Micro-nano satellite multi-pulse maneuvering method considering thrust constraint and J2 perturbation

Abstract

The invention discloses a micro-nano satellite multi-pulse maneuvering method considering thrust constraint and J2 perturbation. The method aims at solving the problems that single speed increment implementation of the optimal maneuvering section of micro thrusters such as micro-nano satellite electric thrust / cold air and the like is limited, the maneuvering precision is reduced due to the fact that a long-term maneuvering orbit is influenced by J2 perturbation, and consequently the multi-satellite formation capture and reconstruction task of a near-circular orbit is difficult to complete. According to the method, the mapping relation between pulses and relative orbit parameters is established based on the relative orbit elements, a supplementary allocation method is provided for thrust constraint to achieve maneuvering pulse splitting, and a pulse sequence is corrected through iteration to compensate for J2 perturbation errors. According to the method, the multi-pulse maneuvering trajectory planning precision under the J2 perturbation influence can be effectively improved, the centimeter-level theoretical control precision can be achieved through 2-3 times of iterative computation, and the method has important practical value in the aerospace field.

Description

Technical Field

[0001] This invention belongs to the field of spacecraft application technology, and specifically relates to a multi-pulse maneuvering method for micro-nano satellites that considers thrust constraints and J2 perturbation. Background Technology

[0002] Complex space formation missions require microsatellites and nanosatellites in multi-satellite systems to perform relative orbital maneuvers during initial formation acquisition and reconfiguration. Pulse-based maneuvering control is the fundamental method for orbital maneuvers. Ichimura et al., in "Optimal impulsive relative orbit transfer along a circular orbit," proposed a minimum open-time fuel pulse strategy to solve the relative orbital transfer problem related to the HCW equations. This algorithm uses three types of in-plane pulses to achieve optimal in-plane reconfiguration and a single-pulse strategy to achieve out-of-plane motion. However, due to the limitations of microsatellite system platform capabilities, especially for microsatellites and nanosatellites equipped with electric propulsion and cold gas micro-propulsion systems, the single stable operating time of their micro-thrusters cannot meet the requirements of traditional orbital maneuvering control methods. Therefore, it is necessary to plan pulse sequences composed of multiple pulses to achieve maneuvering control tasks for microsatellite formations under large-scale configuration acquisition or reconfiguration. Thus, it is urgent to solve multi-pulse solution methods that consider maneuverability constraints while fully considering the impact of orbital perturbations during long-term maneuvering transfers. Summary of the Invention

[0003] This invention proposes a multi-pulse maneuvering method for microsatellites and nanosatellites that considers thrust constraints and J2 perturbations. Considering the finite single velocity increment constraint of microsatellite thrusters and J2 perturbations, a microsatellite maneuvering pulse sequence calculation framework suitable for onboard computing is designed to realize the maneuvering control task of microsatellite formations under large-range configuration capture or reconstruction.

[0004] The technical solution for realizing this invention is as follows: a multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation, comprising the following steps:

[0005] Step 1: Input the initial relative orbital elements of the microsatellite and the target relative orbital elements, calculate the target control relative orbital elements in a near-circular orbit, and proceed to Step 2.

[0006] Step 2: Based on the target control relative orbital elements under near-circular orbit, calculate the ideal normal pulse sequence and the ideal trajectory pulse sequence based on the pulse control model of the relative orbital elements, and proceed to Step 3.

[0007] Step 3: Based on the ideal normal pulse sequence and the ideal track pulse sequence, and according to the single maximum velocity increment constraint of the micro-nano satellite thruster, calculate the normal pulse control sequence and the track pulse control sequence using the supplementary allocation method, and proceed to step 4.

[0008] Step 4: Based on the normal pulse control sequence and the trace pulse control sequence, obtain the relative orbital element endpoint state through the J2 perturbation transfer model, and proceed to step 5.

[0009] Step 5: Calculate the difference between the endpoint state of the relative orbital elements and the target relative orbital elements to obtain the control error. Determine if the error is less than the required control accuracy. If the error is greater than the required control accuracy, update the maneuvering requirement and return to step 3; otherwise, proceed to step 6.

[0010] Step 6: Output the normal pulse control sequence and the track pulse control sequence to complete the multi-pulse maneuver of the microsatellite.

[0011] Compared with the prior art, the significant advantages of this invention are:

[0012] 1) Considering the constraint of limited single velocity increment of micro-nano satellite thrusters, the pulse quantity exceeding the constraint is split into pulse sequences using the supplementary allocation method to meet the limitations of micro-nano satellite platforms.

[0013] 2) Considering the relative motion model of J2 perturbation, the relative orbital element control error can be reduced to the centimeter level by iteratively calculating the corrected pulse sequence and performing 2 to 3 iterations.

[0014] 3) Design a pulse sequence computing framework suitable for spaceborne computing, which greatly simplifies the computing process and improves the real-time performance and reliability of on-orbit missions.

[0015] The present invention will now be described in further detail with reference to the accompanying drawings. Attached Figure Description

[0016] Figure 1 A flowchart for calculating multipulse maneuvers of microsatellites and nanosatellites considering thrust constraints and J2 perturbations.

[0017] Figure 2 The graph shows the change in the number of iterations versus the error.

[0018] Figure 3 This is a diagram of a pulse sequence scheme.

[0019] Figure 4 for Graph of changes with pulse application

[0020] Figure 5 for Graph of changes with pulse application

[0021] Figure 6This is a diagram of the three-dimensional relative motion trajectory under the action of a pulse sequence.

[0022] Figure 7 This is a projection diagram of the XY plane trajectory under the action of a pulse sequence. Detailed Implementation

[0023] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0024] Combination Figure 1 A multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation, comprising the following steps:

[0025] Step 1: Input the initial relative orbital elements of the microsatellite and the target relative orbital elements, and calculate the target control relative orbital elements in a near-circular orbit, as follows:

[0026] The target control relative orbital elements The calculation method is as follows:

[0027] ,

[0028] In the formula, The initial relative orbital elements, ; For the target relative orbital elements, ; Each component is relative to the major semi-axis. Relative average longitude angle Relative eccentricity vector and relative tilt vector The superscript T represents the transpose of the vector, and the subscript... and These represent the initial quantity and the target quantity, respectively. It represents the difference between the target quantity and the initial quantity.

[0029] Proceed to step 2.

[0030] Step 2: Based on the target control relative orbital elements in a near-circular orbit, and using the pulse control model based on the relative orbital elements, calculate the ideal normal pulse sequence and the ideal trajectory pulse sequence, as follows:

[0031] The pulse control model for the relative orbital elements is as follows:

[0032] ,

[0033] In the formula, For the reference orbital semi-axis, For the reference circular orbit's average angular velocity, This represents the average latitude argument of the reference orbit; , and These represent the ideal radial, trace, and normal pulses in the RTN coordinate system, respectively.

[0034] The effect of pulse control on the relative orbital elements shows that the normal pulse... It will only affect the out-of-plane vector. radial pulse With trace pulse It will simultaneously affect the vector within the orbital plane. For control within the orbital plane, it is proposed that only trajectory pulses exist. Control methods.

[0035] The method for calculating the ideal normal pulse sequence is as follows:

[0036] Ideal normal pulse sequence and implementation phase The relative orbital elements controlled by the target The calculation method is as follows:

[0037] ,

[0038] In the formula, To change the relative orbital inclination vector magnitude, phase is implemented. lie in ~ Between; the ideal normal pulse sequence is implemented at a specified phase within one orbital period, and only needs to be implemented once.

[0039] The method for calculating the ideal trajectory pulse sequence is as follows:

[0040] Ideal Track Pulse Sequence and implementation phase The relative orbital elements controlled by the target Calculation, in The ideal trajectory pulse sequence is applied three times within a complete orbital cycle. , and , ;

[0041] Calculate the selected parameters according to the following formula. and parameters :

[0042] ;

[0043] In the formula, To change the magnitude of the relative eccentricity vector, ;

[0044] when At that time, the ideal trajectory pulse sequence for three consecutive cycles is calculated using the following method. , and :

[0045] ,

[0046] at this time:

[0047] ; ; ;

[0048] In the formula, It is the floor function. and As an intermediate variable relating to the orbital period, and An intermediate variable for applying phase to the pulse;

[0049] , and The applied phases are respectively , , :

[0050] , , ,

[0051] In the formula, ;

[0052] when At that time, the ideal trajectory pulse sequence for three consecutive cycles is calculated using the following method. , and :

[0053] ,

[0054] at this time:

[0055] , , .

[0056] , and The applied phase , and They are respectively:

[0057] ; ; ;

[0058] Control the target relative orbital elements Substituting the above methods, we can obtain the ideal normal pulse sequence and the ideal trace pulse sequence.

[0059] Proceed to step 3.

[0060] Step 3: Based on the ideal normal pulse sequence and the ideal track pulse sequence, and according to the constraint of the maximum velocity increment of the micro-nano satellite thruster in a single operation, the normal pulse control sequence and the track pulse control sequence are calculated using the supplementary allocation method, as follows:

[0061] Define the maximum impulse constraint. To represent the maximum velocity increment that a microsatellite can provide in a single operation, the normal pulse control sequence allocation method is as follows:

[0062] When passing through orbit phase Execute normal pulse Realize the relative orbital inclination vector When controlling microsatellites, if the limited thrust system capabilities of microsatellites are taken into account, that is... At phase angle Apply reverse pulse At phase angle Apply a positive pulse To achieve the control quantity required to fulfill the execution needs in multi-track maneuver control, the equivalent formula is expressed as:

[0063] ;

[0064] ;

[0065] ; ; ;

[0066] In the formula, As an intermediate variable relating to the orbital period, For the set of integers, For orbital phase The normal pulse being executed.

[0067] When satisfied When the time comes, the control requirements can be met.

[0068] The normal pulse sequence is calculated using the supplementary allocation method:

[0069] ;

[0070] ;

[0071] To supplement the normal pulse sequence after allocation by the allocation method, i represents the current pulse execution count. For the total number of normal pulses executed, when the phase angle is... At that time, execution and Pulses in opposite directions, when the phase angle is... At that time, execution and Pulses in the same direction; execution is then performed. Number of orbital revolutions required for the subnormal pulse for:

[0072] .

[0073] The method for allocating the trace pulse control sequence is as follows:

[0074] First, design a tracing pulse using two semi-major axis and eccentricity pulses. , achieve and The control requirements are as follows:

[0075] ;

[0076] in, and Indicates two track pulses , The phase angle implemented.

[0077] when and Greater than At this time, configuration needs to be split. and ,make and Execution is performed in a specified phase across multiple orbital cycles to meet the requirements for controlling the pulse quantity and the pulse quantity amplitude constraints.

[0078] For relative average longitude argument When controlling, define Used to describe the trajectory pulse sequence If the relative average longitude angle of control is then:

[0079] ,

[0080] In the formula, i represents the current pulse execution count. This represents the total number of times the trace pulse is executed. It is the expected relative average longitude angle. It is the phase endpoint value of the orbit. Including semi-major axis and eccentricity tracking pulses and relative average longitude amplitude trace pulse ; The execution of this is unavoidable, but it can be mitigated through allocation. Achieve reduction Size.

[0081] According to the supplementary allocation method, the semi-major axis and eccentricity trace pulses Controlled relative average longitude angle As shown in the following formula:

[0082] ,

[0083] in,

[0084] ,

[0085] ,

[0086] In the formula, and They represent and The portion allocated in full, and This indicates the total number of times the trace pulse sequence is executed. and For the first and Sub-pulse control sequence. and They are respectively and Implemented phase angle, p and q represent respectively and Current number of implementations, and They represent and Current implementation phase, , This indicates the number of orbits required to complete a trajectory pulse maneuver mission. and This represents intermediate parameters.

[0087] Proceed to step 4.

[0088] Step 4: Based on the normal pulse control sequence and the trace pulse control sequence, obtain the final state of the relative orbital elements through the J2 perturbation transfer model, as follows:

[0089] The J2 perturbation transmission model is as follows:

[0090] The state transition matrix of the relative orbital elements as a function of time under J2 perturbation is:

[0091] ,

[0092] ; ;

[0093] , , , , ;

[0094] In the formula, Initial relative orbital elements Under the influence of J2 perturbation, after a period of time The relative orbital elements after that, Here is the state transition matrix under the HCW equation. Here is the state transition matrix under J2 perturbation; , , , and Intermediate parameters for convenient calculation, Indicates the eccentricity of the reference orbit. Indicates the inclination angle of the reference orbit. This represents the Earth's average radius. This represents the perturbation constant.

[0095] The first derivatives of each element in the relative orbital elements under the influence of J2 perturbation, calculated according to the above formula, are as follows:

[0096] ;

[0097] Based on the above model, the final state of the relative orbital elements can be obtained by substituting the initial state of the relative orbital elements, the transition time, and the normal pulse control sequence and the trace pulse control sequence.

[0098] Proceed to step 5.

[0099] Step 5: Subtract the relative orbital elements at the endpoint from the target relative orbital elements to obtain the control error. Determine if the error is less than the required control accuracy. If the error is greater than the required control accuracy, update the maneuvering requirements and return to Step 3; otherwise, proceed to Step 6, as detailed below:

[0100] The phase endpoint of the orbit is obtained through the J2 perturbation transfer model. Relative orbital elements endpoint state Controlling error amount for:

[0101] ;

[0102] Error for:

[0103] ;

[0104] Judgment error Is it less than the required control precision? ,when At that time, the demand for updated mobility , ,in Given the current mobility demand, the subscript r indicates the iteration number. Then return to step 3; otherwise, proceed to step 6.

[0105] Step 6: Output the normal pulse control sequence and the track pulse control sequence to complete the multi-pulse maneuver of the microsatellite.

[0106] The above method has a simple calculation process and is suitable for pulse sequence calculation frameworks for spaceborne computing, which can improve the real-time performance and reliability of on-orbit missions.

[0107] Example 1

[0108] Substitute the parameters according to the above method. The value is [400, -10000, 1500, 2000, 3000, -4000] (m). The reference orbit semi-major axis is [0, 0, -50, -40, 30, 70] (m). The maximum pulse constraint is 6878.137 km. The speed is 0.2 m / s, and the phase range of the orbit change is... The proposed method is validated with a rad range of [0, 75.3982]. The simulation results are detailed in [link to simulation results]. Figures 2-7 .

[0109] The absolute value of the precision of the iterative process and the result are as follows: Figure 2 As shown, when the J2 perturbation effect is not considered, the components of the relative orbital elements are... , , and The errors are relatively large, specifically -114.749m, -34.340m, 27.188m, and -139.391m; during each iteration of the calculation, , , and The error will be significantly reduced, and centimeter-level accuracy can be achieved after two iterations. and The value was consistently below 10 during each iteration.-11 The reason is that the J2 perturbation has very little effect on the semi-major axis and inclination of the absolute orbit. Figure 3 The pulse maneuver scheme obtained from the fourth iteration is presented. The red arrows in the figure represent the path pulses. The green arrow represents the normal pulse. The size and direction of the arrows indicate the magnitude and direction of the pulse value. This scheme involves a total of 29 pulse maneuvers, including 9 trajectory pulses. 20 normal pulses It takes a total of 10 orbital cycles to complete all pulse maneuvering operations. Figure 4 and Figure 5 The curves showing the variation of relative orbital elements under the influence of a stellar pulse are presented. Figure 4 of In the change curve, the reverse trace pulse causes The smaller the positive trace pulse makes Increase, 9-times trajectory pulse control It changed from 400m to 0. Figure 5 of In the change curve, the first two reversed trajectory pulses make Lower and make ,at this time The rate of change changes from negative to positive and moves towards the target quantity. Subsequent directional pulses gradually control this. Approaching the target value. Figure 6 and Figure 7 The curves showing the relative position change under 29 pulses are presented. As can be seen from the image, this pulse scheme controls the star trajectory to move from the initial state of drifting away from the original state towards a stable formation flying configuration.

[0110] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention without departing from its spirit and scope should be included within the protection scope of the present invention.

Claims

1. A multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation, characterized in that, The steps are as follows: Step 1: Input the initial relative orbital elements of the microsatellite and the target relative orbital elements, calculate the target control relative orbital elements in a near-circular orbit, and proceed to Step 2; Step 2: Based on the target control relative orbital elements under near-circular orbit, and the pulse control model based on the relative orbital elements, calculate the ideal normal pulse sequence and the ideal trajectory pulse sequence, and proceed to Step 3; Step 3: Based on the ideal normal pulse sequence and the ideal track pulse sequence, and according to the single maximum velocity increment constraint of the micro-nano satellite thruster, calculate the normal pulse control sequence and the track pulse control sequence using the supplementary allocation method, and proceed to step 4; Step 4: Based on the normal pulse control sequence and the trace pulse control sequence, obtain the relative orbital element endpoint state through the J2 perturbation transfer model, and proceed to step 5; Step 5: Calculate the difference between the endpoint state of the relative orbital elements and the target relative orbital elements to obtain the control error. Determine whether the error is less than the required control accuracy. If the error is greater than the required control accuracy, update the maneuver requirement and return to step 3; otherwise, proceed to step 6. Step 6: Output the normal pulse control sequence and the track pulse control sequence to complete the multi-pulse maneuver of the microsatellite.

2. The multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation according to claim 1, characterized in that, In step 1, input the initial relative orbital elements of the microsatellite and the target relative orbital elements, and calculate the target control relative orbital elements, as follows: The target control relative orbital elements The calculation method is as follows: ， In the formula, The initial relative orbital elements, ; For the target relative orbital elements, ; Each component is relative to the major semi-axis. Relative average longitude angle Relative eccentricity vector and relative tilt vector The superscript T represents the transpose of the vector, and the subscript... and These represent the initial quantity and the target quantity, respectively. It represents the difference between the target quantity and the initial quantity.

3. The multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation according to claim 2, characterized in that, In step 2, the pulse control model relative to the orbital elements is as follows: ， In the formula, For the reference orbital semi-axis, For the reference circular orbit's average angular velocity, This represents the average latitude argument of the reference orbit; , and These represent the ideal radial, trace, and normal pulses in the RTN coordinate system, respectively. The effect of pulse control on the relative orbital elements shows that the normal pulse... It will only affect the out-of-plane vector. radial pulse With trace pulse It will simultaneously affect the vector within the orbital plane. For control within the orbital plane, it is proposed that only trajectory pulses exist. Control methods.

4. The multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation according to claim 3, characterized in that, In step 2, based on the target control relative orbital elements in a near-circular orbit, and using a pulse control model based on the relative orbital elements, the ideal normal pulse sequence is calculated, as follows: Ideal normal pulse sequence and implementation phase The relative orbital elements controlled by the target The calculation method is as follows: ， In the formula, To change the relative orbital inclination vector magnitude, phase is implemented. lie in ~ Between; the ideal normal pulse sequence is implemented at a specified phase within one orbital period, and only needs to be implemented once.

5. The multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation according to claim 4, characterized in that, In step 2, based on the target control relative orbital elements in a near-circular orbit, and using a pulse control model based on the relative orbital elements, the ideal trajectory pulse sequence is calculated, as follows: Ideal Track Pulse Sequence and implementation phase The relative orbital elements controlled by the target Calculation, in The ideal trajectory pulse sequence is applied three times within a complete orbital cycle. , and , ; Calculate the selected parameters according to the following formula. and parameters : ； In the formula, To change the magnitude of the relative eccentricity vector, ; Based on the selected parameters and parameters Whether it is greater than 0, substitute it into the target control relative orbital elements. This will give you the ideal trajectory pulse sequence.

6. The multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation according to claim 5, characterized in that, when At that time, the ideal trajectory pulse sequence for three consecutive cycles is calculated using the following method. , and : ， at this time: ； ； ； In the formula, It is the floor function. and As an intermediate variable relating to the orbital period, and An intermediate variable for applying phase to the pulse; , and The applied phases are respectively , , : ， ， ， In the formula, ; Control the target relative orbital elements By substituting the above method, an ideal trace pulse sequence can be obtained.

7. The multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation according to claim 6, characterized in that, when At that time, the ideal trajectory pulse sequence for three consecutive cycles is calculated using the following method. , and : ， at this time: ， ， ； , and The applied phase , and They are respectively: ； ； ； Control the target relative orbital elements By substituting the above method, an ideal trace pulse sequence can be obtained.

8. The multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation according to claim 7, characterized in that, In step 3, based on the ideal normal pulse sequence and the ideal trajectory pulse sequence, and according to the constraint of the maximum velocity increment of the micro-nano satellite thruster in a single operation, the normal pulse control sequence is calculated using the supplementary allocation method, as follows: Define the maximum impulse constraint. To represent the maximum velocity increment that a microsatellite can provide in a single launch, the normal pulse control sequence allocation method is as follows: When passing through orbit phase Execute normal pulse Realize the relative orbital inclination vector When controlling microsatellites, if the limited thrust system capabilities of microsatellites are taken into account, that is... At phase angle Apply reverse pulse At phase angle Apply a positive pulse To achieve the control quantity required to fulfill the execution needs in multi-track maneuver control, the equivalent formula is expressed as: ； ； ； ； ； In the formula, As an intermediate variable relating to the orbital period, For the set of integers, For orbital phase Executed normal pulse; When satisfied Control requirements can be met in a timely manner; The normal pulse sequence is calculated using the supplementary allocation method: ； ； To supplement the normal pulse sequence after allocation by the allocation method, i represents the current pulse execution count. For the total number of normal pulses executed, when the phase angle is... At that time, execution and Pulses in opposite directions, when the phase angle is... At that time, execution and Pulses in the same direction; execution is then performed. Number of orbital revolutions required for the subnormal pulse for: 。 9. The multi-pulse maneuvering method for micro / nano satellites considering thrust constraints and J2 perturbation according to claim 8, characterized in that, In step 3, based on the constraint of the maximum velocity increment of the microsatellite thruster in a single operation, the trajectory pulse control sequence is calculated using the supplementary allocation method, as follows: First, design a tracing pulse using two semi-major axis and eccentricity pulses. , achieve and The control requirements are as follows: ； in, and Indicates two track pulses , The phase angle implemented; when and Greater than At this time, configuration needs to be split. and ,make and Execution is performed in a specified phase across multiple orbital cycles to meet the requirements for controlling the pulse quantity and the pulse quantity amplitude constraints. For relative average longitude argument When controlling, define Used to describe the trajectory pulse sequence If the relative average longitude angle of control is then: ， In the formula, i represents the current pulse execution count. This represents the total number of times the trace pulse is executed. It is the expected relative average longitude angle. It is the phase endpoint value of the orbit. Including semi-major axis and eccentricity tracking pulses and relative average longitude amplitude trace pulse ; The execution of this is unavoidable, but it can be mitigated through allocation. Achieve reduction Size; According to the supplementary allocation method, the semi-major axis and eccentricity trace pulses Controlled relative average longitude angle As shown in the following formula: ， in, ， ， In the formula, and They represent and The portion allocated in full, and This indicates the total number of times the trace pulse sequence is executed. and For the first and Sub-pulse control sequence. and They are respectively and Implemented phase angle, p and q represent respectively and Current number of implementations, and They represent and Current implementation phase, , This indicates the number of orbits required to complete a trajectory pulse maneuver mission. and This represents intermediate parameters.

10. The method for multi-pulse maneuvering of micro / nano satellites considering thrust constraints and J2 perturbation according to claim 9, characterized in that, In step 4, based on the normal pulse control sequence and the trace pulse control sequence, the relative orbital element endpoint state is obtained through the J2 perturbation transfer model, as follows: The J2 perturbation transfer model is as follows: The state transition matrix of the relative orbital elements as a function of time under J2 perturbation is: ， ； ； ， ， ， ， ； In the formula, Initial relative orbital elements Under the influence of J2 perturbation, after a period of time The relative orbital elements after that, Here is the state transition matrix under the HCW equation. Here is the state transition matrix under J2 perturbation; , , , and Intermediate parameters for convenient calculation, Indicates the eccentricity of the reference orbit. Indicates the inclination angle of the reference orbit. This represents the Earth's average radius. Represents the perturbation constant; The first derivatives of each element in the relative orbital elements under the influence of J2 perturbation, calculated according to the above formula, are as follows: ； Based on the above model, the final state of the relative orbital elements can be obtained by substituting the initial state of the relative orbital elements, the transition time, and the normal pulse control sequence and the trace pulse control sequence.

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