An underwater target tracking method based on multivariate skew Laplace distribution

Abstract

The application discloses a kind of underwater target tracking methods based on multivariate skew laplace distribution modeling, the motion state data of underwater target is collected using the mode of sonar sensor, and the target position measurement information is obtained by the mode of coordinate system conversion;Underwater target motion model is established, the state space equation of underwater target is determined, the mathematical characteristics of underwater noise are analyzed, and the measurement model of target under non-gaussian noise is established;Based on multivariate skew laplace distribution, the non-gaussian noise is modeled, the mixed parameters, shape parameters and scale matrix of noise are solved under the variational bayesian framework, the target state and noise covariance matrix are iteratively updated, and the target motion state is iteratively updated using Kalman filtering estimation, after the iteration number, the estimated value of underwater target position and speed and the estimated value of covariance matrix are output.The application has better robustness and estimation accuracy, and does not need to select the degree of freedom parameter, and can be better applied to the tracking of target.

Description

Technical Field

[0001] This invention relates to target tracking, and more specifically to an underwater target tracking method based on a multivariate skewed Laplace distribution. Background Technology

[0002] Underwater target tracking refers to the continuous estimation and prediction of a target's state (velocity, position, etc.) over time using filtering algorithms and different measurement information (distance, bearing, etc.) obtained from one or more sensors of different types. By rapidly acquiring the position, velocity, acceleration, and other state information of enemy targets, it is possible to effectively defend against and engage invading enemy ships, torpedoes, submarines, and underwater robots.

[0003] During underwater measurements, sensors are susceptible to non-Gaussian noise, causing measurements to deviate from a Gaussian distribution and instead exhibit a skewed, heavy-tailed distribution. This type of noise distribution has large extreme values, negatively impacting tracking performance.

[0004] In practical engineering applications, when facing non-Gaussian noise problems, researchers have proposed Student's t Filter (STF), which utilizes the Student's t distribution to model the heavy-tailed characteristics of noise. However, as the number of degrees of freedom increases, the heavy-tailed characteristics may weaken, affecting the estimation accuracy. To overcome this problem, a Variational Bayesian Student's t Kalman Filter (VB-STKF) is used to jointly estimate the state and degree-of-freedom parameters using a variational Bayesian method. However, for complex heavy-tailed noise and heavy-tailed skewed noise, the Student's t distribution cannot accurately describe the noise model. Therefore, the Gaussian Scale Mixture (GScM) distribution has been proposed, which can achieve better noise modeling accuracy than the Student's t distribution through adaptive learning of distribution parameters. Although it can flexibly model the noise distribution, the filtering implementation time may be long due to the large number of iterations. The design and selection of these filters should be comprehensively considered based on system characteristics, noise characteristics, and real-time requirements to achieve the optimal state estimation effect. Currently, target tracking algorithms under non-Gaussian noise generally adopt a modeling method based on the Student's t-distribution. This method requires the selection of degree-of-freedom parameters and involves a large number of variables, which is not conducive to practical applications. Furthermore, existing algorithms have large errors when dealing with non-Gaussian skewed heavy-tailed noise. Summary of the Invention

[0005] The purpose of this invention is to provide an underwater target tracking method based on a multivariate skewed Laplace distribution.

[0006] The technical solution to achieve the objective of this invention is: an underwater target tracking method based on multivariate skew Laplace distribution modeling, comprising the following steps:

[0007] Step 1: Use a sonar sensor to collect motion data of the underwater target, including radial distance and azimuth. Then, through coordinate system transformation, convert the radial distance and azimuth of the underwater target collected by the sensor from the spherical / polar coordinate system to the Cartesian coordinate system to obtain the target position measurement information.

[0008] Step 2: Establish a motion model of the underwater target, determine the state-space equation of the underwater target, analyze the mathematical characteristics of underwater noise, and establish a measurement model of the target under non-Gaussian noise.

[0009] Step 3: Model non-Gaussian noise based on multivariate skewed Laplace distribution. Under the variational Bayesian framework, solve for the noise mixing parameters, shape parameters, and scale matrix. Iteratively update the target state and noise covariance matrix, and use Kalman filtering to iteratively update the estimated target motion state. After reaching the required number of iterations, output the estimated values ​​of the underwater target position and velocity, as well as the estimated value of the covariance matrix.

[0010] Further, in step 1: Sonar sensors are used to collect motion data of the underwater target, including radial distance and azimuth. Then, through coordinate system transformation, the radial distance and azimuth data collected by the sensors are converted from polar coordinates to Cartesian coordinates to obtain target position measurement information, where:

[0011] Assuming the position of point P in the Cartesian coordinate system is denoted as (x, y), and its position in the polar coordinate system is denoted as (r, θ), the transformation relationship from polar coordinates to rectangular coordinates is as follows:

[0012]

[0013] Further, step 2: Establish a motion model of the underwater target, determine the state-space equation of the underwater target, analyze the mathematical characteristics of underwater noise, and establish a measurement model of the target under non-Gaussian noise, wherein:

[0014] Establish a motion model of the underwater target and determine its state-space equations;

[0015]

[0016] in, The position and velocity of the target at time k are respectively, w k ~N(0,Q) k ), is a process noise sequence at time k that follows a Gaussian distribution. The process noise is zero-mean Gaussian noise with covariance, expressed as:

[0017]

[0018] The K-distribution is used to simulate non-Gaussian noise underwater. The probability density function of the K-distribution is as follows:

[0019]

[0020] Where v is the shape coefficient, a is the scaling coefficient, and K v-1 (x) is a K-type Bessel function of order v-1, and Γ(·) is a Gamma function;

[0021] The measurement model for the target under non-Gaussian noise is as follows:

[0022]

[0023] Among them, z k Measurements of the target at time k, including range and azimuth; v k The non-Gaussian measurement noise is distributed in K.

[0024] Further, step 3: Based on the multivariate skewed Laplace distribution, model the non-Gaussian noise. Within the variational Bayesian framework, solve for the noise mixing parameters, shape parameters, and scale matrix. Iteratively update the target state and the noise covariance matrix, and use Kalman filtering to iteratively update the estimated target motion state. After reaching the required number of iterations, output the estimated values ​​of the underwater target's position and velocity, as well as the estimated value of the covariance matrix, where:

[0025] Measurement noise has a thick-tailed or thick-tailed skewed distribution. Non-Gaussian noise is modeled based on a multivariate skewed Laplace distribution.

[0026]

[0027] Among them, v k For non-Gaussian distributed measurement noise, N(v) k ;λ k β,λ k R k ) is a function with a mean of λ. k β, covariance λ k R k Gaussian probability density function, Ex(λ) k ;a0) represents the exponential probability density function with parameter a0, λ k ,β,R k a0 and a0 represent the mixing parameter, shape parameter, scale matrix, and rate parameter of the measurement noise, respectively;

[0028] According to the Chapman-Komlogorov equation, the one-step prediction probability density function p(x)k |z 1:k ) is represented as:

[0029]

[0030] in, and P k|k-1 Let z represent the one-step prediction of the state at time k and the corresponding prediction error covariance matrix, respectively. 1:k-1 Measurements from time 1 to time k-1;

[0031] Likelihood PDF P(z) k |x k ) represents

[0032] P(z k |x k )=∫N(z k H k x k +λ k β,λ k R k )Ex(λ k ;a0)dλ k

[0033] The inverse Wishart distribution is used as the conjugate prior distribution of the scaling matrix;

[0034] P(R k ) = IW(R k ;v k V k )

[0035] Among them, IW(R) k ;v k V k ) is the degree of freedom parameter v k The inverse scaling matrix is ​​V k The inverse Wishart probability density function;

[0036] Nominal measurement noise covariance matrix for:

[0037]

[0038] Where m is the dimension of the measurement vector;

[0039] The prior distribution of the shape parameters is chosen to be a Gaussian distribution, i.e.

[0040] p(β)=N(β;β,δI m )

[0041] Where β represents the nominal shape parameter of the measurement noise, σ represents the confidence level of the nominal shape parameter, and I m It is the identity matrix;

[0042] The variational Bayesian method is used to jointly infer the mixing parameters, shape parameters, and scale matrix, i.e.

[0043] Use variational Bayesian methods to obtain p(Θ|z) 1:k An approximate posterior probability density function with a free decomposition form, namely

[0044] p(x k ,β,R k ,λ k |z k )≈q(x i )q(λ k )q(β)q(R k )

[0045] Where q(·) represents an approximation of the true posterior probability density function with a free decomposition form, and the optimal solution satisfies the following equation

[0046]

[0047] Where φ is any element in Θ. For the remaining elements, c φ For constant terms;

[0048] Conditional independence in constructing a hierarchical Gaussian state-space model is then established, and the joint probability density function p(Θ,z) is obtained. 1:k ) is decomposed into

[0049] p(Θ,z 1:k )=N(x k ;F k x k-1|k-1 ,P k|k-1 )N(z k Hx k +λ k β,λ k R k )×IW(R k ;v k V k )Ex(λ k ;a0)N(β;β,σI m )p(z 1:k-1 )

[0050] Among them, F k Let P represent the known state transition matrix. k|k-1 Let H represent the prediction error covariance matrix at time k, and H be the measurement matrix.

[0051] Taking the logarithm of both sides, we get

[0052]

[0053] in, The nominal shape parameter represents the measurement noise, and σ is the confidence level used to characterize the nominal parameter.

[0054] (1) Solve for the state variable x k posterior distribution

[0055] Let Θ = x k Substitute into the following formula

[0056]

[0057] The logarithmic expression for the posterior distribution of the state is:

[0058]

[0059] Where, q (i+1) (·) represents an approximation of the true posterior probability density function in the (i+1)th iteration with a free decomposition form. c represents the inverse of the prediction error covariance matrix at time k. x For constant terms;

[0060] Define the corrected measurement noise mean vector covariance matrix for

[0061]

[0062]

[0063] Using Bayes' criterion q (i+1) (x k Update to a Gaussian probability density function, i.e.

[0064]

[0065] The state estimate and its covariance are as follows:

[0066]

[0067] in This is the state estimate at time k in the (i+1)th iteration. Let represent the error covariance matrix at time k in the (i+1)th iteration. Let I be the Kalman gain at time k in the (i+1)th iteration. n It is the identity matrix. This is the corrected mean vector of measurement noise. The corrected measurement noise covariance matrix;

[0068] (2) Update the posterior distribution of the mixture parameters

[0069] Let Θ = λ k Substitute into the following formula

[0070]

[0071] The logarithmic expression for the posterior distribution of the state is:

[0072]

[0073] in for

[0074] By matching the parameters, it was found that its posterior distribution is still a generalized inverse Gaussian distribution, i.e.

[0075]

[0076] Define relevant parameters They are respectively:

[0077]

[0078] According to the properties of the generalized inverse Gaussian distribution, E i+1 (λ k )for

[0079]

[0080] (3) Update the posterior distribution of shape parameters

[0081] Let Θ = β, logq (i+1) (β) can be written in the following form:

[0082]

[0083] Where c β For constant terms,

[0084] in q (i+1) (β) is updated to a Gaussian distribution.

[0085]

[0086] The mean vector and covariance matrix are as follows:

[0087]

[0088] (4) Update the posterior distribution of the scaling matrix

[0089] Let Θ = R k ,logq (i+1) (R k Write it in the following form:

[0090]

[0091] in For constant terms,

[0092]

[0093] Using the properties of the IW distribution, we have

[0094]

[0095] At this point, the implicit variable iterative solution process based on VB has been completed. After multiple iterations, the estimated value of the target state can be obtained.

[0096] An underwater target tracking system based on multivariable skewed Laplace distribution modeling is characterized by implementing the underwater target tracking method based on multivariable skewed Laplace distribution modeling described herein, realizing underwater target tracking based on multivariable skewed Laplace distribution modeling, and consisting of three modules, each executing steps 1 to 3.

[0097] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements the underwater target tracking method based on multivariate skew Laplace distribution modeling, thereby achieving underwater target tracking based on multivariate skew Laplace distribution modeling.

[0098] A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, the underwater target tracking method based on multivariable skew Laplace distribution modeling is implemented, thereby realizing underwater target tracking based on multivariable skew Laplace distribution modeling.

[0099] Compared with existing technologies, the significant advantages of this invention are: 1) By modeling non-Gaussian skew heavy-tail noise, it can well meet the needs of underwater target tracking, making the algorithm possible in future sonar applications. 2) It does not require pre-setting degree-of-freedom parameters and has better tracking performance under non-Gaussian skew noise conditions. Attached Figure Description

[0100] Figure 1This is a flowchart of the underwater target tracking method based on multivariate skewed Laplace distribution according to the present invention.

[0101] Figure 2 This is a graph of the K-distribution noise probability density function.

[0102] Figure 3 These are the root mean square error (RMSE) values ​​of the positions of KF, UCMKF, ST-EKF, GHSST-EKF, and the novel filtering method proposed in this invention.

[0103] Figure 4 These are the root mean square error (RMSE) values ​​of KF, UCMKF, ST-EKF, GHSST-EKF, and the new filtering method proposed in this invention.

[0104] Figure 5 It represents the position ARMSE at different iteration numbers.

[0105] Figure 6 This refers to the ARMSE speed at different iteration counts. Detailed Implementation

[0106] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0107] Combination Figure 1 This invention relates to an underwater target tracking method based on multivariate skewed Laplace distribution modeling, comprising the following steps:

[0108] Step 1: Use a sonar sensor to collect motion data of the underwater target, including radial distance and azimuth. Then, through coordinate system transformation, convert the radial distance and azimuth of the underwater target collected by the sensor from the spherical / polar coordinate system to the Cartesian coordinate system to obtain the target position measurement information.

[0109] Sonar data processing is generally performed in a Cartesian coordinate system, but sonar measurements are often obtained in a polar coordinate system. The difference between polar and Cartesian coordinate systems lies in the different definitions of the coordinates of a point in each system. In a polar coordinate system, the position of a point is represented by radial distance *r* and azimuth angle *θ*. Since the coordinate axes and planes are defined the same in both Cartesian and polar coordinate systems, trigonometric functions can be used to convert data from polar coordinates to Cartesian coordinates.

[0110] Assuming the position of point P in the rectangular coordinate system is denoted as (x, y), and its position in the polar coordinate system is denoted as (r, θ), the transformation relationship from polar coordinates to rectangular coordinates is as follows:

[0111]

[0112] Step 2: Establish a motion model of the underwater target, construct the state-space equation of the underwater target, analyze the mathematical characteristics of underwater noise, and establish a measurement model of the target under non-Gaussian noise.

[0113] In underwater target tracking, establishing an accurate target motion model is crucial for ensuring system performance. To achieve high stealth capabilities, underwater targets typically employ a uniform motion strategy, and its state-space model is shown below:

[0114]

[0115] in, w represents the target's position and velocity, respectively. k ~N(0,Q) k The process noise sequence follows a Gaussian distribution; the process noise is zero-mean Gaussian noise with covariance.

[0116]

[0117] Underwater non-Gaussian noise is typically in the form of a K-distribution. Therefore, the K-distribution is used to simulate underwater skew noise, as it effectively describes the non-uniformity of the echo. The probability density function of the K-distribution is as follows:

[0118]

[0119] Where: v is the shape factor, a is the scaling factor, and K is the weight of the shape factor. v-1 (x) is a v-1 order K-type Bessel function, and Γ(·) is the Gamma function. For most noise, the shape coefficient v ranges from (0.1, ∞). As v approaches ∞, the noise distribution approximates a Rayleigh distribution. For high-resolution, low-friction noise, the value is between (0.1 and 3).

[0120] Therefore, the observation equation is:

[0121]

[0122] Where z k Measurements at time k include the target's distance and azimuth; v k The non-Gaussian measurement noise is distributed in K.

[0123] Step 3: Model non-Gaussian noise based on a multivariate skewed Laplace distribution, and solve for the noise mixing parameter λ within a variational Bayesian framework. k Shape parameter β and scale matrix R kThe target state and noise covariance matrix are iteratively updated, and the estimated target motion state is estimated by using Kalman filtering.

[0124] As can be seen from the observation equation in step 2, the measurement noise exhibits non-Gaussian heavy-tailed skewness. Traditional target tracking algorithms based on Kalman filtering cannot handle non-Gaussian noise. Therefore, a multivariate skewed Laplace distribution is used to approximate this measurement noise distribution. Then, within the variational Bayesian framework, the target state and the noise covariance matrix are iteratively updated. At this point, the estimated target state approximately follows a Gaussian distribution, and the Kalman filter is then used to iteratively update the estimated target motion state.

[0125] Measurement noise has a thick-tailed or thick-tailed skewed distribution, which can be modeled as the following MSL distribution:

[0126]

[0127] Where v k For non-Gaussian distributed measurement noise, N(v) k ;λ k β,λ k R k ) is a function with a mean of λ. k β, covariance λ k R k Gaussian probability density function, Ex(λ) k ;a0) represents the exponential probability density function with parameter a0, λ k ,β,R k a0 and a0 represent the mixing parameter, shape parameter, scale matrix, and rate parameter of the measurement noise, respectively.

[0128] According to the Chapman-Komlogorov equation, the one-step prediction probability density function p(x) k |z 1:k This can be represented as:

[0129]

[0130] in, and P k|k-1 Let z represent the one-step prediction of the state at time k and the corresponding prediction error covariance matrix, respectively. 1:k-1 Measurements from time 1 to time k-1.

[0131] Likelihood PDFP(z) k |x k ) can be represented as

[0132] P(z k |x k )=∫N(z k Hk x k +λ k β,λ k R k )Ex(λ k ;a0)dλ k

[0133] In Bayesian inference, it is necessary to first choose a conjugate prior distribution for the unknown parameters, because conjugation ensures that the posterior distribution has the same functional form as the prior distribution. In Bayesian statistics, the inverse Wishart distribution and the covariance matrix used as a Gaussian distribution are conjugate prior distributions; since the scale matrix is ​​proportional to the covariance matrix in the hierarchical Gaussian form shown in MSL, the inverse Wishart distribution can also be used as the conjugate prior distribution of the scale matrix.

[0134] P(R k ) = IW(R k ;v k V k )

[0135] Among them, IW(R) k ;v k V k ) is the degree of freedom parameter v k The inverse scaling matrix is ​​V k The inverse Wishart probability density function.

[0136] In order to capture prior information about the scaling matrix, R k The mean is set to Right now

[0137]

[0138] Where m is the dimension of the measurement vector, v k The degree of freedom parameter is V. k It is the inverse scaling matrix.

[0139] Assuming the shape parameter β is inaccurate, a standard variational Bayesian method will be used to jointly infer the mixture parameter, shape parameter, scale matrix, and rate parameter. The prior distribution of the shape parameter is chosen to be a Gaussian distribution, i.e.

[0140] p(β) = N(β; β, σI) m )

[0141] Where β represents the nominal shape parameter of the measurement noise, σ represents the confidence level of the nominal shape parameter, and I m It is an identity matrix.

[0142] Based on the above formula, the variational Bayesian method is used to jointly infer the mixing parameters, shape parameters, and scale matrix, i.e.

[0143] To calculate the joint posterior probability density function p(Θ|z) 1:k The constructed hierarchical Gaussian state-space model cannot directly obtain an analytical solution in a closed form of the posterior probability density function. Therefore, the variational Bayesian method is used to obtain p(Θ|z). 1:k An approximate posterior probability density function with a free decomposition form, namely

[0144] p(x k ,β,R k ,λ k |z k )≈q(x i )q(λ k )q(β)q(R k )

[0145] Where q(·) represents an approximation of the true posterior probability density function with a free decomposition form.

[0146] Its optimal solution satisfies the following equation

[0147]

[0148] Where φ is any element in Θ. For the remaining elements, c φ For constant terms

[0149] Conditional independence for constructing a hierarchical Gaussian state-space model, and the joint probability density function p(Θ,z). 1:k ) can be decomposed into

[0150] p(Θ,z 1:k )=N(x k ;F k x k-1|k-1 ,P k|k-1 )N(z k Hx k +λ k β,λ k R k )×IW(R k ;v k V k )Ex(λ k ;a0)N(β;β,σI m )p(z 1:k-1 )

[0151] Among them, F k Let P represent the known state transition matrix. k|k-1 Let H represent the prediction error covariance matrix at time k, and H be the measurement matrix.

[0152] Taking the logarithm of both sides, we get

[0153]

[0154] in, The nominal shape parameter represents the measurement noise, and σ is the confidence level used to characterize the nominal parameter.

[0155] (1) Solve for the state variable x k posterior distribution

[0156] Let Θ = x k Substitute into the following formula

[0157]

[0158] The logarithmic expression for the posterior distribution of the state can be obtained as follows:

[0159]

[0160] Where, q (i+1) (·) represents an approximation of the true posterior probability density function in the (i+1)th iteration with a free decomposition form. c represents the inverse of the prediction error covariance matrix at time k. x This is a constant term.

[0161] Define the corrected measurement noise mean vector covariance matrix for

[0162]

[0163] Using Bayes' criterion q (i+1) (x k It can be updated to a Gaussian probability density function, i.e.

[0164]

[0165] The state estimate and its covariance are as follows:

[0166]

[0167] in This is the state estimate at time k in the (i+1)th iteration. Let represent the error covariance matrix at time k in the (i+1)th iteration. Let I be the Kalman gain at time k in the (i+1)th iteration. n It is the identity matrix. This is the corrected mean vector of measurement noise. This is the corrected measurement noise covariance matrix.

[0168] (2) Update the posterior distribution of the mixture parameters

[0169] Let Θ = λ k Substitute into the following formula

[0170]

[0171] The logarithmic expression for the posterior distribution of the state can be obtained as follows:

[0172]

[0173] in for

[0174] By matching the parameters, it can be found that its posterior distribution is still a generalized inverse Gaussian distribution, that is...

[0175]

[0176] Define relevant parameters They are respectively:

[0177]

[0178] According to the properties of the generalized inverse Gaussian distribution, E i+1 (λ k )for

[0179]

[0180] (3) Update the posterior distribution of shape parameters

[0181] Let Θ = β, logq (i+1) (β) can be written in the following form:

[0182]

[0183] Where c β For constant terms,

[0184] It can be seen Therefore q (i+1) (β) can be updated to a Gaussian distribution.

[0185]

[0186] The mean vector and covariance matrix are as follows:

[0187]

[0188] (4) Update the posterior distribution of the scaling matrix

[0189] Let Θ = Rk ,logq (i+1) (R k It can be written in the following form:

[0190]

[0191] in For constant terms,

[0192]

[0193] Using the properties of the IW distribution, we have

[0194]

[0195] Step 4: After reaching the required number of iterations, output the estimated values ​​of the underwater target's position and velocity, as well as the estimated value of the covariance matrix.

[0196] Step 3 yields the target state value iterated within the variational Bayesian framework, outputting the underwater target state estimate after N iterations. Estimated values ​​of the sum and covariance matrix

[0197] Example

[0198] The effects of this invention can be further illustrated by the following simulation experiments:

[0199] 1. Simulation conditions

[0200] Constructing the target motion model: Assume the target moves at a constant velocity in two-dimensional space. The motion model is modeled as a constant velocity (CV) model. The measurement equations determine the target's position; therefore, the state matrix xk and the state-space model can be represented as follows:

[0201]

[0202] in, z represents the state vector of the target at time k; k Let w be the measurement vector at time k; k ~N(0,Q) k ), is a process noise sequence that follows a Gaussian distribution, where the process noise is zero-mean Gaussian noise with covariance.

[0203] In this simulation, the filtering method proposed in this invention is compared with KF, UCMKF, ST-EKF, and GHSST-EKF methods. The degree of freedom parameter of STF is set to v = 5, and the initial state matrix and covariance matrix of the filtering method are...

[0204] x 0|0=[10000m,10000m,0m / s,-4m / s]

[0205] P 0|0 =[1000m 2 1000m 2 1000m 2 / s 2 1000m 2 / s 2 ]

[0206] 2. Simulation Content and Result Analysis

[0207] The target's actual trajectory and estimated trajectory are as follows: Figure 2 As shown. To compare the performance of these filters, the root mean square error (RMSE) and the average root mean square error (ARMSE) are defined as follows:

[0208]

[0209] and M = 1000 is the estimated and true locations of the s-th Monte Carlo experiment, and U = 100 is the number of Monte Carlo experiments.

[0210] Figure 3 and Figure 4 The root mean square error of position and root mean square error of velocity are shown for KF, UCMKF, ST-EKF, GHSST-EKF, and the new filter proposed in this invention. It can be seen that the new filtering method proposed in this invention has better estimation accuracy than other filters.

[0211] In summary, for cases where the measurement noise is non-Gaussian, a method for underwater target tracking based on multivariate skewed Laplace distribution modeling under the variational Bayesian (VB) framework for non-Gaussian measurement noise is proposed. Simulation results show that the filtering method has good estimation accuracy and does not require pre-setting of degree-of-freedom parameters, making it applicable to future underwater target tracking.

[0212] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0213] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these modifications and improvements all fall within the protection scope of this application. Therefore, the protection scope of this application should be determined by the appended claims.

Claims

1. An underwater target tracking method based on multivariate skewed Laplace distribution modeling, characterized in that, Includes the following steps: Step 1: Use a sonar sensor to collect motion data of the underwater target, including radial distance and azimuth. Then, through coordinate system transformation, convert the radial distance and azimuth of the underwater target collected by the sensor from the spherical / polar coordinate system to the Cartesian coordinate system to obtain the target position measurement information. Step 2: Establish a motion model of the underwater target, determine the state-space equation of the underwater target, analyze the mathematical characteristics of underwater noise, and establish a measurement model of the target under non-Gaussian noise. Step 3: Model non-Gaussian noise based on multivariate skewed Laplace distribution. Under the variational Bayesian framework, solve for the noise mixing parameters, shape parameters, and scale matrix. Iteratively update the target state and noise covariance matrix, and use Kalman filtering to iteratively update the estimated target motion state. After reaching the required number of iterations, output the estimated values ​​of the underwater target position and velocity, as well as the estimated value of the covariance matrix.

2. The underwater target tracking method based on multivariate skew Laplace distribution modeling according to claim 1, characterized in that, Step 1: Acquire underwater target motion data using sonar sensors, including radial distance and azimuth. Then, transform the acquired radial distance and azimuth data from polar coordinates to Cartesian coordinates to obtain target position measurement information. Assumption point The position of the Cartesian coordinate system is denoted as The position in the polar coordinate system is denoted as The transformation relationship from polar coordinates to rectangular coordinates is as follows: 。 3. The underwater target tracking method based on multivariate skew Laplace distribution modeling according to claim 2, characterized in that, Step 2: Establish a motion model of the underwater target, determine the state-space equation of the underwater target, analyze the mathematical characteristics of underwater noise, and establish a measurement model of the target under non-Gaussian noise, wherein: Establish a motion model of the underwater target and determine its state-space equations; ； in, Let K represent the target's position and velocity at time k, respectively. Let be a process noise sequence at time k that follows a Gaussian distribution. The process noise is zero-mean Gaussian noise with covariance, expressed as: ； The K-distribution is used to simulate non-Gaussian noise underwater. The probability density function of the K-distribution is as follows: ； in, Here, 'a' is the shape factor, and 'a' is the scaling factor. for Step kind function, for function; The measurement model for the target under non-Gaussian noise is as follows: ； in, Measurements of the target at time k, including distance and azimuth; The non-Gaussian measurement noise is distributed in K.

4. The underwater target tracking method based on multivariate skew Laplace distribution modeling according to claim 2, characterized in that, Step 3: Model non-Gaussian noise based on a multivariate skewed Laplace distribution. Within a variational Bayesian framework, solve for the noise's mixing parameters, shape parameters, and scale matrix. Iteratively update the target state and noise covariance matrix, and use Kalman filtering to iteratively update the estimated target motion state. After reaching the required number of iterations, output the estimated values ​​of the underwater target's position and velocity, as well as the estimated value of the covariance matrix. Measurement noise has a thick-tailed or thick-tailed skewed distribution. Non-Gaussian noise is modeled based on a multivariate skewed Laplace distribution. ； in, The measurement noise is non-Gaussian distributed. The mean is covariance is Gaussian probability density function, The parameter is The exponential probability density function, , , and These represent the mixing parameters, shape parameters, scale matrix, and rate parameters of the measurement noise, respectively. Based on the Chapman-Komlogorov equation, the one-step prediction probability density function Represented as: ； in, and They represent The state prediction at time 1 and the corresponding prediction error covariance matrix. From time 1 to The measured value at that moment; Likelihood PDF Represented as: ； The inverse Wishart distribution is used as the conjugate prior distribution of the scaling matrix; ； in, The degree of freedom parameter is The inverse scaling matrix is The inverse Wishart probability density function; Nominal measurement noise covariance matrix for: ； Where m is the dimension of the measurement vector; The prior distribution of the shape parameters is chosen to be a Gaussian distribution, that is: ； in The nominal shape parameter representing the measurement noise. This represents the confidence level regarding the nominal shape parameters. It is the identity matrix; The variational Bayesian method is used to jointly infer the mixing parameters, shape parameters, and scale matrix, i.e. ; Use variational Bayesian methods to obtain An approximate posterior probability density function with a free decomposition form, namely: ； in, An approximation of the true posterior probability density function with a free decomposition form, the optimal solution of which satisfies the following equation ； in for Any element in For the remaining elements, For constant terms; Conditional independence in constructing a hierarchical Gaussian state-space model, then the joint probability density function... Decomposed into: ； in, This represents a known state transition matrix. express The prediction error covariance matrix at time 1. For measurement matrix; Taking the logarithm of both sides, we get: ； in, The nominal shape parameter representing the measurement noise. This is used to characterize the confidence level regarding the nominal parameter; (1) Solving for state variables posterior distribution make Substitute into the following formula: ； The logarithmic expression for the posterior distribution of the state is obtained as follows: ； in, Indicates the first An approximation of the true posterior probability density function in the next iteration, with a free decomposition form. express The inverse of the prediction error covariance matrix at time 1. For constant terms; Define the corrected measurement noise mean vector covariance matrix for: ； ； Using Bayesian criteria Update to a Gaussian probability density function, i.e.: ； The state estimates and their covariances are as follows: ； in For the first iteration State estimation at time 10:00 Indicates the first iteration The error covariance matrix at time t. For the first iteration Kalman gain at time step It is the identity matrix. The corrected mean vector of measurement noise. The corrected measurement noise covariance matrix; (2) Update the posterior distribution of the mixture parameters make Substitute into the following formula ； The logarithmic expression for the posterior distribution of the state is obtained as follows: ； in for ; By matching the parameters, it was found that its posterior distribution is still a generalized inverse Gaussian distribution, that is: ； Define relevant parameters They are respectively: ； According to the properties of the generalized inverse Gaussian distribution for: ； (3) Update the posterior distribution of shape parameters make , Write it in the following form: ； in For constant terms, ; in , Update to Gaussian distribution ； The mean vector and covariance matrix are as follows: ； (4) Update the posterior distribution of the scaling matrix make , Write it in the following form: ； ； in For constant terms, ; ； ； Using the properties of the IW distribution, we have: ； At this point, the implicit variable iterative solution process based on VB has been completed. After multiple iterations, the estimated value of the target state can be obtained.

5. An underwater target tracking system based on multivariate skew Laplace distribution modeling, characterized in that, The underwater target tracking method based on multivariate skew Laplace distribution modeling as described in any one of claims 1-4 is implemented to achieve underwater target tracking based on multivariate skew Laplace distribution modeling. It is divided into three modules, and steps 1 to 3 are executed respectively.

6. A computer device, comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the underwater target tracking method based on multivariate skew Laplace distribution modeling as described in any one of claims 1-4, thereby realizing underwater target tracking based on multivariate skew Laplace distribution modeling.

7. A computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, it implements the underwater target tracking method based on multivariate skew Laplace distribution modeling as described in any one of claims 1-4, thereby realizing underwater target tracking based on multivariate skew Laplace distribution modeling.

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