Moving target source positioning method based on TOA and FOA combined measurement value

Abstract

The invention discloses a moving target source positioning method based on TOA and FOA combined measurement values. The method comprises the following steps: establishing a wireless sensor network system model comprising a target source and a plurality of sensors; constructing a TOA measurement value model and an FOA measurement value model, and respectively obtaining pseudo-linear equations about the TOA and the FOA; unknown variable vectors including target source position, speed, signal propagation speed, clock bias and drift are introduced, the pseudo linear equations are stacked into vector equations, and the vector equations are fused into a first-step vector equation; a weighted least square optimization problem of the first step is constructed and solved, and a first-step closed-form solution is obtained; further constructing a pseudo-linear equation by using a relationship between parameters, introducing a second-step unknown variable vector, and constructing a second-step vector equation; constructing and solving a weighted least square optimization problem in the second step to obtain a closed-form solution in the second step; obtaining estimated values of parameters such as the position and the speed of a target source by combining the two-step solution; the method has the advantages that accurate positioning can be realized under the condition that signal propagation speed, sensor parameter errors and clock information are unknown.

Description

Technical Field

[0001] The present invention belongs to the field of wireless sensor network positioning, and relates to a mobile target source positioning technology, and in particular to a mobile target source positioning method based on TOA (time of arrival) and FOA (frequency of arrival) joint measurement values ​​in the presence of unknown clock bias and unknown clock drift in an environment where both signal propagation speed and sensor parameter errors are unknown. Background Art

[0002] The localization of mobile target sources has been a hot topic in the past decade, and has broad application prospects in the fields of radar monitoring, acoustic wave detection, and wireless sensor networks. Generally speaking, the most common positioning problem is based on time measurement values, including time of arrival (TOA) and time difference of arrival (TDOA). However, a large number of studies have shown that in positioning based on TOA measurements, clock synchronization between the target source and the sensor is required, otherwise unknown clock bias will be generated, which will reduce the performance of the positioning algorithm based on TOA measurements. In order to solve this problem, researchers use pairwise subtraction between TOA measurements to remove the unknown clock bias and simplify the positioning problem to a TDOA measurement model. Although this simplifies the problem, the use of TDOA measurements to avoid the clock synchronization problem between the target source and the sensor comes at the expense of positioning accuracy. Some researchers have found that pairwise subtraction of TOA measurements will increase the measurement noise power by 3dB, and also cause information loss. Therefore, more and more research is focusing on solving the clock synchronization problem between the target source and the sensor while designing a high-precision positioning method. For example, Enyang Xu et al. proposed two convex methods, called the two-step least squares (2LS) algorithm and the minimum-maximum algorithm (MMA), in their results published in IEEE Transactions on Signal Processing to further improve the positioning accuracy when the signal transmission time is not available. However, MMA blurs the true value of the measurement error, resulting in performance loss. For mobile target sources, the frequency of arrival (FOA) measurement value containing speed information can also be used to estimate the speed of the mobile target source and improve the positioning accuracy. In view of this, research on the positioning of mobile target sources dedicated to the joint measurement of TOA and FOA has important practical significance.

[0003] The problem of localizing a mobile target source based on joint TOA and FOA measurements with unknown clock bias and drift is very complex and difficult to solve. For example, Jiong Shi et al. proposed a semi-positive relaxation method for the above mobile target source localization problem in their work published in Signal Processing. This method actually converts the TOA and FOA measurement models into distance and distance rate models by assuming that the signal propagation speed and sensor parameters are accurately known, so as to facilitate derivation. However, in most practical environments, this may not be the case. For example, when the signal propagates underwater or in a solid, the signal propagation speed depends mainly on the propagation medium. In addition, even with GPS technology, it is impossible to obtain accurate sensor parameter information. Therefore, it is necessary to study a method for localizing a mobile target source based on joint TOA and FOA measurements with unknown clock bias and unknown clock drift in an environment where the signal propagation speed is uncertain and the sensor parameter information cannot be accurately obtained. Summary of the invention

[0004] The technical problem to be solved by the present invention is to provide a mobile target source positioning method based on TOA and FOA joint measurement values, which can accurately estimate the position and speed of the mobile target source, as well as the signal propagation speed, clock bias and clock drift when the signal propagation speed and sensor parameter error are unknown and there are unknown clock bias and unknown clock drift.

[0005] The technical solution adopted by the present invention to solve the above technical problems is: a method for positioning a mobile target source based on TOA and FOA joint measurement values, characterized in that the implementation process of the method is: establishing a wireless sensor network system model, in which it is assumed that there is one mobile target source and multiple mobile sensors; constructing a TOA measurement value model related to the TOA measurement values ​​obtained by all mobile sensors and a FOA measurement value model related to the obtained FOA measurement values; obtaining a pseudo-linear equation about the TOA measurement value based on the TOA measurement value model, and obtaining a pseudo-linear equation about the FOA measurement value based on the FOA measurement value model; by introducing a first step unknown variable vector including the position coordinates and velocity coordinates of the mobile target source, the signal propagation speed, the clock bias, and the clock drift, the pseudo-linear equation about the TOA measurement value is stacked into a vector In the form of an equation, the pseudo-linear equations about the FOA measurement values ​​are stacked into the form of a vector equation, and then the two vector equations are merged into a first-step vector equation; on the basis of the first-step vector equation, combined with the weighted least squares criterion, the weighted least squares optimization problem of the first step is constructed, and the closed-form solution of the first step is obtained by solving; using the relationship between the parameters in the first-step unknown variable vector, multiple pseudo-linear equations are constructed, and by introducing the second-step unknown variable vector, multiple pseudo-linear equations are stacked into a second-step vector equation; on the basis of the second-step vector equation, combined with the weighted least squares criterion, the weighted least squares optimization problem of the second step is constructed, and the closed-form solution of the second step is obtained by solving; according to the closed-form solution of the first step and the closed-form solution of the second step, the estimated values ​​of the position coordinates and velocity coordinates of the mobile target source, the signal propagation speed, the clock bias, and the clock drift are obtained.

[0006] The method specifically comprises the following steps:

[0007] Step 1: In the k-dimensional wireless sensor network system model, assume that there is a mobile target source and M mobile sensors. The true value of the position coordinate of the mobile target source and the true value of the velocity coordinate are recorded as x o and The true value of the position coordinate and the true value of the velocity coordinate of the i-th mobile sensor are recorded as and The noisy values ​​of the position coordinates and the noisy values ​​of the velocity coordinates of the i-th mobile sensor are denoted as s i and Where k = 2 or 3, M ≥ 2 × k + 3, 1 ≤ i ≤ M, Δs i The noisy value s represents the position coordinate of the i-th mobile sensor i With the true value The error between Represents the noisy value of the velocity coordinate of the i-th mobile sensor With the true value The error between

[0008] Step 2: In the k-dimensional wireless sensor network system, the mobile target source sends measurement signals to all mobile sensors at the same time. After receiving the measurement signal sent by the mobile target source, each mobile sensor obtains the TOA measurement value and the FOA measurement value divided by the signal carrier frequency, and the TOA true value model is modeled as: The true value model of FOA after dividing by the signal carrier frequency is modeled as: The TOA measurement model is modeled as: The FOA measurement model divided by the signal carrier frequency is modeled as: in, represents the true TOA value associated with the ith mobile sensor, the symbol “|| ||” is the symbol for finding the Euclidean norm, c o It represents the true value of the signal propagation speed during the process from the mobile target source sending the measurement signal to the mobile sensor receiving the measurement signal. represents the true value of the clock offset between the mobile target source and the mobile sensor, represents the true value of FOA related to the ith mobile sensor divided by the signal carrier frequency. The superscript “T” is the transposition symbol. represents the true value of the clock drift between the mobile target source and the mobile sensor, τ i It represents the TOA measurement value obtained by the i-th mobile sensor after receiving the measurement signal sent from the mobile target source. represents the FOA measurement value obtained by the i-th mobile sensor after receiving the measurement signal sent by the mobile target source and divided by the signal carrier frequency, Δτ i Represents τ i The noise present in i Obey Gaussian distribution 0 means Δτ i The mean value, Q τ Denotes Δτ i The covariance matrix of express The noise present in Obey Gaussian distribution 0 means The mean of express The covariance matrix of

[0009] Step 3: Under high signal-to-noise ratio conditions, perform linear transformation on the TOA true value model to obtain: Then and Substituting into equation 3.1 and squaring both sides of the equal sign, and then ignoring the second-order noise term, we get the pseudo-linear equation for the TOA measurement value, expressed as:

[0010]

[0011] Similarly, under high signal-to-noise ratio conditions, the FOA true value model divided by the signal carrier frequency is linearly transformed to obtain: Then substitute equation 3.1 into equation 3.3 to obtain: Then and as well as and Substituting into equation 3.4 and ignoring the second-order noise term, we obtain a pseudo-linear equation for the FOA measurement, expressed as:

[0012] Step 4: Define the true value of the unknown variable vector in the first step Then, we introduce the pseudo-linear equation for TOA measurement, Equation 3.2: The pseudo-linear equations about TOA measurements are stacked into a vector equation form, which is described as: Similarly, in the pseudo-linear equation for the FOA measurement, Equation 3.5, we introduce The pseudo-linear equations about the FOA measurements are stacked into a vector equation form, described as: Among them, the vector h τ The i-th element of is h τ (i), h τ (i)=-||s i || 2 , the matrix G τ The i-th row vector of G τ (i,:), 0 1×k represents a k-dimensional row vector of all zeros, and the vector ε τ =[B1,0 M×M ]Δm+[D1,0 M×Mk ]Δβ, matrix diag{} means finding the diagonal matrix, I M represents the identity matrix of order M, 0 M×M represents an M×M dimensional matrix with all zeros, Δm represents the measurement error vector, Δτ represents the TOA noise vector formed by the noise in the TOA measurements obtained by all mobile sensors, Δτ = [Δτ1, Δτ2, ..., Δτ M ] T , It represents the FOA noise vector formed by stacking the noise existing in the FOA measurement values ​​obtained by all mobile sensors after dividing by the signal carrier frequency, The i-th row vector of matrix D1 is D1(i,:), D1(i,:)=2[0 1×(i-1)k ,(x o -s i ) T ,0 1×(M-i)k ],0 1×(i-1)k represents a (i-1)k-dimensional row vector of all zeros, 0 1×(M-i)k represents a (Mi)k-dimensional row vector of all zeros, 0 M×Mk represents an M×Mk-dimensional matrix with all zeros, Δβ represents the motion sensor error vector, Δs represents the noise vector of the position coordinates of all mobile sensors, which is the error between the noisy values ​​and the true values. Δs = [Δs1, Δs2, ..., Δs M ] T , The mobile sensor velocity coordinate noise vector is a stack of the errors between the noisy values ​​and the true values ​​of the velocity coordinates of all mobile sensors. vector The i-th element of matrix The i-th row vector of vector matrix matrix The i-th row vector of matrix D2 is D2(i,:), The i-th row vector of matrix D3 is D3(i,:), D3(i,:)=[0 1×(i-1)k ,(x o -s i ) T ,0 1×(M-i)k ];

[0013] Step 5: Convert the vector equation about TOA measurement value and the vector equation for the FOA measurement It is fused into a first step vector equation, described as: Among them, the vector matrix vector matrix matrix

[0014] Step 6: Based on the weighted least squares criterion, the positioning problem is transformed into The first step of establishing the weighted least squares optimization problem is described as: Then solve the weighted least squares optimization problem in the first step and get the expression of the weighted least squares solution, that is, Then get the exact value of W1 and substitute the exact value of W1 into the expression of the weighted least squares solution In, get This is the closed-form solution of the first step; then we get On the basis of Substitution In, get and The error vector between Then, at a low noise level, Taking the expectation on both sides of the equality sign, we get Then get The approximation of the covariance matrix is ​​expressed as: in, represents the estimated value of the unknown variable vector in the first step, x represents the estimated value of the position coordinates of the moving target source, represents the estimated value of the velocity coordinate of the moving target source, c represents the estimated value of the signal propagation speed, τ o represents the estimated value of the clock bias, represents the estimated value of clock drift, W1 is the weight matrix, and its expression is E() means to find the expectation, Q m represents the covariance matrix of Δm, Q m =E(ΔmΔm T ), Q β represents the covariance matrix of Δβ, Q β =E(ΔβΔβ T ), cov() means to find the covariance matrix;

[0015] Step 7: The column vector consisting of the 1st to kth parameters in It is an estimate of the position coordinates of the moving target source, expressed as: The column vector consisting of the k+1th to 2kth parameters in It is an estimate of the velocity coordinate of the moving target source, expressed as: The 2k+1th parameter in It is expressed as: The 2k+2th parameter in It is expressed as: The 2k+3th parameter in It is expressed as: The 2k+4th parameter in It is expressed as: The 2k+5th parameter in It is expressed as: Then use formula 7.1 to get: Combining equation 7.1 and equation 7.2 gives: Combining equation 7.3 and equation 7.4, we get: Combining equation 7.4 and equation 7.6, we get: Then, Equation 7.8, Equation 7.9, Equation 7.3, Equation 7.10, Equation 7.5, Equation 7.11, and Equation 7.7 are combined into a system of equations, expressed as: Then remove the second-order noise term in equation 7.12 to obtain a pseudo-linear system of equations, expressed as: Finally, all the pseudo-linear equations in Equation 7.13 are stacked into a second-step vector equation, described as:

[0016] in, express is a column vector consisting of the 1st to kth parameters in , express The column vector consisting of the k+1th to 2kth parameters in , Corresponding representation The 2k+1th parameter, the 2k+2th parameter, the 2k+3th parameter, the 2k+4th parameter, the 2k+5th parameter, represents a k-dimensional row vector with all 1s, and the symbol “⊙” represents the Hadamard product. I k×k represents the k×k dimensional identity matrix, 0 k×k represents a k×k dimensional matrix with all zeros, 0 k×1 represents a k-dimensional column vector with all zeros, 1 1×k represents a k-dimensional row vector with all 1s, that is, vector is the true value of the unknown variable vector introduced in the second step, vector matrix 0 k×5 represents a k×5 dimensional matrix with all zeros, 0 5×k represents a 5×k-dimensional matrix with all zeros, the matrix

[0017] Step 8: Based on the weighted least squares criterion, the positioning problem is transformed into The weighted least squares optimization problem of the second step is established, which is described as: Then solve the weighted least squares optimization problem in the second step and get the expression of the weighted least squares solution, that is, Then get the exact value of W2 and substitute it into the expression of weighted least squares solution: In, get This is the closed-form solution of the second step; then we get and Based on this, the estimated value x of the position coordinates of the moving target source and the estimated value x of the velocity coordinates of the moving target source are obtained. The estimated value of signal propagation speed c and the estimated value of clock bias τ o , the estimated value of clock drift in, represents the estimated value of the unknown variable vector in the second step, W2 is the weight matrix, and its expression is sgn() represents the symbol function, express is a column vector consisting of the 1st to kth parameters in , express The column vector consisting of the k+1th to 2kth parameters in , express The 2k+1th parameter in, express The 2k+2th parameter in, express The 2k+3th parameter in .

[0018] In step 6, the exact value of W1 is obtained by: Let the initial value of W1 be Then Substitute the expression of the weighted least squares solution into In, get The initial value of; then according to the expression of W1, use The initial value of W1 is obtained.

[0019] In step 8, the exact value of W2 is obtained by: The parameter x in c、τ o , Substitute into the expression of W2 to get the initial value of W2; then substitute the initial value of W2 into the expression of the weighted least squares solution In, get The initial value of Substitute the initial value of In the equation, we get x, c、τ o , Then get x, c、τ o , Substitute into the expression of W2 to obtain the exact value of W2.

[0020] Compared with the prior art, the advantages of the present invention are:

[0021] In a wireless sensor network system, when the signal propagation speed, sensor parameter error, and clock parameters are unknown, the TOA measurement model and the FOA measurement model are converted into a series of pseudo-linear equations by introducing auxiliary variables. Then, the corresponding weighted least squares problem is constructed based on the weighted least squares criterion to obtain a closed-form solution for the initial estimate of the unknown variable vector. Then, another weighted least squares problem is constructed using the relationship between the parameters in the unknown variable vector to obtain a more accurate solution. Not only can the position coordinates and velocity coordinates of the mobile target source be estimated, but also the signal propagation speed and clock parameters can be estimated. This provides good positioning accuracy and low computational complexity in environments with small and medium noise.

[0022] Compared with the positioning method using TDOA measurement values, the present invention omits the step of pairwise subtraction of TOA measurement values ​​and combines them with FOA measurement values ​​for estimation. Although the number of unknown quantities is increased, making the problem more complicated, the unknown clock bias and clock drift can be estimated, and the positioning accuracy can be improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 It is a general implementation block diagram of the method of the present invention;

[0024] Figure 2 The figure is a schematic diagram of the deployment of mobile target sources and 10 mobile sensors in a 3D wireless sensor network system;

[0025] Figure 3 The mean square error (MSE) of the estimated value of the position coordinate of the mobile target source obtained by the method of the present invention and the comparative method (TSWLS method under TDOA and FDOA models) varies with the noise power of the measured value. Schematic diagram of the changing curve;

[0026] Figure 4 The mean square error (MSE) of the estimated value of the velocity coordinate of the moving target source obtained by the method of the present invention and the comparative method (TSWLS method under TDOA and FDOA models) varies with the noise power of the measured value. Schematic diagram of the changing curve;

[0027] Figure 5 The mean square error (MSE) of the estimated value of the position coordinate of the mobile target source obtained by the method of the present invention and the comparative method (TSWLS method under TDOA and FDOA models) varies with the noise power of the mobile sensor parameters. Schematic diagram of the changing curve;

[0028] Figure 6 The mean square error (MSE) of the estimated value of the velocity coordinate of the moving target source obtained by the method of the present invention and the comparative method (TSWLS method under TDOA and FDOA models) varies with the noise power of the mobile sensor parameters. Schematic diagram of the changing curve. DETAILED DESCRIPTION

[0029] The present invention is further described in detail below with reference to the accompanying drawings.

[0030] The present invention proposes a method for positioning a mobile target source based on joint measurement values ​​of TOA and FOA, which is divided into two stages. In the first stage, the TOA and FOA measurement models are appropriately transformed by introducing auxiliary variables and approximated as a series of pseudo-linear equations, and then the corresponding weighted least squares problem is established to obtain a closed-form expression for the initial estimate of the unknown variable vector. Since the relationship between the parameters in the unknown variable vector is not considered in the first stage, this estimation is usually not very accurate; in the second stage, the relationship between the parameters in the unknown variable vector is used to construct another weighted least squares problem to obtain the final closed-form solution expression. Since the signal propagation speed, clock bias and clock drift are all unknown, it is less affected by random signal propagation speed, clock bias and clock drift, and can provide more accurate positioning performance. The overall implementation block diagram of the method is shown in the figure. Figure 1As shown in FIG. 1 , the implementation process is as follows: a wireless sensor network system model is established, in which one mobile target source and multiple mobile sensors are assumed to exist; a TOA measurement value model related to the TOA measurement values ​​obtained by all mobile sensors and a FOA measurement value model related to the obtained FOA measurement values ​​are constructed; a pseudo-linear equation about the TOA measurement value is obtained based on the TOA measurement value model, and a pseudo-linear equation about the FOA measurement value is obtained based on the FOA measurement value model; by introducing the first step unknown variable vector including the position coordinates and velocity coordinates of the mobile target source, the signal propagation speed, the clock bias, and the clock drift, the pseudo-linear equation about the TOA measurement value is stacked into the form of a vector equation, and the pseudo-linear equation about the FOA measurement value is stacked into the form of a vector equation. In the form of a program, the two vector equations are merged into a first-step vector equation; on the basis of the first-step vector equation, combined with the weighted least squares criterion, the weighted least squares optimization problem of the first step is constructed, and the closed-form solution of the first step is obtained by solving; using the relationship between the parameters in the first-step unknown variable vector, multiple pseudo-linear equations are constructed, and by introducing the second-step unknown variable vector, multiple pseudo-linear equations are stacked into a second-step vector equation; on the basis of the second-step vector equation, combined with the weighted least squares criterion, the weighted least squares optimization problem of the second step is constructed, and the closed-form solution of the second step is obtained by solving; according to the closed-form solution of the first step and the closed-form solution of the second step, the estimated values ​​of the position coordinates and velocity coordinates of the mobile target source, the signal propagation speed, the clock bias, and the clock drift are obtained.

[0031] The preferred solution is that the method specifically comprises the following steps:

[0032] Step 1: In the k-dimensional wireless sensor network system model, assume that there is a mobile target source and M mobile sensors. The true value of the position coordinate of the mobile target source and the true value of the velocity coordinate are recorded as x o and The true value of the position coordinate and the true value of the velocity coordinate of the i-th mobile sensor are recorded as and However, in practical applications, only the noisy values ​​of the position coordinates and velocity coordinates can be obtained for the mobile sensor. The noisy values ​​of the position coordinates and velocity coordinates of the i-th mobile sensor are denoted as s i and The relationship between the noisy value and the true value is Where k = 2 or 3, M ≥ 2 × k + 3, i is a positive integer, 1 ≤ i ≤ M, Δs i The noisy value s represents the position coordinate of the i-th mobile sensor i With the true value The error between Represents the noisy value of the velocity coordinate of the i-th mobile sensor With the true value The error between .

[0033] Figure 2 A deployment diagram of a mobile target source and 10 mobile sensors in a 3D wireless sensor network system is given.

[0034] Step 2: In the k-dimensional wireless sensor network system, the mobile target source sends measurement signals to all mobile sensors at the same time. After receiving the measurement signal sent by the mobile target source, each mobile sensor obtains the TOA measurement value and the FOA measurement value divided by the signal carrier frequency, and the TOA true value model is modeled as: The true value model of FOA after dividing by the signal carrier frequency is modeled as: In actual application, only the TOA measurement value and the FOA measurement value divided by the signal carrier frequency can be obtained. Therefore, the TOA measurement value model is modeled as: The FOA measurement model divided by the signal carrier frequency is modeled as: in, represents the true TOA value associated with the ith mobile sensor, the symbol “|| ||” is the symbol for finding the Euclidean norm, c o It represents the true value of the signal propagation speed during the process from the mobile target source sending the measurement signal to the mobile sensor receiving the measurement signal. represents the true value of the clock offset between the mobile target source and the mobile sensor, represents the true value of FOA related to the ith mobile sensor divided by the signal carrier frequency. The superscript “T” is the transposition symbol. represents the true value of the clock drift between the mobile target source and the mobile sensor, τ i It represents the TOA measurement value obtained by the i-th mobile sensor after receiving the measurement signal sent from the mobile target source. represents the FOA measurement value obtained by the i-th mobile sensor after receiving the measurement signal sent by the mobile target source and divided by the signal carrier frequency, Δτ i Represents τ i The noise present in i Obey Gaussian distribution 0 means Δτ i The mean value, Q τ Denotes Δτ i The covariance matrix of express The noise present in Obey Gaussian distribution 0 means The mean of express The covariance matrix of .

[0035] Step 3: Under high signal-to-noise ratio conditions, perform linear transformation on the TOA true value model to obtain: Since the true TOA value associated with the i-th mobile sensor It can be expressed as The true value of the position coordinates of the i-th mobile sensor It can be expressed as So then and Substituting into equation 3.1 and squaring both sides of the equal sign, and then ignoring the second-order noise term, we get the pseudo-linear equation for the TOA measurement value, expressed as:

[0036] Similarly, under high signal-to-noise ratio conditions, the FOA true value model divided by the signal carrier frequency is linearly transformed to obtain: Then substitute equation 3.1 into equation 3.3 to obtain: Since the true TOA value associated with the i-th mobile sensor It can be expressed as The true value of FOA associated with the ith motion sensor divided by the signal carrier frequency It can be expressed as The true value of the position coordinates of the i-th mobile sensor It can be expressed as The true value of the velocity coordinate of the i-th mobile sensor It can be expressed as Therefore, and as well as and Substituting into equation 3.4 and ignoring the second-order noise term, we obtain a pseudo-linear equation for the FOA measurement, expressed as:

[0037] Step 4: Define the true value of the unknown variable vector in the first step Then, we introduce the pseudo-linear equation for TOA measurement, Equation 3.2: The pseudo-linear equations about TOA measurements are stacked into a vector equation form, which is described as: Similarly, in the pseudo-linear equation for the FOA measurement, Equation 3.5, we introduce The pseudo-linear equations about the FOA measurements are stacked into a vector equation form, described as: in, contains auxiliary variables formed by coupling multiple unknowns, such as and The symbol “[]” represents the vector, and the symbol “≈” represents approximately equal. The vector h τ The i-th element of is h τ (i), h τ (i)=-||s i || 2 , the matrix G τ The i-th row vector of G τ (i,:), 0 1×k represents a k-dimensional row vector of all zeros, and the vector ε τ =[B1,0 M×M ]Δm+[D1,0 M×Mk ]Δβ, matrix diag{} means to find the diagonal matrix, that is, diag{[τ1,τ2,...,τ M ] represents a diagonal matrix, diag{[τ1,τ2,...,τ M The diagonal elements of ]} are τ1, τ2, ..., τ M , τ1 represents the TOA measurement value obtained by the first mobile sensor, τ2 represents the TOA measurement value obtained by the second mobile sensor, τ M represents the TOA measurement value obtained by the Mth mobile sensor, I M represents the identity matrix of order M, 0 M×M represents an M×M dimensional matrix with all zeros, Δm represents the measurement error vector, Δτ represents the TOA noise vector formed by the noise in the TOA measurements obtained by all mobile sensors, Δτ = [Δτ1, Δτ2, ..., Δτ M ] T , Δτ1 represents the noise in τ1, Δτ2 represents the noise in τ2, and Δτ M Represents τ M The noise present in It represents the FOA noise vector formed by stacking the noise existing in the FOA measurement values ​​obtained by all mobile sensors after dividing by the signal carrier frequency, Represents the FOA measurement value obtained by the first motion sensor divided by the signal carrier frequency The noise present in Represents the FOA measurement value obtained by the second motion sensor divided by the signal carrier frequency The noise present in Represents the FOA measurement value obtained by the Mth mobile sensor divided by the signal carrier frequency The noise in the matrix D1 is D1(i,:), D1(i,:)=2[0 1×(i-1)k ,(x o -s i ) T ,0 1×(M-i)k ],0 1×(i-1)k represents a (i-1)k-dimensional row vector of all zeros, 0 1×(M-i)k represents a (Mi)k-dimensional row vector of all zeros, 0 M×Mk represents an M×Mk-dimensional matrix with all zeros, Δβ represents the motion sensor error vector, Δs represents the noise vector of the position coordinates of all mobile sensors, which is the error between the noisy values ​​and the true values. Δs = [Δs1, Δs2, ..., Δs M ] T ,Δs1,Δs2,...,Δs M Corresponding to the noisy value s1 and the true value representing the position coordinates of the first mobile sensor The error between the noisy value s2 of the position coordinate of the second mobile sensor and the true value The error between, ..., the noisy value s of the position coordinates of the Mth mobile sensor M With the true value The error between The mobile sensor velocity coordinate noise vector is a stack of the errors between the noisy values ​​and the true values ​​of the velocity coordinates of all mobile sensors. Corresponding to the noisy value representing the velocity coordinate of the first motion sensor With the true value The error between the speed coordinates of the second mobile sensor and the noisy value With the true value The error between, ..., the noisy value of the velocity coordinate of the Mth mobile sensor With the true value The error between The i-th element of matrix The i-th row vector of vector matrix represents a diagonal matrix, The diagonal elements of Correspondingly, the FOA measurement values ​​obtained by the first mobile sensor, the second mobile sensor, ..., the Mth mobile sensor after receiving the measurement signal sent by the mobile target source and divided by the signal carrier frequency are represented by the matrix The i-th row vector of matrix D2 is D2(i,:), The i-th row vector of matrix D3 is D3(i,:), D3(i,:)=[0 1×(i-1)k ,(x o -s i ) T ,0 1×(M-i)k ].

[0038] Step 5: Convert the vector equation about TOA measurement value and the vector equation for the FOA measurement It is fused into a first step vector equation, described as: Among them, the vector matrix vector matrix matrix

[0039] Step 6: Based on the weighted least squares criterion, the positioning problem is transformed into The first step of establishing the weighted least squares optimization problem is described as: Then solve the weighted least squares optimization problem in the first step and get the expression of the weighted least squares solution, that is, Then get the exact value of W1 and substitute the exact value of W1 into the expression of the weighted least squares solution In, get This is the closed-form solution of the first step; then we get On the basis of Substitution In, get and The error vector between Then, at a low noise level, Taking the expectation on both sides of the equality sign, we get Then get The approximation of the covariance matrix is ​​expressed as: Among them, min is the minimum value function, represents the estimated value of the unknown variable vector in the first step, and correspond, x represents the estimated value of the position coordinates of the moving target source, represents the estimated value of the velocity coordinate of the moving target source, c represents the estimated value of the signal propagation speed, τ o represents the estimated value of the clock bias, represents the estimated value of clock drift, W1 is the weight matrix, and its expression is E() means to find the expectation, Q m represents the covariance matrix of Δm, Q m =E(ΔmΔm T ), Q β represents the covariance matrix of Δβ, Q β =E(ΔβΔβ T ), cov() means to find the covariance matrix.

[0040] According to the expression of weight matrix W1, it can be found that to obtain weight matrix W1, it is necessary to know the parameter x o , c o , The values ​​of these parameters cannot be obtained directly, so the weight matrix W1 cannot be directly calculated, but it can be obtained through iteration. The exact value of W1 is solved as follows: Let the initial value of W1 be Then Substitute the expression of the weighted least squares solution into In, get The initial value of; then according to the expression of W1, use The initial value of W1 is obtained.

[0041] Step 7: Since in step 6, we ignored The relationship between the parameters in , thus leading to the obtained It may not be optimal. In order to improve the accuracy, you can use The relationship between the parameters in the Through observation, we can find The column vector consisting of the 1st to kth parameters in It is an estimate of the position coordinates of the moving target source, expressed as: The column vector consisting of the k+1th to 2kth parameters in It is an estimate of the velocity coordinate of the moving target source, expressed as: The 2k+1th parameter in It is expressed as: The 2k+2th parameter in It is expressed as: The 2k+3th parameter in It is expressed as: The 2k+4th parameter in It is expressed as: The 2k+5th parameter in It is expressed as: Then use formula 7.1 to get: Equation 7.1 and

[0042] Combining Equation 7.2, we can get:

[0043] Combining equation 7.3 and equation 7.4, we can obtain:

[0044] Combining equation 7.4 and equation 7.6 gives: Then, Equation 7.8, Equation 7.9, Equation 7.3, Equation 7.10, Equation 7.5, Equation 7.11, and Equation 7.7 are combined into a system of equations, expressed as: Then remove the second-order noise term in equation 7.12 to obtain a pseudo-linear system of equations, expressed as: Finally, all the pseudo-linear equations in Equation 7.13 are stacked into a second-step vector equation, described as:

[0045] in, express is a column vector consisting of the 1st to kth parameters in , express The column vector consisting of the k+1th to 2kth parameters in , Corresponding representation The 2k+1th parameter, the 2k+2th parameter, the 2k+3th parameter, the 2k+4th parameter, the 2k+5th parameter, represents a k-dimensional row vector with all 1s, and the symbol “⊙” represents the Hadamard product. I k×k represents the k×k dimensional identity matrix, 0 k×k represents a k×k dimensional matrix with all zeros, 0 k×1 represents a k-dimensional column vector with all zeros, 1 1×k represents a k-dimensional row vector with all 1s, that is, vector is the true value of the unknown variable vector introduced in the second step, vector matrix 0 k×5 represents a k×5 dimensional matrix with all zeros, 0 5×k represents a 5×k-dimensional matrix with all zeros, the matrix

[0046] Step 8: Based on the weighted least squares criterion, the positioning problem is transformed into The weighted least squares optimization problem of the second step is established, which is described as: Then solve the weighted least squares optimization problem in the second step and get the expression of the weighted least squares solution, that is, Then get the exact value of W2 and substitute it into the expression of weighted least squares solution: In, get This is the closed-form solution of the second step; then we get and Based on this, the estimated value x of the position coordinates of the moving target source and the estimated value x of the velocity coordinates of the moving target source are obtained. The estimated value of signal propagation speed c and the estimated value of clock bias τ o , the estimated value of clock drift in, represents the estimated value of the unknown variable vector in the second step, and correspond, W2 is the weight matrix, and its expression is sgn() represents a sign function, which is used to avoid sign ambiguity caused by square root operations. express is a column vector consisting of the 1st to kth parameters in , express The column vector consisting of the k+1th to 2kth parameters in , express The 2k+1th parameter in, express The 2k+2th parameter in, express The 2k+3th parameter in .

[0047] According to the expression of weight matrix W2, it can be found that the parameter x is required to obtain weight matrix W2 o , c o , The values ​​of these parameters cannot be obtained directly, so the weight matrix W2 cannot be directly calculated, but it can be obtained in a similar way to W1. The exact value of W2 is calculated as follows: The parameter x in c、τ o , Substitute into the expression of W2 to get the initial value of W2; then substitute the initial value of W2 into the expression of the weighted least squares solution In, get The initial value of Substitute the initial value of In the equation, we get x, c、τ o , Then get x, c、τ o , Substitute into the expression of W2 to obtain the exact value of W2.

[0048] The feasibility, effectiveness and positioning performance of the method of the present invention are verified by simulation experiments as follows.

[0049] Assume that there is one moving target source and 10 (i.e., M=10) moving sensors in a three-dimensional scene (i.e., k=3), where the true values ​​of the position coordinates of the 10 moving sensors are:

[0050] The true values ​​of the velocity coordinates of the 10 mobile sensors are:

[0051] The true value of the position coordinate of the moving target source is x o =[-370 300 470] T , the true value of the velocity coordinate of the moving target source is The actual value of the clock bias From the interval [0 150] / c o Randomly selected from meters, the true value of the clock drift From the interval [0 15] / c o The true value of the signal propagation speed c is randomly selected from meters per second. o Randomly selected from the interval [1400 1600] m / s.

[0052] Assuming that the power of the noise in the noisy values ​​of the position coordinates and the noisy values ​​of the velocity coordinates of any mobile sensor is the same, their covariance matrix is in, Represents the noise power of a given mobile sensor parameter, vector diag{[b,0.5b]} represents a diagonal matrix, whose diagonal elements are [b,0.5b], and the symbol is the Kronecker product, represents a 3D row vector of all ones.

[0053] Assuming that the noise power in the TOA and FOA measurements obtained by any mobile sensor is the same, their covariance matrix is in, Indicates the noise power of a given measurement, blkdiag{I M ,0.1I M} represents a block diagonal matrix, whose diagonal is the matrix I M and 0.1I M , I M represents the identity matrix of order M.

[0054] The comparison method in the experiment is the two-step weighted least squares method under the TDOA and FDOA measurement models (TSWLS method under the TDOA and FDOA models), and the comparison method is Underwater Source Localization Using TDOA and FDOAMeasurements With Unknown Propagation Speed ​​and Sensor Parameter Errors (Underwater source positioning based on TDOA and FDOA measurements when propagation speed and sensor parameter errors are unknown) published by Zhang et al. in IEEE Access (a journal in the field of communications of the Institute of Electrical and Electronics Engineers (IEEE)). The TDOA measurement values ​​in the comparison method are designed to be obtained by pairwise subtraction of the elements in the set consisting of the TOA measurement values ​​obtained by the mobile sensor in the present invention. Similarly, the FDOA measurement values ​​in the comparison method are designed to be obtained by pairwise subtraction of the elements in the set consisting of the FOA measurement values ​​obtained by the mobile sensor in the present invention. The covariance matrix of the TDOA measurement noise is: Among them, vector J = [-1 (M-1)×1 ,I M-1 ], -1 (M-1)×1 represents an M-1-dimensional column vector with all values ​​being -1, I M-1 represents the identity matrix of order M-1, Q m (1:M,1:M) represents the matrix Q m The covariance matrix of FDOA measurement noise is: Among them, Q m (M+1:2M,M+1:2M) represents the matrix Q m The submatrix consists of the elements from the M+1th row to the 2Mth row and the M+1th column to the 2Mth column.

[0055] The positioning performance of the method of the present invention is tested as the noise power of the measurement value increases.

[0056] Figure 3The mean square error (MSE) of the estimated value of the position coordinate of the moving target source obtained by the method of the present invention and the comparative method (TSWLS method under TDOA and FDOA models) under the above experimental conditions is given along with the measured value noise power. Schematic diagram of the changing curve, Figure 4 The mean square error (MSE) of the estimated velocity coordinates of the moving target source obtained by the method of the present invention and the comparative method (TSWLS method under TDOA and FDOA models) under the above experimental conditions is given along with the measured value noise power. Schematic diagram of the changing curve. The given mobile sensor parameter noise power Fixed at 1 meter. Figure 3 and Figure 4 It can be seen that when the noise power of the measured value is small, the mean square error of the estimated value of the position coordinate and the estimated value of the velocity coordinate of the moving target source obtained by the two methods is closest to the Cramer-Rao lower bound, indicating that both methods have good positioning performance under small and medium noise levels. However, as the noise power of the measured value increases, the performance of both methods gradually deteriorates. Figure 3 As shown, when the measured noise power is greater than 10dB m 2 When the measured noise power is greater than 20 dB m 2 The mean square error deviation of the two methods is Figure 4 This is more evident in the velocity coordinate estimates shown in Figure 4 As shown, when the measured noise power is equal to 25dB m 2 When the mean square error of the method of the present invention is 13.7dB (m / s) lower than that of the comparative method. 2 , indicating that the method of the present invention has superior performance in both the position coordinate estimation and the velocity coordinate estimation of the moving target source. The main reason for the poor performance of the comparison method is that it is sensitive to the noise environment and is easily affected by noise changes.

[0057] The positioning performance of the method of the present invention is tested as the noise power of the mobile sensor parameters increases.

[0058] Figure 5 The mean square error (MSE) of the estimated value of the position coordinates of the moving target source obtained by the method of the present invention and the comparative method (TSWLS method under TDOA and FDOA models) under the above experimental conditions is given along with the noise power of the mobile sensor parameters. Schematic diagram of the changing curve, Figure 6The mean square error (MSE) of the estimated velocity coordinates of the moving target source obtained by the method of the present invention and the comparative method (TSWLS method under TDOA and FDOA models) under the above experimental conditions is given along with the noise power of the mobile sensor parameters. The measured noise power is given here. Fixed at 1 meter. Figure 5 As can be seen in Figure 2, when the mobile sensor parameter noise power is less than 5dB m 2 When , the mean square error of the two methods is close to the Cramer-Rao lower bound, indicating that both methods can provide the most accurate estimation of the moving target source coordinate position in a low noise environment. However, when the noise power of the mobile sensor parameter is greater than 5dB m 2 When the mean square error of the comparison method begins to deviate from the Cramer-Rao lower bound, the proposed method is effective when the noise power of the mobile sensor parameter is greater than 25 dB m 2 The reason why both methods deviate from the Cramer-Rao lower bound is that the second-order noise term is ignored in the process of converting the measurement value model into a pseudo-linear equation. Figure 6 Also demonstrated with Figure 5 Similar results show that the mean square error of the comparison method deviates from the Cramer-Rao lower bound earlier than the mean square error of the method of the present invention, especially under high noise, the difference between the mean square errors of the two methods is the largest. Figure 6 As shown in the figure, when the mobile sensor parameter noise power is equal to 25dB m 2 When the mean square error of the method of the present invention is 7.6dB (m / s) lower than that of the comparative method. 2 This shows that the method of the present invention has excellent positioning performance in an environment with relatively high noise.

[0059] It can be seen from the above simulation results that the method of the present invention has good positioning performance and can well achieve high-precision positioning in a relatively high noise environment.

Claims

1. A method for locating a mobile target source based on combined TOA and FOA measurements, characterized in that The implementation process of the method is as follows: establish a wireless sensor network system model, in which one mobile target source and multiple mobile sensors are assumed to exist; construct a TOA measurement value model related to the TOA measurement values ​​obtained by all mobile sensors and a FOA measurement value model related to the FOA measurement values ​​obtained; obtain a pseudo-linear equation about the TOA measurement value based on the TOA measurement value model, and obtain a pseudo-linear equation about the FOA measurement value based on the FOA measurement value model; By introducing the first step unknown variable vector including the position coordinates and velocity coordinates of the mobile target source, the signal propagation speed, the clock bias, and the clock drift, the pseudo-linear equations about the TOA measurement value are stacked into the form of vector equations, and the pseudo-linear equations about the FOA measurement value are stacked into the form of vector equations, and then the two vector equations are merged into a first step vector equation; on the basis of the first step vector equation, combined with the weighted least squares criterion, the weighted least squares optimization problem of the first step is constructed, and the closed-form solution of the first step is obtained by solving; using the relationship between the parameters in the first step unknown variable vector, multiple pseudo-linear equations are constructed, and by introducing the second step unknown variable vector, multiple pseudo-linear equations are stacked into a second step vector equation; on the basis of the second step vector equation, combined with the weighted least squares criterion, the weighted least squares optimization problem of the second step is constructed, and the closed-form solution of the second step is obtained by solving; According to the closed-form solution of the first step and the closed-form solution of the second step, the estimated values ​​of the position coordinates and velocity coordinates of the mobile target source, the signal propagation speed, the clock bias, and the clock drift are obtained.

2. The method for locating a mobile target source based on a combined TOA and FOA measurement according to claim 1, characterized in that The method specifically comprises the following steps: Step 1: In the k-dimensional wireless sensor network system model, assume that there is a mobile target source and M mobile sensors. The true value of the position coordinate of the mobile target source and the true value of the velocity coordinate are recorded as x o and The true value of the position coordinate and the true value of the velocity coordinate of the i-th mobile sensor are recorded as and The noisy values ​​of the position coordinates and the noisy values ​​of the velocity coordinates of the i-th mobile sensor are denoted as s i and Where k = 2 or 3, M ≥ 2 × k + 3, 1 ≤ i ≤ M, Δs i The noisy value s represents the position coordinate of the i-th mobile sensor i With the true value The error between Represents the noisy value of the velocity coordinate of the i-th mobile sensor With the true value The error between Step 2: In the k-dimensional wireless sensor network system, the mobile target source sends measurement signals to all mobile sensors at the same time. After receiving the measurement signal sent by the mobile target source, each mobile sensor obtains the TOA measurement value and the FOA measurement value divided by the signal carrier frequency, and the TOA true value model is modeled as: The true value model of FOA after dividing by the signal carrier frequency is modeled as: The TOA measurement model is modeled as: The FOA measurement model divided by the signal carrier frequency is modeled as: in, represents the true value of TOA associated with the ith mobile sensor, the symbol "|| ||" is for finding the Euclidean norm symbol, c o It represents the true value of the signal propagation speed during the process from the mobile target source sending the measurement signal to the mobile sensor receiving the measurement signal. represents the true value of the clock offset between the mobile target source and the mobile sensor, represents the true value of FOA related to the i-th mobile sensor divided by the signal carrier frequency. The superscript "T" is the transposition symbol. represents the true value of the clock drift between the mobile target source and the mobile sensor, τ i It represents the TOA measurement value obtained by the i-th mobile sensor after receiving the measurement signal sent from the mobile target source. represents the FOA measurement value obtained by the i-th mobile sensor after receiving the measurement signal sent by the mobile target source and divided by the signal carrier frequency, Δτ i Represents τ i The noise present in i Obey Gaussian distribution 0 means Δτ i The mean value, Q τ Denotes Δτ i The covariance matrix of express The noise present in Obey Gaussian distribution 0 means The mean of express The covariance matrix of Step 3: Under high signal-to-noise ratio conditions, perform linear transformation on the TOA true value model to obtain: Then and Substituting into equation 3.1 and squaring both sides of the equal sign, and then ignoring the second-order noise term, we get the pseudo-linear equation for the TOA measurement value, expressed as: Similarly, under high signal-to-noise ratio conditions, the FOA true value model divided by the signal carrier frequency is linearly transformed to obtain: Then substitute equation 3.1 into equation 3.3 to obtain: Then and as well as and Substituting into equation 3.4 and ignoring the second-order noise term, we obtain a pseudo-linear equation for the FOA measurement, expressed as: Step 4: Define the true value of the unknown variable vector in the first step Then, we introduce the pseudo-linear equation for TOA measurement, Equation 3.2: The pseudo-linear equations about TOA measurements are stacked into a vector equation form, which is described as: Similarly, in the pseudo-linear equation for the FOA measurement, Equation 3.5, we introduce The pseudo-linear equations about the FOA measurements are stacked into a vector equation form, described as: Among them, the vector h τ The i-th element of is h τ (i), h τ (i)=-||s i || 2 , the matrix G τ The i-th row vector of G τ (i,:), 0 1×k represents a k-dimensional row vector of all zeros, and the vector ε τ =[B1,0 M×M ]Δm+[D1,0 M×Mk ]Δβ, matrix diag{} means finding the diagonal matrix, I M represents the identity matrix of order M, 0 M×M represents an M×M dimensional matrix with all zeros, Δm represents the measurement error vector, Δτ represents the TOA noise vector formed by the noise in the TOA measurements obtained by all mobile sensors, Δτ = [Δτ1, Δτ2, ..., Δτ M ] T , It represents the FOA noise vector formed by stacking the noise existing in the FOA measurement values ​​obtained by all mobile sensors after dividing by the signal carrier frequency. The i-th row vector of matrix D1 is D1(i,:), D1(i,:)=2[0 1×(i-1)k ,(x o -s i ) T ,0 1×(M-i)k ],0 1×(i-1)k represents a (i-1)k-dimensional row vector of all zeros, 0 1×(M-i)k represents a (Mi)k-dimensional row vector of all zeros, 0 M×Mk represents an M×Mk-dimensional matrix with all zeros, Δβ represents the motion sensor error vector, Δs represents the noise vector of the position coordinates of all mobile sensors, which is the error between the noisy values ​​and the true values. Δs = [Δs1, Δs2, ..., Δs M ] T , The mobile sensor velocity coordinate noise vector is a stack of the errors between the noisy values ​​and the true values ​​of the velocity coordinates of all mobile sensors. vector The i-th element of matrix The i-th row vector of vector matrix matrix The i-th row vector of matrix D2 is D2(i,:), The i-th row vector of matrix D3 is D3(i,:), D3(i,:)=[0 1×(i-1)k ,(x o -s i ) T ,0 1×(M-i)k ]; Step 5: Convert the vector equation about TOA measurement value and the vector equation for the FOA measurement It is fused into a first step vector equation, described as: Among them, the vector matrix vector matrix matrix Step 6: Based on the weighted least squares criterion, the positioning problem is transformed into The first step of establishing the weighted least squares optimization problem is described as: Then solve the weighted least squares optimization problem in the first step and get the expression of the weighted least squares solution, that is, Then get the exact value of W1 and substitute the exact value of W1 into the expression of the weighted least squares solution In, get This is the closed-form solution of the first step; then we get On the basis of Substitution In, get and The error vector between Then, at a low noise level, Taking the expectation on both sides of the equality sign, we get Then get The approximation of the covariance matrix is ​​expressed as: in, represents the estimated value of the unknown variable vector in the first step, x represents the estimated value of the position coordinates of the moving target source, represents the estimated value of the velocity coordinate of the moving target source, c represents the estimated value of the signal propagation speed, τ o represents the estimated value of the clock bias, represents the estimated value of clock drift, W1 is the weight matrix, and its expression is E() means to find the expectation, Q m represents the covariance matrix of Δm, Q m =E(ΔmΔm T ), Q β represents the covariance matrix of Δβ, Q β =E(ΔβΔβ T ), cov() means to find the covariance matrix; Step 7: The column vector consisting of the 1st to kth parameters in It is an estimate of the position coordinates of the moving target source, expressed as: The column vector consisting of the k+1th to 2kth parameters in It is an estimate of the velocity coordinate of the moving target source, expressed as: The 2k+1th parameter in It is expressed as: The 2k+2th parameter in It is expressed as: The 2k+3th parameter in It is expressed as: The 2k+4th parameter in It is expressed as: The 2k+5th parameter in It is expressed as: Then use formula 7.1 to get: Combining equation 7.1 and equation 7.2, we get: Combining equation 7.3 and equation 7.4, we get: Combining equation 7.4 and equation 7.6, we get: Then, Equation 7.8, Equation 7.9, Equation 7.3, Equation 7.10, Equation 7.5, Equation 7.11, and Equation 7.7 are combined into a system of equations, expressed as: (Equation 7.12); then remove the second-order noise term in Equation 7.12 to obtain a pseudo-linear system of equations, expressed as: (Equation 7.13); Finally, all the pseudo-linear equations in Equation 7.13 are stacked into a second-step vector equation, which is described as: in, express is a column vector consisting of the 1st to kth parameters in , express The column vector consisting of the k+1th to 2kth parameters in , Corresponding representation The 2k+1th parameter, the 2k+2th parameter, the 2k+3th parameter, the 2k+4th parameter, the 2k+5th parameter, represents a k-dimensional row vector with all 1s, and the symbol "⊙" represents the Hadamard product. I k×k represents the k×k dimensional identity matrix, 0 k×k represents a k×k dimensional matrix with all zeros, 0 k×1 represents a k-dimensional column vector with all zeros, 1 1×k represents a k-dimensional row vector with all 1s, that is, vector is the true value of the unknown variable vector introduced in the second step, vector matrix 0 k×5 represents a k×5 dimensional matrix with all zeros, 0 5×k represents a 5×k-dimensional matrix with all zeros, the matrix Step 8: Based on the weighted least squares criterion, the positioning problem is transformed into The weighted least squares optimization problem of the second step is established, which is described as: Then solve the weighted least squares optimization problem in the second step and get the expression of the weighted least squares solution, that is, Then get the exact value of W2 and substitute it into the expression of weighted least squares solution: In, get This is the closed-form solution of the second step; then we get and Based on this, the estimated value x of the position coordinates of the moving target source and the estimated value x of the velocity coordinates of the moving target source are obtained. The estimated value of signal propagation speed c and the estimated value of clock bias τ o , the estimated value of clock drift in, represents the estimated value of the unknown variable vector in the second step, W2 is the weight matrix, and its expression is sgn() represents the symbol function, express is a column vector consisting of the 1st to kth parameters in , express The column vector consisting of the k+1th to 2kth parameters in , express The 2k+1th parameter in, express The 2k+2th parameter in, express The 2k+3th parameter in .

3. The method for locating a mobile target source based on a combined TOA and FOA measurement according to claim 2, characterized in that In step 6, the exact value of W1 is obtained by: Let the initial value of W1 be Then Substitute the expression of the weighted least squares solution into In, get The initial value of; then according to the expression of W1, use The initial value of W1 is obtained.

4. The method for locating a mobile target source based on a combined TOA and FOA measurement according to claim 2 or 3, characterized in that In step 8, the exact value of W2 is obtained by: The parameter x in c、τ o , Substitute into the expression of W2 to get the initial value of W2; then substitute the initial value of W2 into the expression of the weighted least squares solution In, get The initial value of Substitute the initial value of In the equation, we get x, c、τ o , Then get x, c、τ o , Substitute into the expression of W2 to obtain the exact value of W2.

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