A method for designing resonant halo orbits of small thrust spacecraft under thrust direction constraint

Abstract

The application discloses a small-thrust spacecraft resonance halo orbit design method under thrust direction constraint, and belongs to the field of spacecraft orbit design. The application solves the problem of thrust direction constraint in a high eccentricity gravitational field. The method comprises the following steps: halo orbits satisfying a preset resonance relationship with the orbit period of a second gravitational body are calculated in a sun-second gravitational body circular restricted three-body, and initial states of the halo orbits are determined on the two sides of sun-second gravitational body Lagrange points L1 and L2; a first step numerical continuation is performed on the halo orbits, the eccentricity of the second gravitational body orbit is increased from zero to an actual orbit eccentricity, and a non-thrust resonance halo orbit is obtained; and a second step numerical continuation is performed on the non-thrust resonance halo orbit, the thrust acceleration amplitude parameter of the spacecraft is gradually increased from zero to a preset upper limit value or until the resonance halo orbit collides with the surface of the second gravitational body, and a resonance halo orbit under thrust direction constraint is obtained. The application is suitable for spacecrafts mainly propelled by continuous small thrust.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft orbit design, and in particular relates to a method for designing resonant halo orbits for low-thrust spacecraft under thrust direction constraints. Background Technology

[0002] In the classic Circular Restricted Three-Body Problem, various special orbits exist, such as halo orbits, Lyapunov orbits, Lissajous orbits, and distant retrograde orbits (DRO). Except for the Lissajous orbit, the others are periodic orbits, existing continuously as families within the Circular Restricted Three-Body Problem. However, most celestial bodies in the real solar system have orbits with significant eccentricity, especially Mercury, whose orbital eccentricity can exceed 0.2. Under such highly eccentric orbital conditions, the dynamical models constructed using the Circular Restricted Three-Body Problem are insufficient to accurately describe actual systems, necessitating the introduction of the Elliptic Restricted Three-Body Problem for modeling. In the Elliptic Restricted Three-Body Problem, periodic orbits are no longer continuously distributed but exist as discrete resonant orbits, with their orbital periods being integer ratios to the orbital periods of the second primary body.

[0003] Under the influence of external periodic perturbations or the spacecraft's own thrust, the spacecraft's translational point will shift, forming a so-called "artificial translational point." Correspondingly, its orbital morphology will also shift and deform. Recent years have seen significant progress in research on translational point orbits. By introducing solar radiation pressure into the circularly restricted three-body problem of the Earth-Moon system, the original system is transformed into a non-autonomous system. The Earth-Moon Line Steering Law and the Sun-Sail Line Steering Law can be used to design Lyapunov orbits and halo orbits for solar sail spacecraft, respectively. This has been further extended to solar sail pointing strategies outside the XY plane in the circularly restricted three-body problem, yielding richer orbital morphologies. Introducing solar gravity and solar radiation pressure into the Sun-Earth-Moon four-body system allows for analysis of the changing characteristics of the spacecraft's translational point orbit under perturbation. However, these studies remain limited to circular models or simple perturbation scenarios, failing to address the core issue of thrust direction constraints in highly eccentric gravitational fields.

[0004] The three-body system centered on the Sun, consisting of the Sun, the second gravitational body, and the spacecraft, is a common dynamic system in the solar system. This is mainly because the eccentricity of the second gravitational body's orbit makes the circularly restricted three-body problem unsuitable for accurately approximating the dynamic environment. Therefore, the elliptical restricted three-body problem must be used to approximate the dynamic environment. Although some special periodic orbits have been discovered, the characteristics of the periodic orbit of a spacecraft under continuous small thrusts regarding the changes in thrust direction and amplitude still need to be studied. Summary of the Invention

[0005] In view of this, the present invention aims to propose a method for designing resonant halo orbits for low-thrust spacecraft under thrust direction constraints, so as to solve the problem of thrust direction restriction in high eccentricity gravitational fields.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: A method for designing resonant halo orbits for low-thrust spacecraft under thrust direction constraints, the method comprising: Step S1: Calculate the halo orbit that satisfies the preset resonance relationship with the orbital period of the second gravitational body in the circular restricted three-body system of the Sun-Second Gravitational Body, and determine the initial state of the corresponding halo orbit on both sides of the Lagrange points L1 and L2 of the Sun-Second Gravitational Body respectively; Step S2: Perform the first step of numerical extension on the initial state of the halo orbit, increasing the eccentricity of the second gravitational body orbit from zero to the actual orbital eccentricity; thus obtaining the thrustless resonance halo orbit in the Sun-Second Gravitational Elliptical Restricted Three-Body Problem. Step S3: Perform a second numerical extension on the obtained thrustless resonant halo orbit. Gradually increase the thrust acceleration amplitude parameter of the spacecraft from zero to a preset upper limit value or until the resonant halo orbit collides with the surface of the second gravitational body to obtain the resonant halo orbit under thrust direction constraint. The thrust direction constraint of the spacecraft is that the thrust direction always points to the sun, and the thrust amplitude is inversely proportional to the square of the distance from the spacecraft to the sun.

[0007] Furthermore, a preferred method is proposed, wherein the numerical extension is based on a multi-segment shooting method using forward-backward integral splicing, including: The periodic orbit is discretized into N segments with the same launch time, containing a total of N+1 discrete points. For an orbit with a period of T, the flight time of each segment is fixed at T / N. The constraints of the multi-segment shooting method are: the position velocity obtained by forward integration T / 2 of the i-th (i=1,…,N) discrete point is equal to the position velocity obtained by backward integration T / 2 of the i+1 discrete point, and the position velocity of the N+1 discrete point is equal to the position velocity of the 1st discrete point.

[0008] Furthermore, a preferred embodiment is proposed, wherein the preset resonance relationship in step S1 includes at least one of 3:1, 4:1, 5:2 and 7:2, wherein the ratio of the period of the halo orbit to the period of the second gravitational body orbit is 2π / 3, π / 2, 4π / 5 and 4π / 7 respectively.

[0009] Furthermore, a preferred method is proposed, wherein the actual orbital eccentricity in step S2 is determined based on the actual orbital parameters of the second gravitational body, and in the first step of numerical extension, the initial point of the resonance halo orbit is the intersection of the orbit with the XZ plane of the Sun-Second Gravitational Body Center of Mass Rotational Pulsating Coordinate System. The origin of the Sun-Second Gravitational Body Center of Mass Rotational Pulsating Coordinate System is located at the Sun-Second Gravitational Body Center of Mass, the X-axis points towards the second gravitational body, the Z-axis is along the orbital angular momentum direction of the second gravitational body, and the Y-axis forms a right-handed frame.

[0010] Furthermore, a preferred method is proposed, wherein step S2 further includes: based on the initial moment corresponding to the initial point of the resonance halo orbit, the second gravitational body is located at the perihelion or aphelion of its orbit, and the resonance halo orbit is divided into a perihelion group and an aphelion group for extension.

[0011] Furthermore, a preferred embodiment is proposed, wherein the thrust acceleration of the spacecraft in step S3 is: , in, A parameter for measuring the magnitude of acceleration. This is the ratio of the spacecraft's distance from the sun to astronomical units. This is the unit vector from the sun to the spacecraft.

[0012] Furthermore, a preferred embodiment is proposed, wherein the preset upper limit value in step S3 is 5e-7 km / s. 2 If the multi-segment shooting method in numerical extension does not converge, then the numerical extension terminates.

[0013] Based on the same inventive concept, the present invention also proposes a computer device, including a memory and a processor, wherein the memory stores a computer program, and when the processor runs the computer program stored in the memory, the processor executes a method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints as described in any of the preceding claims.

[0014] Based on the same inventive concept, the present invention also proposes a computer-readable storage medium storing a computer program, which, when executed by a processor, performs the steps of a method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints as described in any one of the above-mentioned methods.

[0015] Compared with the prior art, the beneficial effects of the present invention are: First, the method proposed in this invention enables a spacecraft, under the constraint of continuously pointing its thrust direction towards the Sun, to generate and maintain various resonance ratio halo orbits, including 3:1, 4:1, 5:2, 7:2, and 6:2 orbits generated by period-doubling bifurcation, on the Lagrange points L1 and L2 of the second gravitational body. Because these orbits are strictly resonant with the orbital period of the second gravitational body, they can remain stable for a long time in a high eccentricity environment. For example, in the Mercury system, with an eccentricity of 0.2056, the extended orbit maintains periodicity within the range of acceleration amplitude parameter σ from 5e-8 km / s² to 3.7e-7 km / s², solving the orbital instability problem caused by the fixed thrust direction in traditional methods.

[0016] Secondly, the method proposed in this invention can be applied to both circularly restricted three-body problems and elliptically restricted three-body problems, directly serving observation missions targeting a second gravitational body. By adjusting the thrust acceleration amplitude parameter σ, the orbital shape, the shortest distance to the second gravitational body, and the surface coverage can be precisely controlled. For example, in the Mercury system, increasing σ can optimize the orbital coverage of Mercury's surface without changing the thrust direction constraint, overcoming the difficulty of existing technologies that require the introduction of the directional law of solar radiation pressure and cannot dynamically adjust orbital parameters when the thrust amplitude changes.

[0017] Finally, the method proposed in this invention addresses the discrete orbit challenge of highly eccentric celestial bodies through a parameterized extension process. In the first extension step, as the eccentricity increases from zero to its actual value, period-doubling bifurcation is automatically handled to ensure continuous evolution of the orbital morphology. In the second extension step, the increase in the thrust acceleration amplitude parameter σ automatically terminates when collision detection or multi-stage target shooting methods fail to converge, ensuring the safety boundary of the design. This closed-loop control mechanism not only avoids orbital failures caused by model simplification in existing technologies but also reduces computational complexity. Attached Figure Description

[0018] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an undue limitation of the invention. In the drawings: Figure 1 This is a flowchart of a method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints, as described in this invention. Figure 2 This is a schematic diagram of the forward-backward integration multi-segment target shooting method described in this invention; Figure 3 This is a schematic diagram of the rotating pulsating coordinate system of the Sun-Second Gravitational Body Center of Mass as described in this invention; Figure 4 This is a schematic diagram of the resonance halo orbit of the low-thrust spacecraft on the L1 side of the Lagrange point according to the present invention; wherein, Figure 4(a) is the L1-3:1 perihelion group with an acceleration amplitude of 5e-8 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 4 (b) is the L1-3:1 aphelion group, with an acceleration amplitude of 9e-8 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 4 (c) is the L1-5:2 perihelion group, with an acceleration amplitude of 9e-8 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 4 (d) represents the L1-5:2 aphelion group, with an acceleration amplitude of 5e-8 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 5 This is a schematic diagram of the resonance halo orbit of the low-thrust spacecraft on the L2 side of the Lagrange point according to the present invention; wherein, Figure 5 (a) is the L2-3:1 perihelion group with an acceleration amplitude of 2.8e-7 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 5 (b) is the L2-3:1 aphelion group, with an acceleration amplitude of 1.8e-7 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 5 (c) is the L2-4:1 perihelion group, with an acceleration amplitude of 1.3e-7 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 5 (d) represents the L2-4:1 aphelion group with an acceleration amplitude of 5e-8 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 5 (e) is the L2-5:2 perihelion group, with an acceleration amplitude of 9e-8 km / s. 2 Schematic diagram of the time resonance halo orbit; Figure 5 (f) represents the L2-5:2 aphelion group, with an acceleration amplitude of 3.7e-7 km / s. 2 Schematic diagram of the time resonance halo orbit. Detailed Implementation

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. It should be noted that, unless otherwise specified, the embodiments and features in the embodiments of the present invention can be combined with each other, and the described embodiments are only some embodiments of the present invention, not all embodiments.

[0020] Implementation Method 1: A method for designing a resonance halo orbit for a low-thrust spacecraft under thrust direction constraints, as described in this implementation method, includes: Step S1: Calculate the halo orbit that satisfies the preset resonance relationship with the orbital period of the second gravitational body in the circular restricted three-body system of the Sun-Second Gravitational Body, and determine the initial state of the corresponding halo orbit on both sides of the Lagrange points L1 and L2 of the Sun-Second Gravitational Body respectively; Step S2: Perform the first step of numerical extension on the initial state of the halo orbit, increasing the eccentricity of the second gravitational body orbit from zero to the actual orbital eccentricity; thus obtaining the thrustless resonance halo orbit in the Sun-Second Gravitational Elliptical Restricted Three-Body Problem. Step S3: Perform a second numerical extension on the obtained thrustless resonant halo orbit. Gradually increase the thrust acceleration amplitude parameter of the spacecraft from zero to a preset upper limit value or until the resonant halo orbit collides with the surface of the second gravitational body to obtain the resonant halo orbit under thrust direction constraint. The thrust direction constraint of the spacecraft is that the thrust direction always points to the sun, and the thrust amplitude is inversely proportional to the square of the distance from the spacecraft to the sun.

[0021] The method described in this embodiment is applicable to spacecraft that use continuous small thrust as the main propulsion method. Various halo orbits with different resonance ratios can be obtained on the Lagrange points L1 and L2 near the second gravitational body. These halo orbits can exist in the vicinity of the second gravitational body for a long time due to their periodic resonance with the orbit of the second gravitational body.

[0022] Implementation Method Two: This implementation method further defines the method for designing resonant halo orbits of small-thrust spacecraft under thrust direction constraints as described in Implementation Method One. The numerical extension is based on a multi-segment firing method using forward-backward integral splicing, including: The periodic orbit is discretized into N segments with the same launch time, containing a total of N+1 discrete points. For an orbit with a period of T, the flight time of each segment is fixed at T / N. The constraints of the multi-segment shooting method are: the position velocity obtained by forward integration T / 2 of the i-th (i=1,…,N) discrete point is equal to the position velocity obtained by backward integration T / 2 of the i+1 discrete point, and the position velocity of the N+1 discrete point is equal to the position velocity of the 1st discrete point.

[0023] In this embodiment, the number of discrete points N is dynamically adjusted based on the orbital period and convergence to ensure numerical stability.

[0024] This implementation method is explained in conjunction with Implementation Method 1. This implementation method addresses the challenge of designing resonant halo orbits for small-thrust spacecraft under thrust direction constraints in the elliptical restricted three-body problem with the Sun as the first gravitational body, proposing an innovative method based on two-step numerical extension. Existing technologies struggle to accurately model circular restricted three-body problems due to high eccentricity of celestial bodies within the solar system, and existing perturbation introduction techniques fail to address the coordination issue between thrust direction constraints and orbital discretization. The core principle of this implementation method lies in constructing a basic resonant orbit within a circular restricted three-body environment, then gradually extending it to a high eccentricity elliptical restricted three-body environment, and integrating thrust amplitude variations to achieve periodic orbit maintenance under strict thrust direction constraints.

[0025] The method proposed in this embodiment is applicable to spacecraft propelled by continuous low thrust. It generates various resonant halo orbits on the L1 or L2 side of the second gravitational body. Because the orbital period resonates with the main celestial body, it can remain stable for a long time in a high eccentricity environment, significantly extending the observation window. Secondly, the method is compatible with both circularly confined three-body and elliptically confined three-body environments and can be directly used for second gravitational body observation missions. By adjusting the σ value, the orbital shape, the shortest distance to the second gravitational body, and the surface coverage are optimized, improving mission adaptability. Thirdly, the numerical extension process ensures controllable orbital parameters, avoids design failures, and supports perihelion and aphelion group extensions, enhancing the method's versatility.

[0026] Implementation Method 3: This implementation method further defines the design method for a resonant halo orbit of a small-thrust spacecraft under thrust direction constraint as described in Implementation Method 1. The preset resonance relationship in step S1 includes at least one of 3:1, 4:1, 5:2 and 7:2, wherein the ratio of the period of the halo orbit to the period of the second gravitational body orbit is 2π / 3, π / 2, 4π / 5 and 4π / 7 respectively.

[0027] Implementation Method Four: This implementation method further defines the design method for a resonant halo orbit of a small-thrust spacecraft under thrust direction constraints as described in Implementation Method One. In step S2, the actual orbital eccentricity is determined based on the actual orbital parameters of the second gravitational body. In the first step of numerical extension, the initial point of the resonant halo orbit is the intersection of the orbit with the XZ plane of the rotating pulsating coordinate system of the Sun-Second Gravitational Body Center of Mass. The origin of the rotating pulsating coordinate system of the Sun-Second Gravitational Body Center of Mass is located at the Sun-Second Gravitational Body Center of Mass, the X-axis points towards the second gravitational body, the Z-axis is along the orbital angular momentum direction of the second gravitational body, and the Y-axis forms a right-handed frame.

[0028] Implementation Method 5: This implementation method further defines the design method for a small-thrust spacecraft resonant halo orbit under thrust direction constraint described in Implementation Method 4. Step S2 further includes: dividing the resonant halo orbit into a perihelion group and an aphelion group for extension based on the initial moment corresponding to the initial point of the resonant halo orbit when the second gravitational body is located at the perihelion or aphelion of its orbit.

[0029] Implementation Method Six: This implementation method further defines the method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints as described in Implementation Method One. In step S3, the thrust acceleration of the spacecraft is: , in, A parameter for measuring the magnitude of acceleration. This is the ratio of the spacecraft's distance from the sun to astronomical units. This is the unit vector from the sun to the spacecraft.

[0030] Thrust acceleration parameters in this embodiment Used to adjust the shape of the resonant halo orbit, the shortest distance to the second gravitational body, and the surface coverage of the second gravitational body.

[0031] Implementation Method Seven: This implementation method further defines the method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints as described in Implementation Method One. The upper limit value preset in step S3 is 5e-7 km / s. 2 If the multi-segment shooting method in numerical extension does not converge, then the numerical extension terminates.

[0032] Implementation Method 8: A computer device according to this implementation method includes a memory and a processor. The memory stores a computer program. When the processor runs the computer program stored in the memory, the processor executes a method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints according to any one of Implementation Methods 1 to 7.

[0033] Implementation Method Nine: A computer-readable storage medium as described in this implementation method stores a computer program, which, when executed by a processor, performs the steps of a low-thrust spacecraft resonant halo orbit design method under thrust direction constraints as described in any one of Implementation Methods One to Seven.

[0034] Implementation Method 10, see below Figures 1 to 5 This embodiment describes a specific example of the resonance halo orbit design method for a low-thrust spacecraft under thrust direction constraints described in Embodiment 1. It also serves to explain Embodiments 2 through 7. Specifically: For the resonant halo orbit of a low-thrust spacecraft in an elliptical restricted three-body problem with the Sun as the first gravitational body, this implementation proposes a novel orbit design method that enables the spacecraft to maintain the periodicity of its orbit when the thrust direction is directly towards the Sun. The basic process is as follows: Figure 1 As shown: First, in the circularly confined three-body problem involving the Sun and the second gravitational body, the halo orbit whose period resonates with the orbital period of the second gravitational body is calculated. Then, through the numerical extension in the first step, the eccentricity of the second gravitational body's orbit is gradually increased from zero until it matches the actual orbital eccentricity, thus obtaining the thrustless resonant halo orbit in the elliptical confined three-body problem involving the Sun and the second gravitational body. Finally, through the numerical extension in the second step, the magnitude of the spacecraft's thrust acceleration is gradually increased from zero until it reaches a certain amplitude or the resonant halo orbit collides with the surface of the second gravitational body. Thus, the resonant halo orbit under continuous small thrust in the direction directly facing the Sun in the elliptical confined three-body problem involving the Sun and the second gravitational body is designed. The numerical extension used is based on a multi-segment target shooting method using forward-backward integral splicing.

[0035] Multi-segment firing method using forward-backward integral splicing, such as Figure 2 As shown. The specific implementation is as follows: A periodic orbit is discretized into N segments with the same launch time, containing a total of N+1 discrete points. For an orbit with a period of T, the flight time of each segment is fixed at T / N. The constraint condition of the multi-segment shooting method is that the position velocity obtained by forward integration T / 2 of the i-th (i=1,…,N) discrete point is equal to the position velocity obtained by backward integration T / 2 of the i+1 discrete point, and the position velocity of the N+1 discrete point is equal to the position velocity of the 1st discrete point.

[0036] The spacecraft's thrust direction is constrained to always point towards the sun, and the thrust amplitude is inversely proportional to the square of the spacecraft's distance from the sun. The spacecraft's thrust acceleration can be expressed as... ,in A parameter for measuring the magnitude of acceleration. This is the ratio of the spacecraft's distance from the sun to astronomical units. This is the unit vector from the sun to the spacecraft.

[0037] The specific implementation is as follows: Step 1: In the circularly restricted three-body problem involving the Sun and the second gravitational body, calculate the halo orbits with orbital periods satisfying the ratios 3:1, 4:1, 5:2, and 7:2 with respect to the second gravitational body. The periods of the 3:1, 4:1, 5:2, and 7:2 halo orbits are respectively... , , and Corresponding halo orbits exist on both sides of the Lagrange points L1 and L2 of the Sun-Second Gravitational Body. Thus, a total of 8 resonant halo orbits of 3:1, 4:1, 5:2, and 7:2 have been obtained in the circular restricted three-body problem of the Sun-Second Gravitational Body.

[0038] Step 2: Perform the first step's numerical extension on the 3:1, 4:1, 5:2, and 7:2 resonance halo orbits in the circularly restricted three-body problem involving the Sun and the second gravitational body. Increase the eccentricity of the second gravitational body's orbit from zero to its actual eccentricity. Set the initial point of the resonance halo orbit as the intersection of the orbit and the XZ plane of the Sun-second gravitational body's center-of-mass rotational pulsating coordinate system. Based on the initial moment corresponding to the initial point, when the second gravitational body is located at its perihelion or aphelion, divide the resonance halo orbits into a perihelion group and an aphelion group. The Sun-second gravitational body's center-of-mass rotational pulsating coordinate system is defined as follows: Figure 3 As shown, the origin is located at the Sun-second gravitational body's center of mass, the X-axis points towards the second gravitational body, the Z-axis is along the direction of the second gravitational body's orbital angular momentum, and the Y-axis forms a right-handed frame. Taking the Sun-Mercury system as an example, the 3:1 resonance halo orbits of the aphelion group on the L1 side of the Lagrange point undergo period-doubling bifurcation at eccentricities of 0.04 and 0.0089, becoming L1-6:2 resonance halo orbits. Similarly, the 3:1 resonance halo orbits of the aphelion group on the L2 side undergo period-doubling bifurcation at eccentricities of 0.044 and 0.0094, becoming L2-6:2 resonance halo orbits. Extending these four 6:2 resonance halo orbits further in the first step, increasing their eccentricity to 0.2056, we obtain a total of 20 resonance halo orbits (3:1, 4:1, 5:2, 6:2, and 7:2) in the Sun-Mercury elliptic restricted three-body problem. These include 4 orbits in the L1 perihelion group, 6 in the L1 aphelion group, 4 in the L2 perihelion group, and 6 in the L2 aphelion group.

[0039] Step 3: Perform a second numerical extension on the resonance halo orbit obtained in Step 2, increasing the spacecraft's acceleration amplitude parameter from zero to a pre-set upper limit value, which is 5e-7 km / s. 2 If a collision is detected between the resonant halo orbit and the surface of the second gravitational body, the increase in acceleration amplitude is stopped and the numerical extension is terminated; if the multi-segment shooting method does not converge in the numerical extension, the numerical extension is terminated.

[0040] In this embodiment, Mercury is used as the second gravitational body. Taking the Sun-Mercury system as an example, its physical parameters are shown in Table 1: Table 1 Physical parameters of the Sun-Mercury system

[0041] The initial orbital parameters of the resonance halo orbit obtained under the Sun-Mercury circularly confined three-body configuration are shown in Table 2: Table 2 Initial orbital states under the Sun-Mercury circular restricted three-body configuration.

[0042] The orbits in Table 2 were numerically extended in the first step to achieve an eccentricity of 0.2056. The initial orbital parameters of the resonance halo orbits under the Sun-Mercury elliptical restricted three-body system are shown in Table 3. Among them, L1-aphelion group-6:2(A1) corresponds to the orbit obtained from the period-doubling bifurcation at an eccentricity of 0.04, L1-aphelion group-6:2(A2) corresponds to the orbit obtained from the period-doubling bifurcation at an eccentricity of 0.0089, L2-aphelion group-6:2(A3) corresponds to the orbit obtained from the period-doubling bifurcation at an eccentricity of 0.044, and L2-aphelion group-6:2(A4) corresponds to the orbit obtained from the period-doubling bifurcation at an eccentricity of 0.0094.

[0043] Table 3 Initial orbital states under the Sun-Mercury elliptical restricted three-body configuration

[0044] The second step involves numerically extending the orbits in Table 3, gradually increasing the magnitude of the spacecraft's acceleration, resulting in the resonance halo orbit under the Sun-Mercury elliptical restricted three-body condition, as shown below. Figure 4 and Figure 5 As shown, this includes the ability to extend the acceleration amplitude to 5e-8 km / s. 2 The above tracks.

[0045] Those skilled in the art will understand that embodiments of this disclosure can be provided as methods, systems, or computer program products. Therefore, this disclosure can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this disclosure can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0046] This disclosure is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this disclosure. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create a machine for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1The computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to operate in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes. These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0047] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of this disclosure and not to limit its protection scope. Although this disclosure has been described in detail with reference to the above embodiments, those skilled in the art should understand that after reading this disclosure, they can still make various changes, modifications or equivalent substitutions to the specific implementation of the invention, but these changes, modifications or equivalent substitutions are all within the protection scope of the published pending claims.

Claims

1. A method for designing resonant halo orbits for low-thrust spacecraft under thrust direction constraints, characterized in that, The method includes: Step S1: Calculate the halo orbit that satisfies the preset resonance relationship with the orbital period of the second gravitational body in the circular restricted three-body system of the Sun-Second Gravitational Body, and determine the initial state of the corresponding halo orbit on both sides of the Lagrange points L1 and L2 of the Sun-Second Gravitational Body respectively; Step S2: Perform the first step of numerical extension on the initial state of the halo orbit, increasing the eccentricity of the second gravitational body orbit from zero to the actual orbital eccentricity; thus obtaining the thrustless resonance halo orbit in the Sun-Second Gravitational Elliptical Restricted Three-Body Problem. Step S3: Perform a second numerical extension on the obtained thrustless resonant halo orbit. Gradually increase the thrust acceleration amplitude parameter of the spacecraft from zero to a preset upper limit value or until the resonant halo orbit collides with the surface of the second gravitational body to obtain the resonant halo orbit under thrust direction constraint. The thrust direction constraint of the spacecraft is that the thrust direction always points to the sun, and the thrust amplitude is inversely proportional to the square of the distance from the spacecraft to the sun. The numerical extension is based on a multi-segment shooting method using forward-backward integral splicing, including: The periodic orbit is discretized into N segments with the same launch time, containing a total of N+1 discrete points. For an orbit with a period of T, the flight time of each segment is fixed at T / N. The constraints of the multi-segment shooting method are: the position velocity obtained by forward integration T / 2 at the i-th discrete point is equal to the position velocity obtained by backward integration T / 2 at the (i+1)-th discrete point, and the position velocity at the (N+1)-th discrete point is equal to the position velocity at the 1st discrete point, i=1,…,N; The thrust acceleration of the spacecraft in step S3 is: ， in, A parameter for measuring the magnitude of acceleration. This is the ratio of the spacecraft's distance from the sun to astronomical units. This is the unit vector from the sun to the spacecraft.

2. The method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints according to claim 1, characterized in that, The preset resonance relationship in step S1 includes at least one of 3:1, 4:1, 5:2 and 7:2, wherein the ratio of the period of the halo orbit to the period of the second gravitational body orbit is 2π / 3, π / 2, 4π / 5 and 4π / 7 respectively.

3. The method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints according to claim 1, characterized in that, In step S2, the actual orbital eccentricity is determined based on the actual orbital parameters of the second gravitational body. In the first step of numerical extension, the initial point of the resonance halo orbit is the intersection of the orbit with the XZ plane of the Sun-Second Gravitational Body Center of Mass Rotational Pulsating Coordinate System. The origin of the Sun-Second Gravitational Body Center of Mass Rotational Pulsating Coordinate System is located at the Sun-Second Gravitational Body Center of Mass, the X-axis points towards the second gravitational body, the Z-axis is along the orbital angular momentum direction of the second gravitational body, and the Y-axis forms a right-handed frame.

4. The method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints according to claim 3, characterized in that, Step S2 further includes: based on the initial moment corresponding to the initial point of the resonance halo orbit, the second gravitational body is located at the perihelion or aphelion of its orbit, and the resonance halo orbit is divided into a perihelion group and an aphelion group for extension.

5. The method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints according to claim 1, characterized in that, The preset upper limit value in step S3 is 5e-7km / s 2 If the multi-segment shooting method in numerical extension does not converge, then the numerical extension terminates.

6. A computer device, characterized in that: It includes a memory and a processor, wherein the memory stores a computer program, and when the processor runs the computer program stored in the memory, the processor executes a method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints as described in any one of claims 1-5.

7. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, performs the steps of a method for designing a resonant halo orbit for a low-thrust spacecraft under thrust direction constraints as described in any one of claims 1-5.

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