Deep learning assisted variational Bayesian group filtering self-alignment method for low-precision SINS (Strapdown Inertial Navigation System)

Abstract

The invention discloses a deep learning assisted variational Bayesian group filtering self-alignment method for a low-precision SINS (Strapdown Inertial Navigation System), which comprises the following steps of: firstly, regarding gyroscope data obtained by the low-precision SINS as a time sequence by utilizing the principle and the characteristic of sensor measurement; a neural network model adopting a gated cycle unit (GRU) is designed and is used for calibrating and compensating the measurement error of the low-precision SINS. Secondly, a novel variational Bayesian Lie group filtering method is developed to process various errors in the Lie group alignment model, especially state dependent noise. A variational Bayesian method is expanded to a Lie group space to adaptively estimate a measurement noise covariance matrix in real time. And finally, performing an experiment on rotary table equipment to verify and test the performance and advantages of the proposed method. Experimental results show that the initial self-alignment method provided by the invention is obviously superior to the existing method in the aspects of alignment precision and time. According to the method, the real-time requirement in practical application is completely met, and extra equipment and cost are not needed.

Description

Technical Field

[0001] The invention discloses a deep learning assisted variational Bayesian Lie group filtering self-alignment method for low-precision SINS, which belongs to the field of navigation methods and application technologies. Background Art

[0002] With the rapid development of microelectronics technology, computer technology and sensor technology, low-precision strapdown inertial navigation systems (SINS), especially inertial measurement units based on microelectromechanical systems (MEMS), have been widely used in navigation, positioning, robot control and other fields due to their significant advantages such as small size and low cost. As the core component for providing navigation information, its importance is self-evident. Among them, the initial alignment link is particularly critical, which aims to determine the conversion relationship between the navigation coordinate system and the carrier coordinate system at the initial moment. Its accuracy and efficiency directly affect the accuracy and real-time performance of subsequent navigation solutions.

[0003] However, due to the problems of large measurement noise and significant drift, MEMS IMU faces many challenges in practical applications. In the initial alignment stage, the difficulty of achieving self-alignment is increased because the information of the earth's rotation angular velocity is difficult to extract accurately. Traditional self-alignment methods mostly rely on external information or high-precision equipment, but in certain environments, such as areas with dense obstructions or environments with strong magnetic interference, the application of these methods will be limited. In order to overcome these challenges, deep learning technology has been introduced into the field of error compensation of MEMS IMU in recent years. By training deep learning models, the prediction and compensation of MEMS IMU measurement errors can be achieved, thereby improving the accuracy of measurement data. This method can not only handle complex nonlinear relationships, but also extract useful feature information from large amounts of data, providing a new solution for error calibration of low-precision SINS.

[0004] Traditional initial alignment methods mainly rely on the optimal estimation of attitude, and these methods usually use unit quaternions to describe the rotation matrix. However, this description method causes the measurement equation to be nonlinear, increases the complexity of nonlinear filter processing, and makes it difficult to accurately estimate sensor errors. In contrast, special orthogonal groups are more intuitive in attitude representation and can avoid non-uniqueness and singular value problems. When constructing the system model, the initial alignment model based on Lie group description can achieve one-step accurate alignment of the initial attitude matrix.

[0005] In order to solve the problems existing in the self-alignment process of low-precision SINS and further improve the performance of initial alignment, the present invention draws on the properties of Lie groups and the advantages of deep learning, and proposes a deep learning-assisted Lie group filtering algorithm. A neural network model is designed and constructed using a gated recurrent unit (GRU) to calibrate and compensate for the measurement errors of low-precision SINS. An initial alignment model is established based on the Lie group description, and a variational Bayesian Lie group filtering algorithm is proposed to handle various errors in the alignment model. Experiments conducted on a turntable device have proved the feasibility of the algorithm, which can be used as a self-alignment method for low-precision SINS. Summary of the invention

[0006] The self-alignment algorithm of low-precision SINS in a shaking state has high research significance and application value. Since the carrier is easily affected by various interference factors during the initial alignment process, the requirements for the effectiveness of the algorithm are also higher. The purpose of the present invention is to address the problems existing in the existing low-precision SINS self-alignment method: (1) The present invention introduces a deep learning method and designs a neural network structure for calibration and compensation of low-precision SINS measurement errors, which can improve the measurement accuracy. (2) The present invention describes the initial alignment matrix through Lie groups, avoiding the singular value problem of the Euler angle method and the nonlinearity and non-uniqueness problem of the quaternion method, and can be used for self-alignment at any initial angle.

[0007] A deep learning-assisted variational Bayesian Lie group filtering self-alignment method for low-precision SINS is implemented through the following steps:

[0008] Step (1): The SINS strapdown inertial navigation system performs system warm-up preparation, starts the system, and obtains the longitude λ, latitude L, and projection g of the local gravity acceleration in the navigation system of the carrier. n The basic information such as the rotation angular rate information of the carrier system relative to the inertial system output by the gyroscope in the inertial measurement unit IMU is projected on the carrier system. And the load system acceleration information f output by the accelerometer b .

[0009] Step (2): Use the sliding window technology to divide the gyroscope measurement data into a time series data set, and use the GRU-based neural network model created by the present invention to perform error compensation on the gyroscope measurement data.

[0010] The present invention designs an end-to-end neural network model based on GRU, which takes the original measurement data of the gyroscope as input and outputs the data after error compensation. The system consists of three core parts: data preprocessing, neural network model, and loss function.

[0011] In the data preprocessing stage, data standardization is first performed. This step normalizes all data based on the mean and standard deviation of the original data to ensure that all input data are within the same scale range. Subsequently, the standardized data is divided into time series data sets using sliding window technology. In view of the time series dependence and cumulative characteristics of gyroscope errors, the selection of window size needs to comprehensively consider error characteristics and computational efficiency. The present invention selects a window size of 10 and sets the sliding step size to 1. The continuous measurement data of the gyroscope is divided into multiple time series sequences, and the sequence form is:

[0012] s=[x1,x2,x3…,x N ](N=10)#(1)

[0013] Among them, x i (i=1, 2…N) represents the measured angular velocity of the three axes of the gyroscope. The label of each set of data is the 11th set of data y collected by the high-precision equipment. N+1 , thereby constructing a time series dataset suitable for neural network training.

[0014] The core neural network uses GRU to build an end-to-end neural network. The gyroscope measurement data is directly used as the network input, and the output is the gyroscope data after prediction and error compensation. The internal mechanism of the GRU network can deeply explore the complex error patterns hidden in the gyroscope measurement data and automatically perform efficient error compensation. The estimation of the gyroscope error-free data is expressed as:

[0015]

[0016] in, is the angular velocity measured by the gyroscope, is the angular velocity estimate output by the neural network, and f(·) is the function defined by the neural network.

[0017] The loss function is used to measure the accuracy of the prediction results. It quantifies the prediction performance of the neural network model by calculating the deviation or difference between the predicted error-free gyroscope data and the reference data. During the training process, the loss function can monitor and evaluate the prediction accuracy of the model in real time, and optimize the model parameters through the back propagation algorithm to ensure that the model can more accurately capture the error mode of the gyroscope and output error-free measurement data that is closer to the true value. The loss function is defined as:

[0018]

[0019] in, is the predicted value output by the neural network, and y is the corresponding data label.

[0020] Step (3): Preprocess the error-compensated data and establish a linear alignment system model based on Lie group description based on Lie group differential equations.

[0021] The coordinate system in the detailed description of this method is defined as follows:

[0022] The earth coordinate system e system takes the center of the earth as the origin, the X axis is located in the equatorial plane, points from the center of the earth to the prime meridian, and the Z axis points from the center of the earth to the geographic North Pole. The X axis, Y axis and Z axis form a right-handed coordinate system, which rotates with the rotation of the earth;

[0023] Geocentric inertial coordinate system i system, choose the center of the earth as the origin, the X axis is located in the equatorial plane, points from the center of the earth to the vernal equinox, and the Z axis points from the center of the earth to the geographic North Pole. The X axis, Y axis and Z axis form a right-handed coordinate system;

[0024] Navigation coordinate system n system. In this method, the navigation coordinate system is selected as the geographic coordinate system, with the center of gravity of the carrier as the origin, aligned with the east-north-sky coordinate axis, the X axis coincides with the east direction (E), the Y axis coincides with the north direction (N), and the Z axis coincides with the sky direction (U);

[0025] The carrier coordinate system b system represents the coordinate system where the output of the inertial sensor in the strapdown inertial navigation system is located. The carrier center of gravity is taken as the origin, and the X-axis, Y-axis, and Z-axis point to the right along the carrier's horizontal axis, forward along the vertical axis, and upward along the vertical axis respectively;

[0026] The initial navigation coordinate system n(0) represents the navigation coordinate system when the strapdown inertial navigation system is turned on and remains stationary relative to the inertial space during the entire alignment process;

[0027] The initial carrier coordinate system b(0) represents the carrier coordinate system when the strapdown inertial navigation system is turned on and remains stationary relative to the inertial space during the entire alignment process;

[0028] Error navigation coordinate system n′ system, the navigation coordinate system obtained by the attitude estimation algorithm.

[0029] Combining the properties of Lie group and the output of low-precision SINS, an initial alignment model based on Lie group description is established:

[0030] According to the characteristics of the strapdown inertial navigation system, the alignment problem in motion can be transformed into the attitude estimation problem of the carrier. The attitude transformation matrix represents the rotation between the navigation coordinate system n and the carrier coordinate system b. The matrix is ​​a 3×3 orthogonal matrix with a determinant of 1, which just conforms to the properties of the three-dimensional special orthogonal group SO(3) in the Lie group and constitutes the three-dimensional rotation group SO(3):

[0031]

[0032] Where R∈SO(3) represents the element of the three-dimensional rotation group SO(3) used to represent the attitude transformation matrix. represents a 3×3 vector space, the superscript T represents the transpose of the matrix, I represents the three-dimensional identity matrix, and det(R) represents the determinant of the matrix R;

[0033] The alignment problem under the shaking state is transformed into the estimation problem of the attitude transformation matrix R of the carrier described by Lie group. According to the chain rule of real-time attitude matrix decomposition based on Lie group description, the attitude matrix to be calculated is Decomposed into the product form of three matrices in the time domain:

[0034]

[0035] Where t represents time, Represents the attitude matrix of the initial navigation coordinate system relative to the navigation coordinate system at time t. The initial attitude matrix Represents the attitude matrix of the initial carrier coordinate system relative to the initial navigation coordinate system, Represents the attitude matrix of the carrier coordinate system relative to the initial carrier coordinate system at time t;

[0036] According to the kinematic characteristics and Lie group differential equations, the attitude matrix and The time-varying update differential equation is:

[0037]

[0038] in, Represents the attitude matrix of the initial carrier coordinate system relative to the current carrier coordinate system, It represents the projection of the rotation angular rate of the navigation coordinate system relative to the inertial coordinate system in the navigation coordinate system. It represents the projection of the rotation angular rate of the carrier coordinate system output by the gyroscope relative to the inertial coordinate system in the carrier coordinate system. The symbol (·×) represents the operation of converting a three-dimensional vector into an antisymmetric matrix. The operation rules are as follows:

[0039]

[0040] Discretizing formulas (6) and (7), we can get the posture matrix and The iterative update equation is:

[0041]

[0042] From formulas (5)-(10), we can get: and It can be calculated in real time from the IMU sensor data. Just need The value of . Represents the attitude matrix at the initial moment, which does not change with time and is a constant attitude matrix; therefore, the attitude matrix during the alignment process in the motion state The solution problem is transformed into the initial posture matrix based on Lie group description of solving problems;

[0043] There is no translational motion of the load under the shaking base, so the IMU directly measures the gravitational acceleration under the load system and establishes the observation equation through gravity:

[0044]

[0045] Among them, g b It represents the gravitational acceleration under the load system, g n Represents the gravitational acceleration in the navigation coordinate system.

[0046] Considering the influence of sensor output error, the relationship between the measured value and the true value of the gyroscope and accelerometer is expressed as:

[0047]

[0048] in, represents the output of the gyroscope, represents the output of the accelerometer, δω represents the measurement noise of the gyroscope, and δg represents the measurement noise of the accelerometer.

[0049] Integrate both sides of formula (12):

[0050]

[0051] In summary, the initial posture matrix Abbreviated as R, the initial alignment model based on Lie group description is expressed as:

[0052]

[0053] Step (4): In order to solve the error problem faced in the low-precision SINS initial alignment process, especially the state-related noise in model (16), a Lie group filtering method based on variational Bayes is proposed.

[0054] Since the initial state matrix R is a time-invariant matrix, the state one-step prediction is:

[0055]

[0056] The one-step forecast covariance is:

[0057] P k|k-1 =Pk-1|k-1 (18)

[0058] In the absence of error, β and the prediction vector is the same vector in the n system. However, due to the estimation error, the estimated rotation matrix and the actual rotation matrix R k There is a deviation between them, which should be compensated by the following formula:

[0059]

[0060] Most of the existing methods are based on Lie group differential equations for compensation and use or As the new information ε. There are some problems with this. According to the properties of the Lie group, ε should be the rotation Lie algebra corresponding to the compensation rotation matrix, but neither of these two ways of expressing The corresponding rotation Lie algebra will cause the estimation accuracy to decrease. Therefore, according to the definition of the error Lie algebra as the rotation vector, the new information ε can be obtained k The exact definition:

[0061] ε k =θl (20)

[0062]

[0063] Among them, θ represents the rotation angle of the rotation vector, and l represents the rotation axis of the rotation vector.

[0064] The measurement residual error matrix is:

[0065] S k =HP k|k-1 H T +B k (twenty three)

[0066] Among them, B k is the measurement noise covariance matrix. Due to internal and external reasons, the sensor has large errors, and the sensor output noise is coupled with the state variable, making the noise estimation difficult. At this time, using a single fixed measurement noise covariance matrix will degrade the performance of the filter.

[0067] Therefore, this method adjusts B by using the variational Bayesian method. k In order to adaptively estimate the statistical characteristics of the measurement noise, it is necessary to select a suitable prior distribution for the measurement noise. In Bayesian statistics, the Inverse Wishart distribution can be used. e The Wishart (IW) distribution describes the distribution of the measurement noise covariance:

[0068]

[0069] Where μ represents the degree of freedom parameter, U represents the scale matrix, d represents the dimension of the measurement noise covariance matrix B and the scale matrix U, and Γ(·) is the gamma function.

[0070] The estimation of unknown variables is actually the estimation of the posterior distribution P(B k |α 1：k ), when the nonlinear posterior distribution is difficult to obtain, the variational Bayesian method can achieve the approximation of P(B k |α 1：k ). The KL divergence is used to measure the difference between two probability models:

[0071]

[0072] Among them, Q(B k ) represents the approximate posterior distribution.

[0073]

[0074] Let the second term after the equal sign of formula (26) be:

[0075]

[0076] The KL divergence between the two probability models is minimal, which is equivalent to L[Q(B k )] is the largest, followed by L[Q(B k )] is the objective function. At the same time, the measurement noise covariance matrix has a non-negative and symmetric spatial structure. The concept of manifold is used to deal with the specific geometric structure of the measurement noise covariance matrix and derive the natural gradient with respect to the variational parameter.

[0077] The natural gradient is a gradient that takes into account the geometric structure of the parameter space. The natural gradient of the variational parameter is:

[0078]

[0079] in, represents the standard gradient, Fisher information matrix I F describes the local geometric structure of the variational parameters and is defined as:

[0080]

[0081] Next, we derive the natural gradients of the scale matrix U and the degree of freedom parameter μ. First, the gradient of the objective function is:

[0082]

[0083] in,

[0084]

[0085] Find the partial derivatives of the parameters in the variational distribution separately:

[0086]

[0087] in,

[0088]

[0089] The natural gradients of the pseudo-scaling matrix U and the degree of freedom parameter μ are calculated as follows:

[0090]

[0091] The parameter update process is:

[0092]

[0093] R(·) represents the mapping from vector space to manifold space, ensuring that the updated value remains on the manifold. The specific calculation is:

[0094]

[0095] According to the properties of IW distribution, if A~IW(A;γ,Ψ), when λ>d+1:

[0096] E[A -1 ]=(λ-d-1)Ψ -1 (39)

[0097] Then the updated value of the measurement noise covariance matrix is ​​expressed as:

[0098]

[0099] In summary, the measurement noise covariance matrix can be estimated adaptively in real time. The filter gain matrix is:

[0100] K k =P k|k-1 H k T S k -1 (41)

[0101] The update equation for pose estimation is:

[0102]

[0103] The covariance matrix of the pose estimation is:

[0104] Pk|k =(IK k H k ) k|k-1 (43)

[0105] In each filtering step, As The estimated value of and combined with (5), (6) and (7) can obtain the attitude matrix at each discrete moment The best estimate of .

[0106] Step (5): Solve the attitude matrix required by the navigation system Thereby completing the alignment process in the shaking state.

[0107] According to the posture change matrix obtained in the previous step and The navigation attitude matrix can be solved by formula (5) to complete the alignment of the low-precision SINS under the shaking state.

[0108] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0109] (1) The present invention regards the measurement data of the gyroscope in MEMSIMU as a time series by integrating the deep learning method. On this basis, a GRU-based neural network model is designed and constructed to compensate the error of the measurement data of the gyroscope in MEMSIMU, thereby improving the accuracy and reliability of the data.

[0110] (2) Based on the variational Bayesian method, the present invention designs a novel Lie group filtering algorithm specifically for the various error problems existing in the self-alignment model based on the shaking base, especially the state-dependent noise. The algorithm significantly reduces the interference of measurement noise on system performance by dynamically and adaptively adjusting the measurement noise covariance matrix, thereby improving the accuracy and stability of self-alignment. BRIEF DESCRIPTION OF THE DRAWINGS

[0111] Figure 1 It is the flow chart of strapdown inertial navigation system.

[0112] Figure 2 It is a neural network model diagram

[0113] Figure 3 This is a comparison chart of angular velocity error compensation.

[0114] Figure 4 This is the alignment result diagram. DETAILED DESCRIPTION

[0115] The present invention is a deep learning-assisted variational Bayesian Lie group filtering self-alignment method design for low-precision SINS. The specific implementation steps of the present invention are described in detail below in conjunction with the system flow chart of the present invention:

[0116] The beneficial effects of the present invention are as follows:

[0117] (1) The gyroscope error compensation method based on deep learning was tested under the following experimental conditions:

[0118] In step (1), a three-axis turntable is used as the experimental platform, the MEMSIMU is installed on the platform, and the system is preheated. 15 minutes of static gyroscope and turntable angular velocity data are collected, 80% of which is used as the training data set, and the remaining 20% ​​is used as the test data set.

[0119] In step (2), the constructed static training data set is input into the neural network model designed by the present invention. The system model is as follows: Figure 2 As shown in the figure, the model is trained. After the model training is completed, it is applied on the static test data set to verify its performance. The result of angular velocity error compensation is shown in Figure 3 shown.

[0120] (2) The Lie group filter self-alignment method based on variational Bayes is experimented under the following experimental conditions:

[0121] In step (1), set the dynamic experimental conditions, set the turntable's heading angle to rotate at a constant speed of 5° / s in the range of 0° to 360°, set the pitch angle to swing at a speed of 5° / s in the range of -80° to 80°, and keep the roll angle stationary. Collect 15 minutes of gyroscope and turntable angular velocity data and the turntable's attitude change trajectory to construct a dynamic data set. Use the dynamic data after error compensation to perform the alignment experiment.

[0122] In step (1), the initial geographical location is: 118° east longitude, 40° north latitude;

[0123] In step (1), the sensor output frequency is 200 Hz;

[0124] In step (3), the Earth's rotation angular rate is 7.2921158e -5 rad / s;

[0125] In step (3), the time interval T is 0.01s;

[0126] In step (4), the initial value of the Lie group filtering algorithm is

[0127] The experimental results of the method are as follows:

[0128] The experiment lasted for 600 seconds, and the estimated error of the attitude angle was used as the measurement indicator. The experimental results are as follows Figure 4 As shown in the figure, the pitch attitude is aligned in about 65s and converges to 0.05°; the roll attitude is aligned in about 55s and converges to 0.04°; the heading attitude is aligned in about 139s and converges to 0.39°. The experimental results show that this method can quickly and effectively complete the self-alignment task of low-precision SINS in a shaking state.

[0129] The present invention proposes and implements a deep learning assisted variational Bayesian Lie group filtering self-alignment method for low-precision SINS. According to the measurement characteristics of MEMSIMU, a lightweight deep learning model is designed and implemented. The model does not require any external auxiliary equipment to calibrate and compensate for sensor errors. The strategy can effectively step and learn the complex nonlinear error patterns existing in the low-precision SINS inertial sensor, thereby significantly improving the accuracy of gyroscope measurement data. In addition, the present invention also designs an adaptive Lie group filtering algorithm to solve the state-related error problem in the self-alignment model on the shaking base. The algorithm uses a variational Bayesian method to adjust the measurement noise covariance matrix. The self-alignment method proposed by the present invention is suitable for low-precision SINS and effectively improves the stability and reliability of the navigation system.

[0130] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. It should be noted that, for those skilled in the art, several improvements and modifications can be made without departing from the principles of the present invention, and these improvements and modifications should also be considered as the protection scope of the present invention.

Claims

1. A deep learning-assisted variational Bayesian Lie group filtering self-alignment method for low-precision SINS, characterized by: The method is implemented by the following steps: Step (1): The SINS strapdown inertial navigation system performs system warm-up preparation, starts the system, and obtains the longitude λ, latitude L, and projection g of the local gravity acceleration in the navigation system at the location of the carrier. n The basic information such as the rotation angular rate information of the carrier system relative to the inertial system output by the gyroscope in the inertial measurement unit IMU is projected on the carrier system. And the load system acceleration information f output by the accelerometer b ; Step (2): using sliding window technology to divide the gyroscope measurement data into time series data sets, and using a GRU-based neural network model to perform error compensation on the gyroscope measurement data; Design an end-to-end neural network model based on GRU, taking the original measurement data of the gyroscope as input and outputting the data after error compensation; The neural network model consists of three core parts: data preprocessing, neural network model, and loss function; Step (3): preprocessing the error-compensated data, and establishing a linear alignment system model based on Lie group description based on Lie group differential equations; Step (4): In order to solve the error and state-related noise faced in the low-precision SINS initial alignment process, a Lie group filtering method based on variational Bayes is proposed; Step (5): Solve the attitude matrix required by the navigation system Thereby completing the alignment process in the shaking state; According to the posture change matrix obtained in the previous step and Information is used to solve the navigation attitude matrix and complete the alignment of the low-precision SINS under shaking state.

2. According to the deep learning assisted variational Bayesian Lie group filtering self-alignment method for low-precision SINS, it is characterized by: In step (2), in the data preprocessing stage, data standardization is first performed. Based on the mean and standard deviation of the original data, all data are normalized to ensure that all input data are within the same scale range. Subsequently, the standardized data is divided into time series data sets using the sliding window technology. In view of the time series dependence and cumulative characteristics of the gyroscope error, the selection of the window size needs to comprehensively consider the error characteristics and computational efficiency. The window size is selected as 10, and the sliding step is set to 1. The continuous measurement data of the gyroscope is divided into multiple time series sequences, and the sequence form is: S=[x1,x2,x3…,x N ](N=10) (1) Among them, x i (i=1, 2…N) represents the measured angular velocity of the three axes of the gyroscope. The label of each set of data is the 11th set of data y collected by the high-precision equipment. N+1 , thereby constructing a time series data set suitable for neural network training; The core neural network uses GRU to build an end-to-end neural network. The gyroscope measurement data is directly used as the network input, and the output is the gyroscope data after prediction and error compensation. The internal mechanism of the GRU network can deeply explore the complex error patterns hidden in the gyroscope measurement data and automatically perform efficient error compensation. The estimation of the gyroscope error-free data is expressed as: in, is the angular velocity measured by the gyroscope, is the angular velocity estimate output by the neural network, and f(·) is the function defined by the neural network; The loss function is used to measure the accuracy of the prediction results. It quantifies the prediction performance of the neural network model by calculating the deviation or difference between the predicted error-free gyroscope data and the reference data. During the training process, the loss function can monitor and evaluate the prediction accuracy of the model in real time, and optimize the model parameters through the back propagation algorithm to ensure that the model can more accurately capture the error mode of the gyroscope and output error-free measurement data that is closer to the true value. The loss function is defined as: in, is the predicted value output by the neural network, and y is the corresponding data label.

3. According to the deep learning assisted variational Bayesian Lie group filtering self-alignment method for low-precision SINS, it is characterized by: In step (3), the coordinate system is defined as follows: The earth coordinate system e system takes the center of the earth as the origin, the X axis is located in the equatorial plane, points from the center of the earth to the prime meridian, and the Z axis points from the center of the earth to the geographic North Pole. The X axis, Y axis and Z axis form a right-handed coordinate system, which rotates with the rotation of the earth; Geocentric inertial coordinate system i system, choose the center of the earth as the origin, the X axis is located in the equatorial plane, points from the center of the earth to the vernal equinox, and the Z axis points from the center of the earth to the geographic North Pole. The X axis, Y axis and Z axis form a right-handed coordinate system; Navigation coordinate system n system. In this method, the navigation coordinate system is selected as the geographic coordinate system, with the center of gravity of the carrier as the origin, aligned with the east-north-sky coordinate axis, the X axis coincides with the east direction (E), the Y axis coincides with the north direction (N), and the Z axis coincides with the sky direction (U); The carrier coordinate system b system represents the coordinate system where the output of the inertial sensor in the strapdown inertial navigation system is located. The carrier center of gravity is taken as the origin, and the X-axis, Y-axis, and Z-axis point to the right along the carrier's horizontal axis, forward along the vertical axis, and upward along the vertical axis respectively; The initial navigation coordinate system n(0) represents the navigation coordinate system when the strapdown inertial navigation system is turned on and remains stationary relative to the inertial space during the entire alignment process; The initial carrier coordinate system b(0) represents the carrier coordinate system when the strapdown inertial navigation system is powered on and remains stationary relative to the inertial space during the alignment process; Error navigation coordinate system n′ system, the navigation coordinate system obtained by the attitude estimation algorithm; Combining the properties of Lie group and the output of low-precision SINS, an initial alignment model based on Lie group description is established: According to the characteristics of the strapdown inertial navigation system, the alignment problem in motion can be transformed into the attitude estimation problem of the carrier. The attitude transformation matrix represents the rotation between the navigation coordinate system n and the carrier coordinate system b. The matrix is ​​a 3×3 orthogonal matrix with a determinant of 1, which just conforms to the properties of the three-dimensional special orthogonal group SO(3) in the Lie group and constitutes the three-dimensional rotation group SO(3): Where R∈SO(3) represents the element of the three-dimensional rotation group SO(3) used to represent the attitude transformation matrix. represents a 3×3 vector space, the superscript T represents the transpose of the matrix, I represents the three-dimensional identity matrix, and det(R) represents the determinant of the matrix R; The alignment problem under the shaking state is transformed into the estimation problem of the attitude transformation matrix R of the carrier described by Lie group. According to the chain rule of real-time attitude matrix decomposition based on Lie group description, the attitude matrix to be calculated is Decomposed into the product form of three matrices in the time domain: Where t represents time, Represents the attitude matrix of the initial navigation coordinate system relative to the navigation coordinate system at time t. The initial attitude matrix Represents the attitude matrix of the initial carrier coordinate system relative to the initial navigation coordinate system, Represents the attitude matrix of the carrier coordinate system relative to the initial carrier coordinate system at time t; According to the kinematic characteristics and Lie group differential equations, the attitude matrix and The time-varying update differential equation is: in, Represents the attitude matrix of the initial carrier coordinate system relative to the current carrier coordinate system, Represents the projection of the rotation angular rate of the navigation coordinate system relative to the inertial coordinate system in the navigation coordinate system, It represents the projection of the rotation angular rate of the carrier coordinate system output by the gyroscope relative to the inertial coordinate system in the carrier coordinate system. The symbol (·×) represents the operation of converting a three-dimensional vector into an antisymmetric matrix. The operation rules are as follows: Discretizing formulas (6) and (7), we can get the posture matrix and The iterative update equation is: From formulas (5)-(10), we can get: and It is calculated in real time from the IMU sensor data and requires to be solved Just need The value of Represents the attitude matrix at the initial moment; the attitude matrix during the alignment process in the motion state The solution problem is transformed into the initial posture matrix based on Lie group description of solving problems; There is no translational motion of the load under the shaking base, so the IMU directly measures the gravitational acceleration under the load system and establishes the observation equation through gravity: Among them, g b It represents the gravitational acceleration under the load system, g n Represents the gravitational acceleration in the navigation coordinate system; Considering the influence of sensor output error, the relationship between the measured value and the true value of the gyroscope and accelerometer is expressed as: in, represents the output of the gyroscope, represents the output of the accelerometer, δω represents the measurement noise of the gyroscope, and δg represents the measurement noise of the accelerometer; Integrate both sides of formula (12): In summary, the initial posture matrix Abbreviated as R, the initial alignment model based on Lie group description is expressed as:

4. According to the deep learning assisted variational Bayesian Lie group filtering self-alignment method for low-precision SINS, it is characterized by: In step (4), since the initial state matrix R is a time-invariant matrix, the state one-step prediction is: The one-step forecast covariance is: P k|k-1 =P k-1|k-1 (18) In the absence of error, β and the prediction vector is the same vector in the n-frame; however, due to estimation errors, the estimated rotation matrix and the actual rotation matrix R k There is a deviation between them, which is compensated by the following formula: According to the definition of error Lie algebra as rotation vector, we can get the new information ε k The exact definition: e k =θl (20) Among them, θ represents the rotation angle of the rotation vector, and l represents the rotation axis of the rotation vector; The measurement residual error matrix is: S k =HP k|k-1 H T +B k (23) Among them, B k is the measurement noise covariance matrix; B is adjusted by the variational Bayes method k ; In Bayesian statistics, the inverse Wishart IW distribution is used to describe the distribution of measurement noise covariance: Where μ represents the degree of freedom parameter, U represents the scale matrix, d represents the dimension of the measurement noise covariance matrix B and the scale matrix U, and Γ(·) is the gamma function; The estimation of unknown variables is actually the estimation of the posterior distribution P(B k |α 1：k ), when the nonlinear posterior distribution is difficult to obtain, the variational Bayesian method can achieve the approximation of P(B k |α 1：k ); KL divergence is used to measure the difference between two probability models: Among them, Q(B k ) represents the approximate posterior distribution; Let the second term after the equal sign of formula (26) be: The KL divergence between the two probability models is minimal, which is equivalent to L[Q(B k )] is the largest, followed by L[Q(B k )] is the objective function; at the same time, the measurement noise covariance matrix has a non-negative and symmetric spatial structure. The concept of manifold is used to process the specific geometric structure of the measurement noise covariance matrix and derive the natural gradient of the variational parameter; The natural gradient is a gradient that takes into account the geometric structure of the parameter space. The natural gradient of the variational parameter is: in, represents the standard gradient, Fisher information matrix I F Describing the local geometric structure of the variational parameters, it is defined as: Next, the natural gradients of the scale matrix U and the degree of freedom parameter μ are derived respectively; first, the gradient of the objective function is: in, Find the partial derivatives of the parameters in the variational distribution separately: in, The natural gradients of the pseudo-scaling matrix U and the degree of freedom parameter μ are calculated as follows: The parameter update process is: R(·) represents the mapping from vector space to manifold space, ensuring that the updated value remains on the manifold. The specific calculation is: According to the properties of IW distribution, if A~IW(A;γ,Ψ), when λ>d+1: E[A -1 ]=(λ-d-1)Ψ -1 (39) Then the updated value of the measurement noise covariance matrix is ​​expressed as: In summary, the measurement noise covariance matrix can be estimated adaptively in real time; the filter gain matrix is: K k =P k|k-1 H k T S k -1 (41) The update equation for pose estimation is: The covariance matrix of the pose estimation is: P k|k =(I-K k H k )P k|k-1 (43) In each filtering step, As The estimated value of and combined with (5), (6) and (7) can obtain the attitude matrix at each discrete moment The best estimate of .

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