Optimal orbit transfer method with finite thrust for bi-linear tangent law

Abstract

The application discloses a space orbit finite thrust optimal transfer method based on a bilinear tangent law, and comprises the following steps: defining a thrust direction; establishing a dynamic equation containing thrust under a two-body orbit; performing integral calculation by using the dynamic equation containing thrust under the two-body orbit and the thrust direction to obtain virtual terminal state positions and velocities; obtaining a flight time before a given distance pulse application point and a flight time after the distance pulse application point according to terminal state constraints; determining an engine start point corresponding to a pulse transfer time and an engine stop point corresponding to the pulse transfer time; and determining a transfer corresponding motion track and a position and a velocity corresponding to each moment. The application can calculate an optimal transfer strategy corresponding to pulse transfer in compliance with an optimal control viewpoint when a spacecraft is in orbit and is transferred.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft orbital maneuvering technology, and in particular to a bilinear tangent law space orbit finite thrust optimal orbit change method. Background Technology

[0002] During spaceflight, the orbital change engine model used has three types based on thrust characteristics: pulse thrust, finite thrust, and low thrust.

[0003] Pulse thrust is a thrust applied for an extremely short time, and the thrust changes with time as an approximation of a pulse function. Before and after the thrust is applied, the spacecraft's spatial position remains unchanged, but its velocity vector suddenly changes by Δv, with its direction along the thrust vector and its magnitude being Δv. At the same time, the spacecraft's mass suddenly changes by Δm.

[0004] The low-thrust model refers to a spacecraft that uses an electric propulsion system with very low thrust, or utilizes natural forces such as solar radiation pressure for maneuvering. In this model, the spacecraft's acceleration is extremely small, the total time of the orbital maneuver is much longer than the orbital period, and the orbital elements change very little within one orbital period.

[0005] Corresponding to the two thrust models above, the thrust is neither too small nor too small, which is the finite thrust model.

[0006] The characteristics of the finite thrust model and its application experience show that when calculating spacecraft orbital maneuvers using the pulse approximation, the existence of gravity loss is ignored, which is too idealistic and differs significantly from actual engineering design. This difference is even more pronounced when the applied velocity increment pulse is large and the engine thrust is not very large.

[0007] In current spacecraft orbit changes, to mitigate the adverse effects of the change and to better reduce gravity losses, special orbit change segments such as apogee and / or perigee are typically employed. However, with the increasing number and variety of space missions, these traditional orbit design and analysis methods based on pulse-based orbit changes are becoming increasingly inadequate to meet the requirements of future space orbit changes and maneuvers.

[0008] When the pulse orbit change model approximates the orbit change maneuver of a spacecraft, it ignores the existence of gravity loss. Robbins elaborates in detail that pulse orbit change leads to the neglect of gravity loss, and analytically estimates the upper bound of gravity loss when using finite thrust orbit change and orbital maneuver (Robbins, HM, An analytical study of the impulsive approximation, AIAA JOURNAL, 4(8):1417-1423, 1966).

[0009] "Spacecraft Orbital Dynamics and Control (Part 2)" quotes Robbins' analysis process in detail (Spacecraft Orbital Dynamics and Control (Part 2), Aerospace Press, 2002), and points out that in practical engineering, when the thrust is less than 1 / 10 of the gravity, it is best to use numerical integration methods to calculate the gravity loss more accurately. It also briefly indicates that a way to reduce gravity loss is to increase the number of orbital maneuvers and decrease the length of the thrust arc segment for each maneuver.

[0010] Both of the above documents point out the shortcomings of the pulse maneuver model in practical engineering applications. Specifically, the idealized pulse maneuver approximation ignores actual gravitational losses, and the longer the maneuver time, the greater the gravitational loss. However, neither document provides a method for converting orbital maneuvers into finite thrust orbital maneuvers. Furthermore, they only offer rough solutions without providing specific, precise calculation methods. In fact, more precise methods for calculating the optimal maneuver under the finite thrust model have never appeared in publicly available literature.

[0011] The paper "Analysis and Optimization of Perigee Maneuvers with Finite Thrust" details the severe gravity loss problem caused by the presence of finite thrust during the orbital design of my country's first lunar probe satellite. To mitigate this impact, the paper introduces a maneuver scheme that transforms a single large pulse into three special point pulses. This method is commonly used in engineering to minimize gravity loss. However, the paper still does not provide a solution for the optimal orbital maneuver under the finite thrust model (Yang Weilian. Analysis and Optimization of Perigee Maneuvers with Finite Thrust [C]. 11th National Conference on Space and Motion Body Control Technology, Lijiang, Yunnan, 2004, 8).

[0012] The paper "Optimization Design of Finite Thrust Orbit Transfer with Minimum Fuel Consumption" establishes a three-degree-of-freedom motion model for spacecraft orbit transfer in polar coordinates and performs dimensionless processing on the state equation of the transfer orbit. Secondly, it presents the performance index, terminal constraints, and control variable constraints of the orbit transfer optimization problem. Based on this, the Legendre pseudospectral method is used to transform the optimal control problem of finite thrust orbit transfer into a nonlinear programming problem, and the Snopt software package is applied to solve the nonlinear programming problem. (Zhang Wei, Fang Qun, Yuan Jianping, Optimization Design of Finite Thrust Orbit Transfer with Minimum Fuel Consumption, Journal of Northwestern Polytechnical University, 2008, 26(5):616-620). The method described in this paper is used to solve reentry problems in two-dimensional plane motion and cannot be applied to three-dimensional Earth orbit maneuvering problems. While the pseudospectral method itself can be extended to solve optimal orbit change problems in three-dimensional space, the solution of this method depends entirely on the solution performance of nonlinear programming software, such as requiring commercial software like Snopt, resulting in poor universality.

[0013] The article "Saturn V guidance, navigation and targeting" introduces the use of iterative guidance technology for the Saturn V launch vehicle's ascent phase ballistic guidance. The guidance law employed is derived from the bilinear tangent law (Martin DT, Sievers RF, Saturn V guidance, navigation and targeting, J. SPACECRAFT, 4(7):891-898, 1967). The bilinear tangent guidance control law constitutes the core technology of rocket iterative guidance, and to this day, iterative guidance technology is still widely used in launch vehicle guidance around the world.

[0014] The Saturn V and other iterative guidance systems generally employ a simplified two-dimensional bilinear tangent law algorithm within the ballistic plane. For a bilinear tangent law applicable to the three-dimensional ballistic plane, please refer to "Bilinear Tangent Yaw Guidance" (Brusch RG, Bilinear Tangent Yaw Guidance, Guidance and Control Conference, Boulder, Colo, United States; 6-8 Aug. 1979. pp. 250-264). This paper derives the bilinear tangent law in detail, along with its corresponding simplifications and various applications for rocket launch trajectory changes.

[0015] Current iterative guidance technology for launch vehicles, due to various simplifications, can only constrain a maximum of 5 orbital parameters (Chen Xinmin, Yu Menglun. Research on the application of iterative guidance in launch vehicles [J]. Journal of Astronautics, 2003, 24(5): 484-489; Ru Jiaxin. An iterative guidance method for liquid launch vehicles [J]. Science in China Series E: Technological Sciences, 2009, 39(4): 696-706.; Fu Yu, Chen Gong, Lu Baogang, et al. An adaptive iterative guidance method for launch vehicles outside the atmosphere based on optimal analytical solutions [J]. Acta Aeronautica Sinica, 2011, 32(9): 1696-1704.). Therefore, it has long been only applicable to launch vehicle launch and orbit insertion missions. Space orbital maneuvers and orbital transfer missions, due to constraints on 6 orbital parameters (position and velocity), have never absorbed any benefits from bilinear tangent law or iterative guidance technology in this field. Instead, numerical optimization techniques such as pseudospectral methods have been adopted, leading to a different development path. Rocket trajectory optimization and space orbital maneuvering optimization have become two increasingly different algorithms. Summary of the Invention

[0016] The technical problem solved by this invention is to overcome the shortcomings of the prior art and provide a bilinear tangent law-based optimal orbit change method for space orbits with finite thrust. For space orbit change or orbit transfer problems, based on the magnitude of the applied velocity increment pulse and the pulse application time, it provides the engine start-up and shutdown times under the corresponding finite thrust conditions, as well as the thrust direction during the corresponding engine start-up time period. At the same time, it obtains the gravity loss under the corresponding finite thrust orbit change model that is more in line with engineering reality.

[0017] The technical solution of this invention is:

[0018] A bilinear tangent-law-based optimal orbital maneuvering method with finite thrust for space orbits includes:

[0019] Define the thrust direction u(t);

[0020] Establish the dynamic equations for a two-body orbit with thrust;

[0021] By integrating the dynamic equations of the two-body orbit containing thrust and the thrust direction u(t), the virtual terminal position can be obtained. speed

[0022] Based on the virtual terminal status location speed Given terminal state constraints, the flight time t1 before the application point of a given distance pulse and the flight time t2 after the application point of a fixed distance pulse are obtained; the pulse trajectory change time t is determined. c The corresponding engine start-up point and pulse track change time t c The corresponding engine shutdown point; thus obtaining the engine start-up and shutdown time under limited thrust conditions;

[0023] Determine the trajectory corresponding to the orbit change, as well as the position and velocity at each moment.

[0024] Preferably, the thrust direction u(t) is as follows:

[0025]

[0026]

[0027] A=[cosα,sinαcosβ,sinαsinβ] T

[0028] B = [b1, b2, b3] T

[0029] Where u(t) is a unit vector; α and β are the angular components of vector A; b1, b2, and b3 are the three-axis components of vector B in the rectangular coordinate system; and t is the flight time.

[0030] Preferably, the dynamic equations involving thrust in a two-body orbit are as follows:

[0031]

[0032] Where Γu(t) represents the spacecraft acceleration along the thrust direction vector u(t); g(r,t) is the gravitational acceleration at time t, and r is the position at time t.

[0033] Preferably, the terminal state constraint is as follows:

[0034] Furthermore, the Hamiltonian function H of the dynamic system obtained from optimal control theory at the final time t f Equals 0;

[0035] Where, r f v represents the location of the flight engine shutdown point. f The speed at which the flight engine shuts down.

[0036] Preferably, the pulse orbit change time t c The corresponding engine start-up point is equal to t. c -t1.

[0037] Preferably, the pulse orbit change time t c The corresponding engine shutdown point is equal to t c +t2.

[0038] Compared with the prior art, the advantages of the present invention are mainly reflected in the following aspects:

[0039] 1) Based on the pulse size of the orbital transfer, this invention can obtain the engine start-up time under the corresponding finite thrust condition. By using the start-up time length, a more accurate estimate of gravity loss can be obtained, avoiding the disadvantage of excessive estimation error under pulse orbital change conditions.

[0040] 2) Based on the pulse size and pulse point time of the orbit transfer, this invention can determine the engine start-up and shutdown times under the corresponding finite thrust conditions, which is more in line with engineering practice;

[0041] 3) Based on the pulse size and pulse point time of the orbit transfer, this invention can determine the optimal thrust direction for engine start-up and shutdown under the corresponding finite thrust conditions, thereby reducing gravity loss;

[0042] 4) This invention solves the long-standing problem that space orbit change can only obtain pulse orbit change solutions, and cannot obtain accurate finite thrust optimal orbit change solutions. In order to avoid this difficulty, engineering practice has long adopted the orbit change method at special points of apogee and perigee, so as to avoid solving the finite thrust optimal orbit change problem as much as possible. Attached Figure Description

[0043] Figure 1 This is a schematic diagram showing the correspondence between pulse trajectory change and finite thrust trajectory change in this invention; Detailed Implementation

[0044] To address the overly idealistic nature of long-standing pulse orbit change models, this invention designs an optimal orbit change method under finite thrust conditions that better aligns with engineering realities. This method can calculate the optimal orbit change strategy based on pulse orbit change, conforming to the optimal control perspective, during spacecraft on-orbit maneuvers and orbit changes, while simultaneously determining reasonable gravity losses.

[0045] When a spacecraft flies under the gravitational field described by the two-body problem, given a pulse orbit change scheme (the timing of the pulse orbit change or pulse maneuver, the magnitude and direction of the applied velocity increment), based on the assumption of a uniform gravitational field (i.e., considering that the spacecraft's distance from the Earth's center changes very little during the relatively short flight time under finite thrust, it can be reasonably assumed that the magnitude of the central gravitational field does not change with time and is a constant value), the optimal fuel control solution under the finite thrust orbit change model that satisfies optimal control theory is derived. Solving the boundary value problem corresponding to the optimal control that satisfies the orbit start and end conditions before and after the pulse orbit change, the engine start-up and start-up times and optimal thrust direction under the finite thrust model corresponding to the pulse orbit change model can be obtained. Furthermore, by integrating according to the calculated thrust direction, a finite thrust orbit change trajectory that precisely matches the orbit before and after the pulse orbit change can be obtained. Finally, based on the finite thrust orbit change time, a gravity loss that is more consistent with engineering reality can be calculated.

[0046] To better describe the present invention, the following detailed explanation is provided with reference to schematic diagrams and examples. The correspondence between pulse trajectory change and finite thrust trajectory change is shown below. Figure 1 As shown.

[0047] Figure 1 As shown, 1 is the flight trajectory corresponding to the pulse trajectory change, and 2 is the flight trajectory under the finite thrust condition corresponding to the pulse trajectory change.

[0048] The orbital state X before the pulse is applied is known. + (6-dimensional, including three-dimensional position r) + Three-dimensional velocity v + Alternatively, the position and velocity can be converted from the six orbital elements (+ indicates before the pulse is applied), and the applied three-dimensional velocity increment Δv can be used to calculate the orbital state X after the pulse is applied. - (6-dimensional, including three-dimensional position r) - Three-dimensional velocity v - , - indicates after the pulse is applied.

[0049] Defined by pulse trajectory change:

[0050] X + =[r +T ,v +T ] T =[r T ,v +T ] T (1)

[0051] X - =[r -T ,v -T ] T =[r T ,v -T ] T (2)

[0052] v - -v + =Δv (3)

[0053] r = r + =r - (4)

[0054] Define the start-up time of the finite thrust engine as t0; and the shutdown time as t. f The pulse is applied at time t. c .

[0055] Corresponding to the pulse application time t c Limited thrust must be applied before this moment and shut down after this moment; from the power-on point t0 to the pulse application time t c The duration between them is t1, and at the pulse application time t c This continues until the shutdown point t. f The duration is t2.

[0056] The flight time t1 of the finite thrust trajectory change from the start point to the pulse application point:

[0057] t1=t c -t0 (5)

[0058] The flight time t2 from the pulse application point to the shutdown point for finite thrust distance:

[0059] t2=t f -t c (6)

[0060] The equations of motion for spacecraft flight under the influence of Earth's gravitational field are as follows:

[0061]

[0062] In the above formula, X = [r T ,vT ] T Let r be the orbital state, v be the velocity, g(r,t) be the gravitational acceleration, and u be the unit vector of the thrust vector.

[0063] The inverse square gravitational field describing the two-body problem is:

[0064]

[0065] Under thrust, fuel consumption is minimized, and the corresponding dynamic system performance functional is:

[0066]

[0067] Where: T is the engine thrust, 0≤T≤T max Γ = T / m, where m is the mass and Γ is the acceleration of the aircraft.

[0068] Hamiltonian function of the system from optimal control theory:

[0069]

[0070] Where: λ r , λ v Let r and v be the costate variables corresponding to r and v. The last term of the Hamiltonian function H indicates that the constraint thrust direction vector u is a unit vector.

[0071] The adjoint equation yields:

[0072]

[0073]

[0074] According to the principal vector theory, the optimal thrust direction must be parallel to λ. v Define the principal vector p(t) as: p(t) ≡ -λ v (t); At this point, the optimal thrust unit direction u(t) is consistent with the direction of the principal vector p(t):

[0075] u(t)=p(t) / p(t) (13)

[0076] Direct integration of two-body orbits is difficult. To simplify the analysis, the bilinear tangent law assumes that the Earth's gravitational field is uniform during a specific flight time, i.e.:

[0077]

[0078] The corresponding adjoint equation at this point becomes:

[0079]

[0080]

[0081] or:

[0082]

[0083] The corresponding principal vector p(t) is:

[0084]

[0085] Direct double integration yields the general solution for the principal vector:

[0086]

[0087] Here, A and B are three-dimensional vectors, representing integration constants.

[0088] Therefore, the corresponding optimal thrust direction is:

[0089]

[0090] Clearly, from the above equation, if we let t = 0 be the initial time, then the unit direction vector corresponding to the optimal thrust direction at the initial time is:

[0091]

[0092] It is obvious that a three-dimensional unknown unit vector A can be defined by two angles α and β as follows, satisfying the above equation (21):

[0093] A=[cosα,sinαcosβ,sinαsinβ] T (twenty two)

[0094] The unknown vector B is represented by three components:

[0095] B = [b1, b2, b3] T (twenty three)

[0096] As the above analysis shows, the finite thrust optimal trajectory problem corresponding to pulse trajectory change has a total of 7 unknown parameters: the angles α and β related to vector A, the three components of vector B: b1, b2, and b3, and the two time periods t1 and t2. To solve this nonlinear problem, 7 constraint equations are also required.

[0097] For finite thrust flight, given the flight time t1 before the distance pulse is applied, the two-body motion equations can be derived from the initial point [r] before the pulse trajectory change. T ,v +T ] T By inversely integrating and calculating time t1, the position r0 and velocity v0 of the finite thrust flight engine at the start-up point can be obtained.

[0098] At this point, the integral corresponds to the two-body motion equations in free flight:

[0099]

[0100] Similarly, for finite thrust flight, given the flight time t2 after the distance pulse is applied, the two-body motion equations can be derived from the initial point [r]. T ,v -T ] T By calculating time t2 using forward integration, the position r of the shut-off point of the finite thrust flight engine can be obtained. f Speed ​​v f .

[0101] The above equation gives the starting point of the flight engine (r0, velocity v0) and the starting point of finite thrust flight; the starting point of the engine (r) is given by the engine's starting position. f (Known quantity), velocity v at the shutdown point f (Known quantities) The target point for finite thrust flight is given, or the terminal constraint.

[0102] Dynamic equations of a two-body orbit with thrust

[0103]

[0104] and the given optimal thrust direction

[0105]

[0106] By integrating the power-on point position r0 and velocity v0 in the positive direction over time t1+t2, a virtual terminal state position can be obtained. speed

[0107] This forms a set of 6-dimensional nonlinear equations concerning the terminal state constraints:

[0108]

[0109]

[0110] For the optimal control problem of terminal freedom, the Hamiltonian function at the terminal time t f Equal to 0:

[0111]

[0112] Wherein, the subscript f represents the shutdown end time t. f m f The mass of the aircraft at the moment of shutdown; g(r) f ) is r f The gravity vector corresponding to the location.

[0113] The seven unknown parameters—the angles α and β related to vector A, the three components of vector B (b1, b2, and b3), the two time periods t1 and t2, and the seven-dimensional nonlinear constraints formed above—can form a set of nonlinear equations, which can be solved using a nonlinear equation-solving algorithm such as the Levenberg-Marquardt method.

[0114] The solution yields seven parameters: the angles α and β related to vector A, and the three components of vector B: b1, b2, and b3. After two time intervals t1 and t2, the following is obtained:

[0115] 1) Pulse orbit change time t c The corresponding engine start-up point t c -t1;

[0116] 2) Pulse orbit change time t c Corresponding engine shutdown point t c +t2;

[0117] 3) Engine start-up time corresponding to optimal trajectory change with finite thrust: t1+t2;

[0118] 4) Optimal thrust direction corresponding to optimal trajectory change with finite thrust:

[0119] 5) By integrating the dynamic equation (25), we can also obtain the trajectory of the optimal trajectory change with finite thrust and the corresponding position and velocity at each moment.

[0120] While the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the invention. Any person skilled in the art can make possible variations and modifications to the technical solutions of the present invention using the disclosed methods and techniques without departing from the spirit and scope of the invention. Therefore, any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention, without departing from the content of the technical solutions of the present invention, shall fall within the protection scope of the present invention. Where there is no conflict, the embodiments of this application and the technical features thereof can be combined with each other.

[0121] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method for optimal orbit change with finite thrust based on a bilinear tangent law for space orbits, characterized in that, include: Define the thrust direction u(t); Establish the dynamic equations for a two-body orbit with thrust; By integrating the dynamic equations of the two-body orbit containing thrust and the thrust direction u(t), the virtual terminal position can be obtained. speed Based on the virtual terminal status location speed Given terminal state constraints, the flight time t1 before the application point of a given distance pulse and the flight time t2 after the application point of a fixed distance pulse are obtained, and the pulse trajectory change time t is determined. c The corresponding engine start-up point and pulse track change time t c The corresponding engine shutdown point; thus obtaining the engine start-up and shutdown time under limited thrust conditions; Determine the trajectory corresponding to the orbit change, as well as the position and velocity at each moment.

2. The optimal orbit change method for space orbits with finite thrust according to claim 1, characterized in that, The direction of the thrust u(t) is as follows: A=[cosα,sinαcosβ,sinαsinβ] T B=[b1,b2,b3] T Where u(t) is a unit vector; α and β are the angular components of vector A; b1, b2, and b3 are the three-axis components of vector B in the rectangular coordinate system; and t is the flight time.

3. The optimal orbit change method for space orbits with finite thrust according to claim 2, characterized in that, The dynamic equations for a two-body orbit involving thrust are as follows: Where Γu(t) represents the spacecraft acceleration along the thrust direction vector u(t); g(r,t) is the gravitational acceleration at time t, and r is the position at time t.

4. A method for optimal orbit change with finite thrust based on a bilinear tangent law in space orbits, as described in any one of claims 1 to 3, characterized in that... The terminal state constraints are as follows: Furthermore, the Hamiltonian function H of the dynamic system obtained from optimal control theory at the final time t f Equals 0; Where, r f This is the position of the engine shutdown point, v f The speed at which the engine shuts off.

5. The optimal orbit change method for space orbits with finite thrust according to claim 4, characterized in that, Pulse orbit change time t c The corresponding engine start-up point is equal to t. c -t1.

6. The optimal orbit change method for space orbits with finite thrust according to claim 5, characterized in that, Pulse orbit change time t c The corresponding engine shutdown point is equal to t c +t2.

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