A transfer design method from a libration point orbit to a Mars orbit based on lunar gravity assist

Abstract

A transfer design method from libration point orbit to Mars orbit based on lunar gravity assistance, the steps are as follows: in the earth-moon system, a transfer orbit from libration point orbit to the vicinity of the moon is designed; according to the obtained transfer orbit, the position and velocity of the spacecraft in the earth-moon system are obtained, and the position and velocity of the spacecraft in the transfer orbit are converted to the heliocentric coordinate system to obtain the position and velocity of the spacecraft in the heliocentric coordinate system; in the sun-earth-mars multi-body system, according to the position and velocity of the spacecraft in the heliocentric coordinate system, the transfer orbit from the gravity assistance orbit position to Mars is determined. The present application effectively reduces the difficulty of orbit design in the multi-body system by segmenting the complete orbit, and quickly determines the initial value of the design variable by combining the global and local optimization search algorithm, and the design method and result can provide reference for future spacecraft Mars expansion exploration mission.

Description

Technical Field

[0001] This invention relates to a design method for transferring from a translational point orbit to a Mars orbit based on lunar leverage, and more particularly to a method for optimal transfer of translational point orbits for different three-body systems and initial value search for multi-constrained celestial body leverage flight variables, belonging to the field of spacecraft orbit design and optimization. Background Technology

[0002] Planetary exploration has become a hot research area in the aerospace field in recent years, and Mars, as Earth's close neighbor with similar properties, is currently the most studied planet besides Earth. Transfer orbit design and analysis is a crucial part of Mars exploration missions, playing a key role in the design of subsequent subsystems. In recent years, the design of dynamic equilibrium point orbits has been a hot topic in the field of deep space exploration orbit design and optimization. Based on translational point orbits under different systems, missions such as solar activity monitoring, continuous communication on the far side of the moon, and space environment exploration can be achieved. At the same time, the translational point is also the hub for the optimal transfer orbit design of deep space exploration missions. Due to the many advantages of orbits near translational points, how to achieve energy-efficient transfer orbit design around translational point orbits has received increasing attention from research institutions and scholars of major spacefaring nations. Optimal fuel transfer stems from the combined gravitational effects of multiple celestial bodies, and the orbit design process will face the influence of complex dynamic environments. Against this background, research on low-energy transfer design and optimization for translational point orbit missions is necessary.

[0003] To address the optimal transfer design problem from the Earth-Moon translation point to Mars, Kakoi et al. conducted in-depth research on the feasibility of reaching Mars from the Earth-Moon system's L1 and L2 translation point orbits, utilizing the Earth-Moon unstable manifold, the Sun-Earth stable manifold, and the gravitational assistance of celestial bodies. Their research showed that, compared to direct transfer, celestial body-assisted transfer can better reduce fuel consumption. Alonso, using the transfer design between Earth and Mars Halo orbits as a research background, achieved inter-interplanetary Halo orbit pulse transfer using manifolds under an accurate ephemeris model, and optimized the orbit using a two-layer differential correction algorithm. Nakimiya, based on the Mars Halo orbit and the Earth Halo orbit, analyzed the Earth-Mars round-trip orbit design problem, studying long-term and short-term transfer problems under the assumption of coplanar Earth and Mars orbits, and investigating the capture time of invariant manifolds. Domestic scholar Wang Yamin, using the Moon's transfer to the Sun-Earth translation point orbit and the Sun-Earth translation point orbit's transfer to near-Earth asteroids as research background, designed the optimal transfer orbit for the translation point orbit to reach the asteroid using principal vector theory and perturbation methods. Based on periodic orbits and invariant manifolds near the translation point, Wang Shuai proposed a low-energy orbit design method for manned asteroid exploration. He discussed and analyzed the fuel consumption of the entire mission, compared it with actual exploration mission schemes, and verified the effectiveness of the proposed transfer strategy.

[0004] Introducing celestial gravitational pull technology into libration point orbit missions can further reduce fuel consumption during transfer missions. However, the diversity and complexity of the dynamic environment near the celestial body posed a challenge to determining the optimal location of the gravitational pull point for the transfer orbit. The selection of the location of the celestial gravitational pull point significantly impacted the fuel efficiency of the transfer orbit. Secondly, designing a low-energy transfer orbit from the Earth-Moon system's libration point orbit to Mars could serve as an extension mission for the probe, enabling multi-objective missions such as lunar and Martian observations. After completing its designated mission, how a probe navigating in the Earth-Moon system's libration point orbit can transfer to Mars, how to utilize the gravitational pull of the Moon and Earth to alter its trajectory and reduce fuel consumption, how to select the timing of maneuvers, and what characteristics the transfer opportunity should exhibit are all core issues that urgently need to be addressed in interplanetary orbit transfer design. Summary of the Invention

[0005] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a design method for transferring from a translational point orbit to a Mars orbit based on lunar leverage. This method has important reference value for window and transfer orbit design and parameter selection in multi-target missions for lunar and Mars observation.

[0006] The technical solution of this invention is: a method for transferring from a translational point orbit to a Mars orbit based on lunar leverage, comprising:

[0007] In the Earth-Moon system, design a transfer orbit from a translational point orbit to the vicinity of the Moon;

[0008] Based on the obtained transfer trajectory, the position and velocity of the spacecraft in the Earth-Moon system are obtained, and the position and velocity of the spacecraft in the transfer trajectory are transformed into the Sun-Geocene rotating coordinate system to obtain the position and velocity of the spacecraft in the Sun-Geocene rotating coordinate system.

[0009] In the Sun-Earth-Mars multi-body system, the transfer orbit to Mars is determined based on the position and velocity of the spacecraft in the Sun-Geocene rotating coordinate system.

[0010] The designed translational point orbit to the transfer orbit near the moon includes:

[0011] Establish the flight dynamics equations of the spacecraft in the Earth-Moon centroid rotating coordinate system;

[0012] Based on the flight dynamics equations of the aircraft, determine the unstable manifold of the translational point orbit;

[0013] Based on the obtained unstable manifold, the transfer trajectory from the translational point orbit to the lunar leverage point is constructed.

[0014] The origin of the Sun-Earth rotating coordinate system is the center of mass of the Sun-Earth system. The two principal celestial bodies revolve around the center of mass in a circular motion. The x-axis points from the first principal celestial body—the Sun—to the second principal celestial body—the Earth. The z-axis is parallel to the direction of the angular momentum of the Sun-Earth system, and the y-axis is determined by the right-hand rule.

[0015] The establishment of the flight dynamics equations of the spacecraft in the Earth-Moon centroid rotating coordinate system includes:

[0016] The origin of the Earth-Moon barycentric rotating coordinate system is the barycentric point of the Earth-Moon system. The two principal celestial bodies revolve around the barycentric point in a circular motion. The x-axis points from the first principal celestial body - Earth to the second principal celestial body - Moon. The z-axis is parallel to the direction of the angular momentum of the Earth-Moon system, and the y-axis is determined by the right-hand rule.

[0017] In the Earth-Moon center-of-mass rotating coordinate system, the flight dynamics equations of the spacecraft are:

[0018]

[0019] In the formula,

[0020]

[0021] The position and velocity of the spacecraft are given in the Earth-Moon center-of-mass rotating coordinate system. Let be the partial derivatives of U(r) with respect to x, y, and z; μ is the gravitational constant of the three-body system.

[0022] The determination of the unstable manifold of the translational point orbit includes:

[0023] Let matrix M be the value of the state transition matrix Φ(t,t0) over one orbital period T, i.e., M = Φ(t0 + T,t0);

[0024] The differential equation for the state transition matrix is:

[0025]

[0026] 03 and I3 are the 3×3 zero matrix and identity matrix, respectively. This represents the derivative of the state transition matrix with respect to time. Let be the matrix of second-order partial derivatives of the function U(r) with respect to position r;

[0027] When designing an unstable manifold for a translational point orbit, the linear perturbation state variables used are... satisfy:

[0028]

[0029] In the formula, The eigenvectors of matrix M; perturbation dm The value is 200km in the Sun-Earth system and 50km in the Earth-Moon system; the symbol "||·||" represents the modulo value.

[0030] By combining equations (1) and (3), the state transition matrix Φ(t,t0) at any given time is determined, and then the matrix M is obtained by solving the equations. Combined By integrating equation (1), the unstable manifold of the translational point orbit is finally obtained.

[0031] The transfer trajectory from the constructed translational point orbit to the lunar leverage point includes:

[0032] The design strategy of constructing the transition trajectory employs a differential correction algorithm to iteratively adjust the unstable manifold; the design variable set D used in the differential correction algorithm is... hm With the set of constraint equations F hm satisfy:

[0033]

[0034] In the formula, ΔV1 is the velocity increment of the translational point on the track; T hm τ is the flight time from the translational orbit to the lunar perigee; τ is the time parameter for the orbital change point on the translational orbit; the superscript * indicates the expected value of the constraint; and the target value of the orbital altitude relative to the Moon. leverage azimuth angle track angle As the termination condition for orbital integration;

[0035] By incorporating the unstable manifold, the design variable set D is continuously modified in the differential correction algorithm. hm Guarantee the set of constraint equations F hm The trajectory eventually converges to the expected value, and then a transfer trajectory from the translational point orbit to the lunar leverage point that satisfies the constraints is designed.

[0036] The process of obtaining the spacecraft's position and velocity in the Earth-Moon system based on the transfer orbit, and then converting the spacecraft's transfer orbit position and velocity to the Sun-Geocene rotating coordinate system to obtain the spacecraft's position and velocity in the Sun-Geocene rotating coordinate system includes:

[0037] By transforming the coordinates, the state variables of the Earth-Moon system are converted to those of the geocentric ecliptic inertial frame, i.e.:

[0038]

[0039] in, Let the coordinate transformation matrix represent the position and velocity of the spacecraft in the Earth-centered ecliptic inertial frame. Where R x With R z C are the rotation matrices about the x-axis and z-axis, respectively.ij Represents the (i,j)th element of the corresponding matrix, [θ m i m ,Ω m These represent the latitude argument, orbital inclination, and right ascension of the ascending node of the Moon's orbit relative to the ecliptic plane, respectively. These represent the position and velocity of the spacecraft and the Earth in the Earth-Moon barycentric rotational system, respectively.

[0040] The length L in the Earth-Moon system AB Time T AB After normalization under the Sun-Earth system, the state variables of the Earth-Moon system and the Sun-Earth system are transformed to obtain...

[0041]

[0042] In the formula, The position of the spacecraft in the heliocentric ecliptic inertial frame. Let the velocity of the spacecraft be the velocity in the heliocentric ecliptic inertial frame. Finally, the state variables are transformed into the heliocentric geocentric rotating coordinate system, i.e.

[0043]

[0044] In the formula, The position and velocity of the spacecraft in the Sun's geocentric rotational frame. This represents the position and velocity of the Earth in the heliocentric rotational system. Let be the transformation matrix from the geocentric inertial frame to the synoptic frame. These are the corresponding matrix elements;

[0045] Equations (6) to (8) are used to complete the conversion between the spacecraft's position and velocity state variables in the Earth-Moon system and the Sun-Earth system state variables, thereby obtaining the spacecraft's position and velocity in the Sun-Earth center rotating coordinate system.

[0046] The determination of the transfer orbit to Mars by leveraging the orbital change position includes:

[0047] Based on the spacecraft's position and velocity in the geocentric rotating coordinate system, initial values ​​for optimization variables are found using a genetic algorithm. The performance indicators of the genetic algorithm consider fuel consumption and distance constraints.

[0048] minJ GA =||ΔV DSM ||+||v s / c -v Mars ||+||r s / c -r Mars || (10)

[0049] In the formula, J GALet ΔV be the performance index function of the genetic algorithm. DSM For the velocity increment of deep-space maneuvers; v s / c v is the velocity of the probe when it rendezvous with Mars. Mars Let r be the velocity of Mars in its orbit. s / c -r Mars || indicates the relative distance between the probe and Mars;

[0050] After the genetic algorithm determines the initial values ​​of the design variables, the sequential quadratic programming algorithm is used to calculate the optimal solution for the design variables. The objective function and constraints of the sequential quadratic programming algorithm are as follows:

[0051] minJ SQP =||ΔV DSM ||+||v s / c -v Mars || (11)

[0052]

[0053] Among them, J SQP The objective function is R; Mars T is the radius of Mars; MoP and T PM Let represent the flight time from the lunar perigee to the deep-space maneuvering point and from the deep-space maneuvering point to Mars, respectively; D is the optimization variable, satisfying:

[0054] D = [T] MoP ,ΔV DSM ,T PM ,t JD ] T (13)

[0055] In the formula, t JD To search for the optimal transfer time within a given departure time period;

[0056] By combining genetic algorithm and sequential quadratic programming algorithm, i.e., combining equations (11)-(13), the variable design is optimized iteratively, and the transfer orbit from the Moon's orbital change point to Mars in the Sun-Earth-Mars multi-body system is finally determined.

[0057] The advantages of this invention compared to the prior art are:

[0058] 1. This invention effectively reduces the difficulty of track design by segmenting and splicing the complete track design process.

[0059] 2. This invention performs a series of processes, such as global optimization search, on design variables and target constraints, which can effectively determine the appropriate lunar orbit-changing point location and the initial value range of design variables.

[0060] 3. This invention further combines local optimization algorithms to autonomously search and determine the location of deep space maneuver points and transfer flight time, which can effectively complete the design task of transferring from the translational point orbit to the final Mars orbit, and can provide a reference for future spacecraft Mars extended exploration missions. Attached Figure Description

[0061] Figure 1 A schematic diagram of a rotating coordinate system at the center of mass of a three-body system;

[0062] Figure 2 This represents an unstable manifold in the orbit of the Earth-Moon translational point.

[0063] Figure 3 A schematic diagram of the lunar orbital transfer.

[0064] Figure 4 This is the trajectory from the translational point to the leverage point in the Earth-Moon center-of-mass rotation system according to an embodiment of the present invention.

[0065] Figure 5 The flight trajectory of the geological center rotation system on the 2033 window day of this invention; Detailed Implementation

[0066] This invention discloses a method for transferring from a translational point orbit to a Mars orbit based on lunar glide, the specific steps of which are as follows:

[0067] Step 1: In the Earth-Moon system, design a transfer orbit from the translational point orbit to the vicinity of the Moon.

[0068] (1) Establish the flight dynamics equations of the spacecraft in the Earth-Moon centroid rotating coordinate system.

[0069] The origin of the Earth-Moon barycentric rotating coordinate system is the barycenter of the Earth-Moon system. The two principal celestial bodies revolve around the barycenter in circular motion. The x-axis points from the first principal celestial body (Earth) to the second principal celestial body (Moon); 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. Figure 1 As shown.

[0070] The origin of the Sun-Earth rotating coordinate system is the center of mass of the Sun-Earth system. The two principal celestial bodies revolve around the center of mass in a circular motion. The x-axis points from the first principal celestial body—the Sun—to the second principal celestial body—the Earth. The z-axis is parallel to the direction of the angular momentum of the Sun-Earth system, and the y-axis is determined by the right-hand rule.

[0071] In the Earth-Moon center-of-mass rotating coordinate system, the flight dynamics equations of the spacecraft are:

[0072]

[0073] In the formula,

[0074]

[0075] The position and velocity of the spacecraft are given in the Earth-Moon center-of-mass rotating coordinate system. Let be the partial derivatives of U(r) with respect to x, y, and z; μ is the gravitational constant of the three-body system.

[0076] (2) Determine the unstable manifold of the translational point orbit based on the flight dynamics equations of the aircraft.

[0077] Let matrix M represent the values ​​of the state transition matrix Φ(t,t0) over one orbital period T, i.e., M = Φ(t0 + T,t0).

[0078] The differential equation for the state transition matrix is:

[0079]

[0080] 03 and I3 are the 3×3 zero matrix and identity matrix, respectively. This represents the derivative of the state transition matrix with respect to time. Let be the matrix of second-order partial derivatives of the function U(r) with respect to position r. Then, in the design of the unstable manifold for the translational point trajectory, the linear perturbation state variables used are... satisfy:

[0081]

[0082] In the formula, The eigenvectors of matrix M; perturbation d m The distance is 200km in the Sun-Earth system and 50km in the Earth-Moon system. The symbol "||·||" represents the modulo value. (See attached image.) Figure 2 As shown. By simultaneously solving equations (1) and (3), the state transition matrix Φ(t,t0) at any given time can be determined, and then the matrix M can be solved. Ultimately, an unstable manifold for a translational point orbit can be designed.

[0083] (3) Based on the obtained unstable manifold, construct the transfer trajectory from the translational point orbit to the lunar leverage point.

[0084] A transfer orbit segment from the translational point orbit to the lunar leverage point is constructed, and a design strategy of iteratively adjusting the unstable manifold using a differential correction algorithm is employed for orbit design. The set of design variables D used in the differential correction algorithm is... hm With the set of constraint equations F hm satisfy:

[0085]

[0086] In the formula, ΔV1 is the velocity increment of the translational point on the track; T hmτ represents the flight time from the translational orbit to the lunar perigee; τ is the time parameter for the orbital change point on the translational orbit. In the constraint equations, the superscript "*" indicates the expected value of the constraint. The target value is the orbital altitude relative to the Moon. With leverage azimuth track angle This serves as the termination condition for orbital integration.

[0087] By combining the unstable manifold, the design variables are continuously modified in the differential correction algorithm to ensure that the constraints eventually converge to the expected value, thereby designing the transfer trajectory from the translational point orbit to the lunar leverage point that satisfies the constraints.

[0088] Step 2: Based on the transfer orbit obtained in Step 1, the position and velocity of the spacecraft in the Earth-Moon system are converted to the Sun-Geocene rotating coordinate system to obtain the position and velocity of the spacecraft in the Sun-Geocene rotating coordinate system.

[0089] By transforming the coordinates, the state variables of the Earth-Moon system are converted to those of the geocentric ecliptic inertial frame, i.e.:

[0090]

[0091] in, The coordinate transformation matrix represents the position and velocity of the spacecraft in the geocentric ecliptic inertial coordinate system. R x With R z Let C be the rotation matrices about the x-axis and z-axis, respectively. ij Represents the (i,j)th element of the corresponding matrix, [θ m i m ,Ω m These represent the latitude argument, orbital inclination, and right ascension of the ascending node of the Moon's orbit relative to the ecliptic plane, respectively. These represent the position and velocity of the spacecraft and the Earth in the Earth-Moon center-of-mass rotation coordinate system, respectively.

[0092] The length and time units in the Earth-Moon system and the Sun-Earth system are normalized to complete the conversion of length and time units between state quantities in the Earth-Moon system and the Sun-Earth system.

[0093]

[0094] In the formula, The position of the Sun-Earth inertial frame. Given the velocity in the Sun-Earth inertial frame, the state variables are finally transformed into the Sun-Earth center rotating coordinate system.

[0095]

[0096] In the formula, The values ​​represent the position and velocity of the spacecraft in the Sun's geocentric rotating coordinate system. This represents the Earth's position and velocity in the Sun's geocentric rotating coordinate system. This is the transformation matrix from the geocentric inertial coordinate system to the Earth-Moon centroid rotating coordinate system. These are the corresponding matrix elements;

[0097] By using equations (6) to (8), the conversion between the spacecraft's position and velocity state variables in the Earth-Moon system and the Sun-Earth system state variables is completed, and the spacecraft's position and velocity in the Sun-Earth rotating coordinate system are obtained.

[0098] Step 3: In the Sun-Earth-Mars multi-body system, based on the position and velocity in the Sun-Geocene rotating coordinate system in Step 2, design a transfer orbit to Mars by leveraging the orbital change position.

[0099] like Figure 3 As shown, based on the position and velocity in the rotating coordinate system of the Sun's Earth Center, a suitable initial value for the optimization variable is found using a genetic algorithm. The performance index of the genetic algorithm comprehensively considers fuel consumption and distance constraints, i.e.

[0100] minJ GA =||ΔV DSM ||+||v s / c -v Mars ||+||r s / c -r Mars || (10)

[0101] In the formula, J GA Let ΔV be the performance index function of the genetic algorithm. DSM For the velocity increment of deep-space maneuvers; v s / c v is the velocity of the probe when it rendezvous with Mars. Mars Let r be the velocity of Mars in its orbit. s / c -r Mars || indicates the relative distance between the probe and Mars.

[0102] After the genetic algorithm determines the initial values ​​of the design variables, the optimal solution for the design variables is obtained through further calculation using a sequential quadratic programming algorithm. The objective function and constraints of the sequential quadratic programming algorithm are as follows:

[0103] minJ SQP =||ΔV DSM ||+||v s / c -v Mars || (11)

[0104]

[0105] Among them, J SQP The objective function is R;Mars T is the radius of Mars; MoP and T PM Let represent the flight time from the lunar perigee to the deep-space maneuvering point and from the deep-space maneuvering point to Mars, respectively; D is the optimization variable, satisfying:

[0106] D = [T] MoP ,ΔV DSM ,T PM ,t JD ] T (13)

[0107] In the formula, t JD To search for the optimal transfer time within a given departure time period.

[0108] By combining genetic algorithms and sequential quadratic programming algorithms, i.e., combining equations 11-13, and iteratively designing variables, the transfer orbit from the Moon's orbital shift point to Mars in the Sun-Earth-Mars system can be determined.

[0109] To verify the feasibility of the design method, a transfer trajectory from a translational point orbit to Mars was designed. A typical flight path under the 2033 departure window is attached. Figures 4-5 As shown. The initial translational orbit is selected as a Halo orbit with a normal amplitude of 5000 km, and the suitable launch window is in 2033. For the Halo orbit-Mars transfer in 2033, the launch date is April 15, 2033. Lunar maneuvers and deep-space maneuvers will be performed on April 19 and May 30, 2033, respectively, with corresponding velocity increments of 0.728 km / s and 0.285 km / s. The final rendezvous with Mars is on November 19, 2033. The entire transfer mission requires a flight time of 218.4 days.

[0110] Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Any person skilled in the art can make possible changes and modifications to the technical solutions of the present invention based on the above-disclosed technical content without departing from the spirit and scope of the present invention. Therefore, any simple modifications, equivalent changes and modifications made to the above embodiments based on the technical essence of the present invention without departing from the content of the technical solutions of the present invention shall fall within the protection scope of the technical solutions of the present invention.

Claims

1. A method for transferring from a translational point orbit to a Mars orbit based on lunar glide, characterized in that, include: In the Earth-Moon system, design a transfer orbit from a translational point orbit to the vicinity of the Moon; Based on the obtained transfer trajectory, the position and velocity of the spacecraft in the Earth-Moon system are obtained, and the position and velocity of the spacecraft in the transfer trajectory are transformed into the Sun-Geocene rotating coordinate system to obtain the position and velocity of the spacecraft in the Sun-Geocene rotating coordinate system. In the Sun-Earth-Mars multi-body system, the transfer orbit to Mars is determined based on the position and velocity of the spacecraft in the Sun-Geocene rotating coordinate system. The designed translational point orbit to the transfer orbit near the moon includes: Establish the flight dynamics equations of the spacecraft in the Earth-Moon centroid rotating coordinate system; Based on the flight dynamics equations of the aircraft, determine the unstable manifold of the translational point orbit; Based on the obtained unstable manifold, construct the transfer trajectory from the translational point orbit to the lunar leverage point; The origin of the Sun-Earth rotating coordinate system is the center of mass of the Sun-Earth system. The two principal celestial bodies revolve around the center of mass in a circular motion. The x-axis points from the first principal celestial body - the Sun - to the second principal celestial body - the Earth. The z-axis is parallel to the direction of the angular momentum of the Sun-Earth system, and the y-axis is determined by the right-hand rule. The establishment of the flight dynamics equations of the spacecraft in the Earth-Moon centroid rotating coordinate system includes: The origin of the Earth-Moon barycentric rotating coordinate system is the barycentric point of the Earth-Moon system. The two principal celestial bodies revolve around the barycentric point in a circular motion. The x-axis points from the first principal celestial body - Earth to the second principal celestial body - Moon. The z-axis is parallel to the direction of the angular momentum of the Earth-Moon system, and the y-axis is determined by the right-hand rule. In the Earth-Moon center-of-mass rotating coordinate system, the flight dynamics equations of the spacecraft are: （1） In the formula, （2） The position and velocity of the spacecraft are given in the Earth-Moon center-of-mass rotating coordinate system. , , for Partial derivatives with respect to x, y, and z; The gravitational constant of the three-body system; The determination of the unstable manifold of the translational point orbit includes: Let matrix State transition matrix The value taken over one orbital period T, i.e. ; The differential equation for the state transition matrix is: （3） and They are a 3×3 zero matrix and an identity matrix, respectively. This represents the derivative of the state transition matrix with respect to time. For function Regarding location The second-order partial derivative matrix; When designing an unstable manifold for a translational point orbit, the linear perturbation state variables used are... satisfy: （4） In the formula, For matrix eigenvectors; perturbations The value is 200km in the Sun-Earth system and 50km in the Earth-Moon system; the symbol "||·||" indicates the modulo value; By combining equations (1) and (3), the state transition matrix at any given time can be determined. Then, the matrix is ​​obtained by solving. and ; and then combine By integrating with equation (1), the unstable manifold of the translational point orbit is finally obtained; The transfer trajectory from the constructed translational point orbit to the lunar leverage point includes: The design strategy of constructing the transition trajectory employs a differential correction algorithm to iteratively adjust the unstable manifold; the set of design variables used in the differential correction algorithm is... With the set of constraint equations satisfy: （5） In the formula, This represents the velocity increment during orbital change at the translational point. The time it takes to travel from the translational orbit to the lunar perigee; These represent the time parameters for the orbital change point on the translational orbit; the superscript * indicates the desired value of the constraint; and the target value of the orbital altitude relative to the Moon. =100 km, leverage azimuth angle track angle =0 is used as the termination condition for orbital integration; By incorporating unstable manifolds, the set of design variables is continuously modified in the differential correction algorithm. Guarantee constraint equation set Finally, it converges to the expected value, and then designs a transfer trajectory from the translational point orbit to the lunar leverage point that satisfies the constraints. Based on the transfer orbit, the spacecraft's position and velocity in the Earth-Moon system are obtained. These positions and velocities are then transformed into a Sun-Geocene rotating coordinate system to obtain the spacecraft's position and velocity in that coordinate system, including: By transforming the coordinates, the state variables of the Earth-Moon system are converted to those of the geocentric ecliptic inertial frame, i.e.: （6） in, Let the coordinate transformation matrix represent the position and velocity of the spacecraft in the Earth-centered ecliptic inertial frame. ,in and These are the rotation matrices about the x-axis and z-axis, respectively. This represents the (i,j)th element of the corresponding matrix. , , These represent the latitude argument, orbital inclination, and right ascension of the ascending node of the Moon's orbit relative to the ecliptic plane, respectively. These represent the position and velocity of the spacecraft and the Earth in the Earth-Moon barycentric rotational system, respectively. Length under the Earth-Moon system ,time After normalization under the Sun-Earth system, the state variables of the Earth-Moon system and the Sun-Earth system are transformed to obtain... （7） In the formula, The position of the spacecraft in the heliocentric ecliptic inertial frame. Let the velocity of the spacecraft be the velocity in the heliocentric ecliptic inertial frame. Finally, the state variables are transformed into the heliocentric geocentric rotating coordinate system, i.e. （8） In the formula, The position and velocity of the spacecraft in the Sun's geocentric rotational frame. This represents the position and velocity of the Earth in the heliocentric rotational system. Let be the transformation matrix from the geocentric inertial frame to the synoptic frame. These are the corresponding matrix elements; ; By using equations (6) to (8), the conversion between the spacecraft's position and velocity state variables in the Earth-Moon system and the Sun-Earth system state variables is completed, thereby obtaining the spacecraft's position and velocity in the Sun-Geocene rotating coordinate system. The determination of the transfer orbit to Mars by leveraging the orbital change position includes: Based on the spacecraft's position and velocity in the geocentric rotating coordinate system, initial values ​​for optimization variables are found using a genetic algorithm. The performance indicators of the genetic algorithm consider fuel consumption and distance constraints. （10） In the formula, This is a performance metric function for the genetic algorithm. For the speed increment of deep space maneuvers; The speed at which the probe rendezvous with Mars. This represents the speed at which Mars moves in its orbit. This indicates the relative distance between the probe and Mars; After the genetic algorithm determines the initial values ​​of the design variables, the sequential quadratic programming algorithm is used to calculate the optimal solution for the design variables. The objective function and constraints of the sequential quadratic programming algorithm are as follows: （11） （12） in, The objective function is... The radius of Mars; and These represent the flight time from the perigee to the deep space maneuvering point and from the deep space maneuvering point to Mars, respectively. To optimize the variables, the following conditions must be met: （13） In the formula, To search for the optimal transfer time within a given departure time period; By combining genetic algorithm and sequential quadratic programming algorithm, i.e., combining equation (11)-equation (13), the variable design is optimized iteratively, and the transfer orbit from the Moon's orbital change point to Mars in the Sun-Earth-Mars multi-body system is finally determined.

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