Method and device for minimum-fuel-saving double-pulse transfer between space orbits and computer equipment

Abstract

The invention relates to a fuel-saving double-pulse transfer method and device between space orbits and computer equipment, the method expands a fuel-saving double-pulse solution between elliptical orbits in a plane to the space elliptical orbits, and provides a continuation hybrid optimization algorithm for solving by continuously changing the inclination angles of initial and final orbits. The continuation hybrid optimization algorithm comprises the following steps: constructing an initial auxiliary solution, and constructing a discrete continuation sequence of which the orbit inclination angle coefficient is changed from 0 to 1; in each step, the optimal solution corresponding to the orbit inclination angle coefficient in the previous step is used as an initial value, and optimization is carried out by adopting an adaptive step length, so that a fuel-saving two-pulse solution meeting constraint conditions between space orbits is obtained. According to the method, the accuracy and effectiveness of solving the fuel-saving two-pulse orbit transfer problem between the space orbits are improved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of aircraft space orbit maneuvering, and relates to a most fuel-efficient double-pulse transfer method, device and computer equipment between space orbits. BACKGROUND

[0002] Fuel is a key constraint for space missions, especially in scenarios such as deep space exploration and satellite constellation deployment, where fuel cannot be supplemented on orbit. Fuel consumption directly determines the payload size (the less fuel consumed, the more scientific research equipment or functional modules that can be carried) and mission duration. Therefore, the most fuel-efficient orbit transfer has become a core research direction in the field of orbit dynamics, and "double-pulse transfer" is a classic scheme that combines engineering practicality and theoretical solvability.

[0003] The core advantage of pulse transfer is to simplify orbit adjustment to instantaneous velocity increment (V ), without considering the time distribution of continuous thrust, greatly reducing the computational complexity and being suitable for orbit design and cost estimation in the early stage of the mission.

[0004] In pulse transfer, double-pulse (only two velocity adjustments) is one of the minimum pulse numbers to achieve two-point orbit transfer (single-pulse is only applicable to special collinear orbits), achieving a balance between fuel consumption and control complexity, and becoming the mainstream research object.

[0005] As the complexity of space missions increases (such as the cooperation of polar satellites and equatorial satellites, and the orbit transfer of deep space probes), the demand for non-coplanar orbit transfer (with inclination differences between the initial and target orbits) is highlighted. However, the conclusions of coplanar research cannot be directly applied, creating a technical challenge: Core difficulty: Non-coplanar orbit transfer introduces out-of-plane velocity increments, and the inclination of the initial / target orbit significantly affects the optimal distribution of .

[0006] Key to theoretical extension: Research has found that in non-coplanar orbit transfer, the component of the principal vector in the normal direction of the transfer orbit plane is a periodic function. Therefore, the principal vector optimization condition of coplanar orbit transfer can be extended to non-coplanar orbit transfer, but the influence of the initial and final orbits being non-coplanar still needs to be considered.

[0007] Existing solution: To address the calculation difficulties caused by inclination, the author introduces the homotopy method, which gradually narrows the inclination difference between the initial and target orbits to construct an optimized path, converting the complex non-coplanar problem into a series of solvable approximate coplanar problems, and gradually approaching the optimal solution. SUMMARY

[0008] To address the problems in the above traditional method, the application provides a most fuel-efficient double-pulse transfer method, device and computer equipment between space orbits.

[0009] To achieve the above object, the embodiment of the present application adopts the following technical solutions: In one aspect, a most fuel-efficient double-pulse transfer method between spatial orbits is provided, comprising the steps of: According to the given initial orbit and terminal orbit, the transfer starting point of the spacecraft on the initial orbit and the transfer ending point on the terminal orbit are determined, and the most fuel-efficient double-pulse elliptical transfer orbit between the transfer starting point and the transfer ending point is calculated.

[0010] A three-dimensional optimization model is established, wherein the decision vector includes the true anomaly of the transfer starting point on the initial orbit, the true anomaly of the transfer ending point on the terminal orbit, the semi-latus rectum of the transfer orbit, and the orbit inclination coefficient; the objective function is to minimize the total pulse velocity increment; and the orbit inclination coefficient is used to set the initial and final orbit inclinations in the continuation calculation process.

[0011] In the three-dimensional optimization model, the optimality necessary condition based on the principal vector theory in the orbit coordinate system is introduced as a constraint condition; the constraint condition includes that the coefficient of the long-term term in the expression of each component of the principal vector in the transfer orbit coordinate system is 0, the modulus of the principal vector during the orbit transfer process is not greater than 1, the pulse only occurs at the transfer starting point and the transfer ending point, the modulus of the corresponding principal vector is equal to 1 at the time when the first derivative is 0 and the second derivative is less than 0, and the pulse direction is consistent with the principal vector direction.

[0012] The continuation algorithm is used to numerically solve the three-dimensional optimization model to obtain the optimal decision vector, the minimum fuel cost, and the two pulse values; the continuation algorithm is used to solve the constructed initial auxiliary solution and the solution of the discrete continuation sequence of the orbit inclination coefficient changing from 0 to 1; and each step takes the optimal solution corresponding to the orbit inclination coefficient of the previous step as the initial value, and uses an adaptive step to optimize to obtain the most fuel-efficient two-pulse solution between spatial orbits that satisfies the constraint condition.

[0013] In another aspect, a most fuel-efficient double-pulse transfer device between spatial orbits is also provided, which comprises: An orbit model determination module is configured to determine the transfer starting point of the spacecraft on the initial orbit and the transfer ending point on the terminal orbit according to the given initial orbit and terminal orbit, and calculate the most fuel-efficient double-pulse elliptical transfer orbit between the transfer starting point and the transfer ending point. A three-dimensional optimization model construction module, wherein the decision vector includes the true anomaly of the transfer starting point on the initial orbit, the true anomaly of the transfer ending point on the terminal orbit, the semi-latus rectum of the transfer orbit, and the orbit inclination coefficient; the objective function is to minimize the total pulse velocity increment; and the orbit inclination coefficient is used to set the initial and final orbit inclinations in the continuation calculation process. The constraint condition determination module is configured to introduce an optimality necessary condition based on a principal vector theory in an orbital coordinate system as a constraint condition in the three-dimensional optimization model; the constraint condition includes: coefficients of long-term terms in expressions of components of a principal vector in a transfer orbital coordinate system are 0, a principal vector modulus is not greater than 1 during an orbital transfer, impulses only occur at a transfer starting point and a transfer ending point, corresponding principal vector moduli are equal to 1, first-order derivatives are 0, second-order derivatives are less than 0, and impulse directions are consistent with principal vector directions; The minimum-fuel two-impulse transfer solving module is configured to perform numerical solving on the three-dimensional optimization model by using a continuation algorithm to obtain an optimal decision vector, a minimum fuel cost, and two impulse values; the continuation algorithm is used to solve a discrete continuation sequence of orbital inclination coefficients changing from 0 to 1; and each step takes an optimal solution corresponding to an orbital inclination coefficient of a previous step as an initial value, and performs optimization by using an adaptive step length to obtain a minimum-fuel two-impulse solution between space orbits that satisfies the constraint condition.

[0014] In another aspect, a computer device is also provided, which includes a memory and a processor, the memory stores a computer program, and the processor implements the steps of the minimum-fuel two-impulse transfer method between space orbits of any one of the above aspects when executing the computer program.

[0015] One of the above technical solutions has the following advantages and beneficial effects: The minimum-fuel two-impulse transfer method between space orbits, the device, and the computer device have the following advantages and beneficial effects: the method extends a minimum-fuel two-impulse solution between in-plane elliptical orbits to space elliptical orbits, and proposes a continuation hybrid optimization algorithm for solving by continuously changing initial and final orbital inclinations; the continuation hybrid optimization algorithm process includes: constructing an initial auxiliary solution, and constructing a discrete continuation sequence of orbital inclination coefficients changing from 0 to 1; each step takes an optimal solution corresponding to an orbital inclination coefficient of a previous step as an initial value, and performs optimization by using an adaptive step length to obtain a minimum-fuel two-impulse solution between space orbits that satisfies the constraint condition. The method improves the accuracy and effectiveness of solving a minimum-fuel two-impulse orbital transfer problem between space orbits. BRIEF DESCRIPTION OF DRAWINGS

[0016] In order to more clearly illustrate the technical solutions of the embodiments of the present application or the prior art, the following will briefly introduce the drawings needed to be used in the embodiments or the prior art description. Obviously, the drawings in the following description are only some embodiments of the present application, and other drawings can be obtained by those skilled in the art without any creative effort on the basis of these drawings.

[0017] Figure 1 It is a flowchart of the minimum-fuel two-impulse transfer method between space orbits in one embodiment; Figure 2 It is a geocentric coordinate system in one embodimentO-xyz A schematic diagram illustrating the motion of a spacecraft; Figure 3 A schematic diagram of a two-impulse orbit transfer for one embodiment; Figure 4 A schematic diagram of an orbit coordinate system for one embodiment A schematic diagram; Figure 5 A schematic diagram of a two-impulse transfer orbit for one embodiment k A schematic diagram of a minimum fuel two-impulse transfer orbit for one embodiment when Figure 6 A schematic diagram of a two-impulse transfer orbit for one embodiment k A schematic diagram of a minimum fuel two-impulse solution for one embodiment when Figure 7 A schematic diagram of the magnitude of the principal vector associated with a transfer orbit in the A schematic diagram of the magnitude of the principal vector associated with a transfer orbit in the Figure 8 A schematic diagram of the magnitude of the principal vector associated with an initial orbit in the A schematic diagram of the magnitude of the principal vector associated with an initial orbit in the Figure 9 A schematic diagram of the magnitude of the principal vector associated with a final orbit in the A schematic diagram of the magnitude of the principal vector associated with a final orbit in the DETAILED DESCRIPTION

[0018] In order to make the objects, technical solutions and advantages of the present application clearer, the present application will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein merely aim to explain the present application and are not intended to limit the present application.

[0019] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application belongs. The terminology used in the description of the application herein is for describing the specific embodiments only and is not intended to be limiting of the application.

[0020] It should be noted that a reference herein to "an embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment can be included in at least one embodiment of the application. The appearances of the phrase in various places in the specification are not necessarily all referring to the same embodiment, nor are they necessarily mutually exclusive of one another. One of ordinary skill in the art can understand that the embodiments described herein can be combined with one another. The term "and / or" as used herein refers to any combination of one or more of the associated listed items, as well as all possible combinations of the items, and includes these combinations.

[0021] The embodiments of the present application will be described in detail below with reference to the accompanying drawings.

[0022] In one embodiment, as shown in Figure 1 The method for the most fuel-efficient two-impulse transfer between spatial orbits can include the following steps 100 to 103: Step 100: According to the given initial orbit elements and the terminal orbit elements, the transfer starting point on the initial orbit and the transfer ending point on the terminal orbit are determined, and the most fuel-efficient two-impulse elliptical transfer orbit between the transfer starting point and the transfer ending point is calculated.

[0023] Step 101: A three-dimensional optimization model is established; the decision vector includes the true anomaly of the transfer starting point on the initial orbit, the true anomaly of the transfer ending point on the terminal orbit, the semi-major axis of the transfer orbit, and the inclination coefficient; the objective function is to minimize the total impulse velocity increment; the inclination coefficient is used to set the initial and final orbit inclination in the continuation calculation process.

[0024] Step 102: In the three-dimensional optimization model, the optimality necessary conditions based on the principal vector theory in the orbit coordinate system are introduced as constraint conditions; the constraint conditions include: the coefficient of the long-term term in the expression of the principal vector components in the transfer orbit coordinate system is 0, the modulus of the principal vector during the orbit transfer is not greater than 1, the impulse only occurs at the transfer starting point and the transfer ending point, the modulus of the corresponding principal vector is equal to 1, the impulse direction is consistent with the principal vector direction at the time when the first derivative is 0 and the second derivative is less than 0.

[0025] Step 103: The continuation algorithm is used to numerically solve the three-dimensional optimization model to obtain the optimal decision vector, the minimum fuel cost, and the two impulse values; the continuation algorithm is used to solve the initial auxiliary solution and the discrete continuation sequence of the inclination coefficient changing from 0 to 1; and each step takes the optimal solution corresponding to the inclination coefficient of the previous step as the initial value, and uses adaptive step size for optimization to obtain the most fuel-efficient two-impulse solution between spatial orbits that satisfies the constraint conditions.

[0026] The above-mentioned method for the most fuel-efficient two-impulse transfer between spatial orbits extends the most fuel-efficient two-impulse solution between planar elliptical orbits to spatial elliptical orbits, and proposes a continuation hybrid optimization algorithm by continuously changing the initial and final orbit inclination, which includes: constructing the initial auxiliary solution and the discrete continuation sequence of the inclination coefficient changing from 0 to 1; each step takes the optimal solution corresponding to the inclination coefficient of the previous step as the initial value, and uses adaptive step size for optimization to obtain the most fuel-efficient two-impulse solution between spatial orbits that satisfies the constraint conditions. This method improves the accuracy and effectiveness of solving the most fuel-efficient two-impulse orbit transfer problem between spatial orbits.

[0027] In one embodiment, the expression of the position vector and the velocity vector of the transfer starting point in the geocentric equatorial inertial coordinate system in step 100 is: (1) (2) (3) in, , , , , , These are the semi-major axis, eccentricity, orbital inclination, perigee depression angle, ascending node angle, and true anomaly angle of the initial orbit, respectively. , These represent the position and velocity of the spacecraft at its initial transfer point on its initial orbit; , These are the position vector and velocity vector of the starting point in the geocentric equatorial inertial coordinate system, respectively. , These are two fundamental transformation matrices; k This is the track inclination coefficient. The variable introduced for numerical optimization, with the superscript T indicating vector transpose. These are Earth's gravitational parameters.

[0028] The expressions for the position vector and velocity vector of the transfer endpoint in the geocentric equatorial inertial coordinate system are: (4) (5) (6) in, , , , , , These are the semi-major axis, eccentricity, orbital inclination, perigee angle, ascending node angle, and true anomaly angle of the terminal orbit. , These represent the position and velocity of the spacecraft at the endpoint of its transfer in the terminal orbit; , These are the position vector and velocity vector of the transfer endpoint in the geocentric equatorial inertial coordinate system, respectively.

[0029] Specifically, introduce Figure 2 The coordinates shown are in the Earth's inertial system. O-xyz Describes the motion of a spacecraft. Among them, O With the Earth's center of mass as the center and the equatorial plane as the fundamental plane, Ox The axis points from the Earth's center to the vernal equinox. Oz Perpendicular to the fundamentals and pointing towards the North Pole. Oy and Ox andOz It forms a right-handed system.

[0030] For any angle Define the basic transformation matrix and for: (7) (8) The classical orbital elements of the initial orbit are .in, These are the semi-major axis, eccentricity, orbital inclination, perigee depression angle, ascending node angle, and true anomaly angle of the initial orbit, respectively. O-xyz The position of the spacecraft at point I in the coordinate system. and speed As shown in the expressions for the position vector and velocity vector of the starting point in the geocentric equatorial inertial coordinate system.

[0031] The classical orbital elements of the final orbit are .in, , , , , , These are the semi-major axis, eccentricity, orbital inclination, perigee angle, ascending node angle, and true anomaly angle of the terminal orbit. O-xyz The position of the spacecraft at point T in the coordinate system. and speed As shown in the expressions for the position vector and velocity vector of the transfer endpoint in the geocentric equatorial inertial coordinate system.

[0032] In one embodiment, step 101 includes: establishing a three-dimensional optimization model as follows: (9) (10) (11) (12) (13) in, The fuel required for the two-pulse transfer, l For the transfer track semi-circular diameter, , These are the position vector and velocity vector of the transfer endpoint in the geocentric equatorial inertial coordinate system, respectively. , These are the position vector and velocity vector of the starting point in the geocentric equatorial inertial coordinate system, respectively. For a basic transformation matrix, For Earth's gravitational parameters, Let be the angle between the position vectors of the starting point and the ending point in the geocentric equatorial inertial coordinate system. , These represent the velocity increments of the two applied pulses, , Let represent the velocities of the transfer orbit at the start and end points of the transfer, respectively, and let I and T represent the points on the initial and final orbits of the transfer orbit, respectively.

[0033] Specifically, see the two-pulse orbital transfer. Figure 3 Where I and T represent the points on the initial and final orbits of the transfer orbit, respectively. and These represent the velocities at points I and T on the transfer trajectory, respectively. express and The angle formed between them.

[0034] Define separately and The velocity along the oblique axis is: , , (14) in, , , , This represents the Euclidean norm of a vector.

[0035] Assuming a unit vector The orbital plane is perpendicular to the transfer orbit. Considering formula (14). It can be represented as: (15) Around Rotation have to: (16) make l This is the semi-circular diameter of the transfer orbit. and exist O-xyz In a coordinate system, it can be represented as: (17) (18) (19) Among them, in formula (14) and Substituting into formula (16), we can calculate... θ .

[0036] The performance metrics to be minimized are shown in methods (9) to (11).

[0037] Set orbital parameters and Given, true anterior angle and These are free variables. Furthermore, the transition time from I to T... Undetermined. Given and , Can be determined by the transfer track semi-circular diameter l Confirmed. At this point, the problem of finding the most fuel-efficient two-pulse transfer trajectory simplifies to solving for the optimal... , , and l And make the formula (9) Minimum.

[0038] In one embodiment, step 102 includes: defining the principal vector in the transfer orbit coordinate system as... The expressions for each component are: (20) (twenty one) (twenty two) in, Main vector , , The principal vectors are respectively Components in each direction in the orbital coordinate system A , B , C , D , X and Y These are six constants to be determined. e The eccentricity of the transfer orbit, f For the true anomaly angle of the transfer orbit, , ,make For the angle with the true nearest point The corresponding anomalous angle, according to reference 1 (Carter, TE, 1990, New form for the optimal rendezvousequations near a Keplerian orbit. J. Guid. ControlDyn. 13(1), 183-186, http: / / dx.doi.org / 10.2514 / 3.20533), The expression is: (twenty three) (twenty four) The derivative of the principal vector with respect to time has the following components along the coordinate axes of the transfer orbit coordinate system: (25) (26) (27) in, The principal vectors are respectively Each component , , The derivative with respect to time along the axes of the orbital coordinate system. r To transfer the distance between the Earth's center and the orbital center, .

[0039] For the most fuel-efficient two-pulse trajectory in the plane, the necessary condition for optimality derived from principal vector theory is: (28) (29) (30) in, , These are the principal vector magnitudes of the transfer trajectory at the true anomaly angles at the transfer start and transfer end points, respectively. , For respectively , The first derivative, , are the second derivatives of and , respectively, and are the pulse directions corresponding to the two pulses in the transfer orbit coordinate system; , These are the principal vectors of the transfer trajectory at the true anomaly angles at the transfer start and transfer end points, respectively; according to ,make , The principal vectors of the transfer trajectory at the true anomaly angles at the transfer start and transfer end points are: (31) (32) according to The constants related to the principal vector are calculated. A, B, D, X and Y Specifically: (33) in, ,matrix For matrix Moore-Penrose pseudoinverse matrix, matrix for: ; (34) For the optimal solution, the calculated Constraints must be met: (35) in, This is an operation to calculate the magnitude of a vector.

[0040] In one embodiment, the three-dimensional optimization model with introduced constraints is as follows: (36) in, To transfer the true anomaly angle of the starting point on the initial orbit, To transfer the true anomaly angle of the endpoint on the terminal orbit, l For the transfer track semi-circular diameter, The fuel required for the two-pulse transfer, , These represent the velocity changes after applying two pulses, respectively.

[0041] Specifically, to facilitate numerical optimization, the necessary conditions for optimality must first be determined. The specific process of the numerical optimization algorithm includes: First, introduce the transfer trajectory coordinate system shown in Figure 4. .in, Along the direction of the spacecraft's position vector, by O Pointing to S, Pointing in the direction of motion within the orbital plane and perpendicular to , and and Construct a right-handed coordinate system, where I and T are two points on the initial and final trajectories of the transfer trajectory, respectively. and The principal vectors are defined as follows: )along , and The amount.

[0042] For a given value and l The semi-major axis and eccentricity of the transfer orbit are as follows: (37) when When, define and for: (38) (39) in, f Let E be the true anomaly angle of the transfer orbit. f The corresponding near-point angle, then, according to reference 1, in formula (39) The expressions are shown in formulas (23) to (24).

[0043] Define the principal vector Its components are shown in formulas (20) to (22) above.

[0044] make for The derivative with respect to time. Let... along Figure 4 Orbital coordinate system Each axis , and The components are shown in formulas (25) to (27) above.

[0045] When e = 0, , , , , and The expression is: (40) (41) (42) (43) (44) (45) make The main vector magnitude. Defined separately. and for about f The first and second derivatives.

[0046] Separately and Let be the true anomaly angles of the transfer orbit at I and T. , , , , and For the most fuel-efficient two-pulse orbit in a plane, the necessary conditions for optimality derived in reference 2 (Lawden, DF, 1963. Optimal trajectories for space navigation. Butterworths, London, 1-123) are shown in equations (28) and (30).

[0047] Note that in formula (29) This means the corresponding For about f It is a periodic function and does not contain long-term terms. Formula (30) shows that... and for exist The two maxima. For space orbit transfer, the principal vector components in formula (27) or (45) are the two maxima. or Let be the component of the angular momentum along the direction of the transfer orbit, and be about f The periodic function. Therefore, the necessary condition for in-plane optimality derived in reference 3 (Li, J., 2024. Minimum-fuel two-impulse transfer between coplanar noncoaxial elliptical orbits. Adv. Space Res. 73, 143-159. http: / / dx.doi.org / 10.1016 / j.asr.2023.10.008.) also directly applies to space orbit transfer. The only difference between the two is the principal vector magnitude of the space transfer orbit. The expression is more complex.

[0048] The specific process of solving for the constants includes: Definition and at point I O-xyz coordinate system axes Ox , Oy ,and OzThe relevant unit vector is , and Its expression is: (46) Definition and at point I Transfer orbit coordinate system axes , and The relevant unit vector is , and .exist O-xyz In coordinate system, , and The expression is: (47) O-xyz Coordinate system transformation to the local orbital coordinate system corresponding to point I The transformation matrix is: (48) make , and The local orbital coordinate system corresponding to point T is respectively Each axis , and The corresponding unit vector. In O-xyz In coordinate system, , and The expressions are as follows: (49) O-xyz Coordinate system transformation to the local orbital coordinate system corresponding to point T The transformation matrix is: (50) Refer to formulas (2), (5), (17) and (18). and In the local orbital coordinate system at point I The expression is: (51) (52) and In the local orbital coordinate system corresponding to point T The expression is: (53) (54) Then, the corresponding pulse direction and for: (55) (56) In the local orbital coordinate system at point I , It can be represented as: (57) In the local orbital coordinate system at point T , It can be represented as: (58) Substituting formula (57) into formula (52) yields the solution. Substituting formula (58) into formula (53), the solution can be obtained. .

[0049] make and Combining formulas (20) to (22) and (28), and The expression is: (59) (60) Combine formulas (25) to (27) and (28). and It can be represented as: (61) (62) make and According to the necessary condition for optimality, the principal vector at I and T must satisfy: (63) Therefore, constant A , B , D , X and Y satisfy: (64) , (65) Among them, in formula (55) In the matrix formula (34) The Moore-Penrose inverse matrix. For the optimal solution, equation (62) yields... Must meet: (66) When formula (66) holds true, formulas (55), (56), and (63) indicate that... and This has been automatically satisfied. Therefore, the optimality condition of formula (27) can be simplified to: (67) Considering the boundary conditions given by formulas (1) to (6), the optimal control problem in this application is simplified to seeking to satisfy the constraints formulas (66) and (67), and letting formula (9) J minimum value and l The specific details are shown in formula (36).

[0050] visible, That is, the problem studied in reference 3. The problem to be solved in this application.

[0051] It should be noted that the optimization model in formula (36) is the same as that in reference 3. The main difference is that reference 3 optimizes the model by continuously changing the angle between the principal axes of the initial and final elliptical orbits, while this application optimizes it by setting... k From 0 to 1, refer to the orbital inclination angles in formulas (1) to (6). and ,in Therefore, with only a few minor changes, the extension method in reference 5 can be used to solve the optimization model in formula (36).

[0052] In one embodiment, step 103 includes: taking the most fuel-efficient two-pulse solution corresponding to the initial orbit and the terminal orbit having zero orbit inclination as the initial auxiliary solution; dividing the range of orbit inclination coefficient [0,1] into N sub-intervals on average, and taking the interval boundary points as discrete extension sequences; taking the optimal solution corresponding to the orbit inclination coefficient of the previous step as the initial value for each step, and using an adaptive step size for optimization to obtain the most fuel-efficient two-pulse solution between space orbits that satisfies the constraints.

[0053] In a verification embodiment, numerical simulation was used to verify the effectiveness of the proposed method. Table 1 shows the trajectory parameters of the initial and final trajectories. km represents the radius of Earth's orbit; distance is measured in km, and angles are measured in degrees.

[0054] Table 1 Classical orbital elements for initial and final orbits

[0055] (1) Optimal solution k The most fuel-efficient two-pulse transfer trajectory at = 0 is shown in the figure. Figure 5 In this diagram, the thick solid line and the dashed line represent the I-to-T and T-to-I sections of the complete transfer orbit, respectively, where I represents the initial orbit and T represents the terminal orbit. and The value, To satisfy the computational requirements of the three-dimensional optimization model with introduced constraints, let In the three-dimensional optimization model with introduced constraints , Set to: .

[0056] Typical solutions obtained from the continuation process are shown in Tables 2 to 4. It should be noted that, for simplicity, the amplitude and direction of all pulses are not given.

[0057] Table 2 value

[0058] Table 3 value

[0059] k The optimal two-pulse solution for fuel efficiency when = 1 is as follows: Figure 6 As shown. The thick solid line and the dotted line represent the two parts from I to T and from T to I, respectively, forming a complete transfer trajectory.

[0060] Table 4 A , B , C , D , X , Y value

[0061] make , , , , and is the conjugate vector constant associated with the initial trajectory. , , , , and These are the conjugate vector constants associated with the terminal trajectory. Typical values ​​for these constants are shown in Tables 4 and 5. k Taking 1 as an example, the magnitudes of the principal vectors related to the transfer orbit, initial orbit, and terminal orbit in the orbital coordinate system The values ​​below are respectively from Figure 7 to Figure 9 Given. (See Figure 7 to...) Figure 9 The middle circle represents a point where the magnitude of the principal vector is 1. , and These represent the magnitudes of the principal vectors related to the transfer orbit, initial orbit, and terminal orbit, respectively.

[0062] Table 5 , , , , and value

[0063] Table 6: , , , , , value

[0064] It should be understood that, although the above Figure 1 The steps are shown sequentially as indicated by the arrows, but these steps are not necessarily executed in the order indicated by the arrows. Unless otherwise explicitly stated in this document, there is no strict order in which these steps are executed; they can be performed in other orders. Furthermore, the above... Figure 1 At least some of the steps may include multiple sub-steps or multiple stages. These sub-steps or stages are not necessarily executed at the same time, but can be executed at different times. The execution order of these sub-steps or stages is not necessarily sequential, but can be executed in turn or alternately with other steps or at least some of the sub-steps or stages of other steps.

[0065] In one embodiment, a fuel-efficient dual-pulse transfer device between space orbits is also provided, the device comprising: The orbit model determination module is used to determine the transfer start point on the initial orbit and the transfer end point on the terminal orbit based on the given initial orbit elements and terminal orbit elements, and to calculate the position vector and velocity vector of the transfer start point and the transfer end point in the geocentric equatorial inertial coordinate system; the orbit is an elliptical orbit.

[0066] The three-dimensional optimization model construction module is used to establish a three-dimensional optimization model; the decision vector includes the initial true anomaly angle, the terminal true anomaly angle, the transfer orbit semi-circle, and the orbit inclination coefficient; the objective function is to minimize the total pulse velocity increment; the orbit inclination coefficient is used to set the initial and terminal orbit inclination angles in the extension calculation process.

[0067] The constraint determination module is used to introduce the necessary optimality conditions based on the principal vector theory in the orbit coordinate system as constraints in the three-dimensional optimization model. The constraints include: the coefficients of the long-term terms in the expressions of each component of the principal vector at the initial orbit true anomaly angle and the terminal orbit true anomaly angle are 0; the principal vector magnitude is less than or equal to 1 during the orbit transfer process; the pulse only occurs when the derivative of the principal vector magnitude is 0 and the second derivative is less than 0; and the pulse direction is consistent with the principal vector direction.

[0068] The fuel-efficient two-pulse transfer solution module is used to numerically solve the three-dimensional optimization model using an extension algorithm to obtain the optimal decision vector, minimum fuel cost, and two pulse values. The extension algorithm is used to construct an initial auxiliary solution and to construct a discrete extension sequence in which the orbital inclination coefficient changes from 0 to 1. Each step uses the optimal solution corresponding to the orbital inclination coefficient of the previous step as the initial value, and optimizes it using an adaptive step size to obtain the fuel-efficient two-pulse solution between space orbits that meets the constraints.

[0069] In one embodiment, the position vector and velocity vector of the transfer starting point in the geocentric equatorial inertial coordinate system in the orbit model determination module are shown in Equations (1) to (3).

[0070] The position vector and velocity vector of the transfer endpoint in the geocentric equatorial inertial coordinate system are shown in Equations (4) to (6).

[0071] In one embodiment, the three-dimensional optimization model building module is also used to build a three-dimensional optimization model as shown in formulas (9) to (13).

[0072] In one embodiment, step 102 includes: defining the principal vector in the transfer orbit coordinate system as... The expressions for each component are shown in equations (20) to (22). The components of the derivative of the principal vector with respect to time along the coordinate axes of the transfer orbit coordinate system are shown in equations (25) to (27). For the most fuel-efficient two-pulse orbit in the plane, the necessary conditions for optimality derived based on the principal vector theory are shown in equations (28) to (30); according to ,make , The principal vectors of the transfer trajectory at the true anterior angles of the transfer start and transfer end points are shown in formulas (31) and (32); according to The constants related to the principal vector are calculated. A, B, D, X and Y ; Specifically, see formulas (33) to (35).

[0073] In one embodiment, the three-dimensional optimization model that introduces constraints in the constraint determination module is shown in Equation (36).

[0074] In one embodiment, the fuel-efficient two-pulse transfer solution module is further used to take the fuel-efficient two-pulse solution corresponding to the initial orbit and the terminal orbit with zero orbit inclination as the initial auxiliary solution; divide the range of orbit inclination coefficient [0,1] into N sub-intervals on average, and take the interval boundary points as discrete extension sequences; take the optimal solution corresponding to the orbit inclination coefficient of the previous step as the initial value for each step, and use an adaptive step size for optimization to obtain the fuel-efficient two-pulse solution between space orbits that meets the constraints.

[0075] It is understood that for a detailed explanation of the most fuel-efficient dual-pulse transfer device between space orbits, please refer to the corresponding explanations of the various embodiments of the most fuel-efficient dual-pulse transfer method between space orbits above, and will not be repeated here. Each module in the aforementioned most fuel-efficient dual-pulse transfer device between space orbits can be implemented entirely or partially through software, hardware, or a combination thereof. Each module can be embedded in hardware or independently of a device with data processing capabilities, or stored in software in the memory of the aforementioned device, so that the processor can call and execute the operations corresponding to each module. The aforementioned device can be, but is not limited to, various types of data processing computer devices already existing in the art.

[0076] In one embodiment, a computer device is also provided, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps in the above method embodiments.

[0077] It is understood that, in addition to the memory and processor mentioned above, the computer equipment described above also includes other hardware and software components not listed in this specification. The specific components can be determined according to the model of the image processing computer in different application scenarios, and will not be listed and described in detail in this specification.

[0078] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0079] The above embodiments are merely illustrative of several implementation methods of this application, and their descriptions are relatively specific and detailed. However, they should not be construed as limiting the scope of protection of this application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of this application, and all such modifications and improvements fall within the scope of protection of this application.

Claims

1. A fuel-efficient dual-pulse transfer method between space orbits, characterized in that, Including the following steps: Based on the given initial orbital elements and terminal orbit, determine the transfer start point on the initial orbit and the transfer end point on the terminal orbit, and calculate the most fuel-efficient double-pulse elliptical transfer orbit between the transfer start point and the transfer end point; Establish a three-dimensional optimization model; The decision vector includes the true anomaly angle of the starting point on the initial track, the true anomaly angle of the ending point on the terminal track, the semi-circular diameter of the transfer track, and the track inclination coefficient; the objective function is to minimize the total pulse velocity increment; the track inclination coefficient is used to set the initial and final track inclination angles in the extension calculation process. In the three-dimensional optimization model, the necessary condition for optimality based on principal vector theory in the orbital coordinate system is introduced as a constraint condition; The constraints include: the coefficients of the long-term terms in the expressions of the principal vector components in the transfer orbit coordinate system are 0; the magnitude of the principal vector is not greater than 1 during the orbit transfer process; the pulse only occurs at the transfer start and transfer end points, and the corresponding principal vector magnitude is equal to 1; the first derivative is 0 and the second derivative is less than 0; and the pulse direction is consistent with the principal vector direction. The continuation algorithm is used to numerically solve the three-dimensional optimization model to obtain the optimal decision vector, the minimum fuel cost, and the two pulse values. The continuation algorithm is used to solve and construct the initial auxiliary solution, and to construct the solution of the discrete continuation sequence of the orbital inclination coefficient changing from 0 to 1. Each step uses the optimal solution corresponding to the orbital inclination coefficient of the previous step as the initial value, and an adaptive step size is used for optimization to obtain the most fuel-efficient two-pulse solution between space orbits that meets the constraints.

2. The most fuel-efficient dual-pulse transfer method between space orbits according to claim 1, characterized in that, The position vector and velocity vector of the transfer starting point in the geocentric equatorial inertial coordinate system are: ； ； ； ； in, , , , , , These are the semi-major axis, eccentricity, orbital inclination, perigee depression angle, ascending node angle, and true anomaly angle of the initial orbit, respectively. , These represent the position and velocity of the spacecraft at its initial transfer point on its initial orbit; , These are the position vector and velocity vector of the starting point in the geocentric equatorial inertial coordinate system, respectively. , These are two fundamental transformation matrices; k This is the orbital inclination coefficient, with the superscript T indicating vector transpose; The position vector and velocity vector of the transfer endpoint in the geocentric equatorial inertial coordinate system are: ； ； ； ； in, , , , , , These are the semi-major axis, eccentricity, orbital inclination, perigee angle, ascending node angle, and true anomaly angle of the terminal orbit. , These represent the position and velocity of the spacecraft at the endpoint of its transfer in the terminal orbit; , These are the position vector and velocity vector of the transfer endpoint in the geocentric equatorial inertial coordinate system, respectively.

3. The most fuel-efficient dual-pulse transfer method between space orbits according to claim 1, characterized in that, The three-dimensional optimization model is established as follows: ； ； ； ； ； in, The fuel required for the two-pulse transfer, l For the transfer track semi-drilled diameter, , These are the position vector and velocity vector of the transfer endpoint in the geocentric equatorial inertial coordinate system, respectively. , These are the position vector and velocity vector of the starting point in the geocentric equatorial inertial coordinate system, respectively. For a basic transformation matrix, For Earth's gravitational parameters, Let be the angle between the position vectors of the starting point and the ending point in the geocentric equatorial inertial coordinate system. , These represent the velocity increments of the two applied pulses, , denoted as and respectively, representing the velocities of the transfer trajectory at the start and end points of the transfer, and I and T respectively, representing the points on the initial and final tracks of the transfer trajectory.

4. The most fuel-efficient dual-pulse transfer method between space orbits according to claim 1, characterized in that, In the aforementioned three-dimensional optimization model, the necessary conditions for optimality based on principal vector theory in the transfer orbit coordinate system are introduced as constraints, including: Define the principal vector in the transfer orbit coordinate system as The expressions for each component are: ； ； ； in, Main vector , , The principal vectors are respectively Components in each direction in the orbital coordinate system A , B , C , D , X and Y These are six constants to be determined. e The eccentricity of the transfer orbit, f For the true anomaly angle of the transfer orbit, , ,make For the angle with the true nearest point The corresponding near-point angle, The expression is: ； ； The derivative of the principal vector with respect to time has the following components along the coordinate axes of the transfer orbit coordinate system: ； ； ； in, The principal vectors are respectively Each component , , The derivative with respect to time along the axes of the orbital coordinate system. r To transfer the distance between the Earth's center and the orbital center, ; For the most fuel-efficient two-pulse trajectory in the plane, the necessary condition for optimality derived from principal vector theory is: ； ； ； in, , These are the principal vector magnitudes of the transfer trajectory at the true anomaly angles at the transfer start and transfer end points, respectively. , For respectively , The first derivative, , They are respectively , The second derivative, These are the pulse directions corresponding to the two pulses in the transfer orbit coordinate system; , These are the principal vectors of the transfer trajectory at the true anomaly angles at the transfer start and transfer end points, respectively; according to ,make , The principal vectors of the transfer trajectory at the true anomaly angles at the transfer start and transfer end points are: ； according to The constants related to the principal vector are calculated. A, B, D, X and Y Specifically: ； in, ,matrix For matrix Moore-Penrose pseudo-inverse matrix, matrix for: ； For the optimal solution, the calculated Constraints must be met .

5. The most fuel-efficient dual-pulse transfer method between space orbits according to claim 4, characterized in that, The three-dimensional optimization model with introduced constraints is as follows: ； in, Let the initial orbital true anomaly angle be . For the true anomaly angle of the terminal orbit, l For the transfer track semi-drilled diameter, The fuel required for the two-pulse transfer, , These represent the velocity changes after applying two pulses, respectively.

6. The most fuel-efficient dual-pulse transfer method between space orbits according to claim 1, characterized in that, The continuation algorithm was used to numerically solve the three-dimensional optimization model, obtaining the optimal decision vector, minimum fuel cost, and two pulse values, including: The two-pulse solution with the lowest fuel efficiency when the inclination angles of both the initial and terminal orbits are zero is used as the initial auxiliary solution. The range of the orbital inclination coefficient [0,1] is divided into N sub-intervals, and the boundary points of the intervals are used as discrete extension sequences. Each step uses the optimal solution corresponding to the orbital inclination coefficient of the previous step as the initial value, and optimizes it by using an adaptive step size to obtain the most fuel-efficient two-pulse solution between space orbits that satisfies the constraints.

7. A fuel-efficient dual-pulse transfer device between space orbits, characterized in that, include: The orbit model determination module is used to determine the transfer start point on the initial orbit and the transfer end point on the terminal orbit of the spacecraft based on the given initial orbit and terminal orbit, and to calculate the most fuel-efficient double-pulse elliptical transfer orbit between the transfer start point and the transfer end point; The three-dimensional optimization model is a structural module. The decision vector includes the true anomaly angle of the starting point on the initial track, the true anomaly angle of the ending point on the terminal track, the semi-circular diameter of the transfer track, and the track inclination coefficient. The objective function is to minimize the total pulse velocity increment. The track inclination coefficient is used to set the initial and final track inclination angles in the extension calculation process. The constraint determination module is used to introduce the necessary optimality conditions based on the principal vector theory in the orbital coordinate system as constraints in the three-dimensional optimization model. The constraints include: the coefficients of the long-term terms in the expressions of the principal vector components in the transfer orbit coordinate system are 0; the magnitude of the principal vector is not greater than 1 during the orbit transfer process; pulses only occur at the transfer start and transfer end points, with the corresponding principal vector magnitude equal to 1; the first derivative is 0; the second derivative is less than 0; and the pulse direction is consistent with the principal vector direction. The fuel-efficient two-pulse transfer solution module is used to numerically solve the three-dimensional optimization model using an extension algorithm to obtain the optimal decision vector, minimum fuel cost, and two pulse values. The extension algorithm is used to solve the discrete extension sequence of the orbital inclination coefficient from 0 to 1. Each step uses the optimal solution corresponding to the orbital inclination coefficient of the previous step as the initial value, and optimizes it with an adaptive step size to obtain the fuel-efficient two-pulse solution between space orbits that meets the constraints.

8. A computer device, comprising a memory and a processor, characterized in that, The memory stores a computer program, and when the processor executes the computer program, it implements the steps of the least fuel-efficient dual-pulse transfer method between space orbits as described in any one of claims 1 to 6.

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