CN112566239A

CN112566239A - Anchor node selection and deployment method, system, storage medium, equipment and application

Info

Publication number: CN112566239A
Application number: CN202011237151.0A
Authority: CN
Inventors: 齐小刚; 范英盛; 刘立芳
Original assignee: Xidian University
Current assignee: Xidian University
Priority date: 2020-11-09
Filing date: 2020-11-09
Publication date: 2021-03-26
Anticipated expiration: 2040-11-09
Also published as: CN112566239B

Abstract

The invention belongs to the technical field of three-dimensional space trilateral positioning, and discloses a method, a system, a storage medium, equipment and application for selecting and deploying an anchor node; the initial anchor node broadcasts own position information, and the unknown node which collects enough information executes an anchor node selection algorithm; the unknown node which completes positioning updates whether the node to be measured completes positioning, the initial value is 0, the node to be measured is updated to 1 after positioning is completed, the node becomes an anchor node in the next positioning round, and the position information of the node to be measured is broadcasted; and the unknown nodes which do not finish positioning in the current round enter a waiting state, and the positioning is finished after enough information is received in the next round of positioning. According to the method, the anchor node is selected on the hollow sphere with unknown nodes based on the condition number of the tetrahedron, so that the problem of iteration error in trilateral positioning can be effectively suppressed; and the method can also be used for the deployment of anchor nodes in trilateration.

Description

Anchor node selection and deployment method, system, storage medium, equipment and application

Technical Field

The invention belongs to the technical field of three-dimensional space trilateral positioning, and particularly relates to an anchor node selection and deployment method, an anchor node selection and deployment system, a storage medium, equipment and application.

Background

At present: location information is the basis on which many clustered networks, such as robotic networks, car networking, unmanned plane clusters, etc., provide services. The location information includes absolute location information, such as geodetic coordinates, and relative location information, such as a relative coordinate system established with the clustered network service area as a reference frame. One possible scheme for acquiring absolute position information is to install a GPS or BDS receiver for each node of the cluster network, but for the cluster network deployed in indoor, underwater, and regional rejection environments, GPS signals are easily affected by factors such as multipath interference and shielding, and often only a few external nodes can acquire accurate absolute position information. Similarly, the above-described problem also exists in acquiring the relative position information.

Current positioning techniques for obtaining location information can be largely divided into two categories, one based on hop count and the other based on relative measurements. The hop-based positioning algorithm avoids the dependence on complex measurement hardware, but the hop count is used for replacing measurement information to estimate the node position, so that the positioning accuracy cannot be ensured. The method for assisting the network node positioning by using the relative measurement information obtained in the communication interaction process among the cluster network nodes is a new idea for realizing high-precision positioning under the condition of GPS deficiency. The relative measurement information includes relative distance measurement, relative angle measurement, relative velocity measurement, and the like. The invention is only directed to positioning based on relative distance measurements. The currently common ranging techniques mainly include: the radio frequency signal strength indicates the RSSI of the model, the technology cost is low but the ranging precision is unstable; time of arrival measurement model TOA and Ultra Wide Band (UWB). Positioning algorithms based on ranging can be mainly classified into two categories, namely single-point positioning and cooperative positioning. Trilateral positioning or multilateral positioning is the basis of a single-point positioning algorithm, is completely distributed, has a simple algorithm, can be completed only by local communication, and has the problems of difficult starting, early shutdown and iteration error. The positioning algorithm based on the rigid graph relaxes the requirement of positioning a single point, thereby partially solving the problem that the algorithm cannot be started due to the sparse anchor nodes. The rigid graph theory mainly aims at a two-dimensional network, and is poor in applicability to a three-dimensional network. The advantage of cooperative positioning over single-point positioning is that the relative measurement information between the cluster network nodes can be used to give an overall solution to the relative positions of the cluster network nodes. Co-location can be modeled as solving a non-convex optimization problemThe solving method mainly includes an algorithm based on semi-definite programming (SDP) and an algorithm based on MDS (Multi-dimensional Scaling). The SDP-based solving method has the basic idea that the original non-convex optimization problem is relaxed into the convex optimization problem, and then the convex optimization problem is solved through the algorithms such as an interior point method, simulated annealing and the like, the problem of the SDP algorithm is that the calculation complexity is too high, and the calculation complexity still reaches O (n) under the ideal condition³)-O(n⁴). Multidimensional Scaling (MDS) is a data analysis technique, and based on the different methods of solving the pressure function of MDS, the positioning algorithm based on MDS can be divided into two categories, one category is classical MDS and the other category is iterative MDS. Although the positioning algorithm based on MDS can partially solve the problems of starting difficulty and iteration error existing in single-point positioning, the calculation complexity of the algorithm is at least O (n)³) It is still a centralized algorithm in nature.

Through the above analysis, the problems and defects of the prior art are as follows:

(1) the cooperative positioning technology has the problems of high calculation complexity and high communication complexity;

(2) although the computation and communication complexity of the single-point positioning technology is lower than that of the cooperative positioning technology, the problem of iterative error exists;

(3) there is a lack of solutions for deploying anchor nodes in three-dimensional space to improve positioning accuracy.

The difficulty in solving the above problems and defects is: evaluating the geometric shape of the anchor node combination in the three-dimensional space, and selecting the anchor node based on the evaluation; an anchor node deployment method in a three-dimensional space.

The significance of solving the problems and the defects is as follows:

(1) a condition number based tetrahedral shape measurement method is proposed, which is based on algebraic operations. Compared with an evaluation method based on geometric operation, the method is simple in calculation and is suitable for shape measurement of plane triangles;

(2) the method for selecting the anchor nodes on the three-dimensional space unknown node hollow sphere can effectively inhibit iteration errors.

(3) The condition number based tetrahedral shape measurement method can guide the deployment of three-dimensional space anchor nodes.

Disclosure of Invention

Aiming at the problems in the prior art, the invention provides an anchor node selection and deployment method, system, storage medium, device and application.

The invention is realized in such a way that an anchor node selection and deployment method comprises the following steps:

in the initial stage, a node and a neighbor in a network perform distance measurement mutually, and store distance information between the node and the neighbor;

the initial anchor node broadcasts own position information, and the unknown node which collects enough information executes an anchor node selection algorithm;

the unknown node which completes positioning updates whether the node to be measured completes positioning, the initial value is 0, the node to be measured is updated to 1 after positioning is completed, the node becomes an anchor node in the next positioning round, and the position information of the node to be measured is broadcasted;

and the unknown nodes which do not finish positioning in the current round enter a waiting state, and the positioning is finished after enough information is received in the next round of positioning.

Further, the anchor node selection and deployment method rules:

(1) tetrahedral condition number cond formed by four anchor nodes_sum＜cond_max；

(2) The volume of a tetrahedron formed by the four anchor nodes is as large as possible;

(3) the distances from the unknown node to the four anchor nodes are kept as consistent as possible.

Further, the anchor node selection and deployment method selects the anchor node on a hollow sphere with an unknown node as a sphere center, the node to be detected is P, R is the communication radius of the node, R is the inner radius of the hollow sphere, and the red marked node is a neighbor node of P in the hollow sphere.

Further, the anchor node selection and deployment method comprises the steps of firstly setting r, reducing the number of neighbor anchor nodes of P, and selecting on a hollow sphere; for in three-dimensional spaceIf node i selects the most suitable combination among all neighboring anchor nodes, trilateration of

A second combining operation, N (i) is the number of neighbor anchor nodes of the node i; if node i selects the anchor node on the hollow hemisphere, n (i) is reduced to n (i)' ═ n (i) · (R)³-r³)/(R³) (ii) a Anchor nodes are selected on the hollow hemispheroid, so that d can be ensured as much as possible_i(i 1.., 4.) there is no excessive difference, and rule (3) is satisfied.

Further, the anchor node selection and deployment method selects 4 of the anchor nodes on the hollow sphere to position P according to rules (1) and (2). For rule (1), the tetrahedral condition number cond consisting of four anchor nodes_sum＜cond_max(ii) a Wherein cond_maxThe method is characterized in that the method is a threshold value of a tetrahedron condition number, a tetrahedron with the condition number smaller than the threshold value is selected, and an anchor node pair P corresponding to the tetrahedron with the largest volume is selected from all anchor nodes meeting the rules (1) and (3) for positioning.

Further, the anchor node selection and deployment method specifies | | · | |, which represents a two-norm, i.e.

According to the property of the matrix norm, the absolute AB is less than or equal to the absolute A.The absolute B is absolute;

when only ranging errors are present, the errors are reflected on the column vector b, which then becomes:

rewriting the formula AX ═ b as:

A(X+ΔX)＝b+Δb；

where Δ X is a column vector representing the positioning error, Δ b is expressed as:

obtained from a (X + Δ X) ═ b + Δ b:

ΔX＝A^-1(Δb)

let AX ═ b and Δ X ═ A^-1The norm is taken at the two sides of (delta b) simultaneously to obtain 1/| X | | < | | A |/| b | | and | | delta X | < | | A |^-1| | Δ b | |, to obtain:

in the formula, | Δ X |/| X | |, is a relative error of the solution, | Δ b |/| b | |, is a relative error of the column vector b, | | a |^-1The term "| · | | A | |" refers to the condition number of the matrix A, and is used as cond (A) | | | A |^-1Expressed as | · | | a | |; the condition number is a measure of whether a matrix is ill or not. | | A^-1The relative error of | | | Δ X | |/| | X | | | | of | | · | | a | | and | | | Δ b | |/| b | | | will determine the upper bound of the solution;

when the position and the range of the anchor node have errors, the errors are reflected on the coefficient matrix a and the column vector b, and a (X + Δ X) ═ b + Δ b is rewritten as:

Obtaining:

taking norm at two sides to obtain:

substituting | | | AX | | | b | | | into

Obtaining:

by

Obtaining:

observation of

On the right end, the relative error of the solution is the relative error of A, and the upper bound of the relative error of the solution is determined by the conj (A) and the absolute delta b absolute/| b absolute | under the condition that the absolute delta A absolute is smaller;

four coefficient matrices are obtained:

definition 1: four anchor nodes form a tetrahedron in three-dimensional space, and cond (A)₁)+cond(A₂)+cond(A₃)+cond(A₄) Defined as the condition number of four sides, using cond_sumDenotes, cond_sumDepicting the coplanarity degree of the four anchor nodes, and when the four anchor nodes form a regular tetrahedron, cond_sumMinimum value of 8 is obtained, and the minimum condition number of the right-angle tetrahedron is 12.1962; all unknown nodes directly communicate with any anchor node, the distance measurement error is delta d-U (-1,1), the accuracy of position estimation is evaluated by using the average absolute error MAE, and the MAE is defined as follows:

wherein the content of the first and second substances,

refers to the estimated position, X, of the ith node_i＝(x_i,y_i,z_i) Refers to the true position of the ith node;

if the distance is less than the distance, the coordinates of the anchor nodes are accurate, and only the distance measurement error exists, according to the condition that the distance is less than the distance, the distance measurement error is less than the distance^-1(Δb)，

||ΔX||≤||A^-1||·||Δb||；

Will be provided with

The following can be obtained:

Δ d is a random variable and Δ d is U (- ε, ε) since

The value of (c) is small, so:

for a tetrahedron T (T)₁,t₂,t₃,t₄) Determining four coefficient matrixes corresponding to the delta b respectively₁，Δb₂，Δb₃And Δ b₄The position of the first and second electrodes, in fact,

corresponds to Δ b₄；

For the

Taking F norm at two sides simultaneously to obtain:

the same can be obtained:

and

it is also a random variable, since E (Δ D) ═ 0 and D (Δ D) ═ ε²And/3, having:

make it

The mathematical expectation shown takes a minimum, which translates into the following nonlinear programming problem:

wherein

Objective function

The matrix is positively determined,

the method is a convex function, inequality constraints are also convex functions, so that the minimum problem is convex programming, K-T points are necessary to be a global optimal solution, and the K-T condition of the problem is expressed as follows:

ω_i≥0,i＝1,2…5

wherein

Is a gradient operation, and the K-T point is obtained by solving

It is a further object of the invention to provide a computer device comprising a memory and a processor, the memory storing a computer program which, when executed by the processor, causes the processor to perform the steps of:

It is another object of the present invention to provide a computer-readable storage medium storing a computer program which, when executed by a processor, causes the processor to perform the steps of:

Another object of the present invention is to provide an anchor node selection and deployment system implementing the anchor node selection and deployment method, the anchor node selection and deployment system including:

the distance information storage module is used for measuring distance between a node and a neighbor in the network in an initial stage and storing distance information between the node and the neighbor;

the information collection module is used for broadcasting the position information of the initial anchor node and executing an anchor node selection algorithm by the unknown node which collects enough information;

the positioning module of the node to be measured is used for updating the unknown node which completes positioning whether the positioning of the node to be measured is completed or not, the initial value is 0, the node to be measured is updated to 1 after the positioning is completed, the node to be measured becomes an anchor node in the next positioning round, and the position information of the node to be measured is broadcasted;

and the incomplete positioning node processing module is used for entering a waiting state for an unknown node which does not complete positioning in the current round, waiting for receiving enough information in the next round of positioning and completing the positioning.

Another objective of the present invention is to provide an information data processing terminal, where the information data processing terminal is used to implement the anchor node selection and deployment method, and the information data processing terminal is a robot network terminal, an internet of vehicles terminal, or an unmanned aerial vehicle cluster terminal.

By combining all the technical schemes, the invention has the advantages and positive effects that: in the error analysis process of the three-dimensional space trilateral positioning algorithm, the invention constructs a new tetrahedron shape measurement method based on the tetrahedron condition number. The method is based on algebraic operation, is simpler than the traditional method based on geometric operation, and is unified for the shape measurement of a plane triangle and the shape measurement of a space tetrahedron. For trilateral positioning of a three-dimensional space, a method for selecting an anchor node on an unknown node hollow sphere based on a tetrahedral condition number is designed, and the problem of iteration errors in trilateral positioning can be effectively suppressed. Furthermore, the calculation of the condition number of the tetrahedron can be used for deployment of anchor nodes in trilateration.

In the invention, the tetrahedral condition number cond is defined in the error analysis of three-side positioning of the three-dimensional space_sumAnd the method can be used for evaluating the quality of the tetrahedron. Simulation experiments show that cond_sum，θ_minρ and η are similar when assessing tetrahedral quality. cond_sumIs based on algebraic operation, and is unified in evaluating the triangle of two-dimensional plane and the tetrahedron of three-dimensional space. Furthermore, a method TSM for selecting an anchor node on an unknown node hollow hemisphere is provided, simulation experiments show that the method can effectively restrain the problem of iteration errors in single-point positioning, and the positioning accuracy is equivalent to that of an improved LS-6 algorithm. The tetrahedral condition number and TSM algorithm have strong guiding significance for deploying anchor nodes in a three-dimensional space.

Drawings

In order to more clearly illustrate the technical solutions of the embodiments of the present application, the drawings needed to be used in the embodiments of the present application will be briefly described below, and it is obvious that the drawings described below are only some embodiments of the present application, and it is obvious for those skilled in the art that other drawings can be obtained from the drawings without creative efforts.

Fig. 1 is a flowchart of an anchor node selection and deployment method according to an embodiment of the present invention.

Fig. 2 is a schematic structural diagram of an anchor node selection and deployment system according to an embodiment of the present invention;

in the figure: 1. a distance information storage module; 2. an information collection module; 3. a node to be tested positioning module; 4. and an unfinished positioning node processing module.

Fig. 3 is a schematic diagram of single point positioning according to an embodiment of the present invention.

Fig. 4 is a schematic diagram of a trilateration model (3D space) provided by an embodiment of the present invention.

FIG. 5 is a diagram illustrating the relationship between the condition number and the MAE according to the embodiment of the present invention.

FIG. 6 is a schematic illustration of the condition numbers of tetrahedrons provided by an embodiment of the present invention.

Fig. 7 is a schematic diagram of anchor node selection according to an embodiment of the present invention.

Fig. 8 is a flowchart of an implementation of the anchor node selection and deployment method according to the embodiment of the present invention.

FIG. 9(a) is a schematic diagram of an embodiment of the present invention providing no short edge, but four points near coplanarity.

Fig. 9(b) is a schematic diagram of 1 short side provided in the embodiment of the present invention.

Fig. 9(c) is a schematic diagram of 2 short edges provided by the embodiment of the present invention.

Fig. 9(d) is a schematic diagram of 3 short edges provided by the embodiment of the present invention.

Fig. 10 is a schematic diagram of a situation provided by an embodiment of the present invention without a short side.

Fig. 11 is a schematic diagram of a short side according to an embodiment of the present invention.

Fig. 12 is a schematic diagram of two short sides provided by the embodiment of the present invention.

Fig. 13 is a schematic diagram of three short sides provided in the embodiment of the present invention.

Fig. 14(a) is a schematic diagram illustrating anchor nodes randomly distributed around a network according to an embodiment of the present invention.

Fig. 14(b) is a schematic diagram illustrating that anchor nodes are randomly distributed in the network according to the embodiment of the present invention.

Fig. 15(a) is a schematic diagram illustrating that the comparison anchor nodes of the algorithm MAE provided by the embodiment of the present invention are randomly distributed around the network.

Fig. 15(b) is a schematic diagram illustrating that the comparison anchor nodes of the algorithm MAE provided by the embodiment of the present invention are randomly distributed in the network.

Fig. 16(a) is a schematic diagram illustrating that anchor nodes are randomly distributed around a network due to the influence of connectivity on positioning accuracy according to the embodiment of the present invention.

Fig. 16(b) illustrates that the influence of connectivity on the positioning accuracy provided by the embodiment of the present invention is that anchor nodes are randomly distributed in the network.

Detailed Description

In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is further described in detail with reference to the following embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention.

In view of the problems in the prior art, the present invention provides a method, a system, a storage medium, a device and an application for selecting and deploying an anchor node, and the present invention is described in detail below with reference to the accompanying drawings.

As shown in fig. 1, the anchor node selection and deployment method provided by the present invention includes the following steps:

s101: in the initial stage, a node and a neighbor in a network perform distance measurement mutually, and store distance information between the node and the neighbor;

s102: the initial anchor node broadcasts own position information, and the unknown node which collects enough information executes an anchor node selection algorithm;

s103: the unknown node which completes positioning updates whether the node to be measured completes positioning, the initial value is 0, the node to be measured is updated to 1 after positioning is completed, the node becomes an anchor node in the next positioning round, and the position information of the node to be measured is broadcasted;

s104: and the unknown nodes which do not finish positioning in the current round enter a waiting state, and the positioning is finished after enough information is received in the next round of positioning.

Those skilled in the art of the anchor node selection and deployment method provided by the present invention may also implement other steps, and the anchor node selection and deployment method provided by the present invention in fig. 1 is only one specific embodiment.

As shown in fig. 2, the anchor node selection and deployment system provided by the present invention includes:

the distance information storage module 1 is used for measuring distance between a node and a neighbor in a network in an initial stage and storing distance information between the node and the neighbor;

the information collection module 2 is used for broadcasting the position information of the initial anchor node and executing an anchor node selection algorithm by the unknown node which collects enough information;

the positioning module 3 of the node to be measured is used for updating the unknown node which completes positioning whether the positioning of the node to be measured is completed or not, the initial value is 0, the node to be measured is updated to 1 after the positioning is completed, the node to be measured becomes an anchor node in the next positioning round, and the self position information is broadcasted;

and the unfinished positioning node processing module 4 is used for entering a waiting state for an unknown node unfinished in the current round of positioning to wait for receiving enough information in the next round of positioning so as to finish the positioning.

The technical solution of the present invention is further described below with reference to the accompanying drawings.

1. The GPS system and BDS system are also inherently a single point of location. Considering an application as shown in fig. 3, the node to be located is called an unknown node (unknown node); nodes with known locations, called anchor nodes, are used to assist unknown nodes in locating positions. In this application, anchor nodes can only be deployed around the perimeter of the network. Initially, the anchor node broadcasts its own location information, and due to the limitation of communication radius, only the unknown node outside the network obtains enough location information of the anchor node and distance information between the anchor node and the unknown node, thereby completing positioning. And the unknown node which completes positioning is used as an anchor node to assist the positioning of other unknown nodes in the network, and the positioning of the whole network is completed in a continuous iteration mode through the information interaction among the nodes.

Without prior information, determining the location of an unknown node requires at least four anchor nodes in three-dimensional space (two in two-dimensional space). When there are more than four anchor nodes around the unknown node, the position can be estimated using Least Squares (LS). The different ranging technologies have ranging errors and are smallerThe ranging error of (2) also brings about a large position estimation error and an iteration error. In trilateral positioning, assuming that n anchor nodes (n is more than 4) are arranged around an unknown node P, coordinate estimation of P can be obtained by using a least square method

In the experimental process, the invention finds that if traversing n anchor nodes, all possible combinations are found (3D network has

Seed), using trilateration to solve, the ratio is always obtained

More desirable results, of course, can be done because the invention has global information. Error analysis is carried out, and the influence of the ranging error on the positioning precision is found to be mainly dependent on the following two aspects. The first is the relative position of the four anchor nodes, namely the shape of a tetrahedron formed by the four anchor nodes, and the second is the relative position of the unknown node and the four anchor nodes.

The evaluation of tetrahedral quality was originally derived from finite element analysis in three-dimensional space. Tetrahedral meshes are often used in finite elements to solve partial differential equations, with better-shaped tetrahedrons facilitating the solution of the problem, while poorer-shaped tetrahedrons can lead to numerical computation difficulties. The conventional index for evaluating the quality of the tetrahedron is mainly calculated by a geometric method. Three frequently used quality assessment indicators, including the minimum solid angle θ_min(the minimum solidθ_min) Radius ratio ρ (the radius ratio ρ) and average ratio η (the mean ratio η). The three evaluation indexes all need complex geometric operation, and the indexes are difficult to be directly applied to evaluating the quality of the plane triangle. In the process of analyzing the trilateral positioning error in the three-dimensional space, a new index for evaluating the quality of a tetrahedron, namely a tetrahedron condition number cond, is provided from the algebraic point of view_sum. Compared with the evaluation method based on the geometric operation, the evaluation method based on the tetrahedral condition number has the following two advantages.Firstly, in the aspect of calculation, only the coordinates of four vertices of a tetrahedron need to be known to construct a coefficient matrix, and then the condition number is calculated according to the formulas (14) - (17), so that the calculation is simple. The second is that in the aspect of uniformity, the evaluation plane triangle and the spatial tetrahedron are completely uniform, and only the order of the constructed coefficient matrix is different. The main contributions of the present invention are as follows:

(1) in the error analysis process of three-dimensional space trilateral positioning, a tetrahedron condition number is defined, a new method for evaluating tetrahedron quality is constructed, and the method has guiding significance for the deployment of anchor nodes in the three-dimensional space;

(2) a method for selecting an anchor node on an unknown node hollow sphere is provided, and the method can effectively inhibit iteration errors.

2. Three-dimensional space trilateral positioning mathematical model

As shown in FIG. 4, P (x, y, z) represents the coordinates of the unknown node, t₁(x₁,y₁,z₁)、t₂(x₂,y₂,z₂)、t₃(x₃,y₃,z₃) And t₄(x₄,y₄,z₄) Respectively, anchor nodes within the communication radius R of the unknown node.

If the node P to be tested and the four anchor nodes t are known₁、t₂、t₃And t₄Are respectively d₁、d₂、d₃、d₄The coordinates of P can then be obtained by solving the following system of equations:

the commonly used method for solving the equation set (1) is to use the 1-type, 2-type and 3-type equations in the equation set (1) to make difference with the 4-type equations respectively, and eliminate the quadratic term to obtain a linear equation set:

to simplify the discussion, the present invention abstracts the equation set (2) into the form:

AX＝b (3)

where A is the coefficient matrix, X is the coordinates of the unknown node, and b is a set of column vectors.

In practice, ranging is error-bearing. Such as anchor nodes anchor₁The actual Euclidean distance from the node P to be measured is d₁And the observation distance is d₁', error of distance measurement Deltad₁＝d₁-d₁'. In this case the four balls do not intersect at a point as in fig. 3, but at an area. When two spheres intersect, the equations of the two spheres are subtracted to obtain the equation of the plane in which the intersecting circles are located. Essentially, equation set (2) is an equation for three planes, and solving equation set (2) results in the coordinates of the intersection of the three planes. Of course, (2) if there is a unique solution, rank (a) ═ 4 must be satisfied, where rank (a) represents the rank of coefficient matrix a. The present invention always assumes that a is non-singular.

3. Error analysis

When the range error Δ d exists, the coordinates of the point P obtained by solving the equation set (2) are also inaccurate. Even if the range error Δ d is relatively small, the P-point coordinates obtained by different anchor node combinations have relatively large differences. Looking at the examples in table 1: (x, y, z) refers to the coordinates of three anchor nodes, the third column is the actual distance between the node to be measured and the anchor node, and the fourth column is the observed distance with the ranging error.

Table 1 error examples

The exact coordinates of the P point solvable with trilateration based on actual distance are (99.4487, 89.1365, 98.0738), while the P point coordinates resolved based on observed distance are (95.0634; 96.6149; 49.0440). It can be seen that the difference between the actual distance and the observed distance is small, but the solved coordinates, particularly the Z-axis, are approximately 50 meters apart. It is important to investigate which factors will influence the solution of equation set (2).

As can be seen from equations (2) and (3), the elements in the coefficient matrix a are completely determined by the coordinates of the four anchor nodes, i.e., the distribution of the four anchor nodes in the three-dimensional space. The elements in the column vector b are completely determined by the coordinates of the four anchor nodes and the distances from the nodes to be measured to the anchor nodes, that is, the relative positions of the four anchor nodes and the nodes to be measured. According to the invention, error analysis is carried out on the trilateral positioning of the 3D space from two angles of the distribution of the four anchor nodes and the relative positions of the node to be measured and the four anchor nodes. The invention assumes that the coordinates of the four anchor nodes are accurate, i.e. the coefficient matrix a of equation (3) is accurate, only in the case of ranging errors. The present invention discusses the case when the coordinates and range of the four anchor nodes are both in error.

3.1 influence of distribution of Anchor nodes on positioning accuracy

In the present invention, if not specifically stated, the present invention provides that | · | | |, represents a two-norm, that is

According to the property of matrix norm, the invention has | | | AB | | < | | A | | · | | | | | B | | |.

3.1.1 cases where there is only a range error

When only the ranging error exists, the error is mainly reflected on the column vector b, and then the column vector b becomes:

the present invention can rewrite the formula (3) as follows:

A(X+ΔX)＝b+Δb (5)

where Δ X is a column vector representing the positioning error, Δ b can be expressed as:

from the formula (5):

ΔX＝A^-1(Δb) (7)

the norm is taken at the same time at the two sides of (3) and (7) to obtain 1/| X | | less than or equal to | A | |/| b | | and | | delta X | | | less than or equal to | A |^-1I | Δ b | from which we can derive:

in the formula (8), | | | Δ X |/| X | | | is a relative error of the solution, | | Δ b |/| b | | is a relative error of the column vector b, | | | a^-1The term "| · | | A | |" refers to the condition number of the matrix A, and is used as cond (A) | | | A |^-1Expressed as | · | | a |. The condition number is a measure of whether a matrix is ill or not. | | A^-1The relative error of | | | Δ X | |/| X | | | | will be determined by | | · | | | a | | | and | | | Δ b | |/| b | | |.

3.1.2 cases where both anchor node position and ranging are in error

When the position and the range of the anchor node have errors, the errors are mainly reflected on the coefficient matrix A and the column vector b, and the formula (5) is rewritten as follows:

the invention assumes that the | | | delta A | | is smaller and can satisfy | | | A^-1| is | · | | Δ a | < 1. Because if the | | Δ a | | | is relatively large, it means that the coordinate error of the anchor node itself is very large. Substituting formula (3) into formula (9) can obtain:

(10) the norm of both sides of the formula can be obtained:

substituting | | | AX | | | | b | | | into (11) can obtain:

from (12), it can be obtained:

observing the right end of the formula (13), i | Δ a |/| a | |, is the relative error of a, and under the condition that | | Δ a |, is smaller, similar to the discussion in 3.11, the upper bound of the relative error of solution | | Δ X |/| X |, is determined by cond (a) and | | | Δ b |/| | | b |.

3.1.3 Effect of Condition number on positioning accuracy

The relative error | | | Δ X |/| X | | | upper bound of the solution from (8) and (13) depends on the condition number cond (a) of the coefficient matrix. It should be noted that, when solving the equation set (1), the linear equation set (2) and the corresponding coefficient matrix can be obtained by using the equation 1, the equation 2 and the equation 3 in the equation set (1) to respectively make a difference with the equation 4, and by the same reasoning, the linear equation set and the coefficient matrix can also be obtained by using the equation 1, the equation 2 and the equation 4 in the equation set (1) to respectively make a difference with the equation 3, and so on, the invention can obtain four coefficient matrices:

definition 1: four anchor nodes form a tetrahedron in three-dimensional space, and cond (A) is connected in the invention₁)+cond(A₂)+cond(A₃)+cond(A₄) Defined as the condition number of four sides, using cond_sumAnd (4) showing. cond_sumThe coplanarity of the four anchor nodes is characterized. When four anchor nodes form a regular tetrahedron, cond_sumThe minimum value was taken to be 8. By comparison, the minimum condition number of a right-angled tetrahedron is 12.1962. In most cases, the invention is difficult to find the regular tetrahedron formed by four anchor nodes. To illustrate the influence of the condition number on the positioning accuracy, the following simulation experiment is performed: n1000 unknown nodes are randomly deployed in a 100 x 100 network. Two groups of anchor nodes are arranged, and the coordinates of the first group of anchor nodes are respectively as follows: t is t₁(25,25,25)，t₂(75,25,25)，t₃(50,68,25)，t₄(50,40,25+0.5i), i ═ 1,2 … 100. Wherein t is₄The z-axis coordinate of the anchor node is initially taken as 25.5, sampling is carried out once every 0.5, and sampling is carried out 100 times in total, so that the influence of the condition number of tetrahedrons formed by four anchor nodes on the positioning precision is observed. The second group of anchor node coordinates are respectively: t is t₁′(37.5,37.5,37.5)，t₂′(62.5,37.5,37.5)，t₃′(50,60,37.5)，t₄' (50,40,37.5+0.25i), i ═ 1,2 … 100. The second group of anchor nodes differs from the first group of anchor nodes in that the tetrahedron formed by the second group of anchor nodes is smaller in volume. The invention assumes that all unknown nodes can communicate directly with any anchor node, and the distance measurement error is delta d-U (-1, 1). The accuracy of the position estimate is evaluated by the mean absolute error MAE (mean absolute error), which is defined as follows:

wherein the content of the first and second substances,

refers to the estimated position, X, of the ith node_i＝(x_i,y_i,z_i) Refers to the true location of the ith node.

As shown in FIG. 5, the MAE tends to increase with increasing condition number. The MAE of the second group is close to the first group when the first group of anchor nodes and the second group of anchor nodes have similar condition numbersTwice the set, which means that the error of trilateral localization is not completely determined by the condition number of the tetrahedron of anchor nodes. In general, when the condition number is less than 45, the average error of node positioning is within 10. When the tetrahedron shape formed by the four anchor nodes is determined, cond_sumAs determined accordingly. As shown in fig. 5, a tetrahedron T' (T)₁′,t₂′,t₃′,t₄') is a tetrahedron T (T)₁,t₂,t₃,t₄) To an equal ratio abbreviation (e.g. side t)₁t₂Is of length t₁′t₂M times of' M > 1), cond_sum(T)＝cond_sum(T'). Assume four anchor nodes T (T)₁,t₂,t₃,t₄) The corresponding coefficient matrix is A, and the other four anchor nodes T' (T)₁′,t₂′,t₃′,t₄') the corresponding coefficient matrix is A'. Although the condition numbers for A and A' are the same, the effect on Δ X is different. The present invention knows that A is M.A ', then (A')^-1＝MA^-1And from the formula (7), it is apparent that the tetrahedron T (T)₁,t₂,t₃,t₄) Can reduce positioning errors, similar to the conclusions presented in fig. 5. Using the data in Table 1, the condition number of the coefficient matrix corresponding to the four anchor nodes is cond_sumThe large condition number, 438.3616, indicates that the coefficient matrix is ill-conditioned and very sensitive to disturbances. So that the range error is in the centimeter level, while the positioning error is indeed in the tens of meters.

3.2 influence of relative position of Anchor node and node to be measured on positioning accuracy

Assuming that a is smaller, the present invention only discusses the case in 3.1.1, i.e. assuming that the coordinates of the anchor nodes are all accurate, only ranging errors exist. According to the formula (7),

||ΔX||≤||A^-1||·||Δb|| (19)

substituting formula (6) into formula (7) can obtain:

Δ d is a random variable, and Δ d is U (- ε, ε). Due to | Δ d₁ ²-Δd₄ ²The numerical value of | is small, so the present invention assumes:

for a tetrahedron T (T)₁,t₂,t₃,t₄) Four coefficient matrices shown in equation (14) -equation (17) can be determined, which correspond to Δ b, respectively₁，Δb₂，Δb₃And Δ b₄. In fact, the expression (21) corresponds to Δ b₄。

Taking the F norm at the same time for both sides of formula (21) can obtain:

the same can be obtained:

and

as well as random variables. Since E (Δ D) is 0 and D (Δ D) is ε²The invention has the following components:

the invention can model the formula (23) as a process for selecting an anchor node for an unknown node, namely d_iWhen (i ═ 1, …,4) is taken, the minimum value of the formula (23) is desirable. It is to be noted here that if d is_i(i-1, …,4) is small, which means that the volume of the tetrahedron is small, which will enlarge a^-1Of (1). The present invention can assume d₁+d₂+d₃+d₄≥d_minUnder this assumption, to minimize the mathematical expectation shown in (23), equation (23) can be converted into the following nonlinear programming problem:

wherein

Objective function

The matrix is positively determined,

is a convex function, and the inequality constraint is also a convex function, so the minimum problem (24) is convex programming, and the K-T point is necessary to be a global optimal solution. The K-T condition for this problem can be expressed as:

wherein

Is a gradient operation. From the formula (25) can be solved to obtain the K-T point

According to the analysis, the distances between the unknown nodes and the four anchor nodes should be kept as consistent as possible.

4. Anchor node selection algorithm based on error analysis

4.1 Algorithm design rules

Based on error analysis, the anchor node selection algorithm of the three-dimensional space provided by the invention mainly follows the following three rules:

Based on the principle, the invention can select the anchor node on the hollow sphere with the unknown node as the sphere center. As shown in fig. 7, the node to be measured is P, R is the communication radius of the node, R is the inner radius of the hollow sphere, and the red labeled node is the neighbor node of P in the hollow sphere.

Firstly, r is set, the number of the neighbor anchor nodes of P is reduced, and only the hollow sphere shown in FIG. 7 is selected. In this way, on the one hand, the computational complexity is reduced. For trilateration in three-dimensional space, if node i selects the most appropriate combination among all neighboring anchor nodes, it is performed

The second combining operation, N (i), is the number of neighbor anchor nodes for node i. If node i selects the anchor node on the hollow hemisphere, n (i) is reduced to n (i)' ═ n (i) · (R)³-r³)/(R³). On the other hand, the anchor nodes are selected on the hollow hemispheroids, so that d can be controlled as much as possible_iThere is no excessive difference (i 1.., 4), which is mainly to satisfy rule (3).

For the anchor nodes on the hollow sphere as shown in fig. 6, 4 of them can be selected to locate P according to rules (1) and (2). For rule (1), the tetrahedral condition number cond consisting of four anchor nodes_sum＜cond_maxMainly to avoid too high coplanarity of the four anchor nodes. Wherein cond_maxIs a threshold for the condition number of tetrahedrons, and in general, the invention contemplates selecting tetrahedrons having a condition number less than the threshold. In all anchor nodes satisfying the rules (1) and (3), the anchor node P corresponding to the tetrahedron with the largest volume is selected by the method.

4.2 interpretation of parameters

The main parameters of the positioning algorithm of the present invention are shown in table 2:

TABLE 2 parameter and variable description

4.3 Algorithm flow

In the initial stage, the nodes in the network and the neighbors measure distance mutually and store the distance information of the neighbors. The initial anchor node broadcasts its own location information, and the unknown node that has collected sufficient information executes the anchor node selection algorithm, the algorithm flow being shown in fig. 7. The unknown node (i) that has completed positioning updates node (1) and flag1 becomes an anchor node in the next positioning round, and broadcasts its own position information; and the unknown nodes which do not finish positioning in the current round enter a waiting state, and the positioning is finished after enough information is received in the next round of positioning.

It should be noted that, in a relatively sparse network, the waiting time of some unknown nodes in finding the anchor node combinations satisfying the condition of fig. 8 is too long, and therefore, according to the needs of different applications, a waiting time threshold t may be set, and when the unknown nodes reach the waiting time threshold and still do not complete positioning, the condition in the flowchart 8 may be appropriately relaxed. This of course also means a reduction in the positioning accuracy.

The technical effects of the present invention will be described in detail with reference to simulations.

The simulation experiment is divided into three parts, the first part is designed by the invention to compare and evaluate four indexes of tetrahedron quality, cond_sum，θ_minρ and η. The second part is designed with simulation experiments to verify the effectiveness of the anchor node selection algorithm in three-dimensional positioning. The third part is to lay anchor nodes in three-dimensional space according to the tetrahedral condition number. The simulation software is Matlab2017a, and the simulation environment is a windows platform.

1. Condition number estimation of tetrahedron quality

Through the method of designing simulation experiment, theta is illustrated_minWhere ρ and η are "equivalent", cond is also described below by experimental methods in the present invention_sumIs also "equivalent". For poorly formed tetrahedrons, there are mainly four cases, whichThe tetrahedron of middle 9(a) has no short sides, but is "flatter" in shape. Among the methods for evaluating the quality of planar triangles, there is a shortest side length method, which is obviously not applicable to fig. 9(a), the tetrahedron shown in fig. 9(b) has only one short side, the tetrahedron shown in fig. 9(c) has a pair of edges which are both short sides, and the tetrahedron shown in fig. 9(d) has three short sides.

For the four cases as shown in FIG. 9, the present invention sets up four simulation experiments to illustrate θ respectively_minρ, η and cond_sumIs "equivalent". In order to make the dimensions of the four indexes consistent, the value interval is [0,1 ]]Invention pair cond_sumNormalized to make cond_sum′＝8/cond_sum(cond of regular tetrahedron_sumIs 8). For 7(a), the present invention sets up the following simulation experiment. Four vertex coordinates of tetrahedron: t is t₁＝(0,0,0)，t₂＝(1,0,0)，

Wherein u is (0, 1)]When u → 0, the tetrahedron becomes progressively "flattened". As shown in fig. 10, when u tends to 0, all of the four evaluation indexes tend to 0, and the four vertices tend to be coplanar. With the increase of u, the four evaluation indexes are gradually increased and tend to 1, and the graph tends to be a regular tetrahedron. cond_sumThe tendency of variation of (a) and (theta)_minMore closely.

For FIG. 9(b), the invention sets t₁＝(0,0,0)，t₂＝(1,0,0)，

Wherein u is (0, 1)]. The results of the experiment are shown in FIG. 11, with similar conclusions as in FIG. 10.

For FIG. 9(c) the invention sets t₁＝(0,0,0)，t₂＝(1,0,0)，

Wherein u is (0, 1)]. The results of the experiment are shown in FIG. 12.

For FIG. 9(d) the invention sets t₁＝(0,0,0)，t₂＝(u,0,0)，

Wherein u is (0, 1)]. The results of the experiment are shown in FIG. 13.

Through this part of the simulation experiment, cond was used_sumIt is effective to evaluate the quality of tetrahedron.

2. Anchor node selection algorithm based on tetrahedral measurement

Experiment main parameter, N-150 unknown nodes are randomly deployed in a 100 × 100 × 100 network. There are two deployment modes of m-50 anchor nodes, and fig. 14(a) illustrates that the anchor nodes are deployed around the network, which is considered because in many application scenarios, the anchor nodes can only appear around the scenario. The second anchor node deployment is shown in fig. 14(b), and is randomly distributed in the network as the unknown nodes. Node communication radius R epsilon [25,35 ∈ ]]And testing the influence of the connectivity on the positioning precision. The invention sets R to 0.5R, cond_sumAt 50, it should be noted that r and cond_sumAre empirical values obtained through simulation experiments. The distance measurement error is 1 percent of the side length of the network, namely delta d-U (-1, 1). The positioning accuracy was evaluated by MAE of the formula (14). The present algorithm (TSM) is mainly compared to Least Squares (LS) based positioning algorithms. In the experiments of the present invention, the LS-based algorithm was unstable. And if the tetrahedron formed by the four anchor nodes has a poor shape, the coordinate estimation of the unknown node has large deviation, and further large iteration errors are brought to subsequent positioning. To this end, the bookThe LS algorithm is simply improved, and the positioning can be completed only by at least six anchor nodes around an unknown node, and the improved LS algorithm is recorded as LS-6.

2.1 comparison of Algorithm MAE

In this experiment, the present invention takes R ═ 30. The MAE of the algorithm TSM, LS-6, LS all tends to increase with the number of iterations, which is caused by iteration errors. The iteration times of LS are less, but the MAE and the labeling difference are significantly larger than those of TSM and LS-6, which means the importance of the anchor node morphology in trilateration and the effectiveness of TSM and LS-6 in inhibiting the iteration error growth. When the anchor nodes are randomly distributed on the periphery of the network, the overall MAE of the TSM is 1.98m, which is slightly better than 2.2m of LS-6. This is because anchor nodes randomly distributed around the network are more flat in configuration, "selective use" is more efficient than "how much is used", which is seen from the first iteration of fig. 15(a), and the MAE of the first iteration of TSM is smaller than that of LS-6. When anchor nodes are randomly distributed in the network, the MAE of the three algorithms is reduced, wherein the overall MAE of the TSM, the LS-6 and the LS is 1.69m, 1.48m and 5.01m respectively. The MAE of LS-6 is preferred over the TSM because anchor nodes randomly distributed in the network are more evenly configured, "how much" is more efficient than "selective use". When the anchor nodes are randomly distributed in the network, the situation that the anchor nodes distributed at the edge of the network cannot be positioned due to less than 4 neighbor nodes can occur.

2.2 influence of connectivity of nodes on positioning accuracy

In the experiment, 25, 27.5, 30, 32.5 and 35 are taken as R respectively to test the influence of different node average connectivity degrees on the positioning accuracy. Taking the example that the anchor nodes are randomly deployed in the network, the node average degrees corresponding to different communication radii are shown in table 3 below:

TABLE 3 mean degree of nodes

Fig. 16(a) and 16(b) show the effect of different node averages on positioning accuracy. The average connectivity of the nodes increases with the increase of the communication radius, which means that the available information increases for the positioning algorithm based on LS, and the alternative node combinations increase for the TSM. The MAE of all three algorithms tended to decrease, but the MAE and standard deviation of LS were significantly greater than TSM and LS-6. In fig. 16(b), when R is 25, there are less than 4 neighbor nodes with a large number of unknown nodes and the positioning cannot be completed, and the communication radius R starts from 27.5 for the case where the anchor nodes are randomly distributed in the network. From FIGS. 16(a) and 16(b), a comparison of the two algorithms TSM and LS-6 has similar conclusions as 2.1. Generally, compared with the LS algorithm, the positioning accuracy of the TSM is remarkably improved and is equivalent to that of the LS-6 which is simply improved by the invention.

3. Arrangement of anchor nodes in three-dimensional space

Although the positioning accuracy of the TSM is close to LS-5 and the calculation complexity is higher than that of LS-5, the positioning algorithm based on the TSM has certain guiding significance for the deployment of the anchor node. Imagine a scenario where there is a 10 x 4 warehouse and the items within the warehouse need to know the relative coordinates, i.e. establish a relative coordinate system with the warehouse. Four anchor nodes are now ready to be deployed in the warehouse to meet the relative positioning requirements of the items, how these four anchor nodes should be deployed. A rectangular spatial coordinate system is established in a certain solid angle of the warehouse, and four anchor nodes form a tetrahedron under the coordinate system. Three deployment scenarios are readily envisioned as shown in table 4 below:

TABLE 4 deployment scenario for anchor nodes

It can be easily seen which tetrahedron has the same volume and the positioning effect is better? The invention calculates the condition numbers of three tetrahedrons respectively, wherein cond_sum(T₁)＝19.63，cond_sum(T₂)＝21.71，cond_sum(T₃) 14.06. According to the analysis of the invention, the tetrahedron T with the smallest condition number should be selected₃. Is composed ofThe conclusion of the invention is verified, and the following simulation experiment is designed, 5000 unknown nodes are randomly deployed in a space of 10 multiplied by 4, and the deployment of the anchor nodes is as T₁，T₂，T₃. The range error Δ d-U (-0.1,0.1), MAE for the three deployment scenarios are shown in Table 5 below:

TABLE 5 MAE for different deployment scenarios

From Table 5, T₃MAE of is significantly less than T₁And T₂The method also shows the guiding significance of the tetrahedral condition number provided by the invention on the three-dimensional space anchor node deployment. Of course, the concept of the condition number provided by the invention is uniform for two-dimensional planes and three-dimensional stereo, the invention can also define the condition number of two-dimensional plane triangles, and the invention has guiding significance for the deployment of two-dimensional plane anchor nodes.

In the invention, the tetrahedral condition number cond is defined in the error analysis of three-side positioning of the three-dimensional space_sumAnd the method can be used for evaluating the quality of the tetrahedron. The fifth section of simulation experiments shows that cond_sum，θ_minρ and η are similar when assessing tetrahedral quality. cond_sumIs based on algebraic operation, and is unified in evaluating the triangle of two-dimensional plane and the tetrahedron of three-dimensional space. Furthermore, the invention provides a method TSM for selecting an anchor node on an unknown node hollow hemisphere, and simulation experiments show that the method can effectively inhibit the problem of iterative error in single-point positioning, and the positioning precision is equivalent to the improved LS-6 algorithm. The tetrahedral condition number and TSM algorithm have strong guiding significance for deploying anchor nodes in a three-dimensional space.

It should be noted that the embodiments of the present invention can be realized by hardware, software, or a combination of software and hardware. The hardware portion may be implemented using dedicated logic; the software portions may be stored in a memory and executed by a suitable instruction execution system, such as a microprocessor or specially designed hardware. Those skilled in the art will appreciate that the apparatus and methods described above may be implemented using computer executable instructions and/or embodied in processor control code, such code being provided on a carrier medium such as a disk, CD-or DVD-ROM, programmable memory such as read only memory (firmware), or a data carrier such as an optical or electronic signal carrier, for example. The apparatus and its modules of the present invention may be implemented by hardware circuits such as very large scale integrated circuits or gate arrays, semiconductors such as logic chips, transistors, or programmable hardware devices such as field programmable gate arrays, programmable logic devices, etc., or by software executed by various types of processors, or by a combination of hardware circuits and software, e.g., firmware.

The above description is only for the purpose of illustrating the present invention and the appended claims are not to be construed as limiting the scope of the invention, which is intended to cover all modifications, equivalents and improvements that are within the spirit and scope of the invention as defined by the appended claims.

Claims

1. An anchor node selection and deployment method, characterized in that the anchor node selection and deployment method comprises:

2. The anchor node selection and deployment method of claim 1, wherein the rules of the anchor node selection and deployment method are:

3. The anchor node selection and deployment method of claim 2, wherein the anchor node selection and deployment method selects the anchor node on a hollow sphere with the unknown node as the center of the sphere, the node to be tested is P, R is the communication radius of the node, R is the inner radius of the hollow sphere, and the red labeled node is a neighbor node of P in the hollow sphere.

4. The anchor node selection and deployment method of claim 3, wherein the anchor node selection and deployment method first sets r, reduces the number of neighbor anchor nodes of P, selects on a hollow sphere; for trilateration in three-dimensional space, if node i selects the most appropriate combination among all neighboring anchor nodes, it is performed

5. The method of claim 3, wherein the anchor node selection and deployment method selects 4 of the anchor nodes on the hollow sphere according to rules (1) and (2) to locate P, and for rule (1), the tetrahedral condition number cond consisting of four anchor nodes_sum＜cond_max(ii) a Wherein, cond_maxThe method is characterized in that the method is a threshold value of a tetrahedron condition number, a tetrahedron with the condition number smaller than the threshold value is selected, and an anchor node pair P corresponding to the tetrahedron with the largest volume is selected from all anchor nodes meeting the rules (1) and (3) for positioning.

6. The anchor node selection and deployment method of claim 1 wherein the anchor node selection and deployment method specifies that | · | | | represents a two-norm, i.e., that is

rewriting the formula AX ═ b as:

A(X+ΔX)＝b+Δb；

obtained from a (X + Δ X) ═ b + Δ b:

ΔX＝A^-1(Δb)

in the formula, | Δ X |/| X | |, is a relative error of the solution, | Δ b |/| b | |, is a relative error of the column vector b, | | a |^-1The term "| · | | A | | | refers to a matrixCondition number of A, using cond (A) | | A^-1Expressed as | · | | a | |; the condition number is a measure of the morbidity or morbidity of a matrix, | | A^-1The relative error of | | | Δ X | |/| | X | | | | of | | · | | a | | and | | | Δ b | |/| b | | | will determine the upper bound of the solution;

Obtaining:

taking norm at two sides to obtain:

substituting | | | AX | | | b | | | into

Obtaining:

by

Obtaining:

observation of

four coefficient matrices are obtained:

wherein the content of the first and second substances,

||ΔX||≤||A^-1||·||Δb||；

Will be provided with

By substituting Δ X ═ A^-1(Δ b), we can obtain:

Δ d is a random variable and Δ d is U (- ε, ε) since | Δ d₁ ²-Δd₄ ²The value of | is small, so:

corresponds to Δ b₄；

For the

Taking F norm at two sides simultaneously to obtain:

the same can be obtained:

and

make it

wherein

Objective function

The Hesse matrix of (a) is positive,

wherein

Is a gradient operation, and the K-T point is obtained by solving

7. A computer device, characterized in that the computer device comprises a memory and a processor, the memory storing a computer program which, when executed by the processor, causes the processor to carry out the steps of:

8. A computer-readable storage medium storing a computer program which, when executed by a processor, causes the processor to perform the steps of:

9. An anchor node selection and deployment system for implementing the method of any one of claims 1 to 6, wherein the anchor node selection and deployment system comprises:

10. An information data processing terminal, characterized in that the information data processing terminal is used for realizing the anchor node selection and deployment method of any one of claims 1 to 6, and the information data processing terminal is a robot network terminal, a vehicle networking terminal or an unmanned aerial vehicle cluster terminal.