A method for designing a transfer orbit from the earth to the moon based on the jacobi constant classification

Abstract

The application discloses a kind of DRO transfer orbit design method based on Jacobian constant classification method, solve how to design DRO transfer orbit in earth-moon space based on Jacobian constant classification method, belong to transfer orbit design field;Including: based on CRTBP dynamics model and escape judgment condition, determine the transfer orbit of spacecraft running in earth-moon system as target orbit;Based on target Jacobian constant falls into the interval corresponding to Jacobian constant of zero velocity surface, the target orbit is classified;The LEO departure transfer orbit and DRO arrival transfer orbit in the same interval that meet the orbit splicing condition are spliced as candidate feasible solution;Extract candidate feasible solution as initial feasible solution;Using sequential quadratic programming to the optimization variable corresponding to initial feasible solution is corrected, and all convergent earth-moon space DRO transfer orbit parameters are obtained;The application realizes earth-moon space DRO transfer orbit design by Jacobian constant classification method.

Description

Technical Field

[0001] This invention belongs to the field of Earth-Moon space DRO transfer orbit design technology, and relates to a method for designing Earth-Moon space DRO transfer orbits based on the Jacobian constant classification method. Background Technology

[0002] The Earth-Moon system refers to the celestial system formed by the gravitational interaction between the Earth and the Moon. The Moon is Earth's only natural satellite, and the two orbit around a common center of mass (located inside the Earth, about 4.671 kilometers from the Earth's center).

[0003] As humanity's exploration of space deepens, the Earth-Moon space has gradually become the focus of exploration, with the Moon and its orbit becoming the focal point. Dependent orbital (DRO) orbits are a special type of planar periodic symmetric orbit in the planar restricted three-body problem. In the Earth-Moon system, DROs are a family of orbits that retrogradely orbit the Moon. Larger DROs can even cover the Earth-Moon Lagrange points L1 and L2, and they possess long-term stability. Periodic DROs can operate for decades or even centuries without orbital maintenance. Therefore, they are considered important orbits for lunar exploration missions by various countries and may become "transfer stations" in Earth-Moon space transfers, used for the deployment of important missions such as Earth-Moon space stations, communication networks, and space telescopes.

[0004] The design of transfer orbits for entering and exiting the DRO will become an important part of lunar space exploration missions. Transfer modes can generally be divided into direct transfer, lunar-assisted transfer, solar-assisted (also known as weakly stable boundary) transfer, and invariant manifold transfer.

[0005] The selection of stitching points for transfer orbits varies across different models. For lunar-assisted transfer orbits and solar-assisted transfer orbits, the stitching point is usually chosen near the Moon. However, the pulse parameters are very sensitive to the distance of the stitching point from the Moon, and changes in the near-lunar distance of the stitching point may cause abrupt changes in the pulse parameters. For direct transfer orbits and invariant meteor transfer orbits, the stitching point may be located far from the Earth and the Moon, making it impossible to determine the stitching point when performing orbit prediction and coarse stitching. This may result in the rejection of orbits that could be coarsely stitched during coarse stitching screening.

[0006] For direct transfers where the splicing point cannot be directly determined, Lambert's theorem is usually used to solve for three-dimensional out-of-plane transfer trajectories. However, Lambert's theorem cannot be used to solve for lunar or solar gravitational transfers because it does not consider lunar or solar gravity. Furthermore, the Lambert's theorem solution method faces difficulties in guessing initial values ​​when dealing with odd numbers. Although this problem can be overcome using double-layer iterative algorithms and genetic algorithms, it is still impossible to obtain a large number of feasible transfer trajectory solutions.

[0007] For lunar-assisted transfers and solar-assisted transfers with the splicing point located near the Moon, a preliminary screening of the orbital splicing can be performed by dividing the near-lunar distance. For example, in Chao, Peng; Zhang, Hao; Wen Changxuan; Zhu Zhengfan; Gao, Yang. Exploring more solutions for low-energy transfers to lunar distant retrograde orbits[J]. Celestial Mechanics and Dynamical Astronomy, 2022, Vol. 134(1): 1-38, and Peng, Chao; Shang, Yunong; He, Shengmao; Zhu, Zhengfan; Wen, Changxuan. Low-Energy Transfers to Lunar Distant Retrograde Orbits from Geostationary Transfer Orbits[J]. JOURNAL OFSPACECRAFT AND ROCKETS, 2024, Vol. 61(5): 1293-1304, the lunar Hill sphere model is divided into 200 regions based on the radius and angle. Orbits falling into the same Hill sphere region are then coarsely stitched together, which can significantly reduce computational costs. For example, in Zhang Chen and Zhang Hao, "Low-Energy DRO Orbit Insertion Strategy Based on Lunar Leverage [J]. Acta Aeronautica Sinica, 2023, 44(2): 274-285", the orbits reaching the weak stability boundary are first screened by inverse integration of the state reaching the DRO, and then the orbits returning to Earth from the weak stability boundary are screened. The entire orbit design can be completed by calculating the integral of the orbit prediction, and there is no mid-course maneuver. In Zhang Chen, "Lunar-Solar Integrated Leverage DRO Orbit Insertion Strategy Based on Numerical Extension [J]. Journal of Beijing University of Aeronautics and Astronautics, 2024, 50(4): 1176-1186", the orbits reaching the weak stability boundary and returning to the Moon are screened by inverse integration of the DRO, and the splicing point is determined near the Moon. The orbit is coarsely spliced ​​based on the v-infinite matching method, and coarsely spliced ​​from the perspective of the velocity before and after lunar leverage, and then the orbit is corrected. However, the method of using the lunar distance as the selection criterion for orbit splicing has the disadvantage of requiring a large amount of data for screening coarse spliced ​​orbits, and it is only applicable to low-energy transfer orbits where spacecraft fly near the moon with lunar assistance or weak stability boundary. It cannot directly design direct transfer orbits or other types of transfer orbits where spacecraft do not pass near the moon.In the near-lunar distance screening method, each pulse parameter is sensitive to the near-lunar distance, which makes it difficult for the orbit splicing optimization problem to converge. Summary of the Invention

[0008] To address the technical problems of using lunar distance as a selection criterion for orbit stitching, such as the large amount of coarse stitched orbit data required, the inability to directly design direct transfer orbits and other transfer orbit types that do not pass near the Moon, and the sensitivity of various pulse parameters to lunar distance in the lunar distance selection method, leading to difficulties in converging the orbit stitching optimization problem, this invention proposes a lunar-Earth space DRO transfer orbit design method based on the Jacobi constant classification method. This method uses the Jacobi constant classification method to solve the transfer orbit design problem from Low Earth Orbit (LEO) to Distant Retrograde Orbit (DRO) based on pulse maneuvers in lunar space. It overcomes the sensitivity of pulse parameters to lunar distance in the lunar distance selection method, and can classify and stitch the transfer orbit database from the perspective of orbital energy using the Jacobi constant, thereby simplifying the complexity of orbit stitching and enabling the design of transfer orbits of different modes.

[0009] The objective of this invention is specifically achieved through the following technical solutions:

[0010] This invention discloses a method for designing Earth-Moon space DRO transfer orbits based on the Jacobian constant classification method. The method includes:

[0011] Step 1: Combining the CRTBP dynamic model, a seventh-order Runge-Kutta integrator is used to integrate the spacecraft's launch state vector and orbit insertion state vector to obtain the spacecraft's transfer orbit; based on the escape judgment condition, the spacecraft's transfer orbit operating within the Earth-Moon system is determined as the target orbit.

[0012] Step 2: Calculate the target Jacobian constant of the target orbit, and classify the target orbit based on the interval corresponding to the Jacobian constant of the zero velocity surface.

[0013] Step 3: Combining the CRTBP dynamic model, a seventh-order Runge-Kutta integrator is used to integrate the classified target spacecraft departure state vector along the forward time direction to obtain the LEO departure transfer orbit splicing point state; the target spacecraft insertion state vector is integrated along the backward time direction to obtain the DRO arrival transfer orbit splicing point state; the LEO departure transfer orbit and the DRO arrival transfer orbit within the same interval that satisfy the orbit splicing conditions are spliced ​​at the splicing point as candidate feasible solutions;

[0014] Step 4: Plot a scatter plot with the transition time of the candidate feasible solution as the x-axis and the sum of velocity pulses as the y-axis, and divide it into grids. Randomly select candidate feasible solutions as initial feasible solutions in each grid according to a preset ratio.

[0015] Step 5: Construct a trajectory correction model based on the objective function, optimization variables, and constraints. Use sequential quadratic programming to correct the trajectory of the optimization variables corresponding to the initial feasible solution in the input trajectory correction model, and output the objective feasible solution when the sum of velocity pulses is minimized.

[0016] Step 6: Determine whether the number of converged target feasible solutions within the grid meets the iteration conditions. If not, re-extract the initial feasible solutions; if yes, proceed with the iteration until the orbit corrections for all intervals are completed, and obtain all converged Earth-Moon space DRO transfer orbit parameters.

[0017] In step one, the CRTBP dynamic model is as follows:

[0018] ;

[0019] In the formula, This is the output of the CRTBP dynamic model. It is a time variable; For spacecraft in The first derivative of the position vector at time, For spacecraft in The first derivative of the velocity vector at time instant; For spacecraft in The velocity vector at time t, For spacecraft in Position vector at time, For Earth-Moon mass parameters, The distance of the spacecraft from Earth. , The distance of the spacecraft from the moon. ;

[0020] The global accuracy of the seventh-order Runge-Kutta integrator is eighth-order. Error control is based on seventh-order extrapolation. Error estimation is performed by comparing the difference between the eighth-order solution and the seventh-order solution, thereby achieving adaptive step size adjustment.

[0021] In step one, the method for calculating the spacecraft's launch state vector is as follows:

[0022] ;

[0023] ;in,

[0024] ;

[0025] The method for calculating the spacecraft's orbital state vector is as follows:

[0026] ;

[0027] ;

[0028] ;in,

[0029] ;

[0030] Spacecraft transfer orbit for: ;

[0031] In the formula, This is the LEO state vector corresponding to the spacecraft's location at the moment it leaves the LEO; Let i be the semi-major axis of the LEO orbit and i be the inclination angle of the LEO orbit. The right ascension of the ascending node of the LEO is... This is the true perihelion angle of the spacecraft at LEO; This is the LEO position vector corresponding to the spacecraft's location at the moment it leaves LEO. This is the LEO velocity vector corresponding to the spacecraft's location at the moment it leaves the LEO region. This is the spacecraft state vector at the moment of departure from LEO; The angle between the transmitted pulse and the LEO velocity vector. The angle between the transmitted pulse and the LEO orbital plane; Apply the launch pulse to the spacecraft's velocity vector. The size of the transmitted pulse;

[0032] This is the DRO state vector corresponding to the spacecraft's location at the moment of DRO insertion into orbit; This refers to the DRO's inlet phase; This is the DRO position vector corresponding to the spacecraft's location at the DRO moment of orbit insertion. This is the DRO velocity vector corresponding to the spacecraft's position at the moment of DRO insertion into orbit; This represents the spacecraft's state vector at the moment of DRO insertion. The magnitude of the orbital insertion pulse; The angle between the orbital insertion pulse and the DRO velocity vector. The angle between the orbital pulse and the DRO orbital plane; The velocity vector after applying the orbital insertion pulse to the spacecraft; It is a rotation matrix;

[0033] For spacecraft in Position vector at time, For spacecraft in The velocity vector at any given moment.

[0034] In step one, the escape judgment condition is:

[0035] .

[0036] In step two, the target Jacobian constant The calculation method is as follows:

[0037] ;

[0038] The Jacobian constant C at the zero velocity surface is:

[0039] ;

[0040] In the formula, This is the state vector of the target orbit, that is, the position vector and velocity vector of the spacecraft in the target orbit; The position vector of the spacecraft that is tangent to the zero-velocity surface and the DRO.

[0041] In step three, the calculation method for the state of the LEO departure transfer track splicing point is as follows:

[0042] ;

[0043] The calculation method for DRO reaching the transfer track splicing point is as follows:

[0044] ;

[0045] The conditions for track splicing are as follows:

[0046] ;

[0047] In the formula, Let LEO be the state vector of the transfer trajectory splicing point. The transfer time of the LEO departure transfer orbit is obtained by integrating the classified target spacecraft departure state vector along the positive time direction using a seventh-order Runge-Kutta integrator. Let be the position vector of the LEO departure transfer track splicing point. The velocity vector at the LEO departure transfer trajectory splicing point;

[0048] This is the state vector of the DRO upon reaching the transfer orbit splicing point. The transfer time of the DRO to the transfer orbit is the time obtained by integrating the classified target spacecraft's orbital state vector in reverse along time using a seventh-order Runge-Kutta integrator. This is the position vector of the DRO reaching the transfer track splicing point. The velocity vector of the DRO as it reaches the transfer track splicing point;

[0049] This represents the positional difference between the LEO departure transfer track splicing point state and the DRO arrival transfer track splicing point state within the same interval. The speed difference between the LEO starting transfer track splicing point state and the DRO arriving transfer track splicing point state within the same interval.

[0050] In step four, the calculation method for the transfer time and the sum of velocity pulses is as follows:

[0051] ;

[0052] In the formula, TOF represents the transition time; This is the sum of velocity pulses; For splicing pulses; with TOF as the x-axis and... Plot a scatter plot with the vertical axis as the ordinate, and use the TOF with the shortest transition time within that interval. and longest transfer time Divide the boundary into sub-intervals on an average basis, and then divide the sub-intervals into grids at preset intervals.

[0053] In step five, the orbital correction model is as follows:

[0054] ;in,

[0055] The objective function is: ;

[0056] The optimization variables are: ;

[0057] The constraints are: ;

[0058] In the formula, The output value of the objective function. This is the sum of velocity pulses; To optimize the derivative of the variable with respect to time; For inequality constraints, For equality constraints, The lunar position vector. The radius of the moon; To optimize variables, To optimize the boundary of variables, To optimize the lower bound of the variable, To optimize the upper bound of the variable; where,

[0059] .

[0060] In step five, sequential quadratic programming (SQP) is used to correct the trajectory of the optimization variables corresponding to the initial feasible solution in the input trajectory correction model, and the method for outputting the objective feasible solution that minimizes the sum of velocity impulses is as follows:

[0061] ;

[0062] In the formula, This represents the value of the optimization variable that minimizes the objective function of the trajectory correction model, i.e., the feasible solution for the objective when the sum of velocity impulses is minimized. The magnitude of the launch pulse applied by the spacecraft when the sum of velocity pulses is minimized. The angle between the transmitted pulse and the LEO velocity vector is the angle at which the sum of velocity pulses is minimized. The angle between the emitted pulse and the LEO orbital plane when the sum of velocity pulses is minimum. The LEO departure transfer orbit transfer time is when the total sum of velocity pulses is minimized. The magnitude of the orbital insertion pulse vector applied to the spacecraft when the sum of velocity pulses is minimized. The angle between the orbital pulse and the DRO velocity vector when the sum of velocity pulses is minimized. The angle between the insertion pulse and the DRO orbital plane when the sum of velocity pulses is minimum. The transfer time of DRO to the transfer track when the sum of velocity pulses is minimum.

[0063] In step six, the iteration condition is:

[0064] ;

[0065] In the formula, This represents the number of objective feasible solutions that converge within the grid. This represents the number of initially feasible solutions. This represents the number of iterations for the trajectory correction model.

[0066] The beneficial effects of this invention are:

[0067] 1. Calculate the target Jacobian constant of the spacecraft transfer orbit operating in the Earth-Moon system as the target orbit, and classify the target orbits according to the openness of the DRO phase of the zero-velocity surface Jacobian constant. Using the Jacobian constant classification method instead of the near-lunar distance screening method can classify orbits with similar orbital energy before the splicing step from the perspective of orbital energy, so that the candidate orbit solutions obtained by splicing are orbits with similar orbital energy, thereby making the splicing pulse smaller.

[0068] 2. The LEO departure transfer track and the DRO arrival transfer track within the same interval that meet the track splicing conditions are spliced ​​at the splicing point, so that the difference in Jacobian constant between the two transfer tracks is small, thereby reducing the energy difference between the two transfer tracks.

[0069] 3. Plot a scatter plot with the transition time of the candidate feasible solution as the x-axis and the sum of velocity pulses as the y-axis, and divide it into grids. Randomly select candidate feasible solutions as initial feasible solutions in each grid according to a preset ratio, so as to sparse candidate feasible solutions. This saves computation time while preserving the similar dynamic characteristics of the transfer trajectory patterns in each grid.

[0070] 4. Based on the objective function, the boundary of the trajectory correction optimization variables, and the constraints, a trajectory correction model is constructed. The upper and lower bounds of the variables in the optimization problem are reasonably set. Under the premise of not changing the trajectory transfer mode, the trajectory correction problem is constructed based on the constraints of trajectory splicing, so as to retain all transfer modes of the feasible solution candidates for spliced ​​trajectory.

[0071] 5. Sequential quadratic programming was used to correct the orbit of the optimization variables corresponding to the initial feasible solution in the input orbit correction model. The target feasible solution with the minimum sum of velocity pulses of the transfer orbit was obtained by solving the initial feasible solution extracted in each grid. Direct transfer orbit and lunar gravity transfer orbit were designed directly.

[0072] 6. Determining whether the number of feasible solutions to the objective at the time of network convergence satisfies the iteration condition improves the convergence of the orbit splicing optimization problem, thereby improving optimization efficiency and saving computation time. Attached Figure Description

[0073] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0074] Figure 1 This is a schematic diagram of the zero velocity surface and restricted area under different Jacobian constants provided in this invention.

[0075] Figure 2 This is the TOF- of the Jacobian constant interval provided in the example of this invention. Scatter plot diagram.

[0076] Figure 3 The target feasible solution TOF- is provided by the example of this invention. Scatter plot diagram.

[0077] Figure 4 This is a schematic diagram of the direct transfer method provided in the example of the present invention.

[0078] Figure 5 This is a schematic diagram of the inner lunar gravity transfer method provided in this invention.

[0079] Figure 6 This is a schematic diagram of the lunar transfer method provided in this invention.

[0080] Figure 7 This is a schematic diagram of two lunar transfer methods provided in this invention. Detailed Implementation

[0081] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0082] This invention provides a method for designing a Moon-Earth space DRO transfer orbit based on the Jacobian constant classification method. The method includes:

[0083] Step 1: Combining the CRTBP dynamic model, a seventh-order Runge-Kutta integrator is used to integrate the spacecraft's launch state vector and orbit insertion state vector to obtain the spacecraft's transfer orbit; based on the escape judgment condition, the spacecraft's transfer orbit operating within the Earth-Moon system is determined as the target orbit.

[0084] The CRTBP dynamics model represents the time derivatives of variables. The seventh-order Runge-Kutta integrator integrates these time derivatives to deduce the spacecraft's state. In essence, the CRTBP dynamics model is the logic of the seventh-order Runge-Kutta integrator. Integrating the spacecraft's launch and orbital insertion state vectors requires simultaneously inputting the CRTBP dynamics model's output, the spacecraft's initial state, and the integration time into the seventh-order Runge-Kutta integrator.

[0085] Preferably, the spacecraft launch state vector is positively integrated for 5 days, and the spacecraft orbit insertion state vector is negatively integrated for 5 days to determine whether the spacecraft state vector has escaped the Earth-Moon system, thus obtaining the spacecraft transfer orbit composed of the spacecraft launch state vector and the spacecraft orbit insertion state vector operating within the Earth-Moon system (a celestial system composed of the Earth and the Moon).

[0086] Step 2: Calculate the target Jacobian constant of the target orbit. Based on the target Jacobian constant falling within the interval corresponding to the Jacobian constant of the zero velocity surface, classify the target orbit. That is, classify the launch state vector and orbit insertion state vector of the spacecraft operating in the Earth-Moon system according to the openness of the DRO phase of the zero velocity surface.

[0087] Step 3: Combining the CRTBP dynamic model, a seventh-order Runge-Kutta integrator is used to integrate the classified target spacecraft departure state vector along the forward time direction to obtain the LEO departure transfer orbit splicing point state; the target spacecraft insertion state vector is integrated along the backward time direction to obtain the DRO arrival transfer orbit splicing point state; the LEO departure transfer orbit and the DRO arrival transfer orbit within the same interval that satisfy the orbit splicing conditions are spliced ​​at the splicing point as candidate feasible solutions;

[0088] Preferably, combining the CRTBP dynamic model, a seventh-order Runge-Kutta integrator is used to integrate the classified target spacecraft departure state vector along the forward time direction for 20 days to obtain the LEO departure transfer orbit; the target spacecraft insertion state vector is integrated along the backward time direction for 30 days to obtain the DRO arrival transfer orbit; the closest point between the LEO departure transfer orbit and the DRO arrival transfer orbit in the same interval is calculated to obtain the LEO departure transfer orbit splicing point state and the DRO arrival transfer orbit splicing point state.

[0089] Step 4: Plot a scatter plot with the transition time of the candidate feasible solution as the x-axis and the sum of velocity pulses as the y-axis, and divide it into grids. Randomly select candidate feasible solutions as initial feasible solutions in each grid according to a preset ratio.

[0090] Preferably, 10% of the candidate feasible solutions in each grid are randomly selected as the initial feasible solutions.

[0091] Step 5: Construct a trajectory correction model based on the objective function, optimization variables, and constraints. Use sequential quadratic programming to correct the trajectory of the optimization variables corresponding to the initial feasible solution in the input trajectory correction model, and output the objective feasible solution when the sum of velocity pulses is minimized.

[0092] Step 6: Determine whether the number of converged target feasible solutions within the grid meets the iteration conditions. If not, re-extract the initial feasible solutions; if yes, proceed with the iteration until the orbit corrections for all intervals are completed, and obtain all converged Earth-Moon space DRO transfer orbit parameters.

[0093] Preferably, 10% of the candidate feasible solutions are randomly selected again as the initial feasible solutions for trajectory correction.

[0094] In step one, the CRTBP dynamic model is as follows:

[0095] ;

[0096] In the formula, This is the output of the CRTBP dynamic model. It is a time variable; For spacecraft in The first derivative of the position vector at time, For spacecraft in The first derivative of the velocity vector at time instant; For spacecraft in The velocity vector at time t, For spacecraft in Position vector at time, For Earth-Moon mass parameters, The distance of the spacecraft from Earth. , The distance of the spacecraft from the moon. ;

[0097] The global accuracy of the seventh-order Runge-Kutta integrator is eighth-order. Error control is based on seventh-order extrapolation. Error estimation is performed by comparing the difference between the eighth-order solution and the seventh-order solution, thereby achieving adaptive step size adjustment.

[0098] In step one, the method for calculating the spacecraft's launch state vector is as follows:

[0099] ;

[0100] ;in,

[0101] ;

[0102] The method for calculating the spacecraft's orbital state vector is as follows:

[0103] ;

[0104] ;

[0105] ;in,

[0106] ;

[0107] Spacecraft transfer orbit for: ;

[0108] In the formula, This is the LEO state vector corresponding to the spacecraft's location at the moment it leaves the LEO; Let i be the semi-major axis of the LEO orbit and i be the inclination angle of the LEO orbit. The right ascension of the ascending node of the LEO is... This is the true perihelion angle of the spacecraft at LEO; This is the LEO position vector corresponding to the spacecraft's location at the moment it leaves LEO. This is the LEO velocity vector corresponding to the spacecraft's location at the moment it leaves the LEO region. This is the spacecraft state vector at the moment of departure from LEO; The angle between the transmitted pulse and the LEO velocity vector. The angle between the transmitted pulse and the LEO orbital plane; Apply the launch pulse to the spacecraft's velocity vector. The size of the transmitted pulse;

[0109] This is the DRO state vector corresponding to the spacecraft's location at the moment of DRO insertion into orbit; This refers to the DRO's inlet phase; This is the DRO position vector corresponding to the spacecraft's location at the DRO moment of orbit insertion. This is the DRO velocity vector corresponding to the spacecraft's position at the moment of DRO insertion into orbit; This represents the spacecraft's state vector at the moment of DRO insertion. The magnitude of the orbital insertion pulse; The angle between the orbital insertion pulse and the DRO velocity vector. The angle between the orbital pulse and the DRO orbital plane; The velocity vector after applying the orbital insertion pulse to the spacecraft; It is a rotation matrix;

[0110] For spacecraft in Position vector at time, For spacecraft in The velocity vector at any given moment.

[0111] In step one, the escape judgment condition is:

[0112] .

[0113] In step two, the target Jacobian constant The calculation method is as follows:

[0114] ;

[0115] The Jacobian constant C at the zero velocity surface is:

[0116] ;

[0117] In the formula, This is the state vector of the target orbit, that is, the position vector and velocity vector of the spacecraft in the target orbit; The position vector of the spacecraft that is tangent to the zero-velocity surface and the DRO.

[0118] Calculate the Jacobian constant C corresponding to the zero-velocity surface tangent to the DRO phase, and classify different Jacobian constants based on the reachable DRO phase. The corresponding spacecraft launch state vector and spacecraft orbit insertion state vector are classified, and the Jacobian constant of each set of parameters is compared. Determine the Jacobian constant of this set of parameters by comparing the values ​​of the Jacobian constant C at each zero-velocity surface. The Jacobi constant C interval falls within the corresponding zero velocity surface. Based on the openness of the zero velocity surface to the DRO phase, the launch state vector and orbit insertion state vector of the spacecraft operating in the Earth-Moon system are classified.

[0119] In step three, the calculation method for the state of the LEO departure transfer track splicing point is as follows:

[0120] ;

[0121] The spacecraft's launch state vector is integrated along the positive time direction using a seventh-order Runge-Kutta integrator. Duration, obtain the status of the LEO departure transfer orbit splicing point.

[0122] The calculation method for DRO reaching the transfer track splicing point is as follows:

[0123] ;

[0124] The spacecraft's launch state vector is integrated in reverse time using a seventh-order Runge-Kutta integrator. Duration, to obtain the DRO's arrival at the transfer orbit splicing point status.

[0125] The conditions for track splicing are as follows:

[0126] ;

[0127] In the formula, Let LEO be the state vector of the transfer trajectory splicing point. The transfer time of the LEO departure transfer orbit is obtained by integrating the classified target spacecraft departure state vector along the positive time direction using a seventh-order Runge-Kutta integrator. Let be the position vector of the LEO departure transfer track splicing point. The velocity vector at the LEO departure transfer trajectory splicing point;

[0128] This is the state vector of the DRO upon reaching the transfer orbit splicing point. The transfer time of the DRO to the transfer orbit is the time obtained by integrating the classified target spacecraft's orbital state vector in reverse along time using a seventh-order Runge-Kutta integrator. This is the position vector of the DRO reaching the transfer track splicing point. The velocity vector of the DRO as it reaches the transfer track splicing point;

[0129] This represents the positional difference between the LEO departure transfer track splicing point state and the DRO arrival transfer track splicing point state within the same interval. The speed difference between the LEO starting transfer track splicing point state and the DRO arriving transfer track splicing point state within the same interval.

[0130] In step four, the calculation method for the transfer time and the sum of velocity pulses is as follows:

[0131] ;

[0132] In the formula, TOF represents the transition time; This is the sum of velocity pulses; For splicing pulses; with TOF as the x-axis and... Plot a scatter plot with the vertical axis as the ordinate, and use the TOF with the shortest transition time within that interval. and longest transfer time Divide the boundary into sub-intervals on an average basis, and then divide the sub-intervals into grids at preset intervals.

[0133] In step five, the orbital correction model is as follows:

[0134] ;in,

[0135] The objective function is: ;

[0136] The optimization variables are: ;

[0137] The constraints are: ;

[0138] In the formula, The output value of the objective function. This is the sum of velocity pulses; To optimize the derivative of the variable with respect to time; For inequality constraints, For equality constraints, The lunar position vector. The radius of the moon; To optimize variables, To optimize the boundary of variables, To optimize the lower bound of the variable, To optimize the upper bound of the variable; where,

[0139] .

[0140] In step five, sequential quadratic programming (SQP) is used to correct the trajectory of the optimization variables corresponding to the initial feasible solution in the input trajectory correction model, and the method for outputting the objective feasible solution that minimizes the sum of velocity impulses is as follows:

[0141] ;

[0142] In the formula, This represents the value of the optimization variable that minimizes the objective function of the trajectory correction model, i.e., the feasible solution for the objective when the sum of velocity impulses is minimized. The magnitude of the launch pulse applied by the spacecraft when the sum of velocity pulses is minimized. The angle between the transmitted pulse and the LEO velocity vector is the angle at which the sum of velocity pulses is minimized. The angle between the emitted pulse and the LEO orbital plane when the sum of velocity pulses is minimum. The LEO departure transfer orbit transfer time is when the total sum of velocity pulses is minimized. The magnitude of the orbital insertion pulse vector applied to the spacecraft when the sum of velocity pulses is minimized. The angle between the orbital pulse and the DRO velocity vector when the sum of velocity pulses is minimized. The angle between the insertion pulse and the DRO orbital plane when the sum of velocity pulses is minimum. The transfer time of DRO to the transfer track when the sum of velocity pulses is minimum.

[0143] In step six, the iteration condition is:

[0144] ;

[0145] In the formula, This represents the number of objective feasible solutions that converge within the grid. This represents the number of initially feasible solutions. This represents the number of iterations for the trajectory correction model.

[0146] .

[0147] To provide a detailed explanation of the technical solution of this invention, a specific example is provided below:

[0148] The technical solution disclosed in this invention is applied to the design problem of the LEO-DRO transfer orbit based on pulse maneuvering in the Earth-Moon space:

[0149] This example demonstrates the design of the LEO-DRO transfer orbit between the Earth and the Moon using the Jacobi constant classification algorithm. The simulation parameters involved are shown in Table 1.

[0150] Table 1

[0151]

[0152] In the design of the LEO-DRO transfer orbit, the spacecraft is set to launch from LEO, which has an altitude of 200km and an orbital inclination of 28.44°, the same as the lunar orbital inclination.

[0153] Construct initial value databases for the LEO departure segment and the DRO arrival segment. Transmit pulse parameters. The range of values ​​is shown in Table 2. Since commonly used LEO orbits are prograde orbits, the range of values ​​is selected as [0°, 90°]. The right ascension of the ascending node of the Moon is approximately 0.097°, so the right ascension of the ascending node of the LEO is selected to be near the right ascension of the ascending node of the Moon, i.e., [10°, 10°]. The true anomaly angle of the LEO... No constraints are imposed. Since the spacecraft can transfer from Earth to the Moon by applying a LEO transmission pulse greater than 3.0, the selected range is [3.0, 3.6]. In order to ensure that the transmission pulse is always forward of the LEO velocity direction, the azimuth and elevation angles of the transmission pulse are constrained to be in the range of [-45°, 45°].

[0154] Table 2

[0155]

[0156] Approximately eight million sets of parameters are constructed, and then the state of the LEO at the moment the transmit pulse is applied is calculated for each set of parameters. Secondly, the spacecraft velocity state at the moment the launch pulse is applied is calculated based on each set of parameters. This yields the complete spacecraft launch state vector.

[0157] Parameters of the orbital insertion pulse The range and interval of values ​​are shown in Table 3. The DRO phase is discretized into 360 phases, denoted as [0°:1°:360°]. No constraints are imposed on the azimuth and elevation angles of the orbital pulse.

[0158] Table 3

[0159]

[0160] Approximately ten million sets of parameters were constructed, and then the state of the DRO at the moment of rendezvous between the spacecraft and the DRO was calculated for each set of parameters. Then, the velocity vector of the spacecraft at the moment of rendezvous with DRO corresponding to each set of parameters was calculated. This yields the complete spacecraft orbital state vector.

[0161] Using the LEO departure state vector as the initial state value, a 7th-order Runge-Kutta integrator is used for forward integration over 5 days based on the CRTBP dynamic model. Using the DRO entry state vector as the initial state value, a 7th-order Runge-Kutta integrator is used for backward integration over 5 days based on the CRTBP dynamic model. The criterion is whether all points in the trajectory are more than 1.2 AU away from the origin of the Earth-Moon synodic system. If there are points in the trajectory that are more than 1.2 AU away from the origin, the trajectory is considered to have escaped from the Earth-Moon space and is removed from the orbit initial value database.

[0162] The zero-velocity surface and the opening condition of the DRO phase under different Jacobi constants were calculated as follows: Figure 1 The Jacobi constants of the initial value databases for the LEO departure segment and DRO arrival segment were classified, and the values ​​of the Jacobi integral constants of each set of parameters were compared with the values ​​of the zero-velocity surface corresponding to different DRO phase opening conditions. The specific classifications are shown in Table 4. Based on the Jacobi constant classifications, several LEO departure segment initial value sub-databases and DRO arrival segment initial value sub-databases were stored. For each set of parameters in the LEO departure segment initial value sub-database corresponding to the same Jacobi constant interval, a 20-day forward integration was performed using a seventh-order Runge-Kutta integrator based on the CRTBP dynamic model. For each set of parameters in the DRO arrival segment initial value sub-database, a 30-day backward integration was performed using a seventh-order Runge-Kutta integrator based on the CRTBP dynamic model. All trajectories were stored to obtain the LEO departure segment and DRO arrival segment track sub-databases under that Jacobi constant classification.

[0163] Table 4

[0164]

[0165] For each Jacobi constant classification interval, traverse all combinations of the LEO departure segment orbital sub-database and the DRO arrival segment orbital sub-database, and determine the point closest to the other two trajectories. Store the parameters of the two orbital segments that meet the coarse splicing accuracy and the splicing point status into the coarse splicing orbital sub-database to obtain the sub-coarse splicing orbital library composed of candidate feasible solutions for the transfer orbit within the Jacobi constant interval.

[0166] Calculate the time of transition (TOF) and total velocity impulse for each candidate feasible solution in the coarsely stitched orbit library. With the time of transition (TOF) as the x-axis, the total velocity pulse Plot TOF- for the ordinate Scatter plot, see Figure 2 , Figure 2The horizontal axis uses TOF units to represent normalized time. A grid is drawn based on the data distribution of the scatter plot. The transition time is first divided into 35 sub-intervals, and then the sub-intervals are divided into grids at intervals of 0.05 km / s. 10% of the candidate feasible solutions are extracted from each grid as the initial feasible solutions.

[0167] Without altering the transition mode in the coarse stitching, i.e., constraining the boundaries of the optimization variables in the trajectory correction model, the SQP algorithm is used to optimize the initial feasible solution. The first Jacobian constant interval yields 4165 objective feasible solutions, approximately 10% of the candidate feasible solutions. The TOF-... Scatter plot, see Figure 3 , Figure 3 The horizontal axis is in days (TOF). It includes four transfer methods: direct transfer, inner lunar-assisted transfer, outer lunar-assisted transfer, and two lunar-assisted transfers.

[0168] Direct transfer method such as Figure 4 It is formed by splicing together an Earth resonance orbit and an Earth-Moon transfer orbit. This orbit is one of the orbits with relatively small velocity pulses in the direct transfer method, with a transfer time of 13.5021 days and a total velocity pulse of The speed is 3.5728 km / s, of which LEO transmits pulses. The DRO insertion pulse is 2.99 km / s. The speed is 0.1421 km / s, and the spliced ​​pulse is... It is 0.4407 km / s.

[0169] Inner lunar leverage method such as Figure 5 It is formed by splicing together an Earth resonance orbit and an inner lunar gravity-borrowing orbit. This orbit is one of the smaller velocity pulses in the inner lunar gravity-borrowing method, with a transfer time of 17.1717 days and a total velocity pulse. The speed is 4.1069 km / s, of which LEO transmits pulses. The DRO insertion pulse is 3.25 km / s. 0.16 km / s, spliced ​​pulse It is 0.6872 km / s.

[0170] outer lunar orbital leveraging methods such as Figure 6 It is formed by splicing together an Earth resonance orbit and an outer lunar gravity-borrowing orbit. This orbit is one of the outer lunar gravity-borrowing orbits with a relatively small velocity pulse, and the transfer time is 23.528 days, with a total velocity pulse of... The speed is 3.8510 km / s, of which LEO transmits pulses. The DRO insertion pulse was 3.1704 km / s. 0.14 km / s, spliced ​​pulse It is 0.5206 km / s.

[0171] Two lunar transfer methods such as Figure 7 The spacecraft first enters the outer lunar orbit from the Earth's resonance orbit, and then enters the inner lunar orbit from the outer lunar orbit. This orbit is the one with the smaller velocity pulse in the two lunar orbit transfer methods, with a transfer time of 32.4573 days and a total velocity pulse of [missing information]. The speed is 3.5687 km / s, of which LEO transmits pulses. The DRO insertion pulse is 2.97 km / s. 0.16 km / s, spliced ​​pulse It is 0.4587 km / s.

[0172] By dividing the target orbit's Jacobian constant into 16 intervals with 0.01 intervals using two zero-velocity surfaces tangent to the DRO, the Jacobian constant difference between target orbits within each interval is less than 0.01, thus reducing the energy difference between the two transfer orbits. By comparing the transfer time-velocity pulse sum scatter plots of the candidate feasible solutions with those of the target feasible solution, the dynamic characteristics within each grid are largely preserved, saving 50% to 90% of computation time. Reasonable constraints on the boundaries of optimization variables directly limit the transfer mode of the target feasible solution. Furthermore, the upper and lower bounds of the optimization variables can be further constrained based on the grid range of the transfer time-velocity pulse sum scatter plot, thus strictly constraining the dynamic characteristics of the target feasible solution to be identical to those of the candidate feasible solutions. By determining whether the number of target feasible solutions at network convergence satisfies the iteration condition, the convergence of the orbit splicing optimization problem is improved, thereby increasing optimization efficiency. In this example, by correcting the base orbit, various transfer methods were designed, including direct transfer orbits, single lunar-assisted transfer orbits, and double lunar-assisted transfer orbits. Furthermore, the Earth-to-joint-point transfer orbit included two types: Earth-Moon transfer orbits and Earth-resonance orbits. Similarly, the lunar-assisted transfer orbit included two types: prograde lunar-assisted transfer orbits and retrograde lunar-assisted transfer orbits. Therefore, this design method can yield multiple types of transfer orbits, and the appropriate transfer method can be flexibly selected according to mission requirements.

[0173] The beneficial effects of the embodiments of the present invention are:

[0174] 1. Calculate the target Jacobian constant of the spacecraft transfer orbit operating in the Earth-Moon system as the target orbit, and classify the target orbits according to the openness of the DRO phase of the zero-velocity surface Jacobian constant. Using the Jacobian constant classification method instead of the near-lunar distance screening method can classify orbits with similar orbital energy before the splicing step from the perspective of orbital energy, so that the candidate orbit solutions obtained by splicing are orbits with similar orbital energy, thereby making the splicing pulse smaller.

[0175] 2. The LEO departure transfer track and the DRO arrival transfer track within the same interval that meet the track splicing conditions are spliced ​​at the splicing point, so that the difference in Jacobian constant between the two transfer tracks is small, thereby reducing the energy difference between the two transfer tracks.

[0176] 3. Plot a scatter plot with the transition time of the candidate feasible solution as the x-axis and the sum of velocity pulses as the y-axis, and divide it into grids. Randomly select candidate feasible solutions as initial feasible solutions in each grid according to a preset ratio, so as to sparse candidate feasible solutions. This saves computation time while preserving the similar dynamic characteristics of the transfer trajectory patterns in each grid.

[0177] 4. Based on the objective function, the boundary of the trajectory correction optimization variables, and the constraints, a trajectory correction model is constructed. The upper and lower bounds of the variables in the optimization problem are reasonably set. Under the premise of not changing the trajectory transfer mode, the trajectory correction problem is constructed based on the constraints of trajectory splicing, so as to retain all transfer modes of the feasible solution candidates for spliced ​​trajectory.

[0178] 5. Sequential quadratic programming was used to correct the orbit of the optimization variables corresponding to the initial feasible solution in the input orbit correction model. The target feasible solution with the minimum sum of velocity pulses of the transfer orbit was obtained by solving the initial feasible solution extracted in each grid. Direct transfer orbit and lunar gravity transfer orbit were designed directly.

[0179] 6. Determining whether the number of feasible solutions to the objective at the time of network convergence satisfies the iteration condition improves the convergence of the orbit splicing optimization problem, thereby improving optimization efficiency and saving computation time.

[0180] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for designing Earth-Moon space DRO transfer orbits based on the Jacobi constant classification method, characterized in that, The method includes: Step 1: Combining the CRTBP dynamic model, a seventh-order Runge-Kutta integrator is used to integrate the spacecraft's launch state vector and orbit insertion state vector to obtain the spacecraft's transfer orbit; based on the escape judgment condition, the spacecraft's transfer orbit operating within the Earth-Moon system is determined as the target orbit. Step 2: Calculate the target Jacobian constant of the target orbit, and classify the target orbit based on the interval corresponding to the Jacobian constant of the zero velocity surface. Step 3: Combining the CRTBP dynamic model, a seventh-order Runge-Kutta integrator is used to integrate the classified target spacecraft departure state vector along the forward time direction to obtain the LEO departure transfer orbit splicing point state; the target spacecraft insertion state vector is integrated along the backward time direction to obtain the DRO arrival transfer orbit splicing point state; the LEO departure transfer orbit and the DRO arrival transfer orbit within the same interval that satisfy the orbit splicing conditions are spliced ​​at the splicing point as candidate feasible solutions; Step 4: Plot a scatter plot with the transition time of the candidate feasible solution as the x-axis and the sum of velocity pulses as the y-axis, and divide it into grids. Randomly select candidate feasible solutions as initial feasible solutions in each grid according to a preset ratio. Step 5: Construct a trajectory correction model based on the objective function, optimization variables, and constraints. Use sequential quadratic programming to correct the trajectory of the optimization variables corresponding to the initial feasible solution in the input trajectory correction model, and output the objective feasible solution when the sum of velocity pulses is minimized. Step 6: Determine whether the number of converged target feasible solutions within the grid meets the iteration conditions. If not, re-extract the initial feasible solutions; if yes, proceed with the iteration until the orbit corrections for all intervals are completed, and obtain all converged Earth-Moon space DRO transfer orbit parameters.

2. The method as described in claim 1, characterized in that, In step one, the CRTBP dynamic model is as follows: ； In the formula, This is the output of the CRTBP dynamic model. It is a time variable; For spacecraft in The first derivative of the position vector at time, For spacecraft in The first derivative of the velocity vector at time instant; For spacecraft in The velocity vector at time t, For spacecraft in Position vector at time, For Earth-Moon mass parameters, The distance of the spacecraft from Earth. , The distance of the spacecraft from the moon. ; The global accuracy of the seventh-order Runge-Kutta integrator is eighth-order. Error control is based on seventh-order extrapolation. Error estimation is performed by comparing the difference between the eighth-order solution and the seventh-order solution, thereby achieving adaptive step size adjustment.

3. The method as described in claim 2, characterized in that, In step one, the method for calculating the spacecraft's launch state vector is as follows: ； ;in, ； The method for calculating the spacecraft's orbital state vector is as follows: ； ； ;in, ； Spacecraft transfer orbit for: ; In the formula, This is the LEO state vector corresponding to the spacecraft's location at the moment it leaves the LEO; Let i be the semi-major axis of the LEO orbit and i be the inclination angle of the LEO orbit. The right ascension of the ascending node of the LEO is... This is the true perihelion angle of the spacecraft at LEO; This is the LEO position vector corresponding to the spacecraft's location at the moment it leaves LEO. This is the LEO velocity vector corresponding to the spacecraft's location at the moment it leaves the LEO region. This is the spacecraft state vector at the moment of departure from LEO; The angle between the transmitted pulse and the LEO velocity vector. The angle between the transmitted pulse and the LEO orbital plane; Apply the launch pulse to the spacecraft's velocity vector. The size of the transmitted pulse; This is the DRO state vector corresponding to the spacecraft's location at the moment of DRO insertion into orbit; This refers to the DRO's inlet phase; This is the DRO position vector corresponding to the spacecraft's location at the DRO moment of orbit insertion. This is the DRO velocity vector corresponding to the spacecraft's position at the moment of DRO insertion into orbit; This represents the spacecraft's state vector at the moment of DRO insertion. The magnitude of the orbital insertion pulse; The angle between the orbital insertion pulse and the DRO velocity vector. The angle between the orbital pulse and the DRO orbital plane; The velocity vector after applying the orbital insertion pulse to the spacecraft; It is a rotation matrix; For spacecraft in Position vector at time, For spacecraft in The velocity vector at any given moment.

4. The method as described in claim 3, characterized in that, In step one, the escape judgment condition is: 。 5. The method as described in claim 4, characterized in that, In step two, the target Jacobian constant The calculation method is as follows: ； The Jacobian constant C at the zero velocity surface is: ； In the formula, This is the state vector of the target orbit, that is, the position vector and velocity vector of the spacecraft in the target orbit; The position vector of the spacecraft that is tangent to the zero-velocity surface and the DRO.

6. The method as described in claim 5, characterized in that, In step three, the calculation method for the state of the LEO departure transfer track splicing point is as follows: ； The calculation method for DRO reaching the transfer track splicing point is as follows: ； The conditions for track splicing are as follows: ； In the formula, Let LEO be the state vector of the transfer trajectory splicing point. The transfer time of the LEO departure transfer orbit is obtained by integrating the classified target spacecraft departure state vector along the positive time direction using a seventh-order Runge-Kutta integrator. Let be the position vector of the LEO departure transfer track splicing point. The velocity vector at the LEO departure transfer trajectory splicing point; This is the state vector of the DRO upon reaching the transfer orbit splicing point. The transfer time of the DRO to the transfer orbit is the time obtained by integrating the classified target spacecraft's orbital state vector in reverse along time using a seventh-order Runge-Kutta integrator. This is the position vector of the DRO reaching the transfer track splicing point. The velocity vector of the DRO as it reaches the transfer track splicing point; This represents the positional difference between the LEO departure transfer track splicing point state and the DRO arrival transfer track splicing point state within the same interval. The speed difference between the LEO starting transfer track splicing point state and the DRO arriving transfer track splicing point state within the same interval.

7. The method as described in claim 6, characterized in that, In step four, the calculation method for the transfer time and the sum of velocity pulses is as follows: ； In the formula, TOF represents the transition time; This is the sum of velocity pulses; For splicing pulses; with TOF as the x-axis and... Plot a scatter plot with the vertical axis as the ordinate, and use the TOF with the shortest transition time within that interval. and longest transfer time Divide the boundary into sub-intervals on an average basis, and then divide the sub-intervals into grids at preset intervals.

8. The method as described in claim 7, characterized in that, In step five, the orbital correction model is as follows: ;in, The objective function is: ; The optimization variables are: ; The constraints are: ; In the formula, The output value of the objective function. This is the sum of velocity pulses; To optimize the derivative of the variable with respect to time; For inequality constraints, For equality constraints, The lunar position vector. The radius of the moon; To optimize variables, To optimize the boundary of variables, To optimize the lower bound of the variable, To optimize the upper bound of the variable; where, 。 9. The method as described in claim 8, characterized in that, In step five, sequential quadratic programming (SQP) is used to correct the trajectory of the optimization variables corresponding to the initial feasible solution in the input trajectory correction model, and the method for outputting the objective feasible solution that minimizes the sum of velocity impulses is as follows: ； In the formula, This represents the value of the optimization variable that minimizes the objective function of the trajectory correction model, i.e., the feasible solution for the objective when the sum of velocity impulses is minimized. The magnitude of the launch pulse applied by the spacecraft when the sum of velocity pulses is minimized. The angle between the transmitted pulse and the LEO velocity vector is the angle at which the sum of velocity pulses is minimized. The angle between the emitted pulse and the LEO orbital plane when the sum of velocity pulses is minimum. The LEO departure transfer orbit transfer time is when the total sum of velocity pulses is minimized. The magnitude of the orbital insertion pulse vector applied to the spacecraft when the sum of velocity pulses is minimized. The angle between the orbital pulse and the DRO velocity vector when the sum of velocity pulses is minimized. The angle between the insertion pulse and the DRO orbital plane when the sum of velocity pulses is minimum. The transfer time of DRO to the transfer track when the sum of velocity pulses is minimum.

10. The method as described in claim 9, characterized in that, In step six, the iteration condition is: ； In the formula, This represents the number of objective feasible solutions that converge within the grid. This represents the number of initially feasible solutions. This represents the number of iterations for the trajectory correction model.

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