A self-positioning method for underwater autonomous vehicle cluster nodes

Abstract

The present invention discloses a self-positioning method for underwater autonomous vehicle cluster nodes, which belongs to the field of underwater detection technology. The method comprises the following steps: inputting preprocessed acoustic environment feature data into a Bi-LSTM-MLP model for training, generating a predicted sound speed model, and calculating the propagation time using a two-way time difference method; dividing the vertical depth of a water body into several discrete layers using the sound speed model, establishing a relationship between multi-layer propagation angles and sound speeds using Snell's law, and calculating the propagation angle of each layer in combination with the propagation time; calculating the horizontal distance of the propagation path of each layer using the geometric relationship between the horizontal distance and thickness of each layer, and accumulating the horizontal distance of the total propagation path; and determining the spatial position of the AUV by combining the horizontal distance from the underwater AUV to three surface AUVs with depth information using a three-circle positioning method. The present invention adopts the above-mentioned self-positioning method for underwater autonomous vehicle cluster nodes, and significantly improves the adaptability and robustness of the positioning algorithm to complex marine environments.

Description

Technical Field

[0001] The present invention relates to the field of underwater detection technology, and in particular to a self-positioning method for underwater autonomous vehicle cluster nodes. Background Art

[0002] Node positioning technology for autonomous underwater vehicles (AUVs) swarms has attracted considerable attention due to its widespread application in both military and civilian fields. This technology is crucial for achieving high-precision seafloor mapping, deep-sea resource exploration and development, underwater security monitoring, and analyzing the behavior of unknown targets. However, traditional AUV positioning technology is susceptible to interference from factors such as changes in underwater sound speed, signal propagation delays, and refraction of the propagation path, significantly reducing positioning accuracy. Therefore, underwater AUV positioning methods that incorporate the effects of acoustic refraction have become a research hotspot for improving AUV self-positioning accuracy, driving the in-depth development of related technologies.

[0003] Time Difference of Arrival (TDOA) has always been one of the important methods in the field of AUV (autonomous underwater vehicle) positioning. This technology determines the position of the AUV by measuring the time difference between signals arriving at multiple sensors. It not only has high positioning accuracy, but also effectively overcomes the challenges brought by the propagation delay of underwater acoustic signals. In recent years, LiuYing et al. (2023) proposed a TDOA measurement method based on the multipath channel effect of the underwater environment. This method introduces a calibration source to improve positioning performance. Compared with traditional technologies, its mean square error is significantly reduced. In some scenarios, the efficiency exceeds 90% under low noise conditions and exceeds 80% under medium noise conditions.

[0004] In addition, Zhang Tao et al. proposed an AUV positioning method based on a tightly coupled strapdown inertial navigation system (SINS) / long baseline (LBL) algorithm. This method effectively and regularly compensates for the accumulated AUV position error by introducing LBL positioning information centered on the SINS. However, the positioning accuracy of the aforementioned methods is susceptible to the uncertainty of sound propagation speed, which in turn leads to reduced positioning accuracy.

[0005] To address these issues, we propose a self-localization method for underwater AUV cluster nodes based on sound velocity matching and bidirectional time difference fusion. This method innovatively employs a Bi-LSTM-MLP model to accurately model the spatiotemporal variations of sound velocity in complex ocean environments. Based on Snell's law, it constructs a multi-layered relationship between propagation angle and sound velocity, thereby accurately simulating the impact of sound wave refraction on the propagation path. This improvement effectively reduces the interference of sound velocity variations and refraction on positioning accuracy. Summary of the Invention

[0006] The purpose of the present invention is to provide a self-positioning method for underwater autonomous vehicle cluster nodes to solve the problems mentioned in the above background technology.

[0007] To achieve the above object, the present invention provides a method for self-positioning of underwater autonomous vehicle cluster nodes, comprising the following steps:

[0008] S1. Input the pre-processed acoustic environment feature data into the Bi-LSTM-MLP model for training to generate a predicted sound speed model. The Bi-LSTM-MLP model includes a Bi-LSTM layer consisting of two independent LSTM layers and an MLP layer consisting of an input layer, one or more hidden layers, and an output layer.

[0009] S2. Use the two-way time difference method to calculate the propagation time to avoid the error caused by the clock deviation between the surface AUV and the underwater AUV;

[0010] S3. Divide the vertical depth of the water body into several discrete layers using the sound velocity model. Use Snell's law to establish the relationship between the multi-layer propagation angle and the sound velocity, and calculate the propagation angle of each layer in combination with the propagation time.

[0011] S4. Calculate the horizontal distance of each layer's propagation path using the geometric relationship between the horizontal distance and thickness of each layer, and accumulate the total horizontal distance of the propagation path.

[0012] S5. Combining the horizontal distances from the underwater AUV to the three surface AUVs, the spatial position of the AUV is determined by combining the depth information with the three-circle positioning method.

[0013] Preferably, the specific process of S1 is: the input data passes through the Bi-LSTM layer to output the hidden state y, the hidden state y is input to the hidden layer of the MLP layer, and finally the predicted value of the sound speed is output to obtain the predicted sound speed model.

[0014] Preferably, the specific workflow of the Bi-LSTM layer is:

[0015] LSTM consists of a memory unit and three unit gates: input gate, forget gate, and output gate. Its core idea is to model long-term dependencies in time series through a gating mechanism. The workflow is as follows:

[0016] f t =σ(W f ·[h t-1 ,x t ]+b f ) (1)

[0017] i t =σ(W i ·[h t-1 ,x t]+b i ) (2)

[0018]

[0019] o t =σ(W o ·[h t-1 ,x t ]+b o ) (5)

[0020] h t =o t tanh C t ) (6)

[0021] Among them, x t is the input of the current time step; is the candidate memory cell state; f t ,i t , o t are the outputs of the forget, input, and output gates in the LSTM unit; W f , W i , W o , W c The weight matrices corresponding to the forget, input, output gates, and memory units; C t , C t-1 is the memory cell state at the current time step and the previous time step; h t , h t-1 is the hidden state of the current time step and the previous time step; b f , b i , b c , b o is the corresponding deviation;

[0022] Bi-LSTM processes the forward (from past to future) and reverse (from future to past) sequence data through two independent LSTM networks, and finally outputs the concatenation result y of the forward and reverse LSTM hidden states. The specific process is as follows:

[0023]

[0024] in, is the hidden state of the forward LSTM, is the hidden state of the reverse LSTM, and both are calculated using formula (6).

[0025] Preferably, each layer of the MLP layer is composed of a number of neurons (also called nodes), and the calculation formula is:

[0026] a l =fW l ·al-1 +b l (10)

[0027] Among them, a l represents the activation value of the lth layer, W l and b l are weight matrices and biases respectively, and f is a nonlinear activation function;

[0028] The hidden state y enters the MLP hidden layer, and the activation value of each layer is calculated by formula (10), and the predicted value of the final output sound speed is calculated by formula (11)

[0029]

[0030] Among them, W1 and W2 are the weight matrices of the fully connected layer, and b1 and b2 are bias terms.

[0031] Preferably, the specific steps of S2 are as follows:

[0032] S21, the surface AUV sends a signal to the underwater AUV at time T0. After receiving the signal, the underwater AUV returns a response signal after a delay of Δt. The surface AUV receives the response signal at time T1.

[0033] S22: The surface AUV records the signal sending time T0 and the time T1 when the response signal is received. First, the total propagation time of the signal is calculated. The processing delay of the underwater AUV is subtracted from the total propagation time to obtain the actual propagation time of the signal. The formula is as follows:

[0034] T=T1-(T0+Δt)(12)

[0035] Where T is the actual propagation time of the signal.

[0036] Preferably, the expression of Shell's law in S3 is:

[0037]

[0038] Among them, β i is the propagation angle of the i-th layer, v i is the average sound speed in the i-th layer.

[0039] Preferably, the specific process of calculating the horizontal distance of the total propagation path in S4 is:

[0040] x i =d i tanβ i (14)

[0041]

[0042] Among them, x i is the horizontal propagation distance of the sound wave at the i-th position, d i is the thickness of the i-th layer, t i is the propagation time of the sound wave in the i-th layer, x is the total horizontal distance propagated, and t is the total propagation time.

[0043] Preferably, the formula for solving the three-circle positioning method in S5 is:

[0044] Z=(∥xy-A∥-X1) 2 +(∥xy-B∥-X2) 2 +(∥xy-C∥-X3) 2 (19)

[0045]

[0046] Among them, X1, X2, and X3 are the horizontal distances from three surface AUVs to an underwater AUV, xy = (x, y) is the optimal solution for the target position, A = (Ax, Ay), B = (Bx, By), and C = (Cx, Cy) are the three known coordinates of the circle centers, ∥xy-A∥ represents the Euclidean distance from the target position to point A, and Z is the sum of the distances from the target position to the three circle centers.

[0047] Therefore, the present invention adopts the above-mentioned underwater autonomous vehicle cluster node self-positioning method, which has the following beneficial effects:

[0048] (1) The present invention effectively combines sound velocity modeling with positioning algorithms, significantly improving positioning accuracy and system stability in complex underwater environments. By accurately measuring propagation time through sound velocity prediction and two-way time difference method, the adaptability and robustness of the positioning algorithm to complex marine environments are significantly improved.

[0049] (2) The present invention does not require underwater AUVs to carry expensive sensors or positioning systems, which makes it suitable for the efficient positioning needs of AUV clusters.

[0050] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0051] Figure 1 This is a flow chart of a self-positioning method for underwater autonomous vehicle cluster nodes of the present invention;

[0052] Figure 2 This is a Bi-LSTM-MLP model structure diagram of a self-positioning method for underwater autonomous vehicle cluster nodes of the present invention;

[0053] Figure 3 This is a structural diagram of an LSTM model for a self-positioning method of underwater autonomous vehicle cluster nodes according to the present invention;

[0054] Figure 4 A schematic diagram of a two-way time difference method for a self-positioning method of a cluster node of an underwater autonomous vehicle according to the present invention;

[0055] Figure 5 A propagation path diagram of a self-positioning method for underwater autonomous vehicle cluster nodes according to the present invention;

[0056] Figure 6 This is a three-circle intersection positioning diagram of a self-positioning method for underwater autonomous vehicle cluster nodes according to the present invention. DETAILED DESCRIPTION

[0057] The following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but rather merely represents selected embodiments of the present invention. All other embodiments derived by persons of ordinary skill in the art based on the embodiments of the present invention without inventive effort shall fall within the scope of protection of the present invention.

[0058] Example

[0059] like Figure 1 As shown, the present invention provides a self-positioning method for underwater autonomous vehicle cluster nodes, comprising the following steps:

[0060] S1. Input the pre-processed acoustic environment feature data into the Bi-LSTM-MLP model for training to generate a predicted sound speed model. The Bi-LSTM model structure is as follows: Figure 2 As shown in Figure 1, the Bi-LSTM-MLP model includes a Bi-LSTM layer consisting of two independent LSTMs and an MLP layer consisting of an input layer, one or more hidden layers, and an output layer.

[0061] The input data passes through the Bi-LSTM layer to output the hidden state y, which is then input to the hidden layer of the MLP layer. Finally, the predicted value of the sound speed is output to obtain the predicted sound speed model.

[0062] The specific workflow of the Bi-LSTM layer is as follows:

[0063] like Figure 3 As shown in the figure, LSTM consists of a memory unit and three unit gates: input gate, forget gate, and output gate. Its core idea is to model long-term dependencies in time series through a gating mechanism. The workflow is as follows:

[0064] f t =σ(W f ·[h t-1 ,x t ]+b f ) (1)

[0065] i t =σ(W i ·[h t-1 ,x t ]+b i ) (2)

[0066]

[0067] o t =σ(W o ·[h t-1 ,x t ]+b o ) (5)

[0068] h t =o t tanh C t ) (6)

[0069] Among them, x t is the input of the current time step; is the candidate memory cell state; f t ,i t , o t are the outputs of the forget, input, and output gates in the LSTM unit; W f , W i , W o , W c The weight matrices corresponding to the forget, input, output gates, and memory units; C t , C t-1 is the memory cell state at the current time step and the previous time step; h t , h t-1 is the hidden state of the current time step and the previous time step; b f , b i , b c , b o is the corresponding deviation;

[0070] On this basis, Bi-LSTM processes the sequence data in the forward direction (from past to future) and the reverse direction (from future to past) through two independent LSTM networks. Each LSTM network runs independently according to the above process. The final output is combined by connecting the hidden state vectors of the forward and reverse networks, thereby simultaneously capturing the contextual information of each time point in the sequence. The final output is the concatenation result y of the forward and reverse LSTM hidden states. The specific process is as follows:

[0071]

[0072] in, is the hidden state of the forward LSTM, is the hidden state of the reverse LSTM, and the two are calculated by formula (6). y is the final bidirectional LSTM output, that is, the result of splicing the hidden states of the forward and reverse LSTM together.

[0073] Each MLP layer is composed of several neurons (also called nodes), which transmit information through a fully connected network. The core feature of MLP is its ability to extract features of input data through hidden layers and model complex nonlinear relationships using nonlinear activation functions.

[0074] The input data starts from the input layer and passes through the hidden layer and output layer in sequence. The calculation formula for each layer is:

[0075] a l =fW l ·a l-1 +b l (10)

[0076] Among them, a l represents the activation value of the lth layer, W l and b l are the weight matrix and bias respectively, and f is the nonlinear activation function. By changing the type of nonlinear activation function to adapt to the needs of different tasks, the fitting accuracy can be improved.

[0077] The hidden state y enters the MLP hidden layer, and the activation value of each layer is calculated by formula (10), and the predicted value of the final output sound speed is calculated by formula (11)

[0078]

[0079] Among them, W1 and W2 are the weight matrices of the fully connected layer, and b1 and b2 are bias terms.

[0080] S2. The two-way time difference method is used to calculate the propagation time to avoid the error caused by the clock deviation between the surface AUV and the underwater AUV.

[0081] The core idea of ​​the Two-Way Time Difference method is to eliminate the error caused by clock bias in the one-way propagation process through the propagation time difference of two-way signals, thereby achieving high-precision time measurement.

[0082] The basic workflow of the two-way time difference method is as follows: Figure 4 shown.

[0083] S21, the surface AUV sends a signal to the underwater AUV at time T0. After receiving the signal, the underwater AUV returns a response signal after a delay of Δt. The surface AUV receives the response signal at time T1.

[0084] S22: The surface AUV records the signal sending time T0 and the time T1 when the response signal is received. First, the total propagation time of the signal is calculated. The processing delay of the underwater AUV is subtracted from the total propagation time to obtain the actual propagation time of the signal. The formula is as follows:

[0085] T=T1-(T0+Δt)(12)

[0086] Where T is the actual propagation time of the signal.

[0087] Because both the sending and receiving times are recorded by the surface AUV, there's no need to consider synchronization with the underwater AUV, thus avoiding errors caused by clock offsets between the two. By using the two-way time difference method, the system significantly reduces measurement errors caused by clock asynchrony and eliminates the need for complex clock synchronization at each node, resulting in greater robustness and reliability.

[0088] S3. The vertical depth of the water body is divided into several discrete layers through the sound speed model. The relationship between the multi-level propagation angle and the sound speed is established using Snell's law, and the propagation angle of each layer is calculated in combination with the propagation time.

[0089] Snell's law (also known as the law of refraction) describes the refraction law of light waves or other types of waves when they pass from one medium to another. Similarly, when seawater is divided into multiple layers according to depth, each layer of water can be regarded as a different medium due to the differences in physical parameters such as temperature and salinity. Therefore, the propagation path of sound waves in each water layer also shows similar refraction laws, such as Figure 5 shown.

[0090] In the case of sound wave propagation, Shell's law is expressed as:

[0091]

[0092] Among them, β i is the propagation angle of the i-th layer, v i is the average sound speed in the i-th layer.

[0093] S4. Using the geometric relationship between the horizontal distance and thickness of each layer, calculate the horizontal distance of each layer's propagation path, and accumulate the horizontal distance of the total propagation path.

[0094] The specific process of calculating the horizontal distance of the total propagation path is:

[0095] Depend on Figure 5 The geometric relationship shows that:

[0096] x i =d i tanβ i(14)

[0097] The propagation time of each layer is:

[0098]

[0099] The total horizontal distance propagated is:

[0100]

[0101] The total time of propagation is:

[0102]

[0103] Combining the above equations, we can get:

[0104]

[0105] Among them, x i is the horizontal propagation distance of the sound wave at the i-th position, d i is the thickness of the i-th layer, t i is the propagation time of the sound wave in the i-th layer, x is the total horizontal distance propagated, and t is the total propagation time.

[0106] By solving the above equation, the propagation angle β can be obtained i and the total horizontal distance x traveled.

[0107] Considering that the measured propagation time T is the total time for the signal to travel back and forth, during the signal propagation process, the surface AUV will have a relative horizontal displacement relative to the underwater AUV. Let it be Δx, then:

[0108] Δx=x f -x b (twenty one)

[0109] T=T f +T b (twenty two)

[0110] where x f 、T f is the total horizontal distance and time of the signal transmitted from the surface AUV to the underwater AUV, x b 、T b is the total horizontal distance and time that the signal propagates from the underwater AUV to the surface AUV.

[0111] If:

[0112]

[0113] Then, formulas (14) to (18) can be combined to write formulas (21) and (22) as follows:

[0114]

[0115] Among them, β fi , v fi , d fi is the incident angle, average sound velocity, and thickness of the i-th layer during the signal transmission from the surface AUV to the underwater AUV; β bi , ν bi , d bi is the incident angle, average sound velocity, and thickness of the i-th layer during the signal response from the underwater AUV to the surface AUV. By solving formulas (25) and (26), we can get the value of β f1 and β b1 The nonlinear equation of x f and x b .

[0116] S5. Combining the horizontal distances from the underwater AUV to the three surface AUVs, the spatial position of the AUV is determined by combining the depth information with the three-circle positioning method.

[0117] Three circles are constructed on a depth plane using the horizontal distances X1, X2, and X3 from three surface AUVs to an underwater AUV and the depth meter information. The radius of the three circles is the horizontal distance X1, X2, and X3, and the center of the circle is the coordinates of the three surface AUVs, as shown below. Figure 6 As shown in Figure 3, the intersection of the three circles is the position of the underwater AUV.

[0118] In practice, some errors are unavoidable, and the intersection of the three circles may not be unique. Therefore, it is necessary to use optimization methods such as the least squares method to find the optimal solution within a reasonable error range.

[0119] The formula for solving the three-circle positioning method is:

[0120] Z=(∥xy-A∥-X1) 2 +(∥xy-B∥-X2) 2 +(∥xy-C∥-X3) 2 (19)

[0121]

[0122] Among them, X1, X2, and X3 are the horizontal distances from three surface AUVs to an underwater AUV, xy = (x, y) is the optimal solution for the target position, A = (Ax, Ay), B = (Bx, By), and C = (Cx, Cy) are the three known coordinates of the circle centers, ∥xy-A∥ represents the Euclidean distance from the target position to point A, and the same applies to ∥xy-B∥ and ∥xy-C∥. Z is the sum of the distances from the target position to the three circle centers. When the Z value is the smallest, xy is the optimal solution for the target position.

[0123] Therefore, the present invention adopts the above-mentioned self-positioning method of underwater autonomous vehicle cluster nodes, accurately measures the propagation time through sound speed prediction and two-way time difference method, significantly improving the adaptability and robustness of the positioning algorithm to complex ocean environments. The average error of the final positioning is about 1.2 meters, and the maximum positioning error does not exceed 2.5 meters.

[0124] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A self-positioning method for underwater autonomous vehicle cluster nodes, characterized in that: The following steps are involved: S1. Input the preprocessed acoustic environment feature data into the Bi-LSTM-MLP model for training to generate a predicted sound speed model. The Bi-LSTM-MLP model includes a Bi-LSTM layer consisting of two independent LSTMs and an MLP layer consisting of an input layer, one or more hidden layers, and an output layer. S2. Use the two-way time difference method to calculate the propagation time to avoid the error caused by the clock deviation between the surface AUV and the underwater AUV; S3. Divide the vertical depth of the water body into several discrete layers using the sound velocity model. Use Snell's law to establish the relationship between the multi-layer propagation angle and the sound velocity, and calculate the propagation angle of each layer in combination with the propagation time. S4. Calculate the horizontal distance of each layer's propagation path using the geometric relationship between the horizontal distance and thickness of each layer, and accumulate the horizontal distance of the total propagation path; S5. Combining the horizontal distances from the underwater AUV to the three surface AUVs, the spatial position of the AUV is determined by combining the depth information with the three-circle positioning method.

2. The method for self-positioning of underwater autonomous vehicle cluster nodes according to claim 1, characterized in that: The specific process of S1 is as follows: the input data passes through the Bi-LSTM layer to output the hidden state y, the hidden state y is input to the hidden layer of the MLP layer, and finally the predicted value of the sound speed is output to obtain the predicted sound speed model.

3. The method for self-positioning of underwater autonomous vehicle cluster nodes according to claim 2, characterized in that: The specific workflow of the Bi-LSTM layer is as follows: LSTM consists of a memory unit and three unit gates: input gate, forget gate, and output gate. The workflow is as follows: f t =σ(W f ·[h t-1 ,x t ]+b f ) (1) i t =σ(W i ·[h t-1 ,x t ]+b i ) (2) the t =σ(W o ·[h t-1 ,x t ]+b o ) (5) h t =o t ·tanh(C t ) (6) Among them, x t is the input of the current time step; is the candidate memory cell state; f t ,i t , o t are the outputs of the forget, input, and output gates in the LSTM unit; W f , W i , W o , W c The weight matrices corresponding to the forget, input, output gates, and memory units; C t , C t-1 is the memory cell state at the current time step and the previous time step; h t , h t-1 is the hidden state of the current time step and the previous time step; b f , b i , b c , b o is the corresponding deviation; Bi-LSTM processes the forward and reverse sequence data through two independent LSTM networks, and finally outputs the concatenation result y of the forward and reverse LSTM hidden states. The specific process is as follows: in, is the hidden state of the forward LSTM, is the hidden state of the reverse LSTM, and both are calculated using formula (6).

4. The method for self-positioning of underwater autonomous vehicle cluster nodes according to claim 3, characterized in that: The calculation formula for each layer of the MLP layer is: a (l) =f(W (l) ·a (l-1) +b (l) ) (10) Among them, a (l) represents the activation value of the lth layer, W (l) and b (l) are weight matrices and biases respectively, and f is a nonlinear activation function; y enters the MLP hidden layer, and the activation value of each layer is calculated by formula (10), and the predicted value of the final output sound speed is calculated by formula (11) Among them, W1 and W2 are the weight matrices of the fully connected layer, and b1 and b2 are bias terms.

5. The method for self-positioning of underwater autonomous vehicle cluster nodes according to claim 1, characterized in that: The specific steps of S2 are as follows: S21, the surface AUV sends a signal to the underwater AUV at time T0. After receiving the signal, the underwater AUV returns a response signal after a delay of Δt. The surface AUV receives the response signal at time T1. S22: The surface AUV records the signal sending time T0 and the time T1 when the response signal is received. First, the total propagation time of the signal is calculated. The processing delay of the underwater AUV is subtracted from the total propagation time to obtain the actual propagation time of the signal. The formula is as follows: T=T1-(T0+Δt) (12) Where T is the actual propagation time of the signal.

6. The method for self-positioning of underwater autonomous vehicle cluster nodes according to claim 1, characterized in that: The expression of Snell's law in S3 is: Among them, β i is the propagation angle of the i-th layer, v i is the average sound speed in the i-th layer.

7. The method for self-positioning of underwater autonomous vehicle cluster nodes according to claim 6, characterized in that: The specific process of S4 calculating the horizontal distance of the total propagation path is: x i =d i tanβ i (14) Among them, x i is the horizontal propagation distance of the sound wave in layer i, d i is the thickness of the i-th layer, t i is the propagation time of the sound wave in the i-th layer, x is the total horizontal distance propagated, and t is the total propagation time.

8. The method for self-positioning of underwater autonomous vehicle cluster nodes according to claim 7, characterized in that: The formula for solving the three-circle positioning method in S5 is: Z=(||xy-A||-X1) 2 +(||xy-B||-X2) 2 +(||xy-C||-X3) 2 (19) Among them, X1, X2, and X3 are the horizontal distances from three surface AUVs to an underwater AUV, xy = (x, y) is the optimal solution for the target position, A = (Ax, Ay), B = (Bx, By), and C = (Cx, Cy) are the coordinates of the three known circle centers, ||xy-A|| represents the Euclidean distance from the target position to point A, and Z is the sum of the distances from the target position to the three circle centers.

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