Mobile target source localization method based on joint measurement of toa and foa

Abstract

The application discloses a mobile target source positioning method based on TOA and FOA combined measurement values, which establishes a wireless sensor network system model containing one target source and multiple sensors; a TOA and FOA measurement value model is constructed, pseudo-linear equations about TOA and FOA are obtained respectively; an unknown variable vector containing a target source position, a speed, a signal propagation speed, a clock bias and a drift is introduced, the pseudo-linear equations are stacked into a vector equation, and are fused into a first-step vector equation; a first-step weighted least square optimization problem is constructed and solved, a first-step closed-form solution is obtained; further, a pseudo-linear equation is constructed by utilizing a relationship between parameters, a second-step unknown variable vector is introduced, and a second-step vector equation is constructed; a second-step weighted least square optimization problem is constructed and solved, a second-step closed-form solution is obtained; in combination with two-step solutions, parameter estimation values such as a target source position and a speed are obtained; and the method has the advantages that the target source can be accurately positioned under the conditions that the signal propagation speed, sensor parameter errors and clock information are unknown.

Description

TECHNICAL FIELD

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

[0002] Mobile target source positioning is a research hotspot in recent ten years, and has broad application prospects in radar monitoring, sound wave detection and wireless sensor network fields. Generally speaking, the positioning problem based on time measurement value is the most common, including time of arrival (TOA) and time difference of arrival (TDOA). However, a large number of studies have shown that in the positioning based on TOA measurement value, clock synchronization between target source and sensor is required, otherwise unknown clock bias will be generated, which will reduce the performance of the positioning algorithm based on TOA measurement value. In order to solve this problem, researchers use pair-wise subtraction between TOA measurement values to remove the unknown clock bias generated, and simplify the positioning problem to TDOA measurement model. Although this simplifies the problem, the application of TDOA measurement value to avoid clock synchronization between target source and sensor is at the cost of positioning accuracy. Some researchers have found that pair-wise subtraction of TOA measurement values will increase the measurement noise power by 3dB, and also cause information loss. Therefore, more and more researches are focusing on solving the clock synchronization problem between target source and sensor, and designing a high-precision positioning method. For example, Enyang Xu et al. published the results in IEEE Transactions on Signal Processing, proposed two convex methods, respectively named two-step least squares (2LS) algorithm and minimum maximum algorithm (MMA), to further improve the positioning accuracy when the signal transmission time is unavailable. However, the MMA blurs the true value of the measurement error, resulting in performance loss. For mobile target source, the frequency of arrival (FOA) measurement value containing speed information can also be used to estimate the speed of mobile target source and improve the positioning accuracy. In view of this, the research on mobile target source positioning based on TOA and FOA joint measurement values has important practical significance.

[0003] The mobile target source localization problem based on TOA and FOA joint measurements with unknown clock bias and drift is very complex and difficult to solve. For example, Jiong Shi et al. proposed a semi-definite relaxation method in the Signal Processing journal for the above mobile target source localization problem. In fact, the method converts the TOA and FOA measurement model into a distance and distance rate model by assuming that the signal propagation speed and sensor parameters are accurately known, so as to facilitate derivation. However, in most actual environments, the situation may not be the case. For example, when the signal propagates underwater or in solid, the signal propagation speed mainly depends on the propagation medium. In addition, even if GPS technology is used, it is not possible to obtain accurate sensor parameter information. Therefore, it is necessary to study a mobile target source localization method based on TOA and FOA joint 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

[0004] The technical problem to be solved by the present application is to provide a mobile target source localization method based on TOA and FOA joint measurements, 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, in the case of unknown clock bias and unknown clock drift, and unknown signal propagation speed and sensor parameter error.

[0005] The technical scheme adopted by the present application to solve the above technical problems is: a mobile target source positioning method based on TOA and FOA combined measurement values, characterized in that the implementation process of the method is: a wireless sensor network system model is established, and in the model, one mobile target source and multiple mobile sensors are set; a TOA measurement value model related to the TOA measurement values obtained by all the mobile sensors and a FOA measurement value model related to the FOA measurement values obtained by all the mobile sensors are constructed; pseudo-linear equations about the TOA measurement values are obtained on the basis of the TOA measurement value model, and pseudo-linear equations about the FOA measurement values are obtained on the basis of the FOA measurement value model; by introducing a first-step unknown variable vector containing 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 values are stacked into the form of a vector equation, the pseudo-linear equations about the FOA measurement values are stacked into the form of a vector equation, and the two vector equations are fused into a first-step vector equation; on the basis of the first-step vector equation, a first-step weighted least squares optimization problem is constructed in combination with the weighted least squares criterion, and a closed-form solution of the first step is obtained; by utilizing the relationship between the parameters in the first-step unknown variable vector, multiple pseudo-linear equations are constructed, and by introducing a second-step unknown variable vector, the multiple pseudo-linear equations are stacked into a second-step vector equation; on the basis of the second-step vector equation, a second-step weighted least squares optimization problem is constructed in combination with the weighted least squares criterion, and a closed-form solution of the second step is obtained; 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 a k-dimensional wireless sensor network system model, one mobile target source and M mobile sensors are set, and the true values of the position coordinates and the velocity coordinates of the mobile target source are denoted as x o and The true values of the position coordinates and the velocity coordinates of the i-th mobile sensor are denoted as and The noisy values of the position coordinates and the velocity coordinates of the i-th mobile sensor are denoted as s i and wherein, k=2 or 3, M≥2×k+3, 1≤i≤M, Δs i represents the error between the noisy value s i of the position coordinates of the i-th mobile sensor and the true value represents the error between the noisy value ​error between the true value ;

[0008] Step 2: In the k-dimensional wireless sensor network system, the moving target source sends measurement signals to all mobile sensors at the same time, each mobile sensor obtains TOA measurement value and FOA measurement value after receiving the measurement signals sent by the moving target source, and models the TOA true value model as: The FOA true value model after dividing by the signal carrier frequency is modeled as: The TOA measurement value model is modeled as: The FOA measurement value model after dividing by the signal carrier frequency is modeled as: represents the TOA true value related to the i-th mobile sensor, the symbol "|| ||" is the Euclidean norm symbol, c o represents the true value of the signal propagation speed in the process of sending measurement signals by the moving target source to receiving measurement signals by the mobile sensor, represents the true value of the clock bias between the moving target source and the mobile sensor, represents the FOA true value after dividing by the signal carrier frequency related to the i-th mobile sensor, the superscript "T" is the transpose symbol, represents the true value of the clock drift between the moving target source and the mobile sensor, τ i represents the TOA measurement value obtained by the i-th mobile sensor after receiving the measurement signals sent by the moving target source, represents the FOA measurement value after dividing by the signal carrier frequency obtained by the i-th mobile sensor after receiving the measurement signals sent by the moving target source, Δτ i represents the noise existing in τ i represents the noise existing in Δτ i obeys Gaussian distribution 0 represents the mean of Δτ i , Q τ represents the covariance matrix of Δτ i , represents the noise existing in , obeys Gaussian distribution 0 represents the mean of , represents the covariance matrix of ;

[0009] Step 3: Under the condition of high signal-to-noise ratio, linear transformation is performed on the TOA true value model to obtain: Then and ​Substitute into equation 3.1 and square both sides of the equation, and ignore the second order noise terms, to obtain a pseudo-linear equation in terms of TOA measurements, denoted as:

[0010]

[0011] Similarly, under high signal-to-noise ratio conditions, a linear transformation is applied to the FOA true value model divided by the signal carrier frequency, to obtain: Then substitute equation 3.1 into equation 3.3 to obtain: Substitute and into equation 3.4 and ignore the second order noise terms, to obtain a pseudo-linear equation in terms of FOA measurements, denoted as: Step 4: Define a first step unknown variable vector true value

[0012] Then introduce into the pseudo-linear equation in terms of TOA measurements, i.e. equation 3.2. Stack the pseudo-linear equation in terms of TOA measurements into a vector equation form, described as: Similarly, introduce into the pseudo-linear equation in terms of FOA measurements, i.e. equation 3.5. Stack the pseudo-linear equation in terms of FOA measurements into a vector equation form, described as: Where the i-th element of vector h τ is h τ (i), h τ (i) = -||s i || 2 , the i-th row vector of matrix G τ is G τ (i, :), 0 1×k represents a k-dimensional row vector of all zeros, vector ε τ = [B1,0 M×M ]Δm + [D1,0 M×Mk ]Δβ, matrix diag{} represents a diagonal matrix, I M represents an M-order identity matrix, 0 M×M represents an M x M matrix of all zeros, Δm represents a measurement error vector, Δτ represents a TOA noise vector stacked by the noise existing 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 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 sum of the errors 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 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: Transform the vector equation related to TOA measurements into 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 the weighted least squares optimization problem is established, which 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 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 equal 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 finding 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 Expressed as: The 2k+2th parameter in Expressed as: The 2k+3th parameter in Expressed as: The 2k+4th parameter in Expressed as: The 2k+5th parameter in Expressed as: Then use Equation 7.1 to get: Combining Equation 7.1 and Equation 7.2 yields: Combining Equation 7.3 and Equation 7.4 yields: Combining Equation 7.4 and Equation 7.6 yields: Then, Equations 7.8, 7.9, 7.3, 7.10, 7.5, 7.11, and 7.7 are combined into a system of equations, expressed as: Then, the second-order noise term in Equation 7.12 is removed to obtain a pseudo-linear system of equations, which can be expressed as: Finally, all the pseudo-linear equations in Equation 7.13 are stacked into a second-step vector equation, which is described as:

[0016] in, express The 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 of 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 of all 1s, that is, vector is the true value of the unknown variable vector introduced in the second step, vector matrix 0 k×5represents 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 the exact value of W2 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, we obtain 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. Estimated value of signal propagation speed c, estimated value of clock bias τ o , 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 symbolic function, express The 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: setting the initial value of W1 to 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 the expression of W2 into the expression of the initial value of W2, and substitute the initial value of W2 into the expression of the weighted least square solution , to obtain the initial value of ; then substitute the initial value of into the expression of , to obtain the initial value of x, c and τ o , Substitute the obtained x, c and τ o , into the expression of W2, to obtain the accurate value of W2.

[0020] Compared with the prior art, the present application has the advantages of:

[0021] In the wireless sensor network system, when the signal propagation speed, the sensor parameter error and the clock parameter are all unknown, the TOA measurement model and the FOA measurement model are converted into a series of pseudo-linear equations by introducing auxiliary variables, and then a corresponding weighted least square problem is constructed based on the weighted least square criterion, so that the closed-form solution of the initial estimation of the unknown variable vector is obtained, and then another weighted least square problem is constructed by using the relationship between the parameters in the unknown variable vector, so that a more accurate solution is obtained, which can not only estimate the position coordinates and the speed coordinates of the mobile target source, but also estimate the signal propagation speed and the clock parameter, and can provide good positioning accuracy and low calculation complexity in the environment with small and medium noise.

[0022] Compared with the positioning method using the TDOA measurement value, the present application estimates the unknown clock bias and clock drift by jointly estimating the FOA measurement value, although the number of unknown quantities is increased to make the problem more complex, but the positioning accuracy can be improved. BRIEF DESCRIPTION OF DRAWINGS

[0023] Figure 1 is the overall implementation block diagram of the method of the present application;

[0024] Figure 2 is the deployment schematic diagram of a mobile target source and 10 mobile sensors in a 3-dimensional wireless sensor network system;

[0025] Figure 3 is the curve schematic diagram of the mean square error (MSE) of the estimated value of the position coordinates of the mobile target source obtained by the method of the present application and the comparative method (TSWLS method under the TDOA and FDOA models) respectively, with the measurement value noise power changing;

[0026] Figure 4The mean square error (MSE) of the estimated velocity coordinates of the moving target source obtained by the proposed method and the comparative methods (TSWLS method under TDOA and FDOA models) as a function of the measurement noise power The curve diagram of the change;

[0027] Figure 5 The mean square error (MSE) of the estimated position coordinates of the moving target source obtained by the proposed method and the comparative methods (TSWLS method under TDOA and FDOA models) as a function of the moving sensor parameter noise power The curve diagram of the change;

[0028] Figure 6 The mean square error (MSE) of the estimated velocity coordinates of the moving target source obtained by the proposed method and the comparative methods (TSWLS method under TDOA and FDOA models) as a function of the moving sensor parameter noise power The curve diagram of the change. DETAILED DESCRIPTION

[0029] The application will be further described in detail below with reference to the embodiments of the drawings.

[0030] The application proposes a mobile target source positioning method based on TOA and FOA joint measurement values, 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 a corresponding weighted least squares problem is established to obtain the closed-form expression of 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, the estimate is usually not very accurate. In the second stage, another weighted least squares problem is constructed by using the relationship between the parameters in the unknown variable vector to obtain the final closed-form solution expression. Since the signal propagation speed, clock bias and clock drift are all unknown, the influence of random signal propagation speed, clock bias and clock drift is small, and more accurate positioning performance can be provided. The overall implementation block diagram of the method is as follows Figure 1As shown, the implementation process is: a wireless sensor network system model is established, in which one mobile target source and multiple mobile sensors are set; 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 are constructed; pseudo-linear equations about the TOA measurement values are obtained on the basis of the TOA measurement value model, and pseudo-linear equations about the FOA measurement values are obtained on the basis of the FOA measurement value model; by introducing a first-step unknown variable vector containing 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 values are stacked into the form of a vector equation, the pseudo-linear equations about the FOA measurement values are stacked into the form of a vector equation, and the two vector equations are fused into a first-step vector equation; on the basis of the first-step vector equation, a first-step weighted least square optimization problem is constructed in combination with the weighted least square criterion, and a closed-form solution of the first step is obtained; by utilizing the relationship between the parameters in the first-step unknown variable vector, multiple pseudo-linear equations are constructed, and by introducing a second-step unknown variable vector, the multiple pseudo-linear equations are stacked into a second-step vector equation; on the basis of the second-step vector equation, a second-step weighted least square optimization problem is constructed in combination with the weighted least square criterion, and a closed-form solution of the second step is obtained; 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] Preferably, the method specifically comprises the following steps:

[0032] Step 1: in a k-dimensional wireless sensor network system model, one mobile target source and M mobile sensors are set, and the true values of the position coordinates and the velocity coordinates of the mobile target source are correspondingly denoted as x o and The true values of the position coordinates and the velocity coordinates of the i-th mobile sensor are correspondingly denoted as and However, in actual application, only the position coordinates and the velocity coordinates of the mobile sensor can be obtained, and the noisy values of the position coordinates and the velocity coordinates of the i-th mobile sensor are correspondingly denoted as s i and The relationship between the noisy values and the true values is wherein k=2 or 3, M≥2×k+3, i is a positive integer, 1≤i≤M, Δs i represents the error between the noisy value s i of the position coordinates of the i-th mobile sensor and the true value represents the error between the noisy value of the velocity coordinates of the i-th mobile sensor​ error between the real value and the estimated value.

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

[0034] Step 2: In a k-dimensional wireless sensor network system, the mobile target source sends measurement signals to all mobile sensors simultaneously, each mobile sensor obtains a TOA measurement value and a FOA measurement value divided by the signal carrier frequency after receiving the measurement signals sent by the mobile target source, and models the TOA real value model as: The FOA real value model divided 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, so the TOA measurement value model is modeled as: The FOA measurement value model divided by the signal carrier frequency is modeled as: wherein, represents the TOA real value related to the i-th mobile sensor, the symbol "|| ||" is the symbol for calculating the Euclidean norm, c o represents the real value of the signal propagation speed in the process of sending measurement signals by the mobile target source to receiving measurement signals by the mobile sensor, represents the real value of the clock bias between the mobile target source and the mobile sensor, represents the FOA real value divided by the signal carrier frequency related to the i-th mobile sensor, the superscript "T" is the transpose symbol, represents the real value of the clock drift between the mobile target source and the mobile sensor, τ i represents the TOA measurement value obtained by the i-th mobile sensor after receiving the measurement signals sent by the mobile target source, represents the FOA measurement value divided by the signal carrier frequency obtained by the i-th mobile sensor after receiving the measurement signals sent by the mobile target source, Δτ i represents the noise existing in τ i , Δτ i obeys Gaussian distribution 0 represents the mean of Δτ i , Q τ represents the covariance matrix of Δτ i , represents the noise existing in , obeys Gaussian distribution 0 represents the mean of , represents covariance matrix of the TOA real value.

[0035] Step 3: Linear transformation is applied to the TOA real value model under high signal-to-noise ratio (SNR) condition, resulting in: The TOA real value associated with the ith mobile sensor can be expressed as The real value of the position coordinate of the ith mobile sensor can be expressed as Therefore, then and are substituted into Equation 3.1 and squared on both sides of the equation, and the second-order noise term is ignored, resulting in a pseudo-linear equation about the TOA measurement value, expressed as:

[0036] Similarly, linear transformation is applied to the FOA real value model divided by the signal carrier frequency under high signal-to-noise ratio (SNR) condition, resulting in: Then, Equation 3.1 is substituted into Equation 3.3, resulting in: The TOA real value associated with the ith mobile sensor can be expressed as The FOA real value associated with the ith mobile sensor divided by the signal carrier frequency can be expressed as The real value of the position coordinate of the ith mobile sensor can be expressed as The real value of the velocity coordinate of the ith mobile sensor can be expressed as Therefore, then and as well as and are substituted into Equation 3.4 and the second-order noise term is ignored, resulting in a pseudo-linear equation about the FOA measurement value, expressed as:

[0037] Step 4: Define a real value of a first-step unknown variable vector Then, introduce into the pseudo-linear equation about the TOA measurement value, i.e., Equation 3.2 Stack the pseudo-linear equation about the TOA measurement value into the form of a vector equation, described as: Similarly, introduce into the pseudo-linear equation about the FOA measurement value, i.e., Equation 3.5 Stack the pseudo-linear equation about the FOA measurement value into the form of a vector equation, described as: wherein contains auxiliary variables coupled by multiple unknowns, such as and The symbol “[]” is a vector representation symbol, the symbol “≈” means approximately equal, the vector h τ The i-th element of is h τ (i), h τ (i)=-||s i || 2 , 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.e. diag{[τ1,τ2,...,τ M ] represents a diagonal matrix, diag{[τ1,τ2,...,τ M ]} 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 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 the i-th row vector D1(i,:), D1(i,:)=2[01×(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 sum of the errors 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 coordinate of the first mobile sensor The error between the noise 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 and the true value The error between The mobile sensor velocity coordinate noise vector is a stack of 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 and the true value The error between the two, the noise value of the velocity coordinate of the second mobile sensor and the true value The error between, ..., the noisy value of the velocity coordinate of the Mth mobile sensor and the true value The error between the vector The i-th element of matrix The i-th row vector of vector matrix represents a diagonal matrix, The diagonal elements of The matrix represents the FOA measurement value 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. 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: Transform the vector equation related to TOA measurements into 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 the weighted least squares optimization problem is established, which 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 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 equal 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, Qm = E (AmAm T ), Q β represents the covariance matrix of Δβ, Q β = E (ΔβΔβ T ), cov() represents the covariance matrix.

[0040] From the expression of the weight matrix W1, it can be found that the weight matrix W1 cannot be directly obtained because the values of the parameters x o , c o , cannot be directly obtained. However, the weight matrix W1 can be obtained by iteration. The process of obtaining the accurate value of W1 is as follows: let the initial value of W1 be Then substitute into the expression of the weighted least square solution to obtain the initial value of ; and then use the initial value of to obtain the accurate value of W1 according to the expression of W1.

[0041] Step 7: Because the relationship between the parameters in is ignored in step 6, the obtained may not be optimal. In order to improve the accuracy, the relationship between the parameters in can be used to refine the obtained . By observation, it can be found that the column vector composed of the first to kth parameters in is an estimate of the position coordinates of the moving target source, which is expressed as: the column vector composed of the k+1th to 2kth parameters in is an estimate of the velocity coordinates of the moving target source, which is expressed as: the 2k+1th parameter in is expressed as: the 2k+2th parameter in is expressed as: the 2k+3th parameter in is expressed as: the 2k+4th parameter in is expressed as: the 2k+5th parameter in is expressed as: Equation 7.1 and

[0042] Combining Equation 7.2, we can get: Combining Equation 7.3 and Equation 7.4, we can obtain: Combining Equation 7.4 and Equation 7.6 yields: Then, Equations 7.8, 7.9, 7.3, 7.10, 7.5, 7.11, and 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, which can be expressed as: Finally, all the pseudo-linear equations in Equation 7.13 are stacked into a second-step vector equation, which is described as:

[0043] in, express The 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 of 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 of 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

[0044] 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 the exact value of W2 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, we obtain 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. Estimated value of signal propagation speed c, estimated value of clock bias τ o , 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 the sign ambiguity caused by the square root operation. express The 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 .

[0045] According to the expression of the weight matrix W2, it can be found that the parameter x is required to obtain the 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 solved as follows: The parameter x in c、τ o 、 Substitute the expression of W2 into the expression of the initial value of W2, obtain the initial value of W2; then substitute the initial value of W2 into the expression of the weighted least square solution , obtain the initial value of ; then substitute the initial value of into the expression of , obtain x, c, τ o , Substitute the obtained x, c, τ o , into the expression of W2, and obtain the accurate value of W2.

[0046] The feasibility, effectiveness and positioning performance of the method are verified by simulation experiments.

[0047] A three-dimensional scene (i.e., k=3) is set, in which there is one moving target source and 10 (i.e., M=10) moving sensors, wherein the true values of the position coordinates of the 10 moving sensors are respectively:

[0048] The true values of the speed coordinates of the 10 moving sensors are respectively:

[0049] The true value of the position coordinate of the moving target source is x o =[-370 300 470] T , the true value of the speed coordinate of the moving target source is The true value of the clock bias is randomly selected from the interval [0 150] / c o meters, the true value of the clock drift is randomly selected from the interval [0 15] / c o meters / second, and the true value of the signal propagation speed c o is randomly selected from the interval [1400 1600] meters / second.

[0050] It is assumed that the power of the noise existing in the noisy value of the position coordinate and the noisy value of the speed coordinate of any moving sensor is the same, and their covariance matrix wherein represents the given moving sensor parameter noise power, the vector diag{[b,0.5b]} represents a diagonal matrix, diagonal elements of which are [b,0.5b], and the symbol is the Kronecker product, represents a 3-dimensional row vector with all 1s.

[0051] Assuming that the power of the noise present in the TOA and FOA measurements obtained by any mobile sensor is the same, their covariance matrices are where, denotes the given measurement noise power, blkdiag{I M ,0.1I M} denotes a block diagonal matrix with matrices I M and 0.1I M on its diagonal, I M denotes the M x M identity matrix.

[0052] The comparative method in the experiment is the two-step weighted least squares method under the TDOA and FDOA measurement model (TSWLS method under the TDOA and FDOA model), and the comparative method is Underwater Source Localization Using TDOA and FDOA Measurements With Unknown Propagation Speed and Sensor Parameter Errors (underwater source localization based on TDOA and FDOA measurements with unknown propagation speed and sensor parameter errors) disclosed by Zhang et al. in IEEE Access. The TDOA measurement in the comparative method is designed to be obtained by pairwise subtraction of the elements in the set consisting of the TOA measurements obtained by the mobile sensor in the present application, and similarly, the FDOA measurement in the comparative method is designed to be obtained by pairwise subtraction of the elements in the set consisting of the FOA measurements obtained by the mobile sensor in the present application. The covariance matrix of the TDOA measurement noise is where, vector J = [-1 (M-1)×1 , I M-1 ], -1 (M-1)×1 denotes an M-1 dimensional column vector with all -1, I M-1 denotes the M-1 x M-1 identity matrix, Q m (1:M,1:M) denotes the submatrix consisting of the elements from the 1st row to the Mth row and the 1st column to the Mth column in matrix Q m . The covariance matrix of the FDOA measurement noise is where, Q m (M+1:2M,M+1:2M) denotes the submatrix consisting of the elements from the M+1th row to the 2Mth row and the M+1th column to the 2Mth column in matrix Q m .

[0053] The positioning performance of the method of the present application is tested to see how it changes with the increase of the measurement noise power.

[0054] Figure 3 The mean square error (MSE) of the estimated position coordinates of the moving target source obtained by the method of the present application and the comparative method (TSWLS method under TDOA and FDOA models) respectively is given below as a function of the measurement value noise power under the above experimental conditions, and a curve diagram showing the change of the mean square error of the estimated position coordinates of the moving target source obtained by the method of the present application and the comparative method (TSWLS method under TDOA and FDOA models) respectively is given below as a function of the measurement value noise power Figure 4 The mean square error (MSE) of the estimated velocity coordinates of the moving target source obtained by the method of the present application and the comparative method (TSWLS method under TDOA and FDOA models) respectively is given below as a function of the measurement value noise power under the above experimental conditions, and a curve diagram showing the change of the mean square error of the estimated velocity coordinates of the moving target source obtained by the method of the present application and the comparative method (TSWLS method under TDOA and FDOA models) respectively is given below as a function of the measurement value noise power The moving sensor parameter noise power Figure 3 is fixed as 1 meter. From Figure 4 and Figure 3 it can be seen that when the measurement value noise power is small, the mean square error of the estimated position coordinates and the estimated velocity coordinates of the moving target source obtained by the two methods are both closest to the Cramer-Rao lower bound, indicating that both methods have good positioning performance under small and medium noise levels. However, as the measurement value noise power increases, the performance of both methods gradually deteriorates. As shown in Figure 4 , when the measurement value noise power is greater than 10dB m 2 , the mean square error of the comparative method deviates from the Cramer-Rao lower bound in advance, while the mean square error of the method of the present application deviates from the Cramer-Rao lower bound only when the measurement value noise power is greater than 20dB m 2 . The deviation of the mean square error of the two methods is more obvious in the estimation of the velocity coordinates, as shown in Figure 4 , when the measurement value noise power is equal to 25dB m 2 , the mean square error of the method of the present application is 13.7dB (m / s) 2 lower than that of the comparative method, indicating that the method of the present application exhibits superior performance in both the estimation of the position coordinates and the estimation of the velocity coordinates of the moving target source. The main reason for the poor performance of the comparative method is that it is sensitive to the noise environment, so it is easily affected by noise changes.

[0055] The change of the positioning performance of the method of the present application with the increase of the moving sensor parameter noise power is tested.

[0056] Figure 5 The mean square error (MSE) of the estimated position coordinates of the moving target source obtained by the method of the present application and the comparative method (TSWLS method under TDOA and FDOA models) respectively is given below as a function of the moving sensor parameter noise power under the above experimental conditions, and a curve diagram showing the change of the mean square error of the estimated position coordinates of the moving target source obtained by the method of the present application and the comparative method (TSWLS method under TDOA and FDOA models) respectively is given below as a function of the moving sensor parameter noise power 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 from the figure, when the noise power of the mobile sensor parameter 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 25dB 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 showed 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, when the noise power of the mobile sensor parameter 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 a noisy environment.

[0057] 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 noisy environment.

Claims

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

2. The method for locating a moving target source based on TOA and FOA combined measurements according to claim 1 is 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, and record the true value of the position coordinate and the true value of the velocity coordinate of the mobile target source 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 speed 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 and the true value The error between Represents the noisy value of the velocity coordinate of the i-th mobile sensor and 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. 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 value model is modeled as: The FOA measurement model after dividing by the signal carrier frequency is modeled as: in, represents the true TOA value associated with the i-th 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 of 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 the FOA related to the i-th mobile sensor divided by the signal carrier frequency. The superscript "T" is the transpose symbol. represents the true value of the clock drift between the mobile target source and the mobile sensor, τ i 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 divided by the signal carrier frequency, Δτ i Represents τ i The noise in i Obey Gaussian distribution 0 represents 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 this into Equation 3.1 and squaring both sides of the equality sign, and ignoring the second-order noise term, we obtain a pseudo-linear equation for the TOA measurement, expressed as: Similarly, under high signal-to-noise ratio conditions, a linear transformation is performed on the true value model of FOA divided by the signal carrier frequency 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, which is 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, which is described as: Similarly, the pseudo-linear equation for FOA measurement, Equation 3.5, is introduced The pseudo-linear equations about the FOA measurements are stacked into a vector equation form, which is described as: Among them, the vector h τ The i-th element of is h τ (i), h τ (i)=-||s i || 2 , 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 stacking 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 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 0s, 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 sum of the errors 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 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: Transform the vector equation related to TOA measurements into 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 the weighted least squares optimization problem is established, which 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 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 equal 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 finding 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 Expressed as: The 2k+2th parameter in Expressed as: The 2k+3th parameter in Expressed as: The 2k+4th parameter in Expressed as: The 2k+5th parameter in Expressed as: Then use Equation 7.1 to get: Combining Equation 7.1 and Equation 7.2 yields: Combining Equation 7.3 and Equation 7.4 yields: Combining Equation 7.4 and Equation 7.6 yields: Then, Equations 7.8, 7.9, 7.3, 7.10, 7.5, 7.11, and 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 the pseudo-linear equation system, which is 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 The 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 of 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 of 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 the exact value of W2 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, we obtain 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. Estimated value of signal propagation speed c, estimated value of clock bias τ o , 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 symbolic function, express The 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 moving target source based on TOA and FOA combined measurements according to claim 2 is characterized in that In step 6, the exact value of W1 is obtained by: setting the initial value of W1 to 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 moving target source based on TOA and FOA combined measurements 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 obtain 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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