Earth-moon periodic orbit robust retention point design method based on Lyapunov exponent

Abstract

The invention discloses an earth-moon periodic orbit robust retention point design method based on a Lyapunov exponent, and particularly relates to the technical field of space, an equivalent position speed error model is established, an introduced initial speed error is converted into an equivalent position error by using the equivalent position speed error model, and the equivalent position error is calculated. Calculating an initial velocity error Lyapunov index of the earth-moon periodic orbit; according to the earth-moon periodic orbit initial position error Lyapunov index and the earth-moon periodic orbit initial speed error Lyapunov index, establishing an earth-moon periodic orbit-oriented initial position error and initial speed error convergence ball by using a deep neural network; and according to the obtained earth-moon-oriented periodic orbit initial position error and initial speed error convergence ball, designing a robust periodic position holding point with position error convergence, speed error convergence and position and speed error simultaneous convergence by using a traversal method, and searching an error convergence axis, thereby providing a reference for a control target of an orbit periodic position holding strategy.

Description

Technical Field

[0001] This invention relates to the field of space technology, and more specifically to a robust maintenance point design method for Earth-Moon periodic orbits based on the Lyapunov index. Background Technology

[0002] The Earth-Moon periodic orbit is a crucial orbital resource for deep space exploration. A nominal Earth-Moon periodic orbit allows for periodic return to its nominal position, facilitating autonomous rendezvous missions in deep space. However, due to insufficient accuracy in autonomous navigation during deep space exploration, the initial state of an Earth-Moon probe contains position and velocity errors, causing the terminal state of the actual orbit to deviate from the nominal Earth-Moon periodic orbit and disrupting its periodic characteristics. Therefore, robust position-holding points need to be designed based on the evolutionary behavior of the Earth-Moon periodic orbit to ensure that the terminal state converges within the initial error range.

[0003] However, current methods for designing position-keeping points for Earth-Moon periodic orbits do not meet the high-precision design requirements for robust Earth-Moon periodic orbit position-keeping. Existing techniques (Zhang R, Wang Y, Shi Y, Zhang C, Zhang H. Performance analysis of impulsive station-keeping strategies for cis-lunarorbits with the ephemeris model. Acta Astronautica. 2022, 198: 152-160.) consider the non-spherical gravity of the Moon and the pressure perturbation effect of solar radiation to calculate the position-keeping points for Earth-Moon NRHO and DRO orbits, but neglect the influence of navigation and orbit determination errors and relativistic effects, thus failing to guarantee the robustness of the position-keeping points. Existing techniques (Gurfil P. Milankovitch–Lyapunov Geostationary Satellite Stationkeeping. Journal of Guidance, Control, and Dynamics, 2024, 47(11): 2418-2425.) simultaneously consider both navigation and orbit determination position and velocity errors, designing position-keeping points for the orbit. However, these position-keeping points do not consider periodicity, requiring recalculation for each execution of position-keeping control. Existing techniques (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.) consider initial position and velocity errors in specific directions, studying position-keeping points for Earth-Moon periodic orbits under a simplified dynamic model based on the Taylor expansion method. However, in actual missions, the directions of initial position and initial velocity are random, thus failing to accurately describe the convergence direction of position and velocity errors, and even more so failing to reflect the robustness of the position-keeping points. Summary of the Invention

[0004] To address the shortcomings of existing robust position design methods for periodic orbits in deep space exploration, this invention provides a robust position-holding point design method for Earth-Moon periodic orbits based on the Lyapunov exponent. This method solves the problems of low accuracy of the dynamic model considered in existing Earth-Moon periodic orbit position-holding point design methods, failure to consider the periodicity of the position-holding point, and inability to solve for the convergence axis of the position error and velocity error of the position-holding point.

[0005] To achieve the above objectives, the present invention provides the following technical solution: a robust maintenance point design method for Earth-Moon periodic orbits based on the Lyapunov index, comprising the following steps: 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), a robust periodic position holding point with convergence of position error, velocity error, and simultaneous convergence of position and velocity errors is 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.

[0006] Preferably, 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: .

[0007] Preferably, 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. .

[0008] Preferably, 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: .

[0009] This invention addresses high-precision dynamic models and improves the design accuracy of position-holding points in the Earth-Moon periodic orbit. By proposing an equivalent position error model for velocity errors, it achieves the assessment of the convergence of initial position and velocity errors. Furthermore, by utilizing deep neural networks to predict Lyapunov exponents, it enables the rapid generation of position and velocity error convergence spheres, obtaining periodic position-holding points with convergence axes for non-specific directions of position and velocity errors. Compared to existing technologies, the design method of this invention accurately describes the convergence directions of position and velocity errors and reflects the robustness of the position-holding points. Attached Figure Description

[0010] Figure 1 A schematic diagram of the Earth-Moon rendezvous coordinate system provided by this invention; Figure 2A schematic diagram of the Earth-Moon periodic orbit NRHO provided by this invention; Figure 3 A schematic diagram of position velocity error convergence provided by the present invention; Figure 4 A schematic diagram of the convergence sphere for the NRHO orbital position error provided by this invention; Figure 5 A schematic diagram of the convergence sphere for NRHO orbital velocity error provided by this invention; Figure 6 This refers to the end-point error of the position holding point, which does not consider the initial position velocity error in this invention. Detailed Implementation

[0011] The following specific embodiments illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0012] This invention proposes a robust position-preserving point design method for Earth-Moon periodic orbits based on the Lyapunov index, comprising the following steps: 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. Specifically include: 11) Taking the probe operating in a periodic orbit around the Earth and Moon as the object, establish an Earth-Moon synodic coordinate system XYZ. The origin of the coordinate system is the Earth-Moon barycenter; the X vector is the unit vector pointing from the Earth-Moon barycenter to the Moon; the Z direction is the unit vector of the Moon's revolution around the Earth; the Y vector forms a right-handed relationship with the Z and X vectors. This Earth-Moon synodic coordinate system is as follows: Figure 1 As shown.

[0013] 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 is: The velocity vector is .

[0014] Consider the following high-precision Earth-Moon space dynamics model: ; In the formula, , These represent the gravitational constants of the Earth and the Moon, respectively. and These represent the distances of the Earth-Moon probe from the Earth and the Moon, respectively. The angular velocities of the Earth and Moon as they revolve around their Earth-Moon center of mass; , These represent the positions of the Earth and the Moon on the X-axis in the Earth-Moon synodic coordinate system. , , , These include perturbations caused by Earth's non-spherical gravity, solar gravity, solar radiation pressure, and relativistic effects.

[0015] 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 Consider the near-linear lunar halo orbit NRHO at an altitude of 75,000 km in the south, with the following initial state: (1); The periodic trajectory of the NRHO orbit is as follows Figure 2 As shown.

[0016] 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: (2); In periodic orbit Introducing initial position error into the state at time step The error in the initial position affects the next orbital period. The actual end position after as follows: (3); The Lyapunov exponent for the Earth-Moon periodic orbit with respect to initial position error. as follows: (4); 15) Calculate the Jacobi matrices for the initial position and velocity errors of the Earth-Moon periodic orbits, respectively. First, introduce the initial position errors along three axes in the Earth-Moon synodic coordinate system. , , The Jacobi matrix for calculating the initial position error of the Earth-Moon periodic orbit is as follows: (5); In the formula, (6); 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: (7); In the formula, (8); Using the Jacobi matrices of the initial position error and initial velocity error mentioned above, this invention establishes an equivalent position-velocity error model, transforming 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: (9); 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 equivalent position-velocity error model is used to transform it into an equivalent position error. Calculate the impact of this initial velocity error (equivalent position error) on the next orbital period. The actual end position after as follows: (10); The initial velocity error of the Earth-Moon periodic orbit is calculated using the Lyapunov exponent. as follows: (11); 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. Specifically include: 21) Discretize the Earth-Moon periodic orbit based on its orbital period to obtain multiple sets of initial states. .

[0017] 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 .

[0018] The navigation and orbit determination position error considered is Speed ​​error is .

[0019] 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 To obtain the symbol for the Lyapunov index and .

[0020] 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: (12); 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.

[0021] 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. .

[0022] The deep neural network considers a pattern recognition network with 4 hidden layers and 32 nodes per layer. After training, the prediction error of the Lyapunov exponential sign of the position error and velocity error is 0.01286%.

[0023] 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), a robust periodic position holding point with convergence of position error, velocity error, and simultaneous convergence of position and velocity errors is 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.

[0024] Specifically include: 31) Position and velocity error analysis is performed using a convergence sphere based on Lyapunov exponents for position and velocity errors. The sign of the Lyapunov exponents on the position and velocity error convergence sphere is shown. and These represent the convergence directions of the initial position error or velocity error, respectively. The judgment conditions are as follows: (13); 32) Based on the error convergence direction on the position error convergence sphere, this invention uses an ergodic method to design robust periodic position holding points for position error convergence. In this traversal method, the search variable is time. Position of the periodic position , periodic position-preserving speed By iterating through the search points where the Lyapunov exponent is always negative under any initial position error, robust periodic position holding points are obtained. The objective function for the traversal search is as follows: (14); 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: (15); 34) Based on the convergence direction of the position and velocity errors on the convergent sphere, design robust periodic position-holding points where the errors converge using the ergodic method. In this traversal method, the search variable is time. Position of the periodic position , periodic position-preserving speed The position and velocity errors converge as follows: Figure 3 As shown, robust periodic position-holding points with consistently negative Lyapunov exponents under arbitrary initial position and velocity errors are found through traversal searching. The objective function for the traversal search is as follows: (16); 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 for the traversal 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: (17); 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 using unit vectors. The number of negative Lyapunov exponents for corresponding position or velocity errors reaches its maximum. The objective function for the traversal search is as follows: (18); Assume the periodic position of the Earth-Moon NRHO orbit is maintained as follows: (19); The position error convergence sphere of the position holding point in this cycle is as follows: Figure 4 As shown, the velocity error convergence sphere is as follows Figure 5 As shown. For the position error divergence sphere and velocity error divergence sphere, the blue area represents the error divergence point; the red area represents the error convergence point; the green, cyan, and black coordinate axes represent the X, Y, and Z axes of the Earth-Moon rendezvous coordinate system, respectively. It can be seen that not all Lyapunov exponents of the position and velocity errors are negative at the position-holding points of this period. The two orthogonal error convergence axes are obtained by solving the error convergence method. as follows: (20); With convergence axis mutually orthogonal divergence axes as follows: (twenty one); The design accuracy of the periodic position holding point of this invention is illustrated by comparison. Based on the error convergence direction on the orthogonal error convergence axis ring shown in formula (20), and considering the periodic position holding point shown in formula (19), the end position error of the NRHO track after one cycle is calculated. Meanwhile, existing periodic position holding point design methods that do not consider initial position error and velocity error, considering the same vector direction, calculate the end position error of the track after one cycle. The end position error distributions of the two methods are as follows: Figure 6 As shown in the figure. The red line represents the mean value of the end position error.

[0025] The results show that, compared with the traditional method, the end position error of the periodic position holding point obtained by the present invention in the direction of the error convergence axis is on the order of kilometers, with an average error of 6.818 km; the existing periodic position holding point design method has an end position error on the order of hundreds of kilometers in the direction of the error convergence axis, with an average error of 151.631 km; the position error and velocity error convergence axes are extended from three fixed directions to all directions.

[0026] Although the present invention has been described in detail above with general descriptions and specific embodiments, modifications or improvements can be made to it, which will be obvious to those skilled in the art. Therefore, all such modifications or improvements made without departing from the spirit of the present invention fall within the scope of protection claimed by the present invention.

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), a robust periodic position holding point with convergence of position error, velocity error, and simultaneous convergence of position and velocity errors is 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.

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 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: ； Where, ； 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: ； Where, ； 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: 。 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 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. .

4. The robust maintenance point design method for Earth-Moon periodic orbits based on the Lyapunov index as described in claim 3, 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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