Underwater single beacon semi-definite programming positioning method and system under linear acoustic profile condition

Abstract

The invention belongs to the technical field of underwater positioning, and discloses an underwater single beacon semi-definite programming positioning method and system under a linear sound profile condition. According to the method, based on a semi-definite relaxation technology, a positioning problem in a semi-definite programming form is obtained, the semi-definite programming problem is solved through an interior point method, and an initial position solution of single beacon positioning is obtained; and the estimation error of the initial position solution is explicitly expressed, a linear estimation formula of the estimation error is constructed in combination with a virtual long baseline positioning equation, and the estimation error is solved based on weighted least squares, so that the initial position solution is corrected, and a more accurate AUV position estimation solution is obtained. Compared with other single beacon positioning methods which do not consider sound propagation refraction, the method of the invention can obtain a more accurate position calculation result under a linear sound profile condition, and enhances single beacon positioning, especially the practical application capability of the method in deep sea areas.

Description

Technical Field

[0001] The invention belongs to the technical field of underwater positioning, and in particular relates to an underwater single beacon semi-definite planning positioning method and system under linear acoustic profile conditions. Background Art

[0002] Underwater vehicles (AUVs) are widely used in underwater search and rescue, underwater operations, and marine environment data collection. AUVs need to obtain their own position coordinates in real time to ensure the accuracy of motion control and the validity of the collected data. Since electromagnetic wave signals attenuate rapidly underwater, hydroacoustic positioning has become the most widely used and effective positioning method in underwater environments. Compared with the common underwater long baseline positioning and ultra-short baseline positioning technologies, the positioning method that combines a single hydroacoustic beacon with dead reckoning information only requires a single hydroacoustic beacon with a known position to provide positioning assistance, which can greatly reduce the time and economic cost of long baseline positioning, and can obtain greater positioning flexibility, and has therefore attracted widespread attention from researchers.

[0003] The current underwater single beacon positioning method usually assumes that the water acoustic speed is constant, and directly uses the distance between the AUV and the beacon as the observed value. In actual underwater positioning applications, affected by factors such as seawater temperature, salinity and depth, the water acoustic speed varies with space, especially in the depth direction, the sound speed value changes with a large gradient, which will cause obvious sound propagation refraction. Ignoring the influence of sound propagation refraction will cause a large distance calculation error, which will in turn lead to a large single beacon positioning error, affecting the actual application capability of the single beacon system. Studies have shown that for deep-sea environments, the water acoustic speed profile can be approximated as linear, that is, the sound speed value can be approximated as a linear function of depth. In other words, studying the single beacon positioning method under the condition of linear sound speed profile has broad application prospects. By more accurately describing the trajectory of the sound propagation channel under the condition of linear sound profile and the signal propagation time observation model, the error of single beacon positioning can be reduced to a certain extent. Summary of the invention

[0004] In order to overcome the problems existing in the related art, the embodiments disclosed in the present invention provide a method and system for semi-definite planning positioning of a single underwater beacon under linear acoustic profile conditions.

[0005] The technical solution is as follows: a semidefinite planning positioning method for a single underwater beacon under a linear acoustic profile condition, which solves the problem of acoustic propagation refraction caused by the spatial variation of sound speed in the positioning of a single underwater beacon, and obtains an AUV position estimation solution based on a semidefinite relaxation method to solve the single beacon positioning position under a linear acoustic profile condition;

[0006] Based on the obtained pseudo-linear equation of virtual long baseline positioning under linear acoustic profile conditions, a constrained optimization problem is constructed, and a semidefinite programming problem is constructed through semidefinite relaxation to solve it, and the preliminary position solution of single beacon positioning is obtained;

[0007] The estimation error of the preliminary positioning solution of single beacon positioning is explicitly expressed, and based on weighted least squares, the preliminary position solution is corrected using the virtual long baseline positioning equation to obtain the final AUV position estimate.

[0008] Furthermore, before obtaining the pseudo-linear equation of virtual long baseline positioning under the condition of linear acoustic profile, it is necessary to perform the following:

[0009] Take any point in the positioning area as the origin, set the east, north and sky directions as X, Y and Z axes respectively, and establish the underwater single beacon positioning geodetic coordinate system; take the AUV centroid as the origin, set the front, right and bottom directions as x, y and z axes respectively, and establish the AUV body coordinate system;

[0010] According to the speed of the AUV in the body coordinate system detected by the DVL and the attitude of the AUV detected by the attitude sensor, a dead reckoning model is constructed;

[0011] The hydroacoustic beacon periodically emits hydroacoustic signals. The beacon position coordinates are known. The AUV is equipped with a hydrophone to measure the propagation time of the hydroacoustic signal. The sound velocity profile is modeled as a linear sound profile. Combining the known hydroacoustic profile parameter information with the hydroacoustic beacon position coordinates, a hydroacoustic observation model under the linear sound profile condition is constructed to obtain the relationship between the signal propagation time and the AUV position.

[0012] The AUV dead reckoning information is integrated and all acoustic signal propagation time observations within the time window width N are used to construct a virtual long baseline positioning model under linear acoustic profile conditions.

[0013] Furthermore, a dead reckoning model is constructed, including: the relationship between the AUV position variables at time k and time ki (i≥1) in the dead reckoning model is:

[0014]

[0015]

[0016] In the formula, x k-i is the x-direction position coordinate of the AUV at time ki, y k-i is the u-direction position coordinate of the AUV at time ki, z k-i is the z-direction position coordinate of the AUV at time ki, Δt is the discrete time interval, i is the time index variable, and v x,k-t is the x-direction velocity of AUV at time kt, v y,k-t is the y-direction velocity of AUV at time kt, vz,k-t is the z-direction velocity of AUV at time kt;

[0017] The hydroacoustic observation model under the condition of linear acoustic profile is constructed, including: the relationship between the AUV position and the propagation time of the hydroacoustic signal is:

[0018] 0=(x k -x b ) 2 +(y k -y b ) 2 +(z k -z b ) 2 -m k (az k +b)

[0019] The constructed virtual long baseline positioning model under linear acoustic profile conditions includes:

[0020]

[0021] Moving the noise term to the left side of the equal sign gives:

[0022]

[0023] Among them, η i is the noise coefficient of the virtual long baseline positioning equation, defined as:

[0024]

[0025] Furthermore, by introducing auxiliary variables and performing algebraic transformation, the pseudo-linear equation for virtual long baseline positioning is obtained. The steps are as follows:

[0026] Expand the right side of the virtual long baseline positioning equation to obtain:

[0027]

[0028] definition:

[0029]

[0030]

[0031] Then we have:

[0032]

[0033] All N+1 equations are integrated into a vector matrix form:

[0034]

[0035] In the formula, Bk is the noise coefficient matrix, v k is the noise vector of the underwater acoustic signal propagation time, h k ,G k are system model parameters, is the unknown quantity to be estimated, and the definitions of various variables include:

[0036]

[0037] In the formula, is the observed value of the propagation time of the underwater acoustic signal at time kN, v k-N is the propagation time observation noise of the underwater acoustic signal at time kN.

[0038] Further, the steps of obtaining a single beacon positioning preliminary position solution include:

[0039] According to the constructed pseudo linear equation:

[0040]

[0041] The estimation problem is transformed into a constrained optimization problem, namely:

[0042]

[0043] For any vector a, a[m:n] represents the subvector consisting of the mth to nth elements of vector a, and W 1,k is the weight matrix, which is defined as:

[0044]

[0045] In the formula, E(·) represents the expected value of the random variable, and the superscript -1 represents the inverse of the matrix. It is approximately equal to Q k is the observed variance matrix of the underwater acoustic signal propagation time, defined as The specific form is:

[0046]

[0047] In the formula, is the observed variance of the underwater acoustic signal propagation time at time kN, is the observed variance of the underwater acoustic signal propagation time at time k-1, is the observed variance of the propagation time of the underwater acoustic signal at time k;

[0048] The objective function of the optimization problem is transformed into:

[0049]

[0050] Where tr represents the trace of the matrix;

[0051] Defining auxiliary variables Then the constrained optimization problem is transformed into:

[0052]

[0053] Among them, for any matrix A, A[m:n,p:q] represents the submatrix consisting of the elements of the mth to nth rows and the pth to qth columns of matrix A; the last constraint in the above equation is mathematically equivalent to:

[0054]

[0055] rank(Ψ k )=1

[0056] In the formula, rank(Ψ k ) is the matrix Ψ k The rank of the problem is 1, and the rank constraint is removed. The optimization problem is transformed into a semidefinite programming problem, which can be efficiently solved numerically by the interior point method to obtain the optimal solution of the semidefinite programming. and The initial estimates of the AUV position and sound speed are extracted as:

[0057]

[0058] In the formula, is the preliminary estimated value of the AUV position at time k.

[0059] Furthermore, the obtained single beacon positioning preliminary position solution further includes: the noise coefficient matrix B k The calculation depends on the sound velocity value. In the application process, the weight W is first 1,k Set it as the unit matrix and calculate the preliminary least squares solution Substitute the sound velocity value corresponding to this solution into the noise coefficient matrix B k , calculate the weight W 1,k .

[0060] Further, the single beacon positioning preliminary position estimation solution correction method includes:

[0061] The estimated error corresponding to the obtained preliminary position estimate is defined as:

[0062]

[0063] In the formula, x k ,y k ,z k is the real position coordinate of AUV at time k, Δx k for The estimation error, Δy k for The estimation error, Δz k for The estimation error of

[0064] Substituting the above equation into

[0065]

[0066] get:

[0067]

[0068] Ignoring the second-order noise term, the simplification yields:

[0069]

[0070] in,

[0071]

[0072] All N+1 equations are integrated into a vector matrix form:

[0073] B k v k =g k -F k Δp k

[0074] In the formula, g k With F k is the system model parameter, Δη k is the unknown quantity to be estimated, and each variable is defined as follows:

[0075]

[0076] Based on the weighted least squares estimation principle, the unknown quantity Δp is obtained k The weighted least squares estimation solution of for:

[0077]

[0078] Using the estimated error variable The initial estimate of the semidefinite programming is corrected to obtain the final estimate:

[0079]

[0080] In the formula, is the final estimate of the x-position, is the final estimate of the y-direction position, is the final estimate of the z-direction position.

[0081] Another object of the present invention is to provide an underwater single beacon semi-definite planning positioning system under linear acoustic profile conditions, the system implements the underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions, the system includes an AUV, the AUV is equipped with a hydrophone, a DVL, a posture sensor and a CTD; the AUV detects the sound velocity profile parameters of the experimental area through the CTD; the hydroacoustic beacon periodically broadcasts a hydroacoustic signal, the AUV periodically observes its velocity in the body coordinate system through the carried DVL, combines the AUV attitude angle observed by the posture sensor, calculates the AUV velocity in the geodetic coordinate system, performs dead reckoning, and records and stores the dead reckoning data; after receiving the hydroacoustic signal emitted by the hydroacoustic beacon, the AUV obtains the hydroacoustic signal in combination with the known hydroacoustic signal emission time. propagation time; when the AUV receives a certain number of hydroacoustic signals, a virtual long baseline array is constructed in combination with the stored dead reckoning data; a virtual long baseline positioning problem is constructed in combination with the relationship between the hydroacoustic propagation time and the AUV position under the linear acoustic profile; an auxiliary variable is introduced, and a pseudo-linear positioning equation is obtained based on algebraic transformation to construct a constrained optimization problem; based on the semidefinite relaxation technology, a positioning problem in the form of a semidefinite programming is obtained, and the semidefinite programming problem is solved by the interior point method to obtain a preliminary position solution for single beacon positioning; the estimated error of the preliminary position solution is explicitly expressed, and a linear estimation formula of the estimation error is constructed in combination with the virtual long baseline positioning equation, and the estimation error is solved based on the weighted least squares to correct the preliminary position solution and obtain a more accurate AUV position estimation solution.

[0082] Furthermore, the underwater single beacon semi-definite planning positioning system under linear acoustic profile conditions is suitable for deep sea environments, and single beacon positioning is performed by the underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions described in claim 1.

[0083] Furthermore, the underwater single beacon semi-definite planning positioning system under the linear acoustic profile condition is carried on a computer-readable storage medium, and the computer-readable storage medium stores a computer program. When the computer program is executed by the processor, it can realize the functions of the underwater single beacon semi-definite planning positioning system under the linear acoustic profile condition.

[0084] Combining all the above technical solutions, the beneficial effects of the present invention are as follows: the present invention focuses on the problem of sound propagation refraction caused by the spatial variation of sound speed in underwater single beacon positioning, and designs a single beacon positioning position solution method under linear sound profile conditions based on the semidefinite relaxation method. The underwater vehicle (AUV) is equipped with a hydrophone, a Doppler velocimeter, an attitude sensor, and a temperature-salinity-depth meter; before use, the AUV detects the sound speed profile parameters of the experimental area through the temperature-salinity-depth meter; the hydroacoustic beacon broadcasts hydroacoustic signals periodically, and the AUV obtains the speed of the AUV in the geodetic coordinate system through the carried Doppler velocimeter combined with the attitude sensor, performs dead reckoning, and records and stores the dead reckoning data; after receiving the hydroacoustic signal emitted by the hydroacoustic beacon, the AUV obtains the propagation time of the hydroacoustic signal in combination with the known hydroacoustic signal emission time; when the AUV receives a certain number of hydroacoustic signals, it constructs a virtual long baseline in combination with the stored dead reckoning data array; combining the relationship between the underwater acoustic propagation time and the AUV position under the linear acoustic profile, construct a virtual long baseline positioning problem; introducing auxiliary variables, and obtaining a pseudo-linear positioning equation based on algebraic transformation to construct a constrained optimization problem; based on the semidefinite relaxation technology, obtain a positioning problem in the form of a semidefinite programming, and solve the semidefinite programming problem by the interior point method to obtain a preliminary position solution for single beacon positioning; explicitly express the estimated error of the preliminary position solution, and construct a linear estimation formula of the estimated error in combination with the virtual long baseline positioning equation, and solve the estimated error based on the weighted least squares to correct the preliminary position solution and obtain a more accurate AUV position estimation solution. Compared with other single beacon positioning methods that do not consider the refraction of sound propagation, the method of the present invention can obtain more accurate position solution results under the condition of linear acoustic profile, enhance single beacon positioning, especially its practical application capability in deep sea areas.

[0085] The present invention takes into account the refraction of the water acoustic propagation channel that is widely present in real water acoustic positioning applications, and adopts a more reasonable and accurate water acoustic observation model, which can obtain better ranging accuracy and positioning accuracy, improve the validity and value of the data collected by the AUV, and enhance the operation and application value of the AUV itself.

[0086] Affected by the temperature, salinity and depth changes of seawater, the gradient of the water acoustic speed in the depth direction is large, which will cause obvious refraction of the sound propagation channel. This problem is widely present in all underwater positioning methods based on water acoustic ranging, including single beacon positioning. The current common processing method is to ignore the refraction of the channel, or introduce the concept of effective sound speed, and directly regard the effective sound speed as unknown. Ignoring the refraction of the channel will cause ranging errors and positioning errors, while directly considering the effective sound speed as unknown will ignore the valuable physical information of the water acoustic speed profile, which will also cause performance loss and positioning errors. It is urgent to study the single beacon positioning method that takes into account the refraction of sound propagation. The present invention directly and accurately describes the water acoustic observation model under the linear sound profile, directly processes the observation model that takes into account the propagation refraction, and designs a semi-definite programming positioning problem to solve it, which can obtain better position estimation performance. Description of the drawings

[0087] The accompanying drawings herein are incorporated in and constitute a part of the specification, illustrate embodiments consistent with the present disclosure, and together with the description, serve to explain the principles of the present disclosure;

[0088] Figure 1 It is a flow chart of the underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions provided by an embodiment of the present invention;

[0089] Figure 2 It is a schematic diagram of the underwater single beacon positioning geodetic coordinate system and the AUV body coordinate system provided by an embodiment of the present invention;

[0090] Figure 3 It is the AUV motion trajectory diagram;

[0091] Figure 4 The sound velocity profile used for the simulation;

[0092] Figure 5 Comparison of the average positioning errors of different methods. DETAILED DESCRIPTION

[0093] In order to make the above-mentioned objects, features and advantages of the present invention more obvious and easy to understand, the specific embodiments of the present invention are described in detail below in conjunction with the accompanying drawings. In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention can be implemented in many other ways different from those described herein, and those skilled in the art can make similar improvements without violating the connotation of the present invention, so the present invention is not limited by the specific implementation disclosed below.

[0094] The innovation of the present invention is that the present invention innovatively introduces the relationship between horizontal distance and signal propagation time under the condition of linear sound velocity profile, and innovatively derives the hydroacoustic observation model under the condition of linear sound profile. Furthermore, the present invention integrates the dead reckoning information of AUV, and constructs a virtual long baseline positioning method under the condition of linear sound profile under the framework of virtual long baseline positioning. Furthermore, the present invention designs an AUV position solution method based on semidefinite programming: by introducing auxiliary variables, constructing a pseudo-linear positioning equation, and then constructing a semidefinite programming problem based on the semidefinite relaxation method, and obtaining a preliminary solution of the AUV position based on the interior point method; further, by displaying the error of the preliminary solution, bringing it into the virtual long baseline positioning equation to obtain a linear estimate of the error, and estimating the error value based on weighted least squares and correcting the preliminary estimate solution. The method of the present invention takes into account the refraction of the hydroacoustic propagation channel in single beacon positioning, and can obtain more accurate position estimation results, enhancing the practical application capabilities of AUV.

[0095] Aiming at the refraction problem of sound propagation channel caused by the variation of underwater sound velocity with depth under linear sound profile conditions, a single beacon semi-definite programming positioning method considering the refraction of sound propagation is designed to obtain better positioning performance than the method that does not consider the influence of sound propagation refraction.

[0096] Examples, such as Figure 1 As shown, the underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions provided by an embodiment of the present invention includes the following steps:

[0097] S1, take any point in the positioning area as the origin, set the east, north and sky directions as X, Y and Z axes respectively, and establish the underwater single beacon positioning geodetic coordinate system; take the AUV centroid as the origin, and set the front, right and bottom directions as x, y and z axes respectively, and establish the AUV body coordinate system;

[0098] S2, constructing a dead reckoning model based on the velocity of the AUV in the body coordinate system detected by the DVL and the attitude of the AUV detected by the attitude sensor;

[0099] S3, the hydroacoustic beacon periodically emits hydroacoustic signals, the beacon position coordinates are known, the AUV is equipped with a hydrophone, the propagation time of the hydroacoustic signal is measured, the sound velocity profile is modeled as a linear sound profile, and the known hydroacoustic profile parameter information and the hydroacoustic beacon position coordinates are combined to construct a hydroacoustic observation model under the linear sound profile condition, and the relationship between the signal propagation time and the AUV position is obtained;

[0100] S4, integrating AUV dead reckoning information and using all acoustic signal propagation time observations within the time window width N, constructs a virtual long baseline positioning model under linear acoustic profile conditions;

[0101] S5, by introducing auxiliary variables and performing algebraic transformation, the pseudo-linear equation for virtual long baseline positioning is obtained;

[0102] S6, construct a constrained optimization problem, and solve it by constructing a semidefinite programming problem through semidefinite relaxation to obtain a preliminary position solution for single beacon positioning;

[0103] S7, the estimation error of the preliminary positioning solution of single beacon positioning is explicitly expressed, and based on weighted least squares, the preliminary position solution is corrected using the virtual long baseline positioning equation to obtain the final AUV position estimate.

[0104] Exemplarily, in step S1, the established geodetic coordinate system and body coordinate system are as follows: Figure 2 shown.

[0105] Exemplarily, in step S2, the dead reckoning model constructed is as follows:

[0106] The horizontal coordinates of the AUV at time k are expressed as:

[0107]

[0108] Among them, x k represents the x-coordinate at time k, y k represents the y-direction coordinate at time k, z k represents the z-direction coordinate at time k, and the velocity of the AUV detected by DVL at time k in the body coordinate system is recorded as:

[0109]

[0110] The AUV attitude angle detected by the attitude sensor is recorded as:

[0111]

[0112] In the formula, are the forward, rightward and downward speeds of the AUV respectively, θ k ,ψ k They are the roll angle, pitch angle and heading angle of the AUV respectively.

[0113] Combining the velocity of the AUV in the body coordinate system and the AUV attitude angle, the velocity of the AUV in the earth coordinate system at time k can be obtained as:

[0114]

[0115] In the formula, R k is the rotation matrix; 'is defined as:

[0116]

[0117] According to the position and speed of AUV at time k-1, the position coordinates of AUV at time k are obtained as follows:

[0118] P k =P k-1 +v k-1 Δt

[0119] Where Δt represents the discrete time interval of dead reckoning;

[0120] Expressing the above in component form is:

[0121] x k =x k-1 +v x,k-1 Δt

[0122] y k =y k-1 +v y,k-1 Δt

[0123] z k =z k-1 +v z,k-1 Δt

[0124] Then the relationship between the AUV position variables at time k and time ki (i ≥ 1) is obtained as follows:

[0125]

[0126] In the formula, x k-i is the x-direction position coordinate of the AUV at time ki, y k-i is the y-direction position coordinate of the AUV at time ki, z k-i is the z-direction position coordinate of the AUV at time ki, Δt is the discrete time interval, i is the time index variable, and v x,k-t is the x-direction velocity of AUV at time kt, v y,k-t is the y-direction velocity of AUV at time kt, v z,k-t is the z-direction velocity of the AUV at time kt.

[0127] Exemplarily, in step S3, the constructed hydroacoustic observation model under the linear acoustic profile condition is as follows:

[0128] The coordinates of the hydroacoustic beacon are fixed and known, which can be recorded as:

[0129]

[0130] Where P b is the coordinate vector of the hydroacoustic beacon position, x b is the east coordinate of the hydroacoustic beacon, y b is the north coordinate of the hydroacoustic beacon, z bThe celestial coordinates of the hydroacoustic beacon;

[0131] The sound velocity profile is modeled as linear, that is, the sound velocity value is a linear function of the depth, which is specifically expressed as:

[0132] c=az+b

[0133] Where c is the depth at depth z, a is the sound velocity gradient, z is the depth, and b is the sound velocity at the sea surface;

[0134] When the sound velocity profile is known, the sound velocity at the beacon is known and is denoted by c b , the speed of sound at the AUV position at time k is recorded as c k , the propagation time of the underwater acoustic signal at time k is recorded as t k , the relationship between horizontal distance and signal propagation time is:

[0135]

[0136] Among them, r k is the horizontal distance between the AUV and the beacon, and e is the natural base;

[0137] Squaring both sides of the above equation yields:

[0138]

[0139] Simplifying, we get:

[0140]

[0141] Substituting the sound velocity profile parameters into the equation:

[0142]

[0143] definition:

[0144]

[0145] is a virtual observation, in which all parameters are known and their values ​​depend on the signal propagation time observation value and the acoustic profile parameters at the current moment. The relationship between the AUV position and the underwater acoustic signal propagation time is:

[0146] 0=(x k -x b ) 2 +(y k -y b ) 2 +(z k -z b ) 2 -m k (az k +b)

[0147] Exemplarily, in step S4, the constructed virtual long baseline positioning model under the linear acoustic profile condition is as follows:

[0148] The time window width is defined as N, and the propagation time of the hydroacoustic signal received in the time period from kN to k is used to solve the position coordinates of the AUV at time k. For time ki, the relationship between the propagation time of the hydroacoustic signal constructed by the present invention, the speed of sound, and the position of the AUV is as follows:

[0149] 0=(x k-i -x b ) 2 +(y k-i -y b ) 2 +(z k-i -z b ) 2 -m k-i (az k-i +b), for i=1,2…N

[0150] In the formula, m k-i is the virtual observation at time ki;

[0151] Substitute the AUV’s dead reckoning model, that is, the position relationship between time k and time ki, and we get:

[0152] 0=(x k -x b,i ) 2 +(y k -y b,i ) 2 +(z k -z b,i ) 2 -m k-i (az k +b i ), for i=1,2…N

[0153] in,

[0154]

[0155]

[0156] Respectively represent the x, y, z axis position coordinates of the ith virtual beacon and the surface sound velocity corresponding to the ith virtual beacon; further define:

[0157] x b,0 =x b

[0158] y b,0 =yb

[0159] z b,0 =z b

[0160] b0=b

[0161] is the virtual beacon coordinates and surface sound velocity at time k-0, then combined with the observation equation at time k, all N+1 observation equations are comprehensively expressed as:

[0162] 0=(x k -x b,i ) 2 +(y k -y b,i ) 2 +(z k -z b,i ) 2 -m k-i (az k +b i ), for i=1,2…N.

[0163] In real applications, the propagation time of the underwater acoustic signal contains observation noise. The observed value of the propagation time of the underwater acoustic signal is recorded as:

[0164]

[0165] In the formula, is the observed value of the propagation time of the underwater acoustic signal, t k is the actual underwater acoustic signal propagation time, v k is the observation noise of the underwater acoustic signal propagation time, which is modeled as having a mean of 0 and a variance of A Gaussian random variable Performing Taylor expansion and ignoring higher-order terms, we obtain:

[0166]

[0167] That is:

[0168]

[0169] Substituting the above equation into the virtual observation quantity, we can get:

[0170]

[0171] Similarly, for the ki-th observation, we have:

[0172]

[0173] The virtual observation containing noise is defined as:

[0174]

[0175] Then we have:

[0176]

[0177] Substituting it into the observation equation, we get:

[0178]

[0179] Moving the noise term to the left side of the equal sign gives:

[0180]

[0181] Among them, η i is the noise coefficient of the virtual long baseline positioning equation, defined as:

[0182]

[0183] Exemplarily, in step S5, the steps of obtaining the virtual long baseline positioning pseudo linear equation are as follows: expand the right side of the virtual long baseline positioning equation to obtain:

[0184]

[0185] definition:

[0186]

[0187] Then we have:

[0188]

[0189] All N+1 equations are integrated into a vector matrix form:

[0190]

[0191] In the formula, B k is the noise coefficient matrix, v k is the noise vector of the underwater acoustic signal propagation time, h k ,G k are system model parameters, is the unknown quantity to be estimated, and the definitions of various variables include:

[0192]

[0193] In the formula, is the observed value of the propagation time of the underwater acoustic signal at time kN, v k-N is the propagation time observation noise of the underwater acoustic signal at time kN.

[0194] Exemplarily, in step S6, the steps of obtaining the single beacon positioning preliminary position solution are as follows:

[0195] According to the constructed pseudo linear equation:

[0196]

[0197] The estimation problem can be transformed into a constrained optimization problem:

[0198]

[0199] For any vector a, a[m:n] represents the subvector consisting of the mth to nth elements of vector a, and W 1,k is the weight matrix, which is defined as:

[0200]

[0201] In the formula, E(·) represents the expected value of the random variable, and the superscript -1 represents the inverse of the matrix. It is approximately equal to Q k is the observed variance matrix of the underwater acoustic signal propagation time, defined as The specific form is:

[0202]

[0203] In the formula, is the observed variance of the underwater acoustic signal propagation time at time kN, is the observed variance of the underwater acoustic signal propagation time at time k-1, is the observed variance of the propagation time of the underwater acoustic signal at time k;

[0204] The objective function of the optimization problem is transformed into:

[0205]

[0206] Where tr represents the trace of the matrix;

[0207] Defining auxiliary variables Then the constrained optimization problem is transformed into:

[0208]

[0209] For any matrix A, A[m:n,p:q] represents the submatrix consisting of the elements of the mth to nth rows and the pth to qth columns of matrix A; the last constraint in the above equation is mathematically equivalent to:

[0210]

[0211] rank(Ψ k )=1

[0212] In the formula, rank(Ψ k ) is the matrix Ψ k The rank of the problem is 1, and the rank constraint is removed. The optimization problem is transformed into a semidefinite programming problem, which can be efficiently solved numerically by the interior point method to obtain the optimal solution of the semidefinite programming. and The initial estimates of the AUV position and sound speed are extracted as:

[0213]

[0214]

[0215] In the formula, is the preliminary estimated value of the AUV position at time k.

[0216] Noise coefficient matrix B k The calculation depends on the sound velocity value. In the application process, the weight W is first 1,k Set it as the unit matrix and calculate the preliminary least squares solution Substitute the sound velocity value corresponding to this solution into the noise coefficient matrix B k , calculate the weight W 1,k .

[0217] The main function of the underwater single beacon semidefinite planning positioning method under linear acoustic profile conditions is to deal with the problem that the existing methods do not consider the refraction of sound propagation, and can obtain a more accurate positioning solution.

[0218] The semidefinite programming solution method used, the issues addressed, the models established, and the technical improvements made to existing technologies make the positioning solution more accurate.

[0219] Exemplarily, in step S7, the method for correcting the preliminary position estimation solution of single beacon positioning is as follows:

[0220] The present invention innovatively proposes to define the estimation error corresponding to the preliminary position estimation value obtained in step S6 as:

[0221]

[0222] In the formula, Δx k for The estimation error, Δy k for The estimation error, Δz k for The estimated error of x k ,y k ,z k is the real position coordinate of AUV at time k, is the initial position estimate of the AUV at time k; Substitute the above equation into

[0223]

[0224] get:

[0225]

[0226] Ignoring the second-order noise term, the simplification yields:

[0227]

[0228] in,

[0229]

[0230] All N+1 equations are integrated into a vector matrix form:

[0231] B k v k =g k -F k Δp k

[0232] in,

[0233]

[0234] Based on the weighted least squares estimation principle, the unknown quantity Δp is obtained k The weighted least squares estimation solution of for:

[0235]

[0236] Using the estimated error variable The initial estimate of the semidefinite programming is corrected to obtain the final estimate:

[0237]

[0238] In the formula, is the final estimate of the x-position, is the final estimate of the y-direction position, is the final estimate of the z-direction position.

[0239] The pseudo code of the underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions provided by an exemplary embodiment of the present invention is shown in Table 1.

[0240] Table 1 Pseudo code of underwater single beacon semi-definite planning positioning method

[0241]

[0242] Example 2. The underwater single beacon semi-definite planning positioning system under linear acoustic profile conditions provided in the embodiment of the present invention focuses on the sound propagation refraction problem caused by the spatial variation of sound speed in underwater single beacon positioning, and designs a single beacon positioning position solution method under linear acoustic profile conditions based on the semi-definite relaxation method. The AUV is equipped with a hydrophone, a DVL, an attitude sensor and a CTD; the AUV detects the sound speed profile parameters of the experimental area through the CTD; the hydroacoustic beacon periodically broadcasts hydroacoustic signals, and the AUV periodically observes its velocity in the body coordinate system through the DVL carried, and calculates the AUV velocity in the geodetic coordinate system in combination with the AUV attitude angle observed by the attitude sensor, performs dead reckoning, and records and stores the dead reckoning data; after receiving the hydroacoustic signal emitted by the hydroacoustic beacon, the AUV obtains the hydroacoustic signal propagation time in combination with the known hydroacoustic signal emission time; when the AUV receives a certain number of hydroacoustic signals, it constructs a virtual Long baseline array; construct a virtual long baseline positioning problem based on the relationship between the acoustic propagation time under the linear acoustic profile and the AUV position; introduce auxiliary variables, and obtain the pseudo-linear positioning equation based on algebraic transformation to construct a constrained optimization problem; based on the semidefinite relaxation technology, obtain the positioning problem in the form of semidefinite programming, and solve the semidefinite programming problem through the interior point method to obtain the preliminary position solution of single beacon positioning; explicitly express the estimated error of the preliminary position solution, and construct a linear estimation formula of the estimated error based on the virtual long baseline positioning equation, and solve the estimated error based on the weighted least squares to correct the preliminary position solution and obtain a more accurate AUV position estimation solution.

[0243] To further illustrate the relevant effects of the embodiments of the present invention, the following simulation experiments are conducted: This embodiment simultaneously demonstrates the positioning performance of three single beacon positioning methods that do not consider the refraction of sound propagation. The sound velocity value is calculated using the average value of the entire linear sound velocity profile. Combined with the observed propagation time of the hydroacoustic signal, the distance observation value between the AUV and the beacon can be calculated, and the distance is used to estimate the position coordinates of the AUV. Method 1 is based on weighted least squares to solve the virtual long baseline positioning problem (denoted as R-WLS), method 2 is based on the semidefinite programming method in the literature Yanshen Du, Ping Wei, and Huaguo Zhang. Semidefinite programming approaches for source localization problems. In 2014 IEEE International Conference on Communiction Problem-solving, pages 339–342, 2014. to solve the virtual long baseline positioning problem (denoted as R-SDR1), and method 3 is based on the semidefinite programming method in the literature KW Cheung, WK Ma, and HCSo. Accurate approximation algorithm for to a-based maximum likelihood mobile location using semidefinite programming. In 2004 IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 2, pages ii–145, 2004. to solve the virtual long baseline positioning problem (denoted as R-SDR2). The method of the present invention directly uses the underwater acoustic propagation time as the observed quantity and solves it based on the semidefinite programming method, which is abbreviated as T-SDR.

[0244] The simulation parameters are set as follows: AUV dead reckoning interval is 0.1 seconds, acoustic signal transmission interval is 15 seconds, and acoustic beacon is fixed at [0 0 100] T m, and the total simulation time is 500 seconds. The AUV dynamics model is used to generate the spiral diving motion trajectory of the AUV. The control input of the AUV is horizontal rudder 8°, vertical rudder -8°, and propeller speed 1000 rpm. The initial position of the AUV is [100100 10] T m, the forward initial velocity is 1.2 m / s, and the initial velocity in other directions is 0. The AUV motion trajectory obtained by simulation is as follows Figure 3 shown.

[0245] The standard deviation of the simulated DVL velocity observation noise in three directions is [0.01 0.01 0.01] T m / s, the standard deviation of the attitude sensor roll angle and pitch angle observation is 0.03°, the standard deviation of the heading angle observation is 0.3°, and the standard deviation of the underwater acoustic signal propagation time is 100us. The simulated linear sound velocity profile function is c = -0.22z + 1542m / s, and its profile shape is as follows Figure 4 The sound velocity profile used in the simulation is shown here.

[0246] The Bellhop hydroacoustic simulation toolbox is used to generate hydroacoustic sound lines and real hydroacoustic signal propagation time during the AUV movement. The Bellhop model parameters used are shown in Table 2.

[0247] Table 2 Bellhop environmental parameters

[0248] Signal transmission frequency 10KHz depth 200m Seafloor modeling Acoustic elastic half space Minimum and maximum sound angle -90°，90° Maximum sound line distance 1.6 km Maximum sound depth 200 meters Number of beams 9001

[0249] For all methods, the width of the sliding window is set to 15, that is, the current moment and the 15 hydroacoustic ranging observations before the current moment are used to solve the position coordinates of the AUV at the current moment. The proposed semidefinite programming problem, R-SDR1 and R-SDR2 methods are solved by using the CVX toolbox of Matlab software, using the sedumi solver, and the solution accuracy is set to the highest, see the reference Michael Grant and Stephen Boyd.CVX: Matlab software for disciplined convex programming, version 2.0beta.https: / / cvxr.com / cvx, September 2013. and the reference Michael Grant and Stephen Boyd.Graph implementations for nonsmooth convex programs, Recent Advances in Learning and Control (a tribute to M.Vidyasagar), V.Blondel, S.Boyd, and H.Kimura, editors, pages 95-110, Lecture Notes in Control and Information Sciences, Springer, 2008.http: / / stanford.edu / ~boyd / graph_dcp.html. The positioning error and average positioning error of the four methods are used as performance evaluation criteria, and the calculation formulas of positioning error and average positioning error are:

[0250]

[0251] Among them, K represents the number of single beacon positioning settlements. The positioning errors of the four methods are as follows: Figure 5 The average positioning error is shown in Table 3.

[0252] according to Figure 5 As shown in Table 3, the semidefinite programming method proposed in the present invention can obtain better positioning accuracy than the traditional single beacon positioning method that does not consider the refraction of sound propagation, and its positioning error is lower than that of other comparison methods at most times. This shows that considering the refraction of the sound propagation channel has a certain degree of influence on the performance of single beacon positioning, and the method proposed in the present invention can eliminate this influence.

[0253] Table 3 Comparison of average positioning errors of different methods

[0254] Positioning method Average error (meters) T-SDR 0.618450 R-WLS 2.663769 R-SDR1 1.409439 R-SDR2 2.670061

[0255] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any modifications, equivalent substitutions and improvements made by any technician familiar with the technical field within the technical scope disclosed by the present invention and within the spirit and principles of the present invention should be covered within the protection scope of the present invention.

Claims

1. A method for positioning underwater single beacon semi-definite planning under linear acoustic profile conditions, characterized in that: This method solves the problem of sound propagation refraction caused by the spatial variation of sound speed in underwater single beacon positioning. Based on the semidefinite relaxation method, the single beacon positioning position is solved under the condition of linear sound profile to obtain the AUV position estimation solution. Based on the obtained pseudo-linear equation of virtual long baseline positioning under linear acoustic profile conditions, a constrained optimization problem is constructed, and a semidefinite programming problem is constructed through semidefinite relaxation to solve it, and the preliminary position solution of single beacon positioning is obtained; The estimation error of the preliminary positioning solution of single beacon positioning is explicitly expressed, and based on weighted least squares, the preliminary position solution is corrected using the virtual long baseline positioning equation to obtain the final AUV position estimate.

2. The underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions according to claim 1 is characterized in that: Before obtaining the pseudo-linear equation of virtual long baseline positioning under linear acoustic profile conditions, the following operations are required: Take any point in the positioning area as the origin, set the east, north and sky directions as X, Y and Z axes respectively, and establish the underwater single beacon positioning geodetic coordinate system; take the AUV centroid as the origin, set the front, right and bottom directions as x, y and z axes respectively, and establish the AUV body coordinate system; According to the speed of the AUV in the body coordinate system detected by the DVL and the attitude of the AUV detected by the attitude sensor, a dead reckoning model is constructed; The hydroacoustic beacon periodically emits hydroacoustic signals. The beacon position coordinates are known. The AUV is equipped with a hydrophone to measure the propagation time of the hydroacoustic signal. The sound velocity profile is modeled as a linear sound profile. Combining the known hydroacoustic profile parameter information with the hydroacoustic beacon position coordinates, a hydroacoustic observation model under the linear sound profile condition is constructed to obtain the relationship between the signal propagation time and the AUV position. The AUV dead reckoning information is integrated and all acoustic signal propagation time observations within the time window width N are used to construct a virtual long baseline positioning model under linear acoustic profile conditions.

3. The underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions according to claim 2 is characterized in that: Construct a dead reckoning model, including: The relationship between the AUV position variables at time k and time ki (i ≥ 1) in the dead reckoning model is: In the formula, x k-i is the x-direction position coordinate of the AUV at time ki, y k-i is the y-direction position coordinate of the AUV at time ki, z k-i is the z-direction position coordinate of the AUV at time ki, Δt is the discrete time interval, i is the time index variable, and v x，k-t is the x-direction velocity of AUV at time kt, v y，k-t is the y-direction velocity of AUV at time kt, v z,k-t is the z-direction velocity of AUV at time kt; The hydroacoustic observation model under the condition of linear acoustic profile is constructed, including: the relationship between the AUV position and the propagation time of the hydroacoustic signal is: 0=(x k -x b ) 2 +(y k -y b ) 2 +(z k -z b ) 2 -m k (a) k +b) The constructed virtual long baseline positioning model under linear acoustic profile conditions includes: Moving the noise term to the left side of the equal sign gives: Among them, η i is the noise coefficient of the virtual long baseline positioning equation, defined as:

4. The underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions according to claim 1 is characterized in that: By introducing auxiliary variables and performing algebraic transformation, the pseudo-linear equation for virtual long baseline positioning is obtained. The steps are as follows: Expand the right side of the virtual long baseline positioning equation to obtain: definition: Then we have: All N+1 equations are integrated into a vector matrix form: In the formula, B k is the noise coefficient matrix, v k is the noise vector of the underwater acoustic signal propagation time, h k ,G k are system model parameters, is the unknown quantity to be estimated, and the definitions of various variables include: In the formula, is the observed value of the propagation time of the underwater acoustic signal at time kN, v k-N is the propagation time observation noise of the underwater acoustic signal at time kN.

5. The underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions according to claim 1 is characterized in that: The steps of obtaining the single beacon positioning preliminary position solution include: According to the constructed pseudo linear equation: The estimation problem is transformed into a constrained optimization problem, namely: For any vector a, a[m:n] represents the subvector consisting of the mth to nth elements of vector a, and W 1,k is the weight matrix, which is defined as: In the formula, E(·) represents the expected value of the random variable, and the superscript -1 represents the inverse of the matrix. It is approximately equal to Q k is the observed variance matrix of the underwater acoustic signal propagation time, defined as The specific form is: In the formula, is the observed variance of the underwater acoustic signal propagation time at time kN, is the observed variance of the underwater acoustic signal propagation time at time k-1, is the observed variance of the propagation time of the underwater acoustic signal at time k; The objective function of the optimization problem is transformed into: Where tr represents the trace of the matrix; Defining auxiliary variables Then the constrained optimization problem is transformed into: Among them, for any matrix A, A[m:n,p:q] represents the submatrix consisting of the elements of the mth to nth rows and the pth to qth columns of matrix A; the last constraint in the above equation is mathematically equivalent to: rank(Ψ k )=1 In the formula, rank(Ψ k ) is the matrix Ψ k The rank of the problem is 1, and the rank constraint is removed. The optimization problem is transformed into a semidefinite programming problem, which can be efficiently solved numerically by the interior point method to obtain the optimal solution of the semidefinite programming. and The initial estimates of the AUV position and sound speed are extracted as: In the formula, is the preliminary estimated value of the AUV position at time k.

6. The underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions according to claim 1 is characterized in that: The obtained single beacon positioning preliminary position solution further includes: the noise coefficient matrix B k The calculation depends on the sound velocity value. In the application process, the weight W is first 1,k Set it as the unit matrix and calculate the preliminary least squares solution Substitute the sound velocity value corresponding to this solution into the noise coefficient matrix B k , calculate the weight W 1,k .

7. The underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions according to claim 1 is characterized in that: The single beacon positioning preliminary position estimation solution correction method includes: The estimated error corresponding to the obtained preliminary position estimate is defined as: In the formula, x k ,y k ,z k is the real position coordinate of AUV at time k, Δx k for The estimation error, Δy k for The estimation error, Δz k for The estimation error of Substituting the above equation into get: Ignoring the second-order noise term, the simplification yields: in, All N+1 equations are integrated into a vector matrix form: B k v k =g k -F k Δp k In the formula, g k With F k is the system model parameter, Δη k is the unknown quantity to be estimated, and each variable is defined as follows: Based on the weighted least squares estimation principle, the unknown quantity Δp is obtained k The weighted least squares estimation solution of for: Using the estimated error variable The initial estimate of the semidefinite programming is corrected to obtain the final estimate: In the formula, is the final estimate of the x-position, is the final estimate of the y-direction position, is the final estimate of the z-direction position.

8. An underwater single beacon semi-definite planning positioning system under linear acoustic profile conditions, characterized in that: The system implements the underwater single beacon semi-definite planning positioning method under the linear acoustic profile condition according to any one of claims 1 to 7, the system comprises an AUV, the AUV is equipped with a hydrophone, a DVL, a posture sensor and a CTD; the AUV detects the sound velocity profile parameters of the experimental area through the CTD; the hydroacoustic beacon periodically broadcasts a hydroacoustic signal, the AUV periodically observes its velocity in the body coordinate system through the carried DVL, calculates the velocity of the AUV in the geodetic coordinate system in combination with the AUV attitude angle observed by the posture sensor, performs dead reckoning, and records and stores the dead reckoning data; After receiving the hydroacoustic signal transmitted by the hydroacoustic beacon, the AUV obtains the propagation time of the hydroacoustic signal in combination with the known emission time of the hydroacoustic signal; When the AUV receives a certain number of hydroacoustic signals, it constructs a virtual long baseline array in combination with the stored dead reckoning data; Combining the relationship between the acoustic propagation time under the linear acoustic profile and the AUV position, a virtual long baseline positioning problem is constructed; auxiliary variables are introduced, and a pseudo-linear positioning equation is obtained based on algebraic transformation to construct a constrained optimization problem; based on the semidefinite relaxation technology, a positioning problem in the form of a semidefinite programming is obtained, and the semidefinite programming problem is solved by the interior point method to obtain the preliminary position solution for single beacon positioning; The estimation error of the preliminary position solution is explicitly expressed, and the linear estimation formula of the estimation error is constructed by combining the virtual long baseline positioning equation. The estimation error is solved based on weighted least squares to correct the preliminary position solution and obtain a more accurate AUV position estimation solution.

9. The underwater single beacon semi-definite planning positioning system under linear acoustic profile conditions according to claim 8, characterized in that: The underwater single beacon semi-definite planning positioning system under linear acoustic profile conditions is suitable for deep sea environments, and single beacon positioning is performed by the underwater single beacon semi-definite planning positioning method under linear acoustic profile conditions described in claim 1.

10. The underwater single beacon semi-definite planning positioning system under linear acoustic profile conditions according to claim 8, characterized in that: The underwater single beacon semi-definite planning positioning system under the linear acoustic profile condition is carried on a computer-readable storage medium, and the computer-readable storage medium stores a computer program. When the computer program is executed by the processor, it can realize the functions of the underwater single beacon semi-definite planning positioning system under the linear acoustic profile condition.

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