017] Step 2: After the target node transmits the signal, the sensor node acquires the measurement.

[0018] Step 3: Obtain n through analysis i Statistical characteristics;

[0019] Step 4: Since the arctangent function is highly nonlinear, it needs to be approximated by Taylor series expansion. The arctangent function is expanded by Taylor series at the initial point (x0, y0), and only polynomials of the second degree and below are retained. The initial point (x0, y0) is obtained by the least squares method.

[0020] Step 5: Based on the Gaussian assumptions of the AOA measurement model, use maximum likelihood estimation to locate the target.

[0021] For the target location estimate:

[0022]

[0023] Step Six: Perform semidefinite relaxation on the maximum likelihood estimation problem;

[0024] Step 7: Use the CVX solver (interior point method) to solve the final convex optimization expression obtained in Step 6 to obtain the vector g = [x, y, x 2 ,y 2 [xy] T The specific value is obtained by solving the vector g, where the first part is the target x-coordinate and the second part is the target y-coordinate.

[0025] In step two, considering the angle measurement error and array position error, the AOA measurement obtained by the i-th array is... Represented as:

[0026]

[0027] In the formula, θ i Let ε be the true angle of the target relative to the i-th array. i Let ε be the measurement error of the i-th array AOA. i Model as a function with a mean of zero and a variance of . Gaussian white noise, n i Let ε be the error caused by the node position drift in the i-th array. i and s i They are independent of each other.

[0028] In step three, it can be seen from the measurement model that:

[0029]

[0030] Taking the tan sign of both sides of the above equation and using Taylor expansion to apply the same method to sin(θ) i +n i ) and cos(θ) i +n i Approximation:

[0031]

[0032] Solve the following equations:

[0033]

[0034] After ignoring the second-order terms, we get:

[0035]

[0036] Therefore n i It follows a pattern with a mean of zero and a variance of .

[0037] The Gaussian distribution.

[0038] Therefore, the total noise β i =θ i +n i It follows a mean of zero and a variance of:

[0039] The Gaussian distribution.

[0040] In step four, the polynomial is expressed as:

[0041]

[0042] in:

[0043]

[0044] The final simplified Taylor series approximation is expressed as:

[0045]

[0046] in:

[0047]

[0048] In step six, the objective function in the maximum likelihood estimation is rewritten as:

[0049] (θ-c-L1×x-L2×y-L3×x 2 -L4×y 2 -L5×xy) T

[0050] W(θ-c-L1×x-L2×y-L3×x 2 -L4×y 2 -L5×xy)

[0051] =Trace{W(θ-c-(L1×x+L2×y+L3×x 2 +L4×y 2 +L5×xy))}

[0052] (θ-c-(L1×x+L2×y+L3×x 2 +L4×y 2 +L5×xy)) T

[0053] in:

[0054]

[0055]

[0056] Define a new vector g = [x, y, x 2 ,y 2 [xy] T With matrix B = [L1, L2, L3, L4, L5], the objective function is transformed into:

[0057] Trace{W(θ-c-Bg)(θ-c-Bg) T}=Trace{W(Θ+C+BGB T -2Bgθ T +2Bgc T -2cθ T )}

[0058] in:

[0059] Θ=θθ T C = cc T G = gg T

[0060] At this point, the objective function has been transformed into a convex function, but regarding the constraints...

[0061] Θ=θθ T C = cc T G = gg T It was found that the non-convex constraint G = gg still exists. T Therefore, by using semidefinite relaxation, G = gg T Transform into convex constraints Observing the relationship between vectors g and u, we find that the following five implicit constraints hold:

[0062] g(1:2) = u,

[0063] g(3,1)=G(1,1),

[0064] g(4,1)=G(2,2),

[0065] g(5,1)=G(1,2),

[0066] g(5,1)=G(2,1).

[0067] The final convex optimized representation number is:

[0068]

[0069] st

[0070] g(1:2) = u,

[0071] g(3,1)=G(1,1),

[0072] g(4,1)=G(2,2),

[0073] g(5,1)=G(1,2),

[0074] g(5,1)=G(2,1),

[0075] Θ=θθ T C = cc T .

[0076] An electronic device includes: one or more processors; a memory; and one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, and the one or more programs are configured to perform the methods described above.

[0077] A computer-readable storage medium storing program code that can be invoked by a processor to perform the method described above.

[0078] The beneficial effects of this invention fully consider the impact of node position errors on the positioning system. It models the node position errors in the measurement model, performs approximate analysis of the error terms through Taylor expansion, and then uses second-order Taylor approximation of inverse trigonometric functions to solve the problem that it is difficult to solve convex optimization due to its high nonlinearity. Then, the convex optimization relaxation technique transforms the non-convex problem into a convex problem, and finally uses the CVX solver to solve it. This makes the method of this invention more adaptable to node position errors and has higher positioning accuracy. Attached Figure Description

[0079] Figure 1 This is a general implementation block diagram of the present invention.

[0080] Figure 2 This diagram illustrates how the root mean square error changes as the standard deviation of the azimuth measurement noise increases, given that the standard deviation of the node position error is 2. It also shows the changes in the root mean square error of the method of this invention, existing AOA convex optimization target localization methods, existing least squares methods, and the Cramer-Rao lower bound corresponding to the method of this invention.

[0081] Figure 3 This diagram illustrates how the root mean square error changes as the standard deviation of the node position error increases, along with the existing AOA convex optimization target localization method, the existing least squares method, and the Cramer-Rao lower bound corresponding to the method of this invention, all under the condition that the standard deviation of the azimuth measurement noise is 4.

[0082] Figure 4 This diagram illustrates how the root mean square error of the Cramer-Rao lower bound of the present invention, existing AOA convex optimization target localization methods, existing least squares methods, and the present invention's method changes with the increase of the number of nodes, under the conditions that the standard deviation of azimuth measurement noise is 2 and the standard deviation of node position error is 2. Detailed Implementation

[0083] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0084] This invention proposes an AOA convex optimization target localization method under node position error, the overall implementation block diagram of which is shown below. Figure 1 As shown, it includes the following steps:

[0085] Step 1: Establish a Cartesian coordinate system as the reference coordinate system in the wireless sensor network. There are N arrays with position drift and one target with an unknown position. The position coordinates of the i-th array at the time of deployment are...

[0086] v i =(x i ,y i ) T i = 1, ..., N

[0087] The position coordinates of the target to be located are

[0088] u 0 =(x 0 ,y 0 ) T ,

[0089] Each array passively acquires its AOA (Area of ​​Average) measurement with the target. Assuming the node position error term follows a zero-mean Gaussian distribution, the actual but unknown position of the i-th array is modeled as follows:

[0090]

[0091] Where s i =(△x) i ,△y i ) T Gaussian white noise with zero mean, s i covariance matrix Φ i =δ i 2 I2, where I2 represents a 2×2 unit diagonal matrix.

[0092] Step 2: After the target node transmits the signal, the sensor node acquires the measurement. Considering the angle measurement error and the array position error, the AOA measurement acquired by the i-th array is... for:

[0093]

[0094] In the formula, θ i Let ε be the true angle of the target relative to the i-th array. i The measurement error of the i-th array AOA is modeled as a value with zero mean and variance of . Gaussian white noise, n i Let ε be the error caused by the node position drift in the i-th array. i and s i They are independent of each other.

[0095] Step 3: Obtain n through analysis i Statistical characteristics

[0096] According to the measurement model:

[0097]

[0098] Taking the tan inequality of both sides of the above equation, and applying Taylor expansion to sin(θ) respectively i +n i ) and cos(θ) i +n i Approximation:

[0099]

[0100] Solve the following equations:

[0101]

[0102] After ignoring the second-order terms, we get:

[0103]

[0104] Therefore n iIt follows a pattern with a mean of zero and a variance of .

[0105] The Gaussian distribution.

[0106] Therefore, the total noise β i =θ i +n i , follows a mean of zero and a variance of

[0107] The Gaussian distribution.

[0108] Step 4: Since the arctangent function is highly nonlinear, it needs to be approximated by a Taylor series expansion. The arctangent function is expanded using a Taylor series at the initial point (x0, y0) (usually obtained using the least squares method), retaining only polynomials of degree two and below, which is expressed as:

[0109]

[0110] in:

[0111]

[0112] The final simplified Taylor series approximation is expressed as:

[0113]

[0114] in:

[0115]

[0116] Step 5: Based on the Gaussian assumptions of the AOA measurement model, use maximum likelihood estimation to locate the target. For the target location estimate:

[0117]

[0118] Step Six: Perform semidefinite relaxation on the maximum likelihood estimation problem;

[0119] The objective function in maximum likelihood estimation can be rewritten as:

[0120] (θ-c-L1×x-L2×y-L3×x 2 -L4×y 2 -L5×xy) T

[0121] W(θ-c-L1×x-L2×y-L3×x 2 -L4×y 2 -L5×xy)

[0122] =Trace{W(θ-c-(L1×x+L2×y+L3×x 2 +L4×y 2 +L5×xy))}

[0123] (θ-c-(L1×x+L2×y+L3×x 2 +L4×y 2 +L5×xy)) T

[0124] in:

[0125]

[0126] Define a new vector g = [x, y, x 2 ,y 2 [xy] T With matrix B = [L1, L2, L3, L4, L5], the objective function is transformed into:

[0127] Trace{W(θ-c-Bg)(θ-c-Bg) T}=Trace{W(Θ+C+BGB T -2Bgθ T +2Bgc T -2cθ T )}

[0128] in:

[0129] Θ=θθ T C = cc T G = gg T

[0130] At this point, the objective function has been transformed into a convex function, but for the above constraint Θ=θθ T C = cc T G = gg T It was found that the non-convex constraint G = gg still exists. T Therefore, by using semidefinite relaxation, G = gg T Transform into convex constraints Observing the relationship between vectors g and u, we find that the following five implicit constraints hold:

[0131] g(1:2) = u,

[0132] g(3,1)=G(1,1),

[0133] g(4,1)=G(2,2),

[0134] g(5,1)=G(1,2),

[0135] g(5,1)=G(2,1).

[0136] The final convex optimized representation number is:

[0137]

[0138] st

[0139] g(1:2) = u,

[0140] g(3,1)=G(1,1),

[0141] g(4,1)=G(2,2),

[0142] g(5,1)=G(1,2),

[0143] g(5,1)=G(2,1),

[0144] Θ=θθ T C = cc T .

[0145] Step 7: Use the CVX solver (interior point method) to solve the final convex optimization expression obtained in step 6 to obtain the vector g = [x, y, x 2 ,y 2 [xy] T The specific value of the vector g is calculated so that the first part of the vector g is the x-coordinate of the target to be located, and the second part is the y-coordinate of the target to be located.

[0146] The following simulation experiments are used to verify the feasibility, effectiveness, and positioning performance of the method of the present invention.

[0147] Assume there are N=16 sensor nodes and one target in the wireless sensor network. All nodes (including anchor nodes and target nodes) are randomly deployed in a two-dimensional space of 40×40 square meters.

[0148] The performance of the method of the present invention was tested as the standard deviation of the azimuth measurement noise increased.

[0149] Figure 2 The diagram illustrates the changes in the root mean square error (RMSE) of the proposed method, existing AOA convex optimization target localization methods, existing least squares methods, and the Cramer-Rao lower bound corresponding to the proposed method as the standard deviation of the azimuth measurement noise increases, assuming a node position error standard deviation of 2. Figure 2It can be seen that as the standard deviation of azimuth measurement noise increases, the RMSE of all methods shows an upward trend. The performance of the proposed improved algorithm is always better than other methods. When the measurement error is slightly large, the performance of the least squares method drops sharply. This shows that the method of the present invention is more robust and has higher positioning accuracy.

[0150] The performance of the method of the present invention was tested to see how it changes as the standard deviation of the node position error increases.

[0151] Figure 3 The diagram illustrates the changes in the root mean square error (RMSE) of the proposed method, existing AOA convex optimization target localization methods, existing least squares methods, and the Cramer-Rao lower bound corresponding to the proposed method as the standard deviation of the node position error increases, all under the condition that the standard deviation of the azimuth measurement noise is 4. Figure 4 As can be seen, the RMSE of all methods increases with the increase of node position error, and the performance of the original convex optimization method deteriorates rapidly.

[0152] The performance of the method of this invention was tested to see how it changes with the number of nodes.

[0153] Figure 4 The diagram illustrates the changes in the root mean square error of the proposed method, existing AOA convex optimization target localization methods, existing least squares methods, and the Cramer-Rao lower bound corresponding to the proposed method as the number of nodes increases, under the conditions that the standard deviation of azimuth measurement noise is 2 and the standard deviation of node position error is 2. It can be seen that when the number of nodes is small, the performance of the original convex optimization method deteriorates rapidly, while the least squares method and the proposed method show no significant deterioration. The localization accuracy of the proposed method is always better than the other two algorithms.

[0154] In summary, the algorithm proposed in this invention outperforms existing methods and exhibits good stability in all circumstances.

[0155] As can be seen from the simulation results above, the method of the present invention has good performance, can adapt to node position errors and angle errors, and can well meet the requirements of high positioning accuracy.

Claims

1. A method for underwater target fusion localization considering AOA convex optimization under anchor node drift, characterized in that... Includes the following steps: Step 1: Establish a Cartesian coordinate system as the reference coordinate system in the wireless sensor network. There are N arrays with position drift and one target with an unknown position. The position coordinates of the i-th array when it is deployed are: v i =(x i ,y i ) T ,i=1,...,N, The coordinates of the target to be located are: u 0 =(x 0 ,y 0 ) T , Each array passively acquires its AOA (Area of ​​Effect) measurement with the target. Assuming the node position error term follows a Gaussian distribution with zero mean, the actual but unknown position of the i-th array is: The relationship between the actual location and the deployment location is as follows: Where s i =(△x) i ,△y i ) T Gaussian white noise with zero mean, s i covariance matrix Φ i =δ i 2 I2, where I2 represents a 2×2 unit diagonal matrix; Step 2: After the target node transmits the signal, the sensor node acquires the measurement. In step two, considering the angle measurement error and array position error, the AOA measurement obtained by the i-th array is... Represented as: In the formula, θ i Let ε be the true angle of the target relative to the i-th array. i Let ε be the measurement error of the i-th array AOA. i Model as a function with a mean of zero and a variance of . Gaussian white noise, n i Let ε be the error caused by the node position drift in the i-th array. i and s i They are independent of each other; Step 3: Obtain n through analysis i Statistical characteristics; In step three, it can be seen from the measurement model that: Taking the tan sign of both sides of the above equation and using Taylor expansion to apply the same method to sin(θ) i +n i ) and cos(θ) i +n i Approximation: Solve the following equations: After ignoring the second-order terms, we get: Therefore n i It follows a pattern with a mean of zero and a variance of . Gaussian distribution; Therefore, the total noise β i =θ i +n i , follows a mean of zero and a variance of: Gaussian distribution; Step 4: Since the arctangent function is highly nonlinear, it needs to be approximated by Taylor series expansion. The arctangent function is expanded by Taylor series at the initial point (x0, y0), and only polynomials of the second degree and below are retained. The initial point (x0, y0) is obtained by the least squares method. In step four, the polynomial is expressed as: in: The final simplified Taylor series approximation is expressed as: in: Step 5: Based on the Gaussian assumptions of the AOA measurement model, use maximum likelihood estimation to locate the target. For the target location estimate: Step Six: Perform semidefinite relaxation on the maximum likelihood estimation problem; Step 7: Use the CVX solver to solve the final convex optimization expression obtained in Step 6 to obtain the vector g = [x, y, x 2 ,y 2 [xy] T The specific value is obtained by solving the vector g, where the first part is the target x-coordinate and the second part is the target y-coordinate.

2. The AOA convex optimization underwater target fusion localization method considering anchor node drift as described in claim 1, characterized in that: In step six, the objective function in the maximum likelihood estimation is rewritten as: (θ-c-L1×x-L2×y-L3×x 2 -L4×y 2 -L5×xy) T W(θ-c-L1×x-L2×y-L3×x 2 -L4×y 2 -L5×xy) =Trace{W(θ-c-(L1×x+L2×y+L3×x 2 +L4×y 2 +L5×xy))} (θ-c-(L1×x+L2×y+L3×x 2 +L4×y 2 +L5×xy)) T in: Define a new vector g = [x, y, x 2 ,y 2 [xy] T With matrix B = [L1, L2, L3, L4, L5], the objective function is transformed into: Trace{W(θ-c-Bg)(θ-c-Bg) T }=Trace{W(Θ+C+BGB T -2Bgθ T +2Bgc T -2cθ T )} in: Θ=θθ T ,C=cc T ,G=gg T At this point, the objective function has been transformed into a convex function, but the constraint Θ = θ remains unchanged. T C = cc T G = gg T It was found that the non-convex constraint G = gg still exists. T Therefore, by using semidefinite relaxation, G = gg T Transform into convex constraints Observing the relationship between vectors g and u, we find that the following five implicit constraints hold: g(1:2)=u g(3,1)=G(1,1) g(4,1)=G(2,2) g(5,1)=G(1,2) g(5,1)=G(2,1) The final convex optimized representation number is: st g(1:2) = u, g(3,1)=G(1,1), g(4,1)=G(2,2), g(5,1)=G(1,2), g(5,1)=G(2,1), Θ=θθ T ,C=cc T 。 3. An electronic device, characterized in that, include: One or more processors; Memory; One or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs being configured to perform the method as described in any one of claims 1-2.

4. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores program code that can be invoked by a processor to execute the method as described in any one of claims 1-2.

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