CN112346014B

CN112346014B - Multi-base sonar positioning method based on signal arrival time difference

Info

Publication number: CN112346014B
Application number: CN202011007984.8A
Authority: CN
Inventors: 杨树立; 王刚
Original assignee: Ningbo University
Current assignee: Ningbo University
Priority date: 2020-09-23
Filing date: 2020-09-23
Publication date: 2022-06-21
Anticipated expiration: 2040-09-23
Also published as: CN112346014A

Abstract

The invention discloses a multi-base sonar positioning method based on signal arrival time difference, which considers two situations that the true propagation speed of an assumed signal is a known distributed random variable and the true propagation speed of the signal is completely unknown, and constructs measurement models of the two situations; converting the measurement models of the two conditions into linear relation equations, arranging the linear relation equations into a matrix form, and converting the matrix form into a weighted least square problem; converting the weighted least square problem of the two conditions into a non-convex constraint optimization problem, relaxing the non-convex constraint optimization problem into an easily-processed convex problem, and then adding additional constraint to tighten the problem; solving the convex problem after the tightening of the two conditions to obtain the optimal solution of the variable to be optimized, and further obtaining the preliminary estimation value of the target position under the two conditions; optimizing by using the initial estimation value to obtain a final estimation value; the advantage is that the target positioning performance is good.

Description

Multi-base sonar positioning method based on signal arrival time difference

Technical Field

The invention relates to an underwater target positioning technology, in particular to a multi-base sonar positioning method based on signal arrival time difference.

Background

The problem of unknown target positioning has a great deal of application in a plurality of fields such as wireless sensor networks, radars, sonars and the like, and therefore, the problem of unknown target positioning is widely concerned.

In underwater target positioning, because the attenuation speed of radio signals is high when the radio signals are propagated in water, sonar is generally adopted as a transmitter to transmit signals and as a receiver to collect signals, and the signals are in a sound wave frequency range. The underwater target positioning is more difficult than the land target positioning, and is mainly embodied in the following two aspects:

1) in underwater target positioning, the positions of a transmitter and a receiver are not fixed and are time-varying, so that position errors have randomness and time variation;

2) in underwater target positioning, the signal propagation speed is not constant and can change along with the change of water temperature, water depth and salinity.

Both of the above aspects increase the difficulty in achieving time synchronization of the transmitter and the receiver, resulting in significantly reduced underwater target positioning performance, especially for time-based underwater target positioning, which is worse.

In an underwater environment, the multi-base sonar system has more stable performance and more flexible application than the traditional single-base sonar system. In a multi-base sonar system, each signal transmitted by a transmitter reaches a receiver via two propagation paths: a direct path from the transmitter to the receiver and a reflected path that is reflected by the target after transmission. In a multi-base sonar system, the measurements received by the receiver typically include signal time difference of arrival, azimuth, doppler shift, and combinations thereof. For a pair of transmitter and receiver, the position of the transmitter and the position of the receiver are taken as focuses, the sum of the distance from one focus to the target and the distance from the other focus to the target is obtained according to the signal arrival time difference in the measured values received by the receiver, an elliptical track is determined according to the sum of the two focuses and the distances corresponding to the pair of transmitter and receiver, and the position of the target is the intersection point between the elliptical tracks corresponding to the plurality of pairs of transmitter and receiver. However, the highly non-linear relationship between the target position and the measured values makes it difficult to solve for the target position. Currently, various methods of target location are proposed by those skilled in the art that deal with highly non-linear relationships between target location and measured values. Such as: both the Rui and Ho professors developed an effective four-step closed WLS (weighted least squares) estimator, however, this approach may not have good positioning performance or even fail when the noise is large or the transmitter and receiver locations are poorly distributed. Another example is: teaching Jia et al improves the four-step closed WLS estimator described above from two points, one of which is to reduce the four-step approach to a two-step solution, thereby reducing computational complexity, but it only simplifies the steps and has poor or even failed positioning performance in harsh environments or with a small number of transmitters; another approach is to use the generalized trust domain sub-problem (GTRS) approach to improve positioning performance, however, the GTRS approach cannot handle the case of multiple transmitters, which affects its application.

Disclosure of Invention

The technical problem to be solved by the invention is to provide a positioning method of multi-base sonar based on signal arrival time difference, which has good positioning performance under the condition that the real propagation speed of signals is known to be distributed and under the condition that the real propagation speed of signals is completely unknown.

The technical scheme adopted by the invention for solving the technical problems is as follows: a positioning method of multi-base sonar based on signal arrival time difference is characterized by comprising the following steps:

the method comprises the following steps: establishing a plane coordinate system or a space coordinate system as a reference coordinate system; setting a target with 1 unknown coordinate in a reference coordinate system, N receivers and M transmitters; recording the real value of the coordinate position of the target in the reference coordinate system as u^oRecording the real value of the coordinate position of the jth receiver in the reference coordinate system

Recording the real value of the coordinate position of the ith transmitter in the reference coordinate system

The true propagation velocity of the signal in the underwater environment is recorded as c^oThe nominal coordinate position of the original nominal deployment of the jth receiver in the reference coordinate system is recorded as s_jThe nominal coordinate position of the original nominal deployment of the ith transmitter in the reference coordinate system is recorded as t_iThe standard propagation velocity of a signal in an underwater environment is recorded

Wherein N is more than or equal to 1, M is more than or equal to 1, N + M is more than or equal to 4, j is more than or equal to 1 and less than or equal to N, i is more than or equal to 1 and less than or equal to M, and Delta s_jIndicating the position error, at, of the jth receiver_iDenotes the position error, Δ s, of the ith transmitter_jAnd Δ t_iAll obey zero mean and variance of

Has a Gaussian distribution of 20 to sigma_zLess than or equal to 200 m, where Δ c represents the propagation velocity error of the signal, and Δ c follows a zero mean with a variance of

Gaussian distribution of 2. ltoreq. sigma_cLess than or equal to 20 m/s;

step two: the signal transmitted by the transmitter is received by the receiver successively after passing through two paths, namely a direct path and an indirect path reflected by the target; then, ellipse fitting is carried out by using the signal arrival time difference to obtain a measurement model, which is described as:

then s is_jAnd t_iSubstituting into the measurement model, considering two cases that the true propagation velocity of the signal is a random variable with known distribution and the true propagation velocity of the signal is completely unknown, and correspondingly obtaining the measurement model under the condition that the true propagation velocity of the signal is known distribution and the measurement model under the condition that the true propagation velocity of the signal is completely unknown, describing the measurement model under the condition that the true propagation velocity of the signal is known distribution as follows:

the measurement model in the case where the true propagation velocity of the signal is completely unknown is described as:

wherein,

the real time difference between the time spent by the jth receiver after the signal transmitted by the ith transmitter is received by the jth receiver after passing through the indirect path reflected by the target and the time spent by the jth receiver after the signal transmitted by the ith transmitter is received by the jth receiver after passing through the direct path is represented by the symbol "| |", which is used for solving the problem of the time differenceThe Euclidean distance, "s.t." means "constrained to … …", τ_i,jA measured time difference, ε, between the time elapsed for the signal transmitted by the ith transmitter to be received by the jth receiver after passing through the indirect path where the target reflects and the time elapsed for the signal transmitted by the ith transmitter to be received by the jth receiver after passing through the direct path_i,jAnd

are all intermediate variables that are introduced into the reactor,

Δτ_i,jobeying a mean value and a variance of zero

Gaussian distribution of (a), 0.02. ltoreq. sigma_τLess than or equal to 0.2 second,

are all intermediate variables that are introduced into the reactor,

"T" represents the transpose of a vector or matrix, and both the measurement model under the condition of known distribution of the true propagation velocity of the signal and the measurement model under the condition of completely unknown true propagation velocity of the signal are highly nonlinear models;

step three: the method comprises the step of determining the absolute value u in a measurement model under the condition that the real propagation speed of a signal is distributed in a known mode^o-s_j||-||t_i-s_jMoving to the left side of the equation, squaring and expanding two sides of the equation, and omitting a quadratic term in the expansion equation to obtain a linear relation equation under the condition that the real propagation velocity of the signal is known to be distributed, wherein the equation is described as follows:

(ii) a Similarly, | | u in the measurement model with the true propagation velocity of the signal completely unknown^o-s_j||-||t_i-s_jMoving to the left side of the equation, squaring and expanding two sides of the equation, and omitting a quadratic term in the expansion equation to obtain a linear relation equation under the condition that the real propagation speed of the signal is completely unknown, wherein the equation is described as follows:

then, the linear relation equation under the condition of the known distribution of the true propagation velocity of the signal is sorted into a matrix form, and the matrix form under the condition of the known distribution of the true propagation velocity of the signal is obtained: b epsilon-Ay^o(ii) a Similarly, the linear relation equation under the condition that the true propagation velocity of the signal is completely unknown is arranged into a matrix form, and the matrix form under the condition that the true propagation velocity of the signal is completely unknown is obtained:

wherein, B represents the introduced intermediate matrix,

I_Midentity matrix, symbol of dimension M × M

Represents the kronecker product, diag ([ | u)^o-s₁||,||u^o-s₂||,...,||u^o-s_N||]) Represents the vector [ | | u [ ]^o-s₁||,||u^o-s₂||,...,||u^o-s_N||]Each element in turn being a diagonal matrix of diagonal elements, s₁Nominal coordinate position, s, representing the original nominal deployment of the 1 st receiver in the reference coordinate system₂Nominal coordinate position, s, representing the nominal original deployment of the 2 nd receiver in the reference coordinate system_NIndicating the Nth receiver in the reference frameNominal coordinate position of the original nominal deployment, e ═ e_1,1,ε_1,2,...,ε_1,N,ε_2,1,...,ε_M,N]^T，ε_1,1,ε_1,2,...,ε_1,N,ε_2,1,...,ε_M,NAre all according to

Calculated, b is the introduced intermediate vector, b ═ b_1,1,b_1,2,...,b_1,N,b_2,1,...,b_i,j,...,b_M,N]^T，b_1,1,b_1,2,...,b_1,N,b_2,1,...,b_i,j,...,b_M,NAre all the elements in the b, and the element,

a is an introduced intermediate matrix, and A ═ A₁,A₂,...,A_i,...,A_M]^T，A₁,A₂,...,A_i,...,A_MIs an element in the group A and has the following characteristics,

0_i-1a column vector representing dimensions 1 (i-1) and elements all 0, 0_M-iA column vector having dimensions of 1X (M-i) and elements of all 0, τ_i,1A measured time difference, τ, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the 1 st receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the 1 st receiver after passing through the direct path_i,2A measured time difference, τ, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the 2 nd receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the 2 nd receiver after passing through the direct path_i,NA measured time difference, y, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the nth receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the nth receiver after passing through the direct path^oTo be changed from unknownVector of quantities, y^o＝[u^oT,||u^o-t₁||,||u^o-t₂||,...,||u^o-t_M||]^T，t₁Nominal coordinate position, t, representing the original nominal deployment of the 1 st transmitter in the reference coordinate system₂Nominal coordinate position, t, representing the nominal deployment of the 2 nd transmitter in the reference coordinate system_MRepresenting the nominal coordinate position of the original nominal deployment of the mth transmitter in the reference coordinate system,

are all according to

The calculation results in that,

in order to introduce the intermediate vector(s),

are all made of

The elements (A) and (B) in (B),

in order to introduce an intermediate matrix of the matrix,

is composed of

The elements (A) and (B) in (B),

0_M-1a column vector with dimension 1 (M-1) and elements all 0, a scaling constant introduced to avoid numerical problems in the case where the true propagation velocity of the signal is completely unknown, and a ∈ [100,1000 ]]，

Is a vector made up of unknown variables,

denotes c^oA value after scaling down by a factor of alpha;

step four: converting the matrix form in the case of the known distribution of the true propagation velocities of the signals into a weighted least squares problem in the case of the known distribution of the true propagation velocities of the signals, described as:

likewise, the matrix form in the case where the true propagation velocity of the signal is completely unknown is converted into the weighted least squares problem in the case where the true propagation velocity of the signal is completely unknown, which is described as:

then, the weighted least square problem under the condition of the known distribution of the true propagation velocity of the signal is converted into a non-convex constraint optimization problem under the condition of the known distribution of the true propagation velocity of the signal by adding constraint conditions, and the description is as follows:

similarly, the weighted least squares problem in the case where the true propagation velocity of the signal is completely unknown is converted into a non-convex constrained optimization problem in the case where the true propagation velocity of the signal is completely unknown by adding constraint conditions, which is described as:

wherein Q and

are all introduced intermediate matrix, and the initial value of Q is I_MN，

Is initially value of_MN，I_MNRepresenting an identity matrix of dimensions MN x MN, y being AND^oCorresponding to a vector consisting of the variables to be optimized, y ═ u^T,||u-t₁||,||u-t₂||,...,||u-t_M||]^T，

Is prepared by reacting with

Corresponding to the vector consisting of the variables to be optimized,

u represents u^oThe corresponding variable to be optimized is set to be,

to represent

Corresponding variables to be optimized, y (2+ i) represents the 2+ i th element in y, y (1:2) represents a column vector consisting of the 1 st element and the 2 nd element in y,

represent

The number 4 element of (a) is,

to represent

The number 3 element of (a) is,

to represent

The 4+ i th element in (b),

is represented by

The 1 st element and the 2 nd element of (a),

to represent

The 4+ M + i th element in (a);

step five: let Y equal to yy^TThe non-convex constraint optimization problem under the condition of the known distribution of the true propagation velocity of the signal is converted into a corresponding equivalent problem, which is described as follows:

order to

Converting the non-convex constraint optimization problem under the condition that the true propagation speed of the signal is completely unknown into a corresponding equivalent problem, which is described as follows:

wherein Y and

all are introduced intermediate matrix, tr { } represents the trace of matrix, rank () represents the rank of matrix, F and

are all the introduced intermediate matrixes,

y (2+ i ) represents an element of the 2+ i th row and the 2+ i th column in Y, Y (1:2) represents a matrix composed of all elements of the 1 st column to the 2 nd column in the 1 st row to the 2 nd row in Y,

to represent

Row 4+ i and column 4+ i,

is represented by

A matrix of all elements of the 1 st to 2 nd columns in the 1 st to 2 nd rows,

to represent

Row 3 and column 4+ i in (b),

to represent

Row 4 and column 4+ i in (b),

to represent

Row 3 and column 4+ M + i,

to represent

Line 3 of (1)3 columns of elements;

step six: the method comprises the following steps of relaxing an equivalent problem corresponding to a non-convex constraint optimization problem under the condition of known distribution of the true propagation speed of a signal into an easily-processed convex problem by adopting a semi-positive definite relaxation technology, and describing the easily-processed convex problem under the condition of known distribution of the true propagation speed of the signal as follows:

similarly, a semidefinite relaxation technology is adopted to relax an equivalent problem corresponding to the non-convex constraint optimization problem under the condition that the true propagation speed of the signal is completely unknown into an easily-processed convex problem, and the easily-processed convex problem under the condition that the true propagation speed of the signal is completely unknown is described as follows:

step seven: the problem is tightened by adding an additional constraint to the tractable convex problem with a known distribution of true propagation velocities of the signal, resulting in a tightened convex problem with a known distribution of true propagation velocities of the signal, described as:

the problem is tightened by adding additional constraints to the tractable convex problem in the case where the true propagation velocity of the signal is completely unknown, resulting in a tightened convex problem in the case where the true propagation velocity of the signal is completely unknown, described as:

wherein Y (2+ i,2+ j) represents an element in row 2+ i and column 2+ j in Y, Y (1:2,2+ j) represents a matrix composed of all elements in row 1 to column 2+ j in row 2 in Y, Y (2+ j) represents an element in row 2+ j in Y,

to represent

Row 2+ i and column 2+ j in (b),

is represented by

A matrix of all elements of the 2+ j column in the 1 st to 2 nd rows,

to represent

The 2+ j-th element in (b),

to represent

Row 4+ i and column 3 elements in (b),

is represented by

A matrix of all elements of column 3 in row 1 to row 2,

to represent

Row 4+ M + i and column 3 elements,

to represent

Row 4+ i and column 4+ j in (b),

is represented by

A matrix of all elements of column 4+ j in row 1 to row 2,

to represent

The 4+ j th element in (b),

to represent

Row 4+ i and column 4+ M + j in (c),

is represented by

A matrix of all elements of the 4+ M + j column in the 1 st to 2 nd rows,

to represent

The 4+ M + j element of (1);

step eight: solving the convex problem after the clamping under the condition that the real propagation speed of the signal is distributed in a known way to obtain the optimal solution of y, and recording the optimal solution as y^*(ii) a Similarly, solving the clamped convex problem under the condition that the true propagation speed of the signal is completely unknown to obtain

Is given as

Step nine: according to y^*Obtaining u with known distribution of true propagation velocity of signal^oIs given as u₁ ^*，u₁ ^*＝y^*(1: 2); and according to

U in the case where the true propagation velocity of the resulting signal is completely unknown^oAnd

corresponding to u₂ ^*And

wherein, y^*(1:2) represents y^*The 1 st element and the 2 nd element in the vector,

to represent

The 1 st element and the 2 nd element in the vector,

to represent

The 3 rd element in (1);

step ten: u is to be₁ ^*Substitution into

Update the value of Q, and add u₂ ^*And

substitution into

Update in the middle

A value of (d); then at Q and

after the value of (A) is updated, repeatedly executing the step four to the step eight; and then according to y obtained by repeated execution^*Obtaining u with known distribution of true propagation velocity of signal^oIs given as u^* _1,final，u^* _1,final＝y^*(1: 2); and obtained on repeated execution

In the case where the true propagation velocity of the acquired signal is completely unknown u^oIs given as u^* _2,final，

Wherein Q_τ、D_z、D_t、D_sIn order to introduce an intermediate matrix of the matrix,

I_MNan identity matrix with dimensions MN × MN, D_z＝[D_t,D_s]，D_t＝Diag{D_t,1,D_t,2,...,D_t,i,...,D_t,M}，D_t＝Diag{D_t,1,D_t,2,...,D_t,i,...,D_t,MDenotes by D_t,1,D_t,2,...,D_t,i,...,D_t,MConstructing a block diagonal matrix D as diagonal blocks_t，

Are all according to

The calculation results in that,

is represented by

Constructing a block diagonal matrix D as diagonal blocks_s,i，

Are all according to

The calculation results in that,

an identity matrix having a dimension of 2(M + N) × 2(M + N), τ ═ τ_1,1,τ_1,2,...,τ_1,N,τ_2,1,...,τ_M,N]^T，τ_1,1A measured time difference, τ, representing the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 1 st receiver after passing through the indirect path of reflection from the target and the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 1 st receiver after passing through the direct path_1,2A measured time difference, τ, representing the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 2 nd receiver after passing through the indirect path of reflection from the target and the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 2 nd receiver after passing through the direct path_1,NA measured time difference, τ, between the time taken for the signal transmitted by the 1 st transmitter to be received by the Nth receiver after passing through the indirect path where the signal is reflected by the target and the time taken for the signal transmitted by the 1 st transmitter to be received by the Nth receiver after passing through the direct path_2,1Indicating the time between the reflection of the signal from the 2 nd transmitter by the targetThe measured time difference, τ, between the time elapsed for the reception by the 1 st receiver after the path and the time elapsed for the reception by the 1 st receiver of the signal transmitted by the 2 nd transmitter after the direct path_M,NA measured time difference representing a time elapsed for a signal transmitted by the mth transmitter to be received by the nth receiver after passing through the indirect path where the signal is reflected by the target and a time elapsed for a signal transmitted by the mth transmitter to be received by the nth receiver after passing through the direct path.

Compared with the prior art, the invention has the advantages that:

1) the method considers two situations that the true propagation speed of the signal is a random variable with known distribution and the true propagation speed of the signal is completely unknown, and divides a measurement model into two situations, wherein the position errors of a transmitter and a receiver are considered in the measurement model under the situation that the true propagation speed of the signal is known distribution and the measurement model under the situation that the true propagation speed of the signal is completely unknown, and simultaneously, the uncertainty of the true propagation speed of the signal in an underwater environment is considered, so that the method is closer to practical application.

2) The method of the invention positions the target by utilizing the time difference between the time that the signal transmitted by the transmitter is received by the receiver after passing through the indirect path reflected by the target and the time that the signal transmitted by the transmitter is received by the receiver after passing through the direct path, namely the signal arrival time difference, thereby effectively preventing the error caused by the asynchrony between the transmitter and the receiver and greatly improving the positioning performance of the underwater target positioning.

3) The method adopts a semi-positive planning method to process the target positioning, so that the positioning performance of the method can still keep a stable state under the condition of severe conditions (under the condition that the real propagation speed of a signal is completely unknown).

4) The method of the invention still has better positioning performance under the condition of large noise or poor distribution of the positions of the transmitters and the receivers or less quantity of the transmitters and the receivers.

Drawings

FIG. 1 is a block diagram of an overall implementation of the method of the present invention;

FIG. 2 is a schematic diagram of a process of fitting an ellipse to obtain a measurement model using a signal arrival time difference;

FIG. 3 shows the method of the invention (SDR for short) in combination with the four-step WLS method, the GTRS method and the hybrid Cramer-Roman lower bound (HCRLB) at σ, for a single transmitter with a known distribution of the true propagation velocity of the signal_τ0.05 second,. sigma_cStandard deviation sigma of gaussian distribution obeyed by root mean square error along with position errors of transmitter and receiver in 10 random scenes when the number of receivers is 4 and the number of transmitters is 1 is 7.5 m/s_zSchematic diagrams of the change from 20 meters growth to 200 meters;

FIG. 4 shows the method of the invention (SDR for short) in combination with the four-step WLS method, the GTRS method and the hybrid Cramer-Roman lower bound (HCRLB) at σ, for a single transmitter with a known distribution of the true propagation velocity of the signal_z50 m, sigma_cRoot mean square error with delta tau in 10 random scenes when 7.5 m/s, 4 receivers and 1 transmitter are used_i,jStandard deviation sigma of the obeyed Gaussian distribution_τA plot of the increase from 0.02 seconds to 0.2 seconds (reflecting measurement noise);

FIG. 5 shows the inventive method (SDR for short) in combination with the four-step WLS method and the hybrid Cramer-Row lower bound (HCRLB) at σ, in the case where the true propagation velocity of the signal is completely unknown_τ0.05 second,. sigma_cStandard deviation sigma of gaussian distribution obeyed by root mean square error along with position errors of transmitter and receiver in 10 random scenes when the number of receivers is 5 and the number of transmitters is 3_zA schematic representation of the change from 20 meters growth to 200 meters;

FIG. 6 shows the inventive method (SDR for short) in combination with the four-step WLS method and the hybrid Cramer-Roman lower bound (HCRLB), σ, in the case where the true propagation velocity of the signal is completely unknown_z50 seconds, σ_cRoot mean square error with delta tau in 10 random scenes when 7.5 m/s, 5 receivers and 3 transmitters are used_i,jStandard deviation sigma of the obeyed Gaussian distribution_τ(reflecting measurement noise) increasing from 0.02 to 0.2 secondsSchematic representation.

Detailed Description

The invention is described in further detail below with reference to the accompanying examples.

The general implementation block diagram of the positioning method of the multi-base sonar based on the signal arrival time difference, which is provided by the invention, is shown in fig. 1, and the positioning method comprises the following steps:

The true propagation velocity of the signal in the underwater environment is recorded as c^oThe nominal coordinate position of the nominal deployment of the jth receiver in the reference coordinate system is denoted as s_jLet the nominal coordinate position of the original nominal deployment of the ith transmitter in the reference coordinate system be t_iRecording the standard propagation velocity of the signal in the underwater environment as c, since in the underwater environment the true values of the coordinate positions of the transmitter and the receiver are unknown during data collection and vary with the temperature, pressure and salinity of the underwater environment, as reflected in that the true values of the coordinate positions of the transmitter occur randomly in the vicinity of the nominal coordinate position of the original nominal deployment of the transmitter and the true values of the coordinate positions of the receiver occur randomly in the vicinity of the nominal coordinate position of the original nominal deployment of the receiver, there are some values of the propagation velocity of the signal in the underwater environment, which are not known during data collection, and therefore the true values of the coordinate positions of the transmitter and the true values of the coordinate positions of the receiver vary randomly in the vicinity of the nominal coordinate position of the original nominal deployment of the receiver

In underwater environment, the propagation speed of the signal is also caused by the factors of water temperature, water pressure and salinityThe random variation, i.e. the true propagation velocity of the signal is random, is unknown and therefore has

Wherein N is more than or equal to 1, M is more than or equal to 1, N + M is more than or equal to 4, for example, N is 4, M is 1, or N is 5, M is 3, j is more than or equal to 1 and less than or equal to N, i is more than or equal to 1 and less than or equal to M, and Delta s is_jIndicating the position error, at, of the jth receiver_iIndicating the position error, Δ s, of the ith transmitter_jAnd Δ t_iAll obey zero mean and variance of

Gaussian distribution of 2. ltoreq. sigma_cLess than or equal to 20 m/s.

Step two: the signal transmitted by the transmitter is received by the receiver successively after passing through two paths, namely a direct path and an indirect path reflected by the target; then, ellipse fitting is performed by using the signal arrival time difference, as shown in fig. 2, and a measurement model is obtained, which is described as:

due to the presence of measurement noise and the fact that the true values of the coordinate positions of the receiver and of the transmitter and the true propagation velocity of the signal are unknown, the nominal coordinate position of the receiver and of the transmitter are substituted into the measurement model, taking into account both the case where the true propagation velocity of the signal is assumed to be a random variable of known distribution and the case where the true propagation velocity of the signal is completely unknown, thus obtaining two models, namely: then s is_jAnd t_iSubstituting into the measurement model, and considering the case that the true propagation velocity of the signal is a random variable with a known distribution and the case that the true propagation velocity of the signal is completely unknownThe measurement model under the condition and the measurement model under the condition that the true propagation velocity of the signal is completely unknown describe the measurement model under the condition that the true propagation velocity of the signal is known to be distributed as follows:

wherein,

the real time difference between the time spent by the jth receiver after the signal transmitted by the ith transmitter is received by the jth receiver after passing through the indirect path reflected by the target and the time spent by the jth receiver after the signal transmitted by the ith transmitter is received by the jth receiver after passing through the direct path is represented by the symbol "| | |" which is a euclidean distance, s.t. "represents that" is constrained by … … ", τ_i,jA measured time difference, ε, between the time elapsed for the signal transmitted by the ith transmitter to be received by the jth receiver after passing through the indirect path where the target reflects and the time elapsed for the signal transmitted by the ith transmitter to be received by the jth receiver after passing through the direct path_i,jAnd

are all intermediate variables that are introduced into the reactor,

Δτ_i,jobeying a mean value and a variance of zero

are all introduced intermediate changeThe amount of the (B) component (A),

"T" represents the transpose of a vector or matrix, and both the measurement model in the case of a known distribution of the true propagation velocity of the signal and the measurement model in the case of a completely unknown true propagation velocity of the signal are highly nonlinear models.

then, the linear relation equation under the condition of the known distribution of the real propagation velocity of the signal is arranged into a matrix form, and the matrix form under the condition of the known distribution of the real propagation velocity of the signal is obtained: b epsilon ═ B-Ay^o(ii) a Similarly, the linear relation equation under the condition that the true propagation velocity of the signal is completely unknown is arranged into a matrix form, and the matrix form under the condition that the true propagation velocity of the signal is completely unknown is obtained:

wherein, B represents the introduced intermediate matrix,

I_Midentity matrix, symbol of dimension M × M

Represents the kronecker product, diag ([ | u)^o-s₁||,||u^o-s₂||,...,||u^o-s_N||]) Represents the vector [ | | u [ ]^o-s₁||,||u^o-s₂||,...,||u^o-s_N||]Each element in turn being a diagonal matrix of diagonal elements, s₁Nominal coordinate position, s, representing the original nominal deployment of the 1 st receiver in the reference coordinate system₂Nominal coordinate position, s, representing the nominal original deployment of the 2 nd receiver in the reference coordinate system_NA nominal coordinate position representing an original nominal deployment of the nth receiver in the reference coordinate system, epsilon ═ epsilon_1,1,ε_1,2,...,ε_1,N,ε_2,1,...,ε_M,N]^T，ε_1,1,ε_1,2,...,ε_1,N,ε_2,1,...,ε_M,NAre all according to

a is an introduced intermediate matrix, and A ═ A₁,A₂,...,A_i,...,A_M]^T，A₁,A₂,...,A_i,...,A_MIs an element in the group A, and has the following structure,

0_i-1dimension of representationA column vector of 1 × (i-1) and all elements 0, 0_M-iA column vector representing dimensions 1 (M-i) and elements all 0, τ_i,1A measured time difference, τ, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the 1 st receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the 1 st receiver after passing through the direct path_i,2A measured time difference, τ, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the 2 nd receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the 2 nd receiver after passing through the direct path_i,NA measured time difference, y, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the nth receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the nth receiver after passing through the direct path^oFor vectors consisting of unknown variables, y^o＝[u^oT,||u^o-t₁||,||u^o-t₂||,...,||u^o-t_M||]^T，t₁Nominal coordinate position, t, representing the original nominal deployment of the 1 st transmitter in the reference coordinate system₂Nominal coordinate position, t, representing the nominal original deployment of the 2 nd transmitter in the reference coordinate system_MRepresenting the nominal coordinate position of the original nominal deployment of the mth transmitter in the reference coordinate system,

are all based on

The calculation results in that,

in order to introduce the intermediate vector(s),

are all made of

The elements (A) and (B) in (B),

in order to introduce an intermediate matrix of the matrix,

is composed of

The elements in (A) and (B) are selected,

0_M-1a column vector with dimension 1 (M-1) and elements all 0, a scaling constant introduced to avoid numerical problems in the case where the true propagation velocity of the signal is completely unknown, and a ∈ [100,1000 ]]In this embodiment, α is 1000,

is a vector of unknown variables that is,

is shown by c^oThe value after scaling down by a times.

Step four: converting the matrix form in the case of the known distribution of the true propagation velocities of the signals into a Weighted Least Squares (WLS) problem in the case of the known distribution of the true propagation velocities of the signals, described as:

likewise, the matrix form in the case where the true propagation velocity of the signal is completely unknown is converted into a Weighted Least Squares (WLS) problem in the case where the true propagation velocity of the signal is completely unknown, described as:

similarly, the weighted least squares problem in the case where the true propagation velocity of the signal is completely unknown is converted into a non-convex constrained optimization problem in the case where the true propagation velocity of the signal is completely unknown by adding a constraint condition, which is described as:

wherein,

representing the change y such that the objective function (b-Ay)^TQ^-1(b-Ay) min, Q and

are all introduced intermediate matrices, since Q is equal to u desired to be solved^oTIt is the case that it is a related,

with u desired to be solved^oTAnd

are related, and when Q and

is unknown, so to deal with the problem, let Q and

are all I_MNI.e. Q has an initial value of I_MN，

Is initially ofI_MN，I_MNRepresenting an identity matrix of dimensions MN x MN, y being AND^oCorresponding to a vector consisting of the variables to be optimized, y ═ u^T,||u-t₁||,||u-t₂||,...,||u-t_M||]^T，

Is prepared by reacting with

Corresponding to the vector consisting of the variables to be optimized,

u represents u^oThe corresponding variable to be optimized is set to be,

to represent

to represent

The number 4 element of (a) is,

to represent

The number 3 element of (a) is,

to represent

The 4+ i-th element in (b),

is represented by

The 1 st element and the 2 nd element of (a),

to represent

The 4+ M + i th element in (b).

Step five: let Y equal to yy^TConverting the non-convex constraint optimization problem under the condition of the known distribution of the true propagation speed of the signal into a corresponding equivalent problem, which is described as follows:

order to

wherein Y and

all are introduced intermediate matrix, tr { } represents the trace of matrix solving, rank () represents the rank of matrix solving, F and

are all the intermediate matrixes introduced in the process of the preparation,

y (2+ i ) represents an element of 2+ i row and 2+ i column in Y, and Y (1:2) represents a group consisting of 1 st row to 2 nd column in YA matrix of all the elements is formed,

represent

Row 4+ i and column 4+ i,

is represented by

A matrix of all elements of the 1 st to 2 nd columns in the 1 st to 2 nd rows,

to represent

Row 3 and column 4+ i in (b),

to represent

Row 4, column 4+ i of (1)

To represent

Row 3 and column 4+ M + i,

to represent

Row 3, column 3 elements in (a).

Step six: relaxing an equivalent problem corresponding to a non-convex constraint optimization problem under the condition of known distribution of the real propagation speed of the signal into an easily-processed convex problem by adopting a semi-definite relaxation (SDR) technology, and describing the easily-processed convex problem under the condition of known distribution of the real propagation speed of the signal as follows:

step seven: since the rank-1 constraint of Y is left out by the semi-positive relaxation technique, some additional constraints are added to the tractable convex problem in the case of the known distribution of the true propagation velocity of the signal to tighten the problem, that is, the tightened convex problem in the case of the known distribution of the true propagation velocity of the signal is obtained by adding additional constraints to the tractable convex problem in the case of the known distribution of the true propagation velocity of the signal, and the tightened convex problem in the case of the known distribution of the true propagation velocity of the signal is described as follows:

due to semi-positive definite relaxation technology abandoning

Rank 1, therefore adding some additional constraints to the tractable convex problem in the case that the true propagation velocity of the signal is completely unknown, that is, adding additional constraints to the tractable convex problem in the case that the true propagation velocity of the signal is completely unknown, to obtain a clamped convex problem in the case that the true propagation velocity of the signal is completely unknown, which is described as:

to represent

Row 2+ i and column 2+ j in (b),

is represented by

A matrix of all elements of the 2+ j column in the 1 st to 2 nd rows,

to represent

The 2+ j-th element in (b),

to represent

Row 4+ i and column 3 elements in (b),

is represented by

A matrix of all elements of column 3 in row 1 to row 2,

to represent

Row 4+ M + i and column 3 elements,

to represent

Row 4+ i and column 4+ j in (b),

is represented by

A matrix of all elements of column 4+ j in row 1 to row 2,

to represent

The 4+ j-th element in (b),

to represent

Row 4+ i and column 4+ M + j in (b),

is represented by

A matrix of all elements of the 4+ M + j column in the 1 st to 2 nd rows,

to represent

The 4+ M + j element in (1).

Step eight: by using C in matlabSolving the convex problem after the clamping under the condition that the real propagation speed of the signal is distributed by the VX tool box to obtain the optimal solution of y, and recording the optimal solution as y^*(ii) a Similarly, the clamped convex problem under the condition that the real propagation speed of the signal is completely unknown is solved by utilizing a CVX tool box in matlab to obtain

Is given as

In the case where the true propagation velocity of the resulting signal is completely unknown u^oAnd

corresponding to u₂ ^*And

to represent

The 1 st element and the 2 nd element in the vector,

to represent

The 3 rd element in (1).

Step ten: u is to be₁ ^*Substitution into

Updating the value of Q, and adding u₂ ^*And

substitution into

Update in the middle

A value of (d); then at Q and

Wherein Q is_τ、D_z、D_t、D_sIn order to introduce an intermediate matrix of the matrix,

I_MNidentity matrix with dimension MN × MN, D_z＝[D_t,D_s]，D_t＝Diag{D_t,1,D_t,2,...,D_t,i,...,D_t,M}，D_t＝Diag{D_t,1,D_t,2,...,D_t,i,...,D_t,MDenotes by D_t,1,D_t,2,...,D_t,i,...,D_t,MConstructing a block diagonal matrix D as diagonal blocks_t，

Are all according to

The calculation results in that,

is represented by

Constructing a block diagonal matrix D as diagonal blocks_s,i，

Are all according to

The calculation results in that,

an identity matrix having a dimension of 2(M + N) × 2(M + N), τ ═ τ_1,1,τ_1,2,...,τ_1,N,τ_2,1,...,τ_M,N]^T，τ_1,1A measured time difference, τ, representing the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 1 st receiver after passing through the indirect path of reflection from the target and the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 1 st receiver after passing through the direct path_1,2Representing signals transmitted by the 1 st transmitterThe measured time difference, τ, between the time elapsed for the signal reflected by the target to be received by the 2 nd receiver after the indirect path and the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 2 nd receiver after the direct path_1,NA measured time difference, τ, representing the time elapsed for the signal transmitted by the 1 st transmitter to be received by the Nth receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the 1 st transmitter to be received by the Nth receiver after passing through the direct path_2,1A measured time difference, τ, representing the time elapsed for the signal transmitted by the 2 nd transmitter to be received by the 1 st receiver after passing through the indirect path of reflection from the target and the time elapsed for the signal transmitted by the 2 nd transmitter to be received by the 1 st receiver after passing through the direct path_M,NA measured time difference representing the time elapsed for the signal transmitted by the mth transmitter to be received by the nth receiver after passing through the indirect path reflected by the target and the time elapsed for the signal transmitted by the mth transmitter to be received by the nth receiver after passing through the direct path.

To further illustrate the feasibility and effectiveness of the method of the present invention, comparative experiments were conducted on the method of the present invention.

The positioning method for experimental comparison comprises an existing four-step weighted least square method (four-step WLS for short) and an existing generalized confidence domain sub-problem method (GTRS for short).

Fig. 3 shows the method of the invention (SDR for short) in comparison with the four-step WLS method, the GTRS method and the hybrid krame-luo lower bound (HCRLB) at σ, in the case of a known distribution of the true propagation speed of the signal and a single transmitter_τ0.05 second,. sigma_cStandard deviation sigma of gaussian distribution obeyed by root mean square error along with position errors of transmitter and receiver in 10 random scenes when the number of receivers is 4 and the number of transmitters is 1 is 7.5 m/s_zSchematic diagram of the change from 20 meters growth to 200 meters. As can be seen from fig. 3, the four-step WLS method is clearly less excellent than the GTRS method and the inventive method, which deviates from the HCRLB early, whereas the inventive method has better positioning performance than the GTRS method in case the standard deviation of the gaussian distribution obeying the position errors of the transmitter and receiver is sufficiently large.

Fig. 4 shows the method of the invention (SDR for short) in combination with the four-step WLS method, the GTRS method and the hybrid krame-luo lower bound (HCRLB) at σ, in the case of a known distribution of the true propagation speed of the signal and a single transmitter_z50 m, sigma_cRoot mean square error with delta tau in 10 random scenes when 7.5 m/s, 4 receivers and 1 transmitter are used_i,jStandard deviation sigma of the obeyed Gaussian distribution_τ(reflecting measurement noise) increases from 0.02 seconds to 0.2 seconds. As can be seen from FIG. 4, the method of the present invention and the GTRS method are used when the measurement noise is not too large, i.e., σ_τBelow 0.16 seconds, HCRLB accuracy can be achieved, in contrast to the four-step WLS method which deviates from HCRLB early.

By analyzing fig. 3 and fig. 4, it can be known that the method of the present invention has stronger robustness to larger measurement noise and larger position errors of the transmitter and the receiver, and has better positioning performance; the four-step WLS method does not work perfectly with large noise or large position errors; the GTRS method is only suitable for the comparative single transmitter case because it cannot handle the multiple transmitter case.

FIG. 5 shows the inventive method (SDR for short) in combination with the four-step WLS method and the hybrid Cramer-Role lower bound (HCRLB) at σ, in the case where the true propagation velocity of the signal is completely unknown_τ0.05 second,. sigma_cStandard deviation sigma of gaussian distribution obeyed by root mean square error along with position errors of transmitter and receiver in 10 random scenes when the number of receivers is 5 and the number of transmitters is 3_zSchematic diagram of the change from 20 meters growth to 200 meters. FIG. 6 shows the inventive method (SDR for short) in combination with the four-step WLS method and the hybrid Cramer-Rou lower bound (HCRLB), σ_z50 seconds, σ_cRoot mean square error with delta tau in 10 random scenes when 7.5 m/s, 5 receivers and 3 transmitters are used_i,jStandard deviation sigma of the obeyed Gaussian distribution_τ(reflecting measurement noise) increases from 0.02 seconds to 0.2 seconds.

Analyzing fig. 5 and fig. 6, it can be seen that the positioning performance of the method of the present invention is much more stable, the HCRLB accuracy can be achieved under the condition of little noise, and the four-step WLS method can be successfully positioned under the condition of less noise and less position error.

Claims

1. A positioning method of multi-base sonar based on signal arrival time difference is characterized by comprising the following steps:

Recording the true propagation velocity of a signal in an underwater environment as c^oThe nominal coordinate position of the original nominal deployment of the jth receiver in the reference coordinate system is recorded as s_jThe nominal coordinate position of the original nominal deployment of the ith transmitter in the reference coordinate system is recorded as t_iThe standard propagation velocity of a signal in an underwater environment is recorded

Wherein N is more than or equal to 1, M is more than or equal to 1, N + M is more than or equal to 4, j is more than or equal to 1 and less than or equal to N, i is more than or equal to 1 and less than or equal to M, and delta s_jIndicating the position error, at, of the jth receiver_iIndicating the position error, Δ s, of the ith transmitter_jAnd Δ t_iAll obey zero mean and variance of

Gaussian distribution of 2. ltoreq. sigma_cLess than or equal to 20 m/s;

wherein,

the real time difference between the time spent by the jth receiver after the signal transmitted by the ith transmitter is received by the jth receiver after passing through the indirect path reflected by the target and the time spent by the jth receiver after the signal transmitted by the ith transmitter is received by the jth receiver after passing through the direct path is represented by "| | |" which is a euclidean distance, "s.t." which means "is constrained to … …",τ_i,ja measured time difference, ε, between the time elapsed for the signal transmitted by the ith transmitter to be received by the jth receiver after passing through the indirect path where the target reflects and the time elapsed for the signal transmitted by the ith transmitter to be received by the jth receiver after passing through the direct path_i,jAnd

are all intermediate variables that are introduced into the reactor,

Δτ_i,jobeying a mean value and a variance of zero

are all intermediate variables that are introduced into the reactor,

then, the linear relation equation under the condition of the known distribution of the real propagation velocity of the signal is arranged into a matrix form, and the matrix form under the condition of the known distribution of the real propagation velocity of the signal is obtained: b epsilon-Ay^o(ii) a Similarly, the linear relation equation under the condition that the true propagation velocity of the signal is completely unknown is arranged into a matrix form, and the matrix form under the condition that the true propagation velocity of the signal is completely unknown is obtained:

wherein, B represents the introduced intermediate matrix,

I_Midentity matrix, symbol of dimension M × M

Represents kronecker product, diag ([ | | u)^o-s₁||,||u^o-s₂||,...,||u^o-s_N||]) Represents the vector [ | | u [ ]^o-s₁||,||u^o-s₂||,...,||u^o-s_N||]Each element in turn being a diagonal matrix of diagonal elements, s₁Nominal coordinate position, s, representing the original nominal deployment of the 1 st receiver in the reference coordinate system₂Nominal coordinate position, s, representing the nominal original deployment of the 2 nd receiver in the reference coordinate system_NA nominal coordinate position representing an original nominal deployment of the nth receiver in the reference coordinate system, epsilon ═ epsilon_1,1,ε_1,2,...,ε_1,N,ε_2,1,...,ε_M,N]^T，ε_1,1,ε_1,2,...,ε_1,N,ε_2,1,...,ε_M,NAre all based on

0_i-1a column vector representing dimensions 1 (i-1) and elements all 0, 0_M-iA column vector having dimensions of 1X (M-i) and elements of all 0, τ_i,1A measured time difference, τ, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the 1 st receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the 1 st receiver after passing through the direct path_i,2A measured time difference, τ, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the 2 nd receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the 2 nd receiver after passing through the direct path_i,NA measured time difference, y, representing the time elapsed for the signal transmitted by the ith transmitter to be received by the nth receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the ith transmitter to be received by the nth receiver after passing through the direct path^oFor a direction consisting of unknown variablesAmount, y^o＝[u^oT,||u^o-t₁||,||u^o-t₂||,...,||u^o-t_M||]^T，t₁Nominal coordinate position, t, representing the original nominal deployment of the 1 st transmitter in the reference coordinate system₂Nominal coordinate position, t, representing the nominal deployment of the 2 nd transmitter in the reference coordinate system_MRepresenting the nominal coordinate position of the original nominal deployment of the mth transmitter in the reference coordinate system,

are all according to

The calculation results in that,

in order to introduce the intermediate vector(s),

are all made of

The elements (A) and (B) in (B),

in order to introduce an intermediate matrix of the matrix,

is composed of

The elements (A) and (B) in (B),

Is a vector made up of unknown variables,

denotes c^oA value after scaling down by a factor of alpha;

likewise, the matrix form in the case where the true propagation velocity of the signal is completely unknown is converted into a weighted least squares problem in the case where the true propagation velocity of the signal is completely unknown, described as:

wherein Q and

are all introduced intermediate matrix, and the initial value of Q is I_MN，

Is initially value of_MN，I_MNRepresenting an identity matrix of dimensions MN by MN, y being^oCorresponding to a vector consisting of the variables to be optimized, y ═ u^T,||u-t₁||,||u-t₂||,...,||u-t_M||]^T，

Is and is

Corresponding to the vector consisting of the variables to be optimized,

u represents u^oThe corresponding variable to be optimized is set to be,

represent

Corresponding variables to be optimized, y (2+ i) represents the 2+ i element in y, and y (1:2) represents the 1 st element and the 2 nd element in yA column vector of the elements is formed,

to represent

The number 4 element of (a) is,

to represent

The 3 rd element in (a) is,

to represent

The 4+ i th element in (b),

is represented by

The 1 st element and the 2 nd element of (a),

to represent

The 4+ M + i th element in (a);

order to

wherein Y and

are all the intermediate matrixes introduced in the process of the preparation,

to represent

Row 4+ i and column 4+ i,

is represented by

A matrix of all elements of the 1 st to 2 nd columns in the 1 st to 2 nd rows,

to represent

Row 3 and column 4+ i in (b),

represent

Row 4 and column 4+ i in (b),

to represent

Row 3 and column 4+ M + i,

to represent

Row 3, column 3 elements in (1);

step six: relaxing an equivalent problem corresponding to a non-convex constraint optimization problem under the condition of known distribution of the true propagation speed of the signal into an easily-processed convex problem by adopting a semi-positive definite relaxation technology, and describing the easily-processed convex problem under the condition of known distribution of the true propagation speed of the signal as follows:

similarly, a semi-positive definite relaxation technology is adopted to relax the equivalent problem corresponding to the non-convex constraint optimization problem under the condition that the true propagation speed of the signal is completely unknown into an easily-processed convex problem, and the easily-processed convex problem under the condition that the true propagation speed of the signal is completely unknown is described as follows:

step seven: by the ease of handling in the case of a known distribution of the true propagation speed of the signalThe additional constraint is added to the convex problem to tighten the problem, and the tightened convex problem under the condition of the known distribution of the true propagation speed of the signal is obtained, which is described as follows:

wherein Y (2+ i,2+ j) represents an element in row 2+ i and column 2+ j in Y, Y (1:2,2+ j) represents a matrix composed of all elements in row 1 to column 2+ j in row 2 in Y, Y (2+ j) represents an element in column 2+ j in Y,

to represent

Row 2+ i and column 2+ j in (b),

is represented by

A matrix of all elements of the 2+ j column in the 1 st to 2 nd rows,

to represent

The 2+ j-th element in (b),

to represent

Row 4+ i and column 3 elements in (b),

is represented by

A matrix of all elements of column 3 in row 1 to row 2,

to represent

Row 4+ M + i and column 3 elements in (b),

to represent

Row 4+ i and column 4+ j in (b),

is represented by

A matrix of all elements of column 4+ j in row 1 to row 2,

represent

The 4+ j th element in (b),

to represent

Row 4+ i and column 4+ M + j in (b),

is represented by

A matrix of all elements of the 4+ M + j column in the 1 st to 2 nd rows,

to represent

The 4+ M + j element in (a);

step eight: solving the convex problem after the clamping under the condition that the real propagation speed of the signal is distributed in a known way to obtain the optimal solution of y, and recording the optimal solution as y^*(ii) a Similarly, solving the convex problem after the tightening under the condition that the real propagation speed of the signal is completely unknown to obtain

Is recorded as

Step nine: according to y^*Obtaining u with known distribution of true propagation velocity of signal^oIs expressed as u₁ ^*，u₁ ^*＝y^*(1: 2); and according to

corresponding to u₂ ^*And

to represent

The 1 st element and the 2 nd element in the vector,

to represent

The 3 rd element in (1);

step ten: will u₁ ^*Substitution into

Update the value of Q, and add u₂ ^*And

substitution into

Update in the middle

A value of (d); then at Q and

repeatedly executing the steps after the value of (2) is updatedStep four to step eight; and then according to y obtained by repeated execution^*Obtaining u with known distribution of true propagation velocity of signal^oIs given as u^* _1,final，u^* _1,final＝y^*(1: 2); and obtained on repeated execution

Are all according to

The calculation results in that,

is represented by

Constructing a block diagonal matrix D as diagonal blocks_s,i，

Are all according to

The calculation results in that,

I_2(M+N)an identity matrix having a dimension of 2(M + N) × 2(M + N), τ ═ τ_1,1,τ_1,2,...,τ_1,N,τ_2,1,...,τ_M,N]^T，τ_1,1A measured time difference, τ, representing the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 1 st receiver after passing through the indirect path of reflection from the target and the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 1 st receiver after passing through the direct path_1,2A measured time difference, τ, representing the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 2 nd receiver after passing through the indirect path of reflection from the target and the time elapsed for the signal transmitted by the 1 st transmitter to be received by the 2 nd receiver after passing through the direct path_1,NA measured time difference, τ, representing the time elapsed for the signal transmitted by the 1 st transmitter to be received by the Nth receiver after passing through the indirect path where the signal is reflected by the target and the time elapsed for the signal transmitted by the 1 st transmitter to be received by the Nth receiver after passing through the direct path_2,1Indicating that the signal transmitted by the 2 nd transmitter is reflected by the 2 nd transmitter after passing through an indirect path of the targetThe measured time difference, τ, between the time elapsed for reception by the 1 st receiver and the time elapsed for reception by the 1 st receiver after the signal transmitted by the 2 nd transmitter has passed through the direct path_M,NA measured time difference representing a time elapsed for a signal transmitted by the mth transmitter to be received by the nth receiver after passing through the indirect path where the signal is reflected by the target and a time elapsed for a signal transmitted by the mth transmitter to be received by the nth receiver after passing through the direct path.