Energy substitution based aerodynamic assisted deorbiting fuel-optimal trajectory generation method

Abstract

The application discloses a kind of based on energy replacement's aerodynamic auxiliary deorbiting fuel consumption optimal trajectory high-efficiency generation method, belong to aerospace field, including establishing spacecraft aerodynamic auxiliary deorbiting flight dynamics model, extract out the motion equation in longitudinal plane, convert independent variable into energy form variable, obtain the differential equation of height and energy decoupling, parameterization aerodynamic acceleration variation;Design the segmented parameterization expression of path angle about energy, realize the high-precision fitting of optimal path angle variation;Based on fitting function, the expression of terminal energy is derived in different cases, and then the trajectory velocity is predicted;The corresponding speed size is calculated at different height points, and the fuel consumption optimal reference trajectory is efficiently generated, to ensure that the spacecraft is transferred to the target orbit with low fuel consumption.The application provides a kind of based on energy replacement's aerodynamic auxiliary deorbiting fuel consumption optimal trajectory high-efficiency generation method, whether in height drop segment or rising segment can realize the high-efficiency generation of subsequent fuel consumption optimal trajectory.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of aerospace technology, in particular to an energy substitution-based high-efficiency generation method for a fuel-optimal trajectory of aerodynamic-assisted deorbiting. BACKGROUND

[0002] The planetary atmospheric environment is complex and unpredictable, and when a spacecraft is passing through the atmosphere and consuming excess energy by aerodynamic force to reduce its own orbit, it needs to continuously predict and generate subsequent reference trajectories according to the measured state parameters, so as to correct the guidance control instructions according to the reference trajectory information, realize the adjustment of the spacecraft flight trajectory, and ensure that the spacecraft flies out of the atmosphere and transfers to the target orbit with low fuel consumption. At present, there are mainly two methods for generating reference trajectories: numerical method and analytical method. The numerical method is based on the reference control profile and recursively predicts the subsequent trajectory by numerically integrating the dynamic equation set, which has high accuracy but is highly dependent on the model accuracy and the computing power of the equipment, and is currently difficult to be applied online on the spacecraft. In comparison, the analytical method predicts and generates the subsequent trajectory by deriving the analytical expression of the trajectory, which has high computational efficiency and potential for real-time online application. However, the analytical method usually has poor adaptability, and the trajectory prediction accuracy is difficult to guarantee in the face of different task scenarios and parameter uncertainties.

[0003] In the existing high-efficiency trajectory generation method for aerodynamic-assisted deorbiting, the First Technique (see: Cihan I H, Kluever C A. Analytical Earth-Aerocapture Guidance with Near-Optimal Performance[J]. Journal of Guidance, Control, and Dynamics, 2020, 44(1): 45-56.) proposes an analytical method for reference fuel-optimal aerocapture trajectory characteristics, which solves the explicit expression of the atmospheric exit velocity by parameterizing the aerodynamic parameters and the flight angle-height profile. This method can be applied to the trajectory generation and prediction of the ascent phase in the fully aerodynamic deceleration scenario, but its derivation is based on a dynamic model with height as the independent variable, and the height is assumed to change monotonously, so it cannot be applied to the trajectory generation and prediction of the initial descent phase, and it does not have universality and adaptability to different task scenarios. SUMMARY

[0004] The purpose of the present application is to provide an energy substitution-based high-efficiency generation method for a fuel-optimal trajectory of aerodynamic-assisted deorbiting to solve the problems mentioned in the background.

[0005] To achieve the above-mentioned purpose, the present application provides an energy substitution-based high-efficiency generation method for a fuel-optimal trajectory of aerodynamic-assisted deorbiting, comprising the following steps:

[0006] S1, establish a spacecraft aerodynamic assisted de-orbit flight dynamics model considering the effects of celestial body rotation and oblateness, the spacecraft aerodynamic assisted de-orbit flight dynamics model is a three-degree-of-freedom dynamics equation, and a longitudinal plane motion equation is extracted from the three-degree-of-freedom dynamics equation by decoupling;

[0007] S2, make variable substitution, convert the independent variable from time to energy form, obtain a differential equation with height and energy decoupled, and parameterize the change of aerodynamic acceleration with a fitting function with height as the independent variable;

[0008] S3, based on the trajectory characteristics of the fuel-optimal aerodynamic assisted de-orbit, design a segmented parameterization expression of the flight path angle with respect to energy, and realize high-precision fitting of the optimal flight path angle change in different scenarios;

[0009] S4, based on the fitting functions of aerodynamic drag acceleration and flight path angle in S2 and S3, derive the expression of the terminal energy, when the fitting function is determined, the solving process of the terminal state integral expression is not affected by the non-monotonic change of height, and the trajectory speed is predicted by solving the subsequent trajectory energy in the de-orbiting and ascending segments of the aerodynamic assisted maneuver;

[0010] S5, based on the trajectory speed solving method obtained in S4, predict the corresponding speed at different height points along the trajectory, quickly generate an aerodynamic assisted de-orbit reference trajectory, and assist the spacecraft in maneuvering during flight, so that the spacecraft enters the target orbit with low fuel consumption under the premise of meeting the mission constraints.

[0011] Preferably, the content of S1 is as follows:

[0012] The three-degree-of-freedom dynamics equation of the spacecraft during aerodynamic assisted de-orbit flight is as formula (1):

[0013] (1)

[0014] wherein the variables are dimensionless quantities, represents the radial vector from the center of the celestial body to the center of mass of the spacecraft; represents the longitude; represents the latitude; represents the velocity of the spacecraft relative to the celestial body; represents the flight path angle of the relative velocity vector; represents the heading angle; represents the roll angle, which is the control angle during flight; and respectively represent the dimensionless aerodynamic lift acceleration and aerodynamic drag acceleration, as formula (2) and formula (3):

[0015] (2)

[0016] (3)

[0017] where, is the dimensionally scaled atmospheric density; is the dimensionally scaled spacecraft reference area; , are the lift and drag coefficients, respectively; is the dimensionally scaled spacecraft mass; is the celestial equatorial radius; and are the dimensionless radial and latitudinal components of the gravitational acceleration, respectively, as given by equations (4) and (5):

[0018] (4)

[0019] (5)

[0020] where, characterizes the dynamical oblateness of the celestial body; is the celestial body's angular rotation rate;

[0021] The spacecraft's longitudinal motion parameters, altitude , velocity , and flight path angle are extracted from the three-degree-of-freedom dynamic equations (1) to form a simplified longitudinal decoupled motion model:

[0022] (6)

[0023] (7)

[0024] (8).

[0025] Preferably, S2 content is as follows:

[0026] A new independent variable is introduced:

[0027] (9)

[0028] This variable characterizes the negative value of the orbital energy, and its time derivative, neglecting the effects of the celestial body's oblateness and rotation, is given by:

[0029] (10)

[0030] Since it is monotonically increasing during the atmospheric crossing, it is replaced by as the time variable, and the simplified longitudinal decoupled motion model is reduced to:

[0031] (11)

[0032] (12)

[0033] (13)

[0034] Substitute equation (3) and equation (13) into equation (11) to obtain:

[0035] (14)

[0036] During the whole process of aerodynamic-assisted orbit transfer, close to 1 but less than 1, and the change is small, so the constant is used to replace in equation (14), and the variable is separated to obtain:

[0037] (15)

[0038] The variable on the left side of equation (15) is related to the height, and a fitting function is used to parameterize :

[0039] (16)

[0040] wherein is an arbitrary form fitting function that can be explicitly integrated.

[0041] Preferably, the S3 content is as follows:

[0042] The fuel-optimal aerodynamic-assisted de-orbiting has a bang-bang structure of the tilt angle control profile; when the flight path angle is around 0, , according to the optimal - profile characteristics under different scenarios, a segmented expression about is designed to parameterize term:

[0043] (17)

[0044] wherein , , , , , , are the coefficients of the corresponding fitting functions, the values of which are determined by two-point boundary value information; correspond to the bang-bang switching points of the control profile; is the initial flight path angle of the spacecraft. Trajectory angle for aerodynamic assisted de-orbit terminal

[0045] Substitute equation (16) and equation (17) into equation (15) to obtain:

[0046] (18).

[0047] Preferably, S4 is as follows:

[0048] During flight, the spacecraft determines the initial state corresponding to the current time through the navigation measurement system; and substitutes equation (18) into equation (19) to obtain: Integrates to the terminal state When the integral interval does not cross the switching point , the integral expression is respectively:

[0049] (19)

[0050] Wherein, is an explicit integral expression; equation (19) is analyzed in different cases:

[0051] A1, when , equation (19) is rearranged to obtain:

[0052] (20)

[0053] When is known, equation (20) is a one-dimensional nonlinear equation about ;

[0054] A2, when and the trajectory angle decreases, the analytical expression of is obtained from equation (19), which can be quickly solved numerically:

[0055] (21)

[0056] Wherein,

[0057] (22)

[0058] A3, when and the trajectory angle increases, equation (19) is rearranged to obtain:

[0059] (23)

[0060] When is known, equation (23) is a one-dimensional nonlinear equation about , which can be quickly solved numerically;

[0061] A4、When the integral interval contains the switching point, segmented integration is performed:

[0062] (24)

[0063] wherein,

[0064] (25)

[0065] solved by referring to the flowcharts of A2 and A3 ; after obtaining , the terminal velocity is obtained from formula (13).

[0066] Preferably, the content of S5 is as follows:

[0067] From formula (20), formula (21), formula (23) and formula (24), it is known that in the process of aerodynamic auxiliary de-orbiting, under the condition of the current state of the spacecraft and the fitting function, the velocity of each point on the subsequent trajectory is a function of the altitude at the point, denoted as , and a set of altitudes is formed by taking discrete points on the subsequent trajectory:

[0068] (26)

[0069] wherein, is the altitude corresponding to the atmospheric boundary; for each altitude in the set , the velocity at the corresponding altitude is calculated by to form a velocity set:

[0070] (27)

[0071] According to the set , an open-loop reference trajectory for aerodynamic auxiliary de-orbiting is quickly generated.

[0072] Therefore, the present application adopts the above-mentioned high-efficiency generation method of the fuel consumption optimal trajectory for aerodynamic auxiliary de-orbiting based on energy substitution, and has the following beneficial effects:

[0073] (1) By energy substitution and parameter fitting based on the characteristics of the fuel consumption optimal trajectory, the expressions of the trajectory energy and the velocity are derived, the high-efficiency generation of the fuel consumption optimal trajectory is realized, the requirement for the equipment computing power is lower, and the real-time performance of the online trajectory generation is ensured;

[0074] (2) By converting the kinetic independent variable into a full-range monotonically increasing energy form variable, it is ensured that the spacecraft can efficiently generate subsequent trajectories in both the descent or ascent phase, expanding the application interval of the method;

[0075] (3) There is no strict requirement for the specific form of the aerodynamic parameter fitting function, and the fitting function can be updated according to the measured aerodynamic data during the flight of the spacecraft, ensuring that the real-time changes of the aerodynamic parameters can be reflected, reducing the dependence on the aerodynamic model, and improving the flexibility of the method;

[0076] (4) The characteristics of different fuel-optimal trajectories are considered, which can adapt to significant changes in atmospheric environment, entry trajectory angle and target orbit parameters, and thus is suitable for various aerodynamic-assisted deorbiting task scenarios.

[0077] The technical solutions of the present application will be further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF DRAWINGS

[0078] Figure 1 A flow chart of an energy substitution-based aerodynamic-assisted deorbit fuel-optimal trajectory efficient generation method according to an embodiment of the present application;

[0079] Figure 2 Trajectory generated by the present embodiment in the Earth aerodynamic-assisted deorbit scenario (the starting point is located in the descent segment);

[0080] Figure 3 Trajectory generated by the present embodiment in the Mars aerodynamic-assisted deorbit scenario (the starting point is located in the descent segment);

[0081] Figure 4 Trajectory generation comparison in the Earth aerodynamic-assisted deorbit scenario (the starting point is located in the ascent segment);

[0082] Figure 5 Trajectory generation comparison in the Mars aerodynamic-assisted deorbit scenario (the starting point is located in the ascent segment). DETAILED DESCRIPTION

[0083] The following detailed description of the embodiments of the present application provided in the accompanying drawings is not intended to limit the scope of the claimed application, but only represents selected embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative labor are within the scope of protection of the present application.

[0084] Two aerodynamic-assisted deorbit scenarios are considered, and the entry conditions are shown in the following table:

[0085] Table 1 Inertial entry conditions under two scenarios

[0086]

[0087] In the Earth scenario, the target orbit is a circular orbit with an altitude of 200×200 km; in the Mars scenario, the target orbit is an elliptical orbit with an altitude of 3000×300 km. The parameter model of the spacecraft is taken as an example of the Orion Multi-Purpose Crew Vehicle. Referring to Figure 1 , the specific implementation steps of the high-efficiency generation method of the aerodynamic-assisted deorbiting fuel consumption optimal trajectory based on energy substitution are as follows:

[0088] S1, an aerodynamic-assisted deorbiting flight dynamics model of a spacecraft considering the effects of celestial body rotation and oblateness is established, the aerodynamic-assisted deorbiting flight dynamics model of the spacecraft is a three-degree-of-freedom dynamics equation, and a motion equation in a longitudinal plane is extracted from the three-degree-of-freedom dynamics equation.

[0089] The three-degree-of-freedom dynamics equation of the spacecraft in the process of aerodynamic-assisted deorbiting flight is,

[0090] (1)

[0091] wherein the variables are all dimensionless quantities, represents a vector from the center of the celestial body to the center of mass of the spacecraft; represents longitude; represents latitude; represents the speed of the spacecraft relative to the celestial body; represents the track angle of the relative velocity vector of the celestial body; represents the heading angle; represents the roll angle, which is a control angle in the process of flight; and respectively represent the dimensionless aerodynamic lift acceleration and the aerodynamic drag acceleration,

[0092] (2)

[0093] (3)

[0094] wherein, is the atmospheric density with dimensions, is the reference area of the spacecraft with dimensions, , are respectively the lift coefficient and the drag coefficient, is the mass of the spacecraft with dimensions, is the equatorial radius of the celestial body; and are respectively the radial and latitudinal components of the dimensionless gravitational acceleration,

[0095] (4)

[0096] (5)

[0097] where, characterize the celestial body's flattening; is the celestial body's rotation angular velocity. For the convenience of subsequent derivation, the spacecraft's longitudinal motion parameters, i.e., altitude (corresponding to the radial ), velocity and flight path angle are extracted from the three-degree-of-freedom dynamics equation (1) to form a simplified longitudinal decoupled motion model:

[0098] (6)

[0099] (7)

[0100] (8).

[0101] S2, the independent variable is converted from time to energy form, and the differential equation of altitude and energy decoupling is obtained, and the fitting function with altitude as the independent variable is used to parameterize the change of aerodynamic acceleration.

[0102] A new independent variable is introduced:

[0103] (9)

[0104] This variable represents the negative value of the orbital energy, and the influence of the celestial body's flattening and rotation is ignored. Its derivative with respect to time is obtained:

[0105] (10)

[0106] That is During the process of penetrating the atmosphere, it is monotonically increasing, so is used instead of the time variable, and the simplified longitudinal decoupled motion model is:

[0107] (11)

[0108] (12)

[0109] (13)

[0110] Substitute equation (3) and equation (13) into equation (11) to obtain:

[0111] (14)

[0112] Considering that the aerodynamic auxiliary orbit transfer is a whole process, Close to 1 (but less than 1) and varies little, so it can be treated as a constant Instead of the above equation , and separate variables to get:

[0113] (15)

[0114] The variable on the left side of the above equation is mainly related to the height, which can be parameterized by a fitting function : :

[0115] (16)

[0116] where, is an arbitrary form fitting function that can be explicitly integrated, which can be a six-order polynomial function in this embodiment: .

[0117] S3, based on the trajectory characteristics of the optimal fuel consumption aerodynamic assisted descent, the segmented parameterization expression of the flight path angle about energy is designed, and the high-precision fitting of the optimal flight path angle change is realized in different scenarios.

[0118] The fuel consumption optimal aerodynamic assisted descent has a bang-bang structure of the tilt angle control profile in theory. In this embodiment, the flight path angle shows a decreasing trend after the switching point in the earth scenario, while the flight path angle shows an increasing trend after the switching point in the Mars scenario. Considering that the flight path angle is near 0, , according to the optimal - profile characteristics in different scenarios, the segmented expression about is designed to parameterize term:

[0119] (17)

[0120] where, , , , , , , is the coefficient of the corresponding fitting function, the value of which can be determined by two-point boundary information; is the bang-bang switching point of the control profile; is the initial flight path angle of the spacecraft; is the flight path angle of the terminal aerodynamic assisted descent.

[0121] Substitute equation (16), equation (17) into equation (15) to get:

[0122] (18).

[0123] S4. Based on the fitting functions of aerodynamic drag acceleration and track angle in S2 and S3, the expression for terminal energy is derived for each case, thereby predicting the trajectory velocity.

[0124] During flight, the spacecraft can determine its initial state at the current moment through its navigation and measurement system. Relating both sides of the above equation from the initial state... Points to terminal status When the integral interval does not cross the switching point When, the integral expressions are obtained as follows:

[0125] (19)

[0126] in, This is an explicit integral expression. We will analyze the above expression in several cases.

[0127] A1, when Then, rearranging the above formula, we get:

[0128] (20)

[0129] when When known, the above formula is about A univariate nonlinear equation, It can be solved quickly numerically.

[0130] A2, when Furthermore, when the track angle decreases, it can be obtained from equation (19). The parsing expression:

[0131] (twenty one)

[0132] in,

[0133] (twenty two)

[0134] A3, when Furthermore, as the track angle increases, we can obtain the result from equation (19).

[0135] (twenty three)

[0136] when When known, the above formula is about A univariate nonlinear equation, It can be solved quickly numerically.

[0137] A4. When the integration interval contains the switching point, i.e. When doing this, piecewise integration is required:

[0138] (twenty four)

[0139] where,

[0140] (25)

[0141] Thus, the solution can be obtained by referring to the flow of A2 and A3 .

[0142] After obtaining , the terminal velocity can be obtained by formula (13). It can be seen that after the fitting function is determined, the solution process of the above integral is not affected by the height of the non-monotonic change, so whether in the descent phase or the ascent phase of the aerodynamic auxiliary maneuver, the subsequent trajectory velocity can be efficiently solved.

[0143] S5, based on the trajectory velocity efficient solving method obtained in S4, the corresponding velocity of different height points along the trajectory is predicted, and the aerodynamic auxiliary deorbit reference trajectory is quickly generated to assist the spacecraft in maneuvering during flight. Under the premise of meeting the mission constraints, the spacecraft can accurately enter the target orbit with small fuel consumption.

[0144] In the process of aerodynamic auxiliary deorbiting, under the condition of the current state of the spacecraft and the given fitting function, the velocity of each point on the subsequent trajectory will be a function of the height at that point, denoted as . Take discrete points on the subsequent trajectory to form a height set:

[0145] (26)

[0146] where, is the height corresponding to the atmospheric boundary. For each height in the set , the velocity at the corresponding height can be calculated by to form a velocity set:

[0147] (27)

[0148] According to the set , , the open-loop reference trajectory for aerodynamic auxiliary deorbiting can be quickly generated. This reference trajectory can be used in the guidance and control link to assist the spacecraft in maneuvering. Under the premise of meeting the mission constraints, the spacecraft can accurately enter the target orbit with small fuel consumption.

[0149] In this embodiment, the initial state of the spacecraft is first set in the height descent segment, and the subsequent trajectory is generated efficiently, and is compared with the theoretical fuel consumption optimal trajectory obtained by numerical integration of the dynamic equation set (1) based on the bang-bang control structure, and the results in different scenarios are as shown in Figure 2 and Figure 3 It can be seen that in the two scenarios, the generated trajectory of this embodiment can closely approach the theoretical optimal trajectory, indicating that the trajectory generated quickly from the descent segment can maintain a small error with the theoretical optimal trajectory.

[0150] Further, the initial state of the spacecraft is set in the height rise segment, and the subsequent trajectory is generated quickly, and is compared with the theoretical optimal trajectory and the trajectory generated by the prior art method in the rise segment, as shown in Figure 4 and Figure 5 It can be seen that in the two scenarios, compared with the prior art method, the trajectory generated by this embodiment is significantly closer to the theoretical optimal trajectory, indicating that this embodiment can efficiently generate a trajectory close to the theoretical fuel consumption optimal trajectory in significantly different task scenarios, reflecting the flexibility and adaptability of the method, and can lay a solid foundation for the generation of subsequent spacecraft guidance control instructions.

[0151] Therefore, the present application adopts the above-mentioned aerodynamic auxiliary de-orbiting fuel consumption optimal trajectory efficient generation method based on energy substitution, and constructs and derives the related expression of the flight path angle, height and speed in the aerodynamic auxiliary de-orbiting process through variable substitution and optimal profile approximation, which can efficiently generate the subsequent fuel consumption optimal trajectory in the height descent segment or the height rise segment of the spacecraft.

[0152] Finally, it should be noted that: the above embodiments are only used to illustrate the technical solutions of the present application but not to limit it, although the present application has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that: it can still modify or equivalently replace the technical solutions of the present application, and these modifications or equivalent replacements also cannot make the modified technical solutions deviate from the spirit and scope of the technical solutions of the present application.

Claims

1. An energy substitution-based aerodynamic-assisted deorbiting fuel-optimal trajectory efficient generation method, characterized by: S1, establishing a spacecraft aerodynamic-assisted deorbiting flight dynamics model considering the effects of celestial body rotation and oblateness, the spacecraft aerodynamic-assisted deorbiting flight dynamics model being a three-degree-of-freedom dynamics equation, and a longitudinal plane motion equation being extracted from the three-degree-of-freedom dynamics equation by decoupling; S2, performing variable substitution, converting independent variables from time to energy form, obtaining a differential equation with height and energy decoupled, and parameterizing the change of aerodynamic acceleration with a fitting function with height as the independent variable; S3, based on the trajectory characteristics of fuel-optimal aerodynamic-assisted deorbiting, designing a segmented parameterized expression of flight path angle with respect to energy, and realizing high-precision fitting of optimal flight path angle change in different scenarios; S4, based on the fitting functions of aerodynamic drag acceleration and flight path angle in S2 and S3, deriving the expression of terminal energy, when the fitting function is determined, the solving process of the terminal state integral expression is not affected by the non-monotonic change of height, and the trajectory velocity is predicted by solving the subsequent trajectory in the aerodynamic-assisted maneuvering descent and ascent segments; S5, based on the trajectory velocity solving method obtained in S4, predicting the corresponding velocity magnitude at different height points along the trajectory, quickly generating an aerodynamic-assisted deorbiting reference trajectory, assisting the spacecraft in maneuvering during flight, and enabling the spacecraft to enter the target orbit with low fuel consumption under the premise of meeting mission constraints.

2. The energy swap based aerodynamic assistance optimal deorbit-burn trajectory generation method according to claim 1, wherein, The content of S1 is as follows: The three-degree-of-freedom dynamics equation of the spacecraft during aerodynamic-assisted deorbiting flight is as formula (1): (1) where variables are dimensionless quantities, denotes the radial vector from the center of the celestial body to the center of mass of the spacecraft; denotes the longitude; denotes the latitude; denotes the velocity of the spacecraft relative to the celestial body; is the angular velocity of the rotation of the celestial body; denotes the track angle of the velocity vector relative to the celestial body; denotes the heading angle; denotes the bank angle as a control angle during flight; and denote the dimensionless aerodynamic lift acceleration and the aerodynamic drag acceleration, respectively, as in equations (2) and (3): (2) (3) where is the dimensioned atmospheric density; is the dimensioned spacecraft reference area; , are the lift and drag coefficients, respectively; is the dimensioned spacecraft mass; is the celestial equatorial radius; and are the dimensionless radial and latitudinal components of the gravitational acceleration, as in equations (4) and (5): (4) (5) wherein, characterizing the celestial body dynamics oblateness; The spacecraft longitudinal motion parameters, altitude , velocity and flight path angle are extracted from the three-degree-of-freedom dynamics equation (1) to form a simplified longitudinal decoupled motion model: (6) (7) (8)。 3. The method of claim 2, wherein, The content of S2 is as follows: A new independent variable is introduced: (9) This variable represents the negative value of the orbital energy, and the effects of celestial body oblateness and rotation are ignored, and the derivative with respect to time is obtained as: (10) In the course of crossing the atmosphere is monotonically increasing, then instead of the time variable, the simplified longitudinal decoupled motion model is modeled as: ​ (11) (12) (13) Substitute formula (3) and formula (13) into formula (11) to obtain: (14) During the whole process of aerodynamic-assisted orbit transfer, close to 1 but less than 1, and varies little, so it is treated as a constant instead of (14) in the equation and separate the variables to get: (15) The variables on the left side of equation (15) are related to the height, parameterized using a fitting function :​ (16) wherein, is an arbitrary form fitting function that can be explicitly integrated.

4. The method of claim 3, wherein, The content of S3 is as follows: The optimal fuel consumption aerodynamic assisted de-orbiting has a bang-bang structure of the bank angle control profile; when the flight path angle is near 0, , the optimal - profile characteristics under different scenarios, design about piecewise expression to parameterize terms: (17) wherein , , , , , , are coefficients of the respective fitting functions, whose values are determined by the two-point boundary information; are bang-bang switching points of the corresponding control profile; is the initial flight path angle of the spacecraft; is the flight path angle of the aerodynamic assisted de-orbit terminal; Substitute formula (16) and formula (17) into formula (15) to obtain: (18)。 5. The method of claim 4, wherein, The content of S4 is as follows: During the flight, the spacecraft determines the initial state corresponding to the current time through the navigation measurement system; and integrates both sides of formula (18) from the initial state to the terminal state When the integration interval does not span the switching point , the integral expression is obtained as follows: (19) wherein is the explicit integration expression; the analysis of equation (19) is done on a case-by-case basis: A1, when When A1, the arrangement (19) is obtained: (20) When is known, formula (20) is a monadic nonlinear equation in one unknown, which is quickly solved by numerical methods; A2, when and the track angle is decreasing, the analytical expression for is given by equation (19): (21) Where, (22) A3, when and the track angle increases, we have from (19) (23) When is known, formula (23) is a univariate nonlinear equation in x, which is quickly solved numerically; A4. When the integration interval contains a switching point, a segmented integration is performed: (24) Where, (25) Refer to the flow of A2 and A3 for solution ; after obtaining , the terminal velocity is obtained from equation (13).

6. The method of claim 5, wherein, The content of S5 is as follows: From equation (20), equation (21), equation (23) and equation (24), it is known that in the aerodynamic assisted de-orbiting process, the velocity of each point on the subsequent trajectory is a function of the height at the point, denoted as , given the current state of the spacecraft and the fitting function, that is, , and the height set is composed of discrete points on the subsequent trajectory.​ (26) wherein, is the height corresponding to the atmospheric boundary; for the set of each height is calculated from the velocity at the corresponding height, constituting the set of velocities: (27) According to the set , Fast generation of open-loop reference trajectories for aerodynamic assisted de-orbit.

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