Precise celestial coordinate conversion method based on high-order non-singular recursive Chebyshev fitting

Abstract

In order to solve the problem of rapid precise celestial coordinate conversion on a satellite, the invention discloses a precise celestial coordinate conversion method based on high-order non-singular recursive Chebyshev fitting, which is based on discrete coordinate conversion quaternion data generated by ground calculation. And performing efficient fitting on any number of time-quaternion vector data nodes by adopting a Chebyshev polynomial with an unlimited fitting order to realize on-satellite high-precision and efficient continuous time precise celestial coordinate conversion, and outputting the position and the speed of the spacecraft after coordinate conversion. And calculating the precise orbit of the spacecraft based on the converted position and speed of the spacecraft. The precise celestial coordinate conversion method provided by the invention has the characteristics of small operand, high precision and high efficiency, and is suitable for on-satellite real-time calculation.

Description

Technical Field

[0001] This invention mainly relates to the field of aerospace technology, and in particular to a precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting. Background Technology

[0002] When calculating the precise orbit of a spacecraft, the high-precision conversion between the conventional celestial coordinate system and the instantaneous true celestial coordinate system is an important issue, involving the two complex motions of the Earth: nutation and precession. Existing analytical calculation methods are complex, computationally intensive, and consume a lot of resources for real-time operation on the satellite.

[0003] Chebyshev polynomials are a set of orthogonal polynomials widely used in mathematical analysis, exhibiting good numerical stability and approximation capabilities. Data fitting based on Chebyshev polynomials, known as Chebyshev fitting, performs exceptionally well when handling complex nonlinear data. It significantly improves fitting accuracy while effectively reducing gaps caused by polynomial interpolation, and therefore has been widely applied in signal processing, aerospace, image processing, financial analysis, and other fields.

[0004] Solving for Chebyshev fitting coefficients involves inverting a product matrix, the order of which is closely related to the number of fitting nodes. When the number of fitting nodes is large, the available fitting order is also large, resulting in a large product matrix order. Inverting this matrix can easily lead to singularities, making the fitting coefficients unsolvable. Therefore, in current Chebyshev fitting applications, a relatively low fitting order (less than 20) is generally used, dividing the data into multiple segments for fitting. If the volume of data to be fitted is large but the number of fitting segments is required to be as small as possible, it is necessary to overcome the limitations of existing methods on the fitting order, so that the same set of fitting coefficients covers as many data nodes as possible. Simultaneously, it is desirable that the computations involved in the method be as simple as possible, so as to efficiently solve for the fitting polynomial coefficients. Currently, no efficient Chebyshev polynomial fitting method has been found in the literature that can adaptably segment data for any number of nodes and any high order.

[0005] Therefore, the existing Chebyshev polynomial fitting method is still difficult to meet the actual needs when applied to the problem of fast and precise celestial coordinate transformation on satellites. Summary of the Invention

[0006] To address the problem of high-precision and efficient transformation of celestial coordinates on satellites, this invention proposes a high-order non-singular recursive Chebyshev fitting method for precise celestial coordinate transformation. This method efficiently fits any number of "time-quaternion vector" data nodes using Chebyshev polynomials of unrestricted order.

[0007] To achieve the above-mentioned technical objectives, the present invention adopts the following technical solution: A precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting includes the following steps: The input data includes time and the corresponding coordinate transformation quaternion vectors between the protocol celestial coordinate system and the instantaneous true celestial coordinate system. Time is the independent variable data, which is discretely distributed at certain intervals. The coordinate transformation quaternion vectors between the protocol celestial coordinate system and the instantaneous true celestial coordinate system are the function data corresponding to the independent variable. Normalize the independent variable data so that the normalized independent variable range falls within the range of interval; Based on the normalized independent variables, with an order of 1, calculate the coefficient matrix; Based on the coefficient matrix, calculate the product of its transpose and itself, and then find its inverse; Based on the inverse matrix obtained by inverting, the inverse matrix of the product matrix corresponding to the specified order is calculated recursively; Calculate the Chebyshev fitting coefficients based on the inverse matrix of the product matrix corresponding to the specified order; Based on Chebyshev fitting coefficients, the coordinate transformation quaternions and transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system are calculated. Calculate the flat Greenwich Mean Time (FMT) transformation matrix, and multiply the FMT transformation matrix with the transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system to obtain the transformation matrix between the geocentric inertial frame and the geocentric fixed frame. Input the spacecraft's position and velocity vectors, use the transformation matrix between the geocentric inertial frame and the geocentric fixed frame to complete the mutual transformation of the spacecraft's position and velocity in different coordinate systems, output the spacecraft's position and velocity after coordinate transformation, and calculate the spacecraft's precise orbit based on the transformed spacecraft position and velocity.

[0008] The beneficial effects of this invention are: To address the challenge of rapid and precise celestial coordinate transformation on satellites, this paper proposes a high-order, non-singular recursive Chebyshev fitting method for precise celestial coordinate transformation. Based on discrete coordinate transformation quaternion data generated from ground-based calculations, this method efficiently fits any number of "time-quaternion vector" data nodes using Chebyshev polynomials of unrestricted order. This method overcomes the constraints of traditional Chebyshev fitting methods on the number of nodes and polynomial order. The innovative product matrix recursive inversion algorithm solves the problem of high-order matrix singularity in the coefficient calculation process. Furthermore, due to the relatively small computational cost of each recursive step, it can efficiently compute high-order Chebyshev fitting functions. The method itself has no constraints on the physical meaning or numerical magnitude of the independent and function data, making it highly valuable for practical applications.

[0009] This invention achieves high-precision and efficient continuous-time precise celestial coordinate transformation on satellite, outputting the transformed spacecraft position and velocity. Based on the transformed spacecraft position and velocity, precise orbit calculations are performed. The precise celestial coordinate transformation method proposed in this invention features low computational load, high precision, and high efficiency, making it suitable for real-time on-board calculations. Attached Figure Description

[0010] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0011] Figure 1 This is a flowchart of a precise celestial coordinate transformation method using high-order singularity-free recursive Chebyshev fitting in one embodiment; Figure 2 This is a curve of the original data to be fitted used in one embodiment, where: the horizontal axis... The vertical axis represents time (MJD, Modified Julian Date). The quaternion vector component represents the coordinate transformation from the conventional celestial coordinate system to the instantaneous true celestial coordinate system. Figure 3 The quaternion component fitting error obtained in one embodiment (Dimensionless) varies with the order of fitting The change curve. Detailed Implementation

[0012] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0013] Reference Figure 1 One embodiment provides a precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting, comprising the following steps: Input data includes time and the corresponding quaternion vector part of the coordinate transformation between the protocol celestial coordinate system and the instantaneous true celestial coordinate system; Normalize the independent variable data in the input numbers; Based on the normalized independent variables, with an order of 1, calculate the coefficient matrix; Based on the coefficient matrix, calculate the product of its transpose and itself, and then find its inverse; Based on the inverse matrix obtained by inverting, the inverse matrix of the product matrix corresponding to the specified order is calculated recursively; Calculate the Chebyshev fitting coefficients based on the inverse matrix of the product matrix corresponding to the specified order; Based on Chebyshev fitting coefficients, the coordinate transformation quaternions and transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system are calculated. Calculate the flat Greenwich Mean Time (FMT) transformation matrix, and multiply the FMT transformation matrix with the transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system to obtain the transformation matrix between the geocentric inertial frame and the geocentric fixed frame. Input the spacecraft's position and velocity vectors, use the transformation matrix between the geocentric inertial frame and the geocentric fixed frame to complete the mutual transformation of the spacecraft's position and velocity in different coordinate systems, output the spacecraft's position and velocity after coordinate transformation, and calculate the spacecraft's precise orbit based on the transformed spacecraft position and velocity.

[0014] In the input data, time is the independent variable, discretely distributed at certain intervals; the coordinate transformation quaternion vector between the protocol celestial coordinate system and the instantaneous true celestial coordinate system is the function data corresponding to the independent variable, composed of a large number of sine and cosine trigonometric function values ​​superimposed. The independent variable data is denoted as... The function data is denoted as , , This represents the number of independent variable data. The method for normalizing the independent variable data is as follows: ; In the formula, , .

[0015] Next, we normalize the independent variable data in the input numbers so that the normalized independent variable range falls within the range of... Interval.

[0016] Furthermore, the coefficient matrix for: ; In the formula, To fit the order of the polynomial to Chebyshev, For Chebyshev polynomials, , Its recursive formula is: ; When the order hour: ; The formula for calculating the product matrix is: ; because ,and ,visible The rank of is 2. The inverse matrix must exist.

[0017] Based on the inverse obtained The inverse matrix of is calculated recursively up to the inverse matrix of the product matrix corresponding to a specified order. The recursive formula is as follows: ; In the formula, The order of the polynomial fitted to Chebyshev is The product matrix of time, ; ; .

[0018] For the same number of nodes, when the fitting order increases by 1, we have ; in, According to the matrix The definition has ; calculate ,have: ; Right now The expression is indeed a matrix. The reverse.

[0019] order of The formula for calculating the Chebyshev fitting coefficient is: ; In the formula, It is a function data vector.

[0020] In one embodiment, a method is proposed for calculating the coordinate transformation quaternion and transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system, comprising: The design calculation time is , The formula for calculating the quaternion vector component of the coordinate transformation between the conventional celestial coordinate system and the instantaneous true celestial coordinate system at a given time is: ; In the formula, , The fitting coefficient vector The amount; The recursive calculation formula is the same as The recursive calculation formula will not be elaborated here.

[0021] The standard part of a quaternion is: ;

[0022] The coordinate transformation quaternion between the conventional celestial coordinate system and the instantaneous true celestial coordinate system is obtained as follows: .

[0023] Transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system for: .

[0024] Transformation matrix between geocentric inertial frame and geocentric fixed frame ,for: ; in, For the Greenwich Mean Time angle, Let be the transformation matrix from the geocentric inertial frame to the geocentric fixed frame. This is the flat Greenwich Mean Time (GMT) transformation matrix.

[0025] One embodiment proposes that, by inputting the spacecraft's position and velocity vectors, the transformation matrix between the geocentric inertial frame and the geocentric fixed frame can be used to complete the mutual transformation of the spacecraft's position and velocity between different coordinate systems. The method is as follows: ; ; in, Let be the position vector of the spacecraft in the geocentric inertial frame. Let V be the velocity vector of the spacecraft's geocentric inertial frame. For the spacecraft's geocentric fixed-point position vector, For the spacecraft's Earth-to-ground fixed velocity vector, Let be the angular velocity of the geocentric fixed frame relative to the geocentric inertial frame. This is the transformation matrix between the geocentric inertial frame and the geocentric fixed frame.

[0026] To demonstrate the effectiveness of the precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting described in this invention, a specific application example is provided as follows: Reference Figure 2 , is the original data curve to be fitted, where: the horizontal axis The vertical axis represents time (MJD, Modified Julian Date). This represents the quaternion vector component of the coordinate transformation from the conventional celestial coordinate system to the instantaneous true celestial coordinate system. As can be seen from the figure, the relationship between the two is quite complex and is a nonlinear data sequence.

[0027] Step 1, given the nonlinear data to be fitted, such as... Figure 2 As shown, the coordinate transformation quaternion vectors of time and the corresponding conventional celestial coordinate system to the instantaneous true celestial coordinate system are included. Time is the independent variable, which is discretely distributed at 1-day intervals and covers a time range of [60827, 63018]MJD. The coordinate transformation quaternion vectors of the conventional celestial coordinate system to the instantaneous true celestial coordinate system are function data corresponding to the independent variable, which are composed of a large number of sine and cosine trigonometric function values ​​superimposed. Step 2: Normalize the independent variable data so that the normalized independent variable range falls within the range of... interval; Step 3: Based on the normalized independent variables, with an order of 1, calculate the coefficient matrix; Step 4: Based on the coefficient matrix, calculate the product of its transpose matrix and itself, and then find its inverse; Step 5: Based on the inverse matrix obtained by inverting, recursively calculate the inverse matrix of the product matrix corresponding to the specified order; Step 6: Calculate the Chebyshev fitting coefficients based on the inverse of the product matrix; Step 7: Based on the Chebyshev fitting coefficients, within the continuous time period covered by the fitted data, given the required time, use the Chebyshev polynomial to calculate the coordinate transformation quaternion vector and transformation matrix between the protocol celestial coordinate system and the instantaneous true celestial coordinate system at the corresponding time. Step 8: Calculate the flat Greenwich Mean Time (FMT) transformation matrix. Multiply the FMT transformation matrix with the transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system to obtain the transformation matrix between the geocentric inertial frame and the geocentric fixed frame. Step 9: Input the spacecraft's position and velocity vectors, use the transformation matrix between the geocentric inertial frame and the geocentric fixed frame to complete the mutual transformation of the spacecraft's position and velocity in different coordinate systems, output the spacecraft's position and velocity after coordinate transformation, and calculate the spacecraft's precise orbit based on the transformed spacecraft position and velocity.

[0028] Figure 3 The quaternion component fitting error obtained in the example (Dimensionless) varies with the order of fitting The change curve of , where: Indicates the order of fitting. This represents the fitting error. (From...) Figure 3 It can be seen that when the fitting order When the size is small, the fitting error The fitting error is relatively large; as the fitting order increases, the fitting error generally decreases; when the fitting order increases to... When the time is reached, the fitting error is reduced to 9.565e-09. It can be seen that by using the proposed high-order singularity-free recursive Chebyshev fitting method, singularity in the inversion of the product matrix can be effectively avoided, and the fitting order can be much greater than 20.

[0029] This invention proposes a high-precision, non-singular recursive Chebyshev fitting method for precise celestial coordinate transformation in spacecraft, aiming to achieve high-precision and efficient continuous-time celestial coordinate transformation. Addressing the limitation of existing Chebyshev fitting methods' order constraints, this method overcomes the limitations of traditional Chebyshev fitting methods on the number of nodes and polynomial order. An innovative product matrix recursive inversion algorithm solves the singularity problem of high-order matrices in the fitting coefficient calculation process. Furthermore, due to the relatively small computational load of each recursive step, the high-order Chebyshev fitting function can be efficiently calculated, thereby obtaining the transformation matrix between the geocentric inertial frame and the geocentric fixed frame. Using this transformation matrix, the position and velocity of spacecraft in different coordinate systems are transformed, outputting the transformed spacecraft position and velocity. This achieves high-precision, efficient continuous-time celestial coordinate transformation in spacecraft, and the precise orbit of the spacecraft is calculated based on the transformed spacecraft position and velocity.

[0030] On the other hand, the present invention provides a computer device including a memory and a processor. The memory stores a computer program, and the processor executes the computer program to implement the steps of the high-order singularity-free recursive Chebyshev fitting precise celestial coordinate transformation method provided in any of the above embodiments. This computer device can be a server. The computer device includes a processor, a memory, a network interface, and a database connected via a system bus. The processor of the computer device provides computational and control capabilities. The memory of the computer device includes a non-volatile storage medium and internal memory. The non-volatile storage medium stores an operating system, a computer program, and a database. The internal memory provides an environment for the operation of the operating system and computer program in the non-volatile storage medium. The database of the computer device is used to store sample data. The network interface of the computer device is used for communication with external terminals via a network connection.

[0031] On the other hand, the present invention provides a computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, it implements the steps of the high-order non-singular recursive Chebyshev fitting precise celestial coordinate transformation method provided in any of the above embodiments.

[0032] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium. When executed, the computer program can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory may include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory may include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in a variety of forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), RAMbus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

[0033] Matters not covered in this invention are common knowledge.

[0034] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0035] The embodiments described above are merely illustrative of several implementation methods of this application, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and these modifications and improvements all fall within the protection scope of this application.

[0036] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting, characterized in that, Includes the following steps: The input data includes time and the corresponding coordinate transformation quaternion vectors between the conventional celestial coordinate system and the instantaneous true celestial coordinate system. Time is the independent variable data, which is discretely distributed at certain intervals. The coordinate transformation quaternion vectors between the conventional celestial coordinate system and the instantaneous true celestial coordinate system are the function data corresponding to the independent variable. Normalize the independent variable data so that the normalized independent variable range falls within the range of interval; Based on the normalized independent variables, with an order of 1, calculate the coefficient matrix; Based on the coefficient matrix, calculate the product of its transpose and itself, and then find its inverse; Based on the inverse matrix obtained by inverting, the inverse matrix of the product matrix corresponding to the specified order is calculated recursively; Calculate the Chebyshev fitting coefficients based on the inverse matrix of the product matrix corresponding to the specified order; Based on Chebyshev fitting coefficients, the coordinate transformation quaternions and transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system are calculated. Calculate the flat Greenwich Mean Time (FMT) transformation matrix, and multiply the FMT transformation matrix with the transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system to obtain the transformation matrix between the geocentric inertial frame and the geocentric fixed frame. Input the spacecraft's position and velocity vectors, use the transformation matrix between the geocentric inertial frame and the geocentric fixed frame to complete the mutual transformation of the spacecraft's position and velocity in different coordinate systems, output the spacecraft's position and velocity after coordinate transformation, and calculate the spacecraft's precise orbit based on the transformed spacecraft position and velocity.

2. The precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting according to claim 1, characterized in that, The independent variable data is denoted as The function data is denoted as , , To determine the number of independent variable data, the method for normalizing the independent variable data is as follows: ; In the formula, , .

3. The precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting according to claim 2, characterized in that, coefficient matrix for: ； In the formula, To fit the order of the Chebyshev polynomial, For Chebyshev polynomials, , Its recursive formula is: ； When the order hour: 。 4. The precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting according to claim 3, characterized in that, The recursive formula for calculating the inverse matrix of the product matrix up to a specified order is as follows: ； In the formula, To form a product matrix for Chebyshev polynomials of order n. ； 。 5. The precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting according to claim 4, characterized in that, order of The formula for calculating the Chebyshev fitting coefficient is: ； In the formula, It is a function data vector.

6. The precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting according to claim 5, characterized in that, The methods for calculating the coordinate transformation quaternions and transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system include: The design calculation time is , The formula for calculating the quaternion vector component of the coordinate transformation between the conventional celestial coordinate system and the instantaneous true celestial coordinate system at a given time is: ； In the formula, , The fitting coefficient vector The amount, The recursive calculation formula is the same as The recursive calculation formula; The standard part of a quaternion is: ； The coordinate transformation quaternion between the conventional celestial coordinate system and the instantaneous true celestial coordinate system is obtained as follows: .

7. The precise celestial coordinate transformation method based on high-order singularity-free recursive Chebyshev fitting according to claim 6, characterized in that, Transformation matrix between the conventional celestial coordinate system and the instantaneous true celestial coordinate system for: 。 8. The precise celestial coordinate transformation method for high-order singularity-free recursive Chebyshev fitting according to claim 7, characterized in that, Transformation matrix between geocentric inertial frame and geocentric fixed frame ,for: ； Wherein, GMST is the flat Greenwich Mean Time angle. Let A(GMST) be the transformation matrix from the geocentric inertial frame to the geocentric fixed frame, and let A(GMST) be the flat Greenwich time angle transformation matrix.

9. The precise celestial coordinate transformation method for high-order singularity-free recursive Chebyshev fitting according to any one of claims 1 to 8, characterized in that, Input the spacecraft's position and velocity vectors, and use the transformation matrix between the geocentric inertial frame and the geocentric fixed frame to perform mutual transformations of the spacecraft's position and velocity between different coordinate systems. The method is as follows: ； in, Let be the position vector of the spacecraft in the geocentric inertial frame. Let V be the velocity vector of the spacecraft's geocentric inertial frame. For the spacecraft's geocentric fixed-point position vector, For the spacecraft's Earth-to-ground fixed velocity vector, Let be the angular velocity of the geocentric fixed frame relative to the geocentric inertial frame. This is the transformation matrix between the geocentric inertial frame and the geocentric fixed frame.

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