A relative positioning method based on smacof and progressive helmert transformation

Abstract

The application discloses a relative positioning method based on SMACOF and progressive Helmert transformation, and comprises the following steps: calculating the weighted Euclidean distance between nodes to obtain a weighted Euclidean distance matrix; wherein the nodes comprise unknown nodes and anchor nodes; calculating the relative coordinates of all nodes by an MDS positioning algorithm based on the weighted Euclidean distance matrix to obtain a relative coordinate matrix; obtaining an optimal relative coordinate matrix by solving the optimal solution of the relative coordinate matrix through an SMACOF algorithm; mapping the optimal relative coordinate matrix to global coordinates through Helmert transformation to obtain an absolute coordinate matrix; optimizing the absolute coordinate matrix through a progressive correction mechanism to obtain optimal transformation parameters; and obtaining an optimal absolute coordinate matrix by performing Helmert transformation on the optimal relative coordinate matrix again through the optimal transformation parameters, so as to obtain the accurate positions of all nodes.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of positioning and navigation technology, in particular, to a relative positioning method based on SMACOF and progressive Helmert transformation. BACKGROUND

[0002] With the wide application of global positioning system (GPS) and other satellite navigation technologies, positioning technology has been widely applied in many fields, such as unmanned driving, robot navigation, intelligent transportation, Internet of Things, etc. However, the traditional GPS positioning system is limited by factors such as obstacle blocking and signal attenuation in some environments, which affects the positioning accuracy. In this context, relative positioning based on distance measurement has become an important research direction. By measuring the relative distance between devices through wireless communication technology (such as UWB, Wi-Fi, Bluetooth, etc.), the position of the device can be estimated, especially in indoor or weak GPS signal environments.

[0003] Multidimensional scaling (MDS) is a method commonly used for data dimensionality reduction and visualization, which maps high-dimensional data to low-dimensional space while preserving the distance relationship between original data points as much as possible. MDS technology has been widely applied in image processing, data visualization, recommendation systems, and other fields. In multi-point positioning, MDS constructs a distance matrix representing the relative distance between devices and performs dimensionality reduction based on the matrix to provide accurate low-dimensional coordinates for the positioning system. In the problem of multi-point positioning in cluster networks, the basic optimization method of MDS algorithm is to minimize the stress function to map the distance relationship in high-dimensional space to low-dimensional space. However, the basic optimization method has some shortcomings in practical application, especially when there is noise or measurement error in the data, the optimization process of the stress function may be disturbed, resulting in reduced accuracy of the results. The optimization of the stress function usually relies on traditional methods such as gradient descent, but these methods are prone to local optimal solutions and have slow convergence speed and poor accuracy for complex error structures or high-dimensional data. In addition, Helmert transformation is a linear transformation used to convert local coordinate systems to global coordinate systems, commonly used in coordinate conversion in geographic information systems (GIS) and surveying. Its core is to align two coordinate systems through scaling, rotation, and translation. However, in the application process of Helmert transformation, errors may be amplified during the conversion process, especially in the case of uneven node distribution or poor ranging accuracy, errors will significantly affect the final positioning result.

[0004] And the application number is 2024108476147 Chinese invention application discloses a kind of high-precision indoor positioning method based on ultra-wideband, comprising the following steps: randomly placing to be positioned label in base station signal coverage area, signal time difference measurement value of label is collected;Initialization weight matrix, matrix is substituted into the preset positioning algorithm and carries out initial position estimation;The initial coordinate of to be positioned label is substituted into signal time difference equation group and error vector is calculated, then average value is taken;The average value obtained is input into particle swarm algorithm, and the optimal weight matrix calculation formula parameter is obtained;To be positioned label is moved, and signal time difference measurement value of to be positioned label is collected in real time, which is input into the optimal coefficient weight matrix calculation formula, and the weight matrix of current to be positioned label is obtained;The weight matrix of current to be positioned label is substituted into positioning algorithm, and more accurate estimated position of to be positioned label is obtained. SUMMARY

[0005] To solve the technical problems of insufficient precision and high complexity of existing positioning and navigation technology, the present application provides a relative positioning method based on SMACOF and progressive Helmert transformation. The technical solution adopted by the present application is:

[0006] The first aspect of the present application provides a relative positioning method based on SMACOF and progressive Helmert transformation, which comprises:

[0007] Calculate the weighted Euclidean distance between nodes to obtain a weighted Euclidean distance matrix;Wherein the node includes unknown node and anchor node;

[0008] Calculate the relative coordinates of all nodes by MDS positioning algorithm through the weighted Euclidean distance matrix, to obtain a relative coordinate matrix;

[0009] Obtain the optimal relative coordinate matrix by solving the optimal solution of the relative coordinate matrix through SMACOF algorithm;

[0010] Map the optimal relative coordinate matrix to global coordinates by Helmert transformation to obtain an absolute coordinate matrix;

[0011] Optimize the absolute coordinate matrix by progressive correction mechanism to obtain optimal transformation parameters;

[0012] Perform Helmert transformation on the optimal relative coordinate matrix again by optimal transformation parameters to obtain an optimal absolute coordinate matrix, thereby obtaining the accurate position of all nodes.

[0013] As a preferred scheme, the method for calculating the weighted Euclidean distance between nodes to obtain a weighted Euclidean distance matrix comprises:

[0014] In d-dimensional space, there are n nodes, including m anchor nodes and n-m unknown nodes, and the coordinates are defined as follows:

[0015]

[0016] The weighted Euclidean distance between the ith node and the jth node is obtained by weighting the distance between each pair of points as follows:

[0017]

[0018] where w ij is the weight between the ith node and the jth node, and the weight is defined as follows:

[0019]

[0020] where σ k represents the standard deviation of noise in each dimension k;

[0021] The resulting weighted Euclidean distance matrix is as follows:

[0022]

[0023] where δ ij = δ ji and δ ii = 0, and the weighted Euclidean distance matrix D is a symmetric matrix.

[0024] As a preferred solution, the weighted Euclidean distance matrix is calculated by the MDS positioning algorithm to obtain the relative coordinate matrix of all nodes, and the method includes:

[0025] The row sum and column sum of D are made zero by a double centering operation;

[0026] where the definition of the double centering operation is as follows:

[0027]

[0028] where D ⊙m represents the mth Hadamard power of D, and J is the centering matrix and can be represented as:

[0029]

[0030] where I N represents an N×N identity matrix, and 1 N represents an N-element column vector with all elements being 1.

[0031] The EDM of the relative coordinates Y is defined as Δ = [d i,j ] N×N , di,j Let be the distance between the relative coordinates of the i-th node and the relative coordinates of the j-th node.

[0032]

[0033] right The eigenvalue decomposition is performed as follows:

[0034]

[0035] Where ∧=diag(λ1,λ2…λ N ), λ1≥λ2≥…≥λ N for A diagonal matrix of eigenvalues ​​arranged in descending order, Q = [q1, q2, ... q N ]for The eigenvector matrix is ​​composed of the eigenvectors corresponding to the eigenvalues.

[0036] take out The three largest eigenvalues ​​λ1, λ2, and λ3 form a diagonal matrix ∧3 = diag(λ1, λ2, λ3), and the corresponding eigenvectors form an eigenvector matrix Q3 = [q1, q2, q3]. Then, the relative coordinates Y of the node can be expressed as:

[0037]

[0038] Calculate the relative coordinates of the n nodes to obtain the relative coordinate matrix:

[0039]

[0040] As a preferred approach, the method for obtaining the optimal relative coordinate matrix by using the SMACOF algorithm includes:

[0041] The objective function to be optimized is defined as follows:

[0042]

[0043] The objective function is decomposed into three parts:

[0044]

[0045] in, For a constant term, It is a convex quadratic function with a minimum value; the third term is bounded according to the Cauchy-Schwarz inequality.

[0046]

[0047] The Cauchy-Schwarz inequality is as follows:

[0048]

[0049] where z ij is a constant; when y ij = z ij , the inequality in equation (13) becomes an equality, resulting in:

[0050]

[0051] Equation (15) is converted into matrix form:

[0052]

[0053] For η 2 (Y) in equation (16), we have:

[0054]

[0055] where A ij = (e i - e j )(e i - e j ) T , a ii = a jj = 1, a ij = a ji = -1, and the rest are equal to 0, then:

[0056]

[0057] where tr is the trace of the matrix, and matrix V is the Hessian matrix of function η 2 (Y), which has the following form:

[0058]

[0059] For ρ(Y) in equation (16), we have:

[0060]

[0061] where:

[0062]

[0063] then:

[0064]

[0065] In summary:

[0066]

[0067] According to formula (15), we have

[0068]

[0069] By optimizing τ Z (Y) to iteratively solve σ(Y):

[0070]

[0071] Solving (25) has:

[0072] Y * = V -1 B(Z)Z (26)

[0073] Where V -1 is the generalized inverse of V, and the obtained Y * is the optimal solution of the relative coordinate matrix Y.

[0074] As a preferred solution, the method for mapping the optimal relative coordinate matrix to global coordinates by Helmert transformation to obtain the absolute coordinate matrix comprises:

[0075] The Helmert transformation formula is defined as follows:

[0076] C'=T+Ro·S·X (27)

[0077] Where X is the local coordinate of all nodes, C' is the global coordinate of the device, R O is the rotation parameter, S is the scaling parameter, and T is the translation parameter.

[0078] The Helmert transformation includes the following steps:

[0079] First, calculate the centroid nodes of the local and global coordinates of the anchor nodes, let and be the centroid nodes of the relative coordinates and absolute coordinates of the anchor nodes; therefore, and The calculation formula is as follows:

[0080]

[0081] The formula for finding the new coordinate system of the anchor nodes using the centroid nodes is as follows:

[0082]

[0083] Let Ro m be the rotation matrix, which is defined as follows:

[0084]

[0085] Let v1, v2, v3 and e1, e2, e3 be the vectors of the nodes and edges of the graph G, respectively. The computed eigendecomposition, so the rotation matrix R is computed by equation (33):

[0086]

[0087] The scale factor is computed by equation (34):

[0088]

[0089] The translation matrix T is computed by equation (35):

[0090] T = X g ′-SRoX l ′(35)

[0091] Finally, the optimal relative coordinate matrix Y * for all nodes is computed, and the corresponding absolute coordinate matrix C′ is computed by equation (36):

[0092] C′ = T + Ro·S·Y * (36).

[0093] As a preferred solution, the method for optimizing the absolute coordinate matrix by a progressive correction mechanism to obtain optimal transformation parameters comprises:

[0094] After the initial Helmert transformation is completed, the converted positioning error is calculated;

[0095] Define the residual as:

[0096]

[0097] wherein is the actual observed distance value, is the theoretical calculated distance between the converted coordinates;

[0098] By statistical analysis of the residuals ε ij between all nodes, the distribution of the current error is obtained:

[0099] The root mean square error of the overall error is calculated as an error evaluation index:

[0100]

[0101] wherein N is the number of node pairs, according to the error evaluation index, the positioning error of the initial Helmert transformation is evaluated, if the root mean square error does not meet the preset convergence condition, the transformation parameters are adjusted; the transformation parameters include the scale factor S, the rotation matrix R O, and a translation variable T, the transformation parameter adjustment formula is as follows:

[0102]

[0103]

[0104] wherein S o ,R o ,T o represents the current transformation parameter, S n ,R n ,T n represents the transformation parameter after iteration; η, α, γ are learning rates of S o ,R o ,T o , respectively, and control the step length of adjustment; RMSE is the gradient of the root mean square error of the translation vector;

[0105] After each iteration adjustment, the residual error and the root mean square error are recalculated, the current positioning error level is evaluated, when the root mean square error meets the preset convergence condition, the iteration is stopped, and the optimal transformation parameter is obtained; otherwise, the iteration process is repeated, and the transformation parameter is continuously adjusted.

[0106] As a preferred scheme, the preset convergence condition is represented as:

[0107] |ΔRMSE|<ε

[0108] wherein ΔRMSE is the change amount of RMSE, and ε is a given threshold value; when |ΔRMSE|<ε, the iteration is stopped, and the optimal transformation parameter is obtained; otherwise, the iteration process is repeated, and the transformation parameter is continuously adjusted.

[0109] As a preferred scheme, the optimal transformation parameter is used to perform Helmert transformation on the optimal relative coordinate matrix again, to obtain an optimal absolute coordinate matrix, so that the method for obtaining the accurate positions of all nodes comprises:

[0110] The obtained optimal transformation parameter and the optimal relative coordinate matrix are input into formula (36), to obtain an optimal absolute coordinate matrix.

[0111] The second aspect of the present application provides a computer readable storage medium, which stores a computer program, and the computer program is executed by a processor to realize the steps of the foregoing relative positioning method based on SMACOF and progressive Helmert transformation.

[0112] The third aspect of the present application provides a computer device comprising a storage medium, a processor and a computer program stored in the storage medium and executable by the processor, wherein the computer program, when executed by the processor, implements the steps of the relative positioning method based on SMACOF and progressive Helmert transformation.

[0113] Compared with the prior art, the present application has the beneficial effects that:

[0114] The research object of the method of the present application is a plurality of nodes of a cluster network in a multi-dimensional space. A weighted Euclidean distance matrix between all nodes is constructed by using the known position coordinates of anchor nodes and the ranging information between all nodes, and the matrix reflects the relative position relationship between the nodes. The SMACOF algorithm is used to adjust the Euclidean distances between the nodes in the optimization process, to accurately capture and correct the deviations caused by measurement errors or noise, so that the relative coordinates calculated by the MDS are closer to the actual positions. After obtaining the optimized relative coordinate matrix, the progressive Helmert transformation gradually adjusts the scale factor, the rotation matrix and the translation vector in each iteration process according to the residual error between the actual ranging value of the node and the distance calculation value between the transformed coordinates, to optimize the spatial position of the node, until the positioning error converges to an acceptable range. Finally, the optimal scale factor, rotation matrix and translation vector are obtained, and the positions of the unknown nodes in the cluster network are calculated by using the Helmert transformation formula. It can be seen that the method of the present application can accurately estimate the positions of all nodes in the cluster network by constructing a weighted Euclidean distance matrix and using the joint optimization method of the SMACOF algorithm and the progressive Helmert transformation to solve the node positions, and can still maintain high positioning accuracy in a low signal-to-noise ratio or large error environment, thereby providing a high-precision and low-complexity positioning method for the nodes of the cluster network in a multi-dimensional space. BRIEF DESCRIPTION OF DRAWINGS

[0115] Figure 1 A flowchart of the relative positioning method based on SMACOF and progressive Helmert transformation is provided for the present embodiment. DETAILED DESCRIPTION

[0116] The accompanying drawings are only used for illustrative purposes and should not be understood as limiting the present application;

[0117] It should be clear that the described embodiments are only part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative labor fall within the scope of protection of the present application.

[0118] The terminology used in the description of the application herein is for the purpose of describing particular embodiments only and is not intended to be limiting of the application. As used in the description of the application and the appended claims, the singular forms "a", "an" and "the" are intended to include the plural forms as well, unless the context clearly indicates otherwise. It also will be understood that the term "and / or" as used herein refers to and encompasses any and all possible combinations of one or more of the associated listed items.

[0119] The following description refers to the accompanying drawings. Unless otherwise noted, reference to a term in the "singular" encompasses a description of the "plural" unless the context clearly indicates otherwise. For example, reference to "a" or "one" item is a reference to one or more such items unless the context clearly indicates otherwise.

[0120] In addition, in the description of the application, unless otherwise specified, "multiple" refers to two or more. "And / or", describes the association between the associated objects, which means that there can be three relationships, for example, A and / or B, which can represent the existence of A alone, the existence of A and B at the same time, and the existence of B alone. The character " / " generally represents that the associated objects before and after are in an "or" relationship. The application is further described below in conjunction with the drawings and examples.

[0121] The application is further described below in conjunction with the drawings and examples.

[0122] Embodiment 1

[0123] Please refer to Figure 1 The embodiment provides a relative positioning method based on SMACOF and progressive Helmert transformation, and the method comprises the following steps:

[0124] S1: calculating the weighted Euclidean distance between nodes to obtain a weighted Euclidean distance matrix; wherein the nodes include unknown nodes and anchor nodes;

[0125] In one specific embodiment, the method for calculating the weighted Euclidean distance between nodes to obtain a weighted Euclidean distance matrix comprises:

[0126] In a d-dimensional space, there are n nodes, including m anchor nodes and n-m unknown nodes, and the coordinates are defined as follows:

[0127]

[0128] The distance of each pair of points is weighted, and the weighted Euclidean distance between the ith node and the jth node is:

[0129]

[0130] where w ij is the weight between the ith node and the jth node, and the weight is defined as follows:

[0131]

[0132] where σ k represents the standard deviation of noise in each dimension k;

[0133] The resulting weighted Euclidean distance matrix is as follows:

[0134]

[0135] where δ ij = δ ji and δ ii = 0, and the weighted Euclidean distance matrix D is a symmetric matrix.

[0136] S2: Calculate the relative coordinates of all nodes by the MDS positioning algorithm on the weighted Euclidean distance matrix, to obtain a relative coordinate matrix;

[0137] In one specific embodiment, the method of calculating the relative coordinates of all nodes by the MDS positioning algorithm on the weighted Euclidean distance matrix to obtain a relative coordinate matrix includes:

[0138] The row sum and column sum of D are made zero by a double centering operation;

[0139] where the definition of the double centering operation is:

[0140]

[0141] where D ⊙m represents the mth Hadamard power of D, and J is a centering matrix and can be represented as:

[0142]

[0143] where I N represents an N×N identity matrix, and 1 N represents an N-element column vector with all elements being 1;

[0144] The EDM of the relative coordinates Y is defined as Δ = [d i,j ] N×N , di,j Let be the distance between the relative coordinates of the i-th node and the relative coordinates of the j-th node.

[0145]

[0146] right The eigenvalue decomposition is performed as follows:

[0147]

[0148] Where ∧=diag(λ1,λ2…λ N ), λ1≥λ2≥…≥λ N for A diagonal matrix of eigenvalues ​​arranged in descending order, Q = [q1, q2, ... q N ]for The eigenvector matrix is ​​composed of the eigenvectors corresponding to the eigenvalues.

[0149] take out The three largest eigenvalues ​​λ1, λ2, and λ3 form a diagonal matrix ∧3 = diag(λ1, λ2, λ3), and the corresponding eigenvectors form an eigenvector matrix Q3 = [q1, q2, q3]. Then, the relative coordinates Y of the node can be expressed as:

[0150]

[0151] Calculate the relative coordinates of the n nodes to obtain the relative coordinate matrix:

[0152]

[0153] S3: Calculate the optimal relative coordinate matrix by applying the SMACOF algorithm;

[0154] In a specific embodiment, the method for obtaining the optimal relative coordinate matrix by using the SMACOF algorithm to find the optimal solution of the relative coordinate matrix includes:

[0155] The objective function to be optimized is defined as follows:

[0156]

[0157] The objective function is decomposed into three parts:

[0158]

[0159] in, For a constant term, It is a convex quadratic function with a minimum value; the third term is bounded according to the Cauchy-Schwarz inequality.

[0160]

[0161] The Cauchy-Schwarz inequality is given by:

[0162]

[0163] where z ij is a constant; when y ij = z ij , the inequality in equation (13) becomes an equality, giving:

[0164]

[0165] Equation (15) can be converted into a matrix form as:

[0166]

[0167] For η 2 (Y) in equation (16), we have:

[0168]

[0169] where A ij = (e i - e j )(e i - e j ) T , a ii = a jj = 1, a ij = a ji = -1, and the rest are equal to 0, then:

[0170]

[0171] where tr is the trace of a matrix, and the matrix V is the Hessian matrix of the function η 2 (Y) and has the form:

[0172]

[0173] For ρ(Y) in equation (16), we have:

[0174]

[0175] where:

[0176]

[0177] then:

[0178]

[0179] In summary:

[0180]

[0181] According to formula (15), we have:

[0182]

[0183] By optimizing τ Z (Y) to iteratively solve σ(Y):

[0184]

[0185] Solving (25) has:

[0186] Y * = V -1 B(Z)Z (26)

[0187] Where V -1 is the generalized inverse of V, and the obtained Y * is the optimal solution of the relative coordinate matrix Y.

[0188] S4: mapping the optimal relative coordinate matrix to global coordinates by Helmert transformation to obtain an absolute coordinate matrix;

[0189] In one specific embodiment, the method of mapping the optimal relative coordinate matrix to global coordinates by Helmert transformation to obtain an absolute coordinate matrix includes:

[0190] The Helmert transformation formula is defined as follows:

[0191] C' = T + Ro·S·X (27)

[0192] Where X is the local coordinate of all nodes, C' is the global coordinate of the device, R O is the rotation parameter, S is the scaling parameter, and T is the translation parameter;

[0193] The Helmert transformation includes the following steps:

[0194] First, calculate the centroid nodes of the local and global coordinates of the anchor nodes, let and be the centroid nodes of the relative coordinates and absolute coordinates of the anchor nodes; therefore, and The calculation formula is as follows:

[0195]

[0196] The formula for finding the new coordinate system of the anchor nodes using the centroid nodes is as follows:

[0197]

[0198] Let Ro m be the rotation matrix, defined as follows:

[0199]

[0200] Let v1, v2, v3 and e1, e2, e3 be the eigenvectors and eigenvalues calculated from the eigen decomposition, so the rotation matrix is calculated from equation (33):

[0201]

[0202] The scale factor is calculated from equation (34):

[0203]

[0204] The translation matrix T is calculated from equation (35):

[0205] T = X g ' - SRoX l '(35)

[0206] Finally, for all nodes the optimal relative coordinate matrix Y * is calculated from equation (36):

[0207] C' = T + Ro•S•Y * (36).

[0208] S5: optimizing the absolute coordinate matrix by a progressive correction mechanism to obtain optimal transformation parameters;

[0209] In one specific embodiment, the method of optimizing the absolute coordinate matrix by a progressive correction mechanism to obtain optimal transformation parameters comprises:

[0210] After the initial Helmert transformation is completed, the converted positioning error is calculated;

[0211] Define the residual as:

[0212]

[0213] wherein is the actual observed range value, is the theoretically calculated distance between the transformed coordinates;

[0214] By statistical analysis of the residuals ε ij between all nodes, the distribution of the current error is obtained:

[0215] The root mean square error of the overall error is calculated as an error evaluation index:

[0216]

[0217] where N is the number of node pairs, according to the error evaluation index, the positioning error of the initial Helmert transformation is evaluated, if the root mean square error does not meet the preset convergence condition, the transformation parameters are adjusted; the transformation parameters include the scale factor S, the rotation matrix R O , and the translation variable T, the transformation parameter adjustment formula is as follows:

[0218]

[0219]

[0220] where S o ,R o ,T o represent the current transformation parameters, S n ,R n ,T n represent the transformation parameters after iteration; η, α, γ are the learning rates of S o ,R o ,T o , control the step size of adjustment; RMSE is the gradient of the root mean square error of the translation vector;

[0221] After each iteration adjustment, the residual error and the root mean square error are recalculated, the current positioning error level is evaluated, when the root mean square error meets the preset convergence condition, the iteration is stopped, and the optimal transformation parameter is obtained; otherwise, the iteration process is repeated, and the transformation parameter is continuously adjusted.

[0222] In one specific embodiment, the preset convergence condition is represented as:

[0223] |ΔRMSE|<ε

[0224] where ΔRMSE is the change of RMSE, and ε is a given threshold value; when |ΔRMSE|<ε, the iteration is stopped, and the optimal transformation parameter is obtained; otherwise, the iteration process is repeated, and the transformation parameter is continuously adjusted.

[0225] Specifically, ε is set to 10 -5 .

[0226] S6: The optimal absolute coordinate matrix is obtained by performing Helmert transformation on the optimal relative coordinate matrix again through the optimal transformation parameter, so as to obtain the accurate positions of all nodes;

[0227] In one specific embodiment, the method for obtaining the accurate positions of all nodes comprises:

[0228] The optimal transformation parameter and the optimal relative coordinate matrix are input into formula (36) to obtain the optimal absolute coordinate matrix.

[0229] Embodiment 2

[0230] Reference is made to Figure 1 The embodiment provides a relative positioning method based on SMACOF and progressive Helmert transformation, and the method comprises the following steps:

[0231] The weighted Euclidean distances between nodes are calculated to obtain a weighted Euclidean distance matrix, wherein the nodes comprise unknown nodes and anchor nodes;

[0232] The relative coordinates of all nodes are calculated by an MDS positioning algorithm based on the weighted Euclidean distance matrix to obtain a relative coordinate matrix;

[0233] An optimal solution is obtained by an SMACOF algorithm based on the relative coordinate matrix to obtain an optimal relative coordinate matrix;

[0234] The optimal relative coordinate matrix is mapped to a global coordinate by Helmert transformation to obtain an absolute coordinate matrix;

[0235] The absolute coordinate matrix is optimized by a progressive correction mechanism to obtain an optimal transformation parameter;

[0236] The optimal relative coordinate matrix is subjected to Helmert transformation again by the optimal transformation parameter to obtain an optimal absolute coordinate matrix, so that the accurate positions of all nodes are obtained.

[0237] Specifically, it is assumed that there is a UAV group in a three-dimensional space, and part of the UAVs cannot receive satellite signals due to signal shielding or other reasons, so that the UAVs cannot be positioned by the carried GPS. Therefore, cooperative positioning is required by other UAVs. It is assumed that the UAV group has N UAVs, the real coordinate matrix is X = [x1, x2, …, xN], and the optimal relative coordinate matrix is X = [x1, x2, …, xN]. N ] T ∈R N×3 , wherein x i = [a i ,b i ,c i ] T is the three-dimensional coordinates of the ith UAV. The Euclidean distance d ij between the ith UAV and the jth UAV can be expressed as:

[0238]

[0239] The Euclidean distance matrix D between all N drones can be written as:

[0240]

[0241] Where d i,j =d j,i And d i,i =0, therefore D is a symmetric matrix. The drones measure distances from each other; the i-th drone measures the distance between itself and the j-th drone. It can be represented as:

[0242]

[0243] in Distance measurement error that follows a Gaussian distribution The variance of the distance is calculated. After all UAVs have completed ranging, they send the measured distances to the central UAV. The central UAV then uses the received distance measurements to construct an estimated distance matrix. In obtaining the distance matrix Then, the relative coordinate matrix Y = [y1, y2, ..., y3] of all UAVs is obtained by centering and eigenvalue decomposition using the classic MDS algorithm. N ] T ∈R N×3 Then, SMACOF optimization is performed to obtain the optimal solution for the relative coordinate matrix.

[0244] After obtaining the relative coordinates, i.e., the global coordinates, of all drones, it is necessary to transform the global coordinates into absolute coordinates. Assume that the GPS coordinates of M drones in the drone swarm are available, where M ≤ N. When M < 3, only relative positioning of all drones can be achieved; only when M ≥ 3 can the absolute coordinates of all drones be obtained. Let the GPS coordinate matrix of these M drones be G = [g α1 ,g α2 ,…g αM ] T ∈R M×3 , where g αi For the αth i Given the GPS coordinates of M drones, let the global coordinate matrix of these M drones be Z. M =[z β1 ,z β2 ,…z βM ] T ∈R M×3 , where z βi For the βth iThe global coordinates of the UAVs. Initial values of Helmert transformation parameters are obtained by using GPS coordinates and global coordinates of M UAVs. By analyzing the residual error between the actual observation distance value and the distance calculated by relative coordinates, the optimal solution of scale factor S, rotation matrix R and translation variable T is obtained by using progressive correction mechanism. Finally, the absolute coordinates of the UAV group can be obtained by Helmert coordinate transformation of the global coordinates Z O

[0245]

[0246] Finally, the UAV cluster network realizes the positioning of all UAVs.

[0247] Embodiment 3

[0248] A computer readable storage medium, having stored thereon a computer program, the computer program being executed by a processor to implement the steps of the relative positioning method based on SMACOF and progressive Helmert transformation in embodiment 1.

[0249] Embodiment 4

[0250] A computer device, comprising a storage medium, a processor and a computer program stored in the storage medium and executable by the processor, the computer program being executed by the processor to implement the steps of the relative positioning method based on SMACOF and progressive Helmert transformation in embodiment 1.

[0251] Obviously, the above embodiments of the present application are merely exemplary and are not intended to limit the embodiments of the present application. Based on the above description, other different forms of changes or variations can be made by those skilled in the art. It is not necessary or possible to exhaust all embodiments. Any modification, equivalent replacement and improvement within the spirit and principle of the present application shall be included in the protection scope of the claims of the present application.​

Claims

1. A relative positioning method based on SMACOF and progressive Helmert transformation, characterized in that, The method comprises: calculating the weighted Euclidean distance between nodes to obtain a weighted Euclidean distance matrix; wherein the nodes comprise unknown nodes and anchor nodes; calculating the relative coordinates of all nodes by MDS positioning algorithm based on the weighted Euclidean distance matrix to obtain a relative coordinate matrix; solving the optimal solution of the relative coordinate matrix by SMACOF algorithm to obtain an optimal relative coordinate matrix; mapping the optimal relative coordinate matrix to global coordinates by Helmert transformation to obtain an absolute coordinate matrix; optimizing the absolute coordinate matrix by a progressive correction mechanism to obtain optimal transformation parameters; performing Helmert transformation on the optimal relative coordinate matrix again by the optimal transformation parameters to obtain an optimal absolute coordinate matrix, thereby obtaining the accurate positions of all nodes; the method for optimizing the absolute coordinate matrix by the progressive correction mechanism to obtain optimal transformation parameters comprises: after the initial Helmert transformation is completed, calculating the converted positioning error; defining the residual error as follows: wherein is the actual observed range value, is the theoretical calculated distance between the transformed coordinates; By statistical analysis of the residual error between all nodes , the distribution of the current error is obtained: calculating the root mean square error of the overall error as an error evaluation index: wherein, is the number of node pairs, the positioning error of the initial Helmert transformation is evaluated according to an error evaluation index, and if the root mean square error does not meet a preset convergence condition, the transformation parameters are adjusted; the transformation parameters include a scale factor , a rotation matrix , and a translation variable , and the transformation parameter adjustment formula is as follows: in, Indicates the current transformation parameters. Indicates the transformation parameters after iteration; They are The learning rate, and the step size for adjustment; This represents the gradient of the translation vector with respect to the root mean square error; after each iteration adjustment, recalculating the residual error and the root mean square error to evaluate the current positioning error level, and when the root mean square error meets the preset convergence condition, stopping iteration to obtain the optimal transformation parameters; otherwise, repeating the iteration process to continue adjusting the transformation parameters.

2. The relative positioning method based on SMACOF and progressive Helmert transformation according to claim 1, characterized in that, The method for calculating the weighted Euclidean distance between nodes to obtain a weighted Euclidean distance matrix comprises: In The space of dimensionality contains anchor nodes and unknown nodes, whose coordinates are defined as follows: (1) The distance between each pair of points is weighted to obtain a weighted Euclidean distance between the first node and the second node as follows: d = w1d1+ w2d2+ w3d3+ w4d4+ w5d5+ w6d6​ (2) wherein, is the weight between the th node and the th node, the weight being defined by: (3) wherein denotes the standard deviation of the noise in each dimension denotes the standard deviation of the noise in each dimension the obtained weighted Euclidean distance matrix is as follows: (4) wherein and , the weighted Euclidean distance matrix is a symmetric matrix.

3. The relative positioning method based on SMACOF and progressive Helmert transformation according to claim 2, characterized in that, The method for calculating the relative coordinates of all nodes by MDS positioning algorithm based on the weighted Euclidean distance matrix to obtain a relative coordinate matrix comprises: By a second centering operation, the row and column sums are made zero. of zero; wherein the definition of the second centering operation is as follows: (5) wherein denotes the Hadamard power, is a centered matrix and is denoted by: (6) wherein denotes the identity matrix, denotes a column vector with all elements being one. define relative coordinates of the EDM is , the distance between the relative coordinates of the first node and the relative coordinates of the second node (7) right The eigenvalue decomposition is performed as follows: (8) wherein , is a diagonal matrix whose diagonal elements are the eigenvalues of is an eigenvector matrix whose columns are the eigenvectors corresponding to the eigenvalues of extracting the three largest eigenvalues of , , the diagonal matrix consisting of the eigenvector matrix consisting of the corresponding eigenvectors the relative coordinates of the nodes are expressed as: (9) calculating the relative coordinates of the nodes, obtaining a relative coordinate matrix: (10)。 4. The relative positioning method based on SMACOF and progressive Helmert transformation according to claim 3, characterized in that, The method for solving the optimal solution of the relative coordinate matrix by SMACOF algorithm to obtain an optimal relative coordinate matrix comprises: defining the target function to be optimized as follows: (11) decomposing the target function into three parts: (12) where is a constant term, is a convex quadratic function with a minimum value; the third term is bounded according to the Cauchy-Schwarz inequality: (13) The Cauchy-Schwarz inequality is specifically as follows: (14) where is a constant; when the inequality in equation (13) becomes an equality, yielding: (15) convert formula (15) into a matrix form: (16) For equation (16) in , we have: (17) wherein , , the remaining terms equal 0, then: (18) wherein is the trace of the matrix is the Hessian matrix of the function in the form (19) For equation (16) in we have: (20) wherein: (21) then: (22) In summary: (23) According to formula (15), we have: (24) by optimizing iteratively solving : (25) solving (25) has: (26) wherein is the generalized inverse of i.e. the optimal solution of the relative coordinate matrix .

5. The relative positioning method based on SMACOF and progressive Helmert transformation according to claim 4, characterized in that, The method for mapping the optimal relative coordinate matrix to global coordinates by Helmert transformation to obtain an absolute coordinate matrix comprises: defining the following Helmert transformation formula: (27) wherein is a local coordinate of the total nodes, is a global coordinate of the device, is a rotation parameter, is a scaling parameter, is a translation parameter; The Helmert transformation comprises the following steps: First, the centroid nodes of the local and global coordinates of the anchor nodes are calculated, set and are the centroid nodes of the relative and absolute coordinates of the anchor nodes; therefore, and The calculation formula is as follows: (28) (29) The formula for finding the anchor node new coordinate system with the centroid node is as follows: (30) (31) Let is a rotation matrix, defined as follows: (32) Let , , and , , be the eigendecomposition computed by , so the rotation matrix is computed from equation (33): (33) The calculation formula of the scale factor is shown in formula (34): (34) Translation matrix Calculated from equation (35): (35) Finally, the optimal relative matrix of coordinates for all nodes its corresponding absolute matrix of coordinates is calculated using equation (36) (36)。 6. The relative positioning method based on SMACOF and progressive Helmert transformation of claim 1, wherein, The preset convergence condition is expressed as: ＜ wherein is the amount of change, is a given threshold value; when the iteration is stopped and the optimal transformation parameters are obtained; otherwise the iteration process is repeated and the adjustment of the transformation parameters is continued.​ 7. The relative positioning method based on SMACOF and progressive Helmert transformation of claim 1, wherein, The method for performing Helmert transformation on the optimal relative coordinate matrix again by the optimal transformation parameters to obtain an optimal absolute coordinate matrix, thereby obtaining the accurate positions of all nodes comprises: inputting the obtained optimal transformation parameters and the optimal relative coordinate matrix into formula (36) to obtain an optimal absolute coordinate matrix.

8. A computer readable storage medium having stored thereon a computer program, characterized in that: The computer program is executed by a processor to implement the steps of the relative positioning method based on SMACOF and progressive Helmert transformation according to any one of claims 1 to 7.

9. A computer device, comprising: Computer program product, comprising a storage medium, a processor and a computer program stored in the storage medium and executable by the processor, which, when executed by the processor, carries out the steps of a method of relative positioning based on SMACOF and progressive Helmert transformation according to any one of claims 1 to 7.

Images