Positioning method based on underwater distributed system under non-gaussian measurement noise

Abstract

This invention relates to a positioning method based on an underwater distributed system under non-Gaussian measurement noise, belonging to the field of underwater acoustic positioning technology. The method involves each node of the underwater distributed positioning system acquiring azimuth measurements; constructing an AOA positioning model based on the geometric relationship between the target and each node; modeling the AOA measurement noise based on a t-distribution; adaptively estimating the noise scale parameters of each node based on multi-frame AOA measurement data; and constructing an iteratively reweighted generalized least squares solution based on the noise scale parameters to iteratively solve for the target position. This invention introduces the Student-t distribution as the AOA measurement noise model and proposes a SAGLS method based on adaptive noise scale estimation. Addressing the heterogeneous and heavy-tailed noise conditions encountered by underwater distributed systems in AOA measurements, this method independently estimates the noise scale parameters of each node and constructs an adaptive weighting system that weights the noise scale parameters with geometric relationships. This suppresses the interference of the heterogeneous and heavy-tailed underwater noise environment on passive positioning, effectively improving the accuracy and robustness of the positioning system.

Description

Technical Field

[0001] This invention belongs to the field of underwater acoustic positioning technology, specifically relating to a positioning method based on an underwater distributed system under non-Gaussian measurement noise. Background Technology

[0002] Underwater acoustic positioning is a key technology in fields such as coastal surveillance, environmental monitoring, and autonomous operation of marine platforms. Among them, underwater distributed systems, through the fusion of multi-sensor data distributed in a spatial manner, can effectively improve positioning accuracy and system robustness. Their performance is generally superior to monostatic sonar systems, making them an important means of underwater target positioning.

[0003] Compared to Time of Arrival (TOA), Time Difference of Arrival (TDOA), and Received Signal Strength (RSS) techniques, AOA-based positioning methods do not require strict time synchronization and can be obtained through array processing. They are widely used in underwater distributed systems. Traditional AOA-based positioning methods typically rely on the assumption that measurement noise follows a Gaussian distribution and employ algebraic or geometric closed-form estimators to avoid nonlinear optimization processes. Pseudo-linear least squares (LS) methods, in particular, offer high computational efficiency but generally suffer from inherent biases. These methods are sensitive to model mismatches and poorly designed layouts, limiting their applicability in complex environments. Maximum likelihood estimation (MLE), under correctly defined noise models, provides a theoretically sound inference framework and is asymptotically efficient. However, because the objective function is often non-convex and sensitive to linearization, MLE typically relies on iterative solvers and requires appropriate initial values, increasing implementation complexity in practical applications. However, most of these classic estimation methods rely on the Gaussian noise assumption or second-order statistical properties. Under conditions of strong impact noise or other non-Gaussian heavy-tailed conditions, the positioning accuracy of these methods is significantly affected.

[0004] To address the problem of non-Gaussian heavy-tailed noise, existing robust localization research mainly follows different technical paths. M-estimation-based strategies mitigate the impact of large residuals by reconstructing the cost function. Iterative reweighted instrumental variable (IRIV) methods can effectively correct spurious linear biases while suppressing impulse interference. Non-convex sparse optimization methods utilize the sparsity of outliers to cope with extreme non-Gaussian noise scenarios. Essentially, the core processing mechanisms of these methods primarily focus on reducing and suppressing non-Gaussian noise such as impulses, lacking dynamic adaptability to time-varying and spatially heterogeneous noise.

[0005] In underwater environments, multipath effects, scattering, and intermittent interference often lead to non-Gaussian heavy-tailed noise, such as occasional large deviations in azimuth measurements, severely impacting the positioning accuracy of AOA-based methods. Furthermore, in distributed sonar systems, the spatial non-uniformity of the underwater acoustic channel, including variations in sound velocity profiles (SSPs) and local seabed topography, causes changes in ray propagation paths and multipath effects. These environmental differences result in significant fluctuations in azimuth estimation performance at different locations, leading to marked heterogeneity in azimuth measurements between different sensors. This heavy-tailed and heterogeneous non-Gaussian AOA measurement noise severely degrades the positioning performance of traditional estimation methods based on Gaussian assumptions and robust fixed-threshold processing methods.

[0006] To address the impulse interference and independent, heterogeneous AOA measurement noise present in complex underwater environments, this invention introduces Student's t-distribution as the AOA measurement noise model and proposes an iterative reweighted generalized least squares localization method (SAGLS method) based on adaptive noise scale estimation. By utilizing the degrees-of-freedom (DoF) parameter of Student's t-distribution, the heavy-tailed error characteristics are effectively characterized, achieving suppression of large outliers. Based on the Gaussian scale mixture (GSM) representation, an adaptive learning method is employed to learn sensor-specific noise scale parameters, adapting to the heterogeneity of AOA measurement noise among different sensors. Summary of the Invention

[0007] To overcome the shortcomings of existing technologies, this application provides a positioning method based on an underwater distributed system under non-Gaussian measurement noise, belonging to the field of underwater acoustic positioning technology. The underwater distributed positioning system acquires orientation measurements at each node; an AOA positioning model is constructed based on the geometric relationship between the target and each node; the AOA measurement noise is modeled based on a t-distribution; the noise scale parameters of each node are adaptively estimated based on multi-frame AOA measurement data; and an iterative reweighted generalized least squares solution is constructed based on the noise scale parameters to iteratively solve for the target position. This invention introduces the Student-t distribution as the AOA measurement noise model and proposes a SAGLS method based on adaptive noise scale estimation. Addressing the heterogeneous and heavy-tailed noise conditions encountered by underwater distributed systems in AOA measurements, this method independently estimates the noise scale parameters of each node and constructs an adaptive weighting of the noise scale parameters and geometric relationships. This suppresses the interference of the heterogeneous and heavy-tailed underwater noise environment on passive positioning, effectively improving the accuracy and robustness of the positioning system.

[0008] Localization methods based on underwater distributed systems under non-Gaussian measurement noise, such as Figure 1 As shown, the specific steps are as follows:

[0009] Step 1: Obtain the azimuth measurement of the underwater distributed positioning system observation node;

[0010] Step 2: Based on the orientation measurement and the geometric relationship between the target and the observation node, construct the AOA positioning model;

[0011] Step 3: Construct an AOA measurement noise model based on the Student-t distribution;

[0012] Step 4: Based on multi-frame AOA measurement data, adaptively estimate the noise scale parameters of the observation nodes;

[0013] Step 5: Based on the noise scale parameter, construct an iterative reweighted generalized least squares solution and iteratively solve for the target position.

[0014] Furthermore, in step 1, the method for obtaining the azimuth measurement is as follows:

[0015] Step 1.1: The observation nodes of the underwater distributed system use a uniform circular array as acoustic sensors to receive acoustic signals from underwater targets.

[0016] Step 1.2: Use high-gain beamforming to perform azimuth scanning on the acoustic signals received by the observation node to obtain the azimuth spectrum;

[0017] Step 1.3: For the azimuth spectrum obtained by azimuth scanning, take the main lobe peak as the azimuth measurement to obtain the AOA measurement sequence of the target at different times.

[0018] Furthermore, in step 2, the process of constructing the AOA localization model is as follows:

[0019] Step 2.1: A distributed observation system exists within the two-dimensional plane, consisting of N spatially distributed acoustic passive observation nodes; at discrete times k, k=1,…,K, where K is the total observation duration, the i-th observation node is located at known coordinates. x i and y i The x-axis and y-axis coordinates of the i-th observation node are respectively, and the target position is... x and y are the x-axis and y-axis coordinates of the target, respectively, and the azimuth measurement of the i-th observation node. satisfy:

[0020] (1);

[0021] in, For the actual location, For AOA noise measurement, Indicates matrix transpose;

[0022] Step 2.2: Based on the geometric relationship between the underwater distributed system observation nodes and the target, determine the true orientation. Linearization is represented as:

[0023] (2);

[0024] In practice, azimuth measurement Includes AOA measurement noise This leads to an algebraic difference in equation (2); the algebraic difference is defined as a pseudo-linear residual function. :

[0025] (3);

[0026] Step 2.3: Solve the equations of the observation nodes simultaneously to construct the matrix form AOA localization model for the k-th frame:

[0027] (4);

[0028] in, This represents the pseudolinear residual vector of the k-th frame. System matrix. and observation vector They are respectively:

[0029] (5);

[0030] (6);

[0031] in, For the target location; It is the set of real numbers.

[0032] Furthermore, in step 3, the process of constructing the AOA measurement noise model is as follows:

[0033] Step 3.1: The Student-t distribution is used as the AOA measurement noise model, which is also the AOA error model.

[0034] The Student-t distribution is determined by the degree of freedom parameter. Controlling the tail thickness and using noise scale parameters Characterizing the degree of discreteness; assuming that the AOA measurement noise at different times is independent, the noise vector of the k-th frame... The components in the data are independent of each other; the AOA measurement noise of the i-th observation node. It follows a zero-mean Student-t distribution, denoted as AOA measurement noise The probability density function (PDF) is :

[0035] (7);

[0036] in, For gamma function, The noise scale parameter for the AOA measurement of the i-th observation node reflects the degree of noise dispersion. Relationship with variance of Gaussian distribution:

[0037] (8);

[0038] Where Var(·) represents variance;

[0039] Step 3.2, due to the independence assumption, the noise vector of the k-th frame... The joint probability density function is the product of the PDFs of each component:

[0040] (9);

[0041] Step 3.3: Equivalently represent the Student-t distribution as a Gaussian Scale Mixture (GSM) model containing latent variables;

[0042] AOA measurement noise Introducing latent variables The Student-t distribution is decomposed into a hierarchical Gaussian-gamma structure:

[0043] Conditional likelihood distribution: given and Follows a Gaussian distribution with scaled covariance :

[0044] (10);

[0045] Prior distribution of latent variables: It follows a gamma distribution, and its probability density function is... for:

[0046] (11).

[0047] Furthermore, in step 4, the method for adaptively estimating the noise scale parameters of the observation nodes is as follows:

[0048] In underwater passive positioning scenarios, due to the non-uniformity of the ambient noise field where each sensor is located, the observed noise often exhibits significant differences in noise scale. In order to accurately describe this statistical characteristic, a noise model is constructed based on the Student-t distribution using multi-frame AOA measurement data, and the noise scale parameters of each sensor are adaptively learned through the EM algorithm.

[0049] Step 4.1, based on GSM theory, define the complete dataset for observation node i. ,in This is the AOA measurement noise set. Let be the set of unobservable latent variables; let be the set of parameters to be estimated. ;

[0050] Based on the assumption that the noise from each sensor is independent in the spatiotemporal domain, the joint probability density function of all K frames of complete data... Represented as the product of the PDF components:

[0051] (12);

[0052] Step 4.2: Complete the log-likelihood function of all K-frame data, ignoring the AOA measurement noise scale parameter to be estimated. Irrelevant constant terms, objective function Simplified to:

[0053] (13);

[0054] Step 4.3: Use the EM algorithm to alternately execute the expectation step (E-step) and the maximization step (M-step) to solve equation (13);

[0055] In the t-th iteration, the Q-function maximized by the EM algorithm is: :

[0056] (14);

[0057] in, For the log-likelihood function with respect to Expectations;

[0058] Execute E-step:

[0059] Latent variable posterior estimation, in the t-th iteration, based on the current parameter estimates. Calculate latent variables The posterior expectation;

[0060] According to Bayes' theorem, the posterior distribution of latent variables still follows a gamma distribution. Expected value :

[0061] (15);

[0062] in, For the log-likelihood function with respect to Expectations; The square of the normalized noise;

[0063] Execute M-step:

[0064] Noise scale parameter update, causing latent variables Expected value As a weight, upon obtaining the latest weight... Then; maximize the Q function to update the noise scale parameter;

[0065] Let the Q function be about The partial derivative is zero, so the noise scale parameter update formula for the i-th observation node is: :

[0066] (16);

[0067] Step 4.4, EM algorithm convergence judgment:

[0068] Define the convergence metric for the t-th iteration of the EM algorithm. :

[0069] (17);

[0070] Among them, the convergence threshold The number of iterations is determined based on the accuracy requirements; when the requirements are met... Alternatively, if the preset maximum number of iterations is reached, the iteration stops, and the final noise scale estimate of the i-th observation node is output. .

[0071] Furthermore, in step 5, the iterative process of solving for the target position is as follows:

[0072] Noise scale parameter estimates based on each sensor A pseudo-linear maximum likelihood estimator based on the independent heterogeneous Student-t distribution is constructed to solve for the target location;

[0073] Step 5.1, Pseudolinear Residual Modeling:

[0074] The pseudolinear residual of the i-th observation node in the k-th frame AOA measurement noise Nonlinear coupling exists; under the assumption of low noise, a first-order Taylor expansion of the pseudolinear equation yields the pseudolinear residual. Model:

[0075] (18);

[0076] in, Let be the Euclidean distance from the target to the i-th observation node. Represents the l2 norm;

[0077] Therefore, the noise scaling parameter of the pseudolinear domain residual Represented as:

[0078] (19);

[0079] The pseudolinear residual of the i-th observation node in the k-th frame The PDF is :

[0080] (20);

[0081] Step 5.2, construct the objective function for solving the target location:

[0082] Since it is assumed that the observations between each frame and between each sensor are statistically independent, and For target location If the function is , then equation (20) is equivalent to: Therefore, the joint likelihood function of the entire observation dataset Represented as:

[0083] (twenty one);

[0084] Taking the natural logarithm of the joint likelihood function and ignoring the target location of the parameters to be estimated Irrelevant constant terms, maximizing the likelihood function; objective function :

[0085] (twenty two);

[0086] Step 5.3, rewrite the objective function in least squares form:

[0087] Step 5.3.1: Due to the nonlinear characteristics of formula (22), it is difficult to directly minimize formula (22); therefore, by calculating the gradient and setting formula (22) to zero, a comprehensive weight is introduced. :

[0088] (twenty three);

[0089] In the m-th iteration, the fixed target position is: The weight is defined as follows: :

[0090] (twenty four);

[0091] in, Substitution The subsequent weight value, This represents the value of the variable in the m-th iteration. For the generalized Mahalanobis distance; for the k-th frame, the weight matrix is... ;

[0092] Step 5.3.2, due to the weight matrix For target location The nonlinear function directly affects the target position. Joint optimization is extremely difficult; therefore, a two-layer iterative solution architecture is constructed:

[0093] In the m-th iteration, fix the current estimated point. The m-th iteration weight matrix Construct a weighted nonlinear least squares solution:

[0094] (25);

[0095] Define weighted residuals :

[0096] (26);

[0097] Equation (25) is equivalent to the least squares form. :

[0098] (27);

[0099] Step 5.4, Solve for the least squares objective function:

[0100] Step 5.4.1, at the current estimated point Nearby, introduce step size For pseudo-linear weighted residual vector Perform a first-order Taylor expansion:

[0101] (28);

[0102] in, For weighted residuals in Take the value at that location. For pseudolinear residuals with respect to The Jacobian matrix;

[0103] Rearranging equations (26) to (28), we get:

[0104] (29);

[0105] The sign for partial derivatives;

[0106] Step 5.4.2, gives the following... Quadratic approximation model in the sense of Gauss-Newton (GN) solutions in the neighborhood :

[0107] (30);

[0108] By taking the gradient of equation (30) and rearranging it, we obtain the approximate gradient vector corresponding to equation (30). With approximate Hessian matrix :

[0109] (31);

[0110] (32);

[0111] Step 5.4.3: Based on the Levenberg-Marquardt (LM) method, introduce regularization constraints into the GN quadratic approximation model and set the damping factor. The optimization objective is revised to:

[0112] (33);

[0113] Substituting (30) into (33), we can consider the objective function with respect to... To find the first-order optimality condition, we have:

[0114] (34);

[0115] Arrangement of equation (34) yields the LM regularized normal equation with benign characteristics:

[0116] (35);

[0117] in, It is the identity matrix;

[0118] Thus, a trial solution is given. :

[0119] (36);

[0120] Step 5.4.4, define the gain ratio Determine the convergence of the LM method:

[0121] (37);

[0122] based on Execute an adaptive damping strategy: when If the update is accepted, the damping is reduced to accelerate convergence; otherwise, the update is rejected and the damping is increased, and equation (35) is solved again until the acceptance condition is met.

[0123] Step 5.5: Calculate the relative step size and determine the convergence of the target position estimation.

[0124] (38);

[0125] When the relative step size is less than or equal to Or it may terminate when the maximum number of iterations is reached. The number of iterations is determined based on the required accuracy and the computing power of the device, and the final estimate is output. .

[0126] An electronic device, characterized in that it comprises: one or more processors; a memory; one or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs being configured to perform the above-described positioning method based on an underwater distributed system under non-Gaussian measurement noise.

[0127] A computer-readable storage medium, characterized in that the computer-readable storage medium stores program code, which can be called by a processor to execute the above-described positioning method based on an underwater distributed system under non-Gaussian measurement noise.

[0128] Compared with the prior art, the technical effects of the present invention are as follows:

[0129] Existing technologies assume that measurement noise follows a Gaussian distribution or use a fixed threshold for target localization. However, under heavy-tailed and heterogeneous non-Gaussian AOA measurement noise, the localization performance of traditional estimation methods based on the Gaussian assumption and robust localization methods with fixed thresholds degrades significantly. In contrast, the method of this invention is based on the SAGLS method of student-t distributed noise modeling. By processing multi-frame AOA measurement data, it adaptively learns the noise scale parameters of different nodes and incorporates the learned measurement noise scale parameters into the maximum likelihood solution framework to construct an iterative reweighted generalized least squares equation to solve for the target position. The method of this invention effectively suppresses weight mismatch in distributed localization systems under non-Gaussian heavy-tailed noise conditions such as time-varying, heteroscedastic, and shock noise, significantly reducing localization errors and improving robustness.

[0130] Currently, most AOA-based positioning methods rely on Gaussian noise assumptions or second-order statistical properties. In environments with strong impulsive noise or other non-Gaussian heavy-tailed noise, the positioning accuracy of these methods is significantly affected. Existing robust positioning research primarily focuses on weighted suppression of non-Gaussian noise such as impulses, lacking dynamic adaptability to time-varying and spatially heterogeneous noise. For underwater distributed positioning systems, the non-stationarity and spatial differences of the underwater environment lead to heterogeneous heavy-tailed noise, affecting the positioning accuracy of the aforementioned methods. This invention relates to a robust positioning method for underwater distributed systems under non-Gaussian measurement noise conditions. By processing multi-frame AOA measurement data from the distributed system, and modeling noise based on the t-distribution, an iterative reweighted generalized least squares method with adaptive noise scale estimation, called the SAGLS method, is proposed. By independently learning the noise scale parameters of each node, an adaptive weighting based on the noise scale parameters and geometric relationships is constructed, significantly reducing the impact of heterogeneous and heavy-tailed noise on the passive positioning of underwater targets, thus improving positioning accuracy and robustness.

[0131] This invention introduces the Student-t distribution as the AOA measurement noise model and proposes an iterative reweighted generalized least squares (SAGLS) localization method based on adaptive noise scale estimation. Addressing the heterogeneous, heavy-tailed noise conditions in AOA measurements faced by underwater distributed systems, this invention constructs an adaptive weighting system that combines noise scale parameters with geometric relationships by independently estimating the noise scale parameters of each node. This suppresses the interference of the heterogeneous and heavy-tailed underwater noise environment on passive localization, effectively improving the accuracy and robustness of the localization system. Attached Figure Description

[0132] Figure 1 This is an overall flowchart of the present invention;

[0133] Figure 2 This is the simulation experimental setup for the present invention;

[0134] Figure 3 This is the positioning RMSE curve of the present invention as a function of noise degree of freedom parameters;

[0135] Figure 4 This is the positioning RMSE curve of the present invention as a function of noise heterogeneity parameters;

[0136] Figure 5 This is the positioning RMSE curve as a function of noise pollution rate according to the present invention;

[0137] Figure 6 This is a diagram illustrating the experimental situation on the lake for this invention.

[0138] Figure 7 This is the time-frequency diagram of the transmitted signal of the present invention;

[0139] Figure 8 This is a schematic diagram of the experimental setup for the present invention;

[0140] Figure 9 is a time-frequency diagram of the received signal of the present invention;

[0141] Figure 9(a) shows observation node 1; Figure 9(b) shows observation node 2; Figure 9(c) shows observation node 3;

[0142] Figure 10 shows the BTR diagrams of each observation node in this invention;

[0143] Figure 10(a) shows observation node 1; Figure 10(b) shows observation node 2; Figure 10(c) shows observation node 3;

[0144] Figure 11 The direction finding error of each node in this invention;

[0145] Figure 12 shows the localization results of six methods: PLE, WLS, WIV, HPLE, IRIVE, and SAGLS.

[0146] Figure 12(a) shows the PLE localization result; Figure 12(b) shows the WLS localization result; Figure 12(c) shows the WIV localization result; Figure 12(d) shows the HPLE localization result; Figure 12(e) shows the IRIVE localization result; Figure 12(f) shows the SAGLS localization result;

[0147] Figure 13 shows the positioning error and statistical results of the present invention;

[0148] Figure 13(a) shows the positioning error results of the six methods; Figure 13(b) shows the statistical results of the positioning error of the six methods. Detailed Implementation

[0149] The embodiments of the present invention will now be described in detail with reference to the accompanying drawings.

[0150] A three-node underwater distributed system is used for target localization. Each node is equipped with a 12-element circular array. High-gain beamforming is used to process the received signals to estimate the target's azimuth and obtain the AOA measurement value, which is then used for target localization.

[0151] like Figure 2 As shown, the coordinates of each node in the underwater distributed system are as follows: , , They are arranged in a triangular formation. The actual positions of the targets are set to... .

[0152] Assume each node collects 60 frames of angle of arrival measurement data. Both the SAGLS method and the comparative method use a sliding window to process K=15 consecutive frames of data for a single estimation. In actual deployment, the noise degrees of freedom that dominate the measurement error distribution... Typically unknown. Here, the degree of freedom parameters input to the SAGLS algorithm are fixed at... Subsequently, different noise conditions were set, and Monte Carlo simulation experiments were conducted to compare the results with PLE. WLS WIV HPLE and IRIVE By comparing different methods, the robustness under heavy tail noise conditions is evaluated.

[0153] To evaluate the robustness of the algorithm under different degrees of heavy tails, the noise degrees of freedom are... The scale increments from 1 to 10. To reflect the noise differences between different sensors, the scale parameter is set to... The unit is . Figure 3 This demonstrates the effect of root mean square error (RMSE) on 1000 Monte Carlo experiments. The variation curves were compared with the multi-frame Cramer-Rhodes lower bound (CRLB).

[0154] SAGLS method in all All values ​​achieved the lowest RMSE and closely matched CRLB. When At lower (heavy-tailed noise) values, the performance of traditional methods (PLE, WLS, WIV) degrades significantly, while robust methods (HPLE, IRIVE, SAGLS) maintain stability by suppressing outliers. With... Increasing the noise level to approximate Gaussian noise improves the performance of all methods, but SAGLS maintains higher estimation efficiency compared to HPLE and IRIVE, demonstrating full degree of freedom from heavy tails to Gaussian. Excellent robustness within the range.

[0155] To evaluate the algorithm's robustness under different node noise scale differences, we introduce a heterogeneity parameter. The noise standard deviations of the three sensors are respectively set as , , And convert it into a noise scale parameter using equation (8). With the noise degrees of freedom fixed, let... The value varies in unit steps within the range [3, 10]. The root mean square error (RMSE) based on 1000 Monte Carlo experiments is as follows: Figure 4 As shown.

[0156] Figure 4 The results show that, with the heterogeneity of noise scale parameters Enhancement significantly degrades the performance of traditional methods (PLE, WLS, WIV). This is because these methods cannot effectively suppress low signal-to-noise ratio (SNR) AOA measurements from high-noise sensors, resulting in weight mismatch and increased positioning errors. In contrast, robust methods (HPLE, IRIVE, SAGLS) show improved RMSE with... The noise level increases and then decreases, consistent with the trend of CRLB. Among them, the SAGLS method achieved the best performance throughout the entire test range and always closely followed the CRLB curve. This is due to its noise scale parameter estimation, which can accurately characterize the noise characteristics of each sensor, thereby achieving precise fusion and maintaining high positioning accuracy even under conditions of large differences in measurement quality among nodes.

[0157] To evaluate the algorithm's robustness to heavy-tailed noise, the AOA measurement noise was modeled as a Gaussian mixture distribution. The standard deviations of the baseline measurement noise for the three nodes were set to 3°, 5°, and 1.5°, respectively. A contamination ratio was introduced. , indicating probability Measurement noise with 2x variance AOA is present. Contamination ratio. The percentage increased from 5% to 30% in 5% increments. The RMSE was calculated based on 1000 independent Monte Carlo experiments, and the results are as follows: Figure 5 As shown.

[0158] like Figure 5 As shown, when the pollution ratio When the contamination ratio increases from 5% to 30%, the RMSE of traditional methods (PLE, WLS, WIV) rises rapidly, demonstrating high sensitivity to outliers. The fundamental reason is that the quadratic loss function cannot suppress the impact of large residuals. In contrast, HPLE and IRIVE employ robust weighting and iterative reweighting mechanisms, respectively, effectively mitigating the impact of outliers, thus resulting in a more gradual performance degradation. The proposed SAGLS maintains the lowest RMSE across the entire contamination ratio range. Based on adaptive weights derived from t-distribution likelihood, it effectively suppresses anomalous AOA measurements as contamination increases, thereby achieving superior robustness and localization performance under Gaussian mixture outlier conditions.

[0159] like Figure 6 As shown, an underwater distributed system consisting of three nodes was used in a lake experiment at a water depth of 3m, a typical shallow water environment. A bending-tension transducer was used as the sound source to simulate the target signal. The sound source was deployed at a depth of 2m and emitted a single-frequency pulse signal. The transmission time-frequency diagram is shown below. Figure 7 As shown.

[0160] like Figure 8As shown, the three nodes of the underwater distributed system have an approximate triangular geometric configuration. Each observation node has a circular array with 12 elements and a central diameter of 242 mm. The deployment depth during the experiment was 2 m.

[0161] The time-frequency characteristics of the received signals at each node are shown in Figure 9. For the received data, high-gain beamforming is used for azimuth scanning to obtain the target azimuth measurement sequence, and a target time-azimuth history (BTR) diagram is given, as shown in Figure 10. Based on the relative geometric relationship between each receiving node and the target, the azimuth measurement error is calculated, such as... Figure 11 As shown in the figure. The results indicate that the orientation error sequence exhibits intermittent large deviations and abrupt changes, while the AOA noise at different nodes differs, which is reflected in the differences in noise scale at different nodes.

[0162] Based on the AOA measurement data of each node, the localization performance of PLE, WLS, WIV, HPLE, IRIVE, and SAGLS methods was compared. Due to limited experimental data, all methods used a sliding window of K=5 consecutive AOA measurements for target position estimation. Figure 12 shows the target position estimation results obtained by fusing AOA measurements from each localization method. In the figure, the red pentagram represents the true target position, the triangle represents the actual sensor position, and the blue, orange, and green correspond to sensors 1 to 3, respectively. The position estimation values ​​of each method are marked with solid dots. Figure 13 shows the localization error at different times.

[0163] Experimental results reveal significant performance differences among the methods, as shown in Figure 13. The PLE algorithm has an RMSE of 31.15m, reflecting its extreme sensitivity to heavy-tailed noise in the measured data. The WIV method effectively corrects for pseudo-linear bias using instrumental variables, achieving an RMSE of 19.60m. In contrast, WLS exhibits better accuracy with an RMSE of 16.63m. Notably, despite employing a robust Huber kernel function, HPLE's RMSE is 19.96m, slightly inferior to WLS and WIV. This performance degradation is attributed to the Huber operator's reliance on a globally uniform threshold, which fails to effectively distinguish local accuracy differences between different sensors, leading to Fisher information loss. The IRIVE algorithm, through iterative weight updates, outperforms WLS with an RMSE of 15.6m. SAGLS demonstrates the best performance, with an RMSE of only 12.06m and an average error of 9.52m. Compared to the suboptimal IRIVE method, SAGLS improves positioning accuracy by approximately 22.7%.

Claims

1. A localization method based on an underwater distributed system under non-Gaussian measurement noise, characterized in that, Includes the following steps: Step 1: Obtain the azimuth measurement of the underwater distributed positioning system observation node; Step 2: Based on the orientation measurement and the geometric relationship between the target and the observation node, construct the AOA positioning model; Step 3: Construct an AOA measurement noise model based on the Student-t distribution; Step 4: Based on multi-frame AOA measurement data, adaptively estimate the noise scale parameters of the observation nodes; Step 4.1, based on GSM theory, define the complete dataset for observation node i. ,in Let N be the set of AOA measurement noise, i∈N, N≥3. Let be the set of unobservable latent variables; let be the set of parameters to be estimated. ; Based on the assumption that the noise from each sensor is independent in the spatiotemporal domain, the joint probability density function of all K frames of complete data... Represented as the product of the PDF components: （12）； Step 4.2: Complete the log-likelihood function of all K-frame data, ignoring the AOA measurement noise scale parameter to be estimated. Irrelevant constant terms, objective function Simplified to: （13）； Step 4.3: Use the EM algorithm to alternately execute E-step and M-step to solve equation (13); In the t-th iteration, the Q-function maximized by the EM algorithm is: : （14）； in, For the log-likelihood function with respect to Expectations; Execute E-step: Latent variable posterior estimation, in the t-th iteration, based on the current parameter estimates. Calculate latent variables The posterior expectation; According to Bayes' theorem, the posterior distribution of latent variables still follows a gamma distribution. Expected value : （15）； in, For the log-likelihood function with respect to Expectations; The square of the normalized noise; Execute M-step: Noise scale parameter update, causing latent variables Expected value As a weight, upon obtaining the latest weight... Then; maximize the Q function to update the noise scale parameter; Let the Q function be about The partial derivative is zero, so the noise scale parameter update formula for the i-th observation node is: : （16）； Step 4.4, EM algorithm convergence judgment: Define the convergence metric for the t-th iteration of the EM algorithm. : （17）； Among them, the convergence threshold The number of iterations is determined based on the accuracy requirements; when the requirements are met... Alternatively, if the preset maximum number of iterations is reached, the iteration stops, and the final noise scale estimate of the i-th observation node is output. ; Step 5: Based on the noise scale parameter, construct an iterative reweighted generalized least squares solution and iteratively solve for the target position.

2. The positioning method based on an underwater distributed system under non-Gaussian measurement noise according to claim 1, characterized in that, In step 1, the method for obtaining the azimuth measurement is as follows: Step 1.1: The observation nodes of the underwater distributed system use a uniform circular array as acoustic sensors to receive acoustic signals from underwater targets. Step 1.2: Use high-gain beamforming to perform azimuth scanning on the acoustic signals received by the observation node to obtain the azimuth spectrum; Step 1.3: For the azimuth spectrum obtained by azimuth scanning, take the main lobe peak as the azimuth measurement to obtain the AOA measurement sequence of the target at different times.

3. The positioning method based on an underwater distributed system under non-Gaussian measurement noise according to claim 1, characterized in that, In step 2, the process of constructing the AOA localization model is as follows: Step 2.1: A distributed observation system exists within the two-dimensional plane, consisting of N spatially distributed acoustic passive observation nodes; at discrete times k, k=1,…,K, where K is the total observation duration, the i-th observation node is located at known coordinates. x i and y i The x-axis and y-axis coordinates of the i-th observation node are respectively, and the target position is... x and y are the x-axis and y-axis coordinates of the target, respectively, and the azimuth measurement of the i-th observation node. satisfy: （1）； in, For the actual location, For AOA noise measurement, Indicates matrix transpose; Step 2.2: Based on the geometric relationship between the underwater distributed system observation nodes and the target, determine the true orientation. Linearization is represented as: （2）； In practice, azimuth measurement Includes AOA measurement noise This leads to an algebraic difference in equation (2); the algebraic difference is defined as a pseudo-linear residual function. : （3）； Step 2.3: Solve the equations of the observation nodes simultaneously to construct the matrix form AOA localization model for the k-th frame: （4）； in, The pseudo-linear residual vector of the k-th frame; system matrix. and observation vector They are respectively: （5）； （6）； in, For the target location; It is the set of real numbers.

4. The positioning method based on an underwater distributed system under non-Gaussian measurement noise according to claim 1, characterized in that, In step 3, the process of constructing the AOA measurement noise model is as follows: Step 3.1: The Student-t distribution is used as the AOA measurement noise model, which is also the AOA error model. The Student-t distribution is determined by the degree of freedom parameter. Controlling the tail thickness and using noise scale parameters Characterizing the degree of discreteness; assuming that the AOA measurement noise at different times is independent, the noise vector of the k-th frame... The components in the data are independent of each other; the AOA measurement noise of the i-th observation node. It follows a zero-mean Student-t distribution, denoted as AOA measurement noise The PDF is : （7）； in, For gamma function, The noise scale parameter for the AOA measurement of the i-th observation node reflects the degree of noise dispersion. Relationship with variance of Gaussian distribution: （8）； Where Var(·) represents variance; Step 3.2, due to the independence assumption, the noise vector of the k-th frame... The joint probability density function is the product of the PDFs of each component: （9）； Step 3.3, the Student-t distribution is equivalently represented as a GSM containing latent variables; AOA measurement noise Introducing latent variables The Student-t distribution is decomposed into a hierarchical Gaussian-gamma structure: Conditional likelihood distribution: given and Follows a Gaussian distribution with scaled covariance : （10）； Prior distribution of latent variables: It follows a gamma distribution, and its probability density function is... for: （11）。 5. The positioning method based on an underwater distributed system under non-Gaussian measurement noise according to claim 1, characterized in that, In step 5, the iterative process of solving for the target position is as follows: Step 5.1, Pseudolinear Residual Modeling: The pseudolinear residual of the i-th observation node in the k-th frame AOA measurement noise There is nonlinear coupling; Under the assumption of low noise, a first-order Taylor expansion of the pseudolinear equation yields the pseudolinear residual. Model: （18）； in, Let be the Euclidean distance from the target to the i-th observation node. Represents the l2 norm; Therefore, the noise scaling parameter of the pseudolinear domain residual Represented as: （19）； The pseudolinear residual of the i-th observation node in the k-th frame The PDF is : （20）； Step 5.2, construct the objective function for solving the target location: Since it is assumed that the observations between each frame and between each sensor are statistically independent, and For target location If the function is , then equation (20) is equivalent to: Therefore, the joint likelihood function of the entire observation dataset Represented as: （21）； Taking the natural logarithm of the joint likelihood function and ignoring the target location of the parameters to be estimated Irrelevant constant terms, maximizing the likelihood function; objective function : （22）； Step 5.3, rewrite the objective function in least squares form: Step 5.3.1: By calculating the gradient and setting formula (22) to zero, a comprehensive weight is introduced. : （23）； In the m-th iteration, the fixed target position is: The weight is defined as follows: : （24）； in, Substitution The subsequent weight value, This represents the value of the variable in the m-th iteration. For the generalized Mahalanobis distance; for the k-th frame, the weight matrix is... ; Step 5.3.2, construct a two-layer iterative solution architecture: In the m-th iteration, fix the current estimated point. The m-th iteration weight matrix Construct a weighted nonlinear least squares solution: （25）； Define weighted residuals : （26）； Equation (25) is equivalent to the least squares form. : （27）； Step 5.4, Solve for the least squares objective function: Step 5.4.1, at the current estimated point Nearby, introduce step size For pseudo-linear weighted residual vector Perform a first-order Taylor expansion: （28）； in, For weighted residuals in Take the value at that location. For pseudolinear residuals with respect to The Jacobian matrix; Rearranging equations (26) to (28), we get: （29）； The sign for partial derivatives; Step 5.4.2, gives the following... Quadratic approximation model of Gaussian-Newton solution in the neighborhood under the GN sense : （30）； By taking the gradient of equation (30) and rearranging it, we obtain the approximate gradient vector corresponding to equation (30). With approximate Hessian matrix : （31）； （32）； Step 5.4.3: Based on the LM method, introduce regularization constraints into the GN quadratic approximation model and set the damping factor. The optimization objective is revised to: （33）； Substituting (30) into (33), we can consider the objective function with respect to... To find the first-order optimality condition, we have: （34）； Arrangement of equation (34) yields the LM regularized normal equation with benign characteristics: （35）； in, It is the identity matrix; Thus, a trial solution is given. : （36）； Step 5.4.4, define the gain ratio Determine the convergence of the LM method: （37）； based on Execute an adaptive damping strategy: when If the update is accepted, the damping is reduced to accelerate convergence; otherwise, the update is rejected and the damping is increased, and equation (35) is solved again until the acceptance condition is met. Step 5.5: Calculate the relative step size and determine the convergence of the target position estimation. （38）； When the relative step size is less than or equal to Or it may terminate when the maximum number of iterations is reached. The number of iterations is determined based on the required accuracy and the computing power of the device, and the final estimate is output. .

6. An electronic device, characterized in that, include: One or more processors; Memory; One or more programs, wherein the one or more programs are stored in the memory and configured to be executed by the one or more processors, the one or more programs being configured to perform the positioning method as described in any one of claims 1 to 5.

7. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores program code that can be invoked by a processor to execute the positioning method as described in any one of claims 1 to 5.

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