Spacecraft single impulse absolute reachable set solution method based on differential algebraic technique

Abstract

The application discloses a spacecraft single-pulse absolute reachable set solving method based on differential algebra technology and belongs to the technical field of on-orbit motion of space spacecrafts; firstly, a geocentric inertial coordinate system, a geocentric geodetic rotating coordinate system and a cylindrical coordinate system are established, and a dynamics model of on-orbit motion of the spacecraft in the cylindrical coordinate system is established; then, the earth shape perturbation, the solar pressure perturbation and the third body gravity perturbation of the sun and the moon are converted into the cylindrical coordinate system for representation, and a dynamics model considering multiple perturbation forces is obtained; finally, the absolute reachable set is solved in the dynamics model considering multiple perturbation forces based on Jet Transport technology; the dynamics model considering multiple perturbation forces is established in the cylindrical coordinate system, the absolute reachable set is solved by using Jet Transport technology based on the dynamics equation, and the absolute reachable set is simulated by using C++ and MATLAB; compared with the traditional Monte Carlo method, the solving method based on Jet Transport technology is high in accuracy and calculation efficiency.

Description

Technical Field

[0001] This invention belongs to the field of on-orbit motion technology of spacecraft, and relates to a method for solving the absolute reachability domain of a single pulse in a spacecraft based on differential algebra technology. Background Technology

[0002] In recent years, with the increase in space activities and the continuous rise in the number of spacecraft in orbit, orbital space has begun to become congested, and available orbital resources are gradually decreasing. To ensure the sustainable development of limited orbital resources, mission spacecraft need to perform on-orbit operations. For mission spacecraft, their own maneuverability and the maneuverability of the target are fundamental to their operational missions. Therefore, the concept of absolute reachability is introduced to characterize the maneuverability of spacecraft, providing a theoretical basis and applied research for on-orbit mission spacecraft.

[0003] There are two existing methods for solving the absolute reachability domain: one is to ignore the influence of perturbation and solve the absolute reachability domain analytically, but the accuracy of the obtained absolute reachability domain is low; the other is to use numerical methods, but the computational efficiency is low and cannot meet the requirements of real-time online applications of spacecraft. Summary of the Invention

[0004] The purpose of this invention is to solve the technical problems of low accuracy and low efficiency in solving the absolute reachability domain in the prior art, which makes it difficult to meet the requirements of real-time online applications of spacecraft, and to provide a method for solving the absolute reachability domain of spacecraft based on differential algebra technology.

[0005] To achieve the above objectives, the present invention employs the following technical solution: This invention provides a method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques, comprising the following steps: S1. Establish the geocentric inertial coordinate system, the geocentric-ground-fixed rotating coordinate system, and the cylindrical coordinate system, and establish the dynamic model of the spacecraft's on-orbit motion in the cylindrical coordinate system; S2, respectively transforms the perturbation of Earth's shape, the perturbation of solar radiation pressure, and the perturbation of the gravitational force of the Sun, Moon and Third Body into cylindrical coordinates for representation, and obtains a dynamic model that considers multiple perturbation forces; S3 solves for the absolute reachability domain in a dynamic model considering multiple perturbations based on Jet Transport technology.

[0006] Furthermore, the origin of the geocentric inertial coordinate system is located at the Earth's center of mass. ; The axis points from the Earth's center of mass to the vernal equinox; The Earth's rotation axis is perpendicular to the equatorial plane and points from the Earth's center of mass towards the Earth's North Pole; shaft and axis, The axes form a right-handed coordinate system; The origin of the geocentric-solid rotating coordinate system is located at the Earth's center of mass. , The axis is parallel to the Earth's axis and points from the Earth's center of mass toward the North Pole. The axis points from the Earth's center of mass to the intersection of the prime meridian and the equator; Axis perpendicular to plane, and shaft and The axes form a right-handed coordinate system; The establishment of the cylindrical coordinate system includes: the spacecraft moving in orbit to a point in the geocentric inertial coordinate system. In the geocentric inertial coordinate system Projection points on the plane polar coordinates are and , Point Time distance, , and It formed points Cylindrical coordinates.

[0007] Furthermore, the establishment of the dynamic model of the spacecraft's on-orbit motion in cylindrical coordinates specifically includes the following steps: During the spacecraft's operation in orbit, with zero forces acting on it, the dynamic equations in the geocentric inertial coordinate system are expressed as follows: (1) In the formula, , is the Earth's gravitational constant. This represents the spacecraft's position vector in the geocentric inertial coordinate system. Introducing a cylindrical coordinate system, the components of the spacecraft's position vector in the cylindrical coordinate system include the radius. Azimuth and height Based on the relationship between cylindrical and Cartesian coordinate systems, the on-orbit motion equations of the spacecraft in cylindrical coordinates are obtained as follows: (2) In the formula, , and This represents the components of the velocity vector in cylindrical coordinates.

[0008] Furthermore, the process of deriving the on-orbit motion equations of the spacecraft in cylindrical coordinates based on the relationship between cylindrical and Cartesian coordinate systems specifically includes the following steps: The equations of motion for the spacecraft in the geocentric inertial coordinate system are expressed as follows: (3) In the formula, Represents the Earth's gravitational constant; The spacecraft's position vector in Cartesian coordinates is decomposed into: (4) In the formula, , and These are the spacecraft position vectors. The components of the three coordinate axes in the geocentric inertial coordinate system; Introducing a cylindrical coordinate system, the components of the spacecraft's position vector in the cylindrical coordinate system include the radius. Azimuth and height Based on the relationship between cylindrical coordinates and Cartesian coordinates, we obtain: (5) The equations of motion for the spacecraft in cylindrical coordinates are as follows: (6) In the formula, This represents the acceleration of a spacecraft in cylindrical coordinates, including the maneuvering acceleration exerted by the spacecraft itself and the perturbation acceleration experienced by the spacecraft. According to the relation in equation (5), we get: (7) by The independent variable is, i.e. Based on equations (6) and (7), the on-orbit motion equations of the spacecraft in cylindrical coordinates are established as follows: (8) In the formula, The radius of the component of the spacecraft's position vector in cylindrical coordinates; The azimuth angle of the spacecraft's position vector in cylindrical coordinates; The altitude is the position vector of the spacecraft in cylindrical coordinates.

[0009] Furthermore, the Earth shape perturbation in S2 is represented in cylindrical coordinates, specifically including the following steps: Earth's shape perturbation acceleration in the geocentric-solid rotating coordinate system Represented as: (9) In the formula, , and These represent the coefficients in the three directions of the Earth-centered, Earth-fixed rotating coordinate system: (10) In the formula, Represents Earth's radius and topography coefficient. and Given by the EGM96 gravitational model; when The time indicates that only the Earth's central gravity is considered; and They are represented as follows: (11) (12) In the formula, These are the components of the spacecraft's position vector in the Earth-centered, Earth-fixed rotating coordinate system. ; Since equation (9) is established in the geocentric rotating coordinate system, it is necessary to transform the Earth's shape perturbation to the geocentric inertial coordinate system through coordinate transformation. The transformation matrix is: (13) In the formula, Represents the Earth's angular velocity of rotation. This represents the time interval between rotations of the two coordinate systems. Therefore, the perturbation of the Earth's shape in the geocentric inertial coordinate system is represented as: (14) In cylindrical coordinates, the representation of Earth's shape perturbations mainly involves... and Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. and They are represented as follows: (15) (16) Solar radiation pressure perturbation in a geocentric inertial coordinate system is expressed as: (17) In the formula, Indicates the shadow function, Units Solar radiation pressure at a distance It represents the distance from the Earth to the Sun. This represents the solar radiation pressure coefficient. This represents the illuminated cross-sectional area of ​​the spacecraft. It is the mass of the spacecraft. This represents the vector representing the spacecraft's relative position to the sun. This indicates the relative distance between the spacecraft and the sun.

[0010] Furthermore, the solar radiation pressure perturbation in S2 is transformed and represented in cylindrical coordinates, specifically including the following steps: Will Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. Represented as: (18) In the formula, This represents the relative vector between the Earth and the Sun in a geocentric inertial frame of reference. .

[0011] Furthermore, the gravitational perturbation of the Sun and Moon (third bodies) in S2 is transformed and represented in cylindrical coordinates, specifically including the following steps: The gravitational perturbation of the Sun and Moon as a third body in a geocentric inertial coordinate system is expressed as follows: (19) In the formula, Represents the gravitational constant; Indicates the mass of the sun; Indicates the mass of the moon; This represents the position vector of the spacecraft in the geocentric inertial coordinate system. This represents the position vector of the Sun in the geocentric inertial coordinate system. This represents the position vector of the Moon in the geocentric inertial coordinate system. Will , and Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. , and They are represented as follows: (20) In the formula, Represents the radius of the solar vector in cylindrical coordinates; Represents the radius of the lunar vector in cylindrical coordinates; Represents the azimuth angle of the solar vector in cylindrical coordinates; Represents the lunar vector azimuth in cylindrical coordinates; This represents the altitude of the sun in cylindrical coordinates. This represents the height of the moon in cylindrical coordinates.

[0012] Further, S3 includes: Based on the dynamic equations, assume the initial conditions are: (twenty one) At the initial moment, the single pulse applied to the mission spacecraft is represented as: (twenty two) Solving for the absolutely reachable domain is transformed into an initial value problem for an ordinary differential equation, as follows: (twenty three) In the formula, , Indicates initial conditions The initial set of states centered on; The nominal value representing the initial state; The maneuver containing a single pulse represents the deviation value related to the initial state; Starting from the initial moment, the initial state set nominal value in and deviation value Substitute into the function conduct The Taylor series expansion of order X gives the result at a certain moment. of The Taylor polynomial of order 1 is as follows: (twenty four) In the formula, express The result of the Taylor polynomial expansion of order 1, Represents the zeroth-order term. Indicates deviation value Substitute into function In Taylor series of order, represents the coefficients of the Taylor polynomial expansion.

[0013] Compared with the prior art, the present invention has the following beneficial effects: This invention discloses a method for solving the absolute reachability domain of a spacecraft using single-pulse calculations based on differential algebra. By establishing a dynamic model of the spacecraft's on-orbit motion in a coordinate system, perturbations such as Earth shape perturbation, solar radiation pressure perturbation, and gravitational perturbations from the Sun, Moon, and other third bodies are represented in cylindrical coordinates. The absolute reachability domain is then solved using Jet Transport technology. Due to the use of Jet Transport technology, the solution method achieves high accuracy and computational efficiency, which is beneficial for real-time on-orbit calculation of the spacecraft's absolute reachability domain. Furthermore, it considers multiple perturbations, making it more consistent with engineering realities. This method enables rapid and accurate solutions for the absolute reachability domain of spacecraft under multiple perturbations. Attached Figure Description

[0014] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the embodiments will be briefly introduced below. It should be understood that the following drawings only show some embodiments of the present invention and should not be regarded as a limitation on the scope. For those skilled in the art, other related drawings can be obtained based on these drawings without creative effort.

[0015] Figure 1 This is a flowchart of the method of the present invention; Figure 2 This is a schematic diagram of the spacecraft in orbit according to the present invention; Figure 3 This is a simulation diagram of the absolute reachability domain based on Jet Transport technology in this invention; Figure 4 This is a diagram showing the position error components of the solution method of this invention compared to the traditional Monte Carlo method. Detailed Implementation

[0016] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations.

[0017] Therefore, the following detailed description of the embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0018] It should be noted that similar labels and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures.

[0019] In the description of the embodiments of the present invention, it should be noted that if terms such as "upper," "lower," "horizontal," or "inner" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings, or the orientation or positional relationship commonly used when the product of the invention is in use, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation of the present invention. Furthermore, terms such as "first" and "second" are only used to distinguish descriptions and should not be construed as indicating or implying relative importance.

[0020] Furthermore, the use of the term "horizontal" does not imply that the component must be absolutely horizontal, but rather that it can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal than "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted.

[0021] In the description of the embodiments of the present invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in the present invention according to the specific circumstances.

[0022] The present invention will now be described in further detail with reference to the accompanying drawings: See Figure 1 This invention discloses a method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques, comprising the following steps: S1. Establish the geocentric inertial coordinate system, the geocentric-ground-fixed rotating coordinate system, and the cylindrical coordinate system, and establish the dynamic model of the spacecraft's on-orbit motion in the cylindrical coordinate system; S2, respectively transforms the perturbation of Earth's shape, the perturbation of solar radiation pressure, and the perturbation of the gravitational force of the Sun, Moon and Third Body into cylindrical coordinates for representation, and obtains a dynamic model that considers multiple perturbation forces; S3 solves for the absolute reachability domain in a dynamic model considering multiple perturbations based on Jet Transport technology.

[0023] In one feasible embodiment of the present invention, see [link to relevant documentation]. Figure 2 Establish Earth-Centered Inertial (ECI), Earth-Centered, Earth-Fixed (ECEF) Rotational Coordinate System, and Cylindrical Coordinate System.

[0024] The origin of the geocentric inertial coordinate system is located at the Earth's center of mass. ; The axis points from the Earth's center of mass to the vernal equinox; The Earth's rotation axis is perpendicular to the equatorial plane and points from the Earth's center of mass towards the Earth's North Pole; shaft and axis, The axes form a right-handed coordinate system; The origin of the geocentric-solid rotating coordinate system is located at the Earth's center of mass. , The axis is parallel to the Earth's axis and points from the Earth's center of mass toward the North Pole. The axis points from the Earth's center of mass to the intersection of the prime meridian and the equator; Axis perpendicular to plane, and shaft and The axes form a right-handed coordinate system; The establishment of the cylindrical coordinate system includes: the spacecraft moving in orbit to a point in the geocentric inertial coordinate system. In the geocentric inertial coordinate system Projection points on the plane polar coordinates are and , Point Time distance, , and It formed points Cylindrical coordinates.

[0025] In one feasible embodiment of the present invention, establishing the dynamic model of the spacecraft's on-orbit motion in cylindrical coordinates specifically includes the following steps: During the spacecraft's operation in orbit, with zero forces acting on it, the dynamic equations in the geocentric inertial coordinate system are expressed as follows: (1) In the formula, , is the Earth's gravitational constant. This represents the spacecraft's position vector in the geocentric inertial coordinate system. Introducing a cylindrical coordinate system, the components of the spacecraft's position vector in the cylindrical coordinate system include the radius. Azimuth and height Based on the relationship between cylindrical and Cartesian coordinate systems, the on-orbit motion equations of the spacecraft in cylindrical coordinates are obtained as follows: (2) In the formula, , and This represents the components of the velocity vector in cylindrical coordinates.

[0026] In a feasible embodiment of the present invention, obtaining the on-orbit motion equations of the spacecraft in cylindrical coordinates based on the relationship between cylindrical coordinates and Cartesian coordinates specifically includes the following steps: See Figure 2 Spacecraft orbiting the Earth are represented by black dots. Represents the Earth's center of mass. The dotted line represents the spacecraft's center of mass, and the dashed curve represents the spacecraft's orbit around the Earth. The spacecraft's position vector in the geocentric inertial coordinate system is... The equations of motion for a spacecraft in a geocentric inertial coordinate system are expressed as follows: (3) In the formula, Represents the Earth's gravitational constant; The spacecraft's position vector in Cartesian coordinates is decomposed into: (4) In the formula, , and These are the spacecraft position vectors. The components of the three coordinate axes in the geocentric inertial coordinate system; Introducing a cylindrical coordinate system, the components of the spacecraft's position vector in the cylindrical coordinate system include the radius. Azimuth and height Based on the relationship between cylindrical coordinates and Cartesian coordinates, we obtain: (5) The equations of motion for the spacecraft in cylindrical coordinates are as follows: (6) In the formula, This represents the acceleration of a spacecraft in cylindrical coordinates, including the maneuvering acceleration exerted by the spacecraft itself and the perturbation acceleration experienced by the spacecraft. According to the relation in equation (5), we get: (7) by The independent variable is, i.e. Based on equations (6) and (7), the on-orbit motion equations of the spacecraft in cylindrical coordinates are established as follows:

[0027] In a feasible embodiment of the present invention, the Earth shape perturbation in S2 is transformed into cylindrical coordinates for representation, specifically including the following steps: Earth's shape perturbation acceleration in the geocentric-solid rotating coordinate system Represented as: (9) In the formula, , and These represent the coefficients in the three directions of the Earth-centered, Earth-fixed rotating coordinate system: (10) In the formula, Represents Earth's radius and topography coefficient. and Given by the EGM96 gravitational model; when The time indicates that only the Earth's central gravity is considered; and They are represented as follows: (11) (12) In the formula, These are the components of the spacecraft's position vector in the Earth-centered, Earth-fixed rotating coordinate system. ; Since equation (9) is established in the geocentric rotating coordinate system, it is necessary to transform the Earth's shape perturbation to the geocentric inertial coordinate system through coordinate transformation. The transformation matrix is: (13) In the formula, Represents the Earth's angular velocity of rotation. This represents the time interval between the rotations of the two coordinate systems.

[0028] Therefore, the perturbation of the Earth's shape in the geocentric inertial coordinate system can be expressed as: (14) In cylindrical coordinates, the representation of Earth's shape perturbations mainly involves... and Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. and They are represented as follows: (15) (16) Solar radiation pressure perturbation in a geocentric inertial coordinate system can be expressed as: (17) In the formula, Indicates the shadow function, Units Solar radiation pressure at a distance It represents the distance from the Earth to the Sun. This represents the solar radiation pressure coefficient. This represents the illuminated cross-sectional area of ​​the spacecraft. It is the mass of the spacecraft. This represents the vector representing the spacecraft's relative position to the sun. This indicates the relative distance between the spacecraft and the sun.

[0029] In a feasible embodiment of the present invention, the solar radiation pressure perturbation in S2 is transformed into cylindrical coordinates for representation, specifically including the following steps: Will Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. Represented as: (18) In the formula, This represents the relative vector between the Earth and the Sun in a geocentric inertial frame of reference. .

[0030] In a feasible embodiment of the present invention, the gravitational perturbation of the Sun and Moon third bodies in S2 is transformed into cylindrical coordinates for representation, specifically including the following steps: The gravitational perturbation of the Sun and Moon as a third body in a geocentric inertial coordinate system is expressed as follows: (19) In the formula, Represents the gravitational constant; Indicates the mass of the sun; Indicates the mass of the moon; This represents the position vector of the spacecraft in the geocentric inertial coordinate system. This represents the position vector of the Sun in the geocentric inertial coordinate system. This represents the position vector of the Moon in the geocentric inertial coordinate system. Will , and Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. , and They are represented as follows: (20) In the formula, Represents the radius of the solar vector in cylindrical coordinates; Represents the radius of the lunar vector in cylindrical coordinates; Represents the azimuth angle of the solar vector in cylindrical coordinates; Represents the lunar vector azimuth in cylindrical coordinates; This represents the altitude of the sun in cylindrical coordinates. This represents the height of the moon in cylindrical coordinates.

[0031] In one feasible embodiment of the present invention, S3 includes: Based on the dynamic equations, assume the initial conditions are: (twenty one) At the initial moment, the single pulse applied to the mission spacecraft is represented as: (twenty two) Solving for the absolutely reachable domain is transformed into an initial value problem for an ordinary differential equation, as follows: (twenty three) In the formula, , Indicates initial conditions The initial set of states centered on; The nominal value representing the initial state; The maneuver containing a single pulse represents the deviation value related to the initial state; Starting from the initial moment, the initial state set nominal value in and deviation value Substitute into the function conduct The Taylor series expansion of order X gives the result at a certain moment. of The Taylor polynomial of order 1 is as follows: (twenty four) In the formula, express The result of the Taylor polynomial expansion of order 1, Represents the zeroth-order term. Indicates deviation value Substitute into function In Taylor series of order, represents the coefficients of the Taylor polynomial expansion.

[0032] The working principle of this invention is as follows: For complex spacecraft dynamic systems, due to the influence of complex perturbations, it is often difficult to obtain analytical solutions to the dynamic equations established based on the actual on-orbit motion of the spacecraft. Therefore, the method adopted in this invention is a semi-analytical method based on Jet Transport technology, which can quickly and accurately solve the absolute reachability domain of the spacecraft. The basic principle of the semi-analytical method based on Jet Transport technology is given below.

[0033] For an initial value problem of an ordinary differential equation: (25) In the formula, the initial conditions It can be equivalent to a polynomial condition. That is, in the nominal state A set of nearby states. It should be noted that all numerical integration operations can be replaced by polynomial operations in Jet Transport techniques, thus transforming the dynamical system, i.e. At a fixed time step, a set of It is approximated by an expansion polynomial of order 1.

[0034] For the forward Euler integral algorithm based on Taylor series expansion: (26) After one time step, the state vector can be represented as: (27) Need to find the function exist Output the value at that location. yes of The Taylor series expansion vector. This process can be repeated in subsequent steps, thus a dynamical system can be approximated by arbitrary time steps. of Taylor polynomial representation: (28) In the formula, , Let be the coefficients of the Taylor series expansion polynomial. This is the state transition tensor.

[0035] Since the equations for solving the absolute reachable domain of a single pulse in a spacecraft are continuously differentiable, the Taylor series expansion method is mainly used, and the numerical integrator adopts the seventh or eighth order Runge-Kutta (RK78) method with variable step size.

[0036] Depend on The general form of the explicit Runge-Kutta formula, which is a linear combination of function values, is: (29) In the formula, represents the initial slope. This represents the slope at the midpoint of each time interval. , and 'a' represents undetermined coefficients. This indicates the step size. The variable step size RK78 algorithm uses the seventh-order Runge-Kutta formula to provide candidate solutions, with an eighth-order control error for each step. The variable step size RK78 algorithm is an adaptive variable step size numerical solution method for differential equations, with an overall truncation error of... The principle behind its variable step size is as follows: Calculate the first according to equation (29). The "true value" of a step. and low-order valuation Then the error estimate between the two values ​​is: (30) For one step Error estimation exists , Error estimation exists The two estimation errors are related as follows: (31) Based on the original step size, if The required step size can be determined based on equation (30); if Equation (30) determines the safe range within which the step size can be increased, and the step size is varied according to the guidance of this rule.

[0037] Based on Taylor series expansion and integrator design, Jet Transport technology enables semi-analytical solutions of ordinary differential equations of arbitrary order using polynomial approximation, ensuring both high accuracy and computational efficiency. Therefore, this invention introduces Jet Transport technology into the problem of the absolute reachability region of a single-pulse spacecraft, using a polynomial approximation method for solution.

[0038] Based on the dynamic equations, assume the initial conditions are: (twenty one) At the initial moment, the single pulse applied to the mission spacecraft can be represented as: (twenty two) As can be seen from equation (22), the maneuver applied to the mission spacecraft is a random single pulse with a magnitude within a certain range and an arbitrary direction.

[0039] The solution of the absolutely reachable region is transformed into an initial value problem of an ordinary differential equation: (twenty three) In the formula, , Indicates initial conditions The initial set of states centered on The nominal value representing the initial state. The maneuver contains a single pulse, which represents the deviation value related to the initial state. Starting from the initial time, the initial state set... nominal value in and deviation value Substitute into the function conduct The Taylor series expansion of order X gives the result at a certain moment. of Taylor polynomial of order: (twenty four) In the formula express The result of the Taylor polynomial expansion of order 1, Represents the zeroth-order term. Indicates deviation value Substitute into function In Taylor series of order, represents the coefficients of the Taylor polynomial expansion.

[0040] Based on the Taylor series expansion method and the characteristics of Jet Transport technology, it can be seen from equation (24) that The accuracy and the retained order and deviation of the Taylor polynomial. The size is related to the precision requirements, and the order and deviation value can be adjusted according to the precision requirements.

[0041] To solve for the absolute reachability region of a single pulse in a spacecraft, a semi-analytical solution method based on JetTransport technology was simulated using C++ and MATLAB software. The relevant simulation parameters are as follows: Earth radius is... The initial orbital altitude of the mission spacecraft is The initial eccentricity is 0.5, the initial orbital inclination is 0°, the initial perigee argument is 0°, and the initial true perigee angle is 60°. The single pulse size is 1. The solar radiation pressure coefficient is 1.3, and the mass of the mission spacecraft is 3300. The cross-sectional area is 74 .

[0042] See Figure 3 Simulation diagrams of the absolute reachability domain of a spacecraft single pulse obtained by a semi-analytical solution method based on Jet Transport technology are presented. Figure 3 As can be seen, all trajectories intersect at the same point, which is the initial position where the spacecraft applies the single pulse.

[0043] To illustrate the accuracy of the solution method proposed in this invention, the method is compared with the traditional Monte Carlo method. (See [link to relevant documentation]). Figure 4 The paper demonstrates a comparison of the position error components of the method of this invention with those of the traditional Monte Carlo method. The position error components are less than 3m, which is negligible compared to the magnitude of the absolute reachability region. Therefore, the results show that the spacecraft absolute reachability region solution method based on Jet Transport technology has high accuracy.

[0044] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention 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.

[0045] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. 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, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0046] These 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 function 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.

[0047] 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.

[0048] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit it. Although the present invention has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of the present invention. Any modifications or equivalent substitutions that do not depart from the spirit and scope of the present invention should be covered within the scope of protection of the claims of the present invention.

Claims

1. A method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques, characterized in that, Includes the following steps: S1. Establish the geocentric inertial coordinate system, the geocentric-ground-fixed rotating coordinate system, and the cylindrical coordinate system, and establish the dynamic model of the spacecraft's on-orbit motion in the cylindrical coordinate system; S2, respectively transforms the perturbation of Earth's shape, the perturbation of solar radiation pressure, and the perturbation of the gravitational force of the Sun, Moon and Third Body into cylindrical coordinates for representation, and obtains a dynamic model that considers multiple perturbation forces; S3, based on Jet Transport technology, solves for the absolute reachability domain in a dynamic model considering various perturbations, including: Based on the dynamic equations, assume the initial conditions are: （21） In the formula, The radius of the component of the spacecraft's position vector in cylindrical coordinates; The azimuth angle of the spacecraft's position vector in cylindrical coordinates; The altitude is the position vector of the spacecraft in cylindrical coordinates. At the initial moment, the single pulse applied to the mission spacecraft is represented as: （22） Solving for the absolutely reachable domain is transformed into an initial value problem for an ordinary differential equation, as follows: （23） In the formula, , Indicates initial conditions The initial set of states centered on; The nominal value representing the initial state; The maneuver containing a single pulse represents the deviation value related to the initial state; Starting from the initial moment, the initial state set nominal value in and deviation value Substitute into the function conduct The Taylor series expansion of order X gives the result at a certain moment. of The Taylor polynomial of order 1 is as follows: （24） In the formula, express The result of the Taylor polynomial expansion of order 1, Represents the zeroth-order term. Indicates deviation value Substitute into function In Taylor series of order, represents the coefficients of the Taylor polynomial expansion.

2. The method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques according to claim 1, characterized in that, The origin of the geocentric inertial coordinate system is located at the Earth's center of mass. ; The axis points from the Earth's center of mass to the vernal equinox; The Earth's rotation axis is perpendicular to the equatorial plane and points from the Earth's center of mass towards the Earth's North Pole; shaft and axis, The axes form a right-handed coordinate system; The origin of the geocentric-solid rotating coordinate system is located at the Earth's center of mass. , The axis is parallel to the Earth's axis and points from the Earth's center of mass toward the North Pole. The axis points from the Earth's center of mass to the intersection of the prime meridian and the equator; Axis perpendicular to plane, and shaft and The axes form a right-handed coordinate system; The establishment of the cylindrical coordinate system includes: the spacecraft moving in orbit to a point in the geocentric inertial coordinate system. In the geocentric inertial coordinate system Projection points on the plane polar coordinates are and , Point Time distance, , and It formed points Cylindrical coordinates.

3. The method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques according to claim 2, characterized in that, The establishment of the dynamic model of the spacecraft's on-orbit motion in cylindrical coordinates specifically includes the following steps: During the spacecraft's operation in orbit, with zero forces acting on it, the dynamic equations in the geocentric inertial coordinate system are expressed as follows: （1） In the formula, , is the Earth's gravitational constant. This represents the spacecraft's position vector in the geocentric inertial coordinate system. Introducing a cylindrical coordinate system, the components of the spacecraft's position vector in the cylindrical coordinate system include the radius. Azimuth and height Based on the relationship between cylindrical and Cartesian coordinate systems, the on-orbit motion equations of the spacecraft in cylindrical coordinates are obtained as follows: （2） In the formula, , and This represents the components of the velocity vector in cylindrical coordinates; This represents the acceleration of a spacecraft in cylindrical coordinates, including the maneuvering acceleration exerted by the spacecraft itself and the perturbation acceleration experienced by the spacecraft.

4. The method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques according to claim 3, characterized in that, The process of deriving the on-orbit motion equations of the spacecraft in cylindrical coordinates based on the relationship between cylindrical and Cartesian coordinates includes the following steps: The equations of motion for the spacecraft in the geocentric inertial coordinate system are expressed as follows: （3） In the formula, Represents the Earth's gravitational constant; The spacecraft's position vector in Cartesian coordinates is decomposed into: （4） In the formula, , and These are the spacecraft position vectors. The components of the three coordinate axes in the geocentric inertial coordinate system; Introducing a cylindrical coordinate system, the components of the spacecraft's position vector in the cylindrical coordinate system include the radius. Azimuth and height Based on the relationship between cylindrical coordinates and Cartesian coordinates, we obtain: （5） The equations of motion for the spacecraft in cylindrical coordinates are as follows: （6） According to the relation in equation (5), we get: （7） by The independent variable is, i.e. Based on equations (6) and (7), the on-orbit motion equations of the spacecraft in cylindrical coordinates are established as follows: （8）。 5. The method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques according to claim 1, characterized in that, The Earth shape perturbation in S2 is transformed into cylindrical coordinates for representation, specifically including the following steps: Earth's shape perturbation acceleration in the geocentric-solid rotating coordinate system Represented as: （9） In the formula, , and These represent the coefficients in the three directions of the Earth-centered, Earth-fixed rotating coordinate system: （10） In the formula, Represents Earth's radius and topography coefficient. and Given by the EGM96 gravitational model; when The time indicates that only the Earth's central gravity is considered; and They are represented as follows: （11） （12） In the formula, These are the components of the spacecraft's position vector in the Earth-centered, Earth-fixed rotating coordinate system. ; Since equation (9) is established in the geocentric rotating coordinate system, it is necessary to transform the Earth's shape perturbation to the geocentric inertial coordinate system through coordinate transformation. The transformation matrix is: （13） In the formula, Represents the Earth's angular velocity of rotation. This represents the time interval between rotations of the two coordinate systems. Therefore, the perturbation of the Earth's shape in the geocentric inertial coordinate system is represented as: （14） In cylindrical coordinates, the representation of Earth's shape perturbations mainly involves... and Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. and They are represented as follows: （15） （16） Solar radiation pressure perturbation in a geocentric inertial coordinate system is expressed as: （17） In the formula, Indicates the shadow function, Units Solar radiation pressure at a distance It represents the distance from the Earth to the Sun. This represents the solar radiation pressure coefficient. This represents the illuminated cross-sectional area of ​​the spacecraft. It is the mass of the spacecraft. This represents the vector representing the spacecraft's relative position to the sun. This indicates the relative distance between the spacecraft and the sun.

6. The method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques according to claim 5, characterized in that, The solar radiation pressure perturbation in S2 is transformed into cylindrical coordinates for representation, specifically including the following steps: Will Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. Represented as: （18） In the formula, This represents the relative vector between the Earth and the Sun in a geocentric inertial frame of reference. .

7. The method for solving the absolute reachability domain of a spacecraft single pulse based on differential algebra techniques according to claim 6, characterized in that, The gravitational perturbation of the Sun and Moon as third bodies in S2 is transformed into a cylindrical coordinate system for representation, specifically including the following steps: The gravitational perturbation of the Sun and Moon as a third body in a geocentric inertial coordinate system is expressed as follows: （19） In the formula, Represents the gravitational constant; Indicates the mass of the sun; Indicates the mass of the moon; This represents the position vector of the spacecraft in the geocentric inertial coordinate system. This represents the position vector of the Sun in the geocentric inertial coordinate system. This represents the position vector of the Moon in the geocentric inertial coordinate system. Will , and Transform from the geocentric inertial coordinate system to the cylindrical coordinate system. , and They are represented as follows: （20） In the formula, Represents the radius of the solar vector in cylindrical coordinates; Represents the radius of the lunar vector in cylindrical coordinates; Represents the azimuth angle of the solar vector in cylindrical coordinates; Represents the lunar vector azimuth in cylindrical coordinates; This represents the altitude of the sun in cylindrical coordinates. This represents the height of the moon in cylindrical coordinates.

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