CN113627029B - Method for designing track near earth-moon triangle translational point - Google Patents

Method for designing track near earth-moon triangle translational point Download PDF

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CN113627029B
CN113627029B CN202110973579.XA CN202110973579A CN113627029B CN 113627029 B CN113627029 B CN 113627029B CN 202110973579 A CN202110973579 A CN 202110973579A CN 113627029 B CN113627029 B CN 113627029B
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侯锡云
王鹏
辛晓生
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Nanjing University
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    • G06F30/20—Design optimisation, verification or simulation
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
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    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
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Abstract

The invention discloses a method for designing a track near a moon triangle translation point, which comprises the following steps: (1) constructing a semi-analytical solution approximation of the motion near the earth-moon triangle translational points L4 and L5 under the ephemeris model; (2) designing a task orbit which is maintained for a certain time near the earth-moon triangle translational points L4 and L5 under the ephemeris model; (3) and outputting a track design result, namely generating the position and the speed vector of the detector under the geocentric celestial coordinate system and the earth-moon instantaneous convergence coordinate system at certain time intervals. The invention can design the task track which is maintained for a long time from any time, the adopted semi-analytical solution has higher precision, and the invention is suitable for two triangular translation points and is easy for a track designer to directly call.

Description

Method for designing track near earth-moon triangle translational point
Technical Field
The invention relates to the technical field of aerospace, in particular to a method for designing an orbit near a moon-earth triangle translation point.
Background
Theoretical research results under the existing ephemeris model are not complete, and a certain distance is reserved from the practical engineering application requirements. The simplified circular restrictive trisomy problem the orbit designed near the trigonometric pan point will usually diverge very quickly in a real lunar system. Taking the point L4 under the circular limiting trisomy problem as an example, under the simplified circular limiting trisomy problem model, the probe will be fixed at this point and remain unchanged, however, this orbit diverges quickly when operating in a real lunar system and is not suitable as a mission orbit. Briefly: and the track under the circular restrictive three-body problem model can not meet the requirement as the precision of the task track under the actual force model. The double-circle model of the earth-moon system is not dynamically self-consistent, and for collinear translational points, previous work has shown that the model causes distortion phenomena of certain dynamic structures. At present, no relevant research work exists on the triangle translational point, but the essential defect that the dynamics is not self-consistent still influences the applicability of the solution, so that the method has only theoretical research significance. The plane earth-moon-sun three-body model is more consistent with an actual physical system than a non-self-consistent double-circle model, but still causes a certain degree of distortion compared with the result under the ephemeris model. For the low-amplitude dynamics alternative orbit, the instability index given under the ephemeris model is 1 order of magnitude higher than the result under the planar earth-moon-sun trisomy model. This indicates that the instability of the small amplitude unstable orbit in the actual terrestrial system is worse than the result in the simplified planar model. Briefly: compared with the actual earth-moon system, the double circle model or the planar earth-moon-sun model has certain difference in dynamics, so that the designed orbit has qualitative difference compared with the actual operation orbit of the detector, and the long-time task orbit can be designed by taking the orbit as a basic solution and is limited.
Disclosure of Invention
The technical problem to be solved by the invention is to provide a method for designing a track near a Earth-moon triangle translation point, which can design a task track which can be maintained for a long time from any moment, has higher precision of a adopted semi-analytical solution, is suitable for two triangle translation points, and is easy for a track designer to directly call.
In order to solve the technical problem, the invention provides a method for designing a track near a moon-earth triangle translation point, which comprises the following steps:
(1) constructing a semi-analytical solution approximation of the motion near the earth-moon triangle translational points L4 and L5 under the ephemeris model;
(2) designing a task track which is maintained for a certain time near earth-moon triangle translation points L4 and L5 under an ephemeris model;
(3) and outputting a track design result, namely generating the position and the speed vector of the detector under the geocentric celestial coordinate system and the earth-moon instantaneous convergence coordinate system at certain time intervals.
Preferably, in the step (1), constructing a semi-analytic solution approximation of the motion near the earth-moon trigonometric translation points L4 and L5 under the ephemeris model specifically includes the following steps:
(11) establishing a motion equation under a geocentric celestial coordinate system;
(12) aiming at the small amplitude dynamics replacement orbit, the specific processing comprises the following steps:
(a) constructing a quasi-periodic orbit which is maintained for a certain time by adopting a numerical method of multi-point target shooting through a numerical iteration process, wherein the initial value of iteration is any orbit type near a triangular translation point of a circular restrictive three-body problem and needs small-amplitude motion, and before numerical improvement, the orbit is transferred from a convergence coordinate system X-Y-Z to a geocentric celestial coordinate system X-Y-Z;
(b) aiming at the quasi-periodic orbit constructed in the step (a), analyzing the frequencies by adopting a Fast Fourier Transform (FFT) mode, and only extracting specific frequency items and coefficients thereof aiming at the frequencies
Figure BDA0003226558070000021
Wherein ω is1～ω4Is four fundamental frequencies related to the relative movement of three, namely, sun, earth and moon, and the four fundamental frequencies are kept unchanged no matter what orbit is in the earth-moon system, and the extracted terms of the specific frequencies are represented together in the following form
Figure BDA0003226558070000022
Wherein
Figure BDA0003226558070000023
(1) Compared with the orbit under the circular restrictive trisomy problem, the formula is a better dynamic approximation of the real motion under an ephemeris model;
(c) repeating the steps (a) and (b) to gradually improve the frequency combination and the corresponding coefficient in the formula (1) by taking the formula as an initial guess value, and repeating the steps until the coefficient change amount corresponding to each frequency item is smaller than a preset value epsilon under a certain frequency item combination, namely, the coefficient change amount is satisfied
Figure BDA0003226558070000024
Wherein the superscript (m) represents the result of the mth iteration, and at this time, the small-amplitude dynamics substitution orbit is considered to be obtained under a certain precision condition;
(d) after the steps are carried out, a small-amplitude dynamics substitution track is obtained, and a semi-analysis description of the movement near the small-amplitude dynamics substitution track is given by analyzing the change rule of the deviation of the movement near the small-amplitude dynamics substitution track relative to the dynamics substitution track along with time;
(13) aiming at the large-amplitude dynamics replacement orbit, the specific treatment comprises the following steps:
(a) firstly, the initial state quantity and the initial time t are measured under the geocentric convergence coordinate system x-y-z0Gridding in steps, i.e.
Figure BDA0003226558070000031
Figure BDA0003226558070000032
Figure BDA0003226558070000033
Figure BDA0003226558070000034
Figure BDA0003226558070000035
Figure BDA0003226558070000036
Figure BDA0003226558070000037
Wherein h is*Respectively representing the grid step lengths of corresponding quantities, and recording the set of all grid points as N0Wherein, an orbit is generated after any point is integrated, and a shorter integration time upper limit T is set0Go through set N0All tracks of (1), selected at integration time T0Then the points still can stay in the area near the triangle panning point, and the set of the points is recorded as N1It is obvious that
Figure BDA0003226558070000038
(b) For set N1Set a longer upper limit T of the integration time1Go through set N1Is selected at integration time T1Then the points still can stay in the area near the triangle panning point, and the set of the points is recorded as N2It is obvious that
Figure BDA0003226558070000039
(c) Repeating the step (b), gradually screening initial grid nodes by gradually increasing the upper limit of integration time, and finally obtaining nodes which still stay in the area near the triangular translational point within a million-year time scale as alternatives of a stable track;
(d) analyzing the frequency of the selected stable orbit through Fast Fourier Transform (FFT), and only keeping a term in the form of equation (1) as a semi-analytical solution approximation of the large-amplitude dynamics alternative orbit;
(e) like the case of the small-amplitude dynamics alternative orbit, a semi-analytical solution of the large-amplitude dynamics alternative orbit vicinity motion is constructed by analyzing the motion in its vicinity.
Preferably, in step (11), the relative motion is given by JPL DE 405 numerical ephemeris, taking into account the gravity of mass points of the earth, moon and sun; under the earth center celestial coordinate system, the motion equation of the small celestial body (detector) is
Figure BDA0003226558070000041
Wherein GmE,GmM,GmSGravitational constants of the earth, moon and sun, R, RM、RSRespectively are the position vectors of the small celestial body, the moon and the sun under the earth center celestial coordinate system.
Preferably, in the step (2), the step of designing the mission trajectory for maintaining a certain time around the earth-moon trigonometric translation points L4 and L5 under the ephemeris model specifically includes the following steps:
(21) aiming at the small amplitude dynamics replacement orbit, the specific processing comprises the following steps:
(a) selecting a proper time interval to generate a series of track nodes according to the constructed semi-analytical solution, wherein the total time interval of the whole track nodes needs to meet the task time requirement;
(b) combining possible task boundary constraints, adopting a multipoint target shooting method, and performing numerical improvement under an ephemeris model to generate a more accurate target orbit;
(22) and generating an initial orbit value and performing numerical integration on the initial epoch selected according to the constructed semi-analytical solution aiming at the large-amplitude dynamics alternative orbit so as to design a stable dynamics alternative orbit or an orbit nearby the stable dynamics alternative orbit.
The invention has the beneficial effects that: (1) the adopted ephemeris model is more accurate, so that the designed task orbit has higher precision and can meet the task precision requirement; (2) the high-precision semi-analytical solution is benefited, and a task track which is maintained for a long time can be designed from any time by combining a numerical method; (3) the adopted semi-analytical solution has higher precision and is popularized to two triangular translation points, and the solution is easy to be directly called by a rail designer through comparatively systematic arrangement.
Drawings
FIG. 1 is a schematic flow chart of the method of the present invention.
Fig. 2 is a schematic diagram of an orbit image of a task orbit near a small-amplitude dynamic substitution orbit near a point L4 in a geocentric celestial coordinate system.
Fig. 3 is a schematic diagram of an orbit image of a mission orbit near a small-amplitude dynamic substitution orbit near a point L4 under a geodetic instantaneous convergence coordinate system.
FIG. 4 is a schematic diagram of an orbit image of a large-amplitude dynamic alternative orbit near the point L5 in a geocentric celestial coordinate system.
Fig. 5 is a schematic diagram of a track image of a dynamic alternative track with large amplitude near the point L5 designed by the invention under a lunar instantaneous convergence coordinate system.
Detailed Description
As shown in fig. 1, a method for designing a track near a moon triangle panning point includes the following steps:
(1) constructing a semi-analytical solution approximation of the motion near the moon trigonometric translation points L4 and L5 under the ephemeris model;
(2) designing a task orbit which is maintained for a certain time near the earth-moon triangle translational points L4 and L5 under the ephemeris model;
(3) and outputting a track design result, namely generating the position and the speed vector of the detector under the geocentric celestial coordinate system and the earth-moon instantaneous convergence coordinate system at certain time intervals.
For the dynamic alternative orbit of L4 point or L5 point, whether small amplitude or large amplitude, it can be expressed as follows under the geocentric celestial coordinate system:
Figure BDA0003226558070000051
wherein
Figure BDA0003226558070000052
Frequency omega1～ω4Is related to the relative movement of the three parts of the sun, the earth and the moon and keeps unchanged.
If the trajectory is a dynamic substitution trajectory with small amplitude, the trajectory in the vicinity thereof (deviation from the trajectory with small amplitude) can be expressed as follows:
Figure BDA0003226558070000053
where α, β are the offset amplitude parameters in and out of the white-road plane of the relatively small amplitude dynamic surrogate orbit, θ1、θ2Respectively, the argument corresponding to the amplitude alpha, beta.
If the trajectory is a dynamic substitution trajectory with large amplitude, its neighboring trajectories (deviations from the trajectory with large amplitude) can be expressed in the form:
Figure BDA0003226558070000061
where α, β are two amplitude parameters within the white track plane of the relatively large amplitude dynamic replacement trajectory for the trajectory, γ is an offset amplitude parameter outside the white track plane of the relatively large amplitude dynamic replacement trajectory for the trajectory, and θ1,θ2,θ3Respectively, the argument corresponding to the amplitude alpha, beta, gamma. The above form of semi-analytical solution forms the basis of the overall track design.
Example 1:
assuming that the starting epoch of the mission trajectory is 2021 year 3 month 21 day 01 hour 01: 32.0 seconds (greenwich mean time), the maintenance time of the constructed trajectory is 10 years, and it is necessary to construct a small amplitude dynamics near L4 point in place of the trajectory near the trajectory, the amplitude parameter in the above equation is set to 0.005, and β is set to 0.005.
Selecting 915 track nodes in 10-year tracks with the time interval of 4 days according to the semi-analytical solution; and (4) carrying out numerical iteration to design a task orbit by adopting a multipoint shooting method under a complete ephemeris model.
And outputting the designed track node data for analysis and drawing. Fig. 2 and 3 show images of the designed trajectory (maintained for 10 years) in the geocentric celestial coordinate system and the earth-moon instantaneous convergence coordinate system (projection of the trajectory on x-y, x-z, y-z planes and image in three-dimensional space), respectively, according to the output data. In the geocentric celestial coordinate system, the designed orbit near the point L4 has a height equivalent to the height of the moon orbit, and the average angular velocity of motion is the same as the orbital angular velocity of the moon, so the trajectory appears as an orbit around the earth (fig. 2). Under the earth-moon instantaneous convergence coordinate system, the designed track near the point L4 needs to keep the characteristics of moving around the point L4 and not diverge within the mission time (i.e. intersect with the earth-moon line or be more than 2 earth-moon distances away from the earth), so the track is displayed as a track around the point L4 (FIG. 3). The third trajectory shown in fig. 2 is suitable for a space task staying near the point of march L4.
Example 2:
assuming that the start epoch of the mission trajectory is 0.0 minutes 15 minutes at 22 days 3 and 9 months 2025, the maintenance time of the constructed trajectory is 10 years, and a large-amplitude dynamic replacement trajectory itself near the point L5 needs to be constructed, the amplitude parameter in the above equation is set to 0.0, and β is set to 0.0. And according to a semi-analytical solution, giving the position and the speed of the initial epoch moment detector under the geocentric celestial coordinate system, and generating a 10-year task orbit by integration.
And outputting the designed track node data for analysis and drawing. Fig. 4 and 5 show images of the designed trajectory (maintained for 10 years) in the geocentric celestial coordinate system and the earth-moon instantaneous convergence coordinate system (projection of the trajectory on x-y, x-z, y-z planes and image in three-dimensional space), respectively, according to the output data. In the geocentric celestial coordinate system, the height of the orbit near the L5 point is designed to be equivalent to the height of the lunar orbit, and the average angular velocity of motion is the same as the orbital angular velocity of the moon, so the trajectory appears as an orbit around the earth (fig. 4). Under the earth-moon instant convergence coordinate system, the designed track near the point L5 needs to keep the characteristics of moving around the point L5 and not diverge in the mission time (namely, the track intersects with the earth-moon connecting line or is more than 2 earth-moon distances away from the earth), so that the track is displayed as a track around the point L5 (fig. 5). The third trajectory shown in fig. 4 is suitable for a space task staying near the point of march L5.

Claims (3)

1. A method for designing a track near a three-triangle panning point in Earth and moon is characterized by comprising the following steps:
(1) constructing a semi-analytical solution approximation of the motion near the moon trigonometric translation points L4 and L5 under the ephemeris model; the method specifically comprises the following steps:
(11) establishing a motion equation under a geocentric celestial coordinate system;
(12) aiming at the small-amplitude dynamics replacement orbit, the specific processing comprises the following steps:
(a) constructing a quasi-periodic orbit which is maintained for a certain time by adopting a multi-point targeting numerical method through a numerical iteration process, wherein an initial value of iteration is any orbit type near a triangular translation point of a circular restrictive three-body problem and needs small-amplitude motion, and the orbit is transferred to an X-Y-Z earth-centered celestial coordinate system from a converging coordinate system X-Y-Z before numerical improvement;
(b) aiming at the quasi-periodic orbit constructed in the step (a), analyzing the frequencies by adopting a Fast Fourier Transform (FFT) mode, and only extracting specific frequency items and coefficients thereof aiming at the frequencies
Figure FDA0003591196210000011
Wherein omega1～ω4Is four fundamental frequencies related to the relative movement of three, namely, sun, earth and moon, no matter what type of orbit in the earth-moon system, the four fundamental frequencies are kept unchanged, and the extracted terms of the specific frequencies are represented together in the following form
Figure FDA0003591196210000012
Wherein
Figure FDA0003591196210000013
(1) Compared with the orbit under the circular restrictive trisomy problem, the formula is a better dynamic approximation of the real motion under the ephemeris model;
(c) repeating the steps (a) and (b) to gradually improve the frequency combination and the corresponding coefficient in the formula (1) by taking the formula as an initial guess value, and repeating the steps until the coefficient change amount corresponding to each frequency item is smaller than a preset value epsilon under a certain frequency item combination, namely, the coefficient change amount is satisfied
Figure FDA0003591196210000014
The superscript m represents the result of the mth iteration, and at the moment, a small-amplitude dynamic alternative orbit is obtained under a certain precision condition;
(d) after the steps are carried out, a small-amplitude dynamics substitution track is obtained, and a semi-analysis description of the movement near the small-amplitude dynamics substitution track is given by analyzing the change rule of the deviation of the movement near the small-amplitude dynamics substitution track relative to the dynamics substitution track along with time;
(13) aiming at the large-amplitude dynamics alternative orbit, the specific processing comprises the following steps:
(a) firstly, an initial state quantity and an initial time t are measured under a geocentric convergence coordinate system x-y-z0Gridding in steps, i.e.
Figure FDA0003591196210000021
Figure FDA0003591196210000022
Figure FDA0003591196210000023
Figure FDA0003591196210000024
Figure FDA0003591196210000025
Figure FDA0003591196210000026
Figure FDA0003591196210000027
Wherein h is*Respectively representing the grid step lengths of corresponding quantities, and recording the set of all grid points as N0Wherein, an orbit is generated after any point is integrated, and a shorter integration time upper limit T is set0Go through set N0Is selected at integration time T0Then, the points in the area near the triangle translational point can still stay, and the set of the points is recorded as N1It is obvious that
Figure FDA0003591196210000028
(b) For set N1Set a longer upper limit T of the integration time1Go through set N1All tracks of (1), selected at integration time T1Then the points still can stay in the area near the triangle panning point, and the set of the points is recorded as N2It is obvious that
Figure FDA0003591196210000029
(c) Repeating the step (b), gradually screening initial grid nodes by gradually increasing the upper limit of integration time, and finally obtaining nodes which still stay in the area near the triangular translational point within a million-year time scale as alternatives of a stable track;
(d) analyzing the frequency of the selected stable orbit through Fast Fourier Transform (FFT), and only keeping a term in the form of equation (1) as a semi-analytical solution approximation of the large-amplitude dynamics alternative orbit;
(e) constructing a semi-analytical solution of the motion near the large-amplitude dynamics alternative orbit by analyzing the motion near the small-amplitude dynamics alternative orbit, similar to the case of the small-amplitude dynamics alternative orbit;
(2) designing a task orbit which is maintained for a certain time near the earth-moon triangle translational points L4 and L5 under the ephemeris model;
(3) and outputting a track design result, namely generating the position and the speed vector of the detector under the geocentric celestial coordinate system and the earth-moon instantaneous convergence coordinate system at certain time intervals.
2. The method for designing orbits near the Earth-moon trigonometric panning points as claimed in claim 1, wherein in step (11), the gravity of mass points of the Earth, the moon and the sun is taken into account, and the relative motion is given by JPL DE 405 numerical ephemeris; under the earth center celestial coordinate system, the motion equation of the small celestial body is
Figure FDA0003591196210000031
Wherein GmE,GmM,GmSGravitational constants of the earth, moon and sun, R, RM、RSRespectively are the position vectors of the small celestial body, the moon and the sun under the earth center celestial coordinate system.
3. The method for designing an orbit around the earth-moon trigonometric panning point of claim 1, wherein in the step (2), the step of designing the orbit of the task which is maintained for a certain time around the earth-moon trigonometric panning points L4 and L5 under the ephemeris model specifically comprises the following steps:
(21) aiming at the small amplitude dynamics replacement orbit, the specific processing comprises the following steps:
(a) selecting a proper time interval to generate a series of track nodes according to the constructed semi-analytical solution, wherein the total time interval of the whole track nodes needs to meet the task time requirement;
(b) combining possible task boundary constraints, adopting a multipoint target shooting method, and performing numerical improvement under an ephemeris model to generate a more accurate target orbit;
(22) and generating an initial orbit value and performing numerical integration on the initial epoch selected according to the constructed semi-analytical solution aiming at the large-amplitude dynamics alternative orbit so as to design a stable dynamics alternative orbit or an orbit nearby the stable dynamics alternative orbit.
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