Two-dimensional optimal site layout method

Abstract

This invention provides a two-dimensional optimal base station deployment method to improve the positioning accuracy of an N+1 hybrid base station positioning system. The method includes the following steps: S1, determining the communication range of a reference station and judging whether the target is within the communication range of the reference station, and establishing constraints based on the judgment result; S2, determining the noise distribution of the TDOA and AOA values, and calculating the FIM determinant of the target's position; S3, expressing the FIM determinant under different constraints as an optimization problem, and solving the optimization problem to obtain the two-dimensional optimal base station deployment method. This invention, by changing the base station deployment method, obtains a calculated target position with higher positioning accuracy under existing hardware facilities and positioning algorithms, preparing for subsequent measures to be taken against the target.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of passive positioning technology, and particularly provides a two-dimensional optimal station arrangement method for an N+1 hybrid base station positioning system, wherein N+1 base stations are used to measure time difference of arrival (TDOA), and a reference station is used to measure angle of arrival (AOA). BACKGROUND

[0002] In the field of navigation, aviation, aerospace and other fields involving passive positioning technology, target reconnaissance and positioning are very important, and the positioning accuracy of the target is a common index for evaluating the performance of the positioning system. The positioning accuracy upper limit of the N+1 hybrid base station positioning system is higher than that of a single form of TDOA or AOA positioning, but it is limited by hardware facilities. Sometimes, the positioning accuracy still cannot meet the demand under the condition of fully utilizing the measurement values.

[0003] The station arrangement method affects the accuracy of target positioning, and the optimal station arrangement method can further improve the positioning accuracy. Therefore, the present application proposes a two-dimensional optimal station arrangement method based on an N+1 hybrid base station positioning system according to the communication range of the reference station and the number of base stations. SUMMARY

[0004] The present application provides a two-dimensional optimal station arrangement method for solving the station arrangement method with the highest target positioning accuracy under different base station numbers and different communication constraints.

[0005] The two-dimensional optimal station arrangement method provided by the present application is particularly used to improve the positioning accuracy of the N+1 hybrid base station positioning system, wherein N represents N monitoring stations, N is a positive integer greater than 0; 1 represents a reference station, N+1 base stations are used to measure TDOA, and only the reference station is used to measure AOA. The method comprises the following steps:

[0006] S1. Determine the communication range of the reference station, and determine whether the target is within the communication range of the reference station, and establish a constraint condition according to the determination result;

[0007] S2. Determine the noise distribution of the TDOA value and the AOA value, and calculate the FIM determinant of the position of the target;

[0008] S3. Express the FIM determinant under different constraint conditions as an optimization problem, and solve the optimization problem to obtain the two-dimensional optimal station arrangement method.

[0009] Preferably, when the target is within the communication range of the reference station, i.e. r0

[0010] Preferably, when the target is outside the communication range of the reference station, i.e. r0>L, the angular positions of the N monitoring stations are limited by the distance r0between the target and the reference station, and the constraint condition is introduced:

[0011] λ0= 0°, |λ i |≤λ m i=1,...,N;

[0012] where λ i represents the angular position of the i-th base station relative to the target, λ m = arcsin(L / r0)<90° represents the maximum angular position of the N monitoring stations.

[0013] Preferably, the noise distribution of the TDOA values and the AOA values respectively obeys zero-mean Gaussian distribution with variances and .

[0014] Preferably, the FIM matrix of the target position is as follows:

[0015]

[0016] where J and J respectively represent the Jacobian matrix of the TDOA measurement values and the AOA measurement values relative to the target position, -1 represents the inverse of the matrix, T represents the transpose symbol, Q m is the covariance matrix, blkdiag represents the block diagonal matrix, I N and 1 N respectively represent the N-dimensional unit matrix and the N-dimensional all-1 matrix;

[0017] Substituting various parameters, the FIM matrix is represented as:

[0018]

[0019] where A A 13 = A 23 = A 31 = A 32 = 0,

[0020] The expansion of the FIM determinant obtained from the FIM matrix is as follows:

[0021]

[0022] where det represents the function of calculating the determinant of a square matrix.

[0023] Preferably, the expansion of the FIM determinant is introduced with mathematical constraints The simplified formula is as follows:

[0024]

[0025] Preferably, the FIM determinant is solved according to the D optimality criterion, and the solution of the angle position corresponding to the maximum value of the FIM determinant is taken as the two-dimensional optimal station arrangement method.

[0026] Preferably, the target is in the communication range of the reference station, and the formula (1) is simplified to represent the optimization problem as follows:

[0027]

[0028] s.t. λ0=0°;

[0029] The optimization problem is solved to obtain the two-dimensional optimal station arrangement method when the target is in the communication range of the reference station.

[0030] Preferably, the target is out of the communication range of the reference station, and the formula (1) is simplified to represent the optimization problem as follows:

[0031]

[0032] s.t. λ0=0, |λ i |≤λ m i=1,...,N;

[0033] The optimization problem is solved to obtain the two-dimensional optimal station arrangement method when the target is out of the communication range of the reference station.

[0034] Compared with the prior art, the application can achieve the following beneficial effects:

[0035] The application changes the station arrangement method, obtains the calculation value of the target position with higher positioning accuracy under the original hardware facilities and positioning algorithm, and prepares for subsequent measures for the target. BRIEF DESCRIPTION OF DRAWINGS

[0036] Figure 1 It is a mathematical model diagram of the N+1 hybrid base station positioning system according to the embodiment one of the application;

[0037] Figure 2 It is a flow chart of the two-dimensional optimal station arrangement method according to the embodiment one of the application;

[0038] Figure 3 It is a schematic diagram of the two-dimensional optimal station arrangement when r0

[0039] Figure 4It is a schematic diagram of two-dimensional optimal station layout when r0 > L provided by Embodiment 1 of the present invention;

[0040] Figure 5 It is a curve graph showing the variation of the FIM determinant value of a 1+1 hybrid base station positioning system with the angular position λ1 when r0 < L provided by Embodiment 2 of the present invention;

[0041] Figure 6 It is provided by Embodiment 3 of the present invention When, it is a surface graph showing the variation of the FIM determinant value of a 2+1 hybrid base station positioning system with the angular positions λ1 and λ2;

[0042] Figure 7 It is provided by Embodiment 3 of the present invention When, it is a surface graph showing the variation of the FIM determinant value of a 2+1 hybrid base station positioning system with the angular positions λ1 and λ2;

[0043] Figure 8 It is a surface graph showing the variation of the FIM determinant value of a 2+1 hybrid base station positioning system with the angular positions λ1 and λ2 when r0 > L provided by Embodiment 4 of the present invention. Detailed implementation manners

[0044] In the following, embodiments of the present invention will be described with reference to the accompanying drawings. In the following description, the same modules are denoted by the same reference numerals. In the case of the same reference numerals, their names and functions are also the same. Therefore, their detailed descriptions will not be repeated.

[0045] In order to make the objectives, technical solutions and advantages of the present invention clearer and more understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and do not constitute a limitation to the present invention.

[0046] Based on a positioning system that combines multiple TDOA base stations and a single AOA base station, a two-dimensional optimal station layout method is proposed. The application scenario is an N+1 hybrid base station positioning system, where N represents N monitoring stations and N is a positive integer greater than 0; 1 represents a reference station. The base stations are expressed as Where Represents the latitude information, that is, it represents s here i Is three-dimensional information. When i = 0, it represents the reference station; when i ≠ 0, it represents the monitoring station; the target at the unknown position is represented as The reference station S0 and any monitoring station S i Are both used to measure the TDOA value of the target, but only the reference station S0 can measure the AOA value of the target. The measurement noises of TDOA and AOA respectively follow variances of And zero-mean Gaussian distribution with covariance matrix wherein, blkdiag denotes a block diagonal matrix.

[0047] Figure 1 A mathematical model of the N+1 hybrid base station positioning system according to the first embodiment of the present application is shown.

[0048] As shown in Figure 1 , X, Y, Z represent corresponding coordinate axes of a coordinate system, O is the coordinate origin, r i represents the distance between any base station and the target , when i = 0, it represents the distance between the reference station and the target; when i ≠ 0, it represents the distance between the monitoring station and the target, α, β are components of the arrival angle λ, S i and S j represent any two monitoring stations.

[0049] Figure 2 A two-dimensional optimal base station arrangement method according to the first embodiment of the present application is shown.

[0050] As shown in Figure 1 , Figure 2 , the two-dimensional optimal base station arrangement method comprises the following steps:

[0051] S1, determining the communication range of the reference station S0 and the target and judging whether the target is within the communication range of the reference station, and establishing a constraint condition according to the judgment result as follows:

[0052] Figure 3 A two-dimensional optimal base station arrangement method when r0

[0053] As shown in Figure 3 , when the target is within the communication range of the reference station, the selectable angular positions of all monitoring stations are not constrained, and at this time, a constraint condition λ0 = 0° is introduced.

[0054] Figure 4 A two-dimensional optimal base station arrangement method when r0

[0055] As shown in Figure 4 , when the target is outside the communication range of the reference station, the selectable angular positions of all monitoring stations are limited by the communication range of the reference station, and at this time, constraint conditions λ0 = 0, |λ i |≤λ m i = 1,..., N, wherein, λ m = arcsin(L / r0) < 90° is the maximum value of the angular position of all monitoring stations.

[0056] S2, determine the noise distribution of TDOA values of N monitoring stations and AOA values of reference stations, calculate the FIM determinant of the position of the target, solve the FIM determinant according to the D-optimality criterion, and take the solution of the angular position corresponding to the maximum FIM determinant as the two-dimensional optimal station arrangement method. The FIM determinant is as follows:

[0057]

[0058] Wherein, and respectively represent the Jacobian matrix of TDOA measurement value and AOA measurement value relative to the target position, -1 represents the inverse of the matrix, T represents the transpose symbol, Q m is the covariance matrix, blkdiag represents the block diagonal matrix, I N and 1 N respectively represent the N-dimensional unit matrix and the N-dimensional all-1 matrix.

[0059] Substitute the parameters, and the FIM matrix is represented as:

[0060]

[0061] Wherein, A 13 =A 23 =A 31 =A 32 =0,

[0062] The expansion of the FIM determinant obtained from the FIM matrix is as follows:

[0063]

[0064] Wherein, det represents a function for calculating the determinant of a square matrix.

[0065] The constraint condition is introduced to the FIM determinant The simplified formula is as follows:

[0066]

[0067] The D-optimality criterion is one of the commonly used criteria for solving the optimal geometric configuration, and its physical meaning represents the minimum area of the error ellipse. Herein, the D-optimality criterion is expressed in mathematical form, that is, when the FIM determinant is maximum, the angular position of each monitoring station is the optimal station arrangement.

[0068] S3, express the FIM determinant under different constraint conditions as an optimization problem, and solve the optimization problem to obtain a two-dimensional optimal station arrangement method.

[0069] When the target is in the communication range of the reference station, the simplified FIM determinant is expressed as an optimization problem as follows:

[0070]

[0071] s.t. λ0=0°;

[0072] Solving the optimization problem, a two-dimensional optimal station placement method is obtained when the target is in the communication range of the reference station.

[0073] When the target is out of the communication range of the reference station, the simplified FIM determinant is expressed as an optimization problem as follows:

[0074]

[0075] s.t. λ0=0, |λ i |≤λ m i=1,...,N;

[0076] Solving the optimization problem, a two-dimensional optimal station placement method is obtained when the target is out of the communication range of the reference station.

[0077] The following will solve the two-dimensional optimal station placement with the specific number of monitoring stations N as follows:

[0078] When the target is in the communication range of the reference station and N=1, the optimization problem is simplified as:

[0079] The optimal station placement is λ1=π+λ0, and at this time

[0080] When the target is in the communication range of the reference station and N=2, the optimization problem is simplified as:

[0081]

[0082] When , the optimal station placement is λ1=λ2=π, and max(det)=8p2; when , the optimal station placement is λ1=π±cos -1 a m , det=f(a m ), where

[0083] When the target is in the communication range of the reference station and N>3, the optimization problem needs to be analyzed in combination with the specific scene.

[0084] When the target is out of the communication range of the reference station, the optimization problem is simplified as:

[0085] det = k1N 2 (N+1)(1-x) 3 (1+x)+k2N(1-x) 2 ;

[0086] where cos λ1= cos λ2=... = cos λ N = x, i = 1,..., N, At this time, the solution needs to discuss the parity of N as follows:

[0087] When N is odd, the optimal station arrangement is:

[0088]

[0089] When N is even, the optimal station arrangement is:

[0090]

[0091] The following Examples Two, Three and Four will respectively verify the feasibility of the present application in combination with Example One:

[0092] In Example Two, S1, N = 1, the reference station is s0= [r o ,0,0] T m, a monitoring station is s1= [r1cos λ1, r1sin λ1, 0] T m, and the target is u = [0, 0, 0] T m, wherein the monitoring station and the reference station can measure the TDOA of the target, but only the reference station can measure the AOA of the target. The communication range of the reference station L = 200 m, the distance between the target and the reference station r o = 100 m. At this time, the constraint condition is λ0= 0°.

[0093] S2, the measured TDOA value and AOA value are subject to Gaussian white noise distribution, and the noise standard deviation is and σ r = 10 m.

[0094] S3, the FIM determinant after simplification is: i.e. the simplification of the optimization problem.

[0095] The two-dimensional optimal station arrangement method is λ1= π, at this time and Figure 4 are the same.

[0096] Figure 5The figure shows the curve of the FIM determinant of the 1+1 mixed base station positioning system varying with the angular position λ1 when r0<L according to the second embodiment of the present application. As shown in the figure, the calculation result is the same as the result of the curve of the FIM determinant varying with the angular position λ1 shown in Figure 5 Figure 5

[0097] In the third embodiment, S1, N=2, the reference station is s0=[r o ,0,0] T m, the two monitoring stations are s1=[r1cosλ1,r1sinλ1,0] T m and s2=[r2cosλ2,r2sinλ2,0] T m, and the target is u=[0,0,0] T m, wherein the monitoring stations and the reference station can measure the TDOA of the target, but only the reference station can measure the AOA of the target. The communication range of the reference station is L=200m, and the distance between the target and the reference station is r o =100m. At this time, the constraint condition is λ0=0°, λ1+λ2=2π or sinλ i =0, i=1, 2.

[0098] S2, the measured TDOA value and AOA value are subject to Gaussian white noise distribution, and the noise standard deviation is and σ r =10m.

[0099] The simplified FIM determinant is:

[0100]

[0101] According to the D optimality criterion, the solution of the angular position when the FIM determinant takes the maximum value is the two-dimensional optimal station arrangement method.

[0102] S3, the FIM determinant under the constraint condition is expressed as an optimization problem as follows:

[0103]

[0104] s.t. λ0=0, sinλ i =0, i=1, 2.

[0105] Solving the optimization problem, the two-dimensional optimal station arrangement method is obtained:

[0106] When , the two-dimensional optimal station arrangement is λ1=λ2=π, max(det)=0.0028738.

[0107] Figure 6 The figure shows the curve of the FIM determinant of the 1+1 mixed base station positioning system varying with the angular position λ1 when r0<L according to the second embodiment of the present application. As shown in the figure, the calculation result is the same as the result of the curve of the FIM determinant varying with the angular position λ1 shown in​​ The surface of the FIM determinant of the 2+1 hybrid base station positioning system as a function of the angular positions λ1 and λ2 is shown in Figure 2. As shown, the calculation result is the same as that of the surface of the FIM determinant as a function of the angular positions λ1 and λ2 shown in Figure 1. Figure 6 Figure 6 The surface of the FIM determinant of the 2+1 hybrid base station positioning system as a function of the angular positions λ1 and λ2 is shown in Figure 2. As shown, the calculation result is the same as that of the surface of the FIM determinant as a function of the angular positions λ1 and λ2 shown in Figure 1.

[0108] When , the two-dimensional optimal base station arrangement is λ1 = 134°, λ2 = 226° or λ1 = 226°, λ2 = 134°, and det = 5.9e-06.

[0109] Figure 7 Figure 3 shows the surface of the FIM determinant of the 2+1 hybrid base station positioning system as a function of the angular positions λ1 and λ2 according to an embodiment of the present application. The surface of the FIM determinant of the 2+1 hybrid base station positioning system as a function of the angular positions λ1 and λ2 is shown in Figure 2. As shown, the calculation result is the same as that of the surface of the FIM determinant as a function of the angular positions λ1 and λ2 shown in Figure 1. Figure 7 Figure 7 The surface of the FIM determinant of the 2+1 hybrid base station positioning system as a function of the angular positions λ1 and λ2 is shown in Figure 2. As shown, the calculation result is the same as that of the surface of the FIM determinant as a function of the angular positions λ1 and λ2 shown in Figure 1.

[0110] In the embodiment four, S1, N = 2, the reference station is s0 = [r o ,0,0] T m, the two monitoring stations are s1 = [r1cosλ1, r1sinλ1, 0] T m and s2 = [r2cosλ2, r2sinλ2, 0] T m, and the target is u = [0, 0, 0] T m, wherein the monitoring stations and the reference station can measure the TDOA of the target, but only the reference station can measure the AOA of the target. The communication range of the reference station is L = 100 m, and the distance between the target and the reference station is r o = 200 m. At this time, the constraint conditions are λ0 = 0°, |λ1|≤30° and |λ2|≤30°.

[0111] S2, the measured TDOA value and AOA value are subject to Gaussian white noise distribution, and the noise standard deviation is and σ r = 10 m.

[0112] The simplified FIM determinant is:

[0113] det≤k1(1-x) 3 (1+x)+k2(1-x) 2 ;

[0114] wherein, cosλ1 = cosλ2 = cosλ3 = x.

[0115] ​​According to the D-optimality criterion, the solution of the angle position when the FIM determinant takes the maximum value is the optimal stationing method in two dimensions.

[0116] S3, express the FIM determinant under the constraint as an optimization problem as follows:

[0117]

[0118] Solve the optimization problem to obtain the optimal stationing method in two dimensions: λ0=0, λ1=30°, λ2=-30°, max(det)=8.5508e-07.

[0119] Figure 8 A surface showing the FIM determinant of the 2+1 hybrid base station positioning system varying with the angle positions λ1 and λ2 is shown when r0>L according to the fourth embodiment of the present application. As shown in Figure 8 , the calculation result is the same as that of the surface showing the FIM determinant varying with the angle positions λ1 and λ2 shown in Figure 8 .

[0120] Although the embodiments of the present application have been shown and described above, it should be understood that the above embodiments are exemplary and should not be construed as limiting the present application. Those of ordinary skill in the art can make changes, modifications, replacements and variations to the above embodiments within the scope of the present application.

[0121] The specific embodiments of the present application do not constitute a limitation on the scope of protection of the present application. Any various other corresponding changes and modifications made according to the technical concept of the present application should be included in the scope of protection of the claims of the present application.

Claims

1. A two-dimensional optimal siting method for improving the positioning accuracy of an N+1 hybrid base station positioning system, wherein, N represents N monitoring stations, N is a positive integer greater than 0; 1 represents a reference station, and N+1 base stations are used to measure time difference of arrival (TDOA), and only the reference station is used to measure angle of arrival (AOA), comprising the following steps: S1, determining the communication range of the reference station, and judging whether the target is in the communication range of the reference station, and establishing a constraint condition according to the judgment result; S2, determining the noise distribution of TDOA value and AOA value, and calculating the FIM determinant of the target position; S3, expressing the FIM determinant under different constraint conditions as an optimization problem, and solving the optimization problem to obtain a two-dimensional optimal station arrangement method; When the target is outside the communication range of the reference station, i.e. the angular position of the N monitoring stations is limited by the distance of the target from the reference station with the introduction of the constraint condition: ， ； wherein, denotes the angular position of the i-th base station relative to the target, denotes the maximum angular position of the N monitoring stations.

2. The two-dimensional optimal placement of stations method of claim 1, wherein, When the target is within the communication range of the reference station, i.e. The angular position of N monitoring stations is not constrained, introducing a constraint condition: ; wherein L represents the communication range of the reference station, represents the distance between the target and the reference station.

3. The two-dimensional optimal placement of stations method of claim 1 wherein, The noise distribution of TDOA values and AOA values follows zero-mean Gaussian distribution with variances and respectively.

4. The two-dimensional optimal placement of stations method of claim 1 wherein, The FIM matrix of the target position is as follows: ； wherein and denote the Jacobian matrix of TDOA measurements and AOA measurements with respect to the target position, respectively, denotes the inverse of a matrix, denotes the transpose symbol, is a covariance matrix, , , , blkdiag denotes a block diagonal matrix, and denote the N-dimensional identity matrix and the N-dimensional all-one matrix, respectively; Substituting various parameters, the FIM matrix is expressed as: wherein , , , , ; The expansion of the FIM determinant obtained from the FIM matrix is as follows: ； Wherein, det represents a function for calculating the determinant of a square matrix.

5. The two-dimensional optimal placement of stations method of claim 4 wherein, Introducing mathematical constraints on the expansion of the FIM determinant , , which simplifies to （1）。 6. The two-dimensional optimal placement of stations method of claim 1 wherein, Solve the FIM determinant according to the D optimality criterion, and take the solution of the angular position corresponding to the maximum value of the FIM determinant as the two-dimensional optimal station arrangement method.

7. The two-dimensional optimal placement of stations method of claim 5, wherein, When the target is in the communication range of the reference station, equation (1) is simplified to an optimization problem as follows: ； Solve the optimization problem to obtain the two-dimensional optimal station arrangement method when the target is in the communication range of the reference station.

8. The two-dimensional optimal placement of stations method of claim 5, wherein, When the target is outside the communication range of the reference station, equation (1) is simplified to an optimization problem as follows: ； Solve the optimization problem to obtain the two-dimensional optimal station arrangement method when the target is outside the communication range of the reference station.

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