A transfer orbit fast optimization method based on spatial conic splicing

Abstract

The application discloses a transfer orbit fast optimization method based on spatial conic curve splicing, and belongs to the field of deep space exploration. The application expands the plane conic curve splicing method to the optimization of spatial earth-moon transfer orbit on the basis of the plane conic curve splicing method used for orbit transfer, by using coordinate conversion and geometric analysis method to increase the azimuth angle parameters theta and psi used for describing the spatial orientation of the earth-centered launch orbit plane; the application can improve the orbit analysis and optimization efficiency by using the two-body orbit theory to deduce the two-body model and based on the double two-body model splicing with analytical characteristics; the application can realize the fast optimization and solution of the transfer orbit based on the spatial conic curve splicing by establishing the penalty function iterative optimization of four independent optimization quantities, i.e., the earth-moon transfer launch speed increment, the moon influence ball entry point phase, the azimuth angle parameters theta and psi, according to the orbit inclination and the perigee height constraints. The application has the advantages of good convergence, high optimization efficiency and wide application range. The application is suitable for various earth-moon transfer missions and moon flyby orbit optimization.

Description

Technical Field

[0001] This invention relates to a method for splicing space conic sections in the Earth-Moon system, and more particularly to a method for rapid optimization of transfer orbits based on splicing space conic sections, belonging to the field of deep space exploration. Background Technology

[0002] The conic section stitching method is a commonly used approach for solving multibody problems in transfer orbit design. Within a defined sphere of influence (SOI), the multibody problem is decomposed into multiple two-body models consisting of the spacecraft and the central body. Then, the transfer orbits are stitched together from these multiple two-body models into a single, complete orbit. The conic section stitching method can address uncertainties and strong nonlinearities in the orbit design of some multibody models. Therefore, it has been widely applied in early space missions, such as lunar and interplanetary missions.

[0003] Numerous studies have explored the use of conic section splicing to solve the classic Earth-Moon transfer problem from a near-Earth circular orbit to a near-Moon circular orbit. From the 1960s to the 1980s, classic conic section splicing transfer was used in almost all lunar missions, including the Apollo missions. Transfer orbits constructed using the conic section splicing method have shorter flight times, but higher fuel costs. Furthermore, the conic section splicing method can also be extended to the design of lunar landing and lunar flyby orbits.

[0004] However, most current research on conic section splicing methods in the Earth-Moon system is based on planar cases, and studies on three-dimensional conic section splicing in space cannot provide analytical solutions. To address this issue, this patent extends the planar conic section splicing method for Earth-Moon transfer to the spatial case and derives an analytical solution.

[0005] In the developed methods for splicing conic sections

[0006] Prior art [1] (see: Bate Roger R, Mueller Donald D, Jerry E. Fundamentals of astrodynamics.NY[J].1971.) proposed a planar conic section splicing method for Earth-Moon transfer.

[0007] Prior technology [2] (see: Salazar FJT, Macau EEN, Winter O C. Pareto Frontier for the time–energy cost vector to an Earth–Moon transfer orbit using the patched-conic approximation[J].Computational and Applied Mathematics,2015,34(2):461-475.) Based on Bate[1]’s conic section patching method, they studied the optimal time and minimum cost transfer from a near-Earth circular orbit to a near-Moon circular orbit. They used the concept of Pareto optimality and the conic section patching method to find a set of parameters that minimized the mission’s flight time and total fuel consumption ΔV.

[0008] Prior technology [3] (see: Li J, Yang H, Baoyin H. Lunar exploration phase III: Launch window and trajectory design for a lunar lander[J]. Advances in SpaceResearch,2015,56(5):879-892.) developed an analysis and correction model for the transfer trajectory of a lunar probe using the conic section splicing method.

[0009] Prior technology [4] (see: Gagg Filho LA, da Silva Fernandes SA lunar flyby for a tridimensional Earth-to-Earth mission[J]. Acta Astronautica, 2018, 151: 228-242.) constructed the Earth transfer orbit for lunar flyby between non-coplanar orbits at different altitudes, proposed a spatial conic section approximation method related to the two-point boundary value problem, and compared the results of the conic section approximation with the results of CRTBP, showing the consistency between the models. Summary of the Invention

[0010] This invention discloses a rapid optimization method for transfer orbits based on spatial conic section splicing. Building upon the planar conic section splicing method used for orbit transfer, it adds azimuth parameters θ and ψ to describe the spatial orientation of the Earth-centered starting orbit plane using coordinate transformation and geometric analysis, extending the planar conic section splicing method to the optimization of Earth-Moon transfer orbits. A two-body model is derived using two-body orbit theory, and splicing based on a bi-two-body model with analytical properties improves the efficiency of orbit analysis and optimization. Four independent optimization quantities—the Earth-Moon transfer starting velocity increment ΔV0, the lunar influence sphere entry point phase λ1, and the azimuth parameters θ and ψ—are iteratively optimized based on constraints such as orbital inclination and perigee altitude, achieving rapid optimization of transfer orbits based on spatial conic section splicing. This invention has the advantages of good convergence, high analysis and optimization efficiency, and wide applicability. This invention is applicable to various Earth-Moon transfer missions and lunar flyby orbit optimization.

[0011] The objective of this invention is achieved through the following technical solution:

[0012] This invention discloses a rapid optimization method for Earth-Moon transfer orbit based on spatial conic section splicing. On the basis of the planar conic section splicing method used for orbit transfer, the method adds azimuth parameters θ and ψ to describe the spatial orientation of the orbital plane starting from the Earth's center by using coordinate transformation and geometric analysis, thus extending the planar conic section splicing method to the spatial case. The orbital parameters, starting state, and state of the entry point of the influence sphere are determined by four independent optimization quantities: the Earth-Moon transfer velocity increment ΔV0, the phase λ1 of the lunar influence sphere entry point, the azimuth parameter θ, and ψ, which describe the spatial orientation of the Earth-centered starting orbital plane. Here, ψ is the angle between the intersection of the Earth-centered starting orbital plane and the XY plane of the Earth-centered inertial coordinate system relative to the X-axis (counterclockwise is positive), and θ is the angle between the angular momentum of the Earth-centered starting orbit and the Z-axis of the Earth-centered inertial coordinate system. A Earth-centered starting orbit coordinate system and a lunar-centered arrival orbit coordinate system are established. The entry point state is transferred to the lunar-centered inertial coordinate system to determine the hyperbolic orbital parameters within the lunar influence sphere. The state of the perigee in the lunar-centered inertial frame, as well as the orbital inclination and perigee altitude of the hyperbolic orbit, are calculated, and a penalty function is established. The Earth-Moon transfer orbit is obtained by iteratively optimizing the four optimization quantities based on constraints such as the orbital inclination and perigee altitude, achieving rapid optimization of the transfer orbit based on spatial conic section splicing.

[0013] This invention discloses a rapid optimization method for the Earth-Moon transfer orbit based on spatial conic section splicing, comprising the following steps:

[0014] Step 1: Define the geocentric inertial coordinate system. Given the initial values ​​of the four optimization parameters ΔV0, λ1, θ, and ψ, define the geocentric starting orbit coordinate system based on the azimuth angles θ and ψ that describe the spatial orientation of the starting orbit at the geocentric point, and obtain the coordinate transformation matrix between the geocentric starting orbit coordinate system and the geocentric inertial coordinate system.

[0015] Step 1.1: Define the geocentric inertial coordinate system.

[0016] In a geocentric inertial frame of reference, the X-axis is the direction of the line connecting the Earth and the Moon when the spacecraft enters the Moon, the Z-axis is the direction of the normal along the Moon's orbit, and the Y-axis is determined by the right-hand rule.

[0017] Step 1.2: Define the geocentric starting orbit coordinate system and calculate the transformation matrix between the geocentric inertial coordinate system and the geocentric coordinate system.

[0018] The orbital plane of a geocentric starting orbit is determined by two angles: θ and ψ. ψ is the angle between the line of intersection of the geocentric starting orbital plane and the XY plane of the geocentric inertial coordinate system and the X-axis, with counterclockwise being positive. θ is the angle between the angular momentum of the geocentric starting orbit and the Z-axis of the geocentric inertial coordinate system. The orbital plane of the geocentric starting orbit passes through the Moon, influencing the formation of the sphere around O. m The circular cross section S centered on m From the center of the moon to O m The distance is expressed as

[0019] d=Dsinψsinθ, (1)

[0020] In the formula, D is the distance between the Earth and the Moon. Section S m The radius is

[0021]

[0022] In the formula R s Let R be the radius of the sphere affected by the moon. s =66300km.

[0023] The geocentric coordinate system is defined based on the orbital plane of the starting orbit: the origin is located at the Earth's center, X... n Axis along the Earth's center to O m The direction of the line connecting Z n The axis is along the direction of the angular momentum vector, Y n The axis is defined using the right-hand rule. Therefore, Z... n The unit vector is represented as

[0024] k n =[sinψsinθ,-cosψsinθ,cosθ] T (3)

[0025] O m The position coordinates in the geocentric inertial frame are represented as follows:

[0026] O m =[D,0,0] T -d·k n (4)

[0027] Along X n The unit vector is written as

[0028] i n =O m / D n (5)

[0029] Where D n From the Earth's core to O m The distance, D n =||O m ||. Along Y n The unit vector is equal to j n =k n ×i n From X n Y n Z n The transformation matrix from the coordinate system to the XYZ coordinate system is M. n =[i n j n k n ].

[0030] Step 2: Based on the initial value ΔV0 of the Earth-Moon transfer departure velocity increment, calculate the position and velocity of the initial point of the Earth-Moon transfer in the geocentric inertial frame, and determine the orbital parameters of the geocentric departure orbit. Then, define short-range transfer and long-range transfer, and based on the two methods of entering the lunar influence sphere, the phase λ1 of the lunar influence sphere entry point, and the orbital parameters of the geocentric departure orbit, calculate the position and velocity vector at the entry point of the influence sphere in the geocentric departure orbit coordinate system, and transfer the state to the geocentric inertial frame using a rotation matrix.

[0031] Step 2.1: Calculate the position and velocity of the initial point of the Earth-Moon transfer in the Earth-centered inertial frame, and determine the orbital parameters of the starting orbit from the Earth-centered point.

[0032] The initial point of the Earth-Moon transfer has a geocentric distance and velocity in the geocentric inertial frame, respectively.

[0033]

[0034] Among them, R e and μ e Let be the Earth's radius and gravitational coefficient, respectively, and ΔV0 be the velocity increment at the initial point. In the geocentric inertial frame, the orbital parameters of the starting orbit are obtained using the following calculation method. The energy and angular momentum of the starting orbit are expressed as:

[0035]

[0036] This gives us the semi-major axis and eccentricity of the orbit originating from the Earth's center.

[0037]

[0038] in, It is the semi-major path of the orbit starting from the Earth's center.

[0039] Step 2.2: Based on the two methods of entering the lunar influence sphere, namely short-range transfer and long-range transfer, calculate the position and velocity vector at the entry point of the influence sphere in the geocentric starting orbit coordinate system, and transfer it to the inertial frame using a rotation matrix.

[0040] If the starting orbit from the Earth's center is an elliptical orbit, there are two ways to enter the lunar sphere of influence: a short transfer, where the entry point of the lunar sphere of influence occurs before the apogee of the elliptical orbit at the Earth's center; and a long transfer, where the entry point of the lunar sphere of influence occurs after the apogee of the elliptical orbit at the Earth's center. For both cases, the distance and velocity of the lunar sphere of influence entry point relative to Earth are expressed as...

[0041]

[0042] The effects of short-range and long-range transfers on the geocentric velocity vector V at the sphere's entry point n They are respectively represented as

[0043] V n =V1·[sin(φ1-γ1),cos(φ1-γ1),0] T (10)

[0044] V n =V1·[-sin(φ1+γ1),cos(φ1+γ1),0] T (11)

[0045] in

[0046]

[0047] The equations for short-range transfers described above can also be applied to geocentric starting trajectories such as hyperbolas or parabolas, but long-range transfers are only applicable to elliptical trajectories. In the geocentric inertial coordinate system, the vectors affecting the position and velocity at the sphere's entry point are expressed as:

[0048]

[0049] Where λ1 is the phase angle of the point of entry of the lunar influence sphere, and D n From Earth to O m The distance.

[0050] Step 3: Define two infeasible scenarios for conic section splicing transfer based on the lunar influence on the sphere's entry point, eliminate the infeasible scenarios, and calculate the flight time for short-range and long-range orbits based on the geocentric transfer orbit parameters.

[0051] Step 3.1: Define two infeasible cases for conic section splicing and transfer based on R1 and V1 in the geocentric inertial coordinate system:

[0052] 1. No lunar crossing: This occurs when ΔV0 is not large enough, and the spacecraft cannot reach the lunar influence sphere, represented by the following condition: e e <1 and a e (1+e e ) < R1.

[0053] 2. Secondary Crossing: This scenario is determined by r2·v, where r2 is the position vector of the spacecraft's entry point relative to the Moon in the Earth-Moon rotation frame, and v is the velocity vector of the spacecraft's entry point in the Earth-Moon rotation frame. r2 and v are obtained from R1 and V1 through coordinate transformation between the geocentric inertial frame and the Earth-Moon rotation frame. If r2·v ≥ 0, it indicates that the spacecraft is leaving the Moon rather than entering it, making a conic section transfer impractical.

[0054] Step 3.2: Calculate the flight time of the orbit starting from the Earth's center.

[0055] According to the two-body orbit theory, given R1 and V1, the starting orbit from the geocentric point is uniquely determined in the geocentric inertial frame. The initial point is the perigee of the geocentric orbit, and the initial true anomaly angle f0 = 0. The true anomaly angle of the entry point has two different expressions for short-range and long-range transfer orbits. For short-range transfers…

[0056]

[0057] For long-distance transfers

[0058]

[0059] Determine h through step 2.2 e e e f0, by f e Calculate the flight time t of the orbit starting from the Earth's center. e .

[0060]

[0061] Step 4: Define the lunar center inertial coordinate system, transfer the state of the influencing sphere's entry point to the lunar center inertial coordinate system, calculate the orbital parameters of the lunar center's arrival orbit, as well as the altitude and orbital inclination of the perigee relative to the moon, establish the penalty function, and repeat steps 1, 2, 3, and 4. Based on the task constraints, iteratively optimize the four optimization parameters ΔV0, λ1, θ, and ψ to obtain parameters that satisfy the task constraints.

[0062] Step 4.1: Define the lunar-centered inertial coordinate system.

[0063] Define X under lunar inertia m Y m Z m Coordinate system, where X m Y m and Z m The axis coincides with the corresponding axis in the XYZ coordinate system of the geocentric inertial frame. In this coordinate system, the spacecraft's position and velocity vector at the entry point are...

[0064]

[0065] in That is the speed at which the moon orbits the earth.

[0066] Step 4.2: Calculate the orbital parameters of the lunar center's arrival orbit in the lunar center inertial coordinate system.

[0067] The energy and angular momentum of the lunar orbit relative to the lunar inertial frame are expressed as follows:

[0068]

[0069] Where μ m These are the gravitational parameters of the Moon, from which the eccentricity and semi-major axis of the orbit reaching the lunar center are obtained.

[0070]

[0071] Among them, P m =||h m || 2 / μ m This is the semi-major diameter. The distance and velocity of the perigee relative to the Moon, as well as the orbital inclination, are expressed as...

[0072]

[0073] Step 4.3: Establish the penalty function and iteratively optimize the four optimization parameters ΔV0, λ1, θ, and ψ according to the task constraints to obtain the parameters that satisfy the task constraints.

[0074] F(ΔV0,λ1,θ,ψ)=||r pm -r c ||+||i m -i c || (21)

[0075] Where r c For lunar orbital altitude constraints, i c To address the lunar orbit inclination constraint, we solve for the Earth-Moon transfer orbit that satisfies the constraint by optimizing and iterating to reduce the penalty function to zero.

[0076] Step 5: Based on the obtained parameters that satisfy the mission constraints, calculate the complete Earth-Moon transfer orbit. Define the coordinate system of the lunar center arrival orbit and the coordinate system of the perigee from the orbit parameters of the lunar center arrival orbit. Calculate the position and velocity vector of the lunar center arrival orbit in the lunar center inertial frame. Calculate the flight time of the lunar center arrival orbit within the lunar sphere of influence. Calculate the position and velocity vector of the lunar center arrival orbit in the Earth center inertial frame. This yields the Earth-Moon transfer orbit that satisfies the mission constraints, thus achieving rapid optimization of the Earth-Moon transfer orbit.

[0077] Step 5.1: Define the lunar center arrival orbit coordinate system.

[0078] Definition based on R2 and h m The lunar center reaches the orbital coordinate system X h Y h Z h Coordinate system, X h Z h Y h The unit vector is represented as

[0079]

[0080] X h Y h Z h coordinate system to X m Y m Z m The transformation matrix of the coordinate system is M h =[i h j h k h ].

[0081] Step 5.2: Define the lunar coordinate system.

[0082] Define the lunar coordinate system X p Y p Z p Coordinate system, where X p The axis Z runs along the line connecting the center of the moon to the perigee of the orbit. p Axis and Z h The axes are in the same direction, Y p The axis is defined using the right-hand rule. From X... p Y p Z p coordinate system to X h Y h Z h The transformation matrix of the coordinate system is expressed as

[0083]

[0084] Among them, f in It is X h Axis and Xp The included angle between axes is written as

[0085]

[0086] Step 5.3: Calculate the position and velocity vector of the moon's center reaching the orbit in the lunar inertial frame.

[0087] Let f in ∈[-f in ,f in Let ] be the true anomaly angle of the hyperbolic orbit within the Moon's SOI, then the spacecraft in X p Y p Z p The corresponding position and velocity vectors in the coordinate system are represented as follows:

[0088]

[0089] Among them, v r and v u These represent the radial and tangential components of the velocity vector, respectively.

[0090]

[0091] Get in X m Y m Z m The position and velocity vectors of the spacecraft in the coordinate system

[0092]

[0093] Step 5.4: Calculate the flight time of the lunar center's arrival orbit within the lunar sphere of influence.

[0094] The near angle of the entrance point is

[0095]

[0096] The spacecraft's deviated anterior angle corresponding to the true anterior angle f is equal to

[0097]

[0098] The calculated flight time within the lunar SOI is as follows:

[0099]

[0100] Step 5.5: Calculate the position and velocity vector of the moon's orbit in the geocentric inertial frame.

[0101] The position and velocity of the moon in the XYZ coordinate system are respectively written as R. moon =D·[cosα,sinα,0] T and Vmoon =v m [-sinα,cosα,0] T Where α is the lunar rotation angle corresponding to the flight time within the Moon's SOI. Substituting Δt, we obtain the position and velocity of the lunar center's orbit in the XYZ coordinate system.

[0102]

[0103] Step Six: Based on the Earth-Moon transfer orbit that satisfies the mission constraints obtained in Step Five, implement the Earth-Moon orbit transfer that satisfies the mission constraints.

[0104] Beneficial effects:

[0105] 1. This invention discloses a rapid optimization method for Earth-Moon transfer orbits based on spatial conic section splicing. The pulse maneuver applied during the Earth-Moon transfer is equivalent to a tangential maneuver, with the flight path angle at the maneuver point being 0°. The altitude H0 of the near-Earth parking circular orbit is set according to mission requirements. The Earth-Moon transfer orbit spliced ​​by spatial conic sections is determined by four parameters: the Earth-Moon transfer departure velocity increment ΔV0, the phase λ1 of the lunar influence sphere entry point, and the parameters θ and ψ describing the spatial azimuth of the Earth-centered departure orbital plane. Specifically, coordinate transformation and geometric analysis methods are used to add the azimuth parameters θ and ψ to describe the spatial azimuth of the Earth-centered departure orbital plane, extending the planar conic section splicing method to the spatial case, making this invention applicable to the rapid optimization and solution of spatial Earth-Moon transfer orbits. If θ = 0° or θ = 180°, the spatial transfer orbit degenerates into a planar transfer orbit.

[0106] 2. The present invention discloses a rapid optimization method for the Earth-Moon transfer orbit based on spatial conic section splicing. It is derived using two-body orbit theory and based on the splicing of a double two-body model. It has analytical characteristics and has the advantage of speed in orbit analysis and optimization.

[0107] 3. The present invention discloses a rapid optimization method for Earth-Moon transfer orbit based on spatial conic section splicing. By defining the cases of short-range transfer and long-range transfer, and analyzing and excluding two infeasible cases of spatial conic section splicing transfer, the method has wider applicability and better convergence when optimizing the orbit design, thereby improving the efficiency of rapid optimization of Earth-Moon transfer orbit. Attached Figure Description

[0108] Figure 1 This is a schematic diagram of the starting trajectory from the geocentric point in the geocentric inertial coordinate system.

[0109] Figure 2 (a) is a schematic diagram of the short-range transfer orbit from the Earth's center. Figure 2 (b) is a schematic diagram of the long-distance transfer orbit from the Earth's core.

[0110] Figure 3 It is the lunar inertial coordinate system X m Y m Z m Schematic diagram of coordinate system.

[0111] Figure 4 (a) is the coordinate system X of the lunar center reaching the orbital coordinate system. h Y h Z h Coordinate system diagram Figure 4 (b) is the X coordinate system of the perilunar point. p Y p Z p Schematic diagram of coordinate system.

[0112] Figure 5 This is a schematic diagram of the orbit for a calculation example of Earth-Moon transfer in a dimensionless Earth-Moon rotating system.

[0113] Figure 6 This is a flowchart of a rapid optimization method for the Earth-Moon transfer orbit based on spatial conic section splicing disclosed in this invention. Detailed Implementation

[0114] like Figure 6 As shown in the figure, this embodiment discloses a rapid optimization method for the Earth-Moon transfer orbit based on spatial conic section splicing. The specific implementation steps are as follows:

[0115] Taking a short-range transfer as an example, the starting trajectory is a near-Earth circular orbit at an altitude of 200km, and the arrival trajectory is constrained to be a lunar circular orbit at an altitude of 200km and an inclination angle of 20°. The initial values ​​of the optimized design parameters are taken as follows: ΔV0 = 3.13km / s, λ1 = 50°, θ = 0°, and ψ = 0°.

[0116] Step 1: Define the geocentric inertial coordinate system. The initial values ​​of the four optimization parameters are known: ΔV0 = 3.13 km / s, λ1 = 50°, θ = 0°, and ψ = 0°. Define the geocentric starting orbit coordinate system based on the azimuth angles θ and ψ that describe the spatial orientation of the geocentric starting orbit, and obtain the coordinate transformation matrix between the geocentric starting orbit coordinate system and the geocentric inertial coordinate system.

[0117] Step 1.1: Define the geocentric inertial coordinate system.

[0118] Figure 1 This is a schematic diagram of the Earth-centered starting orbit in a geocentric inertial coordinate system. The green sphere represents the Moon's SOI, and the blue line represents the Earth-centered starting orbit. In the geocentric inertial frame, the X-axis is the direction of the line connecting the Earth and the Moon when the spacecraft enters the Moon, the Z-axis is the direction of the normal along the Moon's orbit, and the Y-axis is determined by the right-hand rule.

[0119] Step 1.2: Define the geocentric starting orbit coordinate system and calculate the transformation matrix between the geocentric inertial coordinate system and the geocentric coordinate system.

[0120] As can be seen from the figure, the orbital plane of the starting orbit from the Earth's center is determined by two angles: θ and ψ. ψ is the angle between the line of intersection of the starting orbital plane and the XY plane of the Earth's inertial coordinate system and the X-axis, with the counterclockwise direction being positive. θ is the angle between the angular momentum of the starting orbit and the Z-axis of the Earth's inertial coordinate system.

[0121] exist Figure 1 In the middle, the orbital plane of the Earth-centered orbit passes through the Moon, affecting the formation of the sphere around O. m The circular cross section S centered on m Represented by the red dashed line on the Moon's SOI. From the Moon's center to O m The distance is,

[0122] d=Dsinψsinθ, (32)

[0123] In the formula, D is the distance between the Earth and the Moon, θ is 0°, ψ is 0°, and d = 0 km. Section S m The radius is,

[0124]

[0125] In the formula R s Let R be the radius of the sphere affected by the moon. s =66300km, R is calculated c =66300km.

[0126] The geocentric coordinate system is defined based on the orbital plane of the starting orbit: the origin is located at the Earth's center, X... n Axis along the Earth's center to O m The direction of the line connecting Z n The axis is along the direction of the angular momentum vector, Y n The axis is defined using the right-hand rule. Therefore, Z... n The unit vector is,

[0127] k n =[sinψsinθ,-cosψsinθ,cosθ] T (34)

[0128] k n =[0,0,1] T O m The position coordinates in the geocentric inertial frame are,

[0129] O m =[D,0,0] T -d·k n (35)

[0130] O m =[D,0,0] TAlong X n The unit vector is written as,

[0131] i n =O m / D n (36)

[0132] Where D n From the Earth's core to O m The distance, D n =||O m ||,i n =[1,0,0] T Along Y n The unit vector is equal to j n =k n ×i n From X n Y n Z n The transformation matrix from the coordinate system to the XYZ coordinate system is:

[0133]

[0134] Step 2: Based on the initial value ΔV0 of the Earth-Moon transfer departure velocity increment, calculate the position and velocity of the initial point of the Earth-Moon transfer in the geocentric inertial frame, and determine the orbital parameters of the geocentric departure orbit. Then, define short-range transfer and long-range transfer, and based on the two methods of entering the lunar influence sphere, the phase λ1 of the lunar influence sphere entry point, and the orbital parameters of the geocentric departure orbit, calculate the position and velocity vector at the entry point of the influence sphere in the geocentric departure orbit coordinate system, and transfer the state to the geocentric inertial frame using a rotation matrix.

[0135] Step 2.1: Calculate the position and velocity of the initial point of the Earth-Moon transfer in the Earth-centered inertial frame, and determine the orbital parameters of the starting orbit from the Earth-centered point.

[0136] The initial point of the Earth-Moon transfer, its distance from the Earth's center in the Earth-centered inertial frame, and its velocity are:

[0137]

[0138] Among them, R e and μ e Let R0 be the Earth's radius and gravitational coefficient, respectively, and ΔV0 be the velocity increment at the initial point. In a near-Earth circular orbit at an altitude of 200 km, R0 = 6578 km, V0 = 10.9143 km / s. In the geocentric inertial frame, the orbital parameters of the geocentric starting orbit are obtained using the following calculation method. The energy and angular momentum of the geocentric starting orbit are expressed as follows:

[0139]

[0140] E was calculated e = -1.0346km 2 / s 2 h e =7.1795×10 4 km 2 / s, from which the semi-major axis and eccentricity of the orbit starting from the geocenter are obtained.

[0141]

[0142] in, Given the semi-major diameter of the orbit originating from the Earth's center, a is calculated. e =1.9264×10 5 km, e e =0.9659.

[0143] Step 2.2: Based on the two methods of entering the lunar influence sphere, namely short-range transfer and long-range transfer, calculate the position and velocity vector at the entry point of the influence sphere in the geocentric starting orbit coordinate system, and transfer it to the inertial frame using a rotation matrix.

[0144] Figure 2 X was displayed n Y n Z n The coordinate system shows two types of geocentric orbits, where the blue line represents the geocentric orbit, and the green disk represents the cross-section of the geocentric orbital plane and the lunar SOI. The dashed line represents the remaining portion of the geocentric orbit unaffected by the Moon's gravity. If the geocentric orbit is an elliptical orbit, there are two ways to enter the lunar SOI. For example... Figure 2 In (a), the first type is a short-range transfer, where the entry point is before the apogee. For example... Figure 2 In (b), the second scenario involves a long-range transfer, with the entry point after the apogee. For both scenarios, the distance and velocity of the lunar influence sphere's entry point relative to Earth are...

[0145]

[0146] The calculated value is R1 = 3.4554 × 10⁻⁶. 5 km, V1=0.4879km / s, short-range transfer affects the geocentric velocity vector V at the sphere's entry point. n for,

[0147] V n =V1·[sin(φ1-γ1),cos(φ1-γ1),0] T (41)

[0148] V was calculated n =[0.4061 0.2704 0] T km / s.

[0149] The equations for short-range transfers described above also apply to geocentric starting trajectories such as hyperbolas or parabolas, but long-range transfers are only applicable to elliptical trajectories. In a geocentric inertial coordinate system, the vectors affecting the position and velocity at the sphere's entry point are represented as follows:

[0150]

[0151] Where λ1 is the phase angle of the point of entry of the lunar influence sphere, and D n From Earth to O m The distance was calculated to be R1 = [3.4179 × 10]. 5 5.0789×10 4 0] T km, V1 = [0.4061 0.2704 0] T km / s.

[0152] Step 3: Define two infeasible scenarios for conic section splicing transfer based on the state of the lunar influence sphere entry point, eliminate the infeasible scenarios, and calculate the flight time of the short-range and long-range orbits respectively based on the geocentric transfer orbit parameters.

[0153] Step 3.1: Define two infeasible cases for conic section splicing and transfer based on R1 and V1 in the geocentric inertial coordinate system:

[0154] 1. No lunar crossing: This occurs when ΔV0 is not large enough, and the spacecraft cannot reach the lunar influence sphere, represented by the following condition: e e <1 and a e (1+e e ) < R1, in the example e e =0.9659<1, a e (1+e e ) = 3.7871 × 10 5 R1 = 3.4554 × 10 5 km, which is enough to reach the moon, so there's no need to rule it out.

[0155] 2. Secondary Crossing: This scenario is determined by r2·v, where r2 is the position vector of the spacecraft's entry point relative to the Moon in the Earth-Moon rotation frame, and v is the velocity vector of the spacecraft's entry point in the Earth-Moon rotation frame. r2 and v are obtained from R1 and V1 through coordinate transformation between the geocentric inertial frame and the Earth-Moon rotation frame. If r2·v ≥ 0, it indicates that the spacecraft is leaving the Moon rather than entering it, which is an infeasible conic section transfer. In the example, r2·v = -0.2824 < 0, so it is not a secondary crossing and does not need to be ruled out.

[0156] Step 3.2: Calculate the flight time of the orbit starting from the Earth's center.

[0157] According to the two-body orbit theory, given R1 and V1, the starting orbit from the geocentric point is uniquely determined in the geocentric inertial frame. The initial point is the perigee of the geocentric orbit, and the initial true anomaly angle f0 = 0. The true anomaly angle of the entry point has two different expressions for short-range and long-range transfer orbits. For short-range transfers…

[0158]

[0159] f e = 3.0592 rad, h is determined through step 2.2. e e e f0, by f e Calculate the flight time t of the orbit starting from the Earth's center. e = 3.0766 days.

[0160]

[0161] Step 4: Define the lunar center inertial coordinate system, transfer the state of the influencing sphere's entry point to the lunar center inertial coordinate system, calculate the orbital parameters of the lunar center's arrival orbit, as well as the altitude and orbital inclination of the perigee relative to the moon, establish the penalty function, and repeat steps 1, 2, 3, and 4. Based on the task constraints, iteratively optimize the four optimization parameters ΔV0, λ1, θ, and ψ to obtain parameters that satisfy the task constraints.

[0162] Step 4.1: Define the lunar-centered inertial coordinate system.

[0163] Define X under lunar inertia m Y m Z m coordinate system Figure 3 X under lunar inertia m Y m Z m Coordinate system, where X m Y m and Z m The axes coincide with the corresponding axes in the XYZ coordinate system of the geocentric inertial frame. In this coordinate system, the spacecraft's position and velocity vector at the entry point are,

[0164]

[0165] in The speed at which the moon orbits the earth is R² = [-4.2617 × 10⁻⁶]. 4 5.0789×10 4 0] T km, V2=[0.4061 -0.7541 0] T km / s.

[0166] Step 4.2: Calculate the orbital parameters of the lunar center's arrival orbit in the lunar center inertial coordinate system.

[0167] The energy and angular momentum of the orbit relative to the lunar center inertial frame upon reaching the lunar center are:

[0168]

[0169] Where μ m It is the gravitational parameter of the moon, E. m =0.2929km 2 / s 2 h m =[0 0 1.1516×10 4 ] T km 2 / s, from which the eccentricity and semi-major axis of the lunar center's orbit are obtained.

[0170]

[0171] Among them, P m =||h m || 2 / μ m =2.7048×10 4 km is the semi-major path, a m =8.3705×10 3 km, e m =2.0570. The distance, velocity, and orbital inclination of the perigee relative to the Moon are,

[0172]

[0173] r pm =8.8479×10 3 km, v pm = 1.3015 km / s, i m =0°, which does not meet the constraints of the lunar orbit mission.

[0174] Step 4.3: Establish the penalty function and iteratively optimize the four optimization parameters ΔV0, λ1, θ, and ψ according to the task constraints to obtain the parameters that satisfy the task constraints.

[0175] F(ΔV0,λ1,θ,ψ)=||r pm -r c ||+||i m -i c || (21)

[0176] Where r c For lunar orbital altitude constraints, i cTo constrain the lunar orbit inclination, the Earth-Moon transfer orbit satisfying the constraints was solved by optimizing the penalty function to zero through iteration. The final values ​​obtained were ΔV0 = 3.13 km / s, λ1 = 68.6633°, θ = 1.3989°, and ψ = 0.3672°.

[0177] Calculate the cross-sectional center O from the center of the Moon to the Moon's influence sphere and orbit. m The distance and cross-sectional radius d = 60.1400 km, R c =66299.9727km.

[0178] Calculate from X n Y n Z n Transformation matrix from coordinate system to XYZ coordinate system

[0179]

[0180] The initial point of the Earth-Moon transfer has a distance and velocity of R0 = 6578 km and V0 = 10.9144 km / s in the Earth-centered inertial frame, respectively.

[0181] The initial state of the Earth-Moon transfer in the geocentric inertial frame is R0 = [-6418.8 -1438.5 0] km, V0 = [2.3868 -10.65 0] km / s.

[0182] The energy, angular momentum, semi-major axis, semi-major diameter, and eccentricity of the orbit starting from the Earth's center are E e = -1.03423km 2 / s 2 h e =71794.74km 2 / s,a e =192703.2km, P e =12931.46km, e e =0.9658.

[0183] The distance and velocity of the entry point relative to Earth are R1 = 365536.37 km and V1 = 0.3353 km / s, respectively.

[0184] For the short-range transfer of lunar influence sphere, the entry point has φ1 = 54.0275°, γ1 = 9.7265°, and the position and velocity vectors at the entry point are V1 = [0.2347 0.2394 0.00581] km / s and R1 = [360281.9 61738.7 1451.4] km.

[0185] Calculate the true anomaly angle of the entrance / exit point and the flight time of the geocentric departure orbit, respectively, f. e=3.09089 rad, t e = 3.7264 days.

[0186] X under lunar inertia m Y m Z m In the coordinate system, the spacecraft's position and velocity vector at the entry point are R2 = [-24123 61738.7 1451.4] km and V2 = [0.2347 -0.7789 0.00581] km / s.

[0187] The energy, angular momentum, eccentricity, semi-major axis, and semi-major axis of the lunar orbit are E. m =0.2569km 2 / s 2 h m = [1489.2 480.8 4299.4]km 2 / s,e m =1.2031, P m = 4269.6km, a m =9541.5km.

[0188] The distance and velocity of the perigee relative to the Moon, and the orbital inclination angle r. pm =200km, i m =20°, which meets the constraints of the lunar orbit mission.

[0189] Step 5: Based on the obtained parameters that satisfy the mission constraints, calculate the complete Earth-Moon transfer orbit. Define the coordinate system of the lunar center arrival orbit and the coordinate system of the perigee from the orbit parameters of the lunar center arrival orbit. Calculate the position and velocity vector of the lunar center arrival orbit in the lunar center inertial frame. Calculate the flight time of the lunar center arrival orbit within the lunar sphere of influence. Calculate the position and velocity vector of the lunar center arrival orbit in the Earth center inertial frame. This yields the Earth-Moon transfer orbit that satisfies the mission constraints, thus achieving rapid optimization of the Earth-Moon transfer orbit.

[0190] Step 5.1: Define the lunar center arrival orbit coordinate system.

[0191] Definition based on R2 and h m The lunar center reaches the orbital coordinate system X h Y h Z h coordinate system Figure 4 (a) is based on R2 and h m X h Y h Z h A coordinate system, where the red line represents the lunar center reaching its orbit. X h Z h Y h The unit vector is represented as

[0192]

[0193] X h Y h Z h coordinate system to X m Y m Z m The transformation matrix of the coordinate system is M h =[i h j h k h ],

[0194]

[0195] Step 5.2: Define the lunar coordinate system.

[0196] Define the lunar coordinate system X p Y p Z p Coordinate system, where X p The axis Z runs along the line connecting the center of the moon to the perigee of the orbit. p Axis and Z h The axes are in the same direction, Y p The axis is defined by the right-hand rule. Figure 4 (b) Shows the X on the orbital plane of the lunar center's orbit. p Y p Z p Coordinate system and X h Y h Z h A coordinate system, where the red line represents the orbit from the moon's center. From X... p Y p Z p coordinate system to X h Y h Z h The transformation matrix of the coordinate system is,

[0197]

[0198] Step 5.3: Calculate the position and velocity vector of the moon's center reaching the orbit in the lunar inertial frame.

[0199] Let f in ∈[-f in ,f in Let f be the true anomaly angle of the hyperbolic orbit within the Moon's SOI, f∈[-2.5754,2.5754] rad. Then, the spacecraft at X... p Y p Z p The corresponding position and velocity vectors in the coordinate system are represented as follows:

[0200]

[0201] Among them, v r and v u These represent the radial and tangential components of the velocity vector, respectively.

[0202]

[0203] Substituting f into X, we get m Y m Z m The position and velocity vectors of the spacecraft in the coordinate system.

[0204]

[0205] Step 5.4: Calculate the flight time of the lunar center's arrival orbit within the lunar sphere of influence.

[0206] The approximation angle of the entrance point is F0 = -2.5754 rad.

[0207]

[0208] The deviated anomaly angle of the spacecraft corresponding to the true anomaly angle f is equal to,

[0209]

[0210] The calculated flight time within the lunar SOI is Δt = 0.8144 days.

[0211]

[0212] Step 5.5: Calculate the position and velocity vector of the moon's orbit in the geocentric inertial frame.

[0213] The position and velocity of the moon in the XYZ coordinate system are respectively written as R. moon =D·[cosα,sinα,0] T and V moon =v m [-sinα,cosα,0] T Where α is the lunar rotation angle corresponding to the flight time within the Moon's SOI. Substituting Δt, we obtain the position and velocity of the lunar center's orbit in the XYZ coordinate system.

[0214]

[0215] Step Six: Based on the Earth-Moon transfer orbit that satisfies the mission constraints obtained in Step Five, such as... Figure 5 As shown, a lunar orbit transfer that satisfies mission constraints is achieved.

[0216] The above detailed description illustrates the purpose, technical solution, and beneficial effects of the invention. The above description is merely a specific implementation example of the invention, used to explain the invention, and is not intended to limit the scope of protection of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the invention should be included within the scope of protection of the invention.

Claims

1. A method for fast optimization of a lunar transfer trajectory based on spatial conic splicing, characterized in that: It comprises the following steps, Step 1: define the geocentric inertial coordinate system, and obtain the initial values of the four optimization parameters , , , , according to the azimuth angle describing the spatial orientation of the geocentric launch orbit and define the geocentric launch orbit coordinate system, and obtain the coordinate conversion matrix of the geocentric launch orbit coordinate system and the geocentric inertial coordinate system; The step one implementation method is, Step 1.1: define the geocentric inertial coordinate system; In the geocentric inertial system, the X axis is the connecting line direction of the spacecraft from the earth to the moon when entering the moon, the Z axis is the normal direction along the moon orbit, and the Y axis is determined by the right-hand rule; Step 1.2: define the geocentric departure orbit coordinate system, and calculate the conversion matrix with the geocentric inertial coordinate system; The orbital plane of the geocentric launch orbit is determined by two angles: and where is the angle between the geocentric launch orbit plane and the XY plane of the geocentric inertial coordinate system, positive in the counterclockwise direction, is the angle between the geocentric launch orbit angular momentum and the Z axis of the geocentric inertial coordinate system; the orbital plane of the geocentric launch orbit passes through the lunar sphere of influence forming a circular cross section centered at ; the distance from the lunar center to is given by where D is the distance between the earth and the moon; the radius of the cross section is In the formula R s The radius of the sphere is influenced by the moon; The geocentric launch orbit coordinate system is defined according to the orbit plane of the geocentric launch orbit, with the coordinate origin at the center of the Earth, The axis is along the line from the center of the Earth to The axis is along the direction of the angular momentum vector, The axis is along the direction of the angular momentum vector, The axis is defined by the right-hand rule; then The unit vector of The position coordinates in the geocentric inertial system are expressed as along the unit vector of where is the distance from the center of the earth to , ; the unit vector along is equal to ; from The conversion matrix from the coordinate system to the XYZ coordinate system is ; Step two: initial value of the increment of the launch velocity for the lunar transfer , the position and velocity of the initial point of the lunar transfer in the Earth-centered inertial system are calculated, and the orbital parameters of the Earth-centered launch orbit are determined, then the short-range transfer and the long-range transfer are defined, and the position and velocity vectors of the entry point of the sphere of influence in the Earth-centered launch orbit coordinate system are calculated according to the two ways of entering the sphere of influence of the Moon and the phase of the entry point of the sphere of influence of the Moon and the orbital parameters of the Earth-centered launch orbit, and the state is converted to the Earth-centered inertial system through a rotation matrix. Step three: define two kinds of unfeasible situations of conic curve splicing transfer according to the moon influence ball entry point state, exclude unfeasible situations, and calculate the flight time of short-range and long-range orbits according to the geocentric transfer orbit parameters respectively; Step four: define the lunar center inertial coordinate system, transfer the impact point state of the sphere to the lunar center inertial coordinate system, calculate the orbit parameters of the lunar center arrival orbit and the altitude and orbit inclination of the perilune relative to the moon, establish a penalty function, and repeat steps one, two, three and four to obtain the parameters satisfying the mission constraints , , , iterative optimization is performed to obtain parameters satisfying the mission constraints; Step five: according to the obtained parameters satisfying the task constraints, calculate the complete geolunar transfer orbit, define the moon arrival orbit coordinate system and the perisel coordinate system according to the orbit parameters of the moon arrival orbit, calculate the position and velocity vector of the moon arrival orbit in the moon inertial system, calculate the flight time of the moon arrival orbit in the moon influence ball, calculate the position and velocity vector of the moon arrival orbit in the geocentric inertial system, and obtain the geolunar transfer orbit satisfying the task constraints, that is, realize the rapid optimization of geolunar transfer orbit.

2. The method of claim 1, wherein the method is based on a spatial conic curve splicing. It also comprises step six, which realizes the geolunar orbit transfer satisfying the task constraints according to the geolunar transfer orbit satisfying the task constraints obtained in step five.

3. The fast optimization method for the transfer orbit between the earth and the moon based on the spatial conic splicing according to claim 2, characterized in that: The step two implementation method is, Step 2.1: calculate the position and velocity of the initial point of geolunar transfer in the geocentric inertial system, and determine the orbit parameters of the geocentric departure orbit; The geocentric distance and velocity of the initial point of geolunar transfer in the geocentric inertial system are, wherein, and R and G are the radius of the earth and the gravitational coefficient, respectively, is the velocity increment at the initial point; the orbital parameters of the geocentric launch orbit are obtained in the geocentric inertial system by the following calculation method; the energy and angular momentum of the geocentric launch orbit are expressed as Thus, the semi-major axis and the eccentricity of the geocentric orbit are wherein, is the semiradius of the geocentric orbit; Step 2.2: according to the two ways of short-range transfer and long-range transfer entering the moon influence ball, calculate the position and velocity vector at the influence ball entry point in the geocentric departure orbit coordinate system, and rotate to the inertial system through the rotation matrix; The geocentric departure orbit is an elliptical orbit, and there are two ways to enter the moon influence ball: short-range transfer, the moon influence ball entry point occurs before the apogee of the geocentric segment elliptical orbit; long-range transfer, the moon influence ball entry point occurs after the apogee of the geocentric segment elliptical orbit; for the two cases, the distance and velocity of the moon influence ball entry point relative to the earth are represented as The short and long range transfers affect the earth-centered velocity vector at the ball entry point are represented respectively by wherein The equations above for short-range transfers can also be applied to Hohmann or parabolic Earth-prograde orbits, but long-range transfers are only applicable to elliptical orbits; in the geocentric inertial frame, the position and velocity vector at the sphere entry point are expressed as wherein, is the phase angle of the lunar perturbing sphere entry point, is the distance from the Earth center to the Moon.

4. The method of claim 3, wherein the method is based on a spatial conic curve splicing. The step three implementation method is, Step 3.1: Based on the geocentric inertial coordinate system and Define two infeasible cases for conic section splicing transfer: No lunar crossing: This situation occurs when or If the influence sphere is not large enough, the spacecraft cannot reach the Moon's sphere of influence, as indicated by the following condition: and Secondary crossing: This situation is caused by... Confirmed, among which Let be the position vector of the spacecraft's entry point relative to the Moon in the Earth-Moon rotation frame. Let be the velocity vector of the spacecraft's entry point in the Earth-Moon rotation frame; from the coordinate transformation between the geocentric inertial frame and the Earth-Moon rotation frame... and get and ;if If this indicates that the spacecraft is leaving the moon rather than entering it, then the conic section splicing transfer is not feasible. Step 3.2: calculate the flight time of the geocentric departure orbit; From the two-body orbit theory, we have and Then, the geocentric orbit is uniquely determined in the geocentric inertial system, the initial point is the perigee of the geocentric orbit, and the initial true anomaly The true anomaly of the entry point has two different expressions for short- and long-range transfer orbits. For short-range transfer, For long-range metastasis, Determined by step 2.2 , , Calculated from Flight time of the Earth-centered launch orbit ; 。 5. The fast optimization method for the transfer orbit between the Earth and the Moon based on the spatial conic splicing according to claim 4, characterized in that: The step four implementation method is, Step 4.1: define the moon inertial coordinate system; The definitions of the inertial frame under the moon center The coordinate system, where , and The axes are consistent with the corresponding axes in the XYZ coordinate system under the earth center inertial system; in this coordinate system, the position and velocity vector of the spacecraft at the entry point are where is the velocity of the moon around the earth; Step 4.2: calculate the orbit parameters of the moon arrival orbit in the moon inertial coordinate system; The energy and angular momentum of the moon arrival orbit relative to the moon inertial system are represented as where is the gravitational parameter of the Moon, from which the eccentricity and semi-major axis of the lunar-centered transfer orbit are derived wherein, is the semi-major axis; the distance of the pericynthion from the Moon and the velocity at the pericynthion are expressed as Step 4.3: Establish a penalty function, according to the task constraints on the four optimization parameters , , , Iterative optimization is performed to obtain parameters that meet the task constraints; where is a lunar orbit altitude constraint, is a lunar orbit inclination constraint, the lunar-earth transfer orbit satisfying the constraints is solved by optimizing iterations to reduce the penalty function to zero.

6. The method of claim 5, wherein the method is based on a spatial conic curve splicing. The step five implementation method is, Step 5.1: define the moon arrival orbit coordinate system; The definition is based on and the selenocentric arrival orbit coordinate system coordinate system, , , a unit vector is represented as The transformation matrix from the coordinate system of the camera to the coordinate system of the object is ;​ Step 5.2: define the perisel coordinate system; Defining a perilune coordinate system Coordinate system, wherein The axis is along the direction of the line from the center of the moon to the perilune of the arrival orbit, The axis is opposite to The axis is in the same direction as The axis is defined by the right-hand rule; from The transformation matrix from The coordinate system to wherein is the axis of the the angle between the axes is written as Step 5.3: Calculate the position and velocity vector of the lunar center of mass at the lunar center of mass arrival orbit in the lunar center of mass inertial frame; Let be the true anomaly of the spacecraft's hyperbolic perigee orbit, then the spacecraft's position and velocity vectors in the coordinate system are given by wherein, and denote the radial and tangential components of the velocity vector, respectively The position and velocity vector of the spacecraft in the coordinate system is obtained a coordinate system Step 5.4: Calculate the time of flight of the periselene orbit inside the sphere of influence of the Moon. The pericenter angle of the entry point is The true anomaly The corresponding spacecraft's eccentric anomaly equals The flight time within the lunar SOI is calculated to be Step 5.5: Calculate the position and velocity vector of the selenocentric arrival orbit in the geocentric inertial frame; The position and velocity of the moon in the XYZ coordinate system are written as and respectively, where is the rotation angle of the moon corresponding to the time of flight within the SOI of the moon, , and the position and velocity of the lunar cislunar orbit in the XYZ coordinate system are obtained by substituting . 。

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