Real-time correction method for pointing error of satellite laser communication based on singular value decomposition

Abstract

This invention provides a real-time pointing error correction method for satellite laser communication based on singular value decomposition. It employs a fixed forgetting factor to recursively estimate the covariance matrix calculated from the servo theoretical angle and command angle during the laser communication terminal's capture and tracking state. Then, singular value decomposition is performed on the estimated covariance matrix to find the optimal rotation matrix, minimizing the angle between the vector corresponding to the servo theoretical angle and the vector corresponding to the servo command angle after rotation. This invention applies the optimal rotation matrix to the calculation of the servo theoretical angle of the laser communication terminal, effectively reducing the pointing error. It can compensate for pointing errors caused by thermal deformation and other factors in real time without storing large amounts of historical data. In this invention, the norm of the first column is required to be greater than the norm of the second column during rotation, ensuring the rotation angle is less than 45°, improving numerical stability, and avoiding algorithm oscillations caused by alternating order of magnitudes.

Description

Technical Field

[0001] This invention relates to the field of communication technology, and specifically to a method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition. Background Technology

[0002] Satellite laser communication, as a next-generation satellite communication technology, offers advantages over traditional microwave communication, including higher communication speeds, higher bandwidth, higher security, and smaller terminal size, making it a crucial development direction for satellite communication. Due to the limited field of view of laser communication terminals, high pointing accuracy is required to achieve rapid link establishment and quick re-establishment after link loss. Factors affecting pointing accuracy mainly include satellite positioning errors, platform attitude errors, actuator errors, and thermal deformation errors. During satellite operation, it experiences severe temperature cycles between sunny and shadowed areas, leading to uneven thermal expansion and contraction of the satellite structure, optical platform, and telescope tube. Therefore, after entering orbit, the pointing accuracy of the laser communication terminal is primarily affected by thermal deformation. Thermal deformation error is a slow, drifting, and systematic error induced by a complex temperature field, acting on the optomechanical structure, and ultimately causing the beam to deviate from its intended direction. It is characterized by extremely low frequency, difficulty in direct real-time measurement, and uncertain model predictions, making it difficult to overcome. To overcome thermal deformation error and improve the pointing accuracy of laser communication terminals, real-time correction using on-orbit test data is necessary.

[0003] The servo theoretical angle of a laser communication terminal is calculated from the satellite orbit, attitude, and ground-calibrated installation matrix, representing the theoretical pointing of the laser communication terminal. The command angle under laser communication terminal acquisition and tracking is the true pointing of the target tracked by the laser communication terminal. Accurately estimating the deviation between the two and applying this deviation to the calculation of the servo theoretical angle of the laser communication terminal, making the theoretical pointing closer to the true pointing, can effectively reduce the pointing error of the laser communication terminal. Currently, there are two main methods for estimating the deviation between the servo theoretical angle and the command angle under laser communication terminal acquisition and tracking conditions: Method one uses an iterative method, estimating the deviation through multiple forward iterations with a set step size and precision. This method is computationally intensive, and the result depends on the selected step size and precision, making real-time on-orbit calculation impossible. Method two uses a least squares method, obtaining the deviation by solving normal equations. The least squares method is simple to implement and computationally efficient, but its numerical stability is poor. When the matrix being calculated is not full rank, there is no unique solution, and it is also sensitive to rounding errors. Summary of the Invention

[0004] This invention aims to provide a real-time correction method for pointing errors in satellite laser communication based on singular value decomposition. This method can improve the pointing accuracy of laser communication terminals, reduce the uncertainty area of ​​acquisition, compensate for pointing errors caused by thermal deformation and other reasons in real time, and enable rapid establishment of laser communication links and rapid re-establishment of links after a break.

[0005] This invention provides a method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition, comprising the following steps: S1. Project the servo theoretical angle and servo command angle obtained in real-time on-orbit laser terminal capture and tracking state to obtain a three-dimensional vector. and three-dimensional vectors ; S2, according to and Obtain the covariance matrix When valid data is obtained for the first time, Assigned to the covariance matrix In addition to the valid data obtained the first time, a forgetting factor is used. , For covariance matrix Perform recursive estimation; S3. Use the bilateral Jacobian method to analyze the covariance matrix. Singular value decomposition yields the left singular vector matrix. and right singular vector matrix ; Calculation using Euclidean norm The norm of each column is calculated and checked for zero. For columns with non-zero norms, normalization is performed. For columns with zero norms, Gram-Schmidt complementation is used to complete the orthogonal basis, resulting in the final left singular vector matrix. ; S4. Based on the final left singular vector matrix and right singular vector matrix Obtain the optimal rotation matrix Optimal rotation matrix Used in the calculation of servo theoretical angles for laser communication terminals.

[0006] The present invention provides a method for real-time correction of pointing error in satellite laser communication based on singular value decomposition. In a preferred embodiment, in step S1, the servo theoretical angle is calculated from the orbit and attitude of the local satellite, the orbit of the target satellite, and the installation matrix calibrated on the ground of the laser communication terminal. The servo command angle is calculated from the servo theoretical angle and the angle error detected in real time on the detector; Both the theoretical servo angle and the command servo angle include azimuth and pitch angles, where the azimuth angle is the pointing vector. The projection and The angle between the axes, the pitch angle is the pointing vector. and The included angles and angular angles of the faces all satisfy the right-hand rule.

[0007] The present invention provides a method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition. In a preferred embodiment, in step S1, ; ; in, For the servo theoretical azimuth angle, For the servo theory pitch angle, For servo command azimuth angle, This is the pitch angle specified by the servo command.

[0008] The present invention discloses a method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition. In a preferred embodiment, step S1 further includes a forgetting factor. Initialize and rotate matrix; initialize the rotation matrix as a 3x3 identity matrix. Once measurement data is obtained, the servo theoretical angle and servo command angle are projected into vectors; if no measurement data is obtained, the current rotation matrix is ​​maintained. In step S2, T is the transpose of the matrix; When the first valid data is obtained, ; Apart from the first valid data, , ≥0.9; In step S3, ,in, It is a singular value matrix; In step S4, the optimal rotation matrix is ​​used. Calculate the servo theoretical angle and return to step S1.

[0009] The present invention provides a preferred method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition. for .

[0010] The present invention discloses a real-time pointing error correction method for satellite laser communication based on singular value decomposition. In a preferred embodiment, step S3 uses a finite number of scans (less than or equal to 10) to gradually reduce the covariance matrix even when orthogonality cannot be satisfied through a series of Givens rotations. All column vectors tend to be pairwise orthogonal, ultimately yielding the rotation matrix. After each rotation, the error estimation vector is updated synchronously using the rotation matrix. ; The condition for determining whether two column vectors are orthogonal is that the column norms are arranged in descending order, and the norm of either column is less than the estimation error or the two column vectors are within the allowable error range. If the norms of the two column vectors are not arranged in descending order, then the two column vectors are rotated by 90°. If the two column vectors are not orthogonal, a series of Givens rotations are used to gradually make the column vectors orthogonal.

[0011] The present invention discloses a method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition. In a preferred embodiment, step S3 includes the following steps: S31. Based on the covariance matrix Calculate the error estimation vector The initial value; S32. Set the maximum number of scans in the algorithm iteration. and number of rotations ; S33, Set the number of rotations Initialize to covariance matrix The order of the covariance matrix is ​​obtained by determining the order of the covariance matrix. column pairs ,Will Initialize to 0, ; Calculate the inner product and norm of the current column and determine if the two column vectors are orthogonal. If they are, then calculate the number of rotations. Decrease by 1, do not rotate, and continue calculating the inner product and norm of the next column pair until all column pairs have been traversed. Complete one full scan; If they are not orthogonal, calculate the Givens rotation angle and the rotation covariance matrix. and right singular vector matrix Update the error estimation vector Continue calculating the inner product and norm of the next column pair until all column pairs have been traversed. Complete one full scan; When the scanning and rotation are complete, the right singular vector matrix is ​​obtained. and left singular vector matrix ; S34. Calculate using Euclidean norms The norm of each column is calculated and checked for zero. For columns with non-zero norms, normalization is performed. For columns with zero norms, Gram-Schmidt complementation is used to complete the orthogonal basis, resulting in the final left singular vector matrix. .

[0012] The present invention provides a preferred method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition. The value can be 0 or 1. The values ​​are 1 and 2; In step S31, the error estimation vector The initial value is the machine precision of the double-precision floating-point number. Multiply by the covariance matrix The norm of each column; Step S33 includes the following steps: S331, Set the number of rotations Initialize to 3 to obtain the covariance matrix. column pairs ,Will Initialize to 0, ; Get twice the inner product of the current queue: ; in, ; , and for The corresponding column vector; get norm a and norm b: ; in, Covariance matrix The The largest number in the column covariance matrix The The largest number in the column; The squared difference of the norms is obtained: ; S332, when satisfied When the two column vectors are orthogonal, Indicates "and": in, make The norm of a column vector is not less than The norm of the column vectors is set to ensure that the rotation angle is less than 45°; This indicates that the two column vectors are orthogonal within the allowable error range; and This indicates that a column is dominated by error when the norm of the two columns is less than the accumulated error during the calculation. Error estimation vector No. One data point, Error estimation vector No. One data point; S333. If two column vectors are orthogonal, then the number of rotations will be... Decrease by 1, do not rotate. Proceed to step S337; S334. If the two column vectors are not orthogonal, then the rotation angle is obtained. θ : ; in, ; S335, Applying Givens' rotational covariance matrix : ; in, Covariance matrix No. The vector after column rotation Covariance matrix No. The vector after column rotation Covariance matrix No. Column vectors Covariance matrix No. Column vector; The right singular vector matrix is ​​obtained by applying Givens rotation. Right singular vector matrix The initial value is a 3x3 identity matrix. , and Its corresponding column vector: ; in, A right singular vector matrix No. The vector after column rotation A right singular vector matrix No. The vector after column rotation A right singular vector matrix No. Column vectors A right singular vector matrix No. Column vector; S336, Utilizing rotation angle θ Update error estimation vector : ; in, Error estimation vector No. The rotated data, Error estimation vector No. The rotated data, Error estimation vector No. One data point, Error estimation vector No. One data point; ; S337. Determine if k is greater than 3. If not, return to step S331 to calculate the inner product and norm of the current queue. in the case of, ,judge j Is it greater than 2? If not, Return to step S331 to calculate the inner product and norm of the current queue. If they are, traverse all queues to complete a full scan and increment the scan count by 1. S338, When the number of rotations =0 or the number of scans Once the set maximum value is reached, a singular value decomposition is completed.

[0013] The present invention provides a method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition. In a preferred embodiment, step S332... .

[0014] The present invention discloses a method for real-time correction of pointing errors in satellite laser communication based on singular value decomposition. In a preferred embodiment, step S34 includes the following steps: S341. After iterative calculation in step S33, use the Euclidean norm to calculate... The norms of each column: ; ; in, , and for Column vectors; The norm is: ; in, Left singular vector matrix The first column norm, Left singular vector matrix The largest number in the first column; Left singular vector matrix The second column norm, Left singular vector matrix The largest number in the second column; Left singular vector matrix The norm of the third column, Left singular vector matrix The largest number in the third column; Judgment norm , and For columns with a non-zero norm, normalization is performed: ; in, The values ​​are 0, 1, and 2. Left singular vector matrix A column whose norm is not 0. Left singular vector matrix The norm of the corresponding column; S352. For columns with a norm of 0, use Gram-Schmidt to complete the orthogonal basis: ; in, To obtain the new column vector using Gram-Schmidt, Left singular vector matrix Columns with a mean norm of 0 for The preceding column; when When the norm of the first column is 0, that is, when At that time, no action was taken. ; S353, will Calculate the norm using the Euclidean norm: ; in, The value can be 1 or 2. column vector norm, column vector The maximum value in, column vector The first row of elements for The second row of elements, for The third row of elements; Then to Normalization is performed: ; use Replace the left singular vector matrix The norm of the column vector whose median norm is 0 : ; Depend on , and The diagonal matrix formed by the three norms yields the singular value matrix. : ; The normalized vector Replace the left singular vector matrix The column vector corresponding to a median norm of 0 The final left singular vector matrix is ​​obtained. .

[0015] A real-time pointing error correction method for satellite laser communication based on singular value decomposition (SVD) is a method that recursively estimates the covariance matrix calculated from the servo theoretical angle and command angle under the acquisition and tracking state of the laser communication terminal using a fixed forgetting factor. Then, singular value decomposition is performed on the estimated covariance matrix to find the optimal rotation matrix that minimizes the angle between the vector corresponding to the servo theoretical angle and the vector corresponding to the servo command angle after rotation. The method first projects the servo theoretical angle and command angle obtained in real-time from the laser communication terminal under the acquisition and tracking state to obtain two sets of three-dimensional vectors. Then, a 3x3 covariance matrix is ​​calculated based on these two sets of three-dimensional vectors. If the data is obtained for the first time, the calculated covariance matrix is ​​used directly; otherwise, a fixed forgetting factor is used to recursively estimate the covariance matrix. Next, singular value decomposition is performed on the covariance matrix. Finally, the optimal 3x3 rotation matrix is ​​obtained, which minimizes the angle between the vector corresponding to the servo theoretical angle multiplied by the rotation matrix and the vector corresponding to the servo command angle. Applying the optimal rotation matrix to the servo theoretical angle calculation of the laser communication terminal can effectively reduce the pointing error of the laser communication terminal and compensate for pointing errors caused by thermal deformation and other reasons in real time.

[0016] This on-orbit real-time correction method uses a recursive algorithm to update the covariance matrix in real time. It can automatically track time-varying parameters during satellite operation without storing large amounts of historical data, achieving real-time self-calibration. The singular value decomposition method used in this invention has advantages such as high numerical stability, high accuracy for ill-conditioned matrices, and the ability to handle matrices with incomplete rank. Singular value decomposition of a 3x3 covariance matrix requires minimal computation. This method meets the requirements of limited onboard product resources and high real-time performance, making it particularly suitable for long-term autonomous satellite operation.

[0017] This invention employs a fixed forgetting factor to recursively estimate the covariance matrix calculated from the servo theoretical angle and command angle under the tracking state captured by the laser communication terminal. Then, singular value decomposition is performed on the estimated covariance matrix to find the optimal rotation matrix, which minimizes the angle between the vector corresponding to the servo theoretical angle and the vector corresponding to the servo command angle after rotation. Applying this optimal rotation matrix to the calculation of the servo theoretical angle of the laser communication terminal can effectively reduce the pointing error of the laser communication terminal and compensate for pointing errors caused by thermal deformation and other factors in real time without storing large amounts of historical data.

[0018] The present invention has the following advantages: (1) The singular value decomposition method used in this invention has high numerical stability and high accuracy for ill-conditioned matrices, and can handle matrices with incomplete rank.

[0019] (2) In this invention, when rotating, the norm of the first column is required to be greater than the norm of the second column. This can ensure that the rotation angle is less than 45°, improve the numerical stability, and avoid algorithm oscillation caused by alternating order of size.

[0020] (3) Vector in this invention Accumulated rounding errors are tracked, preventing the algorithm from continuing calculations when errors dominate.

[0021] (4) The present invention has high calculation accuracy, which can reach the machine accuracy, reduce the acquisition uncertainty area of ​​the laser communication terminal, realize the rapid establishment of the laser communication link, and quickly re-establish the link after the link is broken.

[0022] (5) This invention uses a fixed forgetting factor to recursively estimate the covariance matrix, which is simple to calculate and can automatically track the time-varying parameters of the satellite during its on-orbit operation without storing a large amount of historical data, thus achieving real-time self-calibration. This compensates for pointing errors caused by thermal deformation and other factors during the on-orbit operation of the laser communication terminal. No ground intervention is required, which can reduce the pressure on ground telemetry and control and lower manpower costs.

[0023] (6) The present invention performs singular value decomposition on the 3x3 covariance matrix, which has a small computational load and can meet the requirements of limited spaceborne product resources and high real-time performance. Attached Figure Description

[0024] Figure 1 This is a flowchart of a real-time pointing error correction method for satellite laser communication based on singular value decomposition; Figure 2 An angle definition diagram for Example 1 of a real-time pointing error correction method for satellite laser communication based on singular value decomposition; Figure 3Example 1: Singular Value Decomposition (SVD) flowchart of a real-time pointing error correction method for satellite laser communication; Figure 4 Example 1 of a real-time correction method for pointing error in satellite laser communication based on singular value decomposition: Angle diagram of the vector before correction and the corresponding vector of the command angle. Figure 5 Example 1: An angle diagram between the corrected vector and the command angle for a real-time correction method for pointing errors in satellite laser communication based on singular value decomposition. Detailed Implementation

[0025] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Example 1

[0026] A method for real-time correction of pointing error in satellite laser communication based on singular value decomposition includes the following steps: S1. Project the servo theoretical angle and servo command angle obtained in real time by the laser terminal in the orbital acquisition and tracking state to obtain two sets of three-dimensional vectors (servo theoretical angles). The servo theoretical angle is calculated from information such as the orbit and attitude of the local satellite, the orbit of the target satellite, and the installation matrix of the laser communication terminal ground calibration. The servo command angle is calculated from the servo theoretical angle and the angle error detected in real time on the detector; Both servo theoretical angles and servo command angles include azimuth and pitch angles. The azimuth angle is defined as the pointing vector. The projection and The angle between the axes, the pitch angle is defined as the pointing vector. and The included angles and angular angles of the faces all satisfy the right-hand rule; S2. Calculate the 3x3 covariance matrix based on the two sets of three-dimensional vectors obtained in step S1.

[0027] If the data is obtained for the first time, the calculated covariance matrix is ​​used directly; if the data is not obtained for the first time, a fixed forgetting factor is used to recursively estimate the covariance matrix. The fixed forgetting factor weights the new and old information, which can avoid storing a large amount of historical data.

[0028] S3. The 3x3 covariance matrix obtained in step S2 Perform singular value decomposition: .in, It is a left singular vector matrix. It is a singular value matrix. It is a right singular vector matrix; For a 3x3 covariance matrix When performing singular value decomposition, the bilateral Jacobian method is used; Find the 3x3 covariance matrix. When calculating the norm of each column, the numerically stable Euclidean norm is used to prevent overflow or loss of precision when calculating large or small numbers.

[0029] The condition for determining that two column vectors are orthogonal is that the column norms are arranged in descending order, the norm of either column is less than the estimation error, or the two column vectors are within the allowable error range and are already orthogonal. If the norms of two column vectors are not sorted in descending order, then rotate the two column vectors by 90°. When two column vectors are not orthogonal, a series of Givens rotations are used to gradually make the column vectors orthogonal. By employing a finite number of scans, it is ensured that even when orthogonality cannot be satisfied, the covariance matrix can be gradually improved through a series of Givens rotations. All column vectors tend to be pairwise orthogonal, ultimately yielding a rotation matrix with high accuracy. ; After each rotation, the error estimation vector is updated synchronously using the rotation matrix. .

[0030] Calculating the left singular matrix When, for a column with a singular value of 0, its corresponding left singular vector The matrix is ​​"missing". We use Gram-Schmidt to complete the orthogonal basis and form a complete orthogonal matrix.

[0031] S4. Based on the left singular vector matrix obtained in step S3 and right singular vector matrix The optimal rotation matrix can be calculated. .Will When applied to the servo theoretical angle calculation of laser communication terminals, the pointing error of laser communication terminals can be effectively reduced.

[0032] This method can be used not only for bidirectional acquisition and tracking of laser terminals but also for stellar calibration. It is applicable to all scenarios where the theoretical and actual pointing vectors of the laser terminal are known, and the optimal rotation matrix needs to be found to minimize the angle between the theoretical and actual pointing vectors after rotation.

[0033] Figure 1The flowchart of this invention first initializes the forgetting factor and rotation matrix. It then determines whether the servo theoretical angle and command angle in the laser terminal's capture and tracking state have been acquired. If no new data is acquired, the previous rotation matrix remains unchanged; if new data is acquired, the process proceeds as follows... Figure 2 Projecting the angle definition diagram shown, the servo theoretical angle and command angle are converted into vectors. A 3x3 covariance matrix is ​​calculated based on the two sets of 3D vectors. If the data is obtained for the first time, the calculated covariance matrix is ​​used directly; otherwise, a fixed forgetting factor is used to recursively estimate the covariance matrix. Next, singular value decomposition is performed on the covariance matrix. Finally, an optimal 3x3 rotation matrix is ​​obtained, minimizing the angle between the vector corresponding to the servo theoretical angle multiplied by the rotation matrix and the vector corresponding to the servo command angle. Applying the optimal rotation matrix to the calculation of the servo theoretical angle of the laser communication terminal can effectively reduce the pointing error of the laser communication terminal.

[0034] Figure 2 Define an angle diagram, where the azimuth is the pointing vector. The projection and The angle between the axes, the pitch angle is defined as the angle between the pointing vector and the axis. The included angles and angles of the surfaces all satisfy the right-hand rule.

[0035] This invention relates to a high-precision real-time pointing error correction method for satellite laser communication, which specifically includes the following steps: First, initialize the forgetting factor and rotation matrix. Since the pointing accuracy of the laser communication terminal after orbit insertion is mainly affected by slowly changing factors such as thermal deformation, a strong memory, slow tracking forgetting factor is chosen (i.e., the forgetting factor is selected to be close to 1, strongly correlated with historical data, with a large influence from historical data and a small influence from current data, thus resulting in weak tracking ability for current changes), with a range of... The rotation matrix is ​​a 3x3 identity matrix.

[0036] S1. Whether the servo theoretical angle under laser terminal capture and tracking state is obtained. and command angle .in, For the servo theoretical azimuth angle, For the servo theory pitch angle, For servo command azimuth angle, This is the pitch angle specified by the servo command.

[0037] If no new data is obtained, the previous rotation matrix remains unchanged; if new data is obtained, the servo theoretical angle and command angle are converted into vectors respectively, resulting in two sets of three-dimensional vectors. and .in, Obtained by projecting the azimuth and pitch angles from the servo theory. The conversion formula is obtained by projecting the azimuth and pitch angles from the servo commands, and is as follows: ; ; S2. The covariance matrix is ​​obtained by multiplying the 3x1 vector obtained from the servo command angle projection by the transpose of the 3x1 vector obtained from the servo theoretical angle projection. The formula is as follows: ; If this is the first time valid data has been obtained, then the covariance matrix will be... Directly assign to the covariance matrix The formula is as follows: ; If it is not the first time, then a fixed forgetting factor is used for the covariance matrix. The recursive estimation is performed, and the recursive formula is as follows: ; in, The forgetting factor has a value range of 1. Finally, regarding the covariance matrix... Perform singular value decomposition.

[0038] S3. The covariance matrix obtained in step S2 according to Figure 3 The singular value decomposition flowchart is used for decomposition.

[0039] S31, Assuming the covariance matrix As shown below, , and It is its corresponding column vector.

[0040] ; ; First, calculate the error estimation vector. The initial value is the machine precision of the double-precision floating-point number. Multiply by the covariance matrix The norm of each column. The numerically stable Euclidean norm is used to calculate the norm of each column, preventing overflow or precision loss when calculating large or small numbers. The calculation formula is as follows: ; in, , Covariance matrix The largest number in the first column, covariance matrix The largest number in the second column, covariance matrix The largest number in the third column.

[0041] S32. Next, set the maximum number of scans and rotations for the algorithm iteration. The size is determined by the precision; generally, 3 to 6 complete scans are sufficient to obtain the covariance matrix. All column pairs tend to be orthogonal, and the number of scans in this invention The maximum value is chosen to be 6. This is because the covariance matrix... Given a 3x3 matrix, the number of rotations is... Set it to 3.

[0042] S33. Then, traverse the covariance matrix. All column pairs , The value can be 0 or 1. The value can be 1 or 2. .

[0043] Perform the calculations following these steps: S331. Calculate twice the inner product of the current queue using the following formula: ; Calculate using the same method as in the steps above. and The norm of is given by the following formula: ; in, Covariance matrix The The largest number in the column covariance matrix The The largest number in the column.

[0044] The formula for calculating the squared difference of norms is as follows: ; S332. Determine whether two column vectors are orthogonal. The orthogonality condition is as follows: ; In the formula, make sure The norm of a column vector is not less than The norm of the column vector ensures that the rotation angle is less than 45°, improves numerical stability, and avoids algorithm oscillation caused by alternating order of size.

[0045] This indicates that the two column vectors are orthogonal within the allowable error range.

[0046] ; and This indicates that the norm of the two columns is less than the accumulated error during their calculation, meaning that the column is dominated by error. Among these, Error estimation vector No. One data point, Error estimation vector No. Data.

[0047] S333. If two column vectors are orthogonal, then the number of rotations will be... Decrease by 1, do not rotate. Proceed to step S347.

[0048] S334. If the two column vectors are not orthogonal, calculate the rotation angle using the following formula: ; in, .

[0049] S335, Apply Givens rotation to the covariance matrix The formula is as follows: ; in, Covariance matrix No. The vector after column rotation Covariance matrix No. The vector after column rotation Covariance matrix No. Column vectors Covariance matrix No. Column vector.

[0050] Apply Givens rotation to the right singular vector matrix Right singular vector matrix The initial value is a 3x3 identity matrix. , and The formula for its corresponding column vector is as follows: ; in, A right singular vector matrix No. The vector after column rotation A right singular vector matrix No. The vector after column rotation A right singular vector matrix No. Column vectors A right singular vector matrix No. Column vector.

[0051] S336. Update the error estimation vector using rotation. The formula is as follows: ; in, Error estimation vector No. The rotated data, Error estimation vector No. The rotated data, Error estimation vector No. One data point, Error estimation vector No. Data.

[0052] ; S337. Determine if k is greater than 3. If not, return to step S341. in the case of, Determine if j is greater than 3. If not, Return to step S331. If yes, a complete scan is completed after traversing all queues, and the scan count is incremented by 1.

[0053] S338, Finally, when the number of rotations... 3 or the number of scans Once the set maximum value is reached, a singular value decomposition is completed.

[0054] S34, including the following steps: S341. After iterative calculation in step S33, by , and The matrix composed of column vectors is the right singular vector matrix. .Depend on , and The matrix composed of column vectors is the left singular vector matrix. Then, numerically stable Euclidean norms are used to calculate... The norms of each column. Assume the left singular vector matrix. As shown below, , and It is its corresponding column vector.

[0055] ; ; The formulas for calculating the norm of each column are as follows: ; in, Left singular vector matrix The first column norm, Left singular vector matrix The largest number in the first column; Left singular vector matrix The second column norm, Left singular vector matrix The largest number in the second column; Left singular vector matrix The norm of the third column, Left singular vector matrix The largest number in the third column.

[0056] Determine the norm of each column , and Is it zero? For columns with a non-zero norm, normalization is performed, as shown in the formula below: ; in, The values ​​are 0, 1, and 2. Left singular vector matrix A column whose norm is not 0. Left singular vector matrix The norm of the corresponding column.

[0057] S342. For columns with a norm of 0, the Gram-Schmidt complete orthogonal basis is used, as shown in the following formula: ; in, To obtain the new column vector using Gram-Schmidt, Left singular vector matrix Columns with a mean norm of 0 for The preceding column. When When the norm of the first column is 0, that is, when At that time, no action was taken. .

[0058] S343, The new column vector obtained Its norm is calculated using the numerically stable Euclidean norm, as shown in the following formula: ; in, The value can be either 1 or 2. column vector norm, column vector The maximum value in, column vector The first row of elements For its second row of elements, This is its third row element.

[0059] Then to The normalization process is performed using the following formula: ; use Replace the left singular vector matrix The norm of the column vector whose median norm is 0 The formula is as follows: ; Depend on , and The diagonal matrix formed by the three norms yields the singular value matrix. The formula is as follows: ; The normalized vector Replace the left singular vector matrix The column vector corresponding to a median norm of 0 The formula is as follows: ; column vector , and The resulting column vectors constitute the final left singular vector matrix. .

[0060] S4. Obtain the left singular vector matrix as described above. Multiply by the right singular vector matrix The transpose yields the optimal rotation matrix. The formula is as follows: ; The optimal rotation matrix In the calculation of the servo theoretical angle of laser communication terminals, the pointing error of laser communication terminals can be effectively reduced.

[0061] like Figures 4-5 As shown, a set of on-orbit test data was selected for verification, including the forgetting factor. The value is 0.995. Figure 4 It is the angle between the vector of the uncorrected servo theoretical angle projection and the vector of the corresponding command angle projection. Figure 5 The vector representing the projection of the servo theoretical angle is multiplied by the angle between the vector of the optimal rotation matrix calculated using the method in this invention and the vector of the corresponding command angle projection. The horizontal axis represents the number of data points, and the vertical axis represents the vector angle, in microradians. Before correction, the maximum angle between the two vectors was close to 2800 microradians, with an average of approximately 1950 microradians. After correction, the maximum angle between the two vectors was less than 600 microradians, with an average of approximately 300 microradians. This demonstrates that this correction method can effectively reduce the deviation between the servo theoretical angle and the command angle, making the theoretical pointing closer to the actual pointing, and significantly reducing the pointing error of the laser communication terminal. This invention can improve the pointing accuracy of the laser communication terminal, reduce the acquisition uncertainty area of ​​the laser communication terminal, compensate for pointing errors caused by thermal deformation and other reasons in real time, realize the rapid establishment of the laser communication link, and quickly re-establish the link after a break.

[0062] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any equivalent substitutions or modifications made by those skilled in the art within the scope of the technology disclosed in the present invention, based on the technical solution and inventive concept of the present invention, should be covered within the scope of protection of the present invention.

Claims

1. A method for real-time correction of pointing error in satellite laser communication based on singular value decomposition, characterized in that: Includes the following steps: S1. Project the servo theoretical angle and servo command angle obtained in real-time on-orbit laser terminal capture and tracking state to obtain a three-dimensional vector. and three-dimensional vectors ; S2, according to and Obtain the covariance matrix When valid data is obtained for the first time, Assigned to the covariance matrix In addition to the valid data obtained the first time, a forgetting factor is used. , For covariance matrix Perform recursive estimation; S3, regarding the covariance matrix Singular value decomposition yields the left singular vector matrix. and right singular vector matrix ; Calculation using Euclidean norm The norm of each column is calculated and checked for zero. For columns with non-zero norms, normalization is performed. For columns with zero norms, Gram-Schmidt complementation is used to complete the orthogonal basis, resulting in the final left singular vector matrix. ; S4. Based on the final left singular vector matrix and right singular vector matrix Obtain the optimal rotation matrix The optimal rotation matrix In the calculation of servo theoretical angles for laser communication terminals, to make the three-dimensional vector After rotation and with a 3D vector The angle after rotation is the smallest.

2. The method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 1, characterized in that: In step S1, the servo theoretical angle is calculated from the orbit and attitude of the local satellite, the orbit of the target satellite, and the installation matrix calibrated on the ground of the laser communication terminal. The servo command angle is calculated from the servo theoretical angle and the angle error detected in real time on the detector; Both the servo theoretical angle and the servo command angle include azimuth and pitch angles, where the azimuth angle is a pointing vector. The projection and The angle between the axes, the pitch angle is the pointing vector. and The included angles and angular angles of the faces all satisfy the right-hand rule.

3. The method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 1, characterized in that: In step S1, ； ； in, For the servo theoretical azimuth angle, For the servo theory pitch angle, For servo command azimuth angle, This is the pitch angle specified by the servo command.

4. The method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 1, characterized in that: Step S1 also includes a forgetting factor. Initialize and rotate matrix; initialize the rotation matrix as a 3x3 identity matrix. Once measurement data is obtained, the servo theoretical angle and the servo command angle are projected into a vector; if no measurement data is obtained, the current rotation matrix is ​​maintained. In step S2, T is the transpose of the matrix; When the first valid data is obtained, ; Apart from the first valid data, , ≥0.9; In step S3, ,in, It is a singular value matrix; In step S4, the optimal rotation matrix is ​​used. Calculate the rotated vector of the vector corresponding to the servo theoretical angle and return to step S1.

5. The method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 4, characterized in that: for .

6. The method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 1, characterized in that: In step S3, the bilateral Jacobi method is used to analyze the covariance matrix. Perform singular value decomposition; Using a finite number of scans, even when orthogonality cannot be satisfied, the covariance matrix is ​​gradually reduced through a series of Givens rotations. All column vectors tend to be pairwise orthogonal, ultimately yielding the rotation matrix. ; After each rotation, the error estimation vector is updated synchronously using the rotation matrix. ; The condition for determining whether two column vectors are orthogonal is as follows: if the norms of the two columns are arranged in descending order, and the norm of either column is less than the estimation error or the two column vectors are within the allowable error range, then the two column vectors are orthogonal; if the norms of the two column vectors are not arranged in descending order, then the two column vectors are rotated by 90°; if the two column vectors are not orthogonal, a series of Givens rotations are used to gradually make the column vectors orthogonal.

7. The method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 1, characterized in that: Step S3 includes the following steps: S31. Based on the covariance matrix Calculate the error estimation vector Initial value; S32. Set the maximum number of scans in the algorithm iteration. and number of rotations ; S33, Set the number of rotations Initialize to covariance matrix The order of the covariance matrix is ​​used to obtain the covariance matrix. column pairs ,Will Initialize to 0, ; Calculate the inner product and norm of the current column and determine if the two column vectors are orthogonal. If they are, then calculate the number of rotations. Decrease by 1, do not rotate, and continue calculating the inner product and norm of the next column pair until all column pairs have been traversed. Complete one full scan; If they are not orthogonal, calculate the Givens rotation angle and the rotation covariance matrix. and right singular vector matrix Update the error estimation vector Continue calculating the inner product and norm of the next column pair until all column pairs have been traversed. Complete one full scan; When the scanning and rotation are complete, the right singular vector matrix is ​​obtained. and left singular vector matrix ; S34. Calculate using Euclidean norms The norm of each column is calculated and checked for zero. For columns with non-zero norms, normalization is performed. For columns with zero norms, Gram-Schmidt complementation is used to complete the orthogonal basis, resulting in the final left singular vector matrix. .

8. The method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 7, characterized in that: The value can be 0 or 1. The values ​​are 1 and 2; In step S31, the error estimation vector The initial value is the machine precision of the double-precision floating-point number. Multiply by the covariance matrix The norm of each column; Step S33 includes the following steps: S331, Set the number of rotations Initialize to 3 to obtain the covariance matrix. column pairs ,Will Initialize to 0, ; Get twice the inner product of the current queue: ； in, ; , and for The corresponding column vector; get norm a and norm b: ； in, Covariance matrix The The largest number in the column covariance matrix The The largest number in the column; The squared difference of the norms is obtained: ； S332, when satisfied When the two column vectors are orthogonal: in, make The norm of a column vector is not less than The norm of the column vectors is set to ensure that the rotation angle is less than 45°; This indicates that the two column vectors are orthogonal within the allowable error range; and This indicates that a column is dominated by error when the norm of the two columns is less than the accumulated error during the calculation. Error estimation vector No. One data point, Error estimation vector No. One data point; S333. If two column vectors are orthogonal, then the number of rotations will be... Decrease by 1, do not rotate. Proceed to step S337; S334. If the two column vectors are not orthogonal, then the rotation angle is obtained. θ : ； in, ; S335, Applying Givens' rotational covariance matrix : ； in, Covariance matrix No. The vector after column rotation Covariance matrix No. The vector after column rotation Covariance matrix No. Column vectors Covariance matrix No. Column vector; The right singular vector matrix is ​​obtained by applying Givens rotation. Right singular vector matrix The initial value is a 3x3 identity matrix. , and for Column vectors: ； in, A right singular vector matrix No. The vector after column rotation A right singular vector matrix No. The vector after column rotation A right singular vector matrix No. Column vectors A right singular vector matrix No. Column vector; S336, Utilizing rotation angle θ Update error estimation vector : ； in, Error estimation vector No. The rotated data, Error estimation vector No. The rotated data, Error estimation vector No. One data point, Error estimation vector No. One data point; ； S337. Determine if k is greater than 3. If not, return to step S331 to calculate the inner product and norm of the current queue. in the case of, judge j Is it greater than 2? If not, Return to step S331 to calculate the inner product and norm of the current queue. If they are, traverse all queues to complete a full scan and increment the scan count by 1. S338, When the number of rotations =0 or the number of scans Once the set maximum value is reached, a singular value decomposition is completed.

9. A method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 8, characterized in that: In step S332, .

10. A method for real-time correction of pointing error in satellite laser communication based on singular value decomposition according to claim 6, characterized in that: Step S34 includes the following steps: S341. After iterative calculation in step S33, use the Euclidean norm to calculate... The norms of each column: ； ； in, , and for Column vectors; The norm is: ； in, Left singular vector matrix The first column norm, Left singular vector matrix The largest number in the first column; Left singular vector matrix The second column norm, Left singular vector matrix The largest number in the second column; Left singular vector matrix The norm of the third column, Left singular vector matrix The largest number in the third column; Judgment norm , and For columns with a non-zero norm, normalization is performed: ； in, The values ​​are 0, 1, and 2. Left singular vector matrix A column whose norm is not 0. Left singular vector matrix The norm of the corresponding column; S352. For columns with a norm of 0, use Gram-Schmidt to complete the orthogonal basis: ； in, To obtain the new column vector using Gram-Schmidt, Left singular vector matrix Columns with a mean norm of 0 for The preceding column; when When the norm of the first column is 0, that is, when At that time, no action was taken. ; S353, will Calculate the norm using the Euclidean norm: ； in, The value can be 1 or 2. column vector norm, column vector The maximum value in, column vector The first row of elements for The second row of elements, for The third row of elements; Then to Normalization is performed: ； use Replace the left singular vector matrix The norm of the column vector whose median norm is 0 : ； Depend on , and The diagonal matrix formed by the three norms yields the singular value matrix. : ； The normalized vector Replace the left singular vector matrix The column vector corresponding to a median norm of 0 The final left singular vector matrix is ​​obtained. .

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