Lunar transfer robustness design based on lyapunov exponent

Abstract

The application discloses a design method of earth-moon periodic orbit robust maintenance point based on Lyapunov index, and particularly relates to the technical field of space, and establishes an equivalent position-velocity error model, converts the introduced initial velocity error into equivalent position error by using the equivalent position-velocity error model, and calculates the Lyapunov index of the initial velocity error of the earth-moon periodic orbit; according to the Lyapunov index of the initial position error of the earth-moon periodic orbit and the Lyapunov index of the initial velocity error of the earth-moon periodic orbit, a deep neural network is used to establish the convergence and divergence sphere of the initial position error and the initial velocity error of the earth-moon periodic orbit; according to the obtained convergence and divergence sphere of the initial position error and the initial velocity error of the earth-moon periodic orbit, a traversal method is used to design the robust periodic position maintenance point of the position error convergence, the velocity error convergence and the simultaneous convergence of the position and velocity error, and the error convergence axis is searched, so as to provide a reference for the control target of the orbit periodic position maintenance strategy.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of space technology, and in particular to a design method of robust points of a lunar periodic orbit based on Lyapunov exponents. BACKGROUND

[0002] The lunar periodic orbit is an important orbital resource for deep space exploration. The nominal lunar periodic orbit can periodically return to the nominal position, which is convenient for the development of deep space autonomous rendezvous mission. Due to the insufficient accuracy of deep space autonomous navigation, the initial state of the lunar probe has position error and velocity error, which causes the end state of the actual orbit to deviate from the nominal lunar periodic orbit, thereby destroying the periodic characteristics of the orbit. Therefore, it is necessary to design a robust position holding point according to the evolution behavior of the lunar periodic orbit, so that the end state converges within the initial error range.

[0003] However, the current design method of the lunar periodic orbit position keeping point does not meet the high-precision design requirements of robust lunar periodic orbit position keeping. The existing technology (Zhang R, Wang Y, Shi Y, Zhang C, Zhang H. Performance analysis of impulsive station-keeping strategies for cis-lunar orbits with the ephemeris model. Acta Astronautica. 2022, 198: 152-160.) considers the non-spherical gravity of the moon and the perturbation effect of solar pressure, calculates the position keeping point of the lunar NRHO orbit and DRO orbit, but ignores the influence of navigation orbit determination error and relativistic effect, and cannot guarantee the robustness of the position keeping point. The existing technology (Gurfil P. Milankovitch-Lyapunov Geostationary Satellite Stationkeeping. Journal of Guidance, Control, and Dynamics, 2024, 47(11): 2418-2425.) considers the navigation orbit determination position error and velocity error at the same time, designs the position keeping point of the orbit, but the position keeping point does not consider periodicity, and each time the position keeping control is executed, it needs to be recalculated. The existing technology (Gao C, Masdemont J, Gómez G, Yuan J. Low-thrust station-keeping control for lunar near rectilinear halo orbits. Celestial Mechanics and Dynamical Astronomy, 2023, 135(2): 14.) considers the initial position error and velocity error in a specific direction, and studies the lunar periodic orbit position keeping point based on the Taylor expansion method under the simplified dynamics model, but in actual tasks, the direction of the initial position and the initial velocity is random, so it cannot accurately describe the convergence direction of the position error and the velocity error, and cannot reflect the robustness of the position keeping point. SUMMARY

[0004] In view of the shortcomings of the existing deep space exploration periodic orbit robust position, the present application provides a lunar periodic orbit robust keeping point design method based on Lyapunov index, which solves the problems of low accuracy of the dynamics model considered by the existing lunar periodic orbit position keeping point design method, and the periodicity of the position keeping point, the convergence axis of the position error and the velocity error of the position keeping point cannot be solved.

[0005] In order to achieve the above object, the present application provides the following technical scheme: a design method of robust keeping points of earth-moon periodic orbit based on Lyapunov index, comprising the following steps:

[0006] 1) An equivalent position and velocity error model is established by using the initial position error Jacobi matrix and the initial velocity error Jacobi matrix of the earth-moon periodic orbit, the initial velocity error introduced is converted into equivalent position error by using the equivalent position and velocity error model, and the initial velocity error Lyapunov index of the earth-moon periodic orbit is calculated;

[0007] 2) According to the initial position error Lyapunov index and the initial velocity error Lyapunov index of the earth-moon periodic orbit, a deep neural network is used to establish the initial position error and initial velocity error convergence sphere of the earth-moon periodic orbit;

[0008] 3) According to the initial position error and initial velocity error convergence sphere of the earth-moon periodic orbit obtained in step 2), a robust periodic position keeping point with convergent position error, convergent velocity error and convergent position and velocity error is designed by using the traversal method, and an error convergence axis is searched, so as to provide a reference for the control target of the orbit periodic position keeping strategy.

[0009] Preferably, the step 1) specifically comprises:

[0010] 11) A probe running on an earth-moon space periodic orbit is taken as an object, an earth-moon rendezvous coordinate system XYZ is established, the coordinate origin is the earth-moon barycenter, the X vector is a unit vector of the earth-moon barycenter pointing to the moon, the Z direction is a unit vector of the momentum axis of the moon revolving around the earth, and the Y vector forms a right-hand relationship with the Z and X vectors;

[0011] 12) An earth-moon space dynamics model is established , an orbit description mode of the earth-moon space probe is established, wherein the position vector of the earth-moon space probe in the earth-moon rendezvous coordinate system XYZ is , and the velocity vector is ;

[0012] 13) It is assumed that the position vector of the earth-moon space probe at is , the velocity vector is , and the period of the probe orbit is ;

[0013] 14) The initial position error Lyapunov index of the earth-moon periodic orbit is calculated; the terminal position of the nominal earth-moon space periodic orbit after one orbit period is as follows:

[0014] ;

[0015] In the periodic orbit The initial position error is introduced in the state at the moment , the initial position error Affects the actual end position after the next orbit period ; As follows:

[0016] ;

[0017] The Lyapunov exponent of the initial position error of the Earth-Moon periodic orbit As follows:

[0018] ;

[0019] 15) Calculate the Jacobi matrix of the initial position error and the velocity error of the Earth-Moon periodic orbit respectively; first, introduce three-axis initial position error , , in the Earth-Moon conjunction coordinate system XYZ, calculate the Jacobi matrix of the initial position error of the Earth-Moon periodic orbit as follows:

[0020] ;

[0021] In the formula,

[0022] ;

[0023] Wherein, represents the error of the initial position in the X-axis direction, represents the error of the initial position in the Y-axis direction, represents the error of the initial position in the Z-axis direction.

[0024] Then, introduce three-axis initial velocity error , , in the Earth-Moon conjunction coordinate system, calculate the Jacobi matrix of the initial velocity error of the Earth-Moon periodic orbit as follows:

[0025] ;

[0026] In the formula,

[0027] ;

[0028] Utilize the initial position error Jacobi matrix and the initial velocity error Jacobi matrix , establish an equivalent position-velocity error model, convert the initial velocity error into the initial position error, and solve the problem that the convergence and divergence of the initial velocity error cannot be evaluated; the equivalent position-velocity error model is as follows:

[0029] ;

[0030] 16) Calculate the initial velocity error Lyapunov index of the Earth-moon periodic orbit; in the periodic orbit introduce the initial velocity error into the state at the moment , convert the initial velocity error into the equivalent position error by using the equivalent position-velocity error model, and calculate the actual end position after one orbit period under the influence of the initial velocity error as follows:

[0031] ;

[0032] The initial velocity error Lyapunov index of the Earth-moon periodic orbit is as follows:

[0033] .

[0034] Preferably, the step 2) specifically comprises:

[0035] 21) Discretize the Earth-moon periodic orbit according to the orbit period to obtain multiple groups of initial states , wherein, is the time, is the initial position, is the initial velocity; represents the group, and each group has a corresponding position error Lyapunov index and a velocity error Lyapunov index to distinguish each group of initial states;

[0036] 22) According to the actual navigation orbit determination capability of the Earth-moon space probe, calculate the navigation orbit determination initial position error and the initial velocity error of the Earth-moon space probe;

[0037] 23) Discretize the initial position error and the initial velocity error in the direction to obtain multiple groups of states with errors , the position error Lyapunov exponent and the velocity error Lyapunov exponent , the sign of the position error Lyapunov exponent is obtained and the sign of the velocity error Lyapunov exponent wherein, is the initial position error, is the initial velocity error;

[0038] 24) taking time , initial position , initial velocity , initial position error , initial velocity error as inputs, the signs of the position error and velocity error Lyapunov exponents and as outputs, a training sample is formed, and the specific form is as follows:

[0039] ;

[0040] 25) discretizing the initial position error vector and the initial velocity error vector in the direction, using the trained radial basis neural network to predict the sign of the Lyapunov exponent, and using the same to establish the initial position error convergence sphere and the initial velocity error convergence sphere;

[0041] On the initial position error convergence sphere, the spherical point is , and the labeled information is the sign of the position error Lyapunov exponent ; on the initial velocity error convergence sphere, the spherical point is , and the labeled information is the sign of the velocity error Lyapunov exponent .

[0042] Preferably, the step 3) specifically comprises:

[0043] 31) using the position error and velocity error convergence spheres based on the Lyapunov exponent to analyze the position and velocity error; the signs of the Lyapunov exponents on the position error and velocity error convergence spheres and respectively represent the convergence direction of the initial position error or the velocity error; the judgment condition is as follows:

[0044] ;

[0045] 32) according to the error convergence direction on the position error convergence sphere, using the traversal method to design a robust periodic position keeping point of the position error convergence ; where the search variable of the exhaustive search is time , the position of the periodic position preserving point , the velocity of the periodic position preserving point ; the robust periodic position preserving point with Lyapunov exponent being negative for arbitrary initial position error is searched by exhaustive search , the objective function of the exhaustive search is as follows:

[0046] ;

[0047] 33) According to the error convergence direction on the error convergence sphere of the velocity error, the robust periodic position preserving point with velocity error convergence is designed by using the exhaustive search ; where the search variable of the exhaustive search is time , the position of the periodic position preserving point , the velocity of the periodic position preserving point ; the robust periodic position preserving point with Lyapunov exponent being negative for arbitrary initial velocity error is searched by exhaustive search , the objective function of the exhaustive search is as follows:

[0048] ;

[0049] 34) According to the error convergence direction on the error convergence sphere of the position error and the velocity error, the robust periodic position preserving point with error convergence is designed by using the exhaustive search ; where the search variable of the exhaustive search is time , the position of the periodic position preserving point , the velocity of the periodic position preserving point ; the robust periodic position preserving point with Lyapunov exponent being negative for arbitrary initial position error and velocity error is searched by exhaustive search , the objective function of the exhaustive search is as follows:

[0050] ;

[0051] 35) If there is no robust periodic position preserving point with Lyapunov exponent being negative for all error directions, the periodic position preserving point is searched by using the exhaustive search and the error convergence sphere The position error and the velocity error convergence direction of the periodic position preserving point are searched. The initial conditions of the exhaustive search are time , the position of the periodic position preserving point , the velocity of the periodic position preserving point , and the search variable is a three-axis orthogonal unit vector , represent any three unit vectors in the lunar-synodic coordinate system, and the three vectors are mutually orthogonal and satisfy the following conditions ;

[0052] The corresponding initial position error or initial velocity error is as follows:

[0053] ;

[0054] When the unit vector corresponding Lyapunov index is negative, the axis is the error convergence axis, otherwise, it is the error divergence axis, and the three-axis orthogonal unit vectors are searched by traversal: The corresponding position error or velocity error Lyapunov index reaches the maximum, and the target function of the traversal search is as follows:

[0055] .

[0056] The present application is high-precision dynamics model, improve the design accuracy of the earth-moon periodic orbit position keeping point, through the equivalent position error model of velocity error, the initial position error and the initial velocity error convergence is evaluated; Lyapunov index is predicted by using deep neural network, realize the fast generation of position error convergence sphere and velocity error convergence sphere, obtain the periodic position keeping point of non-specific direction position and velocity error convergence axis, compared with prior art, the design method of the present application accurately describes the convergence direction of position error and velocity error, and reflects the robustness of the position keeping point. BRIEF DESCRIPTION OF DRAWINGS

[0057] Figure 1 The earth-moon conjunction coordinate system schematic diagram provided by the present application is shown in the figure;

[0058] Figure 2 The earth-moon periodic orbit NRHO schematic diagram provided by the present application is shown in the figure;

[0059] Figure 3 The position and velocity error convergence schematic diagram provided by the present application is shown in the figure;

[0060] Figure 4 The NRHO orbit position error convergence sphere schematic diagram provided by the present application is shown in the figure;

[0061] Figure 5 The NRHO orbit velocity error convergence sphere schematic diagram provided by the present application is shown in the figure;

[0062] Figure 6 The end error of the position keeping point of the present application without considering the initial position and velocity error. DETAILED DESCRIPTION

[0063] The following embodiments of the present application are explained by way of specific examples, and other advantages and effects of the present application will be readily understood by those skilled in the art from the disclosure. Obviously, the described embodiments are only part of the embodiments of the present application, but not all the embodiments. Based on the embodiments of the present application, all other embodiments obtained by those skilled in the art without creative work fall within the scope of the present application.

[0064] The present application provides a design method of robust position keeping points of earth-moon periodic orbit based on Lyapunov index, comprising the following steps:

[0065] 1) An equivalent position and velocity error model is established by using the initial position error Jacobi matrix and the initial velocity error Jacobi matrix of the earth-moon periodic orbit, the initial velocity error introduced is converted into equivalent position error by using the equivalent position and velocity error model, and the initial velocity error Lyapunov index of the earth-moon periodic orbit is calculated;

[0066] Specifically, it comprises:

[0067] 11) A probe running in an earth-moon space periodic orbit is taken as an object, an earth-moon rendezvous coordinate system XYZ is established, the coordinate origin is the earth-moon barycenter, the X vector is a unit vector pointing from the earth-moon barycenter to the moon, the Z direction is a unit vector of the moon's orbit around the earth, and the Y vector forms a right-hand relationship with the Z and X vectors. The earth-moon rendezvous coordinate system is shown in Figure 1 .

[0068] 12) An earth-moon space dynamics model is established , and an earth-moon space probe orbit description method is established, wherein the position vector of the earth-moon space probe in the earth-moon rendezvous coordinate system is , and the velocity vector is .

[0069] The high-precision earth-moon space dynamics model is considered as follows:

[0070] ;

[0071] In the formula, , represent the gravitational constants of the earth and the moon, respectively; and represent the distances of the earth-moon probe from the earth and the moon, respectively; is the angular velocity of the earth and the moon rotating around the earth-moon barycenter; , are the positions of the earth and the moon in the X axis in the earth-moon rendezvous coordinate system; , , , The perturbations are the earth non-spherical gravity, the sun gravity, the sun light pressure perturbation, the relativistic effect, etc.

[0072] 13) Suppose that the position vector of the earth-moon space probe at is , the velocity vector is , and the period of the probe orbit is . Consider the earth-moon near rectilinear halo orbit NRHO with the south direction height of 75000 km, and its initial state is as follows:

[0073] (1);

[0074] The periodic trajectory of the NRHO orbit is shown in Figure 2 .

[0075] 14) Calculate the initial position error Lyapunov exponent of the earth-moon periodic orbit. The nominal earth-moon space periodic orbit at the end of one orbit period is as follows:

[0076] (2);

[0077] Introduce the initial position error in the state of the periodic orbit , and the actual end position after one orbit period under the influence of the initial position error is as follows:

[0078] (3);

[0079] Then the earth-moon periodic orbit Lyapunov exponent facing the initial position error is as follows:

[0080] (4); 15) Calculate the Jacobi matrix of the earth-moon periodic orbit initial position error and velocity error respectively. First, introduce the three-axis initial position error

[0081] , , in the earth-moon conjunction coordinate system, and the Jacobi matrix of the earth-moon periodic orbit initial position error is calculated as follows:

[0082] (5); In the formula,

[0083]

[0084] (6);

[0085] ​​​Wherein, Error representing the initial position in the X-axis direction, Error representing the initial position in the Y-axis direction, Error representing the initial position in the Z-axis direction;

[0086] Then, the three-axis initial velocity error is introduced under the geocentric lunar coordinate system 、 、 The initial velocity error Jacobi matrix of the geocentric lunar periodic orbit is calculated as follows:

[0087] (7);

[0088] In the formula,

[0089] (8);

[0090] Using the above initial position error and initial velocity error Jacobi matrix, the application establishes an equivalent position and velocity error model, converts the initial velocity error into an initial position error, and solves the problem that the convergence and divergence of the initial velocity error cannot be evaluated. The equivalent position and velocity error model is as follows:

[0091] (9);

[0092] 16) Calculate the Lyapunov index of the initial velocity error of the geocentric lunar periodic orbit. In the state of the periodic orbit The initial velocity error is introduced, which is converted into an equivalent position error by using the equivalent position and velocity error model, and the actual end position after one orbit period under the influence of the initial velocity error (equivalent position error) is calculated as follows:

[0093] (10);

[0094] The Lyapunov index of the initial velocity error of the geocentric lunar periodic orbit is as follows:

[0095] (11);

[0096] 2) According to the Lyapunov index of the initial position error of the geocentric lunar periodic orbit and the Lyapunov index of the initial velocity error of the geocentric lunar periodic orbit, a deep neural network is used to establish a convergence and divergence sphere of the initial position error and the initial velocity error for the geocentric lunar periodic orbit;

[0097] Specifically includes:

[0098] 21) Discretize the initial states of the LEO according to the orbital period .

[0099] 22) Calculate the initial position error and initial velocity error of the navigation orbit determination of the LEO according to the actual navigation orbit determination capability of the LEO . .

[0100] The considered initial position error is and the initial velocity error is .

[0101] 23) Discretize the initial position error and the initial velocity error in the direction, obtain multiple sets of states with errors , calculate the position error Lyapunov index and the velocity error Lyapunov index , obtain the signs of the Lyapunov indexes and .

[0102] 24) Take the time , the initial position , the initial velocity , the initial position error , the initial velocity error as the input, and the signs of the position error and velocity error Lyapunov indexes and as the output, form a training sample, and the specific form is as follows:

[0103] (12);

[0104] 25) Discretize the initial position error vector and the initial velocity error vector in the direction, use the trained radial basis neural network to predict the signs of the Lyapunov indexes, and establish the initial position error convergence sphere and the initial velocity error convergence sphere based on the prediction.

[0105] On the initial position error convergence sphere, the spherical point is , and the labeled information is the sign of the position error Lyapunov index . On the initial velocity error convergence sphere, the spherical point is , and the labeled information is the sign of the velocity error Lyapunov index .

[0106] The deep neural network considers a pattern recognition network with 4 hidden layers and 32 nodes per layer, and the prediction error of the signs of the position error and velocity error Lyapunov indexes after training is 0.01286%.

[0107] 3) According to the initial position error and initial velocity error convergence sphere of the lunar periodic orbit obtained in step 2), a robust periodic position keeping point with position error convergence, velocity error convergence and simultaneous convergence of position and velocity error is designed by using the traversal method, and an error convergence axis is searched, so as to provide a reference for the control target of the periodic position keeping strategy.

[0108] Specifically, it comprises:

[0109] 31) The position and velocity error convergence sphere based on Lyapunov index is used for position and velocity error analysis. The signs of Lyapunov index on the position and velocity error convergence sphere and respectively represent the convergence direction of the initial position error or velocity error. The judgment condition is as follows:

[0110] (13);

[0111] 32) According to the error convergence direction on the position error convergence sphere, a robust periodic position keeping point with position error convergence is designed by using the traversal method . Wherein, the search variables of the traversal method are time , position of the periodic position keeping point , and velocity of the periodic position keeping point . Through the traversal search, the robust periodic position keeping point with Lyapunov index being negative under any initial position error is searched , and the objective function of the traversal search is as follows:

[0112] (14);

[0113] 33) According to the error convergence direction on the velocity error convergence sphere, a robust periodic position keeping point with velocity error convergence is designed by using the traversal method . Wherein, the search variables of the traversal method are time , position of the periodic position keeping point , and velocity of the periodic position keeping point . Through the traversal search, the robust periodic position keeping point with Lyapunov index being negative under any initial velocity error is searched , and the objective function of the traversal search is as follows:

[0114] (15);

[0115] 34) According to the error convergence direction on the position and velocity error convergence sphere, a robust periodic position keeping point with simultaneous convergence of position and velocity error is designed by using the traversal method . Wherein, the search variables of the traversal method are time , position of the periodic position keeping point , the velocity of the periodic position keeping point . The position and velocity errors converge as shown in Figure 3 . By exhaustive search for a robust periodic position keeping point with Lyapunov exponents being negative for any initial position error and velocity error , the objective function of the exhaustive search is as follows

[0116] (16);

[0117] 35) If there is no robust periodic position keeping point with Lyapunov exponents being negative for all error directions, then the periodic position keeping point is searched by using the exhaustive method and the error convergent and divergent sphere to search the convergent direction of the position error and the velocity error. The initial conditions of the exhaustive method are time , the position of the periodic position keeping point , the velocity of the periodic position keeping point , and the search variable is a three-axis orthogonal unit vector , representing any three unit vectors in the lunar-synodic coordinate system, the three vectors are mutually orthogonal and satisfy the following conditions ;

[0118] The corresponding initial position error or initial velocity error is as follows

[0119] (17);

[0120] When the Lyapunov exponent corresponding to the unit vector is negative, the axis is the error convergent axis; otherwise, it is the error divergent axis. By exhaustive search, the number of the three-axis orthogonal unit vectors corresponding to the negative Lyapunov exponents of the position error or the velocity error is maximized, and the objective function of the exhaustive search is as follows

[0121] (18);

[0122] Suppose that the periodic position keeping point of the lunar-synodic NRHO orbit is as follows:

[0123] (19);

[0124] The position error convergent and divergent sphere of the periodic position keeping point is shown in Figure 4 , and the velocity error convergent and divergent sphere is shown in Figure 5The blue area represents the error divergence point, and the red area represents the error convergence point. The green, cyan, and black coordinate axes represent the X, Y, and Z axes of the lunar-synodic coordinate system, respectively. It can be seen that the periodic position keeping point is not negative in all position and velocity error Lyapunov exponents. Two orthogonal error convergence axes As follows:

[0125] (20);

[0126] and the convergence axes are orthogonal to each other As follows:

[0127] (21);

[0128] The design accuracy of the periodic position keeping point of the application is illustrated by comparison. Based on the error convergence direction of the orthogonal error convergence axis ring surface shown in formula (20), the end position error of the NRHO orbit after one period is calculated by considering the periodic position keeping point shown in formula (19). Meanwhile, the same vector direction is considered, and the end position error of the orbit after one period is calculated by using the existing periodic position keeping point design method which does not consider the initial position error and velocity error. The end position error distribution of the two methods is shown in Figure 6 The red line is the average end position error.

[0129] The results show that, compared with the traditional method, the end position error of the periodic position keeping point obtained by the application in the error convergence axis direction is in the order of kilometers, and the error average is 6.818 km; the end position error of the periodic position keeping point designed by the existing method in the error convergence axis direction is in the order of hundreds of kilometers, and the error average is 151.631 km; the position error and velocity error convergence axes are expanded from three fixed directions to all directions.

[0130] Although the application has been fully described in the foregoing by general statement and specific embodiments, some modifications or improvements can be made on the basis of the application, which is obvious to those skilled in the art. Therefore, these modifications or improvements made on the basis of not deviating from the spirit of the application are within the scope of protection claimed by the application.

Claims

1. A robust maintenance point design method for Earth-Moon periodic orbits based on the Lyapunov index, characterized by: The steps include the following: 1) Using the Jacobi matrix of initial position error and initial velocity error of the Earth-Moon periodic orbit, establish an equivalent position and velocity error model. Use the equivalent position and velocity error model to transform the introduced initial velocity error into an equivalent position error, and calculate the Lyapunov exponent of the initial velocity error of the Earth-Moon periodic orbit. 2) Based on the Lyapunov exponent of the initial position error and the Lyapunov exponent of the initial velocity error of the Earth-Moon periodic orbit, a convergence and divergence sphere for the initial position error and initial velocity error of the Earth-Moon periodic orbit is established using a deep neural network. 3) Based on the convergence and divergence spheres of the initial position error and initial velocity error of the Earth-Moon periodic orbit obtained in step 2), robust periodic position holding points with convergence of position error, velocity error, and simultaneous convergence of position and velocity errors are designed using the ergonomic method. The error convergence axis is searched to provide a reference for the control objective of the orbital periodic position holding strategy. Step 1) specifically includes: 11) Taking the probe operating in the Earth-Moon periodic orbit as the object, establish the Earth-Moon synodic coordinate system XYZ, with the origin being the Earth-Moon barycenter, the X vector being the unit vector pointing from the Earth-Moon barycenter to the Moon; the Z direction being the unit vector of the Moon's revolution around the Earth; and the Y vector forming a right-handed relationship with the Z and X vectors. 12) Establish a dynamic model of the Earth-Moon space. A method for describing the orbit of the Earth-Moon space probe is established, wherein the position vector of the Earth-Moon space probe in the Earth-Moon rendezvous coordinate system XYZ is: The velocity vector is ; 13) Assuming the Earth-Moon space probe is in The position vector at time is The velocity vector is The period of the probe's orbit is ; 14) Calculate the Lyapunov exponent of the initial position error for the nominal Earth-Moon periodic orbit; the nominal Earth-Moon periodic orbit is defined as one orbital period. The end position as follows: ； In periodic orbit Introducing initial position error into the state at time step In the initial position error Influence the next orbital period The actual end position after as follows: ； The initial position error of the Earth-Moon periodic orbit is the Lyapunov exponent. as follows: ； 15) Calculate the Jacobi matrices for the initial position and velocity errors of the Earth-Moon periodic orbits, respectively; first, introduce the three-axis initial position errors in the Earth-Moon rendezvous coordinate system XYZ. , , The Jacobi matrix for calculating the initial position error of the Earth-Moon periodic orbit is as follows: ； In the formula, ； in, This represents the error of the initial position in the X-axis direction. Represents the error of the initial position in the Y-axis direction. This represents the error of the initial position in the Z-axis direction; Then, triaxial initial velocity errors are introduced in the Earth-Moon rendezvous coordinate system. , , The Jacobi matrix for calculating the initial velocity error of the Earth-Moon periodic orbit is as follows: ； In the formula, ； Using the initial position error Jacobi matrix and the initial velocity error Jacobi matrix An equivalent position-velocity error model is established to transform the initial velocity error into an initial position error, thus solving the problem of the inability to assess the convergence and divergence of the initial velocity error. The equivalent position-velocity error model is as follows: ； 16) Calculate the Lyapunov exponent for the initial velocity error of the Earth-Moon periodic orbit; in the periodic orbit Introducing initial velocity error into the state at time step The initial velocity error is calculated using an equivalent position-velocity error model. Convert to equivalent position error Calculate the impact of this initial velocity error on the next orbital period. The actual end position after as follows: ； The initial velocity error of the Earth-Moon periodic orbit is calculated using the Lyapunov exponent. as follows: 。 2. The method for designing robust maintenance points for Earth-Moon periodic orbits based on the Lyapunov index as described in claim 1, characterized in that: Step 2) specifically includes: 21) Discretize the Earth-Moon periodic orbit based on its orbital period to obtain multiple sets of initial states. ,in, For time, The initial position, The initial velocity; The groups are represented, and each group has a corresponding Lyapunov index for positional error. And velocity error Lyapunov exponent To distinguish the initial states of each group; 22) Based on the actual navigation and orbit determination capabilities of the Earth-Moon space probe, calculate the initial position error of the Earth-Moon space probe's navigation and orbit determination. and initial velocity error ; 23) Initial position error and initial velocity error Discretize the direction to obtain multiple sets of states with errors. Calculate the Lyapunov exponent for position error. And velocity error Lyapunov exponent Obtain the sign of the Lyapunov exponent for position error. The sign of the Lyapunov exponent for velocity error ,in, This is the initial position error. This refers to the initial velocity error; 24) By time Initial position Initial velocity Initial position error Initial velocity error For input, the signs of the Lyapunov exponents for position error and velocity error are... and To produce the output, training samples are generated, in the following form: ； 25) Discretize the initial position error vector and the initial velocity error vector in the direction, use the trained radial basis neural network to predict the sign of the Lyapunov exponent, and use this to establish the initial position error convergence sphere and the initial velocity error convergence sphere. On the initial position error convergence sphere, the spherical point is... The information marked is the symbol of the Lyapunov exponent for positional error. On the initial velocity error convergence sphere, the spherical point is... The information marked is the symbol of the Lyapunov exponent for velocity error. .

3. The method for designing robust maintenance points for Earth-Moon periodic orbits based on the Lyapunov index as described in claim 2, characterized in that: Step 3) specifically includes: 31) Position and velocity error analysis using a convergence sphere based on Lyapunov exponents; the sign of the Lyapunov exponents on the position and velocity error convergence sphere. and These represent the convergence directions of the initial position error or velocity error, respectively; the judgment conditions are as follows: ； 32) Based on the error convergence direction on the position error convergence sphere, design robust periodic position holding points for position error convergence using the ergodic method. In this traversal method, the search variable is time. Position of the periodic position , periodic position-preserving speed ; By traversing and searching, robust periodic position-holding points with a consistently negative Lyapunov exponent under any initial position error are found. The objective function for the traversal search is as follows: ； 33) Based on the error convergence direction on the velocity error convergence sphere, design robust periodic position holding points for velocity error convergence using the ergodic method. In this traversal method, the search variable is time. Position of the periodic position , periodic position-preserving speed ; By traversing and searching, robust periodic position-holding points with a consistently negative Lyapunov exponent under any initial velocity error are found. The objective function for the traversal search is as follows: ； 34) Based on the convergence direction of the position and velocity errors on the convergence sphere, design robust periodic position holding points where the errors converge simultaneously using the ergodic method. In this traversal method, the search variable is time. Position of the periodic position , periodic position-preserving speed By traversing and searching for robust periodic position-holding points where the Lyapunov exponent is always negative under arbitrary initial position and velocity errors. The objective function for the traversal search is as follows: ； 35) If there is no robust periodic position-holding point where the Lyapunov exponent is negative in all error directions, then the periodic position-holding point can be searched using the ergodic method and the error convergence / divergence sphere. The convergence direction of position and velocity errors; the initial condition of the ergodic method is time. Position of the periodic position , periodic position-preserving speed The search variable is a triaxial orthogonal unit vector. , Let represent any three unit vectors in the Earth-Moon synodic coordinate system. These three vectors are mutually orthogonal and satisfy the following condition: ; The corresponding initial position error or initial velocity error is as follows: ； When the Lyapunov exponent corresponding to the unit vector is negative, the axis is the error convergence axis; otherwise, it is the error divergence axis. A search is performed to orthogonalize the three axes with unit vectors. The number of positions or velocity errors with negative Lyapunov exponents reaches its maximum. The objective function for the traversal search is as follows: 。

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