Improved design method for earth-moon triangular libration point orbits

Abstract

This invention discloses an improved method for designing orbits near the Earth-Moon triangular translation point. Based on the quasi-periodicity of orbits near the Earth-Moon triangular translation point applicable to space missions, this method obtains terms with different periodic frequencies through spectral analysis and multi-target tracking. Then, based on the analytical solution of the orbit near the Earth-Moon triangular translation point under an ephemeris model, the corresponding coordinate and frequency coefficients are fitted. Using the semi-analytical solution of the orbit enables rapid orbit design, thus avoiding the performance overhead of numerical iteration, and allows for the design of orbits with specified amplitudes and phases.

Description

Technical Field

[0001] This invention relates to a method for orbit design based on an ephemeris model, and particularly to a space orbit near the actual Earth-Moon triangular translation point. Background Technology

[0002] The orbital design near the Earth-Moon triangular translation point is a necessary prerequisite for related space missions. The dynamic characteristics of the actual Earth-Moon system near these points are complex, making it difficult to construct a universally applicable analytical solution.

[0003] Previous work has generally employed numerical methods, which have two main problems: first, they cannot design orbits with specified amplitudes and phases; and second, they require significant computational resources. Some works have also attempted to design orbits based on analytical solutions. For example, ZL202110973579.X discloses a semi-analytical solution near the celestial plane. While this solves the two problems mentioned above with numerical design methods, the orbits are still limited to those near the celestial plane and cannot be designed with large out-of-plane amplitudes.

[0004] Therefore, there is an urgent need for a simpler and faster orbit design method to overcome the problem that existing technologies cannot design orbits with specified amplitudes and phases. Summary of the Invention

[0005] Purpose of the invention: This invention provides an improved method for designing orbits near the Earth-Moon triangular translation point to solve the problems of computational consumption in numerical iteration and the inability to specify amplitude and phase, and eliminates the limitation of out-of-plane amplitude in existing design methods.

[0006] Technical solution:

[0007] This invention proposes an improved method for designing orbits near the Earth-Moon triangular translation point, comprising:

[0008] S1: Determine the out-of-plane amplitude components of the required orbit based on the actual space mission requirements. and orbital phase parameters;

[0009] S2: Construct the spacecraft motion equations and orbital semi-analysis solutions under the linear system approximation;

[0010] S3: The amplitude component outside the celestial plane is Under the given conditions, calculate the values ​​of the frequency term coefficients and the free frequency in the semi-analytical solution of the orbit;

[0011] S4: Select the initial orbit epoch and orbit point output time interval, and calculate the semi-analytical solution orbit point sequence;

[0012] S5: Optimize the target orbit of the orbit point sequence using the multi-point shooting method to obtain the optimized orbit point sequence.

[0013] Furthermore, the out-of-plane amplitude component of the white channel orbital phase parameters .

[0014] Furthermore, the spacecraft motion equations described in step S2 include:

[0015] Under an ephemeris model that includes the Earth, Moon, and Sun, the spacecraft's equations of motion are expressed as:

[0016]

[0017] in, This represents the spacecraft's position vector in the geocentric celestial coordinate system. The velocity vector of the spacecraft in the geocentric celestial coordinate system. It is the position vector of the Moon in the geocentric celestial coordinate system. These are the position vectors of the Sun in a geocentric celestial coordinate system. All of these vectors are obtained from the solar system planetary ephemeris. is the Earth's gravitational constant. The gravitational constant of the Sun, is the gravitational constant of the Moon.

[0018] Furthermore, the orbital semi-analysis solution described in step S2 is represented in the Earth-Moon rendezvous coordinate system, and the construction method includes:

[0019] Under the linear system approximation, neglecting the amplitude of specific free components within the ecliptic plane, the spacecraft's equations of motion can be written in the form of a trigonometric series solution:

[0020]

[0021] in, For the orbital point, , Represents the coordinates in the Earth-Moon synodic coordinate system; and The frequency coefficient is related to the out-of-plane amplitude component. Related; The orbital angle is related to the out-of-plane amplitude component of the orbit. and orbital phase parameters Related.

[0022] Further, step S3 calculates the frequency term coefficients. and The possible values ​​for include:

[0023]

[0024] in, and It is a constant. The order of the numerical expansion is indicated, truncated to a specified order; the coefficients of the specified frequency combination terms are obtained through spectral analysis using multiple numerical orbitals with different amplitudes. and Use the above formula to evaluate the coefficient values. and Numerical fitting was performed to obtain different orders. The corresponding values ​​of the frequency term coefficients and .

[0025] Furthermore, the free frequency mentioned in step S3 The expression is:

[0026]

[0027] in, These are constant coefficients, representing the free frequencies extracted by fitting numerical orbits of different amplitudes using the above formula. Values ​​obtain constant coefficients The numerical precision is truncated to a specified order based on the required analytical solution. .

[0028] Furthermore, step S4 also includes:

[0029] The initial orbital epoch is divided by the orbital point output time interval to obtain the number of orbital points. The semi-analytical solution of the spacecraft's motion equations in trigonometric series form is used to calculate the orbital point sequence, which is expressed as follows: ,in For orbital points including coordinates, the subscript indicates the sequence index.

[0030] Furthermore, in the spacecraft motion equations in the trigonometric series solution form, the orbital angles... The expression is:

[0031]

[0032]

[0033] in, The coefficients of the trigonometric series are... Indicates time; The fundamental frequency of the lunar orbit is obtained from the specified ephemeris analysis; For phase angle, The orbital phase parameters determined in step S1, The free frequency is calculated in step S3.

[0034] Furthermore, the fundamental frequencies of the lunar orbit include:

[0035] by The form represents the precession frequency of the lunar orbit's perigee; The frequency representing the true longitude of the Sun's orbit at its apogee. by The form represents the average orbital frequency of the sun; by The form represents the precession frequency of the ascending node of the lunar orbit.

[0036] Furthermore, step S5 also includes:

[0037] The mechanical model selected by the multi-point firing method satisfies the motion equations of the spacecraft in the Earth-Moon rendezvous coordinate system. The corrected orbital point sequence is expressed as follows: , For the corrected orbital points including coordinates, the subscript indicates the sequence index.

[0038] Beneficial effects: Compared with the prior art, the present invention has the following significant advantages:

[0039] (1) The present invention provides initial values ​​for the orbit design process by using the orbit semi-analytical solution, which greatly reduces the time spent on numerical iteration in the orbit design process based on numerical methods, and improves the design efficiency of the orbit related to the obtained semi-analytical solution.

[0040] (2) By using the track amplitude and phase parameters together as fitting parameters to construct the semi-analytical solution of the track, the track amplitude and phase are used as adjustable variables in the track design process, which effectively overcomes the problem that the numerical track design method cannot effectively control the track amplitude and phase, and improves the design control accuracy of the track amplitude and phase parameters.

[0041] (3) By performing spectral analysis and fitting of semi-analytical solutions on a stable family of quasi-periodic orbits with large out-of-plane amplitudes, the problem of limited out-of-plane orbit amplitudes in existing orbit design methods based on analytical solutions has been solved. Large out-of-plane mission orbits can be designed to better meet the needs of various missions.

[0042] (4) Through the construction process of the semi-analytical solution of the orbit, the understanding of the dynamic characteristics of the relevant orbit is enhanced, which is beneficial to the future utilization of the relevant orbit; in addition, a general and effective solution method for the semi-analytical solution of the orbit is verified, which can be used for the construction of semi-analytical solutions of more types of quasi-periodic orbits in the future. Attached Figure Description

[0043] Figure 1 This is a flowchart of the method of the present invention;

[0044] Figure 2These are example diagrams of the target orbit and the optimized orbit generated using the method of this invention. Detailed Implementation

[0045] The present invention will be further explained below with reference to the accompanying drawings and specific embodiments.

[0046] like Figure 1 The diagram illustrates an improved method for designing orbits near the Earth-Moon triangular translation point, comprising the following steps:

[0047] S1: Determine the out-of-plane amplitude components of the required orbit based on the needs of actual space missions or research work. and orbital phase parameters .

[0048] Selected The valid range of values ​​is Selected orbital phase parameters The effective range is Under the ephemeris model, the out-of-plane amplitude orbit near the Earth-Moon triangular translation point is... Within this range, spacecraft can remain stable for extended periods. Spacecraft deployed in dynamically stable orbits can operate stably for longer periods with low control overhead, conserving telemetry and control resources. Other... The range of values ​​is insufficient to obtain an orbit with such stability; because These are free parameters that can be arbitrarily selected during the track design process. In this embodiment, we take... , .

[0049] S2: Construct the spacecraft motion equations and orbital semi-analysis solutions under the linear system approximation.

[0050] The semi-analytical solution for the orbit used in this invention is based on the quasi-periodic property of the orbit near the Earth-Moon triangular translation point, meaning the orbit is mathematically expressed as a trigonometric series solution consisting of linear combinations of several fundamental frequencies. Table 1 shows the linear combinations of the fundamental frequencies for some of the component frequency terms in the semi-analytical solution. Since the frequency combinations of the x-axis and y-axis coordinates differ only in phase, their fundamental frequency combinations are the same and are not listed separately.

[0051] Table 1. Basic frequency linear combination forms of trigonometric series terms (partial)

[0052]

[0053] In the Earth-Moon system, there are four forced fundamental frequencies caused by the motion of the Earth and Moon and the perturbation of the Sun, and three free fundamental frequencies caused by the free motion of the spacecraft orbit. Different linear combinations of these seven fundamental frequencies constitute the frequency values ​​of each frequency term included in the semi-analytical solution of the spacecraft orbit motion.

[0054] In the description of the semi-analytical solution for the orbit in this invention, dimensionless units are used. The length used... ,quality and time dimension The definition is as follows:

[0055] (1)

[0056] in, The average distance between the Earth and the Moon. and The masses of the Earth and the Moon are respectively. is the gravitational constant. The semi-analytical solution used in this invention is obtained under an ephemeris model that includes the Earth, Moon, and Sun. Under the ephemeris model, the spacecraft's equations of motion are expressed as:

[0057] (2)

[0058] in, and These represent the spacecraft's position and velocity vectors in the Geocentric Celestial Reference System (GCRS). and These are the position vectors of the Moon and the Sun in the GCRS coordinate system, obtained from the solar system planetary ephemeris. , and These are the gravitational constants for the Earth, the Sun, and the Moon, respectively.

[0059] The semi-analytical solution in this invention is presented in the Earth-Moon rendezvous coordinate system. Let the position vector of the spacecraft in the Earth-Moon rendezvous coordinate system be... Then, the transformation relationship between the Earth-Moon rendezvous coordinate system and the GCRS coordinate system exists:

[0060] (3)

[0061] in It is the transformation matrix from the Earth-Moon inertial frame to the synoptic coordinate system. It is the instantaneous distance between the Earth and the Moon. (Regarding the matrix...) The calculations referenced the research of Gómez (Gomez, Gerard & Jorba, Angel & Masdemont, Josep & Simó, Carles., 2001, Dynamics and mission design nearlibration points, advanced methods for collinear points. 10.1142 / 4337.).

[0062] Applying the transformation to the equations of motion, we then... The equation of motion is written as:

[0063] (4)

[0064] in, It is a 3-order identity matrix. and These are the position vectors of the Moon and the Sun in the Earth-Moon synodic coordinate system, respectively. Considering the relative motion of the triangular translation point, let the triangular translation point and its relative offset in the Earth-Moon synodic coordinate system be respectively... and ,Right now Therefore, equation (4) is rewritten in the following form:

[0065] (5)

[0066] in:

[0067] (6)

[0068] (7)

[0069] At this point, due to the presence of solar gravitational perturbation, the triangular translation point is no longer an equilibrium point in the ephemeris model. Instead, for any given triangular translation point, there are three dynamical alternative orbits. In the Earth-Moon restricted three-body system perturbed by the Sun, there are four forced fundamental frequencies. These are closely related to the Earth-Moon two-body motion under solar perturbation and do not change with variations in spacecraft orbit design. Dynamic alternative orbits are those containing only these four forced fundamental frequencies.

[0070] Based on the above explanation, it can be concluded that under the ephemeris model, the orbit near the Earth-Moon triangular translation point exhibits a quasi-periodic form combining forced and free frequencies under the perturbation of equation (7), which provides a basis for fitting the subsequent orbital semi-analysis solution. Furthermore, we can further derive the form of the corresponding orbital semi-analysis solution in this embodiment.

[0071] In this embodiment, four fundamental frequency values ​​were obtained by analyzing the lunar orbit in ephemeris DE431:

[0072]

[0073] in, by The form represents the precession frequency of the lunar orbit's perigee; The frequency representing the true longitude of the Sun's orbit at its apogee. by The form represents the average orbital frequency of the sun; by The form represents the precession frequency of the ascending node of the lunar orbit.

[0074] For equation (5), it can be rewritten as:

[0075] (8)

[0076] in:

[0077]

[0078] Expanding equation (8) in the vicinity of the dynamical substitution trajectory, we get:

[0079] (9)

[0080] Introducing time-dependent transformation:

[0081] (10)

[0082] The transformed system can be written as:

[0083] (11)

[0084] According to the quasi-Floquet theory (Jorba, A. and Simó, C., 1996, SIAM Journal on Mathematical Analysis, 27, 1704-1737.), there exists a matrix... Make the matrix in the transformed system If the matrix is ​​constant, then the linear stability of the system depends on the matrix. The eigenvalues ​​of . Consider the linearly stable case, where the matrix . It has purely imaginary eigenvalues: .in, That is, the free frequency of the system, the value of which is related to the free amplitude of the orbit. Using the inverse transformation of transformation (10), the solution of equation (11) is transformed back to the Earth-Moon rendezvous coordinate system. At this time, under the linear system approximation, the motion of the spacecraft can be written in the form of a trigonometric series solution, which satisfies:

[0085] (12)

[0086] in, It is a three-dimensional coordinate system; The expression is:

[0087]

[0088] and , , ; , , The phase factor can be set to a custom value. and These are the coordinate coefficients preceding each trigonometric function term.

[0089] S3: Calculations are made in the... coefficients before each frequency term in the semi-analytical solution of the orbital under the given value and The value of the free frequency Size.

[0090] For the orbit involved in this invention, its independent variable is only the out-of-plane amplitude of the orbit. Furthermore, it is assumed that the free components within the celestial plane... and The amplitude of the oscillation is 0 or very small and can be ignored. Therefore, the above trigonometric series solution (12) degenerates into:

[0091] (13)

[0092] in, , .

[0093] and The coordinate coefficients preceding each trigonometric function term are related to the amplitude outside the orbital plane. Related, written as The series form:

[0094] (14)

[0095] in, and These are constants, obtained from spectral analysis of several known numerical orbits (for details, refer to existing research Hou X.-Y., Xin X., Scheeres D.-J., Wang J., 2015, MNRAS, 454, 4172. doi:10.1093 / mnras / stv2216). The coordinate coefficients of the corresponding frequency combination terms are obtained using spectral analysis on several numerical orbits with different amplitudes. and By using the form of equation (14) to numerically fit these coefficients, we can obtain the results. and The specific value to be taken. This indicates the order of the numerical expansion.

[0096] The numerical orbits were obtained using traditional numerical iteration methods. In this embodiment, the numerical iteration method references existing research on the stability of orbits around the Earth-Moon triangular translation point (Liu M.-L., HouX.-Y., Li B.-S., Li H.-H., 2024, MNRAS, 535, 2619.), employing a dynamically continuous approach to obtain the numerical orbits. Existing research starts with the restricted three-body problem (CRTBP) model, continuously applying the vertical Lyapunov orbits to the ephemeris model, and then performs numerical analysis on the resulting numerical orbits. However, the method of this invention uses a semi-analytical solution in the calculation, eliminating the need for further numerical iteration analysis of the numerical orbits, thus significantly improving computational speed.

[0097] Tables 2-7 provide the embodiments of this example. and The constant coefficients of the series form are truncated to order 7. Order 7 is used here because... exist In the range of values ​​of Two orders of magnitude smaller, meeting the required cutoff condition. Lower orders may not meet the cutoff condition in some ranges, while higher orders may lead to numerical instability. Specifically, Tables 2 and 3 show the results for the x-coordinate... and The coefficients, Tables 4 and 5 are for the y-coordinate. and Coefficients, Tables 6 and 7 respectively, are the coefficients under the z-coordinate. and coefficient.

[0098] Table 2 Coefficient (x-coordinate)

[0099]

[0100] Table 3 Coefficient (x-coordinate)

[0101]

[0102] Table 4 Coefficient (y-coordinate)

[0103]

[0104] Table 5 Coefficient (y-coordinate)

[0105]

[0106] Table 6 Coefficients (z-coordinate)

[0107]

[0108] Table 7 Coefficients (z-coordinate)

[0109]

[0110] At this time, the free frequency The value is only related to Value dependence, expressed in the semi-analytical solution as:

[0111] (15)

[0112] in, It is a constant, obtained by extracting the corresponding numerical free frequencies from numerical orbits of different amplitudes (for details, refer to existing research Hou X.-Y., Xin X., Scheeres D.-J., Wang J., 2015, MNRAS, 454, 4172. doi:10.1093 / mnras / stv2216), fitting the numerical free frequencies to obtain the frequency coefficients. The numerical orbital is the same as the one used when calculating the coordinate coefficients before the trigonometric function terms. The numerical precision is truncated to a specified order based on the required accuracy of the analytical solution; in this embodiment, it is truncated to order 6.

[0113] In summary, equation (15) is used to calculate the selected... Free frequency in the semi-analytical solution of the lower orbit The size is calculated using equation (14) in the selected... Coordinate coefficients before each frequency term in the semi-analytical solution of the orbital under the given value and The value of .

[0114] S4: Select initial orbital epoch Track point output time interval Number of orbital points Calculate the semi-analytical solution orbital point sequence.

[0115] The orbits designed during simulation are often used for further research or analysis. The initial orbit epoch is selected based on the needs of this further research, such as orbit sampling frequency, orbit integration time, and the number of orbit samples. Track point output time interval Number of orbital points The semi-analytical solution orbital point sequence is calculated according to equation (13):

[0116] (13)

[0117] in, , . and The coefficients preceding each trigonometric function term are related to the amplitude outside the ecliptic plane. Related to equation (14) The series form. For the selected track design parameters... and , Substitution and The series expression can be used to obtain the coefficients of the required frequency term. Substitution , The frequency value of the corresponding frequency item can be obtained.

[0118] The obtained semi-analytical solution trajectory point sequence is represented as follows:

[0119] (16)

[0120] in, , is the orbital point calculated by equation (13), and the subscript represents the sequence index.

[0121] S5: Optimize the target orbit of the orbit point sequence using the multi-point shooting method to obtain the optimized orbit point sequence.

[0122] The target trajectory described by the trajectory point sequence in equation (16) is optimized using a multi-point firing method. This multi-point firing method references the research of Gómez (Gómez, G., Masdemont, J. & Simó, C., 1998, J of AstronautSci 46, 135–176.). The mechanical model selected for this multi-point firing method satisfies equation (4), and the final corrected trajectory point sequence is expressed as:

[0123] (17)

[0124] in, , where is the corrected orbital point, and the subscript indicates the sequence index.

[0125] Through the above steps, the trajectory described in this invention provides initial values ​​during the trajectory design process, and then optimizes the trajectory using a multi-point firing method, thereby providing the final trajectory result. For example... Figure 2 The image shows a corrected orbital example, which was generated using the semi-analytical solution method proposed in this invention. The orbit was then optimized using a multi-point target method. As can be seen from the figure, the target orbit (dashed line orbit in the figure) is very close to the refined orbit (solid line orbit in the figure) obtained by the present invention, proving that the semi-analytical solution obtained by the method of the present invention has high accuracy.

[0126] The semi-analytical orbit solution provided in this invention is applicable to orbits near the Earth-Moon L5 point with out-of-plane amplitudes between 0.4 and 0.8. The designed orbits can be used in Earth-Moon space-related space missions. In particular, the semi-analytical orbit solution effectively overcomes the problem that numerical orbit design methods cannot effectively control orbit amplitude and phase, thus improving the design and control accuracy of orbit amplitude and phase parameters.

Claims

1. An improved method of designing an orbit near the Earth-Moon triangular libration point, characterized in that, include: S1: Determine the out-of-plane amplitude components of the required orbit based on the actual space mission requirements. and orbital phase parameters; S2: Construct the spacecraft motion equations and orbital semi-analysis solutions under the linear system approximation; S3: The amplitude component outside the celestial plane is Under the given conditions, calculate the values ​​of the frequency term coefficients and the free frequency in the semi-analytical solution of the orbit; S4: Select the initial orbit epoch and orbit point output time interval, and calculate the semi-analytical solution orbit point sequence; S5: Optimize the target orbit of the orbit point sequence using the multi-point shooting method to obtain the optimized orbit point sequence.

2. The method for designing orbits near the Earth-Moon triangular translation point according to claim 1, characterized in that, The out-of-plane amplitude component orbital phase parameters .

3. The method for designing orbits near the Earth-Moon triangular translation point according to claim 1, characterized in that, The spacecraft motion equations in step S2 include: Under an ephemeris model that includes the Earth, Moon, and Sun, the spacecraft's equations of motion are expressed as: in, This represents the spacecraft's position vector in the geocentric celestial coordinate system. The velocity vector of the spacecraft in the geocentric celestial coordinate system. It is the position vector of the Moon in the geocentric celestial coordinate system. These are the position vectors of the Sun in a geocentric celestial coordinate system. All of these vectors are obtained from the solar system planetary ephemeris. is the Earth's gravitational constant. The gravitational constant of the Sun, is the gravitational constant of the Moon.

4. The method for designing an orbit near the Earth-Moon triangular translation point according to claim 1, characterized in that, The orbital semi-analysis solution described in step S2 is represented in the Earth-Moon rendezvous coordinate system. The construction method includes: Under the linear system approximation, neglecting the amplitude of specific free components within the ecliptic plane, the spacecraft's equations of motion can be written in the form of a trigonometric series solution: in, For the orbital point, , Represents the coordinates in the Earth-Moon synodic coordinate system; and The frequency coefficient is related to the out-of-plane amplitude component. Related; The orbital angle is related to the out-of-plane amplitude component of the orbit. and orbital phase parameters Related.

5. The method for designing an orbit near the Earth-Moon triangular translation point according to claim 4, characterized in that, Step S3: Calculate the frequency term coefficients. and The possible values ​​for include: in, and It is a constant. The order of the numerical expansion is indicated, truncated to a specified order; the coefficients of the specified frequency combination terms are obtained through spectral analysis using several known numerical orbitals of different amplitudes. and Use the above formula to evaluate the coefficient values. and Numerical fitting was performed to obtain different orders. The corresponding values ​​of the frequency term coefficients and .

6. The method for designing an orbit near the Earth-Moon triangular translation point according to claim 4, characterized in that, The free frequency mentioned in step S3 The expression is: in, These are constant coefficients, obtained by extracting numerical free frequencies from several known numerical orbits of different amplitudes and fitting these numerical free frequencies. The value; the numerical precision is truncated to a specified order according to the requirements of the analytical solution. The free frequency in the semi-analytical solution of the orbit can be calculated using the above formula. .

7. The method for designing an orbit near the Earth-Moon triangular translation point according to claim 1, characterized in that, Step S4 further includes: The initial orbital epoch is divided by the orbital point output time interval to obtain the number of orbital points. The semi-analytical solution of the spacecraft's motion equations in trigonometric series form is used to calculate the orbital point sequence, which is expressed as follows: ,in For orbital points including coordinates, the subscript indicates the sequence index.

8. The method for designing orbits near the Earth-Moon triangular translation point according to claim 4 or 7, characterized in that, In the spacecraft motion equations in the trigonometric series solution form, the orbital angles... The expression is: in, The coefficients of the trigonometric series are... Indicates time; The fundamental frequency of the lunar orbit is obtained from the specified ephemeris analysis; The phase angle, The orbital phase parameters determined in step S1, The free frequency is calculated in step S3.

9. The method for designing an orbit near the Earth-Moon triangular translation point according to claim 8, characterized in that, The fundamental frequencies of the lunar orbit include: by The form represents the precession frequency of the lunar orbit's perigee; The frequency representing the true longitude of the Sun's orbit at its apogee. by The form represents the average orbital frequency of the sun; by The form represents the precession frequency of the ascending node of the lunar orbit.

10. The method for designing an orbit near the Earth-Moon triangular translation point according to claim 1, characterized in that, Step S5 further includes: The mechanical model selected by the multi-point firing method satisfies the motion equations of the spacecraft in the Earth-Moon rendezvous coordinate system. The corrected orbital point sequence is expressed as follows: , For the corrected orbital points including coordinates, the subscript indicates the sequence index.

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