A spacecraft orbit transfer method based on back-dragging trajectory

Abstract

The application provides a spacecraft orbit transfer method based on a back-and-forth trajectory, and belongs to the field of spacecraft orbit transfer, and solves the problems of difficult conversion of constraint conditions and large amount of calculation during spacecraft orbit transfer; a trajectory in which a reference spacecraft retreats relative to a target spacecraft during forward movement and then turns to forward movement is taken as a back-and-forth trajectory, and geometric parameters corresponding to the back-and-forth trajectory and orbit transfer in a VVLH coordinate system are obtained; a relationship model of geometric parameters and impulse impulse under the same and different orbit altitudes is respectively constructed based on a CW relative motion equation in the VVLH coordinate system; target constraints in a task requirement are converted into conditional constraints of geometric parameters, and impulse impulse required for orbit transfer of the reference spacecraft is obtained by using the relationship model; the application can convert constraint conditions, reduces the amount of calculation, and improves the efficiency of spacecraft orbit transfer.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft orbit transfer technology, and particularly relates to a spacecraft orbit transfer method based on a loop trajectory. Background Technology

[0002] Spacecraft orbit transfer is a core technology in space missions, involving path design and energy optimization from the initial orbit to the target orbit. Its applications encompass satellite deployment, deep space exploration, and on-orbit servicing. The design methodology must comprehensively consider fuel efficiency, mission requirements, propulsion system characteristics, and celestial mechanics. Among these, the velocity increment application scheme is a core indicator in orbit transfer design. Currently, velocity increment application primarily relies on pulse thrust, specifically the magnitude, direction, and timing of the applied pulse impulse.

[0003] Current design methods for pulse thrust trajectory transfer include: Hohmann transfer and Lambert trajectory transfer based on inertial coordinate systems, as well as transfer design based on numerical iterative solution of motion equations in relative coordinate systems;

[0004] The Hohmann transfer solves the problem of the most fuel-efficient orbital transfer method for spacecraft moving from one circular orbital altitude to another by applying pulses along the velocity direction. However, the Hohmann transfer is more suitable for long-distance transfers from Low Earth Orbit (LEO) to Geosynchronous Orbit (GEO). Due to the harsh application conditions, this design has a limited range of applications and does not have high requirements for transfer accuracy.

[0005] Lambert orbit transfer is a common method for boundary value problems involving spacecraft orbit transfer between two points. This method solves the problem of orbit transfer where a spacecraft moves from one position to another at a given time. However, Lambert orbit transfer is designed for transfer between two fixed positions under inertia. When the position of the target changes over time, such as for orbit transfer between two moving spacecraft, it faces problems such as difficulty in changing constraints and the huge computational cost of continuous iterative calculations.

[0006] The transfer design based on the numerical iterative solution method of the motion equations in the relative coordinate system solves the orbit transfer problem of a servicing spacecraft under specific constraints for another moving spacecraft. For example, application number: 2025100320347, publication number: CN119659984A, invention title: A method for constructing and screening the orbit change velocity increment required for on-orbit servicing of a pulse thrust spacecraft based on the CW equation. This invention constructs the set of orbit change velocity increments required for on-orbit servicing of a spacecraft by solving the CW equation in the relative motion coordinate system, which solves the problem of low accuracy of orbit maneuvering strategy when a pulse thrust spacecraft performs on-orbit servicing for another moving space target. However, its solution still belongs to the form of converting mission constraints into kinematic equation constraints. Its solution process still requires a lot of numerical iterative calculations and does not involve the solution of problems under the constraints of orbital motion of the servicing spacecraft. Summary of the Invention

[0007] For orbit transfer problems with the constraint that the reference spacecraft orbits the target spacecraft, the application conditions of the Hohmann transfer method are no longer met. When using the Lambert orbit transfer method and the numerical iterative solution method of the equations of motion in relative coordinate systems, the transfer design involves problems such as difficulty in converting constraints and large computational load. This invention proposes a spacecraft orbit transfer method based on a loop trajectory. This method uses the geometric relationship of the loop trajectory between the two spacecraft to resolve mission constraints for orbit transfer. It directly transforms the constraints in the mission into the condition constraints of the loop trajectory geometric parameters, reduces the amount of computation, realizes the conversion of constraints, improves the efficiency of the reference spacecraft orbit transfer, and makes mission planning more intuitive and efficient.

[0008] To achieve the above objectives, the present invention is specifically implemented through the following technical solutions:

[0009] A spacecraft orbit transfer method based on a loop trajectory includes:

[0010] Step 1: The trajectory of the reference spacecraft that reverses and then resumes its forward motion relative to the target spacecraft is taken as the loop trajectory, and the geometric parameters corresponding to the loop trajectory and the orbital transfer of the reference spacecraft in the VVLH coordinate system are obtained.

[0011] Step 2: In the VVLH coordinate system, construct the first CW relative motion equation under the same orbital height. Based on the first CW relative motion equation, construct the first relationship model between the geometric parameters and the pulse impulse under the same orbital height.

[0012] Step 3: In the VVLH coordinate system, construct the second CW relative motion equation for different orbital heights. Based on the second CW relative motion equation, construct the second relationship model between the geometric parameters and the pulse impulse for different orbital heights.

[0013] Step four: Convert the target constraints in the mission requirements into conditional constraints of the geometric parameters. Based on the orbital altitude in the mission requirements, use the corresponding first or second relational model to obtain the pulse impulse required for the orbital transfer of the reference spacecraft.

[0014] The beneficial effects of this invention are:

[0015] 1. This invention proposes the concept of a loop trajectory and defines the geometric parameters corresponding to the loop trajectory and the orbital transfer of the reference spacecraft in the VVLH coordinate system.

[0016] 2. The CW relative motion equations are used to describe the relative motion between two spacecraft. They are derived based on the "two-body" motion model and provide a high-precision analytical solution for the relative motion. The calculation method is fast and convenient. Since the CW relative motion equations do not consider the influence of perturbation factors on the trajectory, using the CW relative motion equations to calculate the relative trajectory can eliminate the potential influence of perturbation factors on the loop trajectory. This allows for a more objective study of the impact of pulses applied by the spacecraft on the loop trajectory. Therefore, this invention calculates the CW relative motion equations in the VVLH coordinate system for the loop phenomenon and constructs the first CW relative motion equations for the same orbital altitude and the second CW relative motion equations for different orbital altitudes.

[0017] 3. The reference spacecraft will produce different loop trajectories under different applied pulse impulses. Therefore, based on the first CW relative motion equation, a first relationship model between geometric parameters and pulse impulse is constructed for the same orbital altitude. Based on the second CW relative motion equation, a second relationship model between geometric parameters and pulse impulse is constructed for different orbital altitudes. The construction process adopts a step-by-step approach, starting with the specific and progressing to the general, and from simple to complex. First, according to the simple special case of two spacecraft at the same orbital altitude, the relationship equation between the applied pulse impulse of the reference spacecraft and the geometric parameters of the loop trajectory is studied, resulting in the first relationship model. Second, according to the complex case of two spacecraft at different orbital altitudes, the relationship equation between the applied pulse impulse of the reference spacecraft and the geometric parameters of the loop trajectory is further constructed, resulting in the second relationship model. The obtained first and second relationship models make the orbit transfer problem solution process clear, intuitive, and efficient, and the results reliable and easy to implement.

[0018] 4. This invention transforms the mission requirements in actual orbit design into geometric parameter constraints, thereby obtaining the pulse impulse applied during orbit transfer. This avoids the complex constraint transformation and numerous repetitive calculation steps required in the traditional Lambert orbit transfer method and numerical iterative solution method of motion equations in relative coordinate systems. The technical solution disclosed in this invention reduces the amount of computation, realizes constraint transformation, improves the efficiency of spacecraft orbit transfer, and makes mission planning more intuitive and efficient. Attached Figure Description

[0019] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0020] Figure 1 This is a schematic diagram of the loop trajectory provided by the present invention.

[0021] Figure 2 This is one of the geometric parameter indication diagrams of the loop trajectory provided by the present invention.

[0022] Figure 3 This is the second of the geometric parameter indication diagrams for the loop trajectory provided by this invention.

[0023] Figure 4 This is a schematic diagram showing the association between trajectory points in the VVLH coordinate system and the geocentric inertial coordinate system provided by the present invention.

[0024] Figure 5 This is a schematic diagram showing the change in distance between two spacecraft at the same orbital altitude over time, provided by the present invention.

[0025] Figure 6 This is a schematic diagram of the trajectory of a reference spacecraft in the X-axis and Z-axis planes of the VVLH coordinate system at the same orbital altitude, as provided by this invention.

[0026] Figure 7 This is a schematic diagram showing the relationship between the geometric parameters of the loop trajectory and the pulse impulse at the same track height, as provided by the present invention.

[0027] Figure 8 This is a schematic diagram showing the change of distance between two spacecraft over time at different orbital altitudes, provided by the present invention.

[0028] Figure 9 This is a schematic diagram of the trajectory of a reference spacecraft in the X-axis and Z-axis planes of the VVLH coordinate system at different orbital altitudes, provided by this invention.

[0029] Figure 10 This is a schematic diagram showing the relationship between the geometric parameters of the loop trajectory and the pulse impulse at different track heights provided by the present invention.

[0030] Figure 11 This is a task planning sketch provided by the present invention.

[0031] Figure 12 This is a schematic diagram of the orbital transfer process of spacecraft B, which is provided by the present invention. Detailed Implementation

[0032] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0033] This invention provides a spacecraft orbit transfer method based on a loop trajectory, comprising:

[0034] In the complex and ever-changing trajectories of the relative motion of the two spacecraft, such as Figure 1 As shown, in step one, the trajectory of the reference spacecraft that moves backward relative to the target spacecraft during its forward movement and then turns back to forward is taken as the loop trajectory, and the geometric parameters corresponding to the loop trajectory and the orbital transfer of the reference spacecraft in the VVLH coordinate system are obtained.

[0035] Step 2: In the VVLH coordinate system, construct the first CW relative motion equation under the same orbital height. Based on the first CW relative motion equation, construct the first relationship model between the geometric parameters and the pulse impulse under the same orbital height.

[0036] Step 3: In the VVLH coordinate system, construct the second CW relative motion equation for different orbital heights. Based on the second CW relative motion equation, construct the second relationship model between the geometric parameters and the pulse impulse for different orbital heights.

[0037] Step four: Convert the target constraints in the mission requirements into conditional constraints of the geometric parameters. Based on the orbital altitude in the mission requirements, use the corresponding first or second relational model to obtain the pulse impulse required for the orbital transfer of the reference spacecraft.

[0038] In the orbital design of servicing spacecraft, the mission requirements specify the timeframe and proximity at which the servicing spacecraft (reference spacecraft) will observe the servicing spacecraft (target spacecraft) during which it will be served. When designing the orbit using a loop trajectory, it is only necessary to transform the target constraints in the mission requirements, such as distance and time constraints, into conditional constraints of the loop trajectory's geometric parameters. Then, based on the orbital altitude in the mission requirements, the required pulse impulse for this mission plan can be determined using the relationship between the geometric parameters and the pulse impulse in the corresponding first or second relational model.

[0039] In step one, the geometric parameters corresponding to the orbital transfer of the loop trajectory and the reference spacecraft in the VVLH coordinate system are: loop top, loop bottom, loop endpoint, loop center, loop height, loop width, loop duration, and loop spacing; among which,

[0040] The top of the loop is the intersection point of the relative motion trajectory of the reference spacecraft in the VVLH coordinate system; the intersection point is formed when the relative motion of the reference spacecraft involves both retreating and advancing during its forward motion; it is worth noting that the top of the loop is not necessarily located above the loop trajectory.

[0041] The bottom of the loop is the maximum or minimum point of the Z-axis of the relative motion trajectory of the reference spacecraft in the VVLH coordinate system. The bottom of the loop has the same X-axis coordinate as the top of the loop. At the bottom of the loop, the tangent of the relative motion trajectory of the reference spacecraft is parallel to the X-axis of the VVLH coordinate system, and the Z-axis component of the relative motion velocity is 0. It is worth noting that the bottom of the loop is not necessarily located below the loop trajectory.

[0042] The loop endpoint is the point where the relative velocity of the reference spacecraft in the VVLH coordinate system changes from positive to negative or from negative to positive on the X-axis component. The loop endpoint includes the loop front end and the loop back end. The loop front end is located in front of the direction of relative motion trajectory, and the loop back end is located behind the direction of relative motion trajectory. At the loop endpoint, the tangent of the relative motion trajectory of the reference spacecraft is perpendicular to the X-axis of the VVLH coordinate system, and the X-axis component of the relative velocity is 0.

[0043] The center of the loop is the intersection of the line connecting the top and bottom of the loop and the line connecting the front and back of the loop.

[0044] The return height is the distance from the top of the return pocket to the bottom of the return pocket along the Z-axis.

[0045] The width of the loop is the distance from the front end to the rear end of the loop along the X-axis.

[0046] The loop duration is the length of time that a spacecraft takes from the moment it enters the top of the loop, through the front end, rear end, and bottom of the loop, until it moves back to the top of the loop again.

[0047] The loop spacing is the distance between the centers of two adjacent loops.

[0048] Figure 2 and Figure 3 This is a diagram indicating the geometric parameters of the loop trajectory. Figure 2 and Figure 3 In this context, geometric parameters are simply referred to as top, bottom, endpoint, center, height, width, duration, and gap, where endpoints include the front and back ends.

[0049] Correlating the trajectories of the reference and target spacecraft in the VVLH coordinate system with their trajectories in the geocentric inertial coordinate system (GIS), it was found that in the GIS, the target spacecraft is a circle without eccentricity, while the reference spacecraft is an ellipse with eccentricity. The orbital periods of the two spacecraft deviate slightly. The correlation of the state points in the two coordinate systems is as follows: Figure 4 As shown.

[0050] Figure 4 middle, Point 1 is the initial point of the reference spacecraft, located at the maximum value of the Z-axis in VVLH coordinates, and at its perigee in the Earth's inertial frame. Point ① is located at 0 on the Z-axis in VVLH coordinates, and at the point where it intersects the circular trajectory of the target spacecraft in the Earth's inertial frame. Point ② is located at the minimum value of the Z-axis in VVLH coordinates, and at its apogee in the Earth's inertial frame. Point ③ is located at 0 on the Z-axis in VVLH coordinates, and at another point where it intersects the circular trajectory of the target spacecraft in the Earth's inertial frame. Point ④ is located at the maximum value of the Z-axis in VVLH coordinates, and at its perigee in the Earth's inertial frame. Since the trajectory is cyclical in the Earth's inertial frame, it coincides with point ① in the Earth's inertial frame. Point ⑤ is located at 0 on the Z-axis in VVLH coordinates, and it coincides with point ① in the Earth's inertial frame.

[0051] Depend on Figure 4 Analysis shows that the bottom of the loop is closest to or farthest from Earth in the VVLH coordinate system, thus representing the perigee or apogee of the reference spacecraft in the Earth's inertial frame. In the Earth's inertial frame, the two spacecraft rotate cyclically along their respective trajectories. Within one orbital period, the distance between them increases over time. However, due to deviations in eccentricity and semi-major axis, there is a short period where the distance between the two spacecraft decreases and then increases again. This process is reflected in the VVLH coordinate system as the loop phenomenon. Therefore, step one of this invention defines the geometric parameters corresponding to the loop trajectory and the orbital transfer of the reference spacecraft in the VVLH coordinate system.

[0052] In step two, the cases with the same orbital altitude are as follows: the target spacecraft's orbit is a circular orbit; the reference spacecraft and the target spacecraft have the same orbit but do not overlap in position; initially, the reference spacecraft remains relatively stationary relative to the target spacecraft, and the reference spacecraft achieves a loop trajectory by applying a pulse impulse along the velocity direction.

[0053] In the VVLH coordinate system, the first CW relative motion equation under the same orbital altitude is constructed by combining the CW relative motion equation in the VVLH coordinate system with the initial conditions. The CW relative motion equation in the VVLH coordinate system is:

[0054]

[0055] In the formula, x(t) and z(t) are the positions of the reference spacecraft on the x-axis and z-axis at time t, respectively, and t is the time variable. These are the velocities of the reference spacecraft along the x-axis and z-axis at time t, respectively.

[0056] x0 and z0 are the positions of the reference spacecraft on the x-axis and z-axis at the initial moment. ω represents the initial velocity of the reference spacecraft along the x and z axes, and w represents the angular velocity of the target spacecraft.

[0057] The initial conditions for the same orbital altitude are:

[0058]

[0059] The first CW relative motion equation is:

[0060]

[0061] In the formula, x(t1) represents the position of the reference spacecraft on the x-axis at time t1 under the same orbital altitude, and t1 represents the time variable under the same orbital altitude. t01 is the initial position of the reference spacecraft at the same orbital altitude, Δv1 is the pulse impulse applied by the reference spacecraft at the same orbital altitude, w is the angular velocity of the target spacecraft, and z(t1) is the position of the reference spacecraft on the z-axis at time t1 at the same orbital altitude. The velocity of the reference spacecraft on the x-axis at time t1 under the same orbital altitude; The velocity of the reference spacecraft on the z-axis at time t1 under the same orbital altitude.

[0062] In step two, the method for constructing the first relationship model between the geometric parameters and the pulse impulse under the same orbital height, based on the first CW relative motion equation, is as follows:

[0063] The initial state parameters of the reference spacecraft at the same orbital altitude are shown in Table 1:

[0064] Table 1

[0065]

[0066] When the loop trajectory is at the endpoint, the velocity component in the X-axis direction is 0. Therefore, in the first CW relative motion equation, let x(t1) = 0. From the first CW relative motion equation, we obtain the first equation for the front end of the loop and the first equation for the rear end of the loop, thus obtaining the first equation for the width of the loop. By combining the first equation for the width of the loop with the first CW relative motion equation, we obtain the first equation relating the width of the loop to the pulse impulse.

[0067] When the loop trajectory reaches the bottom, the velocity component in the Z-axis direction is 0, and the coordinate values ​​in the X-axis direction are the same at both the bottom and top moments. The first equation for the top and bottom of the loop is obtained from the first CW relative motion equation, thus obtaining the first equation for the loop height. The first equation for the loop height is combined with the first CW relative motion equation to obtain the first equation relating the loop height to the pulse impulse.

[0068] The loop duration equation is obtained by defining the geometric parameters of the loop duration. The loop duration equation is combined with the first equation of the loop top and the first equation of the loop bottom to obtain the first relationship equation of the loop duration.

[0069] The first equation for the loop spacing is obtained from the geometric parameters of the loop spacing. The first equation for the loop spacing is combined with the first CW relative motion equation to obtain the first equation relating the loop spacing to the pulse impulse.

[0070] The first relationship model between the geometric parameters and the pulse impulse is constructed by the first relationship equations between the loop width and the pulse impulse, the first relationship equations between the loop height and the pulse impulse, the first relationship equations between the loop duration and the loop spacing and the pulse impulse, under the condition of the same track height.

[0071] In step two, at the same track height, the first equation relating the loop width to the pulse impulse is:

[0072]

[0073] The first equation relating the loop height to the pulse impulse is:

[0074]

[0075] The first equation relating the loop duration is:

[0076]

[0077] The first equation relating the loop spacing to the pulse impulse is:

[0078]

[0079] In the formula, W1 is the loop width under the same orbital altitude; Δv1 is the pulse impulse applied by the reference spacecraft under the same orbital altitude; w is the angular velocity of the target spacecraft; H1 is the loop height under the same orbital altitude; T1 is the loop duration under the same orbital altitude; and I1 is the loop spacing under the same orbital altitude.

[0080] In step two, the first equation for the front end of the loop and the first equation for the back end of the loop are:

[0081]

[0082] The first equation for the width of the loop is:

[0083] W1=x(t 后端1 )-x(t 前端1 );

[0084] The first equations for the top and bottom of the loop are:

[0085]

[0086] The first equation for the height of the loop is:

[0087] H1=z(t 兜底1 )-z(t 兜顶1 );

[0088] The equation for the time of the loop is:

[0089] T1=2(t 兜底1 -t 兜顶1 );

[0090] The first equation for the loop spacing is:

[0091]

[0092] In the formula, t 前端1 The time point at the front end of the loop under the same track height conditions; t 后端1 The point in time at the rear end of the loop under the same orbital height; x(t) 前端1 ) t is the same as the track height 前端1 At time t, the reference spacecraft's position on the x-axis is; x(t) 后端1 ) t is the same as the track height 后端1 The reference spacecraft's position on the x-axis is at time t; 兜顶1 The point at the top of the loop under the same track height conditions; t 兜底1 The point at which the loopback and bottom-out occur under the same orbital height conditions; z(t) 兜顶1 ) t is the same as the track height 兜顶1 The reference spacecraft's position on the z-axis is at time t; z(t) 兜底1 ) t is the same as the track height 兜底1 The reference spacecraft's position on the z-axis is at time 2π / w; x1(2π / w) represents the reference spacecraft's position on the x-axis at time 2π / w, assuming the same orbital altitude. t01 represents the initial position of the reference spacecraft on the x-axis at the same orbital altitude.

[0093] In step three, the different orbital altitudes are as follows: the target spacecraft's orbit is a circular orbit; the reference spacecraft and the target spacecraft are coplanar circular orbits but at different altitudes; initially, the reference spacecraft maintains a straight-line flyby state relative to the target spacecraft, and the reference spacecraft achieves a loop trajectory by applying a pulse impulse along the velocity direction;

[0094] In the VVLH coordinate system, the second CW relative motion equations for different orbital altitudes are constructed by combining the CW relative motion equations in the VVLH coordinate system with the initial conditions for different orbital altitudes. The initial conditions for different orbital altitudes are as follows:

[0095]

[0096] The second CW relative motion equation is:

[0097]

[0098] In the formula, x(t2) represents the position of the reference spacecraft on the x-axis at time t2 under different orbital altitudes, and t2 represents the time variable under different orbital altitudes. t02 represents the initial position of the reference spacecraft at different orbital altitudes, Δv2 represents the initial time at different orbital altitudes, and w represents the angular velocity of the target spacecraft. z(t2) represents the position of the reference spacecraft on the z-axis at the initial time under different orbital altitudes; z(t2) represents the position of the reference spacecraft on the z-axis at time t2 under different orbital altitudes. The velocity of the reference spacecraft along the x-axis at time t2 under different orbital altitudes; The velocity of the reference spacecraft on the z-axis at time t2 under different orbital altitudes.

[0099] In step three, the method for constructing a second relationship model between the geometric parameters and the pulse impulse under different orbital altitudes, based on the second CW relative motion equation, is as follows:

[0100] Table 2 shows the initial state parameters of the reference spacecraft at different orbital altitudes:

[0101] Table 2

[0102]

[0103] At the endpoint of the loop trajectory, the velocity component in the X-axis direction is 0. Therefore, in the second CW relative motion equation, let... The second equations of relative motion of the second CW are used to derive the second equations for the front and rear ends of the loop; due to the constraints of trigonometric function values, the following conditions must be met: Only when the endpoint of the loop trajectory is reached will the calculation yield a result, meaning the loop trajectory will appear. If this condition is not met, the loop trajectory will not appear. Simplifying this inequality, we can obtain the constraint conditions for the appearance of the loop trajectory under different track heights as follows: Based on the constraints of the loop trajectory under different track heights, the second equations of the loop front end and the loop back end are combined with the second CW relative motion equation to obtain the second relationship equation between the loop width and the pulse impulse.

[0104] According to the geometric parameters of the loop-and-go trajectory, at the moment of bottoming out, the velocity component in the Z-axis direction is 0. Based on the characteristics of the loop-and-go trajectory, the moment of bottoming out is half the orbital period. The second equation for the loop-back bottom is obtained; the coordinate values ​​in the X-axis direction at the top and bottom times are the same, and the second equation for the loop-back top is obtained from the second CW relative motion equation; the second equation for the loop-back bottom and the second equation for the loop-back top are combined with the second CW relative motion equation to obtain the second equation relating the loop-back height to the pulse impulse.

[0105] The second relational equation for the loop-around time is obtained from the geometric parameters of the loop-around time.

[0106] The second equation for the loop spacing is obtained from the geometric parameters of the loop spacing. The second equation for the loop spacing is combined with the second CW relative motion equation to obtain the second equation relating the loop spacing to the pulse impulse.

[0107] The second relationship model between the geometric parameters and the pulse impulse is constructed by the second relationship equation between the loop width and the pulse impulse, the second relationship equation between the loop height and the pulse impulse, the second relationship equation between the loop duration and the loop spacing and the pulse impulse, respectively, for different track heights.

[0108] In step three, the second equation relating the loop width to the pulse impulse is:

[0109]

[0110] The second equation relating the back-loop height to the pulse impulse is:

[0111]

[0112] The second equation relating the loop duration is:

[0113] T2=2(t 兜底2 -t 兜顶2 );

[0114] The second equation relating the loop spacing to the pulse impulse is:

[0115]

[0116] In the formula, W2 is the loop width under different orbital altitudes, Δv2 is the pulse impulse applied by the reference spacecraft under different orbital altitudes, and w is the angular velocity of the target spacecraft. Let t02 be the initial position of the reference spacecraft on the z-axis at different orbital altitudes; σ is the intermediate variable value for solving the loop width at different orbital altitudes; H2 is the loop height at different orbital altitudes; and z(t02) is the initial time at different orbital altitudes. 兜顶2 ) represents t under different orbital altitudes. 兜顶2 At time t, the reference spacecraft's position on the z-axis is z(t). 兜底2 ) represents t under different orbital altitudes. 兜底2 At time t, the reference spacecraft's position on the z-axis is given. 兜顶2 For the time points t at the top of the loop under different track height conditions, 兜底2 t2 represents the bottoming point of the loop under different orbital altitudes; z(t2) represents the position of the reference spacecraft on the z-axis at time t2 under different orbital altitudes, t2 represents the time variable under different orbital altitudes; T2 represents the loop duration under different orbital altitudes; I2 represents the loop spacing under different orbital altitudes.

[0117] In step three, the second equation for the front end of the loop and the second equation for the back end of the loop are:

[0118]

[0119] The constraints for the loop trajectory under different track heights are:

[0120]

[0121] The second equation, which involves looping back to the bottom, is:

[0122]

[0123] The second equation at the top of the loop is:

[0124]

[0125] The second equation for the loop spacing is:

[0126]

[0127] In the formula, A is the parameter of the first-order term in the second equation at the top of the loop, B is the parameter of the trigonometric function term in the second equation at the top of the loop, and x2(2π / w) is the position of the reference spacecraft on the x-axis at time 2π / w under different orbital altitudes. The position of the reference spacecraft on the x-axis at the initial moment under different orbital altitudes.

[0128] In step three, the second equation at the top of the loop is a transcendental equation, t 兜顶2 The solution is obtained using a numerical method.

[0129] To provide a detailed explanation of the technical solution of the present invention, specific examples are provided below:

[0130] Example 1. Validation of the first relational model:

[0131] Initial conditions for simulation experiment:

[0132] Set the initial time as January 1, 2030, 00:00:00 (UTCG time), and the initial position-velocity state vector of the target spacecraft as [42167000; 0; 0; 0; 3074; 0]. Establish a VVLH coordinate system centered on the target spacecraft. The coordinates of the reference spacecraft in the VVLH coordinate system are [-100000; 0; 0; 0; 0], that is, the reference spacecraft is 100 kilometers behind the target spacecraft in the direction of travel, and the two remain relatively stationary at this time.

[0133] Simulation Experiment 1: Procedure and Data Results

[0134] At 12:00:00, 12 hours later, a reference spacecraft is subjected to pulse impulses of 1 m / s and 2 m / s along its velocity direction. The first CW relative motion equation is used to calculate the position of the reference spacecraft at different times, and the change in distance from the target spacecraft over time. Experimental data are as follows: Figure 5 As shown in the diagram. Data analysis reveals that during the first 0-12 hours, the reference spacecraft did not maneuver, therefore the distance between the two spacecraft remained constant. At the 12-hour mark, the distance changed after the application of the pulse impulse, generally increasing over time, but with some periods showing an initial decrease followed by an increase. Furthermore, the distance change caused by a 2 m / s pulse was greater than that caused by a 1 m / s pulse impulse.

[0135] Calculate the trajectory of the reference spacecraft in the X and Z planes of the VVLH coordinate system under two pulse impulses. Experimental data are as follows: Figure 6 As shown in Table 3, the geometric parameters of the loop trajectory are as follows. Analysis of the data shows that both 1 m / s and 2 m / s pulses produced loop trajectories. The duration of the loop trajectories produced by the two pulse values ​​is the same, but the width, length, and loop spacing of the loop trajectory produced by the 1 m / s pulse are smaller. The distance values ​​of both are half of the geometric parameters corresponding to the loop trajectory produced by the 2 m / s pulse. The simulation data agrees well with the previously derived first relationship model between the loop trajectory geometric parameters and the pulse impulse, verifying the correctness of the first relationship model construction.

[0136] Table 3

[0137]

[0138] Simulation Experiment 2: Procedure and Data Results

[0139] A reference spacecraft was subjected to pulses ranging from 0 to 10 m / s at 0.01 m / s intervals. The first CW relative motion equation was used to calculate the state values ​​of the reference spacecraft at different times. Next, based on the definitions of the geometric parameters of the loop trajectory, the values ​​of loop duration, loop height, loop width, and loop spacing were solved. Finally, the pulse impulses were plotted against each geometric parameter. Experimental data are as follows: Figure 7 As shown in the diagram. Data analysis reveals that the loop width increases linearly and is directly proportional to the pulse size, with a smaller increase than the height. The relationship is W1 = 13.0985Δv1, consistent with the first equation relating loop width to pulse impulse. The loop height also increases linearly and is directly proportional to the pulse size, with a relationship of H1 = 19.4078Δv1, consistent with the first equation relating loop height to pulse impulse. The loop duration remains constant at 9.71 hours, consistent with the first equation relating loop duration. The loop spacing increases linearly and is directly proportional to the pulse size, with a significantly larger increase than the height and width (an order of magnitude larger). The relationship is I1 = 258.3Δv1, consistent with the first equation relating loop spacing to pulse impulse.

[0140] Example 2. Validation of the second relational model:

[0141] Initial conditions for simulation experiment:

[0142] Assume the initial time is January 1, 2030, 00:00:00 (UTCG time). The initial position-velocity state vector of the target spacecraft is [42167000; 0; 0; 0; 3074; 0]. Establish a VVLH coordinate system centered on the target spacecraft. The coordinates of the reference spacecraft in the VVLH coordinate system are [-100000; 0; 100000; 10.9; 0; 0]. That is, the reference spacecraft is 100 kilometers behind and 100 kilometers below the target spacecraft in the direction of travel. At this time, the reference spacecraft is approaching the target spacecraft in a straight line below it at a speed of 10.9 m / s.

[0143] Simulation Experiment 3: Procedure and Data Results

[0144] At 12:00:00, 12 hours later, pulses of 1 m / s and 2 m / s were applied to the reference spacecraft along the velocity direction. The second CW relative motion equation was used to calculate the position of the reference spacecraft at different times, and the change in distance from the target spacecraft over time. The experimental data are as follows: Figure 8 As shown. Two trajectories of the reference spacecraft in the X and Z planes of the VVLH coordinate system were calculated. Experimental data are as follows: Figure 9 As shown in the figure. Analysis of the two sets of data shows that during the 0-12 hour period, the reference spacecraft did not maneuver, and the distance between the two spacecraft initially decreased to 100 kilometers, the difference in orbital altitude, based on the original relative speed, and then gradually increased. At the 12-hour mark, after the application of a pulse impulse, the relative state of the two spacecraft changed. The relative distance corresponding to the 2 m / s pulse value first decreased and then increased, indicating that a loop-like phenomenon occurred. The relative distance corresponding to the 1 m / s pulse value only showed a slower increase, without decreasing, meaning that a loop-like phenomenon did not occur, and its relative motion trajectory was a skipping arch-like forward movement. The geometric parameters of the loop-like trajectory are shown in Table 4.

[0145] Table 4

[0146]

[0147] Simulation Experiment 4: Procedure and Data Results

[0148] A reference spacecraft was subjected to pulses ranging from 0 to 10 m / s at 0.01 m / s intervals. The second CW relative motion equation was used to calculate the state values ​​of the reference spacecraft at different times. Next, based on the definition of the geometric parameters of the loop trajectory, the values ​​of loop duration, loop height, loop width, and loop spacing were solved. Finally, the pulse impulses were plotted against each geometric parameter, and the experimental data are shown below. Figure 10 As shown in the figure, data analysis reveals that when the pulse value is less than 1.53 m / s, the reference spacecraft cannot generate a loop trajectory; its actual trajectory shape is a skipping arch. When the pulse value is greater than 1.53 m / s, the relative motion trajectory begins to loop. The simulation data matches the constraints for the loop trajectory at different orbital altitudes.

[0149] The variation patterns of the geometric parameters of the loop trajectory are as follows:

[0150] The loop width and loop height show similar variation patterns, increasing with the increase of pulse value. The pulse value increases more rapidly in the range of 1.53-3.53 m / s, and the growth rate slows down and increases linearly after 3.53 m / s.

[0151] The variation pattern of the loop-back duration is unique. Initially, the loop-back duration increases rapidly with the increase of pulse impulse. When the pulse value is 3.53 m / s, the loop-back duration reaches its maximum value of 23.5 hours. This duration occupies almost most of the entire orbital cycle, meaning that at this time, most of the trajectory appears in the form of a loop-back trajectory. Subsequently, the loop-back duration begins to gradually decrease with the increase of pulse value.

[0152] The loop-back distance first decreases and then increases with the magnitude of the pulse impulse. At a pulse value of 3.53 m / s, it decreases to a minimum of 25 km. Subsequently, the loop-back distance increases with the pulse value and shows a linear relationship.

[0153] Example 3. Application Case of Orbit Design for Spacecraft Orbit Transfer Based on Loop Trajectory

[0154] Spacecraft A operates in a GEO geostationary orbit, and its servicing spacecraft B is located at an altitude of 100 kilometers below spacecraft A's orbit, moving in a straight line relative to spacecraft A. At a certain moment, a malfunction is detected in spacecraft A, requiring servicing spacecraft B to perform an orbital transfer. After the transfer, it needs to orbit spacecraft A once for inspection before restoring its initial orbital motion. This moment is the initial time, denoted as 0 seconds. At this time, the initial state of servicing spacecraft B in the VVLH coordinate system of spacecraft A is [-200000; 0; 100000; 10.9; 0; 0]. The requirement is that when servicing spacecraft B orbits spacecraft A once, the distance between them should not exceed 100 kilometers and should not be less than 50 kilometers. Using the technical solution disclosed in this invention, the pulse impulse that should be applied during the orbital transfer of servicing spacecraft B is determined. Subsequently, based on the obtained pulse impulse, the time for applying the pulse impulse can be calculated according to mission requirements.

[0155] 4.1 Task Requirements Constraint Analysis

[0156] This case involves a spacecraft orbit transfer problem. Analyzing the mission requirements and constraints, we transform them into geometric parameter constraints for the loop trajectory. We can deduce that after the pulse is applied, the top of the loop of the service spacecraft B should be located 50-100 km below spacecraft A, the bottom of the loop should be located 50-100 km above spacecraft A, the front end of the loop should be located 50-100 km in front of spacecraft A, and the rear end of the loop should be located 50-100 km behind spacecraft A. These constraints can be expressed mathematically as follows:

[0157]

[0158] In the formula, z(t) 兜底2 Let z(t) be the coordinate value of spacecraft B on the Z-axis at the last possible moment. 兜顶2 x(t) represents the coordinates of spacecraft B on the Z-axis at the last possible moment. 后端2 x(t) represents the coordinates of spacecraft B on the X-axis at the backend time. 前端2 ( ) represents the X-axis coordinates of the service spacecraft B at the preceding time. The mission planning sketch is as follows: Figure 11 As shown.

[0159] 4.2 Constraint Transformation and Scheme Design

[0160] Combining the above constraints with the geometric parameters of the loop trajectory and the second relational model of the pulse impulse, we can obtain the second relational model equations under the constraints:

[0161]

[0162] Based on the initial state values ​​of the servicing spacecraft B in the VVLH coordinate system [-200000; 0; 100000; 10.9; 0; 0], the initial parameters of the second relational model equations under the constraints are:

[0163]

[0164] Substituting the initial values ​​of the above equation into the second relational model equations under the constraints, we can find that the range of values ​​for Δv2 that satisfy the constraints is 2.786-2.912. Following the principle of retaining two significant figures and taking the smaller pulse value to save fuel, we choose Δv2 = 2.79 m / s.

[0165] If the task requires calculating the time of the applied pulse impulse, then after determining the magnitude of the pulse impulse, the loop distance I = 21.931 km is calculated. This can be done according to the equation for the pulse application time: t plus1 =t0+(-x0-I / 2) / (3 / 2*w*z0) gives the time t when the pulse is applied. plus1 = t0 + 8282.2s, where t0 represents the initial time. Since the servicing spacecraft needs to complete one orbit and return to its original orbit, a pulse impulse in the same direction but opposite to the first pulse impulse needs to be applied one orbital period after the first pulse impulse is applied. Therefore, this mission plan requires two pulses, as shown in Table 5 (the negative sign indicates the opposite direction along the velocity direction). The trajectory of the servicing spacecraft B's orbital transfer process is as follows... Figure 12 As shown.

[0166] Table 5

[0167]

[0168] The beneficial effects of the embodiments of the present invention are:

[0169] 1. This invention proposes the concept of a loop trajectory and defines the geometric parameters corresponding to the loop trajectory and the orbital transfer of the reference spacecraft in the VVLH coordinate system.

[0170] 2. The CW relative motion equations are used to describe the relative motion between two spacecraft. They are derived based on the "two-body" motion model and provide a high-precision analytical solution for the relative motion. The calculation method is fast and convenient. Since the CW relative motion equations do not consider the influence of perturbation factors on the trajectory, using the CW relative motion equations to calculate the relative trajectory can eliminate the potential influence of perturbation factors on the loop trajectory. This allows for a more objective study of the impact of pulses applied by the spacecraft on the loop trajectory. Therefore, this invention calculates the CW relative motion equations in the VVLH coordinate system for the loop phenomenon and constructs the first CW relative motion equations for the same orbital altitude and the second CW relative motion equations for different orbital altitudes.

[0171] 3. The reference spacecraft will produce different loop trajectories under different applied pulse impulses. Therefore, based on the first CW relative motion equation, a first relationship model between geometric parameters and pulse impulse is constructed for the same orbital altitude. Based on the second CW relative motion equation, a second relationship model between geometric parameters and pulse impulse is constructed for different orbital altitudes. The construction process adopts a step-by-step approach, starting with the specific and progressing to the general, and from simple to complex. First, according to the simple special case of two spacecraft at the same orbital altitude, the relationship equation between the applied pulse impulse of the reference spacecraft and the geometric parameters of the loop trajectory is studied, resulting in the first relationship model. Second, according to the complex case of two spacecraft at different orbital altitudes, the relationship equation between the applied pulse impulse of the reference spacecraft and the geometric parameters of the loop trajectory is further constructed, resulting in the second relationship model. The obtained first and second relationship models make the orbit transfer problem solution process clear, intuitive, and efficient, and the results reliable and easy to implement.

[0172] 4. This invention transforms the mission requirements in actual orbit design into geometric parameter constraints, thereby obtaining the pulse impulse applied during orbit transfer. This avoids the complex constraint transformation and numerous repetitive calculation steps required in the traditional Lambert orbit transfer method and numerical iterative solution method of motion equations in relative coordinate systems. The technical solution disclosed in this invention reduces the amount of computation, realizes constraint transformation, improves the efficiency of spacecraft orbit transfer, and makes mission planning more intuitive and efficient.

[0173] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A spacecraft orbit transfer method based on a loop trajectory, characterized in that, include: Step 1: The trajectory of the reference spacecraft that reverses and then resumes its forward motion relative to the target spacecraft is taken as the loop trajectory, and the geometric parameters corresponding to the loop trajectory and the orbital transfer of the reference spacecraft in the VVLH coordinate system are obtained. Step 2: In the VVLH coordinate system, construct the first CW relative motion equation under the same orbital height. Based on the first CW relative motion equation, construct the first relationship model between the geometric parameters and the pulse impulse under the same orbital height. Step 3: In the VVLH coordinate system, construct the second CW relative motion equation for different orbital heights. Based on the second CW relative motion equation, construct the second relationship model between the geometric parameters and the pulse impulse for different orbital heights. Step four: Convert the target constraints in the mission requirements into conditional constraints of the geometric parameters. Based on the orbital altitude in the mission requirements, use the corresponding first or second relational model to obtain the pulse impulse required for the orbital transfer of the reference spacecraft.

2. The spacecraft orbit transfer method based on a loop trajectory as described in claim 1, characterized in that, In step one, the geometric parameters corresponding to the orbital transfer of the loop trajectory and the reference spacecraft in the VVLH coordinate system are: loop top, loop bottom, loop endpoint, loop center, loop height, loop width, loop duration, and loop spacing; among which, The top of the loop is the intersection point of the relative motion trajectory of the reference spacecraft in the VVLH coordinate system; the intersection point is formed when the relative motion of the reference spacecraft involves both retreat and forward movement during its forward motion. The bottom loop is the maximum or minimum point of the Z-axis of the relative motion trajectory of the reference spacecraft in the VVLH coordinate system. The bottom loop and the top loop have the same X-axis coordinate. At the bottom loop, the tangent of the relative motion trajectory of the reference spacecraft is parallel to the X-axis of the VVLH coordinate system, and the Z-axis component of the relative motion velocity is 0. The loop endpoint is the point where the relative velocity of the reference spacecraft in the VVLH coordinate system changes from positive to negative or from negative to positive on the X-axis component. The loop endpoint includes the loop front end and the loop back end. The loop front end is located in front of the direction of relative motion trajectory, and the loop back end is located behind the direction of relative motion trajectory. At the loop endpoint, the tangent of the relative motion trajectory of the reference spacecraft is perpendicular to the X-axis of the VVLH coordinate system, and the X-axis component of the relative velocity is 0. The center of the loop is the intersection of the line connecting the top and bottom of the loop and the line connecting the front and back of the loop. The return height is the distance from the top of the return pocket to the bottom of the return pocket along the Z-axis. The width of the loop is the distance from the front end to the rear end of the loop along the X-axis. The loop duration is the length of time that a spacecraft takes from the moment it enters the top of the loop, through the front end, rear end, and bottom of the loop, until it moves back to the top of the loop again. The loop spacing is the distance between the centers of two adjacent loops.

3. The spacecraft orbit transfer method based on a loop trajectory as described in claim 2, characterized in that, In step two, the cases with the same orbital altitude are as follows: the target spacecraft's orbit is a circular orbit; the reference spacecraft and the target spacecraft have the same orbit but do not overlap in position; initially, the reference spacecraft remains relatively stationary relative to the target spacecraft, and the reference spacecraft achieves a loop trajectory by applying a pulse impulse along the velocity direction. In the VVLH coordinate system, the first CW relative motion equation under the same orbital height condition is constructed as follows: In the formula, x(t1) represents the position of the reference spacecraft on the x-axis at time t1 under the same orbital altitude, and t1 represents the time variable under the same orbital altitude. t01 is the initial position of the reference spacecraft at the same orbital altitude, Δv1 is the pulse impulse applied by the reference spacecraft at the same orbital altitude, w is the angular velocity of the target spacecraft, and z(t1) is the position of the reference spacecraft on the z-axis at time t1 at the same orbital altitude. The velocity of the reference spacecraft on the x-axis at time t1 under the same orbital altitude; The velocity of the reference spacecraft on the z-axis at time t1 under the same orbital altitude.

4. The spacecraft orbit transfer method based on a loop trajectory as described in claim 3, characterized in that, In step two, the method for constructing the first relationship model between the geometric parameters and the pulse impulse under the same orbital height, based on the first CW relative motion equation, is as follows: Under the same track height, the first equation for the front end of the loop and the first equation for the rear end of the loop are obtained from the first CW relative motion equation, and thus the first equation for the width of the loop is obtained. By combining the first equation for the loop width with the first equation for relative motion of CW, we obtain the first equation relating the loop width to the pulse impulse. The first equation for the top and bottom of the loop is obtained from the first CW relative motion equation, thus obtaining the first equation for the loop height; the first equation for the loop height is combined with the first CW relative motion equation to obtain the first equation relating the loop height to the pulse impulse. The loop duration equation is obtained by defining the geometric parameters of the loop duration. The loop duration equation is combined with the first equation of the loop top and the first equation of the loop bottom to obtain the first relationship equation of the loop duration. The first equation for the loop spacing is obtained from the geometric parameters of the loop spacing. The first equation for the loop spacing is combined with the first CW relative motion equation to obtain the first equation relating the loop spacing to the pulse impulse. The first relationship model between the geometric parameters and the pulse impulse is constructed by the first relationship equations between the loop width and the pulse impulse, the first relationship equations between the loop height and the pulse impulse, the first relationship equations between the loop duration and the loop spacing and the pulse impulse, under the condition of the same track height.

5. A spacecraft orbit transfer method based on a loop trajectory as described in claim 4, characterized in that, In step two, the first equation relating the loop width to the pulse impulse is: The first equation relating the loop height to the pulse impulse is: The first equation relating the loop duration is: The first equation relating the loop spacing to the pulse impulse is: In the formula, W1 is the loop width under the same orbital altitude; Δv1 is the pulse impulse applied by the reference spacecraft under the same orbital altitude; w is the angular velocity of the target spacecraft; H1 is the loop height under the same orbital altitude; T1 is the loop duration under the same orbital altitude; and I1 is the loop spacing under the same orbital altitude.

6. The spacecraft orbit transfer method based on a loop trajectory as described in claim 5, characterized in that, In step two, the first equation for the front end of the loop and the first equation for the back end of the loop are: The first equation for the width of the loop is: W1=x(t 后端1 )-x(t 前端1 ); The first equations for the top and bottom of the loop are: The first equation for the height of the loop is: H1=z(t 兜底1 )-z(t 兜顶1 ); The equation for the time of the loop is: T1=2(t 兜底1 -t 兜顶1 ); The first equation for the loop spacing is: In the formula, t 前端1 The time point at the front end of the loop under the same track height conditions; t 后端1 The point in time at the rear end of the loop under the same orbital height; x(t) 前端1 ) t is the same as the track height 前端1 At time t, the reference spacecraft's position on the x-axis is; x(t) 后端1 ) t is the same as the track height 后端1 The reference spacecraft's position on the x-axis is at time t; 兜顶1 The point at the top of the loop under the same track height conditions; t 兜底1 The point at which the loopback and bottom-out occur under the same orbital height conditions; z(t) 兜顶1 ) t is the same as the track height 兜顶1 The reference spacecraft's position on the z-axis is at time t; z(t) 兜底1 ) t is the same as the track height 兜底1 The reference spacecraft's position on the z-axis is at time 2π / w; x1(2π / w) represents the reference spacecraft's position on the x-axis at time 2π / w, assuming the same orbital altitude. t01 represents the initial position of the reference spacecraft on the x-axis at the same orbital altitude.

7. A spacecraft orbit transfer method based on a loop trajectory as described in claim 2, characterized in that, Step 3, different orbital altitude scenarios: the target spacecraft's orbit is a circular orbit; the reference spacecraft and the target spacecraft are coplanar circular orbits but at different altitudes; initially, the reference spacecraft maintains a straight-line flyby state relative to the target spacecraft, and the reference spacecraft achieves a loop trajectory by applying a pulse impulse along the velocity direction; In the VVLH coordinate system, the second CW relative motion equations for different orbital heights are constructed as follows: In the formula, x(t2) represents the position of the reference spacecraft on the x-axis at time t2 under different orbital altitudes, and t2 represents the time variable under different orbital altitudes. t02 represents the initial position of the reference spacecraft at different orbital altitudes, Δv2 represents the initial time at different orbital altitudes, and w represents the angular velocity of the target spacecraft. z(t2) represents the position of the reference spacecraft on the z-axis at the initial time under different orbital altitudes; z(t2) represents the position of the reference spacecraft on the z-axis at time t2 under different orbital altitudes. The velocity of the reference spacecraft along the x-axis at time t2 under different orbital altitudes; The velocity of the reference spacecraft on the z-axis at time t2 under different orbital altitudes.

8. A spacecraft orbit transfer method based on a loop trajectory as described in claim 7, characterized in that, In step three, the method for constructing a second relationship model between the geometric parameters and the pulse impulse under different orbital altitudes, based on the second CW relative motion equation, is as follows: Under different orbital heights, the second equations for the front and rear ends of the loop are obtained from the second CW relative motion equations. Based on the constraints of the loop trajectory under different orbital heights, the second equations for the front and rear ends of the loop are combined with the second CW relative motion equations to obtain the second relationship equation between the loop width and the pulse impulse. Based on the geometric parameter definition of the bottom loop and the characteristics of the bottom loop on the loop trajectory, the second equation of the bottom loop is obtained; the second equation of the top loop is obtained from the second CW relative motion equation; the second equation of the bottom loop and the second equation of the top loop are combined with the second CW relative motion equation to obtain the second relationship equation between the loop height and the pulse impulse. The second relational equation for the loop-around time is obtained from the geometric parameters of the loop-around time. The second equation for the loop spacing is obtained from the geometric parameters of the loop spacing. The second equation for the loop spacing is combined with the second CW relative motion equation to obtain the second equation relating the loop spacing to the pulse impulse. The second relationship model between the geometric parameters and the pulse impulse is constructed by the second relationship equation between the loop width and the pulse impulse, the second relationship equation between the loop height and the pulse impulse, the second relationship equation between the loop duration and the loop spacing and the pulse impulse, respectively, for different track heights.

9. A spacecraft orbit transfer method based on a loop trajectory as described in claim 8, characterized in that, In step three, the second equation relating the loop width to the pulse impulse is: The second equation relating the back-loop height to the pulse impulse is: The second equation relating the loop duration is: T2=2(t 兜底2 -t 兜顶2 ); The second equation relating the loop spacing to the pulse impulse is: In the formula, W2 is the loop width under different orbital altitudes, Δv2 is the pulse impulse applied by the reference spacecraft under different orbital altitudes, and w is the angular velocity of the target spacecraft. Let t02 be the initial position of the reference spacecraft on the z-axis at different orbital altitudes; σ is the intermediate variable value for solving the loop width at different orbital altitudes; H2 is the loop height at different orbital altitudes; and z(t02) is the initial time at different orbital altitudes. 兜顶2 ) represents t under different orbital altitudes. 兜顶2 At time t, the reference spacecraft's position on the z-axis is z(t). 兜底2 ) represents t under different orbital altitudes. 兜底2 At time t, the reference spacecraft's position on the z-axis is given. 兜顶2 For the time points t at the top of the loop under different orbital altitudes, 兜底2 t2 represents the bottoming point of the loop under different orbital altitudes; z(t2) represents the position of the reference spacecraft on the z-axis at time t2 under different orbital altitudes, t2 represents the time variable under different orbital altitudes; T2 represents the loop duration under different orbital altitudes; I2 represents the loop spacing under different orbital altitudes.

10. A spacecraft orbit transfer method based on a loop trajectory as described in claim 9, characterized in that, In step three, the second equation for the front end of the loop and the second equation for the back end of the loop are: The constraints for the loop trajectory under different track heights are: The second equation, which involves looping back to the bottom, is: The second equation at the top of the loop is: The second equation for the loop spacing is: In the formula, A is the parameter of the first-order term in the second equation at the top of the loop, B is the parameter of the trigonometric function term in the second equation at the top of the loop, and x2(2π / w) is the position of the reference spacecraft on the x-axis at time 2π / w under different orbital altitudes. The x-axis position of the reference spacecraft at the initial moment under different orbital altitudes.

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