Underwater single-beacon gauss-newton positioning method and system under linear acoustic profile condition

Abstract

The application belongs to the technical field of underwater positioning, and discloses an underwater single-beacon Gauss-Newton positioning method and system under a linear acoustic profile condition. The method models the observation noise of underwater acoustic signal propagation time as a Gaussian distribution, constructs a maximum likelihood estimation problem, and then converts it into a nonlinear least squares estimation problem. Based on the chain rule, the gradient of the observation with respect to the AUV position coordinates is solved, and based on the Gauss-Newton method, the nonlinear least squares estimation problem is iteratively solved. After the iteration converges, the AUV position coordinates are obtained. Compared with other single-beacon positioning methods that do not consider acoustic propagation refraction, the method can obtain more accurate position calculation results under the linear acoustic profile condition, enhances single-beacon positioning, and especially enhances the practical application ability in deep sea areas.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of underwater positioning, and particularly relates to an underwater single-beacon Gauss-Newton positioning method and system under a linear acoustic profile condition. BACKGROUND

[0002] An autonomous underwater vehicle (AUV) is widely used in underwater search and rescue, underwater operation, and ocean environment data collection. The AUV needs to obtain its own position coordinates in real time to ensure the accuracy of motion control and the effectiveness of the collected data. Due to the rapid attenuation of electromagnetic signals underwater, underwater acoustic positioning has become the most widely used and effective positioning method in underwater environments. Compared with common underwater long baseline positioning and ultra-short baseline positioning technologies, the positioning method combining a single acoustic beacon and dead reckoning information can greatly reduce the time and economic cost of long baseline positioning as it only needs a single position-known acoustic beacon to provide positioning assistance. The beacon can be fixed to the seabed or carried on a surface ship, and the system has high flexibility. In view of the above advantages of single-beacon positioning, it has attracted widespread attention from researchers.

[0003] Current underwater single-beacon positioning methods usually assume that the acoustic speed is constant, and directly use the distance between the AUV and the beacon as an observation. However, in actual underwater positioning applications, the acoustic speed varies with space, especially in the depth direction, the gradient of the acoustic speed value is large, which will cause obvious sound propagation refraction. Ignoring the influence of sound propagation refraction will cause a large distance calculation error, which will further lead to a large single-beacon positioning error, affecting the practical application ability of the single-beacon system. Research shows that for deep sea environments, the acoustic speed profile can be approximated as linear, that is, the acoustic speed value can be approximated as a linear function of depth. That is, studying the single-beacon positioning method under the linear acoustic profile condition has wide application prospects. By more accurately describing the acoustic propagation channel trajectory and signal propagation time observation model under the linear acoustic profile condition, the error of single-beacon positioning can be reduced to some extent. SUMMARY

[0004] To overcome the problems in the related art, the present application discloses a single-beacon Gauss-Newton positioning method and system under a linear acoustic profile condition.

[0005] The technical solution is as follows: a single-beacon Gauss-Newton positioning method and system under a linear acoustic profile condition, comprising:

[0006] S1, taking any point in the positioning area as the origin, and setting the east, north, and sky directions as the X, Y, and Z axes respectively, a single-beacon positioning geodetic coordinate system is established; taking the AUV center of mass as the origin, and setting the front, right, and down directions as the x, y, and z axes respectively, an AUV body coordinate system is established;

[0007] S2, constructing a dead reckoning model according to the velocity of the AUV detected by the DVL and the attitude of the AUV detected by the attitude sensor in the AUV body coordinate system;

[0008] S3, the underwater acoustic beacon periodically emits underwater acoustic signals, the position coordinates of the beacon are known, the AUV is equipped with a hydrophone, the propagation time of the underwater acoustic signals is measured, the sound velocity profile is modeled as a linear sound profile, the underwater acoustic observation model under the linear sound profile condition is constructed by combining the known underwater acoustic profile parameter information and the position coordinates of the underwater acoustic beacon, and the relationship between the signal propagation time and the position of the AUV is obtained;

[0009] S4, fusing the dead reckoning information of the AUV, constructing a virtual long baseline positioning model under the linear sound profile condition by using all the underwater acoustic signal propagation time observations within a time window width N;

[0010] S5, calculating the gradient matrix of the underwater acoustic signal propagation time with respect to the position variable, iteratively solving the positioning problem by using the Gauss-Newton method, and obtaining the position estimate of the AUV after the iteration converges.

[0011] In step S2, the dead reckoning model is constructed, including:

[0012] The relationship between the position variable of the AUV at time k and the position variable of the AUV at time k-i, i≥1, is:

[0013]

[0014] In the formula, x k ,y k ,z k is the true position coordinates of the AUV at time k, x k-i is the x-direction position coordinates of the AUV at time k-i, y k-i is the y-direction position coordinates of the AUV at time k-i, z k-i is the z-direction position coordinates of the AUV at time k-i, Δt is the discrete time interval, i is the time index variable, v x,k-t is the x-direction velocity of the AUV at time k-i, v y,k-t is the y-direction velocity of the AUV at time k-i, and v z,k-t is the z-direction velocity of the AUV at time k-i.

[0015] In step S3, the underwater acoustic observation model under the linear sound profile condition is constructed, and the expression is:

[0016]

[0017] In the formula, P b is the position coordinates vector of the underwater acoustic beacon, x bx is the east coordinate of the underwater acoustic beacon, y b z is the north coordinate of the underwater acoustic beacon, z b z is the zenith coordinate of the underwater acoustic beacon;

[0018] The sound speed profile is modeled as linear, and the sound speed value is a linear function of depth, which is:

[0019] c = az + b

[0020] In the formula, c is the depth at depth z, a is the sound speed gradient, z is the depth, and b is the sound speed at sea level;

[0021] In the case of known sound speed profile, the sound speed at the beacon is known, which is denoted as c b , the sound speed at the position of AUV at time k is denoted as c k , the underwater acoustic signal propagation time at time k is denoted as t k , the relationship between horizontal distance and signal propagation time is:

[0022]

[0023] Where r k is the horizontal distance between AUV and beacon, e is the natural base;

[0024] Square both sides of the above equation to get:

[0025]

[0026] Simplify to get:

[0027]

[0028] Put in the sound speed profile parameters to get:

[0029]

[0030] Definition:

[0031]

[0032] In the formula, m k is a virtual observation, where all parameters are known, and the parameter values depend on the current time signal propagation time observation and sound profile parameters;

[0033] The relationship between AUV position and underwater acoustic signal propagation time is obtained as:

[0034] 0 = (x k -x b ) 2 + (y k -y b ) 2+(z k -z b ) 2 -m k (az k +b)。

[0035] In step S4, a virtual long baseline positioning model under linear acoustic profile condition is constructed, including:

[0036] Defining the time window width as N, the position coordinates of the AUV at time k are solved by using the received underwater acoustic signal propagation time from time k-N to time k, and for time k-i, the established underwater acoustic signal propagation time observation equation is as follows:

[0037] 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),

[0038] for i=1,2…N

[0039] In the formula, m k-i is the virtual observation at time k-i.

[0040] The dead reckoning model of the AUV is brought in, that is, the position relationship between time k and time k-i, to obtain:

[0041] 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

[0042] Wherein,

[0043]

[0044] In the formula, x b,i is the x-axis position coordinate of the i-th virtual beacon, y b,i is the y-axis position coordinate of the i-th virtual beacon, z b,i is the z-axis position coordinate of the i-th virtual beacon, and bi is the surface speed corresponding to the i-th virtual beacon; further, define:

[0045] x b,0 = x b

[0046] y b,0 = y b

[0047] z b,0 = z b

[0048] b0= b

[0049] where x b,0 , y b,0 , z b,0 are the virtual beacon coordinates at time k-0, and b0is the virtual surface speed at time k-0;

[0050] Combining the observation equation at time k, all N+1 observation equations are integrated as:

[0051] 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 ),

[0052] for i = 0, 1, 2…N.

[0053] In step S5, the Gauss-Newton method is used to iteratively solve the positioning problem, including:

[0054] The underwater acoustic signal propagation time contains observation noise, and the underwater acoustic signal propagation time observation value is denoted as:

[0055]

[0056] wherein, is the underwater acoustic signal propagation time observation value, t k is the true underwater acoustic signal propagation time, v k is the observation noise of the underwater acoustic signal propagation time, which is modeled as a Gaussian random variable with mean 0 and variance All N+1 underwater acoustic signal propagation time observations are integrated into a vector All N+1 true underwater acoustic signal propagation times are integrated into a vector tk All N+1 observation noises of underwater acoustic propagation time are synthesized into a vector v k As follows:

[0057]

[0058] The observation noise v k Is modeled as Gaussian noise with mean 0 and variance matrix Q k Then the maximum likelihood estimation problem of single beacon positioning is:

[0059]

[0060] In which, is the maximum likelihood estimation value of AUV position at time k, arg max is the solution of the maximum problem, exp{} is the exponential function, p k is the position of AUV at time k, superscript T represents vector matrix transpose, t k (p k ) is the real underwater acoustic signal propagation time, whose value is a function of position coordinates; variance matrix Q k According to the observation variance of underwater acoustic signal propagation time at each time, we have:

[0061]

[0062] In which, σ k-N is the standard deviation of observation variance at time k-N, σ k-N+1 is the standard deviation of observation variance at time k-N+1, σ k is the standard deviation of observation variance at time k, diag(a) is a diagonal matrix with elements of a as diagonal elements, according to the monotonicity of logarithmic function, the above optimization problem is transformed into:

[0063]

[0064] In which, arg min( ) is the minimum function;

[0065] Definition:

[0066]

[0067] In which, S k is the Choleskey decomposition of , and the optimization problem is transformed into:

[0068]

[0069] This problem is a nonlinear least squares optimization problem, which is solved by Gauss-Newton method as follows:

[0070]

[0071] where, is the position estimate obtained at iteration j+1, is the position estimate obtained at iteration j, is the Gauss-Newton position increment at iteration j, defined as:

[0072]

[0073] where, is the gradient matrix of signal travel time with respect to the position coordinates obtained at iteration j, is the acoustic signal travel time estimated from the AUV position estimated at the current iteration cycle, is defined as:

[0074]

[0075] where, is the partial derivative of the acoustic signal travel time with respect to the x-direction position coordinate, is the partial derivative of the acoustic signal travel time with respect to the y-direction position coordinate, is the partial derivative of the acoustic signal travel time with respect to the z-direction position coordinate, is the x, y, z-direction position estimate obtained at iteration j.

[0076] Further, the gradient matrix of the position coordinates is calculated as follows:

[0077] According to the acoustic beacon observation equation:

[0078] 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 ),

[0079] for i = 0, 1, 2...N

[0080] Taking the partial derivative of both sides with respect to x k , y k , z k , respectively, we have:

[0081]

[0082] where, is the partial derivative of the virtual observation with respect to the acoustic signal propagation time, is the partial derivative of the acoustic signal propagation time with respect to the AUV position, respectively;

[0083] According to the definition of m k-i , we have:

[0084]

[0085] Then we have:

[0086]

[0087] We have:

[0088]

[0089] for i = 0, 1,..., N.

[0090] Further, the acoustic signal propagation time estimated according to the AUV position estimated in the current iteration is calculated by:

[0091] According to the equation:

[0092]

[0093] We have:

[0094] n i α i = (α i - 1) 2

[0095] where n i and α i are auxiliary parameters, which are defined as:

[0096]

[0097] Solving the above quadratic equation, we have:

[0098]

[0099] That is:

[0100]

[0101] According to the above equation, we have two different t k-i , since the real acoustic signal propagation time is positive, we choose the one which is greater than 0; the position coordinates obtained in the jth iteration are brought into the above equation, and the acoustic signal propagation time vector

[0102] Further, the iteration convergence criterion is set as when the two-norm of the position update amount is less than a certain threshold ε, the iteration is considered to be converged, and the single-beacon positioning position estimate value

[0103] Another object of the present application is to provide an underwater single-beacon Gauss-Newton positioning system under linear acoustic profile conditions, which implements the underwater single-beacon Gauss-Newton positioning method under linear acoustic profile conditions, and obtains the AUV position coordinates through a positioning solving method based on the Gauss-Newton algorithm, aiming at the refraction problem of the acoustic propagation channel caused by the change of the underwater acoustic sound speed with depth under linear acoustic profile conditions.

[0104] Further, the system is provided with an AUV, which 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 underwater acoustic beacon periodically broadcasts underwater acoustic signals, the AUV periodically observes the velocity in the body coordinate system through the equipped DVL, and calculates the velocity in the geodetic coordinate system of the AUV 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 underwater acoustic signals transmitted by the underwater acoustic beacon, the AUV obtains the underwater acoustic signal propagation time in combination with the known underwater acoustic signal transmission time; when the AUV receives a certain number of underwater acoustic signals, the AUV constructs a virtual long baseline array in combination with the stored dead reckoning data; in combination with the relationship between the underwater acoustic propagation time under linear acoustic profile and the AUV position, a virtual long baseline positioning problem is constructed; the underwater acoustic signal propagation time observation noise is modeled as a Gaussian distribution, a maximum likelihood estimation problem is constructed, and then it is converted into a nonlinear least squares estimation problem; the gradient of the observation with respect to the AUV position coordinates is solved based on the chain rule, the nonlinear least squares estimation problem is iteratively solved based on the Gauss-Newton method, and after the iteration converges, the AUV position coordinates are obtained.

[0105] With all the technical solutions above, the application has the beneficial effects that the application 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 solving method under the linear sound profile condition based on the Gauss-Newton method. An underwater vehicle (AUV) is equipped with a hydrophone, a Doppler speedometer, an attitude sensor, and a temperature-salinity-depth instrument; the AUV detects the sound speed profile parameters of the experimental area through the temperature-salinity-depth instrument; the underwater acoustic beacon periodically broadcasts underwater acoustic signals, and the AUV records and stores the dead reckoning data by performing dead reckoning through the equipped Doppler speedometer combined with the attitude sensor; after receiving the underwater acoustic signals transmitted by the underwater acoustic beacon, the AUV calculates the underwater acoustic signal propagation time; when the AUV receives a certain number of underwater acoustic signals, the stored dead reckoning data is combined to construct a virtual long baseline array; the relationship between the underwater acoustic propagation time under the linear sound profile and the AUV position is combined to construct a virtual long baseline positioning problem; the observation noise of the underwater acoustic signal propagation time is modeled as a Gaussian distribution to construct a maximum likelihood estimation problem, which is then converted into a nonlinear least squares estimation problem; the gradient of the observation with respect to the AUV position coordinates is solved based on the chain rule, and the nonlinear least squares estimation problem is iteratively solved based on the Gauss-Newton method, and after the iteration converges, the AUV position coordinates are obtained. Compared with other single beacon positioning methods that do not consider sound propagation refraction, the method of the application can obtain more accurate position solving results under the linear sound profile condition, and enhances single beacon positioning, especially its practical application ability in deep sea areas.

[0106] The application considers the widespread sound propagation channel refraction in real underwater acoustic positioning applications, uses a more reasonable and accurate underwater observation model, can obtain better ranging accuracy and positioning accuracy, improves the effectiveness and value of the data collected by the AUV, and enhances the operation and application value of the AUV itself. The application considers the widespread sound propagation channel refraction problem in single beacon positioning.

[0107] Affected by the temperature, salinity and depth of seawater, the sound speed gradient in the depth direction is large, which will cause obvious sound propagation channel refraction, and this problem exists widely in all underwater positioning methods based on underwater sound ranging, including single beacon positioning. The common processing method is to ignore the refraction of the channel and directly use a constant sound speed value, or to introduce the concept of effective sound speed and directly regard the effective sound speed as unknown. Ignoring the channel refraction will cause ranging error and positioning error, and directly regarding the effective sound speed as unknown will ignore the valuable physical information of the sound speed profile, and also cause performance loss and positioning error. Research on single beacon positioning method considering sound propagation refraction is urgently needed. The present application directly describes the underwater sound observation model under the linear sound profile, directly processes the observation model considering propagation refraction, and uses the Gauss-Newton method to iteratively solve the AUV position, so that better position estimation performance can be obtained, and the technical problem that people are eager to solve is solved. BRIEF DESCRIPTION OF DRAWINGS

[0108] The accompanying drawings, which are incorporated into and form 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;

[0109] Figure 1 is a flowchart of the underwater single beacon Gauss-Newton positioning method under the linear sound profile provided by the embodiment of the present application;

[0110] Figure 2 is a schematic diagram of the geodetic coordinate system and the AUV body coordinate system of the underwater single beacon positioning provided by the embodiment of the present application;

[0111] Figure 3 is a simulation trajectory diagram;

[0112] Figure 4 is a sound speed profile diagram used in simulation;

[0113] Figure 5 is a positioning error comparison diagram of different positioning methods;

[0114] Figure 6 is an iteration number diagram of the Gauss-Newton method at each time. DETAILED DESCRIPTION

[0115] In order to make the above-mentioned purposes, features and advantages of the present application more obvious and easy to understand, the specific embodiments of the present application will be described in detail below with reference to the drawings. In the following description, many specific details are set forth in order to provide a thorough understanding of the present application. However, the present application can be implemented in many other ways different from those described herein, and those skilled in the art can make similar improvements without departing from the scope of the present application, so the present application is not limited to the specific embodiments disclosed below.

[0116] The application has the following innovative points: the application introduces the relationship between the horizontal distance and the signal propagation time under the linear sound speed profile condition, and deduces the underwater acoustic observation model under the linear sound profile condition; the application fuses the dead reckoning information of the AUV, and constructs a virtual long baseline positioning method under the linear sound profile condition in the framework of the virtual long baseline positioning; the application designs an AUV position solving method based on the Gauss-Newton method: first, initialize the iteration, calculate the gradient of the underwater acoustic signal propagation time with respect to the AUV position in the current iteration step, construct the gradient matrix, and then calculate the Gauss-Newton descent direction, correct the AUV position coordinates in the current iteration step, and obtain the position coordinates of the AUV after the iteration converges.

[0117] Embodiment 1, as shown in the figure, the underwater single-beacon Gauss-Newton positioning method under the linear sound profile condition provided by the embodiment of the application comprises: Figure 1

[0118] S1, taking any point in the positioning area as the origin, and taking the east, north and sky directions as the X, Y and Z axes respectively, an underwater single-beacon positioning geodetic coordinate system is established; taking the center of the AUV as the origin, and taking the front, right and down directions as the x, y and z axes respectively, an AUV body coordinate system is established;

[0119] S2, a dead reckoning model is constructed according to the AUV body coordinate system velocity detected by the DVL and the AUV attitude detected by the attitude sensor;

[0120] S3, the underwater acoustic beacon periodically emits underwater acoustic signals, the beacon position coordinates are known, the AUV is equipped with a hydrophone, the underwater acoustic signal propagation time is measured, the sound speed profile is modeled as a linear sound profile, the underwater acoustic profile parameter information and the underwater acoustic beacon position coordinates are combined, the underwater acoustic observation model under the linear sound profile condition is constructed, and the relationship between the signal propagation time and the AUV position is obtained;

[0121] S4, the dead reckoning information of the AUV is fused, and a virtual long baseline positioning model under the linear sound profile condition is constructed by using all the underwater acoustic signal propagation time observations in the time window width N;

[0122] S5, the gradient matrix of the underwater acoustic signal propagation time with respect to the position variable is calculated, the Gauss-Newton method is used to iteratively solve the positioning problem, and the position estimation of the AUV is obtained after the iteration converges.

[0123] ​Exemplarily, in step S1, the established geodetic coordinate system and the body coordinate system are as shown in Figure 2

[0124] In step S2, a dead reckoning model is constructed, and the horizontal coordinates of the AUV at time k are denoted as:

[0125]

[0126] wherein x k represents the x-direction coordinate at time k, y k represents the y-direction coordinate at time k, and z k represents the z-direction coordinate at time k, and the velocity of the AUV in the body coordinate system detected by the DVL at time k is denoted as:

[0127]

[0128] The attitude angle of the AUV detected by the attitude sensor is denoted as:

[0129]

[0130] wherein u , v , and w k , ψ k are respectively the forward, rightward, and downward velocities of the AUV, and the roll angle, pitch angle, and heading angle of the AUV.

[0131] Combined with the velocity of the AUV in the body coordinate system and the attitude angle of the AUV, the velocity of the AUV in the geodetic coordinate system at time k can be obtained as:

[0132]

[0133] wherein R k is a rotation matrix, and is defined as:

[0134]

[0135] According to the position and velocity of the AUV at time k-1, the position coordinates of the AUV at time k are obtained as:

[0136] P k = P k-1 + v k-1 Δt

[0137] wherein Δt represents the discrete time interval of the dead reckoning;

[0138] The above is denoted in component form as:

[0139] x k = x k-1 + v x,k-1 ​Δt

[0140] y k = y k-1 + v y,k-1 Δt

[0141] z k = z k-1 + v z,k-1 Δt

[0142] The relationship between the AUV position variables at time k and time k-i (i≥1) is obtained as follows:

[0143]

[0144]

[0145] wherein x k-i is the x-direction position coordinate of the AUV at time k-i, y k-i is the y-direction position coordinate of the AUV at time k-i, z k-i is the z-direction position coordinate of the AUV at time k-i, Δt is the discrete time interval, i is the time index variable, v x,k-t is the x-direction velocity of the AUV at time k-i, v y,k-t is the y-direction velocity of the AUV at time k-i, and v z,k-t is the z-direction velocity of the AUV at time k-i.

[0146] In step S3, a water acoustic observation model under a linear acoustic profile condition is constructed, and the expression is as follows:

[0147]

[0148] wherein P b is a water acoustic beacon position coordinate vector, x b is an eastward coordinate of the water acoustic beacon, y b is a northward coordinate of the water acoustic beacon, and z b is a skyward coordinate of the water acoustic beacon.

[0149] The sound velocity profile is modeled as linear, and the sound velocity value is a linear function of depth, which is as follows:

[0150] c = az + b

[0151] wherein c is the depth at depth z, a is the sound velocity gradient, z is the depth, and b is the sound velocity at sea level.

[0152] In the case where the sound velocity profile is known, the sound velocity at the beacon is known, which is denoted as c b , the sound velocity at the AUV position at time k is denoted as c k , and the water acoustic signal propagation time at time k is denoted as tk The relationship between the horizontal distance and the signal propagation time is:

[0153]

[0154] Wherein, r k is the horizontal distance between the AUV and the beacon, and e is the natural base;

[0155] The present application innovatively squares both sides of the above equation to obtain:

[0156]

[0157] Simplifying obtains:

[0158]

[0159] Substituting the sound speed profile parameter obtains:

[0160]

[0161] Definition:

[0162]

[0163] In the formula, m k is a virtual observation quantity, wherein all the parameters are known, and the parameter values depend on the signal propagation time observation value at the current time and the sound profile parameter;

[0164] The relationship between the AUV position and the underwater acoustic signal propagation time is obtained as:

[0165] 0=(x k -x b ) 2 +(y k -y b ) 2 +(z k -z b ) 2 -m k (az k +b).

[0166] In step S4, a virtual long baseline positioning model under the linear sound profile condition is constructed, including:

[0167] Defining the time window width as N, the underwater acoustic signal propagation time received from k-N to k is used to solve the position coordinates of the AUV at k, and for k-i, the established underwater acoustic signal propagation time observation equation is applied as follows:

[0168] 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

[0169] Where m k-i is the virtual observation at time ki.

[0170] Substitute the AUV’s dead reckoning model, i.e. the position relationship between time k and time ki, and we get:

[0171] 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

[0172] in,

[0173]

[0174] Where x b,i is the x-axis position coordinate of the i-th virtual beacon, y b,i is the y-axis position coordinate of the i-th virtual beacon, z b,i is the z-axis position coordinate of the i-th virtual beacon, b i is the surface sound velocity corresponding to the i-th virtual beacon; further, define:

[0175] x b,0 =x b

[0176] y b,0 =y b

[0177] z b,0 =z b

[0178] b0=b

[0179] Where x b,0 ,y b,0 ,z b,0 is the virtual beacon coordinate at time k-0, b0 is the virtual surface sound speed at time k-0;

[0180] Combining the observation equation at time k, all N+1 observation equations are expressed as:

[0181] 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 ),

[0182] for i = 0, 1, 2…N.

[0183] In step S5, the Gauss-Newton method is used to iteratively solve the positioning problem, including:

[0184] In real applications, the underwater acoustic signal propagation time contains observation noise. Let the underwater acoustic signal propagation time observation value be:

[0185]

[0186] In the formula, is the underwater acoustic signal propagation time observation value, t k is the real underwater acoustic signal propagation time, v k is the observation noise of the underwater acoustic signal propagation time, which is modeled as a Gaussian random variable with mean 0 and variance All N+1 underwater acoustic signal propagation time observations are integrated into a vector All N+1 real underwater acoustic signal propagation times are integrated into a vector t k All N+1 underwater acoustic propagation time observation noises are integrated into a vector v k as follows:

[0187]

[0188]

[0189] The observation noise v k is modeled as Gaussian noise with mean 0 and variance matrix Q k The maximum likelihood estimation problem of single-beacon positioning is:

[0190]

[0191] In the formula, is the maximum likelihood estimate of AUV position at time k, arg max is to solve the maximum problem, exp {} is the exponential function, p k is the position of AUV at time k, superscript T represents the vector matrix transpose, t k (p k ) is the true underwater acoustic signal propagation time, which is a function of position coordinates; variance matrix Q k According to the variance of the underwater acoustic signal propagation time observation at each time, it is obtained as follows:

[0192]

[0193] In the formula, σ k-N is the standard deviation of the observation variance at time k-N, σ k-N+1 is the standard deviation of the observation variance at time k-N+1, σ k is the standard deviation of the observation variance at time k, diag(a) is a diagonal matrix with the elements of a as the diagonal elements, according to the monotonicity of the logarithmic function, the above optimization problem is converted to:

[0194]

[0195] In the formula, arg min() is the minimum function;

[0196] Definition:

[0197]

[0198] Where S k is the Choleskey decomposition of , the optimization problem is converted to:

[0199]

[0200] This problem is a nonlinear least squares optimization problem, which is solved by using Gauss-Newton method, as follows:

[0201]

[0202] Where, is the position estimate obtained by j+1 iteration, is the position estimate obtained by j iteration, is the Gauss-Newton position increment of the jth iteration, which is defined as:

[0203]

[0204] In the formula, is the gradient matrix of signal propagation time to the position coordinates obtained by j iteration, the acoustic signal propagation time estimated from the AUV position estimated at the current iteration cycle, is defined as:

[0205]

[0206] where, is the partial derivative of the acoustic signal propagation time with respect to the x-direction position coordinate, is the partial derivative of the acoustic signal propagation time with respect to the y-direction position coordinate, is the partial derivative of the acoustic signal propagation time with respect to the z-direction position coordinate, is the x, y, z position estimate obtained at the jth iteration.

[0207] The gradient matrix of the position coordinates The calculation method of each element is as follows:

[0208] According to the acoustic beacon observation equation:

[0209] 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 ),

[0210] for i = 0, 1, 2…N

[0211] Take the partial derivative of both sides with respect to x k , y k , z k respectively, to obtain:

[0212]

[0213] where, is the partial derivative of the virtual observation with respect to the acoustic signal propagation time, are the partial derivatives of the acoustic signal propagation time with respect to the AUV position, respectively;

[0214] According to the definition of m k-i , we have:

[0215]

[0216] Then we have:

[0217]

[0218] Obtain:

[0219]

[0220] for i = 0, 1…N.

[0221] The method for estimating the underwater acoustic signal propagation time according to the AUV position estimated in the current iteration cycle is as follows:

[0222] According to the equation:

[0223]

[0224] for i = 0, 1, 2…N

[0225] Obtain:

[0226] n i α i = (α i - 1) 2

[0227] In the formula, n i and α i are auxiliary parameters, and their definitions are as follows:

[0228]

[0229] Solving the above quadratic equation obtains:

[0230]

[0231] That is:

[0232]

[0233] According to the above equation, two different t k-i are obtained, and since the real underwater acoustic signal propagation time is a positive number, select the one greater than 0; the position coordinates obtained in the jth iteration are brought into the above equation to obtain an underwater acoustic signal propagation time vector

[0234] The iteration convergence criterion is set as: when the two-norm of the position update amount is less than a certain threshold ε, it is considered that the iteration converges, and the single-beacon positioning position estimation value is output.

[0235] The pseudo code of the underwater single-beacon Gauss-Newton positioning method under the linear sound profile condition provided in the embodiment of the application is shown in Table 1.

[0236] Table 1 Pseudo code of underwater single-beacon Gauss-Newton positioning method under linear sound profile condition ​

[0237]

[0238]

[0239] Embodiment 2 provides a single-beacon Gauss-Newton positioning system under linear sound speed profile. The system focuses on the refraction problem of acoustic propagation channel under linear sound speed profile, and designs a single-beacon positioning position solution method based on Gauss-Newton method. The AUV of the system 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 underwater acoustic beacon periodically broadcasts underwater acoustic signals, the AUV periodically observes the velocity in the body coordinate system through the DVL carried, combines the attitude angle observed by the attitude sensor, calculates the velocity of the AUV in the geodetic coordinate system, performs dead reckoning, and records and stores the dead reckoning data; after receiving the underwater acoustic signals transmitted by the underwater acoustic beacon, the AUV combines the known underwater acoustic signal transmission time to obtain the underwater acoustic signal propagation time; when the AUV receives a certain number of underwater acoustic signals, the stored dead reckoning data is combined to construct a virtual long baseline array; the relationship between the underwater acoustic propagation time under the linear sound profile and the AUV position is combined to construct a virtual long baseline positioning problem; the observation noise of the underwater acoustic signal propagation time is modeled as a Gaussian distribution to construct a maximum likelihood estimation problem, and then it is converted into a nonlinear least squares estimation problem; based on the chain rule, the gradient of the observation to the AUV position coordinates is solved, the nonlinear least squares estimation problem is iteratively solved based on the Gauss-Newton method, and after the iteration converges, the AUV position coordinates are obtained.

[0240] To further illustrate the effects of the embodiments of the present application, the following simulation experiments are performed: This embodiment shows the positioning performance of three single-beacon positioning methods that do not consider sound propagation refraction, the sound speed value is calculated using the average value of the entire linear sound speed profile, combined with the observed underwater acoustic signal propagation time, the distance observation value between the AUV and the beacon can be calculated, and the distance is used to estimate the AUV position coordinates. 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 semi-definite programming method in the literature Yanshen Du, Ping Wei, and Huaguo Zhang. 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 semi-definite programming method in the literature K.W. Cheung, W.K. Ma, and H.C. So. Accurate approximation algorithm for toa-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 application directly uses the underwater acoustic propagation time as the observation, constructs a nonlinear least squares estimation problem, and solves it based on the Gauss-Newton method, which is simply denoted as T-GN.

[0241] The simulation parameters are set as follows: the AUV dead reckoning interval is 0.1 seconds, the underwater acoustic signal transmission interval is 15 seconds, and the underwater acoustic beacon is fixed at [0 0 100] T m, and the total simulation time is 500 seconds. The AUV's helical diving motion trajectory is generated using the AUV's dynamics model, and the AUV's control input is 8° for the horizontal rudder, -8° for the vertical rudder, and 1000 revolutions per minute for the propeller. The initial position of the AUV is [100 100 10] T m, the initial forward speed is 1.2 m / s, and the initial speed in other directions is 0. The AUV motion trajectory obtained by simulation is as follows: Figure 3The standard deviation of the simulated DVL three direction velocity observation noise is [0.01 0.01 0.01] T m / s, the standard deviation of the attitude sensor roll and pitch angle observation is 0.03°, the standard deviation of the heading angle observation is 0.3°, and the standard deviation of the acoustic signal propagation time is 100us. The simulated linear sound speed profile function is c = -0.22z + 1542m / s, and the profile shape is as shown in Fig. 2. Figure 4

[0242] The Bellhop acoustic simulation toolbox is used to generate the acoustic ray and the real acoustic signal propagation time during the AUV motion. The Bellhop model parameters used are shown in Table 2.

[0243] Table 2 Bellhop environment parameters

[0244] Signal transmission frequency 10 KHz Depth 200m Seafloor modeling Acoustic-elastic half-space Minimum maximum acoustic ray angle -90°，90° Maximum acoustic ray distance 1.6 km Maximum acoustic ray depth 200 m Number of beams 9001

[0245] For all the methods, the width of the sliding window is set to 15, i.e. the current position of the AUV is solved using the current time and the 15 acoustic ranging observations before the current time. For the proposed Gauss-Newton method, the iterative convergence threshold is set to ε = 10 -5 ​, the maximum number of iterations is set to 100. The initial position of the Gauss-Newton algorithm at each moment is set to the AUV coordinate estimated at the previous moment, and the initial position of the AUV at the initial moment is the position of the beacon plus a Gaussian random number with a standard deviation of 1. The R-SDR1 and R-SDR2 methods were solved using the CVX toolbox in Matlab software, using the sedumi solver and setting the solution accuracy to the highest. See Michael Grant and Stephen Boyd. CVX: Matlab software for disciplined convex programming, version 2.0beta. https: / / cvxr.com / cvx, September 2013. and 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 localization error and average localization error of the four methods were used as performance evaluation criteria. The localization error and average localization error were calculated as follows:

[0246]

[0247]

[0248] 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. The number of iterations of the proposed Gauss-Newton method at each moment is shown in Figure 6 shown.

[0249] according to Figure 5 As shown in Table 3, the positioning accuracy of the Gauss-Newton method proposed in the present invention is significantly better than that of the traditional single-beacon positioning method that does not consider the refraction of sound propagation. Its positioning error is lower than that of other comparison methods at most times. This shows that 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.Figure 6 It can be seen that the method of the present application can achieve faster convergence speed, and the maximum iteration number is not more than 10 times. The method of the present application has good real-time performance and can meet the real-time application requirements of single-beacon positioning.

[0250] Table 3 Comparison of average positioning errors of different methods

[0251] Positioning method Mean error (m) T-GN 0.908090 R-WLS 2.396516 R-SDR1 1.518397 R-SDR2 2.374167

[0252] The above describes only the preferred specific embodiments of the present application, but the protection scope of the present application is not limited thereto. Any modification, equivalent replacement and improvement made by those skilled in the art within the technical range disclosed by the present application and within the spirit and principles of the present application shall be encompassed within the protection scope of the present application.

Claims

1. A single underwater beacon Gauss-Newton positioning method under linear acoustic profile conditions, characterized in that: The method includes: S1, with any point in the positioning area as the origin, the east, north, and sky directions are set as X, Y, and Z axes respectively, to establish the underwater single beacon positioning geodetic coordinate system; with the AUV centroid as the origin, the front, right, and bottom directions are set as X, Y, and Z axes respectively, to establish the AUV body coordinate system; S2, constructing a dead reckoning model based on the AUV velocity in the body coordinate system detected by the DVL and the AUV attitude detected by the attitude sensor; S3, the underwater acoustic beacon periodically emits an underwater acoustic signal. The beacon position coordinates are known. The AUV is equipped with a hydrophone to measure the propagation time of the underwater acoustic signal. The sound velocity profile is modeled as a linear acoustic profile. Combining the known underwater acoustic profile parameter information with the position coordinates of the underwater acoustic beacon, an underwater acoustic observation model under the linear acoustic profile condition is constructed to obtain the relationship between the signal propagation time and the AUV position. S4, integrating the AUV dead reckoning information and using all the acoustic signal propagation time observations within the time window width N, constructs a virtual long baseline positioning model under the linear acoustic profile condition; S5, calculates the gradient matrix of the underwater acoustic signal propagation time relative to the position variable, and uses the Gauss-Newton method to iteratively solve the positioning problem. After the iteration converges, the position estimate of the AUV is obtained.

2. The underwater single-beacon Gauss-Newton positioning method under linear acoustic profile conditions according to claim 1, characterized in that: In step S2, a dead reckoning model is constructed, including: The relationship between the AUV position variables at time k and time ki,i≥1 is: Where x k ,y k ,z k is the real position coordinate of AUV at time k, 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, 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.

3. The underwater single-beacon Gauss-Newton positioning method under linear acoustic profile conditions according to claim 1, characterized in that: In step S3, the underwater acoustic observation model under the linear acoustic profile condition is constructed, and the expression is: Where, P b is the coordinate vector of the underwater beacon position, x b is the east coordinate of the acoustic beacon, y b is the north coordinate of the acoustic beacon, z b The celestial coordinates of the hydroacoustic beacon; The sound velocity profile is modeled as linear, and the sound velocity value is a linear function of the depth as follows: v=az+b 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 sea level; When the sound velocity profile is known, the sound velocity at the beacon is known and is denoted as c b , the speed of sound at the AUV position at time k is recorded as c k , the propagation time of underwater acoustic signal at time k is recorded as t k , the relationship between horizontal distance and signal propagation time is: Among them, r k is the horizontal distance between the AUV and the beacon, and e is the natural base; Squaring both sides of the above equation yields: Simplifying to get: Substituting the sound velocity profile parameters into the equation, we get: definition: Where m k is a virtual observation, in which all parameters are known and the value of each parameter depends on the signal propagation time observation value and the sound profile parameter at the current moment; The relationship between the AUV position and the underwater acoustic signal propagation time is obtained as follows: 0=(x k -x b ) 2 +(y k -y b ) 2 +(z k -z b ) 2 -m k (a) k +b)。 4. The underwater single-beacon Gauss-Newton positioning method under linear acoustic profile conditions according to claim 2, characterized in that: In step S4, a virtual long baseline positioning model under the linear acoustic profile condition is constructed, including: Define the time window width as N, use the propagation time of the underwater acoustic signal received from kN to time k, and solve the position coordinates of the AUV at time k. For time ki, the established underwater acoustic signal propagation time observation equation is applied as follows: 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 Where m k-i is the virtual observation quantity at time ki; Substitute the AUV’s dead reckoning model, i.e. the position relationship between time k and time ki, and we get: 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 in, Where x b,i is the x-axis position coordinate of the i-th virtual beacon, y b,i is the y-axis position coordinate of the i-th virtual beacon, z b,i is the z-axis position coordinate of the i-th virtual beacon, b i is the surface sound velocity corresponding to the i-th virtual beacon; further, define: x b,0 =x b and b,0 =and b With b,0 =z b b0=b Where x b,0 ,y b,0 ,z b,0 is the virtual beacon coordinate at time k-0, b0 is the virtual surface sound speed at time k-0; Combining the observation equation at time k, all N+1 observation equations are expressed as follows: 0=(x k -x b,i ) 2 +(y k -y b,i ) 2 +(z k -z b,i ) 2 -m k-i (a) k +b i ), for i=0,1,2…N.

5. The underwater single-beacon Gauss-Newton positioning method under linear acoustic profile conditions according to claim 2, characterized in that: In step S5, the positioning problem is iteratively solved using the Gauss-Newton method, including: The propagation time of underwater acoustic signal contains observation noise, and the observed value of the propagation time of underwater acoustic signal is recorded as: Where, is the observed value of underwater acoustic signal propagation time, t k is the actual underwater acoustic signal propagation time, v k is the observation noise of the underwater acoustic signal propagation time, and its modeling mean is 0 and the variance is Gaussian random variable; all N+1 underwater acoustic signal propagation time observations are combined into a vector The propagation time of all N+1 real underwater acoustic signals is integrated into a vector t k , all N+1 underwater acoustic propagation time observation noises are integrated into a vector v k ,as follows: The observation noise v k Modeled as Gaussian noise with a mean of 0 and a variance matrix denoted as Q k , then the maximum likelihood estimation problem of single beacon positioning is: Where, is the maximum likelihood estimate of the AUV position at time k, arg max is the maximization problem, exp{} is the exponential function, and p k is the position of the AUV at time k, the superscript T represents the vector matrix transpose, t k (p k ) is the real underwater acoustic signal propagation time, and its value is a function of the position coordinates; the variance matrix Q k The variance of the underwater acoustic signal propagation time observation at each moment is obtained as follows: Where, σ k-N is the standard deviation of the observation variance at time kN, σ k-N+1 is the standard deviation of the observation variance at time k-N+1, σ k is the standard deviation of the observed variance at time k, diag(a) is a diagonal matrix with the elements of a as diagonal elements. According to the monotonicity of the logarithmic function, the above optimization problem is transformed into: Where arg min( ) is the minimization function; definition: Among them, S k for The Choleskey decomposition of the optimization problem is transformed into: This problem is a nonlinear least squares optimization problem, and the Gauss-Newton method is used to solve it as follows: in, is the position estimate obtained after j+1 iterations, is the position estimate obtained at the jth iteration, is the Gauss-Newton position increment for the jth iteration, defined as: Where, is the gradient matrix of the signal propagation time to the position coordinates obtained at the jth iteration, is the underwater acoustic signal propagation time calculated based on the AUV position estimated in the current iteration cycle, Defined as: Where, is the partial derivative of the underwater acoustic signal propagation time with respect to the x-direction position coordinate, is the partial derivative of the underwater acoustic signal propagation time with respect to the position coordinate in the y direction, is the partial derivative of the underwater acoustic signal propagation time with respect to the position coordinate in the z direction, is the x, y, and z position estimate obtained at the jth iteration.

6. The underwater single-beacon Gauss-Newton positioning method under linear acoustic profile conditions according to claim 4, characterized in that: Gradient matrix of position coordinates Each element is calculated as follows: According to the underwater acoustic beacon observation equation: 0=(x k -x b,i ) 2 +(y k -y b,i ) 2 +(z k -z b,i ) 2 -m k-i (a) k +b i ), for i=0,1,2…N Both sides of x k ,y k ,z k Calculate the partial derivatives respectively and get: Where, is the partial derivative of the virtual observation with respect to the propagation time of the underwater acoustic signal, are the partial derivatives of the underwater acoustic signal propagation time with respect to the AUV position; According to m k-i The definition of , we get: Then we have: get:

7. The underwater single-beacon Gauss-Newton positioning method under linear acoustic profile conditions according to claim 5, characterized in that: The propagation time of the underwater acoustic signal is calculated based on the AUV position estimated in the current iteration cycle. The method is: According to the equation: get: n i α i ＝(α i -1) 2 Where n i With α i are auxiliary parameters, and their definitions are: Solving the above quadratic equation yields: Right now: According to the above equation, we can get two different t k-i , since the real underwater acoustic signal propagation time is a positive number, select the one greater than 0; substitute the position coordinates obtained in the jth iteration into the above equation to obtain the underwater acoustic signal propagation time vector 8. The underwater single-beacon Gauss-Newton positioning method under linear acoustic profile conditions according to claim 1, characterized in that: Set the iterative convergence criterion as, when the position update amount When the two-norm is less than a certain threshold ε, the iteration is considered to converge and the single beacon positioning position estimate is output 9. An underwater single-beacon Gauss-Newton positioning system under linear acoustic profile conditions, characterized in that: The system implements the underwater single-beacon Gauss-Newton positioning method under linear sound profile conditions as described in any one of claims 1 to 8. The system addresses the problem of sound propagation channel refraction caused by the change of underwater sound velocity with depth under linear sound profile conditions, and obtains the AUV position coordinates through a positioning solution method based on the Gauss-Newton algorithm.

10. The underwater single-beacon Gauss-Newton positioning system under linear acoustic profile conditions according to claim 9, characterized in that: The system includes an autonomous underwater vehicle (AUV) equipped with a hydrophone, a dynamic velocity detector (DVL), an attitude sensor, and a coordinate detection device (CTD). The AUV detects the sound velocity profile parameters of the experimental area using the CTD. The hydroacoustic beacon periodically broadcasts a hydroacoustic signal, and the AUV periodically observes its velocity in the body coordinate system using the DVL it carries. The velocity in the geodetic coordinate system is calculated based on the AUV attitude angle observed by the attitude sensor, and dead reckoning is performed. The dead reckoning data is then recorded and stored. After receiving the underwater acoustic signal transmitted by the underwater acoustic beacon, the AUV obtains the underwater acoustic signal propagation time based on the known underwater acoustic signal transmission time; When the AUV receives a certain number of underwater acoustic signals, it constructs a virtual long baseline array based on the stored dead reckoning data; Combining the relationship between underwater acoustic propagation time and AUV position under linear acoustic profile, a virtual long baseline positioning problem is constructed; the underwater acoustic signal propagation time observation noise is modeled as a Gaussian distribution, and a maximum likelihood estimation problem is constructed, which is then transformed into a nonlinear least squares estimation problem; the gradient of the observation quantity with respect to the AUV position coordinates is solved based on the chain rule, and the nonlinear least squares estimation problem is iteratively solved based on the Gauss-Newton method. After the iterative convergence, the AUV position coordinates are obtained.

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