A calculation method and system for converting orbital Keplerian elements to two-line elements

Abstract

The present invention belongs to the field of aerospace measurement and control technology, and in particular relates to a calculation method and system for converting orbital Kepler elements into double-row elements. The method comprises: utilizing the SGP4 / SDP4 model to iteratively calculate the Kepler elements of a space object's orbit at a specified epoch time to convert orbital quantities in the double-row elements; utilizing extrapolated ephemeris of a certain duration to fit and calculate the orbital semi-major axis correction and drag term in the double-row elements; and correcting the orbital semi-major axis in the double-row elements to generate a set of double-row element orbital parameters. The method of the present invention is relatively simple in conversion calculation and more efficient than orbit determination calculation. It can provide commercial aerospace users with a method for publishing TLE parameters using Kepler elements or position and velocity vectors.

Description

Technical Field

[0001] The present invention belongs to the field of aerospace measurement and control technology, and in particular relates to a method and system for calculating the conversion of orbital Kepler elements into double-row elements. Background Art

[0002] Kepler roots are a set of parameters that describe the spatial motion state of a space object at a given moment, including satellites and space debris. They are commonly expressed as the semi-major axis of the orbit, eccentricity, orbital inclination, right ascension of the ascending node, argument of perigee, and mean anomaly. The Kepler roots at a given moment can be converted to and from position and velocity vectors. Orbit calculation methods based on Kepler roots or position and velocity vectors typically use two methods: analytical and numerical. The analytical method involves solving the satellite orbital motion differential equations to establish an analytical model for orbit calculation. This method is characterized by high computational efficiency, but low accuracy and difficulty in improving it. The numerical method uses ordinary differential numerical methods to solve the satellite orbital motion differential equations for orbit calculation. This method is characterized by high accuracy but low computational efficiency.

[0003] Two-line elements (TLEs) are a set of parameters proposed internationally to describe the orbital motion of space objects. They were first used for cataloging and tracking space objects. TLEs are paired with specialized orbit calculation models (the SGP4 / SDP4 model, a simplified analytical method). The SGP4 (Simplified General Perturbations) model is suitable for orbit calculations of near-Earth spacecraft, while the SDP4 model is an extension of the SGP4 model and is applicable to deep-space targets. SGP4 can accurately predict orbits of space objects with periods less than 225 minutes, while SDP4 primarily predicts orbits of high-orbit and deep-space targets. These models are used for rapid orbit prediction and are characterized by high computational efficiency but low accuracy. Because the SGP4 / SDP4 model has mature general software algorithms and has been widely used in space object monitoring and cataloging, some domestic commercial aerospace companies are adopting TLEs for satellite orbit calculation and prediction, driven by cost and technical difficulties, as well as the free availability of TLE data from relevant websites.

[0004] There are two main sources of TLE parameters: one is from cataloged orbital information of space objects published by relevant international institutions; the other is obtained by converting other forms of orbital state parameters of space objects. Currently, there is uncertainty about the availability of data from relevant international institutions. Conversion from other forms of orbital state parameters involves using orbital predictions to generate ephemeris, then calculating TLE parameters through orbit determination using the SGP4 / SDP4 model. This method is computationally complex and inefficient. Summary of the Invention

[0005] The purpose of the present invention is to overcome the defects of the prior art and propose a method for calculating the conversion of orbital Kepler roots to double row roots.

[0006] In view of this, the present invention proposes a method for calculating the conversion of orbital Kepler elements to double-row elements, comprising:

[0007] Step A: Using the SGP4 / SDP4 model, the Kepler elements of the space object's orbit at the specified epoch are converted into orbital quantities in the double-row elements through iterative calculation;

[0008] Step B: Using the extrapolated ephemeris of a certain length, fit and calculate the orbital semi-major axis correction and the drag term in the double-row elements;

[0009] Step C: Correct the orbital semi-major axis in the double-row element to generate the double-row element orbital parameter set.

[0010] Preferably, the step A comprises:

[0011] Step A1: Get the epoch to be converted Kepler root number of the moment , and the mass and cross-sectional area of the space object; wherein the epoch Kepler root number of the moment , The orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of the ascending node, argument of perigee and mean anomaly, which represent the Kepler roots respectively;

[0012] Defining epochs The orbital elements under the definition of the double row elements of the moment are: ,in, The orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of ascending node, argument of perigee and mean anomaly of the double row elements are represented respectively;

[0013] Set epoch The double row root of the moment defines the orbital root equal , resistance term is 0;

[0014] Step A2: Using the SGP4 / SDP4 model, Calculating epochs Kepler root approximation of time ;

[0015] Step A3: Calculation and The difference ;

[0016] Step A4: Assign to , thus Make corrections;

[0017] Step A5: and Convert them into position and velocity vectors respectively, calculate the position difference between the two, and stop the iteration if the absolute value of the position difference is less than the set threshold or the number of iterations reaches the set value. Otherwise, increase the number of iterations by 1 and go to step A2.

[0018] Preferably, the step B comprises:

[0019] Step B1: Using orbital dynamics method, based on Forecast extrapolated ephemeris, the extrapolated time interval is The extrapolated sampling interval is ,in, is the starting time of the extrapolated time interval, is the end time of the extrapolated time interval, ; jth extrapolated sampling time , The ephemeris of the time is recorded as ;in, Respectively The orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of the ascending node, argument of perigee and mean anomaly of the Kepler root number at the moment;

[0020] Step B2: Using the SGP4 / SDP model, based on Extrapolation calculation Time's ephemeris ;in, Respectively Orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of ascending node, argument of perigee and mean anomaly of the moment double row elements;

[0021] Step B3: Calculate the orbital semi-major axis difference and phase difference at each sampling moment;

[0022] Step B4: Using the phase difference and semi-major axis difference as observation quantities, calculate the orbital semi-major axis correction in the double row elements using the least squares method based on the observation model. and resistance term .

[0023] Preferably, the step B3 comprises: according to the following formula:

[0024]

[0025] Calculate the Time orbit semi-major axis difference and phase difference .

[0026] Preferably, the observation model of step B4 is:

[0027]

[0028] in, is the atmospheric drag perturbation parameter, for Angular position deviation at the moment, is the average angular velocity, satisfying the following formula:

[0029]

[0030]

[0031] ,

[0032]

[0033]

[0034]

[0035] in, is the Earth's gravitational constant, is a parameter related to atmospheric resistance, is the perigee height, is the equatorial radius of the Earth.

[0036] Preferably, the step C comprises:

[0037] Step C1: Correct the semi-major axis in the double row root , and calculate the number of circles NR per day in the double row root;

[0038] Step C2: Combine the obtained ephemeris data and , and obtain the final double-row root orbital parameter set .

[0039] Preferably, the step C1 comprises:

[0040]

[0041]

[0042]

[0043]

[0044]

[0045]

[0046]

[0047] in, is the semi-major axis of the orbit, and are the orbital semi-major axis and mean angular velocity defined by the SGP4 model, 、 、 is the middle value, is pi, is the geodynamic oblateness term.

[0048] In another aspect, the present invention provides a system for converting orbital Kepler elements to double-row elements, comprising:

[0049] Iterative calculation module, used for converting the Kepler roots of the space object orbit at a specified epoch into orbital quantities in double row roots by iterative calculation using the SGP4 / SDP4 model;

[0050] The fitting calculation module is used to calculate the orbital semi-major axis correction and the drag term in the double-row elements using the extrapolated ephemeris of a certain length of time;

[0051] The correction calculation module is used to correct the orbital semi-major axis in the double-row element and generate the double-row element orbital parameter set.

[0052] Compared with the prior art, the advantages of the present invention are:

[0053] This paper adopts a simple calculation method for converting Kepler elements to double-row elements by combining single-point conversion and numerical fitting. First, using the SGP4 / SDP4 model, the Kepler elements at the epoch are converted into orbital quantities in the double-row elements through iterative calculation. Then, using the extrapolated ephemeris of a certain length, the orbital semi-major axis and the drag term in the double-row elements are fitted and corrected to generate the double-row elements. The double-row elements obtained by this method can achieve the same orbit calculation accuracy as the double-row elements obtained by relevant foreign institutions (see Figure 4 ), but the conversion calculation is relatively simple and more efficient than orbit determination calculation, which can provide commercial aerospace users with a method to publish TLE parameters using Kepler roots or position velocity vectors. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 This is a basic processing block diagram of the calculation method of the present invention for converting orbital Kepler elements to double-row elements;

[0055] Figure 2This is a specific flow chart of the method for calculating the conversion of orbital Kepler roots to double-row roots of the present invention;

[0056] Figure 3 is the accuracy of the TLE extrapolated orbit given by this method;

[0057] Figure 4 This is a comparison of the TLE extrapolated orbit accuracy given by this method and the TLE given by relevant foreign institutions. DETAILED DESCRIPTION

[0058] To address the problem of converting Kepler roots (for position and velocity vectors, they can be converted into Kepler roots first) into TLE parameters, the present invention proposes a conversion method that combines single-point conversion and numerical fitting to achieve fast and precision-guaranteed TLE conversion calculation.

[0059] This method first uses the SGP4 / SDP4 model to iteratively convert the Kepler values at a given epoch into orbital quantities in the TLE parameters. The orbit is then extrapolated using the Kepler roots and TLE parameters, respectively. The deviations from the extrapolated orbit are fitted using the least squares method to the orbital semi-major axis and drag term, correcting the orbital semi-major axis in the TLE and generating the TLE parameters. This conversion from Kepler roots to TLE is ultimately accomplished, meeting the needs of orbital calculations when official TLE parameters released by relevant agencies are unavailable, as well as user requests for TLE parameters.

[0060] like Figure 1 Therefore, the main process of the method for converting Kepler roots to double-row roots proposed in the present invention is as follows:

[0061] (1) The input Kepler root number is converted into TLE parameters through single-point iterative calculation. The input Kepler root number is recorded as , the number of orbital elements under TLE definition is , the resistance term is , where a is the semi-major axis of the orbit, e is the orbit eccentricity, and i is the orbit inclination. is the right ascension of the ascending node, is the argument of perigee, M is the mean anomaly, the subscript "k" indicates the Kepler element number, the subscript "TLE" indicates the TLE parameter, and the "0" in the subscript corresponds to the orbital epoch In addition, the unit of length here is meter, and the unit of angle is radian. The specific iteration process is as follows:

[0062] 1) , , iterations=1;

[0063] 2) Using the SGP4 / SDP4 model, calculate Approximate value of Kepler root number at time , the superscript “TLE” indicates the parameters calculated by the SGP4 / SDP4 model through the number of TLE roots;

[0064] 3) ;

[0065] 4) Use Correction : ;

[0066] 5) and Convert them into position and velocity vectors respectively, calculate the position difference between the two, and stop the iteration if the absolute value of the position difference is less than a given threshold (which can be set according to the accuracy requirement) or the number of iterations reaches the maximum value. Otherwise, increase the number of iterations by one and return to step 2);

[0067] (2) Use and Extrapolate the calculated ephemeris and calculate the difference between the calculated ephemeris Correction amount and resistance term , the steps are as follows:

[0068] 1) Using traditional orbital dynamics methods, based on Forecast extrapolated ephemeris, the extrapolated time interval is , the extrapolated sampling interval is ,in, is the starting time of the extrapolated time interval, is the end time of the extrapolated time interval, ; Extrapolated sampling time , The ephemeris of the time is recorded as The extrapolation time interval here can be determined according to the accuracy requirements. Generally, it can be 3 days. The sampling interval is Then you can take 900 seconds;

[0069] 2) Using the SGP4 / SDP model, based on Extrapolate the ephemeris, and the extrapolation time interval is , the extrapolated sampling interval is ; The ephemeris of the time is recorded as ;

[0070] 3) According to the following formula:

[0071]

[0072] Calculate the orbital semi-major axis difference and phase difference at each sampling moment;

[0073] 4) Using the phase difference and semi-major axis difference generated in step 3) as observation quantities, calculate the least squares fitting based on the following observation model: and :

[0074]

[0075] in, is the atmospheric drag perturbation parameter, for Angular position deviation at the moment, is the average angular velocity, satisfying the following formula:

[0076]

[0077]

[0078] ,

[0079]

[0080]

[0081]

[0082] in, is the Earth's gravitational constant, is a parameter related to atmospheric resistance, is the perigee height, is the Earth's equatorial radius, They represent the orbital semi-major axis, right ascension of ascending node, argument of perigee and mean anomaly of Kepler root number at the corresponding moment respectively;

[0083] (3) Correction of the semi-major axis in TLE , calculate the number of laps per day NR in the TLE parameter, and the comprehensive process step (1) given and process step (2) given , the final TLE orbital parameter set can be obtained :

[0084]

[0085]

[0086]

[0087]

[0088]

[0089]

[0090]

[0091] in, is the semi-major axis of the orbit, and are the orbital semi-major axis and mean angular velocity defined by the SGP4 model, 、 、 is the middle value, is pi, is the geodynamic oblateness term.

[0092] The technical solution of the present invention is described in detail below with reference to the accompanying drawings and embodiments.

[0093] Example 1

[0094] like Figure 2 As shown, embodiment 1 of the present invention provides a method for calculating the conversion of orbital Kepler roots to double-row roots, comprising the following steps:

[0095] (1) Initialization: Obtain the Kepler root number of the orbit of the space object to be converted and the corresponding epoch , as well as the mass and cross-sectional area of space objects.

[0096] (2) Order , , iterations = 1;

[0097] (3) Using the SGP4 / SDP4 model, Calculate Kepler orbital elements at time ;

[0098] (4) ;

[0099] (5) Use Correction : ;

[0100] (6) and Convert them into position and velocity vectors respectively, and calculate the position difference between the two. If the absolute value of the position difference is less than a given threshold (here 10m) or the number of iterations reaches the maximum number of iterations, stop iterating and proceed to the next step. Otherwise, increase the number of iterations by 1 and return to step (3).

[0101] (7) Using traditional orbital dynamics methods, based on Forecast extrapolated ephemeris, the extrapolated time interval is The extrapolated sampling interval is 900 seconds, and the extrapolated sampling time is , The ephemeris of the time is recorded as ;

[0102] (8) Using the SGP4 / SDP model, based on Extrapolation calculation Time's ephemeris ;

[0103] (9) According to the following formula:

[0104]

[0105] Calculate the orbital semi-major axis difference and phase difference at each sampling moment;

[0106] (10) Using the phase difference and semi-major axis difference as observation quantities, the least squares method is used to fit the following model: and :

[0107]

[0108] in:

[0109]

[0110]

[0111] ,

[0112]

[0113]

[0114]

[0115] (11) Use the following formula to correct the semi-major axis in TLE , and calculate the number of circles per day in the TLE parameter NR, combined with the previously obtained and , get the final TLE orbit parameter set :

[0116]

[0117]

[0118]

[0119]

[0120]

[0121]

[0122]

[0123] Example 2

[0124] Embodiment 2 of the present invention provides a system for calculating orbital Kepler roots to double-row roots, which is implemented based on the method of embodiment 1 and includes:

[0125] Iterative calculation module, used for converting the Kepler roots of the space object orbit at a specified epoch into orbital quantities in double row roots by iterative calculation using the SGP4 / SDP4 model;

[0126] The fitting calculation module is used to calculate the orbital semi-major axis correction and the drag term in the double-row elements using the extrapolated ephemeris of a certain length of time;

[0127] The correction calculation module is used to correct the orbital semi-major axis in the double-row element and generate the double-row element orbital parameter set.

[0128] It is worth noting that in the embodiment of the above system, the modules included are only divided according to functional logic, but are not limited to the above division, as long as the corresponding functions can be achieved; in addition, the specific names of the functional modules are only for the convenience of distinguishing each other, and are not used to limit the scope of protection of the present invention.

[0129] like Figure 3 Shown is the accuracy of the TLE extrapolated orbit given by this method; Figure 4 This is a comparison of the TLE extrapolated orbit accuracy given by this method and the TLE given by relevant foreign institutions.

[0130] The main innovative features of the present invention are:

[0131] (1) An iterative calculation method for converting single-point Kepler root numbers to TLE was designed;

[0132] (2) A method for improving the semi-major axis and resistance term of TLE parameters by least squares fitting was designed, achieving short-term prediction at the kilometer level (see Figure 3 ) accuracy.

[0133] Finally, it should be noted that the above embodiments are intended only to illustrate the technical solutions of the present invention and are not intended to limit the present invention. Although the present invention has been described in detail with reference to the embodiments, it should be understood by those skilled in the art that modifications or equivalent substitutions to the technical solutions of the present invention do not depart from the spirit and scope of the technical solutions of the present invention and are intended to be encompassed by the claims of the present invention.

Claims

1. A method for calculating the conversion of orbital Kepler elements to double-row elements, comprising: Step A: Using the SGP4 / SDP4 model, the Kepler elements of the space object's orbit at the specified epoch are converted into orbital quantities in the double-row elements through iterative calculation; Step B: Using the extrapolated ephemeris of a certain length, fit and calculate the orbital semi-major axis correction and the drag term in the double-row elements; Step C: Correct the orbital semi-major axis in the double-row element to generate the double-row element orbital parameter set; The step A comprises: Step A1: Get the epoch to be converted Kepler root number of the moment , and the mass and cross-sectional area of the space object, wherein the epoch Kepler root number of the moment , The orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of the ascending node, argument of perigee and mean anomaly, which represent the Kepler roots respectively; Defining epochs The orbital elements under the definition of the double row elements of the moment are: ,in, The orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of ascending node, argument of perigee and mean anomaly of the double row elements are represented respectively; Set epoch The double row root of the moment defines the orbital root equal , resistance term is 0; Step A2: Using the SGP4 / SDP4 model, Calculating epochs Kepler root approximation of time ; Step A3: Calculation and The difference ; Step A4: Assign to , thus Make corrections; Step A5: and Convert them into position and velocity vectors respectively, calculate the position difference between the two, and stop the iteration if the absolute value of the position difference is less than the set threshold or the number of iterations reaches the set value. Otherwise, increase the number of iterations by 1 and go to step A2.

2. The method for calculating the orbital Kepler elements to double row elements according to claim 1, characterized in that: The step B comprises: Step B1: Using orbital dynamics method, based on Forecast extrapolated ephemeris, the extrapolated time interval is The extrapolated sampling interval is ,in, is the starting time of the extrapolated time interval, is the end time of the extrapolated time interval, epoch Moment ;No. j Extrapolated sampling moments , The ephemeris of the time is recorded as ;in, Respectively The orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of the ascending node, argument of perigee and mean anomaly of the Kepler root number at the moment; Step B2: Using the SGP4 / SDP model, based on Extrapolation calculation Time's ephemeris ;in, Respectively Orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of ascending node, argument of perigee and mean anomaly of the moment double row elements; Step B3: Calculate the orbital semi-major axis difference and phase difference at each sampling moment; Step B4: Using the phase difference and semi-major axis difference as observation quantities, calculate the orbital semi-major axis correction in the double row elements using the least squares method based on the observation model. and resistance term .

3. The method for calculating the orbital Kepler elements to double-row elements according to claim 2, characterized in that: The step B3 comprises: according to the following formula: ； Calculate the Time orbit semi-major axis difference and phase difference .

4. The method for calculating the orbital Kepler elements to double row elements according to claim 3, characterized in that: The observation model of step B4 is: ； in, is the atmospheric drag perturbation parameter, for Angular position deviation at the moment, is the average angular velocity, satisfying the following formula: ； ； , ； ； ； ； in, is the Earth's gravitational constant, is a parameter related to atmospheric resistance, is the perigee height, is the equatorial radius of the Earth.

5. The method for calculating the orbital Kepler elements to double-row elements according to claim 4, characterized in that: The step C comprises: Step C1: Correct the semi-major axis in the double row root , and calculate the number of circles per day in the double row root NR ; Step C2: Combine the obtained ephemeris data and , and obtain the final double-row root orbit parameter set .

6. The method for calculating the orbital Kepler elements to double-row elements according to claim 5, characterized in that: The step C1 comprises: ； ； ； ； ； ； in, is the semi-major axis of the orbit, and are the orbital semi-major axis and mean angular velocity defined by the SGP4 model, 、 、 is the middle value, is pi, is the geodynamic oblateness term.

7. A system for calculating orbital Kepler roots to double-row roots, characterized in that: include: Iterative calculation module, used for converting the Kepler roots of the space object orbit at a specified epoch into orbital quantities in double row roots by iterative calculation using the SGP4 / SDP4 model; The fitting calculation module is used to calculate the orbital semi-major axis correction and the drag term in the double-row elements using the extrapolated ephemeris of a certain length of time; and Correction calculation module, used to correct the orbital semi-major axis in the double-row element and generate the double-row element orbital parameter set; The processing of the iterative calculation module includes: Step A1: Get the epoch to be converted Kepler root number of the moment , and the mass and cross-sectional area of the space object, wherein the epoch Kepler root number of the moment , The orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of the ascending node, argument of perigee and mean anomaly, which represent the Kepler roots respectively; Defining epochs The orbital elements under the definition of the double row elements of the moment are: ,in, The orbital semi-major axis, orbital eccentricity, orbital inclination, right ascension of ascending node, argument of perigee and mean anomaly of the double row elements are represented respectively; Set epoch The double row root of the moment defines the orbital root equal , resistance term is 0; Step A2: Using the SGP4 / SDP4 model, Calculating epochs Kepler root approximation of time ; Step A3: Calculation and The difference ; Step A4: Assign to , thus Make corrections; Step A5: and Convert them into position and velocity vectors respectively, calculate the position difference between the two, and stop the iteration if the absolute value of the position difference is less than the set threshold or the number of iterations reaches the set value. Otherwise, increase the number of iterations by 1 and go to step A2.

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