Positioning method based on time difference of arrival of source signals under non-line-of-sight conditions

Abstract

The application belongs to the technical field of source positioning, and more particularly relates to a positioning method based on a time difference of arrival of a source signal under a non-line-of-sight condition. The positioning method is constructed according to a TDOA measurement method to measure a distance difference between a sensor and a target source; a distance difference measurement model is converted into a weighted robust least square problem; the weighted robust least square problem is converted into a semi-positively non-convex optimization problem which can be modified by using S-Lemma; the semi-positively non-convex optimization problem is converted into a convex semi-positively programming problem by using a semi-positively relaxation technology; a semi-positively programming problem after tightening of an inequality constraint is constructed on the basis of the convex semi-positively programming problem; and the position of the signal source and an upper bound of a path NLOS error are jointly estimated by using an iterative method to solve the problem and obtain an optimal estimation value of the position of the target source.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of source positioning, and more particularly relates to a positioning method based on time difference of arrival of source signals under non-line-of-sight conditions. BACKGROUND

[0002] Precise positioning of a signal source is critical for many applications, such as emergency service response, mobile computing and target tracking. Commonly used positioning indicators include time of arrival (TOA) and time difference of arrival (TDOA). In recent years, non-line-of-sight (NLOS) errors have become common in indoor and urban environments, and if not properly handled, can severely degrade source position estimation quality. Therefore, the identification and mitigation of NLOS problems have attracted much attention.

[0003] Previous work tends to precisely model NLOS errors or solve NLOS identification and mitigation problems based on TOA systems. However, since NLOS characteristics are usually related to the environment and time, it is costly to build a model. In addition, TOA measurements require precise synchronization between the source and the sensor, and do not have wide applicability. TDOA methods can be used for passive positioning, do not require synchronization between the sensor and the signal source, and can achieve precise TDOA positioning in wideband positioning systems, and are widely applicable. And a large number of experiments have shown that in dense non-close visual environments, TDOA has higher positioning accuracy than TOA. Therefore, it is necessary to study source positioning based on time difference of arrival of source signals under non-line-of-sight conditions. SUMMARY

[0004] In view of the above problems existing in the prior art, the application provides a positioning method based on time difference of arrival of source signals under non-line-of-sight conditions, which has high positioning accuracy.

[0005] To achieve the above purpose, the application adopts the following technical scheme:

[0006] The positioning method based on time difference of arrival of source signals under non-line-of-sight conditions comprises the following steps,

[0007] Step 1, establishing a coordinate system and initializing the positions of the sensors and the signal source;

[0008] Step 2, constructing a distance measurement model using a TDOA measurement method, i.e., constructing a measurement value of the distance difference between the reference path sensor and other path sensors and the position of the signal source;

[0009] Step 3, converting the distance measurement model into a constrained robust weighted least squares (RWLS) problem.

[0010] Step four, using S-Lemma to transform the weighted robust least squares problem into a non-convex optimization problem which can be modified into semidefinite relaxation (SDR);

[0011] Step five, using semidefinite relaxation technique to transform the non-convex optimization problem into a convex semidefinite programming problem;

[0012] Step six, constructing the semidefinite programming problem with inequality constraints on the basis of the convex semidefinite programming problem;

[0013] Step seven, using the iterative method to jointly estimate the signal source position and the upper bound of the NLOS error to obtain the optimal estimation value of the target source position.

[0014] The further optimization of the technical scheme specifically comprises:

[0015] A k-dimensional positioning scene is established, assuming that a wireless sensor network has N+1 sensors and one unknown signal source, the position of the i-th sensor is represented by s i ∈R k , and the position of the signal source is represented by x∈R k , wherein R k represents a k-dimensional real vector, and s0 represents the position of the reference sensor.

[0016] The further optimization of the technical scheme specifically comprises:

[0017] It is assumed that the measured sensor position is accurate and that there is NLOS signal propagation between the signal source and the sensor, which is measured by the TDOA method and described as:

[0018]

[0019] wherein N is the number of sensors on other paths, t i represents the time difference between the reference path sensor and the i-th path sensor, c represents the speed of light, n i represents the measurement noise, e i is the NLOS error in TDOA measurement, ω0 and ω i respectively represent the non-line-of-sight error on the reference path and the other i sensor paths, and the error values are greater than 0, and ||| is the two-norm identifier. Multiply the speed of light c on both sides of the above equation (1.1) to obtain the measured value of the distance difference between the reference path sensor and the other path sensors to the signal source position, which is described as:

[0020]

[0021] where d i is the distance difference between the reference path sensor and the ith path sensor to the signal source position, and in addition, define a vector n = [n1,...,n N ] T represents the measurement noise on different paths, and define n to follow a Gaussian distribution with mean 0 and covariance , where I N represents an N-order identity matrix, 1 N represents an N-order all-1 matrix, T represents the transpose of a vector or matrix, and σ is the variance of the measurement noise error, and assume that there is a non-line-of-sight error upper bound ρ i such that ω i satisfies 0≤ω i ≤ρ i ;

[0022] It is easy to obtain , whose length is 2ρ i , where ρ0 is the non-line-of-sight error upper bound on the reference path, and according to equation (1.2), it is known that -ω0≤e i ≤ρ i -ω0, whose length is ρ i ; an excessively large upper bound can reduce the positioning performance, therefore, let and substitute it into equation (1.2), to obtain:

[0023]

[0024] where and respectively represent the corrected non-line-of-sight errors on the reference path and the other i sensor paths, is the distance difference between the corrected reference path sensor and the ith path sensor to the signal source position;

[0025] Move the term to the left side of equation (1.3) to obtain:

[0026]

[0027] Square both sides and move the term to the left side to obtain

[0028]

[0029] where r i is the distance difference between the ith sensor and the signal source, and r0 is the distance difference between the reference path sensor and the signal source. Since the second-order noise term is much smaller than the first-order noise term, it can be ignored;

[0030] To facilitate the expression of problem (1.5), let

[0031]

[0032] Substituting into (1.5), we have:

[0033]

[0034] If the weight of the robust least squares problem (RLS) is only related to the unknown NLOS error on the path related to, will cause When the range and amplitude of the change are too large, the model performance is reduced and the positioning effect is poor. Therefore, the variable c is added. i As the weight of the signal source localization problem, the main function is introduced, and then formula (1.5) can be described as:

[0035]

[0036] This technical solution is further optimized, and the step three is specifically as follows:

[0037] The signal source localization problem is transformed into a weighted robust least squares problem, which can be described as:

[0038]

[0039] To further simplify the weighted robust least squares problem, let Then it satisfies Equation (1.7) can be described as:

[0040]

[0041] This technical solution is further optimized, and the step 4 is specifically as follows:

[0042] In order to use S-Lemma to directly transform the above RWLS problem into a non-convex optimization problem that can be modified into SDR, Equation (1.9) can be equivalently transformed into:

[0043]

[0044] in, η=[η1,...,η N ] T , η i Indicates the degree of relaxation of the i-th constraint, min() is the minimization function, max() is the maximization function, and st means "subject to...". This constraint condition shows that for i = 1, ..., N, we have:

[0045]

[0046] That is

[0047]

[0048] According to S-Lemma, So that

[0049]

[0050] Replacing the constraint condition (1.10) in the RWLS problem with formula (1.12), we get

[0051]

[0052] The further optimization of the technical solution is specifically as follows:

[0053] An auxiliary variable Y = yy is introduced T By using the semi-positive relaxation technology, it is relaxed into a convex semi-positive programming problem for solving, and then (1.13) can be converted into

[0054]

[0055] Where y(5) represents the fifth element of y, Y(4,4) represents the fourth row fourth element of Y, Y(3,4) represents the third row fourth element of Y, and rank() represents the rank of a matrix;

[0056] Only rank(Y) = 1 in the problem (1.14) is a non-convex constraint, which can be discarded, and the following constraint is added to tighten the problem:

[0057]

[0058] Where Y(3,3) represents the third row third element of Y, Y(2,2) represents the second row second element of Y, and y(1:2) represents the first to second elements of y;

[0059] Then the problem (1.14) can be described as:

[0060]

[0061] The further optimization of the technical solution is specifically as follows:

[0062] Generally, the SDP problem obtained after relaxation cannot generate the global optimal solution of the original RWLS; in order to avoid local convergence or divergence, a new constraint is introduced to tighten the problem (1.16), and the non-convex equality constraint r i =||x-s i Is relaxed into a convex inequality constraint:||x-s i||≤r i i.e.

[0063] y(3)≥||y(1:2)-s0|| (1.17)

[0064] where y(3) represents the third element of y, and similarly, the equality constraint ω i r i =ω i ||x-s i ||, which can be relaxed to ω i r i ≥ω i ||x-s i ||, i.e.

[0065] y(5)-Y(4,4)≥2||Y(1:2,4)-s0·y(4)|| (1.18)

[0066] Further, based on the known prior information, the following constraints can be obtained: In addition, in order to better deal with the influence of non-line-of-sight error, prevent overestimating non-line-of-sight error, and provide a more accurate error range, let Therefore, it is described as:

[0067]

[0068] where Y(5,5) is the fifth element of the fifth row of Y, and the constraint conditions (1.17), (1.18), (1.19) are substituted into the problem (1.16), to obtain:

[0069]

[0070] Further optimization of the technical solution, the step seven includes using an iterative method to jointly estimate the signal source position and the NLOS error upper bound; the signal source position is updated using the error upper bound obtained in each iteration process under the condition that the maximum number of iterations is specified; the iteration is terminated when the distance between the calculated signal source position and the actual signal source position is less than a specified value or the number of iterations exceeds the maximum number of iterations.

[0071] Unlike the prior art, the above technical solution has the following beneficial effects:

[0072] 1. The existing method needs to know the specific distribution of NLOS error or needs to measure the complete synchronization between the signal source and the sensor using the TOA method. The present application only needs to specify the NLOS error upper bound and does not need the synchronization between the signal source and the sensor, and innovatively uses the calculation method to reach the time difference method for modeling and estimating the signal source position.

[0073] 2. To avoid the overlarge error upper bound reducing the positioning performance, the present application ingeniously subtracts each NOLS error upper bound from

[0074] The new NLOS error upper bound is formed and substituted into the problem for further solving. The method of iteratively updating the NLOS error upper bound and jointly estimating the signal source position and the NLOS error upper bound reduces the influence of the overlarge error upper bound on the positioning performance. In addition, the present application ingeniously uses S-Lemma to eliminate the maximum term in the weighted robust least square problem, and converts the RWLS problem into a non-convex optimization problem which can be modified into a semi-positive definite relaxation.

[0075] 3. The present application adds a variable c i as the weight of the RWLS problem. Avoids the RWLS problem from leading to the decline of the model positioning performance in the case of the NLOS error upper bound changing in a large range and amplitude.

[0076] 4. The present application uses the prior information and the internal relationship between the variables to tighten the convex semi-positive definite programming problem. Compared with the method of only relaxing but not tightening, the present method can effectively obtain the global optimal solution of the original robust least square problem, avoids the local divergence and convergence, and improves the algorithm accuracy. BRIEF DESCRIPTION OF DRAWINGS

[0077] Figure 1 is a flowchart of the positioning method based on the time difference of arrival of source signals under the non-line-of-sight condition;

[0078] Figure 2 is a schematic diagram of the scene in which there are N+1 sensors and source positions under the 2-dimensional condition;

[0079] Figure 3 is a trend graph of the root mean square error (RMSE) of the signal source position estimation varying with other path error upper bounds;

[0080] Figure 4 is a trend graph of the root mean square error (RMSE) of the signal source position estimation varying with other path error upper bounds;

[0081] Figure 5 is a trend graph of the root mean square error (RMSE) of the signal source position estimation varying with the number of sensors;

[0082] Figure 6 is a trend graph of the root mean square error (RMSE) of the signal source position estimation varying with the noise error covariance parameter. DETAILED DESCRIPTION

[0083] To make the technical content, structural features, achieved purposes and effects of the technical scheme clear, the following will be described in detail in combination with specific embodiments and the accompanying drawings. To make the technical content, structural features, achieved purposes and effects of the technical scheme clear, the following will be described in detail in combination with specific embodiments and the accompanying drawings.

[0084] Referring to Figure 1 In a preferred embodiment, a positioning method based on source signal time difference of arrival in non-line-of-sight conditions comprises the following steps:

[0085] Step 1: Establish a coordinate system and initialize the sensor positions and signal source position.

[0086] A k-dimensional positioning scene is established. It is assumed that a wireless sensor network has N+1 sensors and one unknown signal source. The position of the i-th sensor is denoted by s i ∈R k and the position of the signal source is denoted by x ∈ R k . Where R k denotes a k-dimensional real vector, and s0 denotes the reference sensor position. Figure 2 A scene diagram is given in the case of N+1 sensors and signal source positions in 2 dimensions (one of the scene examples with a reference path error upper bound of 0.5 and other path error upper bounds of 6α (α = 0.1,..., 1.2), where the hollow triangle represents the sensor on the other path, the solid triangle represents the sensor on the reference path, and the hollow circle represents the signal source).

[0087] Step 2: Construct a distance measurement model using the TDOA measurement method, i.e., construct the measured value of the distance difference between the reference path sensor and the other path sensor to the signal source position.

[0088] It is assumed that the sensor positions are measured accurately and that there is NLOS signal propagation between the signal source and the sensor. By measuring with the TDOA method, it is described as:

[0089]

[0090] Where N is the number of sensors on the other path, t i is the time difference between the source signal arriving at the reference path sensor and the i-th path sensor, c is the speed of light, n i is the measurement noise, e i is the NLOS error in TDOA measurement, ω0 and ω i represent the non-line-of-sight errors on the reference path and the other i sensor paths, respectively, and the error values are greater than 0, and || || is the two-norm identifier. Multiply both sides of equation (1.1) by the speed of light c to obtain the distance difference between the reference path sensor and the other path sensor to the signal source position, which is described as:

[0091]

[0092] Where d iis the measurement of the distance difference between the reference path sensor and the ith path sensor to the signal source position. In addition, a vector n = [n1,..., nN]Tis defined, where N ] T represents the measurement noise on different paths, and n is defined to follow a Gaussian distribution with mean 0 and covariance , where INis the N-order identity matrix, 1Nis the N-order all-one matrix, T denotes the transpose of a vector or matrix, and σ is the variance of the measurement noise error. It is assumed that there exists a non-line-of-sight error upper bound ρ N such that ω N satisfies 0 ≤ ω i ≤ ρ i . i i .

[0093] It is easy to obtain , whose length is 2ρ i , where ρ0is the non-line-of-sight error upper bound on the reference path. However, according to equation (1.2), it is known that -ω0≤ e i ≤ ρ i - ω0, whose length is ρ i . An excessively large upper bound can reduce the positioning performance. Therefore, let and substitute it into equation (1.2), we obtain:

[0094]

[0095] where and respectively represent the corrected non-line-of-sight errors on the reference path and the other i sensor paths, is the distance difference between the corrected reference path sensor and the ith path sensor to the signal source position.

[0096] Moving the terms in equation (1.3), we obtain:

[0097]

[0098] Square both sides and move the terms, we obtain

[0099]

[0100] where r i is the distance difference between the ith sensor and the signal source, and r0is the distance difference between the reference path sensor and the signal source. Since the second-order noise term is much smaller than the first-order noise term, it can be ignored;

[0101] For convenience of expression of problem (1.5), let

[0102] ​

[0103] Substitute (1.5) into (1.6), we have:

[0104]

[0105] If the weight of the robust least squares problem is only related to the unknown NLOS error on the path, it will cause the model performance to decrease and the positioning effect to be poor in the case of large range and amplitude of variation. Therefore, the variable c is added as the weight of the signal source positioning problem, and the main function is introduced. Then, (1.5) can be described as: i

[0106]

[0107] Step 3: Convert the distance measurement model into a weighted robust least squares problem with constraints.

[0108] Convert the signal source positioning problem into a weighted robust least squares problem, which can be described as:

[0109]

[0110] To further simplify the weighted robust least squares problem, let Then, it satisfies (1.7) can be described as:

[0111]

[0112] Step 4: Convert the weighted robust least squares problem into a non-convex optimization problem that can be modified as a semi-definite relaxation using S-Lemma.

[0113] To convert the above RWLS problem into a non-convex optimization problem that can be modified as SDR using S-Lemma, (1.9) can be equivalently converted as:

[0114]

[0115] where, η=[η1,...,η N ] T , η i represents the relaxation degree of the i-th constraint, min() is the minimization function, max() is the maximization function, and s.t represents "constrained to …". This constraint condition indicates that for i = 1,...,N, we have:

[0116]

[0117] That is,

[0118] ​​​

[0119] According to S-Lemma, such that

[0120]

[0121] Replacing the constraint condition (1.10) in the RWLS problem (1.12) with formula (1.12), we get

[0122]

[0123] Step five, the non-convex optimization problem is converted into a convex semi-definite programming problem by using semi-definite relaxation technique.

[0124] Step four eliminates the maximization part in the original RLS problem (1.8), but problem (1.12) is still a non-convex optimization problem, which is difficult to solve. Therefore, auxiliary variables Y = yy T are introduced, which are relaxed into a convex semi-definite programming problem by using semi-definite relaxation technique, so that (1.13) can be converted into:

[0125]

[0126] where y(5) represents the fifth element of y, Y(4,4) represents the fourth row and fourth element of Y, Y(3,4) represents the third row and fourth element of Y, and rank() represents the rank of the matrix.

[0127] Only rank(Y) = 1 in problem (1.14) is a non-convex constraint, which can be discarded, and the following constraint is added to tighten the problem:

[0128]

[0129] where Y(3,3) represents the third row and third element of Y, Y(2,2) represents the second row and second element of Y, and y(1:2) represents the first to second elements of y.

[0130] Then problem (1.14) can be described as:

[0131]

[0132] Step six, the semi-definite programming problem after tightening the inequality constraint is constructed based on the convex semi-definite programming problem.

[0133] Generally, the SDP problem obtained after relaxation cannot generate the global optimal solution of the original RWLS; to avoid local convergence or divergence, new constraints are introduced to tighten problem (1.16). The non-convex equality constraint r i = ||x-s i||x-s i ||≤r i i.e.:

[0134] y(3)≥||y(1:2)-s0|| (1.17)

[0135] where y(3) represents the third element of y, and similarly, the equality constraint ω i r i =ω i ||x-s i ||, can be relaxed to ω i r i ≥ω i ||x-s i ||, i.e.:

[0136] y(5)-Y(4,4)≥2||Y(1:2,4)-s0·y(4)|| (1.18)

[0137] Further, based on the known prior information, the following constraints can be obtained: In addition, in order to better deal with the influence of non-line-of-sight error, prevent overestimating the non-line-of-sight error, and provide a more accurate error range, let Therefore, it is described as:

[0138]

[0139] where Y(5,5) is the fifth element of the fifth row of Y, and the constraint conditions (1.17), (1.18), (1.19) are substituted into the problem (1.16), to obtain:

[0140]

[0141] Step seven, the signal source position and the path NLOS error upper bound are jointly estimated by using an iterative method to obtain the optimal estimation value of the target source position.

[0142] In order to further reduce the error upper bound and avoid excessive error upper bound to reduce the experimental performance, the signal source position and the NLOS error upper bound are jointly estimated by using an iterative method. The signal source position is updated by using the error upper bound obtained in each iteration process under the condition that the maximum number of iterations is specified. The iteration is terminated until the distance between the calculated signal source position and the actual signal source position is less than a specified value or the number of iterations exceeds the maximum number of iterations.

[0143] In order to verify the feasibility and effectiveness of the method of the application, simulation tests are performed on the method of the application.

[0144] It is assumed that there are at most 13 sensors and 1 signal source in a 2-dimensional space, and their coordinates in a 2-dimensional rectangular coordinate system are s i= [x i ,y i ] T ,i = 1,...,N, t = [m,n] T are randomly generated, where x i ~(0,10) meters, y i ~(0,10) meters, m~(0,10) meters, n~(0,10) meters. The measurement noise error satisfies a Gaussian distribution with mean equal to 0 and covariance , i.e. The NLOS errors ω i ,i = 0,...,N on the reference path and other paths are randomly generated, ω i ~(0,p i ), where p i is the upper bound of the error on the ith path.

[0145] To demonstrate the performance of the method of the present application under different scenarios, simulation tests are conducted to test the performance of the method of the present application under three different conditions of different NLOS error upper bounds, different numbers of sensors, and different measurement noise error variances, and the performance of the method is measured by root mean square error (RMSE). The root mean square error (RMSE) describing the positioning performance under the three conditions is the RMSE obtained by running 2000 Monte Carlo experiments for each of the 10 random scenes, and a trend graph is drawn for visualization. Figure 3 A trend graph of the root mean square error (RMSE) of the signal source position estimation with respect to the upper bound of the error on the other paths is given, where the upper bound of the error on the reference path is 0.5, the upper bound of the error on the other paths is 6a (a = 0.1,..., 1.2), the number of sensors on the other paths is N = 6, and the variance of the measurement noise error is σ = 0.3. Figure 4 A trend graph of the root mean square error (RMSE) of the signal source position estimation with respect to the upper bound of the error on the other paths is given, where the upper bound of the error on the reference path is 6, the upper bound of the error on the other paths is 6a (a = 0.1,..., 1.2), the number of sensors on the other paths is N = 6, and the variance of the measurement noise error is σ = 0.3. Figure 5 A trend graph of the root mean square error (RMSE) of the signal source position estimation with respect to the number of sensors on the other paths is given, where the upper bound of the error on the reference path is 0.5, the upper bound of the error on the other paths is 6, the number of sensors on the other paths is N = [4, 5, 6, 7, 8], and the variance of the measurement noise error is σ = 0.3. Figure 6The trend chart of the root mean square error (RMSE) noise error covariance parameter variation of the signal source position estimation of the method of the application under the condition that the upper bound of the reference path error is 0.5, the upper bound of the other path error is 6, the number of sensors on the other path N is 6, and the variance σ of the measurement noise error is 0.1,...,0.6 can be given. Figure 3 , Figure 4 It can be seen that when the upper bound of the NLOS error increases, the RMSE of the method of the application increases, but the maximum value does not exceed 2.5, and the positioning accuracy is relatively accurate. Figure 5 It can be seen that when the number of sensors on the other path N increases, the RMSE value of the method of the application significantly decreases, which means that when the number of sensors increases, the positioning accuracy of the method of the application is significantly improved. In addition, when the measurement noise error increases, the RMSE of the method of the application takes the minimum value at σ=0.3, but the maximum deviation of the RMSE under different σ values does not exceed 0.015. This shows that the size of the measurement noise has little effect on the positioning accuracy of the method.

[0146] It should be noted that in this paper, relational terms such as first and second are used only to distinguish one entity or operation from another entity or operation, and do not necessarily require or imply that there is any such actual relationship or order between these entities or operations. Moreover, the terms "include", "contain" or any other variants thereof are intended to cover non-exclusive inclusion, so that the process, method, article or terminal device including a series of elements not only includes those elements, but also includes other elements not explicitly listed or inherent to such process, method, article or terminal device. Without more limitations, the elements defined by the statement "include" or "contain" do not exclude the presence of other elements in the process, method, article or terminal device including the elements. In addition, in this paper, "greater than", "less than", "exceed" and the like are understood as not including the number; "above", "below", "within" and the like are understood as including the number.

[0147] Although the above embodiments have been described, those skilled in the art can make further changes and modifications to the embodiments once they know the basic creative concept, so the above description is only an embodiment of the application and does not limit the patent protection scope of the application, and any equivalent structure or equivalent process transformation using the content of the application specification and drawings, or direct or indirect application in other related technical fields, are also included in the patent protection scope of the application.

Claims

1. A positioning method based on the time difference of arrival of source signals under non-line-of-sight conditions, characterized in that: The following steps are included: Step 1: Establish a coordinate system and initialize the sensor position and signal source position; Step 2: Use the TDOA measurement method to build a distance measurement model, that is, to build the measurement value of the distance difference between the reference path sensor and other path sensors and the signal source position; Step 3: Convert the distance measurement model into a constrained weighted robust least squares problem; Step 4: Use S-Lemma to transform the weighted robust least squares problem into a non-convex optimization problem that can be corrected to a semi-definite relaxation. Step 5: Use semi-definite relaxation technology to transform the non-convex optimization problem into a convex semi-definite programming problem; Step 6: Based on the convex semi-positive definite programming problem, construct a semi-positive definite programming problem with tightened inequality constraints; Step 7: Use an iterative method to jointly estimate the signal source position and the upper bound of the path NLOS error to obtain the optimal estimate of the target source position.

2. The positioning method based on the time difference of arrival of source signals under non-line-of-sight conditions according to claim 1, characterized in that: The step 1 specifically includes: Establish a k-dimensional positioning scenario. Assume that a wireless sensor network has N+1 sensors and 1 unknown signal source. The position of the i-th sensor is s i ∈R k Indicates that the signal source position is x∈R k Indicates that, R k represents a k-dimensional real vector, and s0 represents the reference sensor position.

3. The positioning method based on the time difference of arrival of source signals under non-line-of-sight conditions according to claim 2, characterized in that: The second step specifically includes: Assuming that the sensor position is accurately measured and there is NLOS signal propagation between the signal source and the sensor, the TDOA method is used to measure, which is described as: Where N is the number of sensors in other paths, t i represents the time difference between the source signal reaching the reference path sensor and the i-th path sensor, c represents the speed of light, n i represents the measurement noise, e i is the NLOS error in TDOA measurement, ω0 and ω i They represent the non-line-of-sight errors on the reference path and the other i sensor paths, and the error values ​​are all greater than 0. ‖‖ is the two-norm identifier. Multiplying both sides of the above equation in equation (1.1) by the speed of light c, we get the measured value of the distance difference between the reference path sensor and the other path sensors and the signal source position, which is described as: Among them, d i is the distance difference between the reference path sensor and the i-th path sensor to the signal source position. In addition, a vector n=[n1,...,n N ] T Represents the measurement noise on different paths, and defines n to obey the mean value of 0 and the covariance of Gaussian distribution, where I N represents the N-order unit matrix, 1 N represents an N-order all-one matrix, T represents the transpose of a vector or matrix, σ is the variance of the measurement noise error, and it is assumed that there is an upper bound of the non-line-of-sight error ρ i Make ω i Satisfying 0≤ω i ≤ρ i ; Easy to obtain Its length is 2ρ i , where ρ0 is the upper bound of the non-line-of-sight error on the reference path. However, according to formula (1.2), -ω0≤e i ≤ρ i -ω0, whose length is ρ i ; Too large an upper bound may reduce the positioning performance, so let Substituting it into formula (1.2), we get: in, and denote the non-line-of-sight errors on the corrected reference path and the i-th sensor path, respectively. is the distance difference between the corrected reference path sensor and the i-th path sensor to the signal source position; By transposing the terms in equation (1.3), we can obtain: Square both sides and move the terms, and we get Among them, r i is the distance difference between the i-th sensor and the signal source, r0 is the distance difference between the reference path sensor and the signal source, due to the second-order noise term It is much smaller than the first-order noise term and can be ignored; To express the problem (1.5) conveniently, let Substituting into (1.5), we have: If the weights of the robust least squares problem are only related to the unknown NLOS errors on the path related to, will cause When the range and amplitude of the change are too large, the model performance is reduced and the positioning effect is poor. Therefore, the variable c is added. i As the weight of the signal source localization problem, the main function is introduced, and then formula (1.5) can be described as:

4. The positioning method based on the time difference of arrival of source signals under non-line-of-sight conditions according to claim 3, characterized in that: The step three is as follows: The source signal localization problem is transformed into a weighted robust least squares problem, which is described as: To further simplify the weighted robust least squares problem, let Then it satisfies Equation (1.7) can be described as:

5. The positioning method based on the time difference of arrival of source signals under non-line-of-sight conditions according to claim 4, characterized in that: The step 4 is specifically as follows: In order to use S-Lemma to directly transform the above RWLS problem into a non-convex optimization problem that can be modified into SDR, Equation (1.9) can be equivalently transformed into: in, η i Indicates the degree of relaxation of the i-th constraint, min() is the minimization function, max() is the maximization function, and st means "subject to...". The constraint condition shows that for i = 1, ..., N, we have: Right now According to S-Lemma, Make Substituting formula (1.12) for the constraint condition (1.10) in the RWLS problem, we get 6. The positioning method based on the time difference of arrival of source signals under non-line-of-sight conditions according to claim 5, characterized in that: The step five is specifically as follows: Introduce auxiliary variable Y=yy T , using the semi-positive definite relaxation technique, it is relaxed into a convex semi-positive definite programming problem in order to solve it, then (1.13) can be transformed into: Where y(5) represents the fifth element of y, Y(4,4) represents the fourth element of the fourth row of Y, Y(3,4) represents the fourth element of the third row of Y, and rank() represents the rank of the matrix; In problem (1.14), the only non-convex constraint is rank(Y) = 1, which can be discarded and the following constraint added to tighten the problem: Among them, Y(3,3) represents the third element of the third row of Y, Y(2,2) represents the second element of the second row of Y, and y(1:2) represents the first to second elements of y; Then problem (1.14) can be described as:

7. The positioning method based on the time difference of arrival of source signals under non-line-of-sight conditions according to claim 6, characterized in that: The step six is ​​specifically as follows: Usually, the SDP problem obtained after relaxation cannot generate the global optimal solution of the original RWLS. To avoid local convergence or divergence, new constraints are introduced to tighten the problem (1.16), using the second-order cone constraint to relax the non-convex equality constraint r i =||xs i || to convex inequality constraints: ||xs i ||≤r i ,Right now: y(3)≥||y(1:2)-s0|| (1.17) Where y(3) represents the third element of y. Similarly, the equality constraint ω i r i =ω i ||xs i ||, which can be relaxed to ω i r i ≥ω i ||xs i ||, that is: y(5)-Y(4,4)≥2||Y(1:2,4)-s0·y(4)|| (1.18) Furthermore, based on known prior information, the following constraints can be obtained: In addition, in order to better handle the impact of non-line-of-sight errors, prevent overestimation of non-line-of-sight errors, and provide a more accurate error range, So it is described as: Where Y(5,5) is the fifth element in the fifth row of Y. Substituting constraints (1.17), (1.18), and (1.19) into problem (1.16), we obtain:

8. The positioning method based on the time difference of arrival of source signals under non-line-of-sight conditions according to claim 1, characterized in that: The step seven includes jointly estimating the signal source position and the NLOS error upper bound using an iterative method; Under the specified maximum number of iterations, the signal source position is updated using the upper bound of the error obtained in each iteration; The iteration is terminated when the distance between the calculated signal source position and the actual signal source position is less than the specified value or the number of iterations exceeds the maximum number of iterations.

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