A search method for periodic solutions of relative motion in a restricted three-body model with coplanar orbits

Abstract

This invention belongs to the field of aerospace technology, specifically relating to a search method for the periodic solution of relative motion under a co-orbital restricted three-body model. The method includes the following steps: S1, establishing the relative motion dynamics equations under the co-orbital restricted three-body model; S2, solving the analytical solution of the relative motion dynamics equations in S1 using ordinary differential equation theory, and constructing an optimization problem with the initial relative motion state as the optimization variable and minimizing the amplitude of the long-period term of the analytical solution as the optimization index; S3, optimizing using a genetic algorithm with a step-by-step optimization method; the optimization problem in S2 yields the periodic solution of relative motion. The method proposed in this invention utilizes the characteristics of the co-orbital restricted three-body problem, fully analyzes the correspondence between the analytical solution of the relative dynamics equations and the actual physical motion, and has strong engineering practical significance. Furthermore, the proposed two-step optimization based on the genetic algorithm improves the solution speed to a certain extent and avoids the result from getting trapped in local extrema.

Description

Technical Field

[0001] This invention belongs to the field of aerospace technology, specifically relating to a search method for the periodic solution of relative motion under a co-orbital restricted three-body model. Background Technology

[0002] Since the first detection of gravitational wave signals in 2016, the world has begun using gravitational waves for cosmological research. However, as research has deepened, traditional ground-based gravitational wave detection devices have become increasingly unable to meet the detection requirements due to limitations such as ground noise and arm length. Therefore, a series of space missions using satellite formations for gravitational wave detection have been proposed, the most representative of which is the Laser Interferometer Space Antenna (LISA) mission.

[0003] The LISA mission requires three spacecraft to form an equilateral triangle formation within the solar system to detect gravitational waves. To achieve this configuration, LISA constructed relative circular orbits around a virtual center of the formation (located on Earth's orbit, 20° behind Earth) based on the CW equations, and evenly distributed the three spacecraft within these orbits. Based on the formation mechanism of the LISA mission, and considering that the Sun, Earth, and the virtual center of the formation all orbit within the ecliptic plane, the three-body dynamics problem can be viewed as a special type of co-orbital restricted three-body problem. Furthermore, according to the LISA mission's formation design, although the orbital planes of the three spacecraft do not coincide with the ecliptic plane, the angles between them are very small. Therefore, the three-body dynamics problem of the Sun, Earth, and spacecraft can still be approximated as a co-orbital restricted three-body problem.

[0004] Considering the unique characteristics of the co-orbital restricted three-body problem, some scholars have used the co-orbital restricted three-body problem model to study the orbital design problem in the LISA mission. This study derived an approximate analytical solution for the spacecraft's orbital radius from the perspective of the absolute orbit and gave an approximate expression for the formation arm length. However, this study investigates relative motion from the perspective of the absolute orbit, making the physical meaning of the research unclear. Summary of the Invention

[0005] The purpose of this invention is to provide a search method for the relative motion periodic solution under a co-orbital restricted three-body model, thereby providing a new research approach for the orbit optimization field of the LISA mission.

[0006] This invention is achieved through the following technical solution:

[0007] A method for searching the periodic solution of relative motion under a coordinating restricted three-body model includes the following steps:

[0008] S1. Establish the linear relative motion dynamics equations under the co-orbital restricted three-body model;

[0009] S2. Solve the analytical solution of the linear relative motion dynamics equation in S1 using the theory of ordinary differential equations, and construct an optimization problem with the initial state of relative motion as the optimization variable and the minimum amplitude of the long period term of the analytical solution as the optimization index.

[0010] S3. Use a genetic algorithm to optimize the optimization problem in S2 using a step-by-step optimization method to obtain the solution of the relative motion period.

[0011] Furthermore, S1 includes the following steps:

[0012] S1.1 Based on the relevant theories of the restricted three-body problem, construct a co-orbital restricted three-body model for a single spacecraft and obtain the dimensionless dynamic equations of any spacecraft in this coordinate system;

[0013] S1.2 Based on the dimensionless dynamic equations of any spacecraft in this coordinate system, the nonlinear relative motion dynamic equations under the co-orbital restricted three-body model are obtained;

[0014] S1.3 Linearize the nonlinear relative motion dynamics equations under the co-orbital restricted three-body model to obtain the linear relative motion dynamics equations under the co-orbital restricted three-body model.

[0015] Furthermore, S1 specifically includes the following steps:

[0016] S1.1 According to the relevant theories of the restricted three-body problem, in a synoptic coordinate system with the Sun as the origin, the line connecting the Sun to the Earth as the x-direction, and the direction perpendicular to the x-direction and pointing towards the Earth as the y-direction, the dimensionless dynamic equation of any spacecraft in this coordinate system can be written as:

[0017]

[0018] in:

[0019] Ω=Ω0+μΩ' (2)

[0020]

[0021] R 2 =X 2 +Y 2 ,Δ 2 =R 2 -2X+1 (4)

[0022] In the formula, μ represents the ratio of Earth's mass to the Sun's mass; R represents the distance between the spacecraft and the Sun; X represents the X-coordinate of the spacecraft's position vector in the synoptic coordinate system, and Y represents the Y-coordinate of the spacecraft's position vector in the synoptic coordinate system.

[0023] S1.2 For a two-spacecraft formation in the ecliptic plane, the primary star orbits the Earth and is phase-dependent (θ) behind the Earth, while the secondary star orbits near the primary star. Therefore, according to equation (1), the dynamic equation describing the secondary star relative to the primary star in the synoptic coordinate system is:

[0024]

[0025] In the formula, δX = X2 - X1, δY = Y2 - Y1, where subscript 1 represents the primary star and subscript 2 represents the secondary star;

[0026] S1.3 To ensure the validity of the linearized model, if the relative distance between the secondary star and the primary star does not exceed D... max Then, equation (5) is linearized as follows:

[0027]

[0028] Equation (6) is a linear time-varying dynamic system. If we ignore the three-body perturbation of the Earth to the host star and only consider the gravitational pull of the Sun, then X1 and Y1 are a set of constants related to θ. Equation (6) can be further simplified to:

[0029]

[0030] In the formula, This represents the position and velocity of the secondary star relative to the primary star, and:

[0031]

[0032]

[0033] Equation (7) is the relative motion dynamics model under the constructed coordinating restricted three-body model.

[0034] Furthermore, S2 includes the following steps:

[0035] S2.1 Using the theory of ordinary differential equations, give the general solution form of the linear relative motion dynamics model under the co-orbital restricted three-body model;

[0036] S2.2 According to the eigenvalue form of the system matrix of the differential equation system, the general solution form of the linear relative motion dynamics model under the co-orbital restricted three-body model can be divided into three cases;

[0037] S2.3 For the three cases, respectively establish the periodic solution search optimization problem of the linear relative motion dynamics model under the co-orbital restricted three-body model.

[0038] Furthermore, S2 specifically includes the following steps:

[0039] S2.1 For a system of linear time-invariant differential equations, Suppose that the eigenvalues ​​of matrix C contain q real eigenvalues ​​λ. j j = 1, ..., q and 2p complex eigenvalues ​​a j ±b j i,j=1,…,p, and the eigenvector corresponding to the real eigenvalue is v. j ,j=1,…,q; the eigenvectors corresponding to the complex eigenvalues ​​are u j ±w j i,j=1,…,p;If the vector group {v1,...,v...} q ,u1±w1i,...,u p ±w p If i} are linearly independent, then the system of linear time-invariant differential equations The general solution is:

[0040]

[0041] In the formula, m j ,k j ,h j Let Z(t0) be a set of real constants related to the initial value Z(t0), which satisfy the following relationship:

[0042]

[0043] S2.2, According to the relevant theories in S2.1, if If a solution exists, its general solution has the following three possible forms:

[0044] The first type has four distinct real eigenvalues ​​in matrix A1+μA2.

[0045] The second type is matrix A1+μA2, which has two distinct real eigenvalues ​​and a pair of conjugate complex eigenvalues.

[0046] The third type is that matrix A1+μA2 has two pairs of different conjugate complex eigenvalues;

[0047] S2.3. For the three cases in S2.2, provide... The general solution form.

[0048] Furthermore, regarding the first scenario in S2.2, at this time... The general solution is in the form of:

[0049]

[0050] At this point, by observing the form of the general solution, it can be concluded that the system has only two non-periodic forms: asymptotic stability or divergence. Therefore, the system does not have a periodic solution in this case.

[0051] Furthermore, regarding the second scenario in S2.2, at this time... The general solution is in the form of:

[0052]

[0053] In equation (13), M1 and All are 4-dimensional column vectors, and the specific expressions for each element are as follows:

[0054] By observing the form of equation (13), when a1≠0 and if appropriate relative initial conditions are selected so that m j The minimum value is obtained, at which point the system's solution becomes a periodic solution. Therefore, the following optimization problem is constructed:

[0055]

[0056] In the inequality constraints of the above equation, D max This is the maximum permissible distance between the primary and secondary stars, thus ensuring... This linearization result will not fail, and the solution of equation (15) is a set of periodic solution conditions.

[0057] Furthermore, regarding the third scenario in S2.2, at this time... The general solution is in the form of:

[0058]

[0059] In equation (16), M j and All are 4-dimensional column vectors, and the specific expressions for each element are as follows:

[0060]

[0061] Based on the form of equation (16), there are three specific cases:

[0062] 1) When both a1 and a2 are not 0, the solution of the system consists of two aperiodic terms, and there is no condition for a periodic solution.

[0063] 2) If either a1 or a2 is 0, the solution to the system consists of one periodic term and one aperiodic term. Therefore, following the approach in S2.4, the following optimization problem is constructed:

[0064]

[0065] 3) When both a1 and a2 are 0, the solution of the system consists of two periodic terms with different frequencies, and the system must be in periodic motion. However, under the combined "effect" of the two periodic motions with different frequencies, the relative motion of the star will exhibit an irregular form. Therefore, considering the requirement of regularity in the subsequent configuration design, only one periodic motion is selected to be retained. Thus, the following optimization problem is constructed:

[0066]

[0067] Furthermore, S3 specifically includes the following steps:

[0068] S3.1 Input the relevant parameters for the genetic algorithm, including population size L, number of generations N, and crossover probability P. J and the probability of mutation P B ;

[0069] S3.2 Select the initial condition for relative motion X(t0) = [0,0,0,0] T With the relative velocity fixed and the relative position used as the optimization variable, the optimization problem constructed in S2 is solved using a genetic algorithm to obtain the optimization result. and

[0070] S3.3 Selecting the initial conditions for relative motion With relative positions fixed and relative velocities used as optimization variables, a genetic algorithm is employed to solve the optimization problem constructed in S2, yielding the optimization results. and

[0071] S3.4. Based on S3.2 and S3.3, the initial conditions for the relative motion periodic solution under the final co-orbital restricted three-body model are as follows:

[0072] Compared with the prior art, the present invention has the following beneficial technical effects:

[0073] This invention proposes a search method for periodic solutions to relative motion under a co-orbital restricted three-body model. Utilizing the unique characteristics of the co-orbital restricted three-body problem, the dynamic equations of relative motion under this model are derived. The corresponding analytical solutions are given using ordinary differential equation theory. Based on this, an optimization problem for searching periodic solutions is constructed by considering the correlation between the analytical solution form and the actual motion in different cases. Finally, a two-step solution is obtained using a genetic algorithm. The proposed method leverages the characteristics of the co-orbital restricted three-body problem, fully analyzes the correspondence between the analytical solutions of the relative dynamic equations and the actual physical motion, and has strong engineering practical significance. Furthermore, the proposed two-step optimization based on the genetic algorithm improves the solution speed to a certain extent and avoids the results getting trapped in local optima. This invention provides ideas and references for the orbit design and optimization of subsequent space gravitational wave detection projects. Attached Figure Description

[0074] Figure 1 This is a flowchart of the relative motion period solution search method under the coordinating restricted three-body model of the present invention;

[0075] Figure 2 This is a flowchart of the relative motion modeling method under the coordinating restricted three-body model of the present invention;

[0076] Figure 3 This is a flowchart of the method for constructing a periodic solution search optimization problem based on the analytical solution of the relative motion dynamics equations of the present invention;

[0077] Figure 4 This is a flowchart of the relative motion periodicity solution distribution search method based on genetic algorithm in the co-orbital restricted three-body model of the present invention;

[0078] Figure 5 This is a schematic diagram of the co-orbital restricted three-body problem of the present invention;

[0079] Figure 6 This is an optimization result of the specific implementation S3.2 of the present invention, which shows the changes in the position and velocity of the slave star relative to the master star over time;

[0080] Figure 7 This is an optimization result of the specific implementation S3.2 of the present invention, which gives the trajectory of the satellite relative to the host star;

[0081] Figure 8 This is an optimization result of specific embodiment S3.3 of the present invention, showing the changes in the position and velocity of the slave star relative to the master star over time. The results show that both the relative position and relative velocity are divergent, and the relative position satisfies the preset maximum distance D. max Constraints;

[0082] Figure 9This is the optimization result of the specific implementation S3.3 of the present invention, which gives the motion trajectory of the secondary star relative to the primary star. The results show that the unique closed curve of the relative motion trajectory proves the correctness of the search results. Detailed Implementation

[0083] To make the objectives, technical solutions, and advantages of the present invention clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention; that is, the described embodiments are only a part of the embodiments of the present invention, and not all of them.

[0084] The components described and illustrated in the accompanying drawings and embodiments of this invention can be arranged and designed in various different configurations. Therefore, the detailed description of the embodiments of the invention provided in the following drawings is not intended to limit the scope of the claimed invention, but merely to illustrate one selected embodiment of the invention. All other embodiments obtained by those skilled in the art based on the accompanying drawings and embodiments of this invention without inventive effort are within the scope of protection of this invention.

[0085] The features and performance of the present invention will be further described in detail below with reference to embodiments.

[0086] like Figure 1 As shown, this invention discloses a method for searching the periodic solution of relative motion under a coordinating restricted three-body model, comprising the following steps:

[0087] S1. Establish the linear relative motion dynamics equations under the co-orbital restricted three-body model, such as... Figure 2 As shown, it includes the following steps:

[0088] S1.1 According to the relevant theories of the restricted three-body problem, such as Figure 5 As shown, in a synoptic coordinate system with the Sun as the origin, the line connecting the Sun to the Earth as the x-direction, and the direction perpendicular to the x-direction and pointing towards the Earth as the y-direction, the dimensionless dynamic equation of any spacecraft in this coordinate system can be written as:

[0089]

[0090] in:

[0091] Ω=Ω0+μΩ' (2)

[0092]

[0093] R 2 =X 2 +Y 2 ,Δ 2 =R 2 -2X+1 (4)

[0094] In equation (2), μ = 3.0359 × 10 -6 This represents the ratio of the Earth's mass to the Sun's mass.

[0095] S1.2, by Figure 5 For a two-spacecraft formation in the ecliptic plane, where the primary star orbits Earth and is θ = 20° behind Earth, and the secondary star orbits near the primary star, then according to equation (1), the dynamic equation of the secondary star relative to the primary star in the synoptic coordinate system can be described as follows:

[0096]

[0097] In the formula, δX = X2 - X1, δY = Y2 - Y1, where subscript 1 represents the primary star and subscript 2 represents the secondary star;

[0098] S1.3, if the relative distance between the secondary star and the primary star is much smaller than the distance between the primary star and the Sun, then the nonlinear equation (5) can be linearized as follows:

[0099]

[0100] It can be seen that for a linear time-varying dynamic system like equation (6), if the three-body perturbation of the Earth to the host star is ignored and only the gravitational force of the Sun is considered, then X1=cos20°=0.9397 and Y1=sin20°=-0.3420 are a set of constant values, and equation (6) can be further written as:

[0101]

[0102] In the formula, This represents the position and velocity of the secondary star relative to the primary star, and:

[0103]

[0104]

[0105] Equation (7) is the linear relative motion dynamics model under the constructed coordinating restricted three-body model.

[0106] The expression for (A1+μA2) can be obtained by calculation:

[0107]

[0108] S2. Construction of the periodic solution search optimization problem based on the analytical solution of the linear relative motion dynamics equation, such as... Figure 3 As shown, it includes the following steps:

[0109] S2.1 According to the theory of ordinary differential equations, for a system of linear time-invariant differential equations... Suppose that the eigenvalues ​​of matrix A contain q real eigenvalues ​​λ. j j = 1, ..., q and 2p complex eigenvalues ​​a j ±b j i,j=1,…,p and the eigenvector corresponding to each eigenvalue is v. j j = 1, ..., q and u j ±w j If i,j=1,…,p, and the vector group {v1,...,v...} q ,u1±w1i,...,u p ±w p If i} are linearly independent, then the system of linear time-invariant differential equations The general solution can be written as:

[0110]

[0111] In the formula, m j ,k j ,h j Let Z(t0) be a set of real constants related to the initial value Z(t0), which satisfy the following relationship:

[0112]

[0113] S2.2, The eigenvalues ​​and corresponding eigenvectors of matrix A1+μA2 are obtained by calculating the eigenvalues ​​and eigenvectors of matrix (10):

[0114]

[0115]

[0116]

[0117]

[0118] It can be seen that matrix A1+μA2 has two pairs of conjugate complex roots, corresponding to the third case;

[0119] For the third case in S2.2, the general solution of equation (7) can be written as:

[0120]

[0121] In the formula, M j and Both are 4-dimensional column vectors, and the specific expressions for each element are as follows:

[0122] Since both a1 and a2 are 0, the following optimization problem can be constructed:

[0123]

[0124] S3. For example Figure 4 As shown, a genetic algorithm is used to search for the periodic solution of the relative motion step by step:

[0125] S3.1 Input the relevant parameters for the genetic algorithm, including population size L = 40, number of generations N = 20000, and crossover probability P. J =0.9, mutation probability P B =0.1;

[0126] S3.2 Select the initial condition for relative motion X(t0) = [0,0,0,0] T With the relative velocity fixed and the relative position used as the optimization variable, the optimization problem constructed in S2 is solved using a genetic algorithm to obtain the optimization result. and , Figure 6 and Figure 7 The changes in the position and velocity of the slave star relative to the host star over time, as well as the trajectory of the slave star relative to the host star, are presented under the optimized results of S3.2. Figure 6 and Figure 7 This indicates that the optimization results of S3.2 can be regarded as a preliminary solution for the relative motion period, but further optimization is still needed;

[0127] S3.3 Select the initial conditions for relative motion: X(t0) = [-0.0091, -0.0250, 0, 0] T With the relative positions fixed and the relative velocities used as the optimization variables, the optimization problem constructed in S2 is solved using a genetic algorithm to obtain the optimization results. and Figure 8 and Figure 9 The changes in the position and velocity of the slave star relative to the host star over time, as well as the trajectory of the slave star relative to the host star, are presented under the optimized results of S3.3. Figure 8 and Figure 9 This indicates that the final optimization result of S3.3 is a set of periodic solutions for relative motion;

[0128] S3.4. Based on S3.2 and S3.3, the initial conditions for the relative motion periodic solution under the co-orbital restricted three-body model are X(t0) = [-0.0091, -0.0250, -0.0125, 0.0046]. T .

[0129] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A method for searching the relative motion periodic solution of the restricted three-body problem under the co-orbital constraint, characterized in that, Comprising the following steps: S1, establishing a linear relative motion dynamics equation under the co-orbit restricted three-body model; S2, using the theory of ordinary differential equations to solve the analytical solution of the linear relative motion dynamics equation in S1, and constructing an optimization problem with the relative motion initial state as the optimization variable and the minimum amplitude of the long-period term of the analytical solution as the optimization index; S3, using genetic algorithm to optimize the optimization problem in S2 by step-by-step optimization method to obtain the relative motion periodic solution; S1 comprises the following steps: S1.1, according to the relevant theory of restricted three-body problem, constructing a co-orbit restricted three-body model of a single spacecraft, obtaining the dimensionless dynamics equation of any spacecraft in the coordinate system; S1.2, according to the dimensionless dynamics equation of any spacecraft in the coordinate system, obtaining the nonlinear relative motion dynamics equation under the co-orbit restricted three-body model; S1.3, linearizing the nonlinear relative motion dynamics equation under the co-orbit restricted three-body model to obtain the linear relative motion dynamics equation under the co-orbit restricted three-body model; S1 specifically comprises the following steps: S1.1, According to the theory of restricted three-body problem, in the conjunction coordinate system with the sun as the origin, the line from the sun to the earth as the direction, the line perpendicular to the direction and pointing to the direction of the earth's motion as the direction, the dimensionless dynamic equation of any spacecraft in the coordinate system is written as: （1） Wherein: （2） （3） （4） wherein R represents the distance between the spacecraft and the sun; X represents the X coordinate of the position vector of the spacecraft in the synodic coordinate system, and Y represents the Y coordinate of the position vector of the spacecraft in the synodic coordinate system. S1.2, for a two-spacecraft formation in the ecliptic plane, the primary spacecraft is in a geosynchronous orbit and lags the Earth in phase angle The dynamics of the secondary spacecraft in the Hill frame is described by equation (1) as （5） wherein , wherein subscript 1 denotes the primary star and subscript 2 denotes the secondary star; S1.3, to ensure the effectiveness of the linearized model, if the relative distance from the star and the primary star does not exceed then equation (5) is linearized as: （6） Equation (6) is a linear time-varying dynamics system. If the three-body perturbation of the Earth on the primary star is ignored and only the gravitational force of the Sun is considered, then and are a set of constants related to Equation (6) is further simplified as （7） wherein represents the position and velocity of the star relative to the primary star, and: （8） （9） Equation (7) is the relative motion dynamics model under the constructed co-orbit restricted three-body model.

2. The method of claim 1, wherein, S2 comprises the following steps: S2.1, using the theory of ordinary differential equations, giving the general solution form of the linear relative motion dynamics model under the co-orbit restricted three-body model; S2.2, according to the eigenvalue form of the system matrix of differential equation group, the general solution form of the linear relative motion dynamics model under the co-orbit restricted three-body model is divided into three cases; S2.3, for the three cases, the periodic solution search optimization problem of the linear relative motion dynamics model under the co-orbit restricted three-body model is established respectively.

3. The method of claim 2, wherein, S2 specifically comprises the following steps: S2.1, for a linear constant differential equation group, Assume that the eigenvalues of matrix include real eigenvalues and complex eigenvalues , and the eigenvectors corresponding to the real eigenvalues are ; the eigenvectors corresponding to the complex eigenvalues are ; if the vector group is linearly independent, the general solution of the linear constant differential equation group is: （10） wherein is a set of and initial values a real constant related to the relation: （11） S2.2, According to the relevant theory of S2.1, if If there is a solution, the form of its general solution exists in the following three cases: The first, matrix has 4 different real eigenvalues; Second, matrix has 2 distinct real eigenvalues and 1 pair of complex conjugate eigenvalues; Third, matrix has 2 pairs of distinct conjugate complex eigenvalues; S2.

3. For the three cases in S2.2, give the general solution form of ​ 4. The method of claim 3, wherein, For the first case in S2.2, at this time The general solution is: （12） At this time, by observing the form of the general solution, it is concluded that the system only exists in two non-periodic forms: asymptotic stability or divergence, so the system does not exist periodic solution under this condition.

5. The method of claim 3, wherein, For the second case in S2.2, this time The general solution is （13） In formula (13), and are 4-dimensional column vectors, whose elements are given by （14） By observing the form of equation (13), we have that and if by choosing the appropriate relative initial conditions we have min, where the solution of the system is a periodic solution, so we construct the optimization problem as follows: （15） In the inequality constraint of the above formula is the maximum distance allowed between the primary and the secondary, thus ensuring This linearization result does not fail, and the solution of equation (15) is a set of periodic solution conditions.

6. The method of claim 3, wherein, For the third case in S2.2, this time The general solution is （16） In formula (16), and are 4-dimensional column vectors, whose elements are given by （17） According to the form of equation (16), it is specifically divided into the following 3 cases: 1) and are not both zero, in which case the solution of the system consists of two non-periodic terms, and the condition for the existence of a periodic solution does not exist; 2) and one of the is zero, in which case the solution of the system consists of a periodic term and a non-periodic term, and therefore, following the ideas in S2.4, the optimization problem is constructed as follows: （18） 3) and are both 0, the solution of the system is composed of two periodic terms with different frequencies, the system must be a periodic motion, but the relative motion of the star will show irregular form under the joint "action" of two different frequency periodic motions, so considering the regularity requirement of subsequent configuration design, only one periodic motion is selected to be retained, and thus the following optimization problem is constructed: （19）。 7. The method of claim 1, wherein, S3 specifically comprises the following steps: S3.1, input the related parameters of the genetic algorithm, the related parameters including population size , evolution generation number , crossover probability , and mutation probability ; S3.2, selecting the initial condition of relative motion , fixing the relative velocity unchanged, taking the relative position as the optimization variable, using the genetic algorithm to solve the optimization problem constructed in S2, and obtaining the optimization result and ; S3.3, selecting the initial conditions of relative motion , fixing the relative position unchanged to take the relative velocity as the optimization variable to solve the optimization problem constructed in S2 by using the genetic algorithm to obtain the optimization result and ; S3.

4. The initial conditions of the relative motion periodic solution under the final coplanar restricted three-body model are obtained according to S3.2 and S3.

3. .

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