Method for characterizing orbit parameters of earth-moon collinear translation point

Abstract

The invention relates to an earth-moon collinear translation point orbit parameter characterization method. The method comprises the following steps: describing a circular restrictive three-body problem as a chaotic Hamiltonian power system, and converting a motion equation into a new coordinate system to obtain a Hamiltonian function of a corresponding system after coordinate system transformation; performing series expansion on nonlinear terms in the Hamiltonian function by utilizing a Legendre expansion method, and taking quadratic terms of the Hamiltonian function in a polynomial form as a linear model of a translation point in the circular restrictive three-body problem; transforming the linearization model into a real standard shape according to a real linear symplectic transformation matrix, and performing complex transformation on the central part of the real standard shape to obtain a linear complex standard shape; and carrying out regular transformation on a nonlinear term of the linear complex standard shape, and defining a local action-angle variable according to a motion mode of a translation point by referring to a linear integrable part of a final Hamiltonian function to describe the motion of the spacecraft on the central manifold so as to realize parameter representation. By adopting the method, track parameter characterization can be realized.

Description

Technical Field

[0001] This application relates to the technical field of space target cataloging, and particularly to a method for characterizing the orbital parameters of the Earth-Moon collinear libration point orbit. Background Art

[0002] In recent years, with the exploration activities of the moon carried out by various countries, the number of Earth-Moon space targets has been increasing year by year. Due to its special position and dynamic characteristics, the Earth-Moon libration point is an important resource for future Earth-Moon space development. The ARTEMIS mission was launched in 2007, deploying lunar exploration spacecraft in the Lissajous orbits at the Earth-Moon L1 and L2 points; CE5-T1 was launched in 2014 and entered the Lissajous orbit at the Earth-Moon L2 point to carry out exploration tasks; Queqiao entered the Halo orbit at the Earth-Moon L2 point in 2018 to carry out communication relay tasks; the CAPSTONE mission verified the autonomous navigation technology of the Lunar Reconnaissance Orbiter in the NRHO orbit, reducing the dependence on ground control; in addition, NASA's GATEWAY mission was launched in 2024, and various robotic and human lunar landing missions will be gradually realized. The development of the Earth-Moon space will lead to the future libration point area becoming crowded. In order to improve the Earth-Moon space situation awareness ability, it is necessary to explore an efficient and intuitive target cataloging method for the Earth-Moon space libration point area.

[0003] Space target cataloging refers to the process of systematically identifying, tracking, recording, and classifying space objects in orbit. These space objects include satellites, space debris, abandoned spacecraft, and other natural and artificial celestial bodies, etc. The purpose of space target cataloging is to maintain a detailed database to record information such as the orbital parameters, size, shape, and use of these objects, so as to facilitate space situation awareness, collision warning, space traffic management, etc. The existing space target cataloging methods mainly form a recognized standard format based on the Keplerian orbital elements: Two-Line Orbital Element (TLE). In addition, there are also methods using regular Delaunay variables to describe orbital motion. Considering the perturbed two-body problem, the Keplerian orbital elements are no longer integrals of motion, and at this time, the method of circular orbital elements can also be considered.

[0004] Due to the strong gravitational forces from the Earth and the Moon in the Earth-Moon space, mission analysts cannot easily rely on typical two-body Keplerian orbital elements to determine the orbit of a spacecraft. To better describe the motion of an object at the Earth-Moon libration points, the circular restricted three-body problem (CRTBP) model is often used, but this causes great difficulties in cataloging objects at the Earth-Moon libration points. On the one hand, the two-body problem is integrable, while the restricted three-body problem is non-integrable, making it impossible to extract more first integrals in the restricted three-body problem in Cartesian coordinates except for energy conservation. Thus, theoretically, it does not have a cataloging basis similar to the two-body problem. On the other hand, there is no generally recognized parameterization description method for the special dynamic structures near the libration points, such as periodic orbits, quasi-periodic orbits, hyperbolic invariant manifolds, etc. It is difficult to directly correlate the physical characteristics, such as amplitude and period, with the CRTBP dynamics, and orbital parameter characterization is required. Summary of the Invention

[0005] Based on this, in view of the above technical problems, it is necessary to provide a method for characterizing the orbit parameters of the Earth-Moon collinear libration points that can achieve orbital parameter characterization.

[0006] A method for characterizing the orbit parameters of the Earth-Moon collinear libration points, the method comprising:

[0007] Obtain the motion equation of the spacecraft in the rendezvous coordinate system under the assumption of the circular restricted three-body problem, describe the circular restricted three-body problem as a chaotic Hamiltonian dynamical system, transform the motion equation to a new coordinate system, and obtain the Hamiltonian function of the corresponding system after the coordinate transformation;

[0008] Use the Legendre expansion method to perform a series expansion on the non-linear terms in the Hamiltonian function to obtain a Hamiltonian function in polynomial form; use the quadratic term of the Hamiltonian function in polynomial form as the linearized model of the libration point in the circular restricted three-body problem;

[0009] According to the real linear symplectic transformation matrix, transform the linearized model into a real canonical form and perform a complex transformation on the central part of the real canonical form to obtain a linear complex canonical form; perform a canonical transformation on the non-linear terms of the linear complex canonical form to decouple the hyperbolic unstable direction and the central direction of the central manifold, and obtain the final Hamiltonian function;

[0010] Refer to the linearly integrable part of the final Hamiltonian function, define local action-angle variables according to the motion mode of the libration point to describe the motion of the spacecraft on the central manifold, and construct a mapping relationship between the CRTBP coordinates and the characterization parameters to achieve parameter characterization.

[0011] The above method for characterizing the orbit parameters of the Earth-Moon collinear libration points constructs the motion equation of the spacecraft in the rendezvous coordinate system under the assumption of the circular restricted three-body problem into a chaotic Hamiltonian dynamical system, providing a systematic framework for deeply analyzing the motion of the spacecraft. Through the Hamiltonian function, the variation laws of key physical quantities such as the energy and momentum of the spacecraft's dynamical characteristics are organically integrated, so as to comprehensively and accurately describe the dynamic behavior of the system. By means of the Legendre expansion method, the non-linear terms in the Hamiltonian function are expanded into series, and the complex non-linear system is transformed into a polynomial form, so that it is clear how the terms of different powers in the system affect the motion. The quadratic term of the polynomial form Hamiltonian function is selected to construct the libration point linearization model, which not only reflects the main characteristics of the system motion near the libration point, but also greatly reduces the analysis difficulty, enabling the complex three-body problem to be approximated by a relatively simple linear model near the libration point. The linearization model is transformed into a real canonical form by using a real linear symplectic transformation matrix, clearly showing the key characteristics such as the stability of the system. The central part is subjected to a complex transformation to obtain the linear complex canonical form, further simplifying the structure. By means of a canonical transformation, the hyperbolic unstable direction and the central direction of the center manifold are decoupled, and the complex system dynamical characteristics are decomposed into relatively independent parts, facilitating an in-depth understanding of the motion law of the spacecraft in different directions and accurately grasping its motion state. Finally, referring to the linearly integrable part of the final Hamiltonian function, local action-angle variables are defined according to the motion mode of the libration point, and the motion of the spacecraft on the center manifold is intuitively described by using their clear physical meanings. The mapping relationship between the CRTBP coordinates and the characterization parameters is constructed, successfully transforming the abstract dynamical model into specific and observable orbit parameters, providing a practical method for cataloging the targets in the Earth-Moon space libration point region and space situation awareness, and having important practical application value. BRIEF DESCRIPTION OF THE DRAWINGS

[0012] Figure 1 FIG. is an application scenario diagram of a method for characterizing the orbit parameters of the Earth-Moon collinear libration points in an embodiment;

[0013] Figure 2 FIG. is a schematic diagram of coordinate translation transformation in an embodiment. DETAILED DESCRIPTION OF THE EMBODIMENTS

[0014] In order to make the objectives, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present application and are not used to limit the present application.

[0015] In one embodiment, as Figure 1 shown, a method for characterizing the orbit parameters of the Earth-Moon collinear libration points is provided, including the following steps:

[0016] Step 102: Obtain the equations of motion of the spacecraft in the rendezvous coordinate system under the assumption of the circular restricted three-body problem. Describe the circular restricted three-body problem as a chaotic Hamiltonian dynamical system, transform the equations of motion to a new coordinate system, and obtain the Hamiltonian function of the corresponding system after the coordinate transformation.

[0017] Under the assumption of the circular restricted three-body problem (CRTBP), the equations of motion of the spacecraft in the rendezvous coordinate system can be expressed as:

[0018]

[0019] where ρ = (X, Y, Z) T , and the length, time, and mass in the formula are normalized with the Earth-Moon distance, the reciprocal of the angular velocity of the Moon's revolution, and the sum of the Earth-Moon masses as the benchmarks respectively. Ω is the equivalent potential energy, expressed as

[0020]

[0021] where r1 and r2 are the distances from the spacecraft to the Earth and the Moon respectively, and μ = 0.012150568 is the normalized mass of the Moon. To facilitate the study of the dynamics near the libration points and make the series expansion have better numerical properties, the origin of the coordinate system is translated from the center of mass of the Earth-Moon system to the corresponding collinear libration point L i , as shown in Figure 2 , and the length dimension is further normalized with the distance γ i from the libration point L i to the nearest large celestial body. Let X i be the position of the collinear libration point L i in the rendezvous coordinate system. Then, the coordinate (x, y, z) in the new coordinate system and the coordinate (X, Y, Z) in the original rendezvous coordinate system have the following transformation relationship:

[0022]

[0023] The Hamiltonian function of the corresponding system after the coordinate transformation is

[0024]

[0025] where the momentum of the system is defined as

[0026]

[0027] Step 104: Use the Legendre expansion method to expand the non-linear terms in the Hamiltonian function into a series, and obtain the Hamiltonian function in polynomial form; take the quadratic term of the Hamiltonian function in polynomial form as the linearized model of the libration point in the circular restricted three-body problem.

[0028] In order to perform a Lie transformation on the system to transform it into the Birkhoff–Gustavson normal form, it is necessary to perform a series expansion on the non-linear terms \((1 - \mu) / r_1\) and \(\mu / r_2\) in the Hamiltonian function (4) to make it into the form of a polynomial. Here, the Legendre expansion method is applied:

[0029]

[0030] where, P n is the Legendre polynomial of order n. By expanding the non-linear terms through the above transformation method, the Hamiltonian function can be transformed into the form of a polynomial of order n:

[0031]

[0032] where, H n is a homogeneous polynomial of order n, c n (μ) is a real coefficient, and its value depends on μ and the selected libration point L i .

[0033] The linearized model near the libration point can be given by the quadratic term of the Hamiltonian function (7), that is

[0034]

[0035] Step 106, transform the linearized model into a real normal form according to the real linear symplectic transformation matrix and perform a complex transformation on the central part of the real normal form to obtain a linear complex normal form; perform a canonical transformation on the non-linear terms of the linear complex normal form to decouple the hyperbolic unstable direction and the central direction of the central manifold, and obtain the final Hamiltonian function.

[0036] The linearized dynamics of the libration point in CRTBP can be transformed into a real normal form (real normal form) of Equation (8) through a real linear symplectic transformation matrix:

[0037]

[0038] The specific transformation process is as follows: The canonical equations of the system corresponding to Equation (8) are:

[0039]

[0040] where, J and M are defined by the following equations:

[0041]

[0042] I3 is the 3rd order identity matrix, then the characteristic polynomial of the linear system (10) is

[0043]

[0044] Let η = λ 2 , then the roots of p(λ) = 0 are

[0045]

[0046] Since c2 > 1, thus η1 < 0, η2 > 0, η3 > 0, which also indicates that the collinear libration points have the dynamical structure of saddle×center×center. Define

[0047]

[0048] Suppose the matrix formed by scaling with the eigenvectors u i , v i of M is

[0049]

[0050] In order for the real linear symplectic transformation matrix C to satisfy the discriminant condition of the symplectic matrix: C T JC = J, the numerical values of the corresponding scaling coefficients s1, s2, s3 can be calculated. To sum up, the expression of the real linear symplectic transformation matrix is as follows:

[0051]

[0052] Therefore, through the transformation

[0053] (xyzp x p y p z ) T = C(q1q2q3p1p2p3) T (18)

[0054] The Hamiltonian function can be transformed into the real normal form (9). From the canonical equations, it can be known that the linearized dynamical equation corresponding to H2 is:

[0055]

[0056] The solutions of this system of equations are:

[0057]

[0058] Further transform H2 into the complex normal form. Since the saddle part already has a suitable form, only the complex transformation of the center part is needed here:

[0059]

[0060] Through the transformation, the Hamiltonian function (9) is further reduced to the linear complex normal form:

[0061] H2=λq1p1+iω p q2p2+iω v q3p3 (22)

[0062] Now we can define the action-angle variable: I j =q j p j ,θ j =arctan(q j / p j ). Therefore, the linear standard form is integrable with the integral of motion directly corresponding to the central manifold. In these action-angle variables, the equation of motion becomes a constant action and a linearly varying angle. In order to analyze the characteristics of the family of collinear libration point orbits, it is necessary to decouple the hyperbolic unstable direction of the central manifold from the central direction while maintaining the CRTBP dynamic characteristics as much as possible. Here, it is necessary to perform a canonical transformation on the nonlinear terms of the Hamiltonian function. The following describes how to use the Lie transformation method to deal with the high-order terms of the Hamiltonian function.

[0063] Simplification to the central manifold is a semi-standard form processing method that can simplify and analyze the nonlinear terms of the Hamiltonian function. This process requires a canonical transformation of the variables. A canonical transformation is a transformation that can maintain the form of the Hamiltonian canonical equation unchanged. Compared with the 6-dimensional differential equations used to study CRTBP, only one-dimensional Hamiltonian functions need to be studied after the canonical transformation. Usually, it is difficult to obtain a canonical transformation, and the generating function corresponding to the transformation needs to be calculated. The main methods for determining the generating function are implicit transformation (von Zeipel transformation) and explicit transformation (Lie transformation). In order to facilitate coordinate transformation, the Lie transformation method is used here.

[0064] Lie transform is a commonly used technique that can simplify the dynamics while maintaining accuracy. The theory of Lie transform relies on the following two facts: first, if (q, p) transforms to (Q, P) with the phase flow of the Hamiltonian system H over time t, then (q, p) → (Q, P) is a canonical transformation; second, assuming that a function f is a function of the phase flow (q, p) of the Hamiltonian system H, denoted by f(q, p), then the derivative of the function f with respect to t can be expressed as:

[0065]

[0066] The calculation of the Poisson bracket {F,G} is defined as

[0067]

[0068] Similarly, the second-order derivative of function f can be expressed as

[0069]

[0070] Denote the \(n\)-th Poisson bracket of the function \(f\) in the Hamiltonian system \(H\) as Then the \(n\)-th derivative of \(f\) can be expressed as

[0071]

[0072] Suppose there exists a Hamiltonian system \(G\) to be constructed, which can transform the original Hamiltonian function \(H(q,p)\) of the system into and further endow it with certain special properties (such as eliminating certain special terms). This process can be regarded as the canonical coordinates at \(t = 0\) moving to \((q,p)\) at \(t = 1\) under the action of the Hamiltonian system \(G\). By performing a Taylor series expansion at \(t = 0\), we can obtain the explicit transformation of:

[0073]

[0074] where \(i = 1,2,3\), and the Hamiltonian function of the corresponding system is transformed into

[0075]

[0076] In this Lie transformation, the Hamiltonian system \(G\) is also called the generating function. It can be proved that if we want to achieve the inverse transformation we only need to select the generating function \(-G\). Therefore, the core problem of the Lie transformation is how to select an appropriate generating function to transform the Hamiltonian function of the system into a more concise form. In this problem, it is to eliminate the high-order terms in the Hamiltonian function. The following introduces how to achieve this process.

[0077] Since we seek to decouple the hyperbolic unstable direction and the center direction in the high-order terms of the Hamiltonian function, we will choose generating functions \(G\) of different orders to keep the integrity of the saddle point part \(q_1p_1\) in the high-order terms of the Hamiltonian function of the system. First, consider how to deal with the third-order terms in the Hamiltonian function after performing the complex transformation (21). According to Equation (28), it can be seen that the transformed third-order terms are

[0078]

[0079] where \(G_3\) represents that the generating function \(G\) is a third-order homogeneous polynomial. Since \(H_2\) and \(H_3\) can be obtained from the original Hamiltonian function (7) after the complex transformation, and \(H_2\) is in the linear canonical form, so let we can get

[0080]

[0081] where \(h\) i.j is the coefficient of \(H_3\): The purpose of this application is to decouple the hyperbolic direction and the central direction. Therefore, only specific terms in need to be eliminated. The method of eliminating terms is not unique. Here, we choose to keep the saddle point part as a whole q1p1, that is, eliminate the terms where the powers of q1 and p1 are not equal. Therefore, when calculating the generating function G3 according to (30), adding the restriction condition i1≠j1 can achieve the above purpose. After determining the generating function G3, the third-order terms of the transformed Hamiltonian function already meet the requirement that the powers of each q1 and p1 are equal. However, the terms higher than the third order will become new forms due to the transformation of G3. Specifically, it can be calculated from formula (28):

[0082]

[0083] Next, only need to determine G n (n≥4) in turn to process higher-order terms to separate the hyperbolic unstable direction and the central direction. Similarly, the calculation method of the high-order generating function is

[0084]

[0085] where i1≠j1. Implement the above process to N-order accuracy, and the Hamiltonian function will be transformed into

[0086]

[0087] where R N (q,p) is the remainder higher than N order, is the term between the second order and the N order, and all are in the form of q1p1. For the convenience of analysis, perform the inverse transformation of the complex transformation (21) on formula (33), and the complex variables can be converted into real variables. So far, the decoupling of the hyperbolic unstable direction and the central direction within N-order accuracy has been achieved by performing a canonical transformation on the Hamiltonian function.

[0088] Step 108: Refer to the linearly integrable part of the final Hamiltonian function, define local action-angle variables according to the motion mode of the libration point to describe the motion of the spacecraft on the central manifold, and construct the mapping relationship between the CRTBP coordinates and the characterization parameters to achieve parameter characterization.

[0089] On the basis of completing the decoupling in the previous section, refer to the linearly integrable part of the Hamiltonian function and define local action-angle variables according to the motion mode of the libration point. Different methods are used to define the central integral and the saddle point integral:

[0090]

[0091] Among them, the subscript c represents the center motion mode, and s represents the saddle motion mode. For collinear libration points, their motion mode is saddle × center × center. Therefore, for the definition of the motion integrals and angular variables (I1, θ1, I2, θ2, I3, θ3), it can be defined as (I s , θ s ) to define (I1, θ1), (I c , θ c ) to define (I2, θ2) and (I3, θ3). However, when selecting the parameters for orbit characterization, it is more inclined to retain q1 and p1 rather than characterizing them with their motion integrals and angular variables, mainly for the following reasons: First, the definition of the angular variable of the saddle point will involve the generation of complex variables. Although the quadrants of its complex variables contain certain physical meanings, they are relatively abstract in practical applications. For specific references, please refer to the description in Appendix

[26] ; Second, the above physical meanings can be completely characterized by q1 and p1 alone, that is, the degree to which the spacecraft cuts into the unstable and stable manifolds. It can be seen from the solution (20) of the linearized dynamics of collinear libration points that the change of q1 shows an exponential increase, and the change of p1 shows an exponential decrease. Their changes correspond to the trend of the target moving along the invariant manifold: when the target leaves the libration point along the unstable manifold, the value of q1 will increase rapidly; when the target enters the vicinity of the libration point along the stable manifold, the value of p1 will approach 0 infinitely. And q2, p2, q3, p3 correspond to the motion law of the target in the central manifold. Their motion integrals and angular variables can be defined as (I c , θ c ) and have certain physical meanings: I2 represents the motion amplitude of the target in the XY plane of the rendezvous coordinate system, I3 represents the motion amplitude of the target in the Z direction, and θ2, θ3 have physical meanings similar to phases. When θ2, θ3 have specific relationships, periodic / quasi-periodic orbits will be generated. Therefore, the following transformation is made:

[0092]

[0093] where j = 2, 3. The Hamiltonian function at this time can be expressed as

[0094] H = H2(q1p1, I2, I3) + H N (q1p1, I2, I3, θ2, θ3) + R N (36)

[0095] where R N is the remainder of higher than N order, and H Nis a term between the second order and the Nth order. To sum up, the cataloging parameters of the target are selected as (q1, p1, I2, θ2, I3, θ3). According to the previously derived process, the mapping relationship between the state of the target in the rendezvous coordinate system and the cataloging parameters can be constructed:

[0096]

[0097] The process of mutual transformation is further summarized as follows:

[0098]

[0099] Among them, A represents the process of coordinate translation and scaling of the coordinate system shown in formula (3), B represents the process of constructing the Hamiltonian system and redefining the generalized coordinates and generalized momenta shown in formula (5), C represents the process of transforming the linear term of the Hamiltonian function into the real canonical form shown in formula (18), D represents the complex transformation process shown in formula (21), G n represents the process of eliminating the nth-order specific term through the canonical transformation shown in (32), and E represents the process of defining the characterization parameters according to (35). Since each of the above transformation processes is reversible, the transformation process from the characterization parameters to the rendezvous coordinate system coordinates can be similarly deduced to achieve parameter characterization.

[0100] The above method for characterizing the orbit parameters of the Earth-Moon collinear libration points. In this application, the motion equation of the spacecraft in the rendezvous coordinate system under the assumption of the circular restricted three-body problem is constructed into a chaotic Hamiltonian dynamical system, providing a systematic framework for in-depth analysis of the spacecraft motion. Through the Hamiltonian function, the variation laws of the dynamic characteristics of the spacecraft, such as key physical quantities like energy and momentum, are organically integrated, thus comprehensively and accurately describing the dynamic behavior of the system. By using the Legendre expansion method to expand the non-linear terms in the Hamiltonian function into a series, the complex non-linear system is transformed into a polynomial form, and it can be clearly known how the terms of different powers in the system affect the motion. The quadratic term of the polynomial-form Hamiltonian function is selected to construct the libration point linearization model, which not only reflects the main characteristics of the system motion near the libration point but also greatly reduces the analysis difficulty, enabling the complex three-body problem to be approximated by a relatively simple linear model near the libration point. The linearized model is transformed into a real canonical form by using a real linear symplectic transformation matrix, clearly showing the key characteristics such as the stability of the system. A complex transformation is performed on the central part to obtain the linear complex canonical form, further simplifying the structure. By using the canonical transformation to decouple the hyperbolic unstable direction and the central direction of the center manifold, the complex system dynamic characteristics are decomposed into relatively independent parts, facilitating an in-depth understanding of the motion laws of the spacecraft in different directions and accurately grasping its motion state. Finally, referring to the linearly integrable part of the final Hamiltonian function, local action-angle variables are defined according to the motion mode of the libration point, and the physical meaning of the variables is used to intuitively describe the motion of the spacecraft on the center manifold. The mapping relationship between the CRTBP coordinates and the characterization parameters is constructed, successfully transforming the abstract dynamic model into specific and observable orbit parameters, providing a practical method for the cataloging of targets in the Earth-Moon space libration point region and space situation awareness, and having important practical application value.

[0101] In one of the embodiments, the motion equation is transformed into a new coordinate system, and the Hamiltonian function of the corresponding system after the coordinate system transformation is obtained, including:

[0102] The Hamiltonian function of the corresponding system after the motion equation is transformed into a new coordinate system is:

[0103]

[0104] Where r1 and r2 are the distances from the spacecraft to the Earth and the Moon respectively, μ = 0.012150568 is the normalized mass of the Moon, and the momentum of the system is defined as:

[0105]

[0106] (x, y, z) represents the coordinates in the new coordinate system.

[0107] In one of the embodiments, the Legendre expansion method is used to perform a series expansion on the non - linear terms in the Hamiltonian function, and a Hamiltonian function in polynomial form is obtained, including:

[0108] The Legendre expansion method is used to perform a series expansion on the non - linear terms in the Hamiltonian function, and the Hamiltonian function in polynomial form is:

[0109]

[0110] where P n is the n - th order Legendre polynomial, H n is the n - th order homogeneous polynomial, c n (μ) is a real - valued coefficient,

[0111]

[0112] In one of the embodiments, according to the real linear symplectic transformation matrix, the linearized model is transformed into a real canonical form, including:

[0113] According to the real linear symplectic transformation matrix, the linearized model is transformed into a real canonical form as:

[0114]

[0115] where, (xyzp x p y p z ) T = C(q1q2q3p1p2p3) T , C represents the real linear symplectic transformation matrix, λ, ω p 、ω v represent the coefficients of the terms in the canonical form after symplectic transformation.

[0116] In one of the embodiments, a complex transformation is performed on the central part of the real canonical form to obtain a linear complex canonical form, including:

[0117] A complex transformation is performed on the central part of the real canonical form, and the linear complex canonical form obtained is:

[0118] H2 = λq1p1 + iω p q2p2 + iω v q3p3.

[0119] In one of the embodiments, a canonical transformation is performed on the non - linear terms of the linear complex canonical form to decouple the hyperbolic unstable direction and the central direction of the central manifold, and the final Hamiltonian function is obtained, including:

[0120] A canonical transformation is performed on the non - linear terms of the linear complex canonical form to decouple the hyperbolic unstable direction and the central direction of the central manifold, and the final Hamiltonian function is

[0121]

[0122] Among them, H2 represents a linear complex normal form, and R N (q, p) is the remainder term of order greater than N, is the term between the second order and the Nth order, and both are in the form of q1p1. R N .

[0123] In one of the embodiments, local action-angle variables are defined according to the motion mode of the libration point to describe the motion of the spacecraft on the central manifold, including:

[0124] Define local action-angle variables according to the motion mode of the libration point, and define the central integral and the saddle-point integral as

[0125]

[0126] Among them, the subscript c represents the central motion mode, and s represents the saddle-point motion mode;

[0127] For the collinear libration points, their motion mode is saddle-point × central × central. Therefore, for the definition of the motion integral and the angle variables (I1, θ1, I2, θ2, I3, θ3), the definition method of (I s , θ s ) is used to define (I1, θ1), (I c , θ c ) is used to define (I2, θ2) and (I3, θ3). q2, p2, q3, p3 correspond to the motion law of the target on the central manifold. The motion integral and the angle variables thereof can be defined by (I c , θ c ). And q1 and p1 respectively characterize the degrees of the spacecraft cutting into the unstable manifold and the stable manifold. I2 represents the motion amplitude of the target in the XY plane of the rendezvous coordinate system, I3 represents the motion amplitude of the target in the Z direction. θ2 and θ3 have the physical meaning of phase. When θ2 and θ3 have a specific relationship, periodic / quasi-periodic orbits will be generated; the physical meaning of the phase includes the variation law of the generalized coordinate and the generalized momentum;

[0128] Therefore, perform a transformation on q2, p2, q3, p3 to obtain the Hamiltonian function describing the motion of the spacecraft on the central manifold.

[0129] In one of the embodiments, the transformation of q2, p2, q3, p3 is:

[0130]

[0131] Among them, j = 2, 3.

[0132] In one of the embodiments, transformations are performed on q2, p2, q3, and p3 to obtain a Hamiltonian function that describes the motion of the spacecraft on the central manifold, including:

[0133] After performing transformations on q2, p2, q3, and p3, the Hamiltonian function that describes the motion of the spacecraft on the central manifold is:

[0134] H = H2(q1p1, I2, I3) + H N (q1p1, I2, I3, θ2, θ3) + R N

[0135] where H2 represents the linear complex normal form, and R N (q, p) is a remainder term of higher than Nth order, is a term between the 2nd order and the Nth order, and both are in the form of q1p1. R N .

[0136] In one of the embodiments, a mapping relationship between the CRTBP coordinates and the characterization parameters is constructed, including:

[0137] Select the catalog parameters of the target as (q1, p1, I2, θ2, I3, θ3), and construct a mapping relationship between the state of the target in the rendezvous coordinate system and the catalog parameters:

[0138]

[0139] where (X, Y, Z) represents the position of the spacecraft in the rendezvous coordinate system, represents the velocity of the spacecraft in the rendezvous coordinate system.

[0140] It should be understood that although Figure 1 the steps in the flowchart of Figure 1 are shown sequentially according to the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless otherwise clearly stated in this article, the execution of these steps has no strict order limit, and these steps can be executed in other orders. Moreover,

[0141] At least a part of the steps in can include multiple sub-steps or multiple stages. These sub-steps or stages are not necessarily executed at the same time, but can be executed at different times. The execution order of these sub-steps or stages is not necessarily sequential, but can be executed alternately or in turn with at least a part of other steps or sub-steps or stages of other steps.

[0141] The technical features of the above embodiments can be combined arbitrarily. For the sake of brevity of description, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, it should be considered to be within the scope described in this specification.

[0142] The embodiments described above merely represent several implementation manners of the present application. Their descriptions are relatively specific and detailed, but they should not be construed as limitations on the scope of the invention. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present application, several modifications and improvements can still be made, and these all fall within the protection scope of the present application. Therefore, the protection scope of the present application shall be subject to the appended claims.

Claims

1. A method for characterizing orbital parameters of a collinear libration point between the Earth and the Moon, characterized in that: The method comprises: Obtain the motion equation of the spacecraft in the rendezvous coordinate system under the assumption of a circular restricted three-body problem, describe the circular restricted three-body problem as a chaotic Hamiltonian dynamic system, transform the motion equation into a new coordinate system, and obtain the Hamiltonian function of the corresponding system after the coordinate system transformation; The nonlinear terms in the Hamiltonian function are expanded in series by using the Legendre expansion method to obtain a Hamiltonian function in polynomial form; the quadratic terms of the Hamiltonian function in polynomial form are used as a linearized model of the translation point in the circular restricted three-body problem; The linearized model is transformed into a real standard form according to a real linear symplectic transformation matrix and a central part of the real standard form is complex transformed to obtain a linear complex standard form; a nonlinear term of the linear complex standard form is canonically transformed to decouple the hyperbolic unstable direction of the central manifold from the central direction to obtain a final Hamiltonian function; Referring to the linearly integrable part of the final Hamiltonian function, the local action-angle variable is defined according to the motion mode of the libration point to describe the motion of the spacecraft on the central manifold, and the mapping relationship between the CRTBP coordinates and the characterization parameters is constructed to realize parameter characterization.

2. The method according to claim 1, characterized in that The motion equation is transformed into a new coordinate system to obtain the Hamiltonian function of the corresponding system after the coordinate system transformation, including: The motion equation is transformed into the new coordinate system, and the Hamiltonian function of the corresponding system after the coordinate system transformation is obtained: Among them, r1 and r2 are the distances from the spacecraft to the earth and the moon, respectively, μ = 0.012150568 is the normalized mass of the moon, and the momentum of the system is defined as: (x, y, z) represents the coordinates in the new coordinate system.

3. The method according to claim 2, characterized in that The nonlinear terms in the Hamiltonian function are expanded in series using the Legendre expansion method to obtain a Hamiltonian function in polynomial form, including: The nonlinear terms in the Hamiltonian function are expanded in series using the Legendre expansion method to obtain a Hamiltonian function in polynomial form: Among them, P n is the nth-order Legendre polynomial, H n is a homogeneous polynomial of order n, c n (μ) is a real coefficient, 4. The method according to claim 2, characterized in that: Transforming the linearized model into a real standard form according to a real linear symplectic transformation matrix includes: The linearized model is transformed into a real standard form according to the real linear symplectic transformation matrix: Among them, (xyzp x p y p z ) T =C(q1 q2 q3 p1 p2 p3) T , C represents the real linear symplectic transformation matrix, λ, ω p ,ω v Represents the coefficients of the standard form terms after the symplectic transformation.

5. The method according to claim 4, characterized in that A central part of the real standard form is subjected to a complex transformation to obtain a linear complex standard form, including: The central part of the real standard form is complex transformed to obtain the linear complex standard form: H2=λq1p1+iω p q2p2+iω v q3p3.

6. The method according to claim 4, characterized in that The nonlinear terms of the linear complex standard form are subjected to canonical transformation to achieve decoupling of the hyperbolic unstable direction of the central manifold from the central direction, and the final Hamiltonian function is obtained, including: The nonlinear terms of the linear complex standard form are canonically transformed to decouple the hyperbolic unstable direction of the central manifold from the central direction, and the final Hamiltonian function is obtained as Among them, H2 represents the linear complex standard form, R N (q,p) is a remainder greater than order N, is a term between 2nd and Nth order, and all are of the form q1p1R N .

7. The method according to claim 1, characterized in that According to the motion mode of the libration point, the local action-angle variables are defined to describe the motion of the spacecraft on the central manifold, including: According to the motion mode of the translation point, the local action-angle variable is defined, and the central integral and saddle point integral are defined as follows: Among them, the subscript c represents the center motion mode, and s represents the saddle point motion mode; For collinear translation points, the motion pattern is saddle point × center × center, so the definition of motion integral and angular variables (I1, θ1, I2, θ2, I3, θ3) is adopted (I s ,θ s ) to define (I1,θ1), (I c ,θ c ) to define (I2,θ2) and (I3,θ3), q2,p2,q3,p3 correspond to the motion law of the target in the central manifold, and its motion integral and angle variable can be used (I c ,θ c ), and q1 and p1 represent the degree of the spacecraft's entry into the unstable manifold and the stable manifold respectively, I2 represents the motion amplitude of the target in the XY plane of the conjunction coordinate system, I3 represents the motion amplitude of the target in the Z direction, θ2, θ3 have the physical meaning of the phase, when θ2, θ3 have a specific relationship, a periodic / quasi-periodic orbit will be generated; the physical meaning of the phase includes the changing laws of generalized coordinates and generalized momentum; Therefore, by transforming q2, p2, q3, and p3, we obtain the Hamiltonian function that describes the motion of the spacecraft on the central manifold.

8. The method according to claim 7, characterized in that The method further comprises: Transform q2, p2, q3, p3 into: Among them, j=2,3.

9. The method according to claim 8, characterized in that Transform q2, p2, q3, p3 to obtain the Hamiltonian function describing the motion of the spacecraft on the central manifold, including: By transforming q2, p2, q3, and p3, we get the Hamiltonian function describing the motion of the spacecraft on the central manifold: H=H2(q1p1,I2,I3)+H N (q1p1,I2,I3,θ2,θ3)+R N Among them, H2 represents the linear complex standard form, R N (q,p) is a remainder greater than order N, is a term between 2nd and Nth order, and all are of the form q1p1R N .

10. The method according to claim 9, characterized in that Construct the mapping relationship between CRTBP coordinates and characterization parameters, including: The target catalog parameters are selected as (q1, p1, I2, θ2, I3, θ3), and the mapping relationship between the target state in the rendezvous coordinate system and the catalog parameters is constructed: Among them, (X, Y, Z) represents the position of the spacecraft in the rendezvous coordinate system, Represents the velocity of the spacecraft in the rendezvous coordinate system.

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