CN112954637B - Target positioning method under condition of uncertain anchor node position - Google Patents

Abstract

The present invention provides a target positioning method when the anchor node position is uncertain. After obtaining the TOA measurement information between the anchor node and the target node and the target node, the target function is listed, and the error amount of the anchor node is introduced. After the objective function is vectorized, the S-process is used to eliminate the anchor node error vector, and a convex constraint is introduced at the same time. Finally, the problem is transformed into a solvable convex problem through transformation and relaxation. When the positioning system is affected by the environment, resulting in a large deviation between the real position of the anchor node and the measured position, the method can still estimate the target position more accurately, and has strong robustness and practicability.

Description

A target localization method under the uncertainty of anchor node position

technical field

The invention relates to a target positioning method, belongs to the field of signal processing, and is suitable for a self-positioning system in which multiple anchor nodes locate multiple target nodes in a wireless sensor network.

Background technique

The flexibility, wide coverage, and ease of deployment of wireless sensor networks (WSNs) have attracted extensive attention in the past few years. Generally, a wireless sensor network consists of a group of low-cost and low-power sensor nodes distributed in a certain spatial range, which can be used to perform common signal processing tasks, such as detection, localization, and target tracking and monitoring of target state changes . Among them, target positioning is one of the most basic and important tasks in wireless sensor networks, and the acquisition of many physical quantities needs to be based on the premise of clear node positions. Many types of sensor measurements can be used for target location, such as Received Signal Strength (RSS), Angle of Arrival (AOA), Time of Arrival (TOA), Time Difference of Arrival (TDOA). Among the target localization methods based on these types of measurements, TOA and TDOA-based target localization schemes achieve a good balance between localization performance and computational complexity. These two localization methods can effectively avoid deploying expensive sensors like the AOA-based localization scheme, and can also effectively reduce the large localization error caused by the RSS-based localization method.

In the TOA positioning problem, there is often the problem that the anchor node and the target node to be positioned are out of synchronization. There are two main methods to eliminate the impact of time asynchrony. One is to convert the TOA positioning problem into a TDOA positioning problem; the other is to use dual Process TOA, that is, the anchor node and the target node perform two-way information exchange, and the time stamp corresponding to the anchor node and the target node is used to solve the positioning problem. In addition, most of the traditional models assume that the position of the anchor node is accurately known, but in practical applications, this condition is difficult to achieve, because due to the influence of the environment, the real position of the anchor node and the position obtained by measurement will always exist. error. For example, even if the position of a water buoy node is obtained using GPS in advance, the node will drift due to the influence of ocean currents during the positioning process, causing its real position to deviate from the measured position. If certain measures are not taken, the positioning performance will be degraded.

The difference between the real position coordinates of the anchor node and the measured position coordinates of the anchor node is the anchor node error vector. At present, there are two modeling methods for the anchor node error vector. One is to assume that the vector follows a Gaussian distribution with zero mean. . In practical applications, the latter case is more practical, because the maximum value of the anchor node error norm is easier to estimate than the Gaussian covariance matrix under different circumstances. However, most of the existing target localization methods do not consider the anchor node position error due to environmental factors, or assume that the anchor node position error vector obeys a Gaussian distribution with zero mean. In practice, the statistical distribution of errors is often difficult to obtain, and the maximum value of the anchor node error vector modulus is easier to obtain by estimation. Among the existing methods in which the maximum value of the error vector modulus is known, Xu et al. proposed a method in which multiple anchor nodes locate a single target node and only assume that the maximum value of the error vector modulus of the anchor node is known. The method only considers the positioning scene of a single node, and needs to add a penalty item. In the actual operation process, the penalty factor corresponding to the penalty item needs to be adjusted manually, which increases the computational complexity and reduces the practicability.

SUMMARY OF THE INVENTION

In order to overcome the deficiencies of the prior art, the present invention provides a method for locating a target when the position of the anchor node is uncertain, which only needs to know the maximum value of the modulus of the error vector of the anchor node, and does not need to know the statistical distribution of the error in advance, The degree of strictness of prior information requirements is weakened, and there is no penalty item that needs to be manually adjusted, which improves the practicability.

The technical scheme adopted by the present invention to solve its technical problem comprises the following steps:

The first step is to obtain TOA measurements between nodes in the sensor network and model the anchor node error;

In the second step, the anchor node error term is introduced into the original TOA measurement of the communication between the anchor node and the target node;

The third step is to transform the anchor node error term, and use the S-process to eliminate the anchor node error vector;

The fourth step is to transform the problem into a convex problem by transforming and relaxing the objective function and constraints;

The fifth step is to solve the convex optimization problem to obtain an estimate of the target position.

其中i,j分别表示编号为i的锚节点与编号为j的目标节点，c为信号传播速度，t_ij表示锚节点i与目标节点j相互通信所得到的时间量测，r_ij表示锚节点i与目标节点j之间的距离，e_ij表示锚节点i与目标节点j之间通信的加性高斯白噪声，e_ij服从均值为零、方差为

的高斯分布；目标节点之间相互通信获得的TOA量测表达式为

其中j和j'分别表示编号为j和j'的目标节点，且j≠j'，t'_jj'表示目标节点j与目标节点j'相互通信所得到的时间量测，r'_jj'表示目标节点j与目标节点j之间的距离，e'_jj'表示目标节点j与目标节点j之间通信的加性高斯白噪声，e'_jj'服从均值为零、方差为

的高斯分布；锚节点与目标节点之间的距离与节点位置之间的关系为r_ij＝||x_i-x_j||₂；目标节点与目标节点之间的距离与节点位置之间的关系为r'_jj'＝||y_j-y_j'||₂；锚节点真实位置与量测得到的位置之间的关系为

式中

表示量测得到的锚节点位置，ξ_i是锚节点误差向量，ξ_i的模的最大值小于等于一个已知的常数ε。In the first step, there are M anchor nodes with known positions but errors and N target nodes to be located in the sensor network, M≥3, and the real coordinates of the M anchor nodes are x ₁ , x ₂ ,… ,x _M , the coordinates of the N target nodes to be estimated are y ₁ , y ₂ ,...,y _N ; the TOA measurement set is obtained through the communication between nodes, including the TOA amount of communication between the anchor node and the target node The TOA measurement of the communication between the anchor node and the target node; the TOA measurement expression obtained from the communication between the anchor node and the target node is:

where i and j represent the anchor node numbered i and the target node numbered j respectively, c is the signal propagation speed, _tij represents the time measurement obtained by the mutual communication between the anchor node i and the target node _j , and rij represents the anchor node The distance between i and the target node j, e _ij represents the additive white Gaussian noise of the communication between the anchor node i and the target node j, e _ij obeys the mean value of zero and the variance is

The Gaussian distribution of ; the TOA measurement expression obtained by mutual communication between target nodes is:

where j and j' represent the target nodes numbered j and j' respectively, and j≠j', t'_jj' represents the time measurement obtained by the mutual communication between target node j and target node j', r'_jj' represents The distance between the target node j and the target node j, e'_jj' represents the additive white Gaussian noise of the communication between the target node j and the target node j, e'_jj' obeys the mean zero and the variance is

The Gaussian distribution of ; the relationship between the distance between the anchor node and the target node and the node position is r _ij =||x _i -x _j || ₂ ; The relationship is r'_jj' =||y _j -y _j' || ₂ ; the relationship between the real position of the anchor node and the measured position is

in the formula

Represents the measured anchor node position, ξ _i is the anchor node error vector, and the maximum value of the modulus of ξ _i is less than or equal to a known constant ε.

其中o(||ξ_i||)表示锚节点误差向量模||ξ_i||的高阶无穷小量；令

则推得|δ_ij|≤ε，其中

表示锚节点i与目标节点j之间的距离，与r_ij不同，这里的使用的锚节点i的坐标是已知的带有误差的锚节点位置，δ_ij表示锚节点i与目标节点j之间的模的误差值；则最终锚节点与目标节点之间通信的TOA量测表示为

The second step uses the first-order Taylor expansion to further transform the relationship between the real position of the anchor node and the measured position into

where o(||ξ _i ||) represents the higher-order infinitesimal of the anchor node error vector modulo ||ξ _i ||; let

Then it can be deduced that |δ _ij |≤ε, where

Represents the distance between anchor node i and target node j. Different from r _ij , the coordinates of anchor node i used here are the known anchor node positions with errors, and δ _ij represents the distance between anchor node i and target node j. The error value of the modulus between the final anchor node and the target node is expressed as

将第二步中的锚节点误差项代入，得到min-max次优化问题

经过对目标函数的向量化，以及通过应用S-过程消除锚节点误差向量，原始次优化问题被转化为

其中，λ和μ是为了转化锚节点误差向量新引进的常数，≥表示矩阵正定，I表示单位矩阵，Tr(·)表示求括号里面矩阵的迹，d₁＝[d₁₁,d₁₂,…,d_MN]^T为锚节点与目标节点通信的带噪声的距离量测，d₂＝[d'₁₂,d'₁₃,…,d'_N,N-1]^T为目标节点之间通信的带噪声的距离量测，

为所有的带噪声的距离量测集合；定义

为锚节点与目标节点之间通信的真实距离量测，r₂＝[r'₁₂,r'₁₃,…,r'_N,N-1]^T为目标节点之间通信的真实距离量测，

为所有的真实距离量测集合，

G＝blkdiag(G₁,G₂)，Γ＝1_(M+N-1)×N，diag{·}表示对角矩阵，矩阵的对角线元素为大括号中的元素，blkdiag{·}表示分块对角矩阵，1_(M+N-1)×N表示具有(M+N-1)×N个元素全为1的列向量，γ[·],r[·]表示引用向量γ,r中的元素；同时定义中间变量

γ_jj'以及向量γ₁,γ₂和γ，

γ₂＝[γ₁₂,γ₁₃,…,γ_N,N-1]^T，

The third step uses the maximum likelihood estimation method to estimate the position of the target. Let _Xu =[y ₁ ,y ₂ ,...,y _N ] be the set of unknown target node coordinates, X _a =[x ₁ ,x ₂ ,...,x _K ] is the set of coordinates of all anchor nodes, while defining d _ij =t _ij ×c, d'_jj'=t'_jj' ×c, d _ij and d'_jj' are anchor nodes i and d'jj' respectively. The distance measurement value with noise between target nodes j, and between target nodes j and j'(j≠j'); according to the existing conditions, the original objective function to be optimized is expressed as

Substitute the anchor node error term in the second step to get the min-max optimization problem

After vectorizing the objective function, and removing the anchor node error vector by applying an S-process, the original suboptimization problem is transformed into

Among them, λ and μ are newly introduced constants to transform the anchor node error vector, ≥ indicates that the matrix is positive definite, I indicates the identity matrix, Tr( ) indicates the trace of the matrix in the brackets, d ₁ =[d ₁₁ ,d ₁₂ ,… ,d _MN ] ^T is the distance measurement with noise for the communication between the anchor node and the target node, d ₂ =[d' ₁₂ ,d' ₁₃ ,...,d' _N,N-1 ] ^T is the communication between the target nodes distance measurement with noise,

is the set of all noisy distance measurements; define

is the real distance measurement of the communication between the anchor node and the target node, r ₂ =[r' ₁₂ ,r' ₁₃ ,...,r' _N,N-1 ] ^T is the real distance measurement of the communication between the target nodes,

is the set of all true distance measurements,

G=blkdiag(G ₁ , G ₂ ), Γ=1 _(M+N-1)×N , diag{·} represents a diagonal matrix, and the diagonal elements of the matrix are the elements in curly brackets, blkdiag{·} Represents a block diagonal matrix, 1 _(M+N-1)×N represents a column vector with (M+N-1)×N elements all 1, γ[ ], r[ ] represents the reference vector γ , elements in r; also define intermediate variables

γ _jj' and the vectors γ ₁ , γ ₂ and γ,

γ ₂ =[γ ₁₂ ,γ ₁₃ ,...,γ _N,N-1 ] ^T ,

优化问题的最终形式为：The fourth step described introduces the intermediate matrix

The final form of the optimization problem is:

Among them, _Yu [w, v] represents the element of the wth row and _vth column of the reference matrix Yu, and w and v must be integers;

Y _u [j,N+1:N+l] represents all elements of the jth row and N+1 to N+l columns of the matrix Yu _u , _Yu [N+1:N+l,N+1:N +l] represents all elements in the N+1 to N+1 rows and N+1 to N+1 columns of the matrix _{Yu, and X u (} :,j) represents the column composed of all the elements in the jth column of the matrix X _u vector.

The beneficial effect of the present invention is that it is only assumed that the maximum value of the modulus of the error vector of the anchor node is known, which is more in line with the actual situation. After obtaining the TOA measurement information between the anchor node and the target node and the target node, the objective function is listed, and the error amount of the anchor node is introduced. After the objective function is vectorized, the S-process is used to eliminate the anchor node error vector, and a convex constraint is introduced at the same time. Finally, the problem is transformed into a solvable convex problem through transformation and relaxation. When the positioning system is affected by the environment, resulting in a large deviation between the real position of the anchor node and the measured position, the method can still estimate the target position more accurately, and has strong robustness and practicability.

Description of drawings

Figure 1 is a block diagram of the target localization method when the anchor node position is uncertain.

Figure 2 is a graph of method performance versus TOA measurement noise variance.

Figure 3 is a graph of method performance versus the maximum value of the anchor node error vector norm.

Figure 4 is a graph of method performance versus the number of anchor nodes.

Figure 5 is a graph showing the relationship between method performance and the number of target nodes to be located.

Detailed ways

The present invention will be further described below with reference to the accompanying drawings and embodiments, and the present invention includes but is not limited to the following embodiments.

Aiming at the problem that the position of the anchor node is affected by the environment and causes an error with the measured position in the actual positioning problem, a positioning method based on the error of the position of the anchor node is proposed. Select the l-dimensional positioning scene, l=2 or 3. Assume that there are M anchor nodes with known positions but errors in the sensor network, M ≥ 3, and N nodes are the target nodes to be located. Let the real coordinates of the M anchor nodes with known positions be x ₁ , x ₂ ,...,x _M respectively, and the coordinates of the N target nodes to be estimated are y ₁ , y ₂ ,..., y _N .

The main steps of the present invention are as follows:

Step 1: Obtain raw TOA measurements and model anchor node errors

The TOA measurement set is obtained through the communication between nodes, and the set consists of two parts, namely the TOA measurement of the communication between the anchor node and the target node and the TOA measurement of the communication between the target nodes. The TOA measurement expression obtained from the communication between the anchor node and the target node is:

的高斯分布。where i and j are both positive integers, representing the node numbered i and the node numbered j, respectively. c is the speed of signal propagation, t _ij represents the time measurement obtained by the mutual communication between anchor node i and target node j, r _ij represents the distance between anchor node i and target node j, and e _ij represents anchor node i and target node j Additive white Gaussian noise for communication between

Gaussian distribution.

The TOA measurement expression obtained by the mutual communication between target nodes is:

的高斯分布。where j and j' represent the target nodes numbered j and j' respectively, and j≠j', t'_jj' represents the time measurement obtained by the mutual communication between target node j and target node j', r'_jj' represents The distance between the target node j and the target node j, e'_jj' represents the additive white Gaussian noise of the communication between the target node j and the target node j, it obeys the mean of zero, and the variance is

Gaussian distribution.

The relationship between the distance between the anchor node and the target node and the node position is:

r _ij =||x _i -x _j || ₂

i=1,2,...,M,j=1,2,...,N

where || || ₂ represents the 2-norm of the vector.

Similarly, the relationship between the distance between the target node and the target node and the node position is:

r'_jj' =||y _j -y _j' || ₂

j=1,2,...,N,j'=1,2,...,N

Due to the influence of environmental factors, there will be errors between the measured anchor node position and its real position. The relationship between the anchor node real position and the measured position is:

表示量测得到的锚节点位置，ξ_i是锚节点误差向量，它的模的最大值小于一个已知的常数ε，即in the formula

Represents the measured anchor node position, ξ _i is the anchor node error vector, and the maximum value of its modulus is less than a known constant ε, that is

||ξ _i ||≤ε

Step 2: Introduce the anchor node error term in the original TOA measurement of the communication between the anchor node and the target node

Using the first-order Taylor expansion, the relationship between the real position of the anchor node and the measured position can be further transformed into:

i=1,2,...,M,j=1,2,...,N

where o(||ξ _i ||) represents the higher-order infinitesimal of the anchor node error vector modulo ||ξ _i ||.

make

It can be deduced that:

|δ _ij |≤ε

表示锚节点i与目标节点j之间的距离，与r_ij不同，这里的使用的锚节点i的坐标是已知的带有误差的锚节点位置。δ_ij表示锚节点i与目标节点j之间的模的误差值，符号“||”为求绝对值符号。in

Represents the distance between anchor node i and target node j. Different from r _ij , the coordinates of anchor node i used here are the known anchor node positions with errors. δ _ij represents the error value of the modulus between the anchor node i and the target node j, and the symbol "||" is the symbol for finding the absolute value.

Then the TOA measurement of the communication between the final anchor node and the target node can be expressed as:

Step 3: Transform the anchor node error term

The maximum likelihood estimation method is used to estimate the location of the target. Let X _u =[y ₁ ,y ₂ ,...,y _N ] be the set of unknown target node coordinates, and X _a =[x ₁ ,x ₂ ,...,x _K ] be the set of all anchor node coordinates. Also define:

d _ij =t _ij ×c,i=1,2,...,M,j=1,2,...,N

d'_jj'=t'_jj'×c,j=1,2,…,N,j'=1,2,…,N

d _ij and d'_jj' are the noisy distance measurements between anchor node i and target node j, and between target node j and j'(j≠j'), respectively.

According to the existing conditions, the original objective function to be optimized can be expressed as:

Substitute the anchor node error term in the second step to get the min-max optimization problem:

subject to

r'_jj' =||y _j -y _j' ||,j=1,2,…,M,j'=1,2,…,N

Among them, min represents the minimum, max represents the maximum, and ∑ represents the sum. subject to means "subject to".

The above sub-optimization problem is a nonlinear and non-convex problem, and the form of the objective function is the form of element summation, and the objective function contains the node error term δ _ij , which makes the sub-optimization problem very difficult to solve. Therefore, the objective function is firstly vectorized, and the vectorized objective function and constraints are given. At this time, the anchor node error term is converted into a vector, and it is easier to process a single element when processing only one vector. ; After the vectorization is completed, the S-procedure is used to eliminate the anchor node error vector. By introducing two constant variables, the anchor node error term is eliminated and transformed into a convex constraint.

After vectorizing the objective function and eliminating the anchor node error vector by applying an S-process, the original suboptimization problem is transformed into:

表示开平方根，λ和μ是为了转化锚节点误差向量新引进的常数，≥表示矩阵正定，I表示单位矩阵，Tr(·)表示求括号里面矩阵的迹。且d₁＝[d₁₁,d₁₂,…,d_MN]^T为锚节点与目标节点通信的带噪声的距离量测，d₂＝[d'₁₂,d'₁₃,…,d'_N,N-1]^T为目标节点之间通信的带噪声的距离量测，

为所有的带噪声的距离量测集合；定义

为锚节点与目标节点之间通信的真实距离量测，r₂＝[r'₁₂,r'₁₃,…,r'_N,N-1]^T为目标节点之间通信的真实距离量测，

为所有的真实距离量测集合。

G＝blkdiag(G₁,G₂)，Γ＝1_(M+N-1)×N。其中，diag{·}表示对角矩阵，矩阵的对角线元素为大括号中的元素，blkdiag{·}表示分块对角矩阵，1_(M+N-1)×N表示具有(M+N-1)×N个元素全为1的列向量，γ[·],r[·]表示引用向量γ,r中的元素。同时定义中间变量

γ_jj'以及向量γ₁,γ₂和γ：in,

Indicates the square root, λ and μ are newly introduced constants to transform the anchor node error vector, ≥ indicates that the matrix is positive definite, I indicates the identity matrix, and Tr(·) indicates the trace of the matrix in the parentheses. And d ₁ =[d ₁₁ ,d ₁₂ ,...,d _MN ] ^T is the noisy distance measurement between the anchor node and the target node, d ₂ =[d' ₁₂ ,d' ₁₃ ,...,d' _{N, N-1} ] ^T is the noisy distance measurement for communication between target nodes,

is the set of all noisy distance measurements; define

is the real distance measurement of the communication between the anchor node and the target node, r ₂ =[r' ₁₂ ,r' ₁₃ ,...,r' _N,N-1 ] ^T is the real distance measurement of the communication between the target nodes,

Set for all true distance measurements.

G=blkdiag(G ₁ , G ₂ ), Γ=1 _(M+N-1)×N . Among them, diag{·} represents a diagonal matrix, and the diagonal elements of the matrix are the elements in the curly brackets, blkdiag{·} represents a block diagonal matrix, and 1 _(M+N-1)×N represents a matrix with (M+N-1)×N N-1)×N column vector with all 1 elements, γ[·], r[·] represents the elements in the reference vector γ, r. Also define intermediate variables

γ _jj' and the vectors γ ₁ , γ ₂ and γ:

Similar processing in the past is only for single-node positioning problems, but the current processing is for multi-node positioning problems, and the form of the processed objective function is also different from that of other methods. Other similar methods often assume the entire objective function. are smaller than a constant μ, so the first half of Tr[G(Γγ ^T -2dr ^T )] is also put into the convex constraint obtained by the S-process, and the current processing method is to keep the first half of the objective function, Don't put it into constraints. Compared with the previous method, this processing reduces the degree of relaxation and optimizes the structure of the objective function.

Step 4: Transform the problem into a convex problem

After the third step of transformation, the anchor node error term is eliminated, but the problem is still not a solvable convex problem. At this time, through a series of transformation and relaxation of the objective function and constraints, the problem is transformed into a convex problem. The relaxation method used is a positive semi-definite relaxation method. Finally, the objective function is transformed into a convex function, and the constraints are transformed into convex constraints, thereby making the problem solvable.

To transform the problem into a convex problem, an intermediate matrix Yu is _introduced :

Through the connection between the intermediate matrix and the variables to be optimized γ, X _u , and appropriate relaxation, all constraints are transformed into convex constraints, and the objective function is also a convex function, and the problem is transformed into a solvable convex problem.

The final form of the optimization problem is:

Among them, _Yu [w,v] refers to the element in the wth row and _vth column of the reference matrix Yu, and _w and v must be integers; _Yu [j, N+1:N+l] means the matrix Yu jth Row, all elements in columns N+1 to N+l, _Yu [N+1:N+l,N+1:N+l] represents matrix Yu _Row N+1 to N+l, Nth All elements from +1 to N+l columns, X _u (:,j) represents a column vector composed of all elements in the jth column of the matrix X _u .

In similar algorithms before, a penalty term is often added to the objective function, that is, a small positive number (penalty factor) is multiplied by all elements and or traces of a matrix to be optimized. This method requires manual selection of the penalty. The inappropriate selection of the penalty factor will lead to a decrease in the performance of the algorithm, so the practicability is not strong, and the present invention does not add a penalty item, thereby avoiding the manual selection of the penalty factor and increasing the practicability.

Step 5: Solve the convex optimization problem to get an estimate of the target position

The convex optimization problem obtained in the fourth step is solved by the CVX convex optimization toolbox in MATLAB to obtain the estimation of the target position.

In the embodiment of the present invention, the positioning problem in two-dimensional space is considered, and in an area of 1200m×1200m, there are 4 anchor nodes with known positions and 3 target nodes to be positioned, ie M=4, N=3. Let the real coordinates of the four anchor nodes with known positions be x ₁ , x ₂ , x ₃ , and x ₄ respectively, and the coordinates of the N target nodes to be estimated are y ₁ , y ₂ , and y ₃ .

Step 1: Obtain raw TOA measurements and model anchor node errors

The TOA measurement set is obtained through communication between nodes. Among them, the TOA measurement expression obtained from the communication between the anchor node and the target node is:

的高斯分布。i and j are both positive integers, representing the node numbered i and the node numbered j, respectively. c is the speed of signal propagation, t _ij represents the time measurement obtained by the mutual communication between anchor node i and target node j, r _ij represents the distance between anchor node i and target node j, and e _ij represents anchor node i and target node j Additive white Gaussian noise for communication between

Gaussian distribution.

The TOA measurement expression obtained by the mutual communication between target nodes is:

的高斯分布。where j and j' represent the target nodes numbered j and j' respectively, and j≠j', t'_jj' represents the time measurement obtained by the mutual communication between target node j and target node j', r'_jj' represents The distance between the target node j and the target node j, e'_jj' represents the additive white Gaussian noise of the communication between the target node j and the target node j, it obeys the mean of zero, and the variance is

Gaussian distribution.

The relationship between the distance between the anchor node and the target node and the node position is:

r _ij =||x _i -x _j || ₂

i=1,2,3,4,j=1,2,3

where || || ₂ represents the 2-norm of the vector.

Similarly, the relationship between the distance between the target node and the target node and the node position is:

r'_jj' =||y _j -y _j' || ₂ ,j=1,2,3,j'=1,2,3

Due to the influence of environmental factors, there will be errors between the measured anchor node position and its real position. The relationship between the anchor node real position and the measured position is:

表示量测得到的锚节点位置，ξ_i是锚节点误差向量，它的模的最大值小于一个已知的常数ε，即in the formula

Represents the measured anchor node position, ξ _i is the anchor node error vector, and the maximum value of its modulus is less than a known constant ε, that is

||ξ _i ||≤ε

Here ε is set to 0.5m.

Step 2: Introduce the anchor node error term in the original TOA measurement of the communication between the anchor node and the target node

Using the first-order Taylor expansion, the relationship between the real position of the anchor node and the measured position can be further transformed into:

i=1,2,3,4,j=1,2,3

where o(||ξ _i ||) represents the higher-order infinitesimal of the anchor node error vector modulo ||ξ _i ||.

make

then there are

which is

|δ _ij |≤ε

表示锚节点i与目标节点j之间的距离，与r_ij不同，这里的使用的锚节点i的坐标是已知的带有误差的锚节点位置。δ_ij表示锚节点i与目标节点j之间的模的误差值，符号“||”为求绝对值符号。in

Represents the distance between anchor node i and target node j. Different from r _ij , the coordinates of anchor node i used here are the known anchor node positions with errors. δ _ij represents the error value of the modulus between the anchor node i and the target node j, and the symbol "||" is the symbol for finding the absolute value.

Then the TOA measurement of the communication between the final anchor node and the target node can be expressed as:

Step 3: Transform the anchor node error term

The maximum likelihood estimation method is used to estimate the location of the target. Let X _u =[y ₁ , y ₂ , y ₃ ] be the set of unknown target node coordinates, and X _a =[x ₁ , x ₂ , x ₃ , x ₄ ] be the set of all anchor node coordinates. Also define:

d _ij =t _ij ×c,i=1,2,3,4,j=1,2,3

d'_jj'＝t'_jj'×c,j＝1,2,3,j'＝1,2,3

d _ij and d'_jj' are the noisy distance measurements between anchor node i and target node j, and between target node j and j'(j≠j'), respectively.

According to the existing conditions, the original objective function to be optimized can be expressed as:

Substitute the anchor node error term in the second step to get the min-max optimization problem:

subject to

r'_jj' =||y _j -y _j' ||,j=1,2,3,j'=1,2,3

Among them, min represents the minimum, max represents the maximum, and ∑ represents the sum. subject to means "subject to".

Next, vectorize the objective function. For a single TOA measurement, the anchor node error satisfies |δ _ij |≤ε, then the vectorized anchor node error vector satisfies:

表示开平方根。同样地，定义d₁＝[d₁₁,d₁₂,…,d₄₃]^T为锚节点与目标节点通信的带噪声的距离量测，d₂＝[d'₁₂,d'₁₃,…,d'₃₂]^T为目标节点之间通信的带噪声的距离量测，

为所有的带噪声的距离量测集合；定义

为锚节点与目标节点之间通信的真实距离量测，r₂＝[r'₁₂,r'₁₃,…,r₃'₂]^T为目标节点之间通信的真实距离量测，

为所有的真实距离量测集合。同时定义中间变量

γ_jj'以及向量γ₁,γ₂和γ：where δ=[δ ₁₁ ,δ ₁₂ ,...,δ ₄₃ ] ^T is the set of anchor node modulo errors of M anchor nodes communicating with N target nodes,

represents the square root. Similarly, define d ₁ =[d ₁₁ ,d ₁₂ ,...,d ₄₃ ] ^T is the noisy distance measurement between the anchor node and the target node, d ₂ =[d' ₁₂ ,d' ₁₃ ,...,d _'32 ] ^T is a noisy distance measurement for communication between target nodes,

is the set of all noisy distance measurements; define

is the real distance measurement of the communication between the anchor node and the target node, r ₂ =[r' ₁₂ ,r' ₁₃ ,...,r ₃ ' ₂ ] ^T is the real distance measurement of the communication between the target nodes,

Set for all true distance measurements. Also define intermediate variables

γ _jj' and the vectors γ ₁ , γ ₂ and γ:

Then the original suboptimization problem can be transformed into:

subject to

γ[3(i-1)+j]=r ² [3(i-1)+j], i=1,2,3,4,j=1,2,3

γ[12+3(j-1)+j']=r ² [12+3(j-1)+j'],j=1,2,3,j'=1,2,3

r[12+3(i-1)+j]=||y _j -y _j' ||,j=1,2,3,j'=1,2,3

G＝blkdiag(G₁,G₂)，Γ＝1_(3+4-1)×3。其中，diag{·}表示对角矩阵，矩阵的对角线元素为大括号中的元素，blkdiag{·}表示分块对角矩阵，1_(3+4-1)×3表示具有18个元素全为1的列向量，γ[·],r[·]表示引用向量γ,r中的元素。in

G=blkdiag(G ₁ , G ₂ ), Γ=1 _(3+4-1)×3 . Among them, diag{·} represents a diagonal matrix, the diagonal elements of the matrix are the elements in the curly brackets, blkdiag{·} represents a block diagonal matrix, and 1 _(3+4-1)×3 means that there are 18 elements A column vector of all 1s, γ[·], r[·] represents the element in the reference vector γ, r.

有如下结论成立：Eliminate max in min-max optimization problem by introducing a constant factor μ, i.e. for

The following conclusions are established:

δ ^T G ₁ δ-2δ ^T G ₁ (d ₁ -r ₁ )≤μ

which is:

表示充分条件。in

represents a sufficient condition.

The above formula can be further expressed as:

By applying the S-process, the above recurrence relation is finally transformed into a convex constraint:

which is:

表示“存在”，≥表示矩阵半正定。where λ and μ are newly introduced constants to transform the anchor node error vector,

means "exist", and ≥ means that the matrix is positive semi-definite.

At the same time, the objective function to be optimized can be transformed into:

Tr[G(Γγ ^T -2dr ^T )]+μ

In the formula, Tr(·) means to find the trace of the matrix in the brackets.

Step 4: Transform the problem into a convex problem

After the third step of transformation, the anchor node error term is eliminated, but the problem is still not a convex problem. At this time, through a series of transformation and relaxation of the objective function and constraints, the problem is transformed into a convex problem. The relaxation method used is a positive semi-definite relaxation method. Finally, the objective function is transformed into a convex function, and the constraints are transformed into convex constraints, thereby making the problem solvable.

To transform the problem into a convex problem, an intermediate matrix Yu is _introduced :

Then the relationship between the elements in the matrix Yu _u and the existing constraints is:

i=1,2,3,4,j=1,2,3

γ[12+3(j-1)+j']=Y _u [j,j]+Y _u [j',j']-Y _u [j,j']-Y _u [j',j]

j=1,2,3,j'=1,2,3

Among them, _Yu [j, j] represents the element of the jth row and the _jth column of the reference matrix Yu. The above two constraints are transformed into convex constraints. At the same time, relax the other two equality constraints into inequality constraints, we get:

γ[3(i-1)+j]≥r ² [3(i-1)+j], i=1,2,3,4,j=1,2,3

γ[12+3(j-1)+j']≥r ² [12+3(j-1)+j'],j=1,2,3,j'=1,2,3

The other convex _constraints hidden in the matrix Yu are:

y _j =[Y _u [j,4],Y _u [j,5]] ^T ,j=1,2,3

Y _u ≥0 ₃₊₂

This converts all constraints into convex constraints, the objective function is also a convex function, and the problem is transformed into a solvable convex problem.

The final form of the optimization problem is:

Step 5: Solve the convex optimization problem to get an estimate of the target position

The convex optimization problem obtained by the fourth step can be solved by MATLAB's CVX toolbox, the tool is Sedumi. By programming in the MATLAB environment and inputting the objective function and constraints, the estimation of the target position can be obtained.

Claims (3)

1. a target positioning method under the uncertain situation of anchor node position, is characterized in that, comprises the following steps: The first step is to obtain TOA measurements between nodes in the sensor network and model the anchor node error; In the first step, there are M anchor nodes with known positions but errors and N target nodes to be located in the sensor network, M≥3, and the real coordinates of the M anchor nodes are x ₁ , x ₂ ,… ,x _M , the coordinates of the N target nodes to be estimated are y ₁ , y ₂ ,...,y _N ; the TOA measurement set is obtained through the communication between nodes, including the TOA amount of communication between the anchor node and the target node The TOA measurement of the communication between the anchor node and the target node; the TOA measurement expression obtained from the communication between the anchor node and the target node is:

where i and j represent the anchor node numbered i and the target node numbered j respectively, c is the signal propagation speed, _tij represents the time measurement obtained by the mutual communication between the anchor node i and the target node _j , and rij represents the anchor node The distance between i and the target node j, e _ij represents the additive white Gaussian noise of the communication between the anchor node i and the target node j, e _ij obeys the mean value of zero and the variance is

The Gaussian distribution of ; the TOA measurement expression obtained by mutual communication between target nodes is:

where j and j' represent the target nodes numbered j and j' respectively, and j≠j', t'_jj' represents the time measurement obtained by the mutual communication between target node j and target node j', r'_jj' represents The distance between the target node j and the target node j, e'_jj' represents the additive white Gaussian noise of the communication between the target node j and the target node j, e'_jj' obeys the mean zero and the variance is

The Gaussian distribution of ; the relationship between the distance between the anchor node and the target node and the node position is r _ij =||x _i -x _j || ₂ ; The relationship is r'_jj' =||y _j -y _j' || ₂ ; the relationship between the real position of the anchor node and the measured position is

in the formula

Represents the measured anchor node position, ξ _i is the anchor node error vector, and the maximum value of the modulus of ξ _i is less than or equal to a known constant ε. In the second step, the anchor node error term is introduced into the original TOA measurement of the communication between the anchor node and the target node; The third step is to transform the anchor node error term, and use the S-process to eliminate the anchor node error vector; The third step uses the maximum likelihood estimation method to estimate the position of the target. Let _Xu =[y ₁ ,y ₂ ,...,y _N ] be the set of unknown target node coordinates, X _a =[x ₁ ,x ₂ ,...,x _K ] is the set of coordinates of all anchor nodes, while defining d _ij =t _ij ×c, d'_jj'=t'_jj' ×c, d _ij and d'_jj' are anchor nodes i and d'jj' respectively. The distance measurement value with noise between target nodes j, and between target nodes j and j'(j≠j'); according to the existing conditions, the original objective function to be optimized is expressed as

Substitute the anchor node error term in the second step to get the min-max optimization problem

After vectorizing the objective function, and removing the anchor node error vector by applying an S-process, the original suboptimization problem is transformed into

where λ and μ are newly introduced constants to transform the anchor node error vector,

Indicates that the matrix is positive definite, I represents the identity matrix, Tr(·) represents the trace of the matrix in the brackets, d ₁ =[d ₁₁ ,d ₁₂ ,...,d _MN ] ^T is the distance between the anchor node and the target node with noise for communication d ₂ =[d' ₁₂ ,d' ₁₃ ,...,d' _N,N-1 ] ^T is the distance measurement with noise for communication between target nodes,

is the set of all noisy distance measurements; define

is the real distance measurement of the communication between the anchor node and the target node, r ₂ =[r′ ₁₂ ,r′ ₁₃ ,...,r' _N,N-1 ] ^T is the real distance measurement of the communication between the target nodes,

is the set of all true distance measurements,

G=blkdiag(G ₁ , G ₂ ), Γ=1 _(M+N-1)×N , diag{·} represents a diagonal matrix, and the diagonal elements of the matrix are the elements in curly brackets, blkdiag{·} Represents a block diagonal matrix, 1 _(M+N-1)×N represents a column vector with (M+N-1)×N elements all 1, γ[ ], r[ ] represents the reference vector γ , elements in r; also define intermediate variables

γ _jj' and the vectors γ ₁ , γ ₂ and γ,

γ ₂ =[γ ₁₂ ,γ ₁₃ ,...,γ _N,N-1 ] ^T ,

The fourth step is to transform the problem into a convex problem by transforming and relaxing the objective function and constraints; The fifth step is to solve the convex optimization problem to obtain an estimate of the target position. 2. the target positioning method under the uncertain situation of anchor node position according to claim 1, is characterized in that, described second step utilizes first-order Taylor expansion formula to further transform the relationship between anchor node real position and measurement position for

where o(||ξ _i ||) represents the higher-order infinitesimal of the anchor node error vector modulo ||ξ _i ||; let

Then it can be deduced that |δ _ij |≤ε, where

Represents the distance between anchor node i and target node j. Different from r _ij , the coordinates of anchor node i used here are the known anchor node positions with errors, and δ _ij represents the distance between anchor node i and target node j. The error value of the modulus between the final anchor node and the target node is expressed as

3. the target positioning method under the uncertain situation of anchor node position according to claim 1, is characterized in that, described 4th step introduces intermediate matrix

The final form of the optimization problem is:

subject to

γ[(i-1)N+j]≥r ² [(i-1)N+j] γ[MN+(j-1)N+j']≥r ² [MN+(j-1)N+j']

γ[MN+(j-1)N+j']=Y _u [j,j]+Y _u [j',j']-Y _u [j,j']-Y _u [j',j] X _u (:,j)＝Y _u [j,N+1:N+l] Y _u [N+1:N+1,N+1:N+ ₁ ]=I1

Among them, _Yu [w,v] refers to the element in the wth row and _vth column of the reference matrix Yu, and _w and v must be integers; _Yu [j, N+1:N+l] means the matrix Yu jth Row, all elements in columns N+1 to N+l, _Yu [N+1:N+l,N+1:N+l] represents matrix Yu _Row N+1 to N+l, Nth All elements from +1 to N+l columns, X _u (:,j) represents a column vector composed of all elements in the jth column of the matrix X _u .

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