Time difference of arrival positioning method based on stochastic robust least squares

Abstract

The application discloses a time difference positioning method based on random robust least square, mainly solves the threshold effect of the existing time difference positioning method when the measurement error increases, and leads to the problem of sharp deterioration of positioning accuracy, and the implementation scheme is as follows: a plurality of observation stations and noise measurement errors are set, the time difference positioning equation is obtained by using the position parameters and the measurement errors, an auxiliary variable is defined, the auxiliary variable is introduced into the time difference positioning equation to establish a constraint equation, the constraint equation is processed by using a random robust least square method to obtain a cost function, the cost function is solved by using a least square method to obtain a target position, a new constraint equation is established according to the internal relation of the auxiliary variable elements, the new constraint equation is solved by using a weighted least square method to obtain a positioning error, the target position is linearly corrected by using the positioning error to obtain a final positioning result, and the target positioning is completed. The application reduces the positioning error, improves the positioning accuracy and the robustness of positioning, and can be used in aerospace, navigation and electronic warfare.

Description

Technical Field

[0001] This invention belongs to the field of passive positioning technology, and further relates to a time difference positioning method that can be used in aerospace, navigation and electronic warfare. Background Technology

[0002] Time difference positioning (TDRP) is based on the time difference in arrival times of a target radiation source signal at different observation stations. It constructs a time difference observation equation to calculate the location of the radiation source. It has advantages such as simple equipment implementation, high real-time performance, high positioning accuracy, and fewer constraints in practical use. It has wide applications in aerospace, navigation, and electronic warfare fields.

[0003] In 1994, H.O.K.C. and Y.T.C.H. proposed the well-known two-step weighted least squares (TSWLS) algorithm. Its implementation involves: 1) introducing the distance between the reference station and the target as an auxiliary variable to transform the nonlinear equation of time difference positioning into a linear equation; 2) assuming the auxiliary variable is independent of the target radiation source, using the weighted least squares (WLS) method to solve for the auxiliary variable and the target position; 3) using the weighted least squares (WLS) method again to solve the correlation and obtain the final estimate of the target position. This method has an analytical solution, no initialization problem, and low computational cost, but it suffers from poor stability, significant measurement noise threshold effect, and large positioning deviation.

[0004] To address the issue of significant positioning errors in the two-step weighted least squares (TSWLS) algorithm, Professor Guo Fucheng's team proposed a TDOA positioning method based on positioning error correction in 2015. The implementation scheme is as follows: 1) Introducing intermediate variables to transform the nonlinear estimation problem into a linear one; 2) Using weighted least squares (WLS) to obtain estimates of the target position and intermediate variables; 3) Performing a first-order Taylor expansion on the target position estimate obtained in step 1) to reconstruct a linear equation regarding the positioning error; 4) Using weighted least squares (WLS) to estimate the positioning error and correct the target position estimate to obtain the final positioning solution. This method improves positioning accuracy by avoiding nonlinear calculations, but when measurement noise is high, the positioning error surges, leading to inaccurate positioning. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and propose a time difference positioning method based on stochastic robust least squares, so as to reduce positioning error and improve positioning accuracy and robustness without increasing time complexity.

[0006] To achieve the above objectives, the technical solution of the present invention includes the following:

[0007] (1) Set up one reference observation station, multiple auxiliary observation stations, and noise measurement error, using the location parameter s1 of the reference observation station and the location parameter s of the auxiliary observation stations. i and measurement error Δτ i1 Thus, the time difference positioning equation containing quadratic error is obtained;

[0008] (2) Introduce the auxiliary variable θ1 into the above time difference positioning equation to establish the constraint equation of the quadratic term error with respect to the auxiliary variable θ1:

[0009] ξ1=h1-G1θ1+ΔG1θ1+ΔG2r

[0010] Where h1 is a constant vector, G1 is a constant matrix, ΔG1 is the perturbation of G1, ΔG2 is the perturbation of h1, r is the distance difference vector, and ξ1 is the quadratic error.

[0011] (3) Introduce the stochastic robust least squares method into the constraint equation of the auxiliary variable θ1, solve for its expectation, and obtain the cost function J(θ1) for the auxiliary variable θ1:

[0012]

[0013] In the formula, J(·) represents the cost function. It is a constant matrix. Let be a constant vector, (·) T Indicates matrix transpose;

[0014] (4) Solve the cost function J(θ1) using the least squares LS method to obtain the target position estimate.

[0015] (5) Place the auxiliary variable θ1 in the estimated value Perform a Taylor expansion at the position to construct the constraint equations for the positioning error θ2:

[0016] ε2=h2-G2θ2

[0017] In the formula, h2 is a constant vector, G2 is a constant matrix, and ε2 is the difference term between h2 and G2θ2;

[0018] (6) Solve the constraint equations for the positioning error θ2 using the weighted least squares (WLS) method to obtain the estimated positioning error value.

[0019] (7) Use positioning error estimates For the target estimate Linear correction is performed to obtain the final positioning result.

[0020] Compared with the prior art, the present invention has the following advantages:

[0021] 1. This invention introduces the idea of ​​Stochastic Robust Least Squares (SRLS) into the time difference positioning equation, establishes an equation for the quadratic noise term with respect to the estimated quantity and the linear noise term, utilizes the statistical characteristics of noise to take the expectation of the cost function, and uses this expectation as a new cost function to process it, thus transforming the nonlinear problem into a stochastic robust least squares problem. Therefore, it is more in line with the actual scenario, can effectively deal with the threshold problem, and improve positioning performance.

[0022] 2. This invention utilizes the quadratic term of noise to solve the weight matrix, eliminating the need for updating the weight matrix, thus reducing positioning error and improving positioning accuracy.

[0023] 3. This invention utilizes the inherent relationship between the target location and auxiliary variables, performs a first-order Taylor expansion of the auxiliary variables at the estimated target location, and constructs a constraint equation for the positioning error. This not only avoids nonlinear operations involving square roots but also solves the problem of rank deficiency in analytical algorithms, further reducing the positioning error and improving the positioning accuracy and robustness. Attached Figure Description

[0024] Figure 1 This is a flowchart illustrating the implementation of the present invention;

[0025] Figure 2 This is a schematic diagram of the time difference positioning principle in this invention;

[0026] Figure 3 This is a simulation diagram of a positioning scenario for an instantaneous target using the present invention and different existing positioning algorithms;

[0027] Figure 4 To use the present invention and existing different positioning algorithms in Figure 3 A simulation of the instantaneous target positioning error in the given scenario;

[0028] Figure 5 The image shows a simulation of a moving target location using this invention and different existing location algorithms.

[0029] Figure 6 To use the present invention and existing different positioning algorithms in Figure 4 The positioning error results of the simulated moving target in the scenario are shown in the figure.

[0030] Figure 7 The simulation results show the positioning error distribution using the present invention and different existing positioning algorithms under the same station configuration.

[0031] Figure 8 These are schematic diagrams of different station deployment configurations in this invention;

[0032] Figure 9 For the present invention in Figure 8 Positioning error distribution diagram under different station configurations. Detailed Implementation

[0033] The embodiments and effects of the present invention will be described in further detail below with reference to the accompanying drawings.

[0034] Reference Figure 1 The implementation steps for this example are as follows:

[0035] Step 1: Set up M (M≥4) observation stations and noise measurement errors, and construct the time difference positioning equation.

[0036] Reference Figure 2 This example sets up one reference observation station and M-1 auxiliary observation stations. The noise measurement error follows a mean of 0 and a variance of . Given a Gaussian distribution, let the position parameter of the reference station be s1 and the position parameter of the auxiliary station be s2. i (i = 2, 3, ..., M), measurement error is Δτ i1 Substituting these parameters into the existing time difference positioning equation, we obtain a time difference positioning equation that includes quadratic error:

[0037]

[0038] Where c is the electromagnetic wave propagation speed, u represents the target position to be determined, and r i1 =||s i -u||-||s1-u|| represents the distance difference between the target and the reference station and between the target and each auxiliary observation station, r1 o Indicates the distance from the target to the reference observation station, (·) T represents the matrix transpose, and ||·|| represents the 2-norm.

[0039] Step 2: Establish constraint equations.

[0040] 2.1) Constructing the auxiliary variable θ1:

[0041] θ1=[u T ,r1 o ] T

[0042] Where u is a 3×1 vector representing the target position to be determined; r1 o This represents the distance from the target to the reference observation station, (·). T Indicates matrix transpose;

[0043] 2.2) Introduce the auxiliary variable θ1 into the time difference positioning equation. <1> Establish the constraint equation for the quadratic error with respect to the auxiliary variable θ1:

[0044] 2.2.1) For equation <1> After rearranging and simplifying, we obtain the following equation:

[0045]

[0046] 2.2.2) According to the equation <2> ,make The quadratic error term ξ1 is obtained:

[0047] ξ1=[ξ 21 ,ξ 31 ,…,ξ M1 ] T <3>

[0048] 2.2.3) According to the equation <2> ,make We obtain the constant vector h1:

[0049] h1 = [h 21 ,h 31 ,…,h M1 ] T <4>

[0050] 2.2.4) Equivalence <2> Includes distance r1 o By merging and rearranging the terms at the target position u, we obtain the constant matrix G1:

[0051]

[0052] 2.2.5) According to the equation <2> Let Δr i1 =cΔt i1 Δr is obtained:

[0053] Δr=2[Δr 21 ,Δr 31 ,…,Δr M1 ] T <6>

[0054] 2.2.6) According to formula <6> Equivalence <2> The last two terms are rewritten to obtain ΔG1 and ΔG2, which are expressed as follows:

[0055] ΔG1=2[0 (M-1)×3 ,Δr] <7>

[0056] ΔG2=2diag(Δr) <8>

[0057] Where diag(·) denotes diagonalization;

[0058] 2.2.7) Using the constants obtained from the above calculations, according to the formula... <2> By considering the relationships between the terms, we obtain the constraint equations for the auxiliary variable θ1:

[0059] ξ1=h1-G1θ1+ΔG1θ1+ΔG2r <9>

[0060] Step 3: Establish the cost function.

[0061] 3.1) Calculate the weighting matrix W1:

[0062] 3.1.1) Solve for the expected value E{ξ1} of the quadratic error term ξ1, based on the fact that the measurement error follows a mean of zero and a variance of... Given a Gaussian distribution, we obtain E{ξ1}:

[0063] E{ξ1}=[diag(Q t )] T <10>

[0064] Among them, Q t Let E(·) represent the covariance matrix of the noise measurement error, and let E(·) represent the expectation. T The transpose of a matrix is ​​represented by `diag(·)`, and diag(·) represents diagonalization.

[0065] 3.1.2) Based on the idea of ​​weighted least squares, the expression for the weighted matrix W1 is defined as follows:

[0066] W1=(E{(ξ1-E{ξ1})(ξ1-E{ξ1}) T}) -1 <11>

[0067] in,(·) -1 Represents the inverse of a matrix;

[0068] 3.1.3) The formula <9> Japanese style <10> Substitution <11> Expanding and simplifying the weighted matrix W1, we obtain the following equation:

[0069] W1=(2Q t ⊙Q t ) -1 <12>

[0070] 3.2) Based on the idea of ​​weighted least squares, the cost function J1(θ1) of the auxiliary variable θ1 is defined as:

[0071] J1(θ1)=(ξ1-E{ξ1}) T W1(ξ1-E{ξ1}) <13>

[0072] Where J1(·) represents the cost function;

[0073] 3.3) The formula <9> Japanese style <10> Substitution <13> And by taking its expectation, we obtain the expectation E{J1(θ1)} of the cost function J1(θ1):

[0074]

[0075] 3.4) Solving for the matrix in the expectation E{J1(θ1)} and Expectations and

[0076] 3.4.1) Transpose ΔG1 Multiply the weighted matrices W1 and ΔG2, and calculate their expectation to obtain...

[0077]

[0078] Where diag(·) denotes diagonalization;

[0079] 3.4.2) Transform ΔG1 and its transpose matrix Multiply by the weighted matrix W1 and calculate its expectation to obtain...

[0080]

[0081] Where tr(·) represents the trace of the matrix;

[0082] 3.5) Formula <10> ,Mode <15> Japanese style <16> Attached <14> Expanding and simplifying the expectation E{J1(θ1)}, we obtain a new cost function J(θ1):

[0083]

[0084]

[0085]

[0086] Step 4: Calculate the target location estimate

[0087] 4.1) Command Form <17> Taking the derivative with respect to the auxiliary variable θ1, we obtain the derivative of the cost function J(θ1) with respect to the auxiliary variable θ1:

[0088]

[0089] in, To express differentiation, It is a constant matrix. It is a constant vector;

[0090] 4.2) Command Form <20> The value is equal to zero, thus obtaining the least squares estimate of the auxiliary variable θ1.

[0091]

[0092] 4.3) Based on the inherent relationship between the auxiliary variable θ1 and the target position u, use the estimated value Obtain the target location estimate The estimated value It is by The vector consisting of the first three elements is represented as follows:

[0093]

[0094] Step 5: Establish the constraint equations for the positioning error θ2.

[0095] 5.1) Based on auxiliary variables θ1 and distance r1 o The inherent connection, using the estimated value The distance estimate r1 is obtained, which is derived from the estimated value. The fourth element is represented as follows:

[0096]

[0097] 5.2) Set the distance estimate r1 within the estimated value Performing a first-order Taylor expansion at this point, we obtain:

[0098]

[0099] in,

[0100] 5.3) According to the formula <24> Establish the constraint equations for the positioning error θ2:

[0101] 5.3.1) Let θ2 = Δu, and substitute it into the formula. <24> and in the formula <24> Subtracting the left and right sides, we obtain the error term ε2:

[0102]

[0103] 5.3.2) The formula <25> In this context, r1-||u-s1|| can be rewritten as a constant vector h2:

[0104]

[0105] Among them, 0 3×1 Represents a 3×1 zero matrix;

[0106] 5.3.3) The formula <25> In Rewritten as a constant matrix G2:

[0107]

[0108] Among them, I 3×1 This represents a 3×1 identity matrix;

[0109] 5.3.4) Using the constants obtained from the above calculations, the constraint equation for the positioning error θ2 is obtained:

[0110] ε2=h2-G2θ2. <28>

[0111] Step 6: Calculate the estimated positioning error.

[0112] 6.1) Define the difference Δθ1 between the estimated target position and the true target position:

[0113] Δθ1=[Δu T ,Δr1] T

[0114] in Δr1 represents the difference between the estimated target location and the actual target location, Δr1 = r1 - r1 o This represents the difference between the estimated distance and the actual distance from the target to the reference observation station;

[0115] 6.2) Calculate the estimation error Δθ1:

[0116] 6.2.1) Using auxiliary variable θ1 and estimated value Calculate the estimation error Δθ1:

[0117]

[0118] 6.2.2) The formula <21> Substitution <29> The estimation error Δθ1 is obtained as follows:

[0119]

[0120] 6.2.3) The formula <10> , <13> , <14> , <18> and <19> Substitution <30> and the formula <30> After expansion and simplification, the estimation error Δθ1 is obtained:

[0121] Δθ1=ζ+Δζ <31>

[0122]

[0123]

[0124] Where, r o Q represents the distance from the target to each auxiliary observation station. t The measurement error Δτ is represented by i1 The covariance matrix, where Δr represents the error of the first term, and diag(·) denotes diagonalization;

[0125] 6.3) For example <31> Calculate the expectation to obtain the expectation E{Δθ1} of the estimation error Δθ1:

[0126]

[0127] 6.4) Calculate the error covariance matrix

[0128] 6.4.1) Define the estimated value Error covariance matrix The expression is:

[0129]

[0130] 6.4.2) The formula <30> Japanese style <34> Substitution <35> The covariance matrix is ​​obtained.

[0131]

[0132] in,

[0133] 6.5) Solve for the matrix covariance matrix respectively. In the matrix ΔζΔζ T sum matrix ΔrΔζ T The expected value E{ΔrΔζ T} and E{ΔζΔζ T}:

[0134] 6.5.1) Using formula <32> Multiply the transposes of matrices Δr and Δζ, and take their expected value to obtain matrices Δr and Δζ. T The expected value E{ΔrΔζ T}:

[0135]

[0136] 6.5.2) Using formula <33> Multiply the matrix Δζ and its transpose, and then take the expectation of the product to obtain the matrix Δζ. T Expected E{ΔζΔζ T}:

[0137]

[0138] 6.6) Calculate the weighted matrix W2:

[0139] 6.6.1) Based on the idea of ​​weighted least squares, the expression for the weighted matrix W2 is defined as follows:

[0140]

[0141] 6.6.2) The formula <28> Attached <39> The weighted matrix W2 is obtained as follows:

[0142] W2=(B2cov(θ1)B2 T ) -1 (36)

[0143] Where B2 represents a 4×4 identity matrix;

[0144] 6.6) Based on the weighted least squares approach, using the weighting matrix W2, the constant matrix G2, and the constant vector h2, the estimated value of the positioning error θ2 is obtained.

[0145]

[0146] Step 7: Calculate the target's true position u to complete the target localization.

[0147] Using the positioning error estimate For the target estimate Perform linear correction to obtain the true target position u:

[0148]

[0149] in, This represents an estimated value of the target's location. This represents the estimated positioning error.

[0150] The effects of this invention can be further illustrated by the following simulation experiments.

[0151] I. Simulation Conditions

[0152] Condition 1: Set the location scene as follows Figure 3 As shown, there are 5 observation stations in this scenario, with a baseline length of 2.5 km between any two adjacent observation stations. The first observation station is used as the reference station, and the others are used as auxiliary reference stations. Two targets to be determined are set: a near-range target close to the observation stations and a far-range target further away. The initial measurement error is set to σ. t =10ns, and gradually increase in 10ns intervals, and perform 1000 simulation experiments at each error sampling point.

[0153] The location parameters of the observation station are shown in Table 1.

[0154] The target location parameters are shown in Table 2:

[0155] Table 1. Location parameters of observation stations (unit: km)

[0156] Observation station name x y z Observation Station 1 0 0 0 Observation Station 2 0 0 2.5 Observation Station 3 2.165 0 -1.25 Observation Station 4 -1.083 1.875 -1.25 Observation Station 5 -1.083 -1.875 -1.25

[0157] Table 2 Target Location Parameters (Unit: km)

[0158] Target Name x y z Target 1 3 2.5 2 Target 2 12 10 5

[0159] Condition 2: Set the location scenario as follows Figure 5 As shown in Table 1, there are 5 observation stations in this scenario. Two of these stations are moving targets, each 5 km above the ground, moving in uniform circular motion around the z-axis with radii of 17 km and 2.5 km respectively. The measurement error is σ. t =20ns. Whenever the azimuth angle changes by 10°, a point is selected on each of the two target motion trajectories to conduct a 1000-second simulation experiment.

[0160] Condition 3: Set up 6 observation stations. The location parameters of the reference observation station are (0.773km, 2.378km, 0km). The location parameters of the 5 auxiliary observation stations are (-2.023km, 1.47km, 0km), (-2.023km, -1.47km, 0km), (0.773km, -2.378km, 0km), (2.5km, 0km, 0km), and (0km, 0km, 2.5km). The noise measurement error is 200ns. Divide the airspace x = -20km to 20km, y = -20km to 20km, and altitude 5km into 100×100 grids with equal intervals. Each grid point represents a target. Perform 1000 simulation experiments on each target.

[0161] Condition 4: Set up 3 site configurations as follows Figure 8 As shown, where, Figure 8 (a) shows a schematic diagram of configuration 1. Figure 8 (b) shows a schematic diagram of configuration 2. Figure 8 (c) shows a schematic diagram under configuration 3. There are 5 observation stations under each configuration. Observation station 1 is used as the reference observation station, and the rest are used as auxiliary observation stations. The noise measurement error is 20 ns. The airspace with x = -20km to 20km, y = -20km to 20km, and an altitude of 5km is divided into 100×100 grids at equal intervals. Each grid point represents a target. 1000 simulation experiments are performed on each target.

[0162] The location parameters of the observation stations with different configurations are shown in Table 3:

[0163] Table 3. Location parameters of observation stations with different configurations (unit: km)

[0164] Station configuration name Configuration 1 Configuration 1 Configuration 3 Observation Station 1 (0,0,0) (2.5,5.2,0) (0,0,0) Observation Station 2 (0,5,0) (-2.5,2.5,0) (-5,0,0) Observation Station 3 (-4.33,-2.5,0) (-2.5,-2.5,0) (5,0,0) Observation Station 4 (4.33,-2.5,0) (2.5,-2.5,0) (0,5,0) Observation Station 5 (0,0,2.5) (0,0,2.5) (0,0,2.5)

[0165] II. Simulation Content

[0166] Simulation 1: Under condition 1 above, the present invention and two existing positioning algorithms, TSWLS and SRLS, were used to simulate the positioning of an instantaneous target, and the positioning error curves of each were obtained, as shown below. Figure 4 As shown. Among them. Figure 4 (a) shows the positioning curve for locating a nearby target. Figure 4 (b) shows the positioning curve for locating distant targets.

[0167] from Figure 4 As can be seen, when the noise measurement error is large, the positioning error of this invention is closest to the Cramero lower bound (CRLB) compared with the existing algorithm, which has higher positioning accuracy and no threshold effect.

[0168] Simulation 2: Under condition 2 above, the present invention and two existing positioning algorithms, TSWLS and SRLS, were used to simulate the positioning of a moving target, and the positioning error curves of each were obtained, as shown below. Figure 6 As shown. Among them, Figure 6 (a) shows the positioning curve for locating a moving target at close range. Figure 6 (b) shows the positioning curve for locating a distant moving target.

[0169] from Figure 6 As can be seen, compared with existing algorithms, the positioning error of this invention fluctuates more smoothly with the change of the target position azimuth angle, and has stronger positioning robustness.

[0170] Simulation 3: Under condition 3 above, positioning simulations were performed using the present invention and two existing positioning algorithms, TSWLS and SRLS, under the same station configuration. The spatial distribution maps of their respective positioning errors were obtained, as shown below. Figure 7 .in, Figure 7 (a) shows the distribution of the geometrical factor of precision (GDOP). Figure 7 (b) shows the spatial distribution of positioning errors of the TSWLS positioning algorithm. Figure 7 (c) shows the spatial distribution of positioning errors of the SRLS positioning algorithm. Figure 7 (d) represents the spatial distribution diagram of the positioning error of the present invention.

[0171] from Figure 7 As can be seen, compared with existing algorithms, the positioning error of this invention is more evenly distributed in all directions, and the positioning error is closest to the geometric positioning factor (GDOP), resulting in stronger positioning robustness and higher positioning accuracy.

[0172] Simulation 4: Under condition 4 above, positioning simulations were performed using the present invention in three different station configurations. The spatial distribution diagrams of the positioning error of the present invention under different configurations were obtained, as shown below. Figure 9 .in, Figure 9(a) shows the spatial distribution of positioning error of the present invention in configuration 1. Figure 9 (b) shows the spatial distribution of positioning error of the present invention in configuration 2. Figure 9 (c) shows the spatial distribution of positioning error of the present invention in configuration 3.

[0173] from Figure 9 As can be seen, in configuration 1, the positioning error is more evenly distributed in all directions, and the positioning error is closest to the geometric positioning factor GDOP, resulting in stronger positioning robustness and higher positioning accuracy.

[0174] The simulation results above show that the present invention can effectively reduce the adverse effects of random noise on positioning, solve the threshold effect problem, and improve the robustness and accuracy of positioning.

Claims

1. A time-difference positioning method based on stochastic robust least squares, characterized in that, Includes the following steps: (1) Set up one reference observation station, multiple auxiliary observation stations, and noise measurement error, and use the location parameters of the reference observation station. Auxiliary observation station location parameters and measurement error Thus, the time difference positioning equation containing quadratic error is obtained; (2) Auxiliary variables Introducing the above time difference positioning equation, we establish the quadratic error with respect to the auxiliary variable. Constraint equations: ， in, It is a constant vector. It is a constant matrix. yes The disturbance yes The disturbance It is the distance difference vector. It is a quadratic error; (3) Introduce the stochastic robust least squares method into auxiliary variables. The constraint equations are solved, and the expected value is obtained to determine the information about the auxiliary variables. Cost function : ， In the formula, Represents the cost function, It is a constant matrix. A constant vector, Indicates matrix transpose; (4) Solve the cost function using the least squares (LS) method. The target location estimate is obtained. ; (5) Add auxiliary variables In the estimated value Perform a first-order Taylor expansion to construct a function relating the positioning error. Constraint equations: ， In the formula, It is a constant vector. It is a constant matrix. yes and The difference term; (6) Use the weighted least squares (WLS) method to solve for the positioning error. The constraint equations are used to obtain the estimated positioning error. The implementation is as follows: 6a) Define the difference between the estimated target position and the actual target position as: , in This represents the difference between the estimated target location and the actual target location. This represents the difference between the estimated distance and the actual distance from the target to the reference observation station; 6b) Using auxiliary variables and its least squares estimate The estimated error was obtained. : ， in, Indicates about The constant matrix, Indicates about constant vector, This represents the quadratic error. Indicates about constant matrix, Indicates about constant vector, express The disturbance express The disturbance Represents a weighted matrix. Indicates the distance difference. Represents the inverse of a matrix. Represents the mathematical expectation. Indicates matrix transpose; 6c) constant matrix and constant vector and and Expanding on the expected value, and substituting it into the error in 6b), The equation, after simplification, is obtained as follows: ， ， ， in, This represents the distance from the target to each auxiliary observation station. Indicates measurement error The covariance matrix, This represents the error of the first-order term. ; 6d) Solve for the estimated error respectively ,matrix and matrix Expectations , and : ， ， ， in, This represents the sum of the number of reference observation stations and auxiliary observation stations; 6e) Utilizing errors , , and To obtain the estimated value Error covariance matrix : ， 6f) Using the covariance matrix The weighted matrix is ​​obtained. : ， in, express unit array; 6h) Using a weighted matrix constant matrix and constant vector The positioning error was obtained. The estimated value : ； (7) Use the positioning error estimate For the target estimate Linear correction is performed to obtain the final positioning result.

2. The method according to claim 1, characterized in that, In step (1), the error containing the quadratic term is obtained. The time difference positioning equation is expressed as follows: ， in, The speed of electromagnetic wave propagation. Indicates the target location to be determined. This represents the distance difference between the target and the reference station, and between the target and each auxiliary observation station. Indicates the distance from the target to the reference observation station. Indicates matrix transpose. It represents the 2-norm.

3. The method according to claim 1, characterized in that, Auxiliary variables introduced in step (2) , yes The vector is represented as follows: ， in, yes The vector represents the target location to be determined. This represents the distance from the target to the reference observation station. This indicates the matrix transpose.

4. The method according to claim 1, characterized in that, In step (3), the random robust least squares method will be introduced. Enter auxiliary variables The constraint equations are solved to obtain the expected value, as follows: 3a) Utilizing the quadratic term error based on the weighted least squares approach and weighted matrix , get about Cost function : ， in, Represents the cost function, Represents the mathematical expectation; 3b) Calculate the quadratic term error The expected weighted matrix is ​​obtained. : ， in, Indicates noise measurement error The covariance matrix, Represents the inverse of a matrix; 3c) Utilizing auxiliary variables constraint equations and weighting matrices For the cost function Expectation: ， in, It is a constant vector. It is a constant matrix. yes The disturbance yes The disturbance It is the distance difference vector; 3d) Solve for the cost function separately middle and Expectations: ， ， in, , Represents the trace of a matrix; 3e) Utilization , and cost function The expectation is to obtain a new cost function. : ， in: It is a constant matrix. Let them be constant vectors, represented as follows: ， 。 5. The method according to claim 1, characterized in that, In step (4), the least squares (LS) method is used to solve the problem. valence function The implementation is as follows: 4a) Using the cost function For auxiliary variables Differentiate: ， in, To express differentiation, It is a constant matrix. It is a constant vector; 4b) Setting the derivative of 4a) to zero, we obtain the auxiliary variable. Least squares estimate : ， Among them, the least squares estimate yes ; 4c) Based on auxiliary variables and target location The inherent connection, using the estimated value Obtain the target location estimate Estimated value It is by The vector consisting of the first three elements is represented as follows: 。 6. The method according to claim 1, characterized in that, Step (7) uses the positioning error estimate To the eye Standard estimate Linear correction is performed using the following formula: ， in, Indicates the target location. This represents an estimated value of the target's location. This represents the estimated positioning error.

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