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

Abstract

The invention provides an AOA convex optimization underwater target fusion positioning method considering anchor node drifting, a rectangular coordinate system is established in a wireless sensor network as a reference coordinate system, N arrays with position drifting and a target with an unknown position exist in total, and after a target node transmits a signal, the sensor node obtains measurement; taylor series expansion is carried out on an arc tangent function at an initial point, only polynomials of two or below are reserved, a target is positioned by using maximum likelihood estimation according to Gaussian hypothesis of an AOA measurement model, and positive semi-definite relaxation is carried out on a maximum likelihood estimation problem; and solving to obtain a target coordinate. According to the method, better adaptability to node position errors is achieved, and higher positioning precision is achieved.

Description

Technical Field

[0001] The present invention relates to the field of underwater target positioning, and in particular to an underwater target positioning fusion method, and in particular to a wireless sensor network target positioning method based on AOA (Angle-of-Arrival) measurement when there is a node position error. Background Art

[0002] Since the 21st century, all countries have been strengthening the research, development and application of marine science and technology to enhance international marine competitiveness. Therefore, the importance of maintaining marine rights and interests has become increasingly prominent. Underwater Acoustic Sensor Network (UASN) based on acoustic information interaction has been widely used in recent years.

[0003] Target positioning is an important function of underwater acoustic sensor networks. According to the different measurement information obtained by the nodes, the target positioning methods of wireless sensor networks can be divided into: time of arrival (TOA), time difference of arrival (TDOA), angle of arrival (AOA), received signal strength (RSS) and combined multiple measurement methods. Among these measurements, RSS is very sensitive to multipath effects and is not suitable for underwater acoustic environments; TDOA is obtained by subtracting TOA measurements. This operation will cause measurement noise to superimpose and cause a 3dB performance loss; TOA measurement requires clock synchronization between the target and the node, which is not easy to achieve underwater, especially in non-cooperative target positioning. AOA measurement can often obtain better positioning performance and is more suitable for positioning in underwater acoustic environments.

[0004] In the past few years, many sensor network target positioning methods based on AOA have been proposed, such as triangulation positioning, maximum likelihood estimation and least squares estimation. However, the triangulation positioning method cannot effectively utilize the information obtained by the sensor and has large deviations. The maximum likelihood estimation method is likely to fall into the local optimum. When the error is relatively large, the least squares method may have large deviations due to linear approximation. Patent CN117686974A proposes a convex optimization target positioning method of AOA, which has good performance, but this method does not take into account the impact of the position error of the sensor network node, and the position error of the underwater acoustic sensor network node is ubiquitous.

[0005] In summary, the target positioning method based on AOA has better performance than other positioning methods, but the existing target positioning method based on AOA still has many imperfections, so it is necessary to conduct in-depth research on the target positioning method based on AOA. Summary of the invention

[0006] In order to overcome the shortcomings of the prior art, the present invention provides an AOA convex optimized underwater target fusion positioning method considering anchor node drift. This method fully considers the influence of node position error when modeling, and proposes a more accurate maximum likelihood problem, so that the positioning method has better adaptability to node position error and achieves the purpose of accurately positioning the target position.

[0007] The technical solution adopted by the present invention to solve the technical problem comprises the following steps:

[0008] Step 1: Establish a rectangular coordinate system as a reference coordinate system in the wireless sensor network. There are N arrays with position drift and a target with unknown position. The position coordinates of the i-th array when deployed are:

[0009] v i =(x i ,y i ) T ,i=1,…,N,

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

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

[0012] Each array acquires the AOA measurement value between itself and the target in a passive way. Assuming that the node position error term follows a zero-mean Gaussian distribution, the actual but unknown position of the i-th array is:

[0013]

[0014] The relationship between the actual position and the deployment position is:

[0015]

[0016] where s i =(△x i ,△y i ) T is a zero-mean Gaussian white noise, s i The 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 obtains the measurement;

[0018] Step 3: Get n through analysis i Statistical properties of

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

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

[0021] Estimate the target position:

[0022]

[0023] Step 6: Perform semi-positive definite 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 first digit of the vector g obtained is the target horizontal coordinate, and the second digit is the target vertical coordinate.

[0025] In step 2, the AOA measurement obtained by the i-th array is considered to be the angle measurement error and the array position error. It is expressed as:

[0026]

[0027] In the formula, θ i is the true angle of the target relative to the i-th array, ε i is the AOA measurement error of the i-th array, ε i Modeled as a function with mean zero and variance Gaussian white noise, n i is the error of the i-th array caused by the node position drift, assuming ε i and i Are independent of each other.

[0028] In step 3, it can be known from the measurement model that:

[0029]

[0030] Take tan on both sides of the above equation and use Taylor expansion to respectively calculate sin(θ i +n i ) and cos(θ i +n i ) to approximate:

[0031]

[0032] Solve the following equation:

[0033]

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

[0035]

[0036] Therefore n i It has a mean of zero and a variance of

[0037] Gaussian distribution.

[0038] Therefore, the total noise β i =θ i +n i , and the variance is zero:

[0039] Gaussian distribution.

[0040] In step 4, 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 6, 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 And the matrix B = [L1, L2, L3, L4, L5], so 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 for the constraints

[0061] θ=θθ T ,C=cc T ,G=gg T , we find that there is still a non-convex constraint G = gg T , so using semi-positive definite relaxation G = gg T Transform to convex constraints Observe the relationship between vectors g and u and 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 optimization expression 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 comprises: 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 execute the method as described above.

[0077] A computer-readable storage medium stores program codes, and the program codes can be called by a processor to execute the method as described above.

[0078] The beneficial effects of the present invention fully consider the influence of node position error on the positioning system, model the node position error in the measurement model, perform approximate analysis on the error term through Taylor expansion, and then use the second-order Taylor approximation inverse trigonometric function to solve the problem that it is difficult to perform convex optimization solution due to its high nonlinearity. Then, the convex optimization relaxation technique is used to transform the non-convex problem into a convex problem, and finally the CVX solver is used to solve it, so that the method of the present invention has better adaptability to the node position error and has higher positioning accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0079] Figure 1 It is an overall implementation block diagram of the present invention.

[0080] Figure 2 It is a schematic diagram of the change of the root mean square error of the method of the present invention, the existing AOA convex optimization target positioning method, the existing least squares method and the Cramer-Rao lower bound corresponding to the method of the present invention as the standard deviation of the azimuth measurement noise increases under the condition that the standard deviation of the node position error is 2;

[0081] Figure 3 It is a schematic diagram of the change of the root mean square error of the method of the present invention, the existing AOA convex optimization target positioning method, the existing least squares method and the Cramer-Rao lower bound corresponding to the method of the present invention as the standard deviation of the node position error increases under the condition that the standard deviation of the azimuth measurement noise is 4;

[0082] Figure 4 It is a schematic diagram of the change of the root mean square error of the method of the present invention, the existing AOA convex optimization target positioning method, the existing least squares method and the Cramer-Rao lower bound corresponding to the method of the present invention as the number of nodes increases under the conditions that the standard deviation of the azimuth measurement noise is 2 and the standard deviation of the node position error is 2. DETAILED DESCRIPTION

[0083] The present invention is further described below in conjunction with the accompanying drawings and embodiments.

[0084] The present invention provides an AOA convex optimization target positioning method under node position error, and its overall implementation block diagram is as follows: Figure 1 As shown, it includes the following steps:

[0085] Step 1: Establish a rectangular coordinate system as a reference coordinate system in the wireless sensor network. There are N arrays with position drift and a target with unknown position. The position coordinates of the i-th array when deployed 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 acquires the AOA measurement value between itself and the target in a passive way. Assuming that the node position error term follows a zero-mean Gaussian distribution, the actual but unknown position of the i-th array is modeled as

[0090]

[0091] where s i =(△x i ,△y i ) T is a zero-mean Gaussian white noise, s i The 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 a signal, the sensor node obtains the measurement, taking into account the angle measurement error and array position error. The AOA measurement obtained by the i-th array for:

[0093]

[0094] In the formula, θ i is the true angle of the target relative to the i-th array, ε i is the AOA measurement error of the i-th array, modeled as a zero mean and variance Gaussian white noise, n i is the error of the i-th array caused by the node position drift, here we assume that ε i and i Are independent of each other.

[0095] Step 3: Get n through analysis i The statistical properties of

[0096] From the measurement model, we can know that:

[0097]

[0098] Take tan on both sides of the above equation and use Taylor expansion to calculate sin(θ i +n i ) and cos(θ i +n i ) to approximate:

[0099]

[0100] Solve the following equation:

[0101]

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

[0103]

[0104] Therefore n iIt has a mean of zero and a variance of

[0105] Gaussian distribution.

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

[0107] Gaussian distribution.

[0108] Step 4: Since the inverse tangent function is highly nonlinear, it needs to be approximated by Taylor series expansion. Perform Taylor series expansion on the inverse tangent function at the initial point (x0, y0) (generally obtained by the least squares method), and only retain the polynomials of quadratic and below, which can be 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 hypothesis of the AOA measurement model, use maximum likelihood estimation to locate the target. Estimate the target position:

[0117]

[0118] Step 6: Perform semi-positive definite relaxation on the maximum likelihood estimation problem;

[0119] The objective function in maximum likelihood estimation is 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 And the matrix B = [L1, L2, L3, L4, L5], so 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 constraints Θ = θθ T ,C=cc T ,G=gg T , we find that there is still a non-convex constraint G = gg T , so using semi-positive definite relaxation G = gg T Transform to convex constraints Observe the relationship between vectors g and u and 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 optimization expression 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 first digit of the vector g obtained is the horizontal coordinate of the target to be located, and the second digit is the vertical coordinate of the target to be located.

[0146] The feasibility, effectiveness and positioning performance of the method of the present invention are verified by simulation experiments as follows.

[0147] Assume that there are N=16 sensor nodes and one target in the wireless sensor network, and 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 A schematic diagram of the change of the root mean square error of the method of the present invention, the existing AOA convex optimization target positioning method, the existing least squares method and the Cramer-Rao lower bound corresponding to the method of the present invention as the standard deviation of the azimuth measurement noise increases is given under the condition that the standard deviation of the node position error is 2; Figure 2It can be seen that with the increase of the standard deviation of the azimuth measurement noise, the RMSE of various methods shows an upward trend. The performance of the proposed improved algorithm is always better than other methods. When the measurement error is slightly larger, the performance of the least squares method drops sharply, which 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 as the standard deviation of the node position error increased.

[0151] Figure 3 A schematic diagram of the change of the root mean square error of the Cramer-Rao lower bound corresponding to the method of the present invention, the existing AOA convex optimization target positioning method, the existing least squares method, and the method of the present invention as the standard deviation of the node position error increases is given under the condition that the standard deviation of the azimuth measurement noise is 4. Figure 4 It can be seen that with the increase of node position error, the RMSE of all methods increases, and the performance of the original convex optimization method deteriorates rapidly.

[0152] The performance of the method of the present invention is tested as the number of nodes increases.

[0153] Figure 4 A schematic diagram of the change in the root mean square error of the method of the present invention, the existing AOA convex optimization target positioning method, the existing least squares method and the Cramer-Rao lower bound corresponding to the method of the present invention as the number of nodes increases is given under the conditions that the standard deviation of the azimuth measurement noise is 2 and the standard deviation of the 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 deterioration of the least squares method and the method proposed in the present invention is not obvious, and the positioning accuracy of the method of the present invention is always better than that of the other two algorithms.

[0154] In summary, the algorithm proposed in the present invention is superior to the existing methods in any case and has good stability.

[0155] It can be seen from the above simulation results that the method of the present invention has good performance, can adapt to node position error and angle error, and can well meet the demand for high-precision positioning.

Claims

1. An AOA convex optimization underwater target fusion positioning method considering anchor node drift, characterized in that The steps include: Step 1: Establish a rectangular coordinate system as a reference coordinate system in the wireless sensor network. There are N arrays with position drift and a target with unknown position. The position coordinates of the i-th array when deployed are: v i =(x i ,y i ) T ,i=1,…,N, The position coordinates of the target to be located are: u 0 =(x 0 ,y 0 ) T , Each array acquires the AOA measurement value between itself and the target in a passive way. Assuming that the node position error term follows a zero-mean Gaussian distribution, the actual but unknown position of the i-th array is: The relationship between the actual position and the deployment position is: where s i =(△x i ,△y i ) T is a zero-mean Gaussian white noise, s i The 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 obtains the measurement; Step 3: Get n through analysis i Statistical properties of Step 4: Since the inverse tangent function is highly nonlinear, it needs to be approximated by Taylor series expansion. The inverse tangent function is expanded by Taylor series at the initial point (x0, y0), and only polynomials of quadratic or lower are retained. The initial point (x0, y0) is obtained by the least squares method. Step 5: Based on the Gaussian hypothesis of the AOA measurement model, use maximum likelihood estimation to locate the target. Estimate the target position: Step 6: Perform semi-positive definite 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 first digit of the vector g obtained is the target horizontal coordinate, and the second digit is the target vertical coordinate.

2. The AOA convex optimization underwater target fusion positioning method considering anchor node drift according to claim 1 is characterized in that: In step 2, the AOA measurement obtained by the i-th array is considered to be the angle measurement error and the array position error. It is expressed as: In the formula, θ i is the true angle of the target relative to the i-th array, ε i is the AOA measurement error of the i-th array, ε i Modeled as a function with mean zero and variance Gaussian white noise, n i is the error of the i-th array caused by the node position drift, assuming ε i and i Are independent of each other.

3. The AOA convex optimization underwater target fusion positioning method considering anchor node drift according to claim 2 is characterized in that: In step 3, it can be known from the measurement model that: Take tan on both sides of the above equation and use Taylor expansion to respectively calculate sin(θ i +n i ) and cos(θ i +n i ) to approximate: Solve the following equation: Ignoring the second-order terms, we get: Therefore n i It has a mean of zero and a variance of Gaussian distribution of Therefore, the total noise β i =θ i +n i , and the variance is zero: Gaussian distribution.

4. The AOA convex optimization underwater target fusion positioning method considering anchor node drift according to claim 3 is characterized in that: In step 4, the polynomial is expressed as: in: The final simplified Taylor series approximation is expressed as: in:

5. The AOA convex optimization underwater target fusion positioning method considering anchor node drift according to claim 4 is characterized in that: In step 6, 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 And the matrix B = [L1, L2, L3, L4, L5], so 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 for the constraint condition Θ = θθ T ,C=cc T ,G=gg T , we find that there is still a non-convex constraint G = gg T , so using semi-positive definite relaxation G = gg T Transform to convex constraints Observe the relationship between vectors g and u and 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 optimization expression 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 .。 6. 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, and the one or more programs are configured to execute the method according to any one of claims 1 to 5.

7. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores program codes, and the program codes can be called by a processor to execute the method according to any one of claims 1 to 5.

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