Trajectory splicing method for transferring from earth-moon three-body orbit to moon orbit by small thrust

Abstract

The application discloses a trajectory splicing method for small-thrust transfer from a geocentric three-body orbit to a circumlunar orbit, and belongs to the field of aerospace. The application is realized by the following method: a geocentric circular restricted three-body model and a lunar barycenter two-body model are adopted, and a splicing point suitable for the circular restricted three-body model and the two-body model is selected according to a transition orbit; a small-thrust orbit solving process is divided into three solving stages, i.e., a three-body transfer stage, a two-body transfer stage and a matching stage, according to the spatial position of the splicing point; and the three solving stages are solved by a hybrid method with the shortest time as an optimization target, so that the small-thrust transfer trajectory is solved in sections, and the trajectory of the three solving stages is spliced to obtain the shortest-time small-thrust trajectory from an initial orbit to a target orbit. The application realizes accurate docking in the spatial position and velocity, can ensure that a spacecraft is smoothly transferred between different orbit stages, reduces the calculation error of the geocentric two-body dynamics model, and further ensures the safety of the spacecraft and the success of a mission.

Description

TECHNICAL FIELD

[0001] The present application relates to a spacecraft small-thrust trajectory generation method, in particular to a trajectory splicing method for small-thrust transfer from a geocentric libration point orbit to a circumlunar orbit, and belongs to the field of aerospace. BACKGROUND

[0002] The design of geocentric libration point orbits is an important part of human space activities. Compared with traditional chemical propulsion technology, the specific impulse of electric propulsion is several times or even dozens of times that of chemical propulsion, reducing the fuel mass carried by the thruster, and the precise adjustment of thrust direction and size also improves the attitude control accuracy. Small-thrust transfer method has gradually become a major current research hotspot, but the dynamics equation of spacecraft in geocentric space is represented by the restricted three-body model, and the dynamics system is strongly nonlinear. It is difficult to converge when transferring from the geocentric gravity interaction region to the circumlunar orbit using the restricted three-body problem. In view of this problem, the present application studies the small-thrust time-optimal transfer trajectory design of spacecraft in the complex gravity environment of geocentric space. Based on the transition orbit selection assumption, the mixed method is used to solve the optimization problem with the shortest time as the optimization index, and finally the time-optimal small-thrust trajectory problem is solved.

[0003] For the small-thrust trajectory design problem of geocentric system long-distance retrograde orbit, according to the characteristics of resonant orbit, the mixed method is used to design the time-optimal small-thrust transfer trajectory from the Earth to the L2 point of the geocentric libration point orbit by combining the invariant flow in the restricted three-body problem. [1] (see Ozimek M T, Howell K C. Low-Thrust Transfers in the Earth-Moon System, Including Applications to Libration Point Orbits [J]. Journal of Guidance Control and Dynamics, 2010, 33(2): 533-549.) and homotopy method is used to extend the thrust amplitude. The time-optimal problem under small thrust is obtained by gradually reducing the thrust amplitude from the time-optimal problem under large thrust. However, there is a lack of invariant manifold near the geocentric long-distance retrograde orbit, so the convergence of this method is poor. Different initial value orbits need to be redesigned, and the robustness is poor, so it is difficult to have high applicability in space engineering tasks.

[0004] Considering the complex dynamic environment of the Earth-Moon system in different spatial positions, a three-stage method is proposed in the prior art [2] (Bion, L, Pierson, et al. Three-stage approach to optimal low-thrust Earth-moon trajectories [J]. Journal of Guidance, Control, and Dynamics, 1994, 17(6): 1275-1275.) to calculate the minimum fuel orbit from the low Earth orbit (LEO) to the low lunar orbit (LLO). The maximum energy Earth escape and Moon capture problem based on two-body dynamics, the time-optimal problem based on restricted three-body dynamics, and finally the fuel-optimal Earth-Moon transfer problem are solved by using the hybrid method. The influence sphere boundary is used for dynamic segmentation, and the matching at the splicing point is very rough, which cannot meet the accurate splicing of the two-body dynamics problem and the restricted three-body problem, and this method cannot meet the requirements of more realistic space missions. SUMMARY

[0005] The purpose of the present application is to provide a trajectory splicing method for small-thrust transfer from the Earth-Moon three-body orbit to the circumlunar orbit, which selects a splicing point suitable for the circular restricted three-body model and the two-body model according to the transition orbit; divides the small-thrust orbit solving process into three solving stages of three-body transfer segment, two-body transfer segment and matching segment according to the spatial position of the splicing point, and solves the small-thrust transfer trajectory by using the hybrid method to take the shortest time as the optimization target, splices the trajectories of the three solving stages to obtain the shortest time small-thrust trajectory from the initial orbit to the target orbit, that is, to realize the trajectory splicing of small-thrust transfer from the Earth-Moon three-body orbit to the circumlunar orbit.

[0006] The purpose of the present application is realized by the following technical solutions.

[0007] The trajectory splicing method for small-thrust transfer from the Earth-Moon three-body orbit to the circumlunar orbit disclosed in the present application comprises the following steps:

[0008] Step one: Establish the rotating coordinate system of the Earth-Moon system and the lunar central inertial system to describe the circular restricted three-body problem and the lunar central two-body problem, respectively, and construct the dynamic equations of the small-thrust spacecraft motion in the circular restricted three-body model and the lunar central two-body model. Given the radius of the splicing sphere, when designing the small-thrust transfer problem in the Earth-Moon gravity interaction zone to the circumlunar orbit, the transfer segment will be divided into three stages, i.e., the three-body transfer segment, the two-body transfer segment, and the matching segment, according to the spatial position of the spacecraft relative to the lunar center; the transition orbit selection condition is given, and the splicing point position is determined according to the given splicing sphere, the starting orbit parameters, and the target orbit parameters.

[0009] The rotating coordinate system of the Earth-Moon system is established, denoted as O emr x emr y emr z emr , the origin O emr of the coordinate system is located at the common center of mass of the Earth-Moon system, the coordinate axis x emr points from the common center of mass to the center of mass of the Moon, x emr y emr is the coordinate plane, i.e., the relative motion plane of the two main celestial bodies, and the coordinate axis z emr is along the direction of the angular momentum, y emr forms a right-hand rule with the first two coordinate axes.

[0010] The dynamic equation of the small-thrust spacecraft motion is expressed in vector form:

[0011]

[0012] In the above equation, the spacecraft state quantity is x=[x,y,z,v x ,v y ,v z ,m] T , f(x,t)=[v x ,v y ,v z ,a x ,a y ,a z ,0] T , (x,y,z) is the position quantity, (v x ,v y ,v z ) is the velocity quantity, and a x , a y , and a z correspond to the acceleration components of the spacecraft on the three axes of the Earth-Moon rotating system, and m is the mass of the spacecraft. B(x,t) is a 7×4 control vector matrix. u=[T x ,T y ,Tz ,T] T For the control vector, T x T y and T z Let T represent the thrust components along the three axes of the Earth-Moon rotation system, and let T be the magnitude of the thrust provided by the thruster, satisfying the constraints. Triaxial component of acceleration a x a y a z The specific expression for B(x,t) is as follows:

[0013]

[0014] Among them 0 m×n I represents an m x n zero matrix. 3×3 Let r1 and r2 represent the normalized distances of the spacecraft from Earth and the Moon, respectively, as a 3×3 identity matrix:

[0015]

[0016] The lunar central inertial coordinate system (MCI) is denoted as O. mci x mci y mci z mci Origin of coordinate system mci Located at the Moon's center of mass, x mci The axis points to the vernal equinox of the moon, x mci y mci The plane coincides with the Earth's equatorial plane, z eci The axis points to the North Pole along the Moon's rotation axis, y mci axis and x mci axis, z mci The axis satisfies the right-hand rule. Under the improved vernal equinox model, the dynamic equations of motion for a low-thrust spacecraft are expressed in matrix form:

[0017]

[0018] The state variables consist of the six-dimensional improved vernal equinox root number and the mass m, where x = [p, f, g, h, k, L, m]. T M(x,t) is a 7×1 column vector, and D(x,t) is a 7×4 matrix. The corresponding specific expression is:

[0019]

[0020]

[0021] Where, μ c I is the gravitational constant of the central gravitational body.sp Specific impulse provided to the thruster, g0=9.8 m / s 2 is the Earth's sea level weight acceleration.

[0022] U = [T r , T t , T n , T] T is the control vector, satisfying the constraint represents in the RTN coordinate system with the spacecraft center of mass as the origin, the R axis is defined as the radial axis pointing from the central celestial body to the spacecraft, the N axis is the direction of the angular momentum of motion, and the T axis constitutes the right-hand rule.

[0023] After the spacecraft enters the lunar sphere of influence, the transfer trajectory constructed by the Earth-centered two-body model has obvious limitations, especially in the mission of transferring the spacecraft to a circumlunar orbit. Due to the significant effect of the moon's gravity on the spacecraft in the near-moon region, the integration process of the Earth-centered two-body model will bring significant errors. In order to reduce the significant errors brought by the integration process of the Earth-centered two-body model, the spherical surface distance from the moon center is A times the moon radius. Therefore, for the transfer of the Earth-moon gravity region orbit to the circumlunar orbit, the dynamics model of the spacecraft in different space intervals needs to be distinguished. According to the spatial position of the spacecraft relative to the moon center, the transfer segment is divided into three stages: three-body transfer segment, two-body transfer segment, and matching segment. The matching segment refers to the re-solution of the trajectory end error of the three-body transfer segment and the two-body transfer segment. The low-thrust orbit located in the splicing sphere in space belongs to the two-body transfer segment, and the spacecraft dynamics integration model is the moon-centered two-body model; except for the splicing sphere, the part belongs to the Earth-moon gravity interaction region in space, which is the three-body transfer segment and the matching segment, and the dynamics model uses the Earth-moon circular restricted three-body model. By using the moon-centered two-body model instead of the Earth-centered two-body model for integration in the two-body transfer segment, and using the circular restricted three-body model instead of the Earth-centered two-body model for integration in the three-body transfer segment and the matching segment, the dynamics integration results are more consistent with the spacecraft's motion trajectory in the space environment, thereby reducing the significant errors brought by the integration process of the Earth-centered two-body model.

[0024] Further, according to the splicing sphere and the target orbit parameters, a transition orbit selection condition is constructed, and the splicing point for trajectory splicing is selected using the transition orbit selection condition:

[0025] Condition 1: The splicing orbit plane should be consistent or approximately consistent with the target orbit plane, that is, in the moon-centered inertial system, the orbital inclination of the two should be approximately the same.

[0026] Condition 2: The splicing orbit semi-major axis is given as AR m Within this radius range, due to the effect of the moon's gravity, the low-thrust orbit presents the characteristics of dense multi-circle distribution, and this radius is an approximate index for distinguishing the integration of the spacecraft under different dynamics models.

[0027] Condition 3: The transition orbit is consistent with the target orbit on the apsidal line, that is, the right ascension of the ascending node of the two is the same. In order to save fuel consumption, the eccentricity of the initial orbit to the target orbit is not adjusted by using small thrust during the transfer process.

[0028] According to condition 3, the orbit elements of the transition orbit in the lunar center of inertia system determined by the target orbit are represented as: AE0=[a0,e0,i0,Ω0,~,~] T Further according to the provisions, for the transition orbit of the initial orbit, the orbit must have the same orbital inclination and the right ascension of the ascending node as the orbit in the lunar center of inertia system, and the eccentricity of the orbit is e0, then AE1=[a1,e1,i1,Ω1,~,~] T The splicing point position on the transition orbit for the connection of the three-body transfer and the two-body transfer is determined through the intersection of the two transition orbits on the splicing sphere, so as to realize the accurate splicing of the trajectory in space position and velocity.

[0029] Step two: According to the Pontryagin extremum principle, the co-state variable is introduced to construct the spacecraft system Hamilton function of the time optimal performance index, and the co-state equation is obtained according to the spacecraft system Hamilton function, which is taken as the constraint condition of the transfer orbit; The initial orbit and the transition orbit constraint condition is given; According to the initial orbit constraint condition, the transition orbit constraint condition and the transfer orbit constraint condition, combined with the dynamic equation of the small thrust spacecraft motion in the circular restricted three-body model and the control constraint constructed in step one, the optimal control problem of small thrust transfer trajectory is constructed with the shortest time as the optimization objective. The optimal control problem P0 of small thrust transfer trajectory is solved numerically, and the trajectory of three-body transfer stage in the earth-moon rotating system is obtained.

[0030] According to the Pontryagin extremum principle, the co-state variable The spacecraft system Hamilton function of the time optimal performance index is constructed as:

[0031]

[0032] According to the circular restricted three-body dynamic equation (1) and Take The co-state equation is obtained:

[0033]

[0034] Where, And The specific expression of the non-zero elements in the 3x3 gradient matrix is as follows:

[0035]

[0036] The time-optimal low-thrust transfer problem from a given departure orbit to a rendezvous point is constructed in the circular restricted three-body model. The specific method is as follows:

[0037] According to the initial and final positions of the transfer orbit, the integral time start point t0 and the integral time end point t f The state of the spacecraft is constrained.

[0038] The initial state x(t0) of the spacecraft is given by the departure point of the departure orbit, and the endpoint constraint of the start point is x ini The constraint is expressed as:

[0039]

[0040] Where r ini and v ini are the position and velocity of the departure orbit at t0 in the rotating lunar system.

[0041] Define x(t f ) as the orbit state of the spacecraft in the lunar orbit in the lunar inertial system after integration at t f The spacecraft's entry point needs to be aligned with the state of the transition orbit in the lunar inertial system x trans Therefore, the terminal state constraint is expressed as:

[0042] F(t f ,x(t f ))=x(t f )-x trans =0 (14)

[0043] The control of the spacecraft is provided by a small-thrust thruster, and the control constraints correspond to the equality constraints and inequality constraints.

[0044]

[0045] 0≤T≤T max (16) where T max is the maximum thrust, T x , T y and T z are the components of the thrust in the three-axis direction of the rotating system.

[0046] To avoid the spacecraft encountering singular points during integration, the moon-impact constraint condition is introduced:

[0047]

[0048] Thus, the time-optimal transfer problem from the start point to the rendezvous point is constructed:

[0049]

[0050] When solving the optimal control problem P0 numerically, the upper and lower bounds of the optimization variables are set as the artificial interval. For the mass adjoint variable, after normalization, the initial state point of the spacecraft in the lunar-sun rotating system is The corresponding value is 1.

[0051] The initial value of the normalized adjoint variable is randomly targeted by the Monte Carlo method, and the optimal control problem P0 of the low-thrust transfer trajectory is solved by the nonlinear programming solver to obtain the initial value of the trajectory in the lunar-sun rotating system in the three-body transfer phase

[0052] Step three: The initial guess of the low-thrust transfer time is obtained by equivalent time scaling. The departure orbit, transition orbit and moon-impact constraint conditions are given. According to the Pontryagin maximum principle, the adjoint variable is introduced to construct the spacecraft system Hamilton function of the time optimal performance index. The adjoint equation is obtained from the spacecraft system Hamilton function, which is used as the constraint condition of the transfer orbit. According to the departure orbit constraint condition, the transition orbit constraint condition, the moon-impact constraint and the transfer orbit constraint condition, combined with the dynamic equation of the low-thrust spacecraft motion in the lunar-sphere two-body model constructed in step one and the control constraint, the low-thrust transfer trajectory optimal control problem is constructed with the shortest time as the optimization objective. The low-thrust transfer trajectory optimal control problem P1 is solved numerically to obtain the trajectory in the lunar-sphere two-body transfer phase.

[0053] In the lunar-sphere two-body problem model, the initial state of the spacecraft is AE final , and the mass is assumed to be m0. The corresponding spacecraft terminal state of the problem solution is AE0. The state quantity is converted into the position and velocity quantity of the restricted three-body model, which can be obtained as and When the thruster is fully powered at the maximum thrust amplitude T max , the equivalent transfer time t1 from the target orbit to the transition orbit is solved according to the velocity difference:

[0054]

[0055] The transfer time initial guess Δt1 = kt1 is obtained by scaling the equivalent transfer time t1, where k is the amplification coefficient, and k = 1.3.

[0056] The initial state x(t0) of the spacecraft is specified by the target orbit point x final , and the constraint is represented as:

[0057]

[0058] where r final and v final are the terminal orbit at t fThe state x(t) represents the position and velocity of the spacecraft in the lunar-centered inertial frame.

[0059] The definition of x(t f ) is the state of the circumlunar orbit in the lunar-centered inertial frame corresponding to the spacecraft at time t f . The entry point of the spacecraft needs to be aligned with the state of the transition orbit in the lunar-centered inertial frame x'(t trans ), so the terminal state constraint is expressed as:

[0060] F(t f ,x(t f ))=x(t f )-x'(t trans )=0 (21)

[0061] Considering that the terminal orbit is very close to the moon, the moon-impact constraint should be added during the integration process with thrust control, that is, the semi-major axis of the orbit at any time is greater than the normalized moon radius 1, i.e., x(a)>1. For the two-body dynamics equation, according to the Pontryagin extremum principle, the co-state variable λ x is introduced corresponding to the state variable x(t x ), where λ p =[λ f ,λ g ,λ h ,λ k ,λ L ] T , and the time-optimal system Hamiltonian function is constructed:

[0062] where is the 6x3 matrix in the upper left corner of D, is expressed as:

[0063]

[0064] According to the Pontryagin minimum principle, the optimal control direction of the thrust is full thrust when , and the optimal control direction of the thrust is zero when

[0065]

[0066] According to the Lagrange principle, the differential equations of the co-state variable are as follows:

[0067]

[0068] Solving the time-optimal small-thrust orbit problem contains 7 optimization variables, which are the six-dimensional co-state variable λ x (t0) corresponding to the initial time t0 and the terminal time tf Since the costate variable has no specific physical meaning, the upper and lower bounds of each optimization variable are given artificially in the simulation process. The time-optimal problem P1 under the lunar central two-body model is expressed as:

[0069]

[0070] The homotopy process starts from the thrust amplitude of 1 N and gradually reduces the thrust amplitude. After each thrust reduction, the converged costate variable value of the previous solution is used to solve the small-thrust trajectory optimization problem with reduced thrust. The small-thrust trajectory optimization problem is repeatedly solved until a converged solution that satisfies the spacecraft parameters is obtained, and the corresponding two-body model trajectory is calculated.

[0071] Step four: The trajectory of the two-body transfer phase in the lunar central inertial system solved in step three is converted to the trajectory in the restricted three-body model through the coordinate conversion relationship. The state parameters of the trajectory end splicing point in the restricted three-body model are updated, and the splicing point x′ trans in the lunar central two-body model

[0072] Convert the trajectory value of the lunar central two-body model solved in step three to the geolunar restricted three-body model. Let the position and velocity quantities described by the Cartesian coordinate system in the restricted three-body model be X1 = [x, y, z, v x , v y , v z ] T , and the improved equinox orbit elements in the two-body model be X2 = [p, f, g, h, k, L] T . The conversion formula between vector X2 and vector X1 is as follows:

[0073]

[0074] where s 2 = 1 + h 2 + k 2 .

[0075] Get the trajectory initial value of the second stage in the geolunar rotating system Perform reverse processing to get the trajectory value equivalent to the positive integral of the transition orbit to the target orbit According to the conversion relationship, the x′ trans in the lunar central two-body model becomes

[0076] Step five: Take the splicing point obtained in step four as the terminal constraint, and the trajectory end splicing point xtrans As the initial value constraint, the small-thrust transfer trajectory optimal control problem under the circular restricted three-body model constructed in step one is solved, and the small-thrust transfer trajectory optimal control problem is solved. The trajectory of the three-body transfer stage in the earth-moon rotating system in step two and the transfer trajectory of the matching segment in step five are spliced to obtain the trajectory after splicing the three-body transfer stage and the matching segment, and the spliced trajectory and the trajectory under the restricted three-body model obtained in step four are further spliced to obtain the shortest-time small-thrust trajectory from the initial orbit to the target orbit, that is, the trajectory splicing from the earth-moon three-body orbit small-thrust transfer to the circumlunar orbit is realized.

[0077] Due to the difference in dynamic integration, and x trans between them, the transfer trajectory between x and x is solved using the earth-moon restricted three-body dynamics model, and the hybrid method is used for optimization.

[0078] The endpoint constraint of the optimization problem P0 is adjusted as follows:

[0079]

[0080] Since the small-thrust transfer between the splicing points is a transfer problem near the transition orbit, there is no need to add the moon-impact constraint. The optimization problem P2 between two points is constructed:

[0081]

[0082] By solving the above optimization problem, the transfer trajectory between x and x is obtained, the trajectory of the three-body transfer stage in the earth-moon rotating system in step two and the transfer trajectory of the matching segment in step five are spliced to obtain the trajectory after splicing the three-body transfer stage and the matching segment, and the spliced trajectory and the trajectory under the restricted three-body model obtained in step four are further spliced to obtain the shortest-time small-thrust trajectory from the initial orbit to the target orbit, that is, the trajectory splicing from the earth-moon three-body orbit small-thrust transfer to the circumlunar orbit is realized. trans and x T between them, the transfer trajectory between x and x is solved using the earth-moon restricted three-body dynamics model, and the hybrid method is used for optimization.

[0083] Beneficial effects:

[0084] 1. The trajectory splicing method from the earth-moon three-body orbit small-thrust transfer to the circumlunar orbit disclosed in the application takes the shortest time as the optimization index, solves the small-thrust trajectory optimization problem in sections, realizes precise docking in space position and velocity, can ensure smooth transition of the spacecraft between different orbit segments, reduce the calculation error of the earth-centered two-body dynamics model, and further ensure the safety of the spacecraft and the success of the mission.

[0085] ​​2. The trajectory splicing method for small-thrust transfer from a geocentric three-body orbit to a circumlunar orbit, which adopts a geocentric circular restricted three-body model and a lunar-centered two-body model, can switch models according to the dynamics of a spacecraft in different space intervals, improves the adaptability and accuracy of trajectory optimization, and can more accurately handle the dynamics problem in the geolunar gravity interaction zone after the spacecraft enters the lunar sphere of influence.

[0086] 3. The trajectory splicing method for small-thrust transfer from a geocentric three-body orbit to a circumlunar orbit, which uses different kinematic equations according to different dynamic characteristics of the transfer segment. In the two-body transfer segment, the spacecraft state is described using improved equinox parameters, avoiding the dramatic changes in parameters under position and velocity description, and increasing the convergence of the hybrid method in the solving process; in the three-body transfer segment, the spacecraft state is described using position and velocity parameters, which does not affect the convergence of the hybrid method while improving the trajectory splicing accuracy of small-thrust transfer from a geocentric three-body orbit to a circumlunar orbit. BRIEF DESCRIPTION OF DRAWINGS

[0087] Figure 1 is a flow chart of the trajectory splicing method for small-thrust transfer from a geocentric three-body orbit to a circumlunar orbit according to the present application.

[0088] Figure 2 is a schematic diagram of solving in the three-body transfer segment of the method.

[0089] Figure 3 is a schematic diagram of solving in the two-body transfer segment of the method.

[0090] Figure 4 is a schematic diagram of matching in the method.

[0091] Figure 5 is a time-optimal small-thrust transfer trajectory diagram for a distant retrograde orbit to a circumlunar orbit according to the method.

[0092] Figure 6 is a local enlarged view of the time-optimal small-thrust transfer trajectory diagram for a distant retrograde orbit to a circumlunar orbit according to the method. DETAILED DESCRIPTION

[0093] In order to better illustrate the purposes and advantages of the present application, the following further illustrates the content of the application in combination with the drawings and examples.

[0094] Example 1:

[0095] In order to verify the feasibility of the method, the geocentric rotational system initial value is selected as X0 = [0.8089, 0, 0, 0, 0.5154, 0] TThe Distant Retrograde Orbit (DRO) and the Low Lunar Orbit (LLO) are used to simulate the actual engineering task, and the parameters in the lunar center inertial system are shown in Table 1.

[0096] Table 1 Related parameters of the Low Lunar Orbit

[0097]

[0098] As Figure 1 shown, the trajectory splicing method for small-thrust transfer from the Earth-Moon three-body orbit to the Low Lunar Orbit is disclosed, and the specific implementation steps are as follows:

[0099] Step 1: Establish the Earth-Moon rotation coordinate system for describing the circular restricted three-body problem and the lunar center inertial system for describing the lunar center two-body problem, and construct the dynamic equation of small-thrust spacecraft motion in the circular restricted three-body model and the dynamic equation of small-thrust spacecraft motion in the lunar center two-body model. Given the radius of the splicing sphere, when designing the transfer from the Earth-Moon gravity interaction zone to the Low Lunar Orbit for small-thrust transfer problem, the transfer segment will be divided into three stages of three-body transfer segment, two-body transfer segment and matching segment according to the spatial position of the spacecraft relative to the lunar center; the transition orbit selection condition is given, and the splicing point position is determined according to the given splicing sphere, starting orbit parameters and target orbit parameters.

[0100] The Earth-Moon rotation coordinate system EMR (Earth Moon Rotation) is established, denoted as O emr x emr y emr z emr , the coordinate origin O emr is located at the common center of mass of the Earth-Moon system, the coordinate axis x emr points from the common center of mass to the center of mass of the Moon, x emr y emr coordinate plane is the relative motion plane of the two main celestial bodies, and the coordinate axis z emr is along the direction of the angular momentum, y emr forms a right-hand rule with the first two coordinate axes.

[0101] The dynamic equation of small-thrust spacecraft motion is expressed in vector form:

[0102]

[0103] In the above formula, the spacecraft state quantity is x = [x, y, z, v x , v y , v z , m] T , f(x, t) = [v x , vy ,v z ,a x ,a y ,a z ,0] T , (x, y, z) is the position vector, (v x ,v y ,v z ) is the velocity vector, and a x ,a y and a z are the acceleration components of the spacecraft in the lunar orbiting system, and m is the mass of the spacecraft. B(x, t) is a 7x4 control vector matrix. u = [T x ,T y ,T z ,T] T is the control vector, T x ,T y and T z are the thrust components along the three axes of the lunar orbiting system, and T is the thrust provided by the thruster, satisfying the constraint The specific expressions of the acceleration three-axis components a x ,a y ,a z and B(x, t) are as follows:

[0104]

[0105] where 0 m×n represents a zero matrix of m rows and n columns, I 3×3 represents a 3x3 identity matrix, r1 and r2 represent the normalized distances from the spacecraft to the Earth and the Moon, respectively, and are expressed as:

[0106]

[0107] The Moon Central Inertial (MCI) coordinate system is denoted as O mci x mci y mci z mci , with the origin O mci located at the center of mass of the Moon, the x mci axis pointing to the vernal equinox of the Moon, the x mci y mci plane coinciding with the Earth equatorial plane, the z eci axis pointing to the north pole along the Moon's rotation axis, and the y mci axis satisfying the right-hand rule with the x mci and z mci axes. Under the improved vernal equinox model, the dynamics equation of the small thrust spacecraft motion is expressed in matrix form as:

[0108]

[0109] where the state variables are composed of the six-dimensional modified equinoctial elements and the mass m, x = [p, f, g, h, k, L, m] T , M(x, t) is a 7x1 column vector, and D(x, t) is a 7x4 matrix. The corresponding specific expressions are:

[0110]

[0111] where μ c is the gravitational constant of the central celestial body, I sp is the specific impulse provided by the thruster, g0=9.8 m / s 2 is the sea level weight acceleration of the Earth.

[0112] U = [T r , T t , T n , T] T is the control vector, satisfying the constraint represents in the RTN coordinate system with the spacecraft center as the origin, the R axis is defined as the radial axis pointing from the central celestial body to the spacecraft, the N axis is the direction of the angular momentum, and the T axis forms a right-hand rule.

[0113] After the spacecraft enters the lunar sphere of influence, the transfer trajectory constructed by the Earth-centered two-body model has obvious limitations, especially in the mission of transferring the spacecraft to a circumlunar orbit. Due to the significant influence of the Moon's gravity on the spacecraft in the near-Moon region, the integration process of the Earth-centered two-body model will bring significant errors. In order to reduce the significant errors brought by the integration process of the Earth-centered two-body model, the spherical surface distance from the Moon center is A times the Moon radius. Therefore, for the transfer of the Earth-Moon gravity region orbit to the circumlunar orbit, the dynamics model of the spacecraft in different spatial intervals needs to be distinguished. According to the spatial position of the spacecraft relative to the Moon center, the transfer segment is divided into three-body transfer segment, two-body transfer segment and matching segment. The matching segment refers to the re-solution of the trajectory end error of the three-body transfer segment and the two-body transfer segment. The low-thrust orbit located in the matching sphere in space belongs to the two-body transfer segment, and the spacecraft dynamics integration model is the Moon-centered two-body model; except for the matching sphere, the part belongs to the Earth-Moon gravity interaction region in space, i.e. the three-body transfer segment and the matching segment, and the dynamics model uses the Earth-Moon circular restricted three-body model. By using the Moon-centered two-body model instead of the Earth-centered two-body model for integration in the two-body transfer segment, and by using the circular restricted three-body model instead of the Earth-centered two-body model for integration in the three-body transfer segment and the matching segment, the dynamics integration result is more consistent with the spacecraft's motion trajectory in the space environment, thereby reducing the significant errors brought by the integration process of the Earth-centered two-body model.

[0114] Further, the transition orbit selection condition is constructed according to the splicing spherical surface and the target orbit parameter, and the splicing point for trajectory splicing is selected by using the transition orbit selection condition:

[0115] Condition 1: The splicing orbit surface should be consistent or approximately consistent with the target orbit surface, that is, in the lunar center inertia system, the orbit inclinations of the two should be approximately the same.

[0116] Condition 2: The splicing orbit semi-major axis is given as AR m Within this radius range, due to the action of the lunar gravity, the low-thrust orbit presents the characteristics of dense multi-turn distribution, and the radius is an approximate index for distinguishing the integration of the spacecraft under different dynamic models.

[0117] Condition 3: The transition orbit is consistent with the target orbit on the arch line, that is, the ascending node right ascension of the two is the same. In order to save fuel consumption, the eccentricity of the initial orbit to the target orbit is not adjusted during the transfer process.

[0118] According to condition 3, the orbit elements of the transition orbit in the lunar center inertia system determined by the target orbit are represented as: AE0=[a0,e0,i0,Ω0,~,~] T Further, according to the regulation, for the transition orbit of the starting orbit, the orbit must have the same orbit inclination and ascending node right ascension as the starting orbit in the lunar center inertia system, and the eccentricity of the orbit is e0, then AE1=[a1,e1,i1,Ω1,~,~] T The splicing point position on the transition orbit for the three-body transfer and two-body transfer connection is determined through the intersection of the two transition orbits on the splicing spherical surface, so as to realize the accurate splicing of the trajectory in space position and velocity.

[0119] Step two: According to the Pontryagin extremum principle, the spacecraft system Hamilton function of the time optimal performance index is constructed by introducing the co-state variable, and the co-state equation is obtained according to the spacecraft system Hamilton function, which is taken as the constraint condition of the transfer orbit; the starting orbit and the transition orbit constraint condition are given; according to the starting orbit constraint condition, the transition orbit constraint condition and the transfer orbit constraint condition, the dynamic equation of the low-thrust spacecraft motion under the circular restricted three-body model and the control constraint are constructed in combination with the dynamic equation of the low-thrust spacecraft motion under the circular restricted three-body model and the control constraint constructed in step one, and the shortest time is taken as the optimization objective to construct the low-thrust transfer trajectory optimal control problem. The low-thrust transfer trajectory optimal control problem P0 is solved numerically to obtain the trajectory of the three-body transfer stage in the lunar-rotating system.

[0120] According to the Pontryagin extremum principle, the spacecraft system Hamilton function of the time optimal performance index is constructed by introducing the co-state variable The spacecraft system Hamilton function of the time optimal performance index is constructed as:

[0121]

[0122] According to the circular restricted three-body dynamics equation (1) and Take Get the co-state equation:

[0123]

[0124] Where, and The specific expression of the non-zero elements in the 3x3 gradient matrix is as follows:

[0125]

[0126] The construction of the optimal time orbit problem of small thrust to the splicing point under the circular restricted three-body model is as follows:

[0127] According to the start and end positions of the transfer orbit, the integral time starting point t0 and the integral time ending point t f Constraints are made on the state quantities of the spacecraft.

[0128] The initial state x(t0) of the spacecraft is given by the DRO specified starting point, and the constraint is expressed as:

[0129]

[0130] Where, r DRO and v DRO are the position and velocity of the DRO orbit at t0 in the lunar rotating system.

[0131] Define x(t f ) as the orbit state of the spacecraft in the lunar inertial system after integration at t f The spacecraft's entry point needs to be aligned with the state quantity Therefore, the terminal state constraint is expressed as:

[0132]

[0133] The control of the spacecraft is provided by the small thrust thruster, and the control constraints correspond to the equality constraints and inequality constraints.

[0134]

[0135] 0≤T≤T max (16) Where T max is the maximum thrust, T x , T y and T z are the components of the thrust in the three-axis direction of the rotating system.

[0136] To avoid the spacecraft encountering singularities in the integration process, the lunar impact constraint condition is introduced:

[0137]

[0138] The time-optimal transfer problem from the starting point to the splicing point is constructed as follows:

[0139]

[0140] When the problem P0 is simulated numerically, the upper and lower bounds of the optimization variables are set to [-1 × 10 8 , 1 × 10 8 ], and for the mass adjoint variable, after normalization, the initial state point of the spacecraft corresponds to a value of 1.

[0141] The Monte Carlo method is used to randomly target the normalized initial value of the adjoint variable, and the nonlinear programming solver is used to solve the small-thrust transfer trajectory optimal control problem P0, obtaining the trajectory initial value of the three-body transfer phase in the Earth-Moon rotating system

[0142] Step three: The initial value of the small-thrust transfer time is obtained by equivalent time scaling. The departure orbit, transition orbit, and lunar impact constraint condition are given. According to the Pontryagin maximum principle, the adjoint variable is introduced to construct the spacecraft system Hamilton function of the time-optimal performance index. The adjoint equation is obtained from the spacecraft system Hamilton function, which is used as the constraint condition of the transfer orbit. According to the departure orbit constraint condition, the transition orbit constraint condition, the lunar impact constraint, and the transfer orbit constraint condition, combined with the dynamic equation of the small-thrust spacecraft motion in the lunar two-body model constructed in step one and the control constraints, the small-thrust transfer trajectory optimal control problem is constructed with the shortest time as the optimization objective. The small-thrust transfer trajectory optimal control problem P1 is solved numerically to obtain the trajectory of the two-body transfer phase in the lunar inertial system.

[0143] In the lunar two-body problem model, the initial state of the spacecraft is AE LLO , and the mass is assumed to be m0. The corresponding spacecraft terminal state of the problem solution is AE HLO . The state quantity is converted to the position and velocity quantity corresponding to the restricted three-body model, which can be obtained as and Assuming that the thruster is full-thrust at the maximum thrust amplitude T max , the equivalent transfer time t1 of the spacecraft from the terminal orbit to the transition orbit is solved according to the velocity difference:

[0144]

[0145] The transfer time initial value guess Δti = kti is obtained by scaling the equivalent transfer time ti, where k is the scaling factor, and k = 1.3.

[0146] The initial state x(t0) of the spacecraft is given by the target orbit specified point x LLO The constraint is expressed as:

[0147]

[0148] where r LLO and v LLO are the position and velocity of the terminal orbit at time t f in the lunar center inertial system.

[0149] Let x(t f ) be the orbit state of the spacecraft in the lunar center inertial system corresponding to the orbit at time t f after integration. The entry point of the spacecraft needs to be aligned with the state quantity x′ trans of the transition orbit in the lunar center inertial system, so the terminal state constraint is expressed as:

[0150] F(t f , x(t f )) = x(t f )-x HLO = 0 (21)

[0151] Considering that the terminal orbit is very close to the moon, the moon-impact constraint should be added during the integration process with thrust control, and the orbit semi-major axis at any time is greater than the normalized moon radius 1, i.e., x(a) > 1. For the two-body dynamics equation, according to the Pontryagin extremum principle, the co-state variable λ x is introduced corresponding to the state quantity, where λ x = [λ p , λ f , λ g , λ h , λ k , λ L ] T The time-optimal system Hamiltonian function is constructed:

[0152]

[0153] where is the 6x3 matrix in the upper left corner of D, is expressed as:

[0154]

[0155] According to Pontryagin minimum principle, the optimal control direction of the full thrust satisfies The following can be obtained

[0156]

[0157] According to the Lagrange principle, the differential equations of the co-state variables are as follows:

[0158]

[0159] Solving the time-optimal low-thrust orbit problem contains 7 optimization variables, which are the six-dimensional co-state variables λ x (t0) corresponding to the initial time t0 and the terminal time t f , respectively. Since the co-state variables have no specific physical meaning, the upper and lower bounds of each optimization variable are given as (-1x10 8 , 1x10 8 ) in the simulation process. Solving the time-optimal problem P1 under the lunar central two-body model is expressed as:

[0160]

[0161] The homotopy process starts from a thrust amplitude of 1N and gradually reduces the thrust amplitude. After each thrust reduction, the co-state variables of the previous solution are used to iteratively solve the low-thrust trajectory optimization problem under the reduced thrust. Repeat the solution of the low-thrust trajectory optimization problem until a convergent solution that satisfies the spacecraft parameters is obtained, and the corresponding two-body model trajectory is calculated.

[0162] Step four: convert the trajectory of the lunar central inertial system under the two-body transfer phase solved in step three to the trajectory under the restricted three-body model through the coordinate conversion relationship, update the state parameters of the trajectory end splicing point in the restricted three-body model under the Earth-Moon rotating system, and change the splicing point x trans in the lunar central two-body model to the splicing point

[0163] Convert the trajectory value of the lunar central two-body model solved in step three to the Earth-Moon restricted three-body model, and denote the position and velocity quantities described by the Cartesian coordinate system in the restricted three-body model as X1 = [x, y, z, v x , v y , v z ] T and the improved equinox orbit elements in the two-body model as X2 = [p, f, g, h, k, L] T . The conversion formula between vector X2 and vector X1 is as follows:

[0164]

[0165] Among them, s 2 =1+h 2 +k 2 .

[0166] The initial values ​​of the trajectory in the second stage in the Earth-Moon rotating frame were obtained. Reverse the process to obtain the trajectory value, which is equivalent to the positive integral of the transition trajectory towards the target trajectory. Corresponding to the transformation relationship, x′ under the lunar two-body model trans Become Corresponding to the transformation relationship, x under the lunar two-body model HLO Become

[0167] Step 5: Connect the splicing points obtained in Step 4 As a terminal constraint, the trajectory end splicing point x of the three-body transfer stage in the Earth-Moon rotating system obtained in step two is... trans As initial constraints, the optimal control problem of the small-thrust transfer trajectory under the circular restricted three-body model constructed in step one is solved. The trajectory of the three-body transfer stage under the Earth-Moon rotating system in step two and the matching segment transfer trajectory in step five are spliced ​​together to obtain the trajectory after splicing the three-body transfer stage and the matching segment. The spliced ​​trajectory is further spliced ​​together with the trajectory under the restricted three-body model obtained in step four to obtain the shortest time small-thrust trajectory from the initial orbit to the target orbit, that is, to realize the trajectory splicing from the Earth-Moon three-body orbit to the lunar orbit with small thrust.

[0168] Due to the difference in kinetic integrals and x trans Errors exist; a restricted three-body dynamics model based on the Earth and Moon is used to solve the problem. and x trans The transfer trajectories between them were determined and optimized using a hybrid method.

[0169] The endpoint constraints of the optimization problem P0 are adjusted as follows:

[0170]

[0171] Since the small thrust transfer between the splicing points is a transfer problem near the transition trajectory, there is no need to add lunar impact constraints. Construct the optimization problem P2 between the two points:

[0172]

[0173] By solving the above optimization problem, we obtain and x transThe trajectory of the step two is spliced with the trajectory of the step five, and the trajectory of the step four is spliced with the trajectory of the step five, so as to obtain the shortest time small-thrust trajectory from the initial orbit to the target orbit, and realize the precise docking in the space position and the speed.

[0174] The above description is only a specific embodiment of the present application, and is not used to limit the protection scope of the present application. Any modification, equivalent replacement, improvement, etc. within the spirit and principle of the present application shall be included in the protection scope of the present application.

Claims

1. A trajectory stitching method for transferring from a lunar elliptic orbit to a lunar halo orbit, the method comprising: The method comprises the following steps: ​ Step one: establish the lunar-synchronous coordinate system describing the circular restricted three-body problem and the lunar-centered inertial system describing the lunar-centered two-body problem, respectively construct the dynamics equation of the small-thrust spacecraft motion under the circular restricted three-body model and the dynamics equation of the small-thrust spacecraft motion under the lunar-centered two-body model; given the radius of the splicing sphere, when designing the small-thrust transfer problem in the region of the joint action of the earth and the moon gravity to the circumlunar orbit transfer, the transfer section will be divided into three stages, i.e. the three-body transfer stage, the two-body transfer stage and the matching stage, according to the spatial position of the spacecraft relative to the lunar center; give the transition orbit selection condition, and determine the splicing point position according to the given splicing sphere, the starting orbit parameters and the target orbit parameters; Step two: according to the Pontryagin extremum principle, introduce the co-state variable to construct the spacecraft system Hamilton function of the time optimal performance index, obtain the co-state equation according to the spacecraft system Hamilton function, and take the co-state equation as the constraint condition of the transfer orbit; given the starting orbit and the transition orbit constraint condition; according to the starting orbit constraint condition, the transition orbit constraint condition and the transfer orbit constraint condition, combine the dynamics equation of the small-thrust spacecraft motion under the circular restricted three-body model and the control constraint constructed in step one, take the shortest time as the optimization objective, and construct the small-thrust transfer trajectory optimal control problem; solve the small-thrust transfer trajectory optimal control problem P0 numerically, and obtain the trajectory of the three-body transfer stage in the lunar-synchronous coordinate system; Step three: obtain the small-thrust transfer time initial value guess through equivalent time scaling, given the starting orbit, the transition orbit and the moon-impact constraint condition, according to the Pontryagin extremum principle, introduce the co-state variable to construct the spacecraft system Hamilton function of the time optimal performance index, obtain the co-state equation according to the spacecraft system Hamilton function, and take the co-state equation as the constraint condition of the transfer orbit; according to the starting orbit constraint condition, the transition orbit constraint condition, the moon-impact constraint and the transfer orbit constraint condition, combine the dynamics equation of the small-thrust spacecraft motion under the lunar-centered two-body model and the control constraint constructed in step one, take the shortest time as the optimization objective, and construct the small-thrust transfer trajectory optimal control problem; solve the small-thrust transfer trajectory optimal control problem P1 numerically, and obtain the trajectory of the two-body transfer stage in the lunar-centered inertial system; Step four: convert the trajectory of the two-body phase in the lunar-centered inertial system to the trajectory in the restricted three-body model through the coordinate conversion relationship, update the state parameters of the splicing point at the end of the trajectory in the restricted three-body model in the lunar- earth rotating system, and convert the splicing point x′ trans in the lunar-centered two-body model into the splicing point Step five: the splicing point of the trajectory of the matching segment in step four and the trajectory of the three-body transfer phase in step two is obtained As the terminal constraint, the splicing point x of the trajectory of the three-body transfer phase in step two in the lunar rotating coordinate system is obtained trans As the initial value constraint, the optimal control problem of the small-thrust transfer trajectory in the circular restricted three-body model constructed in step one is solved, and the optimal control problem of the small-thrust transfer trajectory is solved; the trajectory of the three-body transfer phase in step two and the matching segment transfer trajectory in step five are spliced to obtain the trajectory after splicing the three-body transfer phase and the matching segment, and the spliced trajectory and the trajectory in the restricted three-body model obtained in step four are further spliced to obtain the shortest-time small-thrust trajectory from the initial orbit to the target orbit, that is, the trajectory splicing from the small-thrust transfer from the lunar three-body orbit to the circumlunar orbit is realized.

2. The transfer trajectory stitching method from a LEO to a halo orbit according to claim 1, wherein: The implementation method of step one is, An Earth-Moon rotating coordinate system, EMR, is established, denoted as O emr x emr y emr z emr , with its origin O emr located at the common center of mass of the Earth-Moon system, its x emr axis pointing from the common center of mass to the center of mass of the Moon, its x emr y emr coordinate plane being the plane of relative motion of the two principal celestial bodies, and its z emr axis being along the direction of the angular momentum of rotation, with y emr forming a right-hand rule with the first two axes. The dynamics equation of the small-thrust spacecraft motion is expressed in vector form: The spacecraft state quantity in the above formula is x = [x, y, z, v x ,v y ,v z ,m] T , f(x, t) = [v x ,v y ,v z ,a x ,a y ,a z ,0] T , (x, y, z) is a position quantity, (v x ,v y ,v z ) is a velocity quantity, and a x , a y , and a z correspond to acceleration components of the spacecraft on three axes of the earth-moon rotating system, and m is the mass of the spacecraft; B(x, t) is a 7x4 control vector matrix; u = [T x ,T y ,T z ,T] T is a control vector, T x , T y , and T z are thrust components along three axes of the earth-moon rotating system, T is the thrust size provided by a thruster, and satisfies the constraint The specific expressions of the acceleration three-axis components a x , a y , a z , and B(x, t) are as follows: where 0 m×n represents an m x n zero matrix, I 3×3 represents a 3 x 3 identity matrix, and r1and r2represent the normalized distances of the spacecraft from the Earth and the Moon, respectively, given by The lunar central inertial coordinate system (MCI) is denoted as O. mci x mci y mci z mci Origin of coordinate system mci Located at the Moon's center of mass, x mci The axis points to the vernal equinox of the moon, x mci y mci The plane coincides with the Earth's equatorial plane, z mci The axis points to the North Pole along the Moon's rotation axis, y mci axis and x mci axis, z mci The axes satisfy the right-hand rule; under the improved vernal equinox model, the dynamic equations of motion for a low-thrust spacecraft are expressed in matrix form: where the state variables are composed of the six-dimensional modified equinox root number and the mass m, x = [p, f, g, h, k, L, m] T , M(x, t) is a 7 1 column vector, and D(x, t) is a 7 4 matrix; the corresponding specific expression is: where μ c is the gravitational constant of the central gravitating body, I sp is the specific impulse provided by the thruster, g0= 9.8 m / s 2 is the Earth sea level weight acceleration, s 2 = 1 + h 2 + k 2 ; U = [T r ,T t ,T n ,T] T is the control vector, satisfying the constraint R represents the RTN coordinate system with the spacecraft center as the origin, R axis is defined as the radial axis pointing to the direction from the central celestial body to the spacecraft, N axis is the direction of angular momentum, and T axis constitutes the right-hand rule; The patched sphere surface is A times the radius of the moon from the center of the moon; therefore, for the transfer from the earth-moon gravity zone orbit to the lunar orbit, the dynamics model of the spacecraft in different space zones needs to be distinguished; the transfer segment is divided into three stages according to the space position of the spacecraft relative to the moon center, i.e., the three-body transfer segment, the two-body transfer segment and the matching segment; the matching segment refers to the re-solution of the trajectory end error of the three-body transfer segment and the two-body transfer segment; the small-thrust orbit in the patched sphere surface is the two-body transfer segment, and the dynamics integral model of the spacecraft is the moon center two-body model; except for the patched sphere surface, the part belongs to the earth-moon gravity zone in space, i.e., the three-body transfer segment and the matching segment, and the dynamics model adopts the earth-moon circular restricted three-body model; by using the moon center two-body model instead of the earth center two-body model for integration in the two-body transfer segment, and by using the circular restricted three-body model instead of the earth center two-body model for integration in the three-body transfer segment and the matching segment, the dynamics integral result is more consistent with the motion trajectory of the spacecraft in the space environment, and thus the significant error caused by the integral process of the earth center two-body model is reduced; According to the patched sphere surface and the target orbit parameters, a transition orbit selection condition is constructed, and the patched point for trajectory splicing is selected by using the transition orbit selection condition: Condition 1: The patched orbit plane should be consistent or approximately consistent with the target orbit plane, i.e., in the moon center inertial system, the orbit inclination angles of the two should be approximately the same; Condition 2: The semi-major axis of the patched orbit is given as AR m ; within this radius, the low-thrust orbit presents a dense multi-turn distribution due to the lunar gravity, which is an approximate indicator to distinguish the spacecraft integrations under different dynamical models; Condition 3: The transition orbit and the target orbit are consistent on the arch line, i.e., the ascending node right ascensions of the two are the same; in order to save fuel consumption, the eccentricity of the initial orbit to the target orbit is not adjusted by using small thrust in the transfer process; According to condition 3, the orbit elements of the transition orbit determined by the target orbit in the lunar center inertial system are represented as: AE0=[a0,e0,i0,Ω0,~,~] T Further according to the regulation, for the transition orbit of the departure orbit, the orbit must have the same orbital inclination and ascending node right ascension as the orbit plane of the departure orbit in the lunar center inertial system, and the eccentricity of the orbit is e0, then AE1=[a1,e1,i1,Ω1,~,~] T The splicing point position for the three-body transfer and the two-body transfer on the transition orbit is determined through the intersection of the two transition orbits on the splicing sphere, so as to realize the accurate splicing of the trajectories in the space position and the speed.

3. The transfer trajectory stitching method from a LEO to a halo orbit according to claim 2, wherein: The implementation method of step two is as follows: According to the Pontryagin extremum principle, the costate variable is introduced The Hamilton function of the spacecraft system with time-optimal performance index is constructed as According to the circular restricted three-body dynamics equation (1) and Taking The co-state equation is obtained: wherein and is a 3x3 gradient matrix, with the specific expressions of the non-zero elements being: The time-optimal orbit problem of the starting orbit to the patched point under the circular restricted three-body model is constructed, and the specific method is as follows: Depending on the start and end position of the transfer orbit, it is necessary to have a start of integration time t0 and an end of integration time t f Constraints are made on the state quantities of the spacecraft; The initial state x(t0) of the spacecraft is given by the departure orbit specifying the departure point, the endpoint constraint of which is x ini The constraint is then expressed as: where r ini and v ini are the position and velocity of the departure orbit at time t0in the geocentric inertial system. Definition of x(t) f ) as the state of the circumlunar orbit in the lunar-centered inertial frame at the time t f at which the spacecraft passes through. The entry point of the spacecraft needs to be aligned with the state of the transfer orbit in the lunar-centered inertial frame x trans , thus the terminal state constraint is expressed as: F(t f ,x(t f )) = x(t f )- x trans = 0 (14) The control of the spacecraft is provided by the small-thrust thruster, and the control constraints correspond to the equality constraints and the inequality constraints; 0≤T≤T max (16) Wherein, T max T represents the maximum thrust. x ,T y and T z These represent the magnitudes of the thrust components along the three axes of the rotating system; In order to avoid the spacecraft encountering singular points in the integral process, a moon-impact constraint condition is introduced: The time-optimal transfer problem from the starting point to the patched point is constructed: When solving the optimal control problem P0 of low-thrust transfer trajectory numerically, the upper and lower bounds of the optimization variables are set as artificial intervals. For the mass adjoint variable, the initial state point of the spacecraft is normalized, and the corresponding value is 1. 1; The normalized co-state variable initial value is randomly targeted by using the Monte Carlo method, and the optimal control problem P0 of the small-thrust transfer trajectory is solved by a nonlinear programming solver to obtain the trajectory initial value of the three-body transfer phase in the rotating lunar-terrestrial system 4. The transfer trajectory stitching method from a LEO to a halo orbit according to claim 3, wherein: The implementation method of step three is as follows: Under the model of the lunar central two-body problem, the initial state of the spacecraft is AE final , and the mass is assumed to be m0; the corresponding final state of the spacecraft is AE0after solving the problem; the state quantity is converted into the position and velocity quantity of the restricted three-body model, which can obtain and When the thruster is fully powered with the maximum thrust amplitude T max , the equivalent transfer time t1of the spacecraft from the target orbit to the transition orbit is solved according to the velocity difference: The equivalent transfer time t1 is scaled to obtain the initial value guess Δt1=k t1, wherein k is the amplification coefficient; The initial state x(t0) of the spacecraft is given by the target orbit specification point x final The constraints are then expressed as: where r final and v final are the position and velocity of the end orbit at time t f in the lunar center inertial system; Define x(t) f ) represents the spacecraft after integration at t f The orbital state of the lunar orbit in the lunar inertial frame at the corresponding moment; the spacecraft's insertion point needs to be aligned with the state variable x′ of the transition orbit in the lunar inertial frame. trans Therefore, the terminal state constraint is expressed as: F(t f ,x(t f )) = x(t f )-x′ trans = 0 (21) Considering the close distance between the end orbit and the moon, the constraint condition of the moon should be added in the integral process of the thrust control, and the semi-major axis of the orbit at any time is greater than the normalized moon radius 1, that is, x(a) > 1; for the two-body dynamics equation, according to the Pontryagin extremum principle, the co-state variable λ corresponding to the state variable is introduced x wherein Construct the Hamilton function of the time optimal system: wherein is the top-left 6x3 matrix in D, is represented as: According to the Pontryagin minimum principle, the optimal control direction of the thrust satisfies is obtained According to the Lagrange principle, the differential equations of the covariant variables are as follows: The time-optimal low-thrust orbit problem contains 7 optimization variables, which are the six-dimensional co-state variables λ x (t0) corresponding to the initial time t0 and the final time t f Since the co-state variables have no specific physical meaning, the upper and lower bounds of each optimization variable are given artificially in the simulation process; the time-optimal problem P1 under the lunar two-body model is expressed as: The thrust amplitude homotopy continuation method is adopted, the homotopy process starts from the thrust amplitude of 1N, and the thrust amplitude is gradually reduced; after each thrust reduction, the covariant variable value converged in the previous solution is used to iteratively solve the small-thrust trajectory optimization problem under the reduced thrust; the small-thrust trajectory optimization problem is repeatedly solved until a converged solution satisfying the spacecraft parameters is obtained, and the corresponding two-body model trajectory is calculated.

5. The transfer trajectory stitching method from a LEO to a halo orbit according to claim 4, wherein: The implementation method of step four is as follows: Convert the trajectory value of the lunar center two-body model to the geocentric lunar restricted three-body model, and record the position and velocity value described by the Cartesian coordinate system in the restricted three-body model X2 = [p, f, g, h, k, L] is the improved vernal equinox orbit root number in the two-body model T Equivalent conversion; the conversion formula of vector X2 and vector X1 is as follows: The second stage trajectory initial value in the moon-earth rotating system is obtained The inverse sequence is processed to obtain the trajectory value corresponding to the positive integration of the target orbit at the transition orbit Corresponding to the conversion relationship, the x′ of the moon-centered two-body model trans becomes 6. The transfer trajectory stitching method from a LEO to a halo orbit according to claim 5, wherein: The implementation method of step five is as follows: Due to the difference in dynamics integration, and x trans there is an error between, use the earth-moon restricted three-body dynamics model to solve and x trans between the transfer orbit, and use the hybrid method to optimize; The endpoint constraints of the optimization problem P0 are adjusted as follows: Since the small-thrust transfer between the patched points is a transfer problem near the transition orbit, the moon-impact constraint does not need to be added; the optimization problem P2 between two points is constructed: By solving the above optimization problem, the transfer orbit between x and x trans is obtained, the trajectory of the step two lunar three-body transfer phase in the lunar rotating coordinate system is spliced with the matching segment transfer trajectory of step five, the spliced trajectory of the three-body transfer phase and the matching segment is further spliced with the trajectory obtained in step four under the restricted three-body model, the shortest time small thrust trajectory from the initial orbit to the target orbit is obtained, and precise docking in space position and velocity is realized; the shortest time small thrust trajectory from the initial orbit to the target orbit is obtained, that is, the trajectory splicing from the small thrust transfer of the lunar three-body orbit to the lunar orbit is realized.

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