Design method for realizing transfer orbit from earth to sun-earth L5 point by utilizing golden star leveraging

Abstract

The invention provides a design method for realizing a transfer orbit from the earth to a sun-earth L5 point by utilizing a golden star leveraging force, and belongs to the technical field of spacecraft transfer orbit design. The method comprises the following steps: constructing a spacecraft dynamic model obtained by splicing a dynamic model of a circular restrictive three-body problem and a dynamic model of a two-body problem; based on a kinetic model, applying a first speed pulse to a discrete point on a departure orbit surrounding the earth, and recording a golden star crossing point of forward integration of the departure orbit; applying a second speed pulse to discrete points on the target orbit around the sun-earth L5 point, and recording a golden star crossing point reversely integrated by the target orbit; and obtaining a transfer track combination corresponding to the same golden star phase. And based on the transfer trajectory combination, calculating a golden star leveraging parameter so as to construct a transfer orbit for transferring from the earth to the sun-earth L5 point by utilizing golden star leveraging. The gravity of the golden star is utilized for boosting, and efficient orbit transfer from the earth to the sun-earth L5 point can be achieved.

Description

Technical Field

[0001] The present invention belongs to the technical field of spacecraft transfer orbit design, and in particular relates to a transfer orbit design method for realizing the transfer from the Earth to the Sun-Earth L5 point by leveraging the force of Venus. Background Art

[0002] The Sun-Earth L5 point, located in Earth's orbit, forms a 60-degree angle with the Earth and the Sun. It is a key strategic observation location that has yet to be fully utilized by international exploration. Exploration in this area has significant scientific significance, high engineering feasibility, and an excellent input-output ratio. Observations from the Sun-Earth L5 point, combined with near-Earth observation data, enable three-dimensional reconstruction of solar activity phenomena, providing key information for revealing the physical mechanisms of solar eruptions. Furthermore, this observation point can detect solar activity approaching Earth 4-5 days in advance and track Earth-facing solar eruptions in real time, revolutionizing the accuracy and timeliness of space weather forecasts.

[0003] Orbit transfer is a fundamental requirement for space missions, especially transfers to deep space targets such as the L5 point between the Sun and the Earth. Researching efficient transfer orbit design methods is crucial for reducing fuel consumption, lowering mission costs, and increasing the flexibility and adaptability of spacecraft. Currently, there are few transfer methods from the Earth to the Sun-Earth triangular libration point. Brian proposed a dual-pulse transfer and low-thrust transfer method based on a two-body model, using orbital rendezvous to analyze the pulses required for transfers with different transfer times. However, this method only considers the gravitational influence of the Sun, and does not consider the gravitational influence of the Earth and Venus. For the L5 point that exists in the restricted three-body problem, such simplifications may lead to inaccurate results. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of existing oversimplified orbit transfer technology models by proposing a transfer trajectory design method that leverages Venus to achieve the Earth-Sun L5 point. This invention utilizes the gravitational pull of Venus to achieve efficient orbital transfer from Earth to the Sun-Earth L5 point. Leveraging Venus's gravity not only effectively reduces the transfer pulse required, ensuring the required velocity pulse is within engineering requirements, but also significantly increases the value and potential returns of scientific exploration during a Venus flyby, thereby enhancing the scientific output of the entire space mission.

[0005] The embodiment of the present invention proposes a method for designing a transfer orbit from the Earth to the Sun-Earth L5 point by leveraging Venus, including:

[0006] Constructing a spacecraft dynamics model, the spacecraft dynamics model being obtained by combining a dynamics model for a circular restricted three-body problem and a dynamics model for a two-body problem; wherein the circular restricted three-body problem considers only the gravitational influences of the Sun and the Earth in the region outside the gravitational influence sphere of Venus, and the two-body problem considers only the gravitational influence of Venus in the region inside the gravitational influence sphere of Venus;

[0007] Based on the dynamic model, applying a set first velocity pulse to discrete points on a starting orbit around the Earth, and recording Venus crossing points obtained by forward integration of the starting orbit;

[0008] Based on the dynamic model, a set second velocity pulse is applied to discrete points on a target orbit around the Sun-Earth L5 point, and a Venus crossing point is recorded by reverse integration of the target orbit;

[0009] Based on the Venus crossing point obtained by forward integration of the departure orbit and the Venus crossing point obtained by reverse integration of the target orbit, a transfer trajectory combination corresponding to the same Venus phase is obtained;

[0010] Based on the transfer trajectory combination, the Venus leverage parameters are calculated to construct a transfer orbit from the Earth to the Sun-Earth L5 point using the leverage of Venus.

[0011] In a specific embodiment of the present invention, the spacecraft dynamics model includes:

[0012] The dynamic model of the circular restricted three-body problem is expressed as follows:

[0013]

[0014] Among them, in the Sun-Earth rotating coordinate system OXYZ, the origin O is located at the center of mass of the Sun and the Earth. The Sun and the Earth move in a circle around the center of mass. The X-axis points along the Sun to the Earth, the Z-axis is parallel to the direction of the system's angular momentum, and the Y-axis is determined by the right-hand rule. x is the coordinate component of the spacecraft's position in the X direction; y is the coordinate component of the spacecraft's position in the Y direction; z is the coordinate component of the spacecraft's position in the Z direction.

[0015] is the first derivative of the coordinate component of the spacecraft position in the X direction with respect to time t; is the first derivative of the coordinate component of the spacecraft position in the Y direction with respect to time t; is the first derivative of the coordinate component of the spacecraft position in the Z direction with respect to time t;

[0016] is the second-order derivative of the coordinate component of the spacecraft position in the X direction with respect to time t; is the second-order derivative of the coordinate component of the spacecraft position in the Y direction with respect to time t; is the second-order derivative of the coordinate component of the spacecraft position in the Z direction with respect to time t;

[0017] is the partial derivative in the X direction; is the partial derivative in the Y direction; is the partial derivative in the Z direction;

[0018] Ω is the pseudo potential of the system, which is expressed as follows:

[0019]

[0020] Where R1 and R2 are the distances from the spacecraft to the sun and the earth respectively:

[0021]

[0022] Where μ = m2 / (m1+m2) is the mass parameter, m1 and m2 are the masses of the sun and the earth respectively;

[0023] The dynamic equation of the two-body problem is expressed as follows:

[0024]

[0025] Where r is the position vector from the center of mass of Venus to the spacecraft; is the second-order derivative of r with respect to time t, i.e., the acceleration vector of the spacecraft; G is Newton's gravitational constant; m v is the mass of Venus; r s is the distance from the center of mass of Venus to the spacecraft.

[0026] In a specific embodiment of the present invention, recording the Venus crossing point forward integrated from the departure orbit includes:

[0027] The starting orbit is discretized uniformly, and a first velocity pulse Δv1 is applied to each discrete point of the starting orbit along the velocity direction. The forward integration is performed for a period of time under the dynamic model of the circular restricted three-body problem, and the moments of all intersections of the transfer trajectory from the starting orbit and the Venus trajectory during this period are recorded. and the state of the intersection The superscript d represents the starting orbit; n1 is the number of the discrete point of the starting orbit; the subscript i is the number of the intersection point obtained by the transfer orbit crossing the Venus orbit starting from this discrete point; S is the state quantity, each state Contains the position and velocity corresponding to the intersection point.

[0028] In a specific embodiment of the present invention, recording the Venus crossing point by reverse integration of the target orbit includes:

[0029] The target orbit is discretized uniformly, and a second velocity pulse Δv2 along the maximum stretching direction is applied to each discrete point of the target orbit. In the dynamic model of the circular restricted three-body problem, the reverse integration is performed for a period of time, and the moments of all intersections between the transfer trajectory starting from the starting orbit and the Venus trajectory during this period are recorded. The state of the intersection The superscript g represents the target orbit; n2 is the target orbit discrete point number; subscript j is the intersection point number of the transfer orbit starting from this discrete point crossing the Venus orbit. Contains the position and velocity corresponding to the intersection point.

[0030] In a specific embodiment of the present invention, the process of determining the maximum stretching direction is as follows:

[0031] The dynamic model of the circular restricted three-body problem (1) is rewritten as:

[0032]

[0033] Where, is the state quantity corresponding to the spacecraft in the 6-dimensional state space; represents the time derivative of the state quantity S, that is, the instantaneous rate of change of the spacecraft in the 6-dimensional state space,

[0034] f(S) is the dynamic evolution function of the spacecraft in the 6-dimensional state space, and its expression is as follows:

[0035]

[0036] Let the autonomous system start time t0 = 0, S0 is the given initial state quantity, is the motion state of the spacecraft at S = S0 at time t0 = 0; where x0, y0, z0 are the initial position components of the spacecraft in the Sun-Earth rotating coordinate system at time t0; is the initial velocity component of the spacecraft in the Sun-Earth rotating coordinate system at time t0;

[0037] According to formula (5),

[0038] make is the state transfer matrix at time t, then:

[0039]

[0040] Where I is the 6×6 identity matrix, is the derivative of the state transfer matrix Φ(t) with respect to time t; is the partial derivative of f(S) with respect to the state S, i.e. the Jacobian matrix; is the Jacobian matrix in the initial state;

[0041] make Is a single-value matrix, denoted as STM:

[0042]

[0043] Among them, Φ r,r ,Φ r,v ,Φ v,r ,Φ v,v The 3×3 matrix blocks at the upper left, upper right, lower left, and lower right of the single-value matrix represent the partial derivatives of the spacecraft state quantity obtained after the integration time T to the state quantity at the initial moment, that is:

[0044]

[0045] Where p f =(x f ,y f ,z f ) is the position vector of the spacecraft after the integration time T, x f ,y f ,z f are the coordinate components of the spacecraft position in the X, Y, and Z directions at the integration time T, respectively; is the velocity vector of the spacecraft after integration time T, are the coordinate components of the spacecraft velocity in the X, Y, and Z directions over the integration time T; p0 = (x0, y0, z0) is the position vector of the spacecraft at the initial moment; is the velocity vector of the spacecraft at the initial moment;

[0046] Take the matrix of the final state change caused by the initial velocity in the STM Denoted as Φ rv,v ;

[0047] For Φ rv,v Perform singular value decomposition:

[0048] UΣV H =Φ rv,v (9)

[0049] Where U is Φ rv,v The left singular matrix of rv,v The right singular matrix, V H represents the conjugate matrix of V; Σ is Φ rv,v The singular value matrix of Σ; the id-th element on the diagonal of the matrix V represents the size of the contraction or stretch along the direction of the id-th column vector of the matrix V, where id is the serial number; the first column vector of V represents the direction of maximum stretching.

[0050] In a specific embodiment of the present invention, obtaining the transfer trajectory combination corresponding to the same Venus phase includes:

[0051] Let angle θ be the angle between the line through the point of intersection of Venus's path and the Sun and the X-axis:

[0052]

[0053] Using formula (10), calculate The corresponding angle is denoted as θ d ,calculate The corresponding angle is denoted as θ a , record satisfies θ d =θ a All by and The combination is recorded as As a combination of transfer tracks corresponding to the same Venus phase.

[0054] In a specific embodiment of the present invention, it also includes:

[0055] Construct a hyperbolic coordinate system with its origin at the intersection of the extensions of the spacecraft's entry and exit from Venus's gravitational sphere, the ξ-axis pointing from Venus toward perihelion, and the η-axis located in the spacecraft's motion plane, perpendicular to the ξ-axis and in the same direction as the spacecraft's velocity at perihelion.

[0056] In the hyperbolic coordinate system, the spacecraft has a residual velocity Enter the impact ball, with the remaining speed Leave the impact ball;

[0057] The distance between the spacecraft perigee and the mass center of Venus is r m That is the distance from the gold point, The rotation angle relative to the η axis is ν - , The rotation angle relative to the η axis is ν + , total rotation angle ν all =ν - +ν + ;

[0058] Construct a heliocentric inertial coordinate system, where the origin of the heliocentric inertial coordinate system is located at the sun, and x in the heliocentric inertial coordinate system is i The y axis is the direction of the sun pointing to Venus at t = 0, i The axis lies in the plane of Venus' orbit and is perpendicular to the x i direction, the positive direction is along the direction of Venus's advance; the speed of the spacecraft entering the impact sphere in the heliocentric inertial system The speed of the detector leaving the impact sphere in the heliocentric inertial system v pis the velocity of Venus in the heliocentric inertial system; converting the velocity in the Sun-Earth rotating coordinate system to the heliocentric inertial system, we get

[0059] According to the hyperbola, the relationship between the residual speed and the total angle is:

[0060]

[0061] Perigold distance r m The expression is as follows:

[0062]

[0063] Among them, μ p is the gravitational constant of Venus;

[0064] but:

[0065]

[0066] Solve for r m , then the velocity pulse Δv3 required near the gold point is:

[0067]

[0068] To ensure that the spacecraft does not collide with Venus, the height h of the spacecraft from the surface of Venus at perigee is p Greater than the set distance threshold h s ,Right now:

[0069] h p =(r m -r v )>h s (15)

[0070] Among them, r v is the planetary radius of Venus;

[0071] From the transfer trajectory combinations corresponding to the same Venus phase, select and retain the combinations that satisfy formula (15), and obtain all possible combinations that use Venus to borrow the force by applying a velocity pulse of Δv1 from the departure orbit and a velocity pulse of Δv2 from the target orbit. After the combination screening is completed, record the corresponding Δv1, Δv2, r m and Δv3, to construct a transfer orbit from the Earth to the Sun-Earth L5 point using the force of Venus.

[0072] The characteristics and beneficial effects of the present invention are:

[0073] The present invention proposes a method for designing a transfer orbit from the Earth to the Sun-Earth L5 point by leveraging the force of Venus. This method uses a spliced ​​model of the circular restricted three-body problem and the two-body problem as the dynamic model of the spacecraft. This model more comprehensively and accurately considers the gravitational effects of the Sun, Earth, and Venus. Accurate dynamic calculations make the orbit design more scientific, and can effectively predict and adjust the orbital dynamics of the spacecraft, ensuring the stability of the spacecraft's orbital transfer under a real gravitational field. In addition, the present invention innovatively considers the use of Venus's gravity to assist, which not only reduces the fuel consumption required for the transfer, but also allows additional scientific observations due to the flyby of Venus, increasing the scientific returns of the space mission. This orbit design not only demonstrates a high degree of innovation, but also has the feasibility of practical application, providing an effective technical solution for solving the problem of orbital transfer from the Earth to the Sun-Earth L5 point. BRIEF DESCRIPTION OF THE DRAWINGS

[0074] Figure 1 This is an overall flow chart of a method for designing a transfer orbit from the Earth to the Sun-Earth L5 point using Venus as a lever, according to an embodiment of the present invention.

[0075] Figure 2 Schematic diagram of the Sun-Earth rotating coordinate system in a specific embodiment of the present invention.

[0076] Figure 3 This is a schematic diagram of Venus' leverage affecting the sphere in a specific embodiment of the present invention. DETAILED DESCRIPTION

[0077] This invention proposes a method for designing a transfer trajectory from Earth to the Sun at L5, leveraging the force of Venus. This invention is further described below with reference to the accompanying drawings and specific examples. The various parameters listed are merely exemplary embodiments of the invention. It should be noted that those skilled in the art may make various improvements and modifications without departing from the principles of the invention, and such improvements and modifications are considered within the scope of protection of this invention.

[0078] The embodiment of the present invention proposes a method for designing a transfer orbit from the Earth to the Sun-Earth L5 point by leveraging Venus, including:

[0079] Constructing a spacecraft dynamics model, the spacecraft dynamics model being obtained by combining a dynamics model for a circular restricted three-body problem and a dynamics model for a two-body problem; wherein the circular restricted three-body problem considers only the gravitational influences of the Sun and the Earth in the region outside the gravitational influence sphere of Venus, and the two-body problem considers only the gravitational influence of Venus in the region inside the gravitational influence sphere of Venus;

[0080] Based on the dynamic model, applying a set first velocity pulse to discrete points on a starting orbit around the Earth, and recording Venus crossing points obtained by forward integration of the starting orbit;

[0081] Based on the dynamic model, a set second velocity pulse is applied to discrete points on a target orbit around the Sun-Earth L5 point, and a Venus crossing point is recorded by reverse integration of the target orbit;

[0082] Based on the Venus crossing point obtained by forward integration of the departure orbit and the Venus crossing point obtained by reverse integration of the target orbit, a transfer trajectory combination corresponding to the same Venus phase is obtained;

[0083] Based on the transfer trajectory combination, the Venus leverage parameters are calculated to construct a transfer orbit from the Earth to the Sun-Earth L5 point using the leverage of Venus.

[0084] In a specific embodiment of the present invention, the method for designing a transfer orbit using Venus to achieve the Earth-Sun L5 point has the following overall process: Figure 1 As shown, the following steps are included:

[0085] 1) Constructing a spacecraft dynamics model, wherein the spacecraft dynamics model is obtained by combining a dynamics model of a circular restricted three-body problem and a dynamics model of a two-body problem.

[0086] In this embodiment, a combination of a circular restricted three-body problem and a two-body problem is used as the spacecraft dynamics model, based on the different primary gravitational sources governing spacecraft motion at each stage. Specifically, in the region outside Venus's gravitational sphere of influence, a Sun-Earth circular restricted three-body problem is employed, considering only the gravitational influences of the Sun and Earth, i.e., the Sun-Earth-spacecraft three-body problem. In the region within Venus's gravitational sphere of influence, only the gravitational influence of Venus is considered, i.e., the Venus-spacecraft two-body problem. The gravitational sphere of influence refers to the region in a multi-body problem where the gravitational influence of one celestial body on other objects in its surrounding space dominates. The radius of Venus's gravitational sphere of influence is 616,277 km. This means that within the spherical region with Venus at its center and a radius of 616,277 km, the spacecraft is considered solely to be subject to the gravitational pull of Venus.

[0087] Furthermore, the dynamic model of this embodiment considers the circular restricted three-body problem. Figure 2 Schematic diagram of the sun-earth rotating coordinate system in a specific embodiment of the present invention. Figure 2As shown, the origin O of the Sun-Earth rotating coordinate system OXYZ is located at the center of mass of the two main celestial bodies (i.e., the Sun and the Earth), and the Sun and the Earth perform circular motion around the center of mass. The X-axis points from the Sun to the Earth, the Z-axis is parallel to the direction of the system's angular momentum, and the Y-axis is determined by the right-hand rule. Assume that the masses of the Sun and the Earth are m1 and m2, respectively. For ease of study, this embodiment adopts dimensionless processing. The following physical quantities are taken as unit 1: the unit mass is the sum of the masses of the two main celestial bodies, i.e., (m1+m2), denoted as MU; the unit distance is the distance between the Sun and the Earth, denoted as LU, which is also equal to the astronomical unit AU; the unit time makes the Sun and the Earth rotate around their common center of mass with a period of 2π, denoted as TU. Define μ=m2 / (m1+m2) as the only mass parameter of the system. By calculation, it can be seen that the Sun is located at (-μ,0,0), the Earth is located at (1-μ,0,0), and the Sun-Earth L5 point is located at (-μ,0,0). The dynamic model of the circular restricted three-body problem is shown in formula (1):

[0088]

[0089] Where x is the coordinate component of the spacecraft position in the X direction; y is the coordinate component of the spacecraft position in the Y direction; and z is the coordinate component of the spacecraft position in the Z direction.

[0090] It is the first derivative of the coordinate component of the spacecraft position in the X direction with respect to time t, that is, the velocity component in the X direction; It is the first derivative of the coordinate component of the spacecraft position in the Y direction with respect to time t, that is, the velocity component in the Y direction; It is the first-order derivative of the coordinate component of the spacecraft position in the Z direction with respect to time t, that is, the velocity component in the Z direction.

[0091] is the second-order derivative of the coordinate component of the spacecraft position in the X direction with respect to time t; is the second-order derivative of the coordinate component of the spacecraft position in the Y direction with respect to time t; It is the second-order derivative of the coordinate component of the spacecraft position in the Z direction with respect to time t.

[0092] is the partial derivative in the X direction; is the partial derivative in the Y direction; is the partial derivative in the Z direction.

[0093] Ω is the pseudo potential of the system, which is expressed as follows:

[0094]

[0095] Where R1 and R2 are the distances from the spacecraft to the sun and the earth respectively:

[0096]

[0097] In the sphere of influence of Venus’ gravity, aerospace is only affected by the gravity of Venus. A simplified dynamic model can be established as shown in formula (4):

[0098]

[0099] Where r is the position vector from the center of mass of Venus to the spacecraft; is the second-order derivative of r with respect to time t, and its physical meaning is the acceleration vector of the spacecraft; G is Newton's gravitational constant, which is approximately 6.67430×10 -11 m 3 / (kg·s 2 );m v is the mass of Venus; r s is the distance from the center of mass of Venus to the spacecraft.

[0100] 2) Based on the dynamic model established in step 1), velocity pulses are applied to discrete points on the departure orbit, and the Venus crossing points are recorded as the forward integral of the departure orbit.

[0101] In this embodiment, the departure orbit can be set to any orbit around the earth. In engineering practice, a near-earth circular orbit around the earth with an orbital altitude of 200km is generally selected as the departure orbit. The departure orbit is discretized uniformly. The number of discrete points depends on the calculation accuracy. Generally, 1000 to 2000 points can be selected. A velocity pulse Δv1 along the velocity direction is applied to each discrete point of the departure orbit. There is no requirement for the size of the velocity pulse. It can be limited according to the mission requirements. It is forward integrated for a period of time under the dynamic model of the circular restricted three-body problem (this time can also be determined by the mission requirements. Generally speaking, it is acceptable within 5 years). Record the time of all intersections of the transfer trajectory starting from the departure orbit and the trajectory of Venus during this period. and the state of the intersection The superscript d represents the starting orbit; n1 is the number of the discrete point of the starting orbit; the subscript i is the number of the intersection point obtained by the transfer orbit crossing the Venus orbit starting from this discrete point; S is the state quantity, each state Contains the position and velocity corresponding to the intersection point.

[0102] 3) Based on the dynamic model established in step 1), velocity pulses are applied to discrete points on the target orbit, and the Venus crossing points obtained by reverse integration of the target orbit are recorded.

[0103] In this embodiment, the target orbit can be set as an orbit around the Sun-Earth L5 point, and a short-period orbit is generally selected as the target orbit. In a specific embodiment of the present invention, the selected target orbit is a short-period orbit with an orbital amplitude of 0.1 AU. The target orbit is discretized uniformly. The number of discrete points also depends on the calculation accuracy, and generally 1000 to 2000 points are sufficient. The number of discrete points in the target orbit can be different from the number of discrete points in the starting orbit.

[0104] Since the dynamic model of the circular restricted three-body problem (Equation (1)) does not explicitly contain the independent variable time t, corresponding to the autonomous system, in order to simplify the expression, Equation (1) is rewritten as:

[0105]

[0106] Where, is the state quantity (i.e. position and velocity) corresponding to the spacecraft in the 6-dimensional state space. It represents the time derivative of the state quantity S, which represents the instantaneous rate of change of the spacecraft in the 6-dimensional state space: f(S) is the dynamic evolution function of the spacecraft in the 6-dimensional state space, and its specific expression is as follows:

[0107]

[0108] Let the autonomous system start time t0 = 0, S0 is the given initial state quantity, is the motion state of the spacecraft under S = S0 at time t0 = 0. Where x0, y0, z0 are the initial position components of the spacecraft in the Sun-Earth rotating coordinate system at time t0 (i.e., the coordinate components in the X, Y, and Z directions respectively); is the initial velocity component of the spacecraft in the Sun-Earth rotating coordinate system at time t0 (i.e., the first-order derivatives of the spacecraft's coordinate components in the X, Y, and Z directions with respect to time t). In this embodiment, S0 is the state of the target orbit discrete point.

[0109] From formula (5), we can know that definition is the state transfer matrix at time t, which can be given by the integral formula, namely:

[0110]

[0111] Where I is the 6×6 identity matrix, is the derivative of the state transfer matrix Φ(t) with respect to time t; is the partial derivative of f(S) with respect to the state S, also known as the Jacobian matrix. This is the Jacobian matrix in the initial state.

[0112] definition It is a single-value matrix, denoted as STM. The STM matrix can be written as follows:

[0113]

[0114] Among them, Φ r,r ,Φ r,v ,Φ v,r ,Φ v,v The 3×3 matrix blocks at the upper left, upper right, lower left, and lower right of the single-value matrix represent the partial derivatives of the spacecraft state quantity obtained after the integration time T to the state quantity at the initial moment, that is:

[0115]

[0116] Where p f =(x f ,y f ,z f ) is the position vector of the spacecraft after the integration time T, x f ,y f ,z f are the coordinate components of the spacecraft position in the X, Y, and Z directions at the integration time T, respectively; is the velocity vector of the spacecraft after integration time T, are the coordinate components of the spacecraft velocity in the X, Y, and Z directions for the integration time T, respectively; p0 = (x0, y0, z0) is the position vector of the spacecraft at the initial moment, and x0, y0, z0 are the coordinate components of the spacecraft position in the X, Y, and Z directions at the initial moment; is the velocity vector of the spacecraft at the initial moment, are the coordinate components of the spacecraft velocity in the X, Y, and Z directions at the initial moment.

[0117] The single value matrix STM provides a mapping of the change from the initial state to the final state. To determine the velocity change direction that is most conducive to the target orbit state change, that is, the maximum stretching direction, take the matrix of the final state change caused by the initial velocity in the STM Denoted as Φ rv,v This shows that the disturbance of the initial velocity causes the change of the final position and velocity. rv,v Performing a singular value decomposition can provide information about the direction of the initial velocity disturbance:

[0118] UΣV H =Φ rv,v (9)

[0119] Where U is Φ rv,v The left singular matrix of rv,v The right singular matrix, V H represents the conjugate matrix of V; Σ is Φrv,v The singular value matrix of Σ is ∑. The idth element on the diagonal of Σ represents the amount of contraction or stretch along the direction of the idth column vector of the matrix V, where id is the sequence number. Therefore, the first column vector of V represents the direction of maximum stretch.

[0120] In this embodiment, a velocity pulse Δv2 along the direction of maximum stretching is applied to each discrete point of the target orbit. The size of this velocity pulse is unlimited and can usually be limited according to the mission requirements. It has no numerical relationship with Δv1. Inversely integrate for a period of time under the dynamic model of the circular restricted three-body problem (this time can also be determined by the mission requirements, generally within 5 years is acceptable, and this time can be different from the integration time of step 2). Record the moments of all intersections of the transfer trajectory starting from the departure orbit and the Venus trajectory during this period. The state of the intersection The superscript g represents the target orbit; n2 is the target orbit discrete point number; subscript j is the intersection point number of the transfer orbit starting from this discrete point crossing the Venus orbit. Contains the position and velocity corresponding to the intersection point.

[0121] 4) Based on the results of steps 2) and 3), obtain the transfer trajectory combination corresponding to the same Venus phase.

[0122] In this embodiment, the angle θ is defined as the angle between the line connecting the intersection of the orbit of Venus and the Sun and the X-axis, ranging from 0° to 360°, with counterclockwise being positive. It can be obtained according to formula (10):

[0123]

[0124] Use formula (10) to calculate all the The corresponding angle θ d and step 3) obtained The corresponding angle θ a , then record the d =θ a All by and The combination is denoted as P= As a combination of transfer tracks corresponding to the same Venus phase.

[0125] 5) Based on the result of step 4), calculate the Venus leverage parameters to construct a transfer orbit from the Earth to the Sun-Earth L5 point using Venus leverage.

[0126] Based on all the combinations P obtained in step 4), this embodiment assumes that at the junction of the three-body problem and the two-body problem, the spacecraft is only subject to the gravity of Venus after entering the gravitational influence sphere of Venus. Due to the leveraged flight, the spacecraft trajectory is a hyperbola. Figure 3 This is a schematic diagram of the influence of Venus's leverage on the sphere in a specific embodiment of the present invention. Figure 3 As shown, the origin of the hyperbolic coordinate system is located at the intersection of the extension lines of the spacecraft entering and exiting the Venus gravitational influence sphere, the ξ axis points from Venus to the perihelion, and the η axis is located in the spacecraft motion plane, perpendicular to the ξ axis, and in the same direction as the spacecraft's velocity at the perihelion. In the hyperbolic coordinate system, the spacecraft moves at the residual velocity. Enter the impact ball, with the remaining speed Leave the impact sphere. Define the distance between the spacecraft perigee and the mass center of Venus as r m ,definition The rotation angle relative to the η axis is ν - , The rotation angle relative to the η axis is ν + , total rotation angle ν all =ν - +ν + .

[0127] Define the heliocentric inertial coordinate system with its origin at the sun, x i The y axis is the direction of the sun pointing to Venus at t = 0, i The axis lies in the plane of Venus' orbit and is perpendicular to the x i The direction is positive along the direction of Venus's advance. The speed of the spacecraft entering the impact sphere in the heliocentric inertial system Similarly, the speed of the detector leaving the impact ball in the heliocentric inertial system is v p is the velocity of Venus in the heliocentric inertial system. Converting the velocity in the Sun-Earth rotating coordinate system to the heliocentric inertial system, we can get

[0128] According to the hyperbola residual speed total angle relationship is:

[0129]

[0130] Perigold distance r m It can be expressed as formula (12), μ p is the gravitational constant of Venus;

[0131]

[0132] Therefore, we can get:

[0133]

[0134] The implied distance r from the gold point in equation (13) can be solved by Matlab's fzero function. m The velocity pulse Δv3 required near the gold point is:

[0135]

[0136] To ensure that the spacecraft does not collide with Venus, the height h from the surface of Venus to the spacecraft at perigee is p Should be at least greater than the set distance threshold h s , which is 200 km in one embodiment of the present invention. Therefore, by further screening and retaining combinations that satisfy equation (15) from the results of step 4), all feasible combinations of applying a velocity pulse of Δv1 from the departure orbit and a velocity pulse of Δv2 from the target orbit, utilizing the force of Venus, can be obtained.

[0137] h p =(r m -r v )>h s (15)

[0138] Among them, r v is the radius of Venus.

[0139] After the combination screening is completed, record the corresponding Velocity pulse Δv1, velocity pulse Δv2, near-gold distance r m and the velocity pulse Δv3 required at perigee. Using these parameters, a transfer orbit from Earth to the Sun-Earth L5 point can be constructed using the force leveraged by Venus. In practical engineering applications, the transfer orbit obtained using the method described in this embodiment can be tested against an actual ephemeris model to obtain a transfer orbit that meets actual force constraints.

[0140] The method for designing a transfer orbit from the Earth to the Sun-Earth L5 point using Venus's leverage in this embodiment is based on a more accurate splicing model and uses Venus's leverage to effectively reduce the speed pulse required for transfer, meeting engineering requirements.

[0141] Next, a specific example is used to further illustrate the track design result obtained by the method described in this embodiment. In this embodiment, Δv1 = 3.5 km / s, Δv2 = 2.5 km / s, and the total transfer pulse Δv can be obtained. all The transfer time TOF is shown in Table 1.

[0142] Δv all =Δv1+Δv2+Δv3 (16)

[0143] TOF=t1+|t2| (17)

[0144] Among them, t1 is the integral time from the departure orbit to Venus in the combination, that is, t2 is the integral time from the target orbit to Venus in the combination, that is, Note that the trajectory from the target orbit to Venus is obtained by backward integration, so t2 is a negative number.

[0145] It can be seen that the total velocity pulse and transfer time of the transfer are both within the engineering constraints, proving the feasibility of the transfer orbit design method proposed in this embodiment for using Venus to leverage the Earth to Sun-Earth L5 point.

[0146] Table 1 Total transfer pulses and transfer schedule for a specific embodiment of the present invention

[0147]

Claims

1. A design method for achieving orbital transfer from the Earth to the Sun-Earth L5 point using Venus's force, characterized in that: include: Constructing a spacecraft dynamics model, the spacecraft dynamics model being obtained by combining a dynamics model for a circular restricted three-body problem and a dynamics model for a two-body problem; wherein the circular restricted three-body problem considers only the gravitational influences of the Sun and the Earth in the region outside the gravitational influence sphere of Venus, and the two-body problem considers only the gravitational influence of Venus in the region inside the gravitational influence sphere of Venus; Based on the dynamic model, applying a set first velocity pulse to discrete points on a starting orbit around the Earth, and recording Venus crossing points obtained by forward integration of the starting orbit; Based on the dynamic model, a set second velocity pulse is applied to discrete points on a target orbit around the Sun-Earth L5 point, and a Venus crossing point is recorded by reverse integration of the target orbit; Based on the Venus crossing point obtained by forward integration of the departure orbit and the Venus crossing point obtained by reverse integration of the target orbit, a transfer trajectory combination corresponding to the same Venus phase is obtained; Based on the transfer trajectory combination, the Venus leverage parameters are calculated to construct a transfer orbit from the Earth to the Sun-Earth L5 point using the leverage of Venus.

2. The method according to claim 1, characterized in that The spacecraft dynamics model includes: The dynamic model of the circular restricted three-body problem is expressed as follows: Among them, in the Sun-Earth rotating coordinate system OXYZ, the origin O is located at the center of mass of the Sun and the Earth. The Sun and the Earth move in a circle around the center of mass. The X-axis points along the Sun to the Earth, the Z-axis is parallel to the direction of the system's angular momentum, and the Y-axis is determined by the right-hand rule. x is the coordinate component of the spacecraft's position in the X direction; y is the coordinate component of the spacecraft's position in the Y direction; z is the coordinate component of the spacecraft's position in the Z direction. is the first derivative of the coordinate component of the spacecraft position in the X direction with respect to time t; is the first derivative of the coordinate component of the spacecraft position in the Y direction with respect to time t; is the first derivative of the coordinate component of the spacecraft position in the Z direction with respect to time t; is the second-order derivative of the coordinate component of the spacecraft position in the X direction with respect to time t; is the second-order derivative of the coordinate component of the spacecraft position in the Y direction with respect to time t; is the second-order derivative of the coordinate component of the spacecraft position in the Z direction with respect to time t; is the partial derivative in the X direction; is the partial derivative in the Y direction; is the partial derivative in the Z direction; Ω is the pseudo potential of the system, which is expressed as follows: Where R1 and R2 are the distances from the spacecraft to the sun and the earth respectively: Where μ = m2 / (m1+m2) is the mass parameter, m1 and m2 are the masses of the sun and the earth respectively; The dynamic equation of the two-body problem is expressed as follows: Where r is the position vector from the center of mass of Venus to the spacecraft; is the second-order derivative of r with respect to time t, i.e., the acceleration vector of the spacecraft; G is Newton's gravitational constant; m v is the mass of Venus; r s is the distance from the center of mass of Venus to the spacecraft.

3. The method according to claim 2, characterized in that The record of the Venus crossing point positively integrated from the departure orbit includes: The starting orbit is discretized uniformly, and a first velocity pulse Δv1 is applied to each discrete point of the starting orbit along the velocity direction. The forward integration is performed for a period of time under the dynamic model of the circular restricted three-body problem, and the moments of all intersections of the transfer trajectory from the starting orbit and the Venus trajectory during this period are recorded. and the state of the intersection The superscript d represents the starting orbit; n1 is the number of the discrete point of the starting orbit; the subscript i is the number of the intersection point obtained by the transfer orbit crossing the Venus orbit starting from this discrete point; S is the state quantity, each state Contains the position and velocity corresponding to the intersection point.

4. The method according to claim 3, characterized in that The recording of the Venus crossing point by reverse integration of the target orbit includes: The target orbit is discretized uniformly, and a second velocity pulse Δv2 along the maximum stretching direction is applied to each discrete point of the target orbit. Inverse integration is performed for a period of time under the dynamic model of the circular restricted three-body problem, and the moments of all intersections between the transfer trajectory starting from the starting orbit and the Venus trajectory during this period are recorded. The state of the intersection The superscript g represents the target orbit; n2 is the target orbit discrete point number; subscript j is the intersection point number of the transfer orbit starting from this discrete point crossing the Venus orbit. Contains the position and velocity corresponding to the intersection point.

5. The method according to claim 4, characterized in that The process of determining the maximum stretching direction is as follows: The dynamic model of the circular restricted three-body problem (1) is rewritten as: Where, is the state quantity corresponding to the spacecraft in the 6-dimensional state space; represents the time derivative of the state quantity S, that is, the instantaneous rate of change of the spacecraft in the 6-dimensional state space, f(S) is the dynamic evolution function of the spacecraft in the 6-dimensional state space, and its expression is as follows: Let the autonomous system start time t0 = 0, S0 is the given initial state quantity, is the motion state of the spacecraft at S = S0 at time t0 = 0; where x0, y0, z0 are the initial position components of the spacecraft in the Sun-Earth rotating coordinate system at time t0; is the initial velocity component of the spacecraft in the Sun-Earth rotating coordinate system at time t0; According to formula (5), make is the state transfer matrix at time t, then: Where I is the 6×6 identity matrix, is the derivative of the state transfer matrix Φ(t) with respect to time t; is the partial derivative of f(S) with respect to the state S, i.e. the Jacobian matrix; is the Jacobian matrix in the initial state; make Is a single-value matrix, denoted as STM: Among them, Φ r,r ,Φ r,v ,Φ v,r ,Φ v,v The 3×3 matrix blocks at the upper left, upper right, lower left, and lower right of the single-value matrix represent the partial derivatives of the spacecraft state quantity obtained after the integration time T to the state quantity at the initial moment, that is: Where p f =(x f ,y f ,z f ) is the position vector of the spacecraft after the integration time T, x f ,y f ,z f are the coordinate components of the spacecraft position in the X, Y, and Z directions at the integration time T, respectively; is the velocity vector of the spacecraft after integration time T, are the coordinate components of the spacecraft velocity in the X, Y, and Z directions over the integration time T; p0 = (x0, y0, z0) is the position vector of the spacecraft at the initial moment; is the velocity vector of the spacecraft at the initial moment; Take the matrix of the final state change caused by the initial velocity in the STM Denoted as Φ rv,v ; For Φ rv,v Perform singular value decomposition: USV H =Φ rv,v (9) Where U is Φ rv,v The left singular matrix of rv,v The right singular matrix, V H represents the conjugate matrix of V; Σ is Φ rv,v The singular value matrix of Σ; the id-th element on the diagonal of the matrix V represents the size of the contraction or stretch along the direction of the id-th column vector of the matrix V, where id is the serial number; the first column vector of V represents the direction of maximum stretching.

6. The method according to claim 5, characterized in that The step of obtaining a transfer trajectory combination corresponding to the same Venus phase includes: Let angle θ be the angle between the line through the point of intersection of Venus's path and the Sun and the X-axis: Using formula (10), calculate The corresponding angle is denoted as θ d ,calculate The corresponding angle is denoted as θ a , record satisfies θ d =θ a All by and The combination is recorded as As a combination of transfer tracks corresponding to the same Venus phase.

7. The method according to claim 6, characterized in that Also includes: Construct a hyperbolic coordinate system with its origin at the intersection of the extensions of the spacecraft's entry and exit from Venus's gravitational sphere, the ξ-axis pointing from Venus toward perihelion, and the η-axis located in the spacecraft's motion plane, perpendicular to the ξ-axis and in the same direction as the spacecraft's velocity at perihelion. In the hyperbolic coordinate system, the spacecraft has a residual velocity Enter the impact ball, with the remaining speed Leave the impact ball; The distance between the spacecraft perigee and the mass center of Venus is r m That is the distance from the gold point, The rotation angle relative to the η axis is ν - , The rotation angle relative to the η axis is ν + , total rotation angle ν all =ν - +ν + ; Construct a heliocentric inertial coordinate system, where the origin of the heliocentric inertial coordinate system is located at the sun, and x in the heliocentric inertial coordinate system is i The y axis is the direction of the sun pointing to Venus at t = 0, i The axis lies in the plane of Venus' orbit and is perpendicular to the x i direction, the positive direction is along the direction of Venus's advance; the speed of the spacecraft entering the impact sphere in the heliocentric inertial system The speed of the detector leaving the impact sphere in the heliocentric inertial system v p is the velocity of Venus in the heliocentric inertial system; converting the velocity in the Sun-Earth rotating coordinate system to the heliocentric inertial system, we get According to the hyperbola, the relationship between the residual speed and the total angle is: Perigold distance r m The expression is as follows: Among them, μ p is the gravitational constant of Venus; but: Solve for r m , then the velocity pulse Δv3 required near the gold point is: To ensure that the spacecraft does not collide with Venus, the height h of the spacecraft from the surface of Venus at perigee is p Greater than the set distance threshold h s ,Right now: h p =(r m -r v )>h s (15) Among them, r v is the planetary radius of Venus; From the transfer trajectory combinations corresponding to the same Venus phase, select and retain the combinations that satisfy formula (15), and obtain all possible combinations that use Venus to borrow the force by applying a velocity pulse of Δv1 from the departure orbit and a velocity pulse of Δv2 from the target orbit. After the combination screening is completed, record the corresponding Δv1, Δv2, r m and Δv3, to construct a transfer orbit from the Earth to the Sun-Earth L5 point using the force of Venus.

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