Rocket last-stage deorbit attitude angle optimizing method, device, equipment and medium
By predicting and optimizing the attitude angle when the Rocket's last stage is de-orbit, the problem that the attitude angle affects the effect but is difficult to optimize when the Rocket's last stage is de-orbit, achieving the best off-orbit effect at the last stage of the Rocket.
Patent Information
- Application Number
- CN202510061266.5
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-01-15
- Publication Date
- 2025-02-28
- Estimated Expiration
- 2045-01-15
AI Technical Summary
How to find the optimal attitude angle when the rocket is de-orbited from orbit at the last stage to achieve better off-orbit effect and solve the problem that attitude angle affects the off-orbit effect but is difficult to optimize in the prior art.
By obtaining the initial position, velocity vector and target de-orbiting duration of the rocket when the last stage is de-orbiting, predicting the de-orbiting trajectory at different target attitude angles, determining the candidate attitude angle that meets the measurement and control conditions and far-field safety conditions, and finally determining the optimal attitude angle.
The attitude angle optimization of the rocket's last stage is achieved when the rocket is derailed, the off-orbit effect is improved, and the trajectory after derailment is ensured that the trajectory after derailment meets the height threshold, measurement and control conditions and far-field safety conditions.
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Figure CN119533214A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of guidance and control technology, and in particular to a method, device, equipment and medium for optimizing the attitude angle of a rocket's final stage off-orbit. Background Art
[0002] Developing and implementing space debris mitigation technology is the main way to protect the future space environment, and deorbit control technology is an important part of space debris mitigation technology. The deorbiting of the last stage of a launch vehicle includes two methods: active deorbiting and passive deorbiting. Compared with passive deorbiting technology, active deorbiting technology achieves the purpose of deorbiting a spacecraft by consuming its own energy.
[0003] In related technologies, in order to reduce space debris, active deorbiting technology is used to achieve deorbiting of the rocket's final stage. The attitude angle of the rocket's final stage when it is deorbiting directly affects the deorbiting effect. Therefore, how to find the optimal attitude angle of the rocket's final stage when it is deorbiting is a technical problem that needs to be solved urgently. Summary of the invention
[0004] In view of the above problems, the present invention is proposed to provide a method, device, equipment and medium for optimizing the attitude angle of the rocket's final stage when deorbiting, which solves the above problems. The optimal attitude angle that meets the measurement and control conditions and far-field safety conditions can be predicted based on the first position and the first velocity vector of the rocket's final stage in the orbital coordinate system when deorbiting, as well as the target deorbiting time, to ensure that the rocket's final stage can have a better deorbiting effect when deorbiting at the optimal attitude angle.
[0005] In a first aspect, the present invention provides a method for optimizing the attitude angle of a rocket's final stage off-orbit, the method comprising:
[0006] Obtaining a first position and a first velocity vector of the rocket's final stage when it leaves the orbit, and a target off-orbit duration of the rocket's final stage in a launch coordinate system; the launch coordinate system is a coordinate system established with the launch point of the rocket's final stage leaving the orbit as the origin and the launch direction as the horizontal axis direction;
[0007] Predicting that the rocket final stage will deorbit at the first position with the first velocity vector and different target attitude angles, and the second position and second velocity vector of the rocket final stage in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration; the target attitude angle uses the orbital coordinate system as a reference coordinate system and includes a target yaw angle, a target roll angle and a target pitch angle; the orbital coordinate system is a coordinate system established with the center of mass of the rocket final stage as the origin and the direction of the first velocity vector as the horizontal axis; the earth-fixed coordinate system is a coordinate system established with the center of mass of the earth as the origin;
[0008] Determine the perigee height of the first first circle after the rocket final stage leaves orbit according to the second position corresponding to each target attitude angle and the corresponding second velocity vector;
[0009] Determine the difference between the perigee height of each of the first first laps and a preset height threshold, and determine the candidate attitude angles by taking the target attitude angle corresponding to the difference being less than or equal to the preset difference threshold;
[0010] The optimal attitude angle is determined from candidate attitude angles that meet preset measurement and control conditions and far-field safety conditions.
[0011] Optionally, before obtaining the first position and the first velocity vector of the rocket's final stage in the launch coordinate system when the final stage leaves the orbit, and the target departure time of the rocket's final stage, the method further includes:
[0012] Obtaining the remaining amount of explosives when the rocket's final stage leaves orbit and the amount of explosives consumed per unit time during the process of leaving orbit;
[0013] The ratio of the remaining amount of medicine to the consumed amount of medicine is determined as the target off-orbit duration.
[0014] Optionally, the predicting that the rocket final stage deorbits at the first position with the first velocity vector and different target attitude angles, and the second position and second velocity vector of the rocket final stage in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration, includes:
[0015] Assigning an initial attitude angle for the rocket final stage when leaving orbit; the initial attitude angle includes an initial yaw angle, an initial roll angle and an initial pitch angle, and the initial pitch angle is 180°; the initial attitude angle uses the orbital coordinate system as a reference coordinate system;
[0016] Predicting that the rocket final stage deorbits at the first position with the initial attitude angle and the first velocity vector, and the rocket final stage is at a third position and a third velocity vector in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration;
[0017] Determining the second perigee height of the rocket final stage after leaving orbit according to the third position and the third velocity vector;
[0018] If the second first-circle perigee height is less than the height threshold, it is predicted that the rocket's final stage will deorbit at the first position with the first velocity vector and different target attitude angles, and when the deorbit duration reaches the target deorbit duration, the rocket's final stage will be at the second position and second velocity vector in the Earth-fixed coordinate system.
[0019] Optionally, after determining the second first perigee height of the rocket final stage after leaving orbit according to the third position and the third velocity vector, the method further includes:
[0020] If the second first-circle perigee height is greater than or equal to the height threshold, the remaining amount of charge when the rocket's final stage leaves orbit is increased.
[0021] Optionally, the measurement and control condition is that after the last stage of the rocket leaves orbit, the angle between the pointing direction of the space-based antenna on the last stage of the rocket and the target straight line is less than or equal to a preset space-based beam angle, and the target straight line is the line between the last stage of the rocket and the Tianlian satellite.
[0022] Optionally, the far-field safety condition is that after the rocket's last stage leaves orbit, the minimum distance between the rocket's last stage and the target satellite is greater than or equal to a preset distance threshold, and the target satellite is a satellite sent into a predetermined orbit by the rocket's last stage.
[0023] Optionally, determining the optimal attitude angle from candidate attitude angles that meet preset measurement and control conditions and far-field safety conditions includes:
[0024] Obtaining a historical attitude angle of the rocket's final stage before it leaves orbit; the historical attitude angle uses the orbital coordinate system as a reference coordinate system;
[0025] Calculating an angle difference between the historical attitude angle and a preferred attitude angle, wherein the preferred attitude angle is a candidate attitude angle that satisfies a preset measurement and control condition and a far-field safety condition;
[0026] The preferred attitude angle corresponding to the minimum angle difference is determined as the optimal attitude angle.
[0027] In a second aspect, the present invention provides a device for optimizing the attitude angle of a rocket's final stage off-orbit, the device comprising:
[0028] An acquisition module is used to acquire a first position and a first velocity vector in a launch coordinate system when the rocket's final stage leaves the orbit, as well as a target off-orbit duration of the rocket's final stage; the launch coordinate system is a coordinate system established with the launch point of the rocket's final stage leaving the orbit as the origin and the launch direction as the horizontal axis direction;
[0029] A prediction module, used for predicting that the rocket final stage deorbits at the first position with the first velocity vector and different target attitude angles, and predicting the second position and the second velocity vector of the rocket final stage in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration; the target attitude angle uses the orbital coordinate system as a reference coordinate system and includes a target yaw angle, a target roll angle and a target pitch angle; the orbital coordinate system is a coordinate system established with the center of mass of the rocket final stage as the origin and the direction of the first velocity vector as the horizontal axis; the earth-fixed coordinate system is a coordinate system established with the center of mass of the earth as the origin;
[0030] A first determination module is used to determine the perigee height of the first first circle after the rocket's final stage leaves the orbit according to the second position corresponding to each target attitude angle and the corresponding second velocity vector;
[0031] A second determination module is used to respectively determine the difference between the perigee height of each of the first first laps and a preset height threshold, and determine the candidate attitude angle by taking the target attitude angle corresponding to the difference being less than or equal to the preset difference threshold;
[0032] The third determination module is used to determine the optimal attitude angle from the candidate attitude angles that meet the preset measurement and control conditions and far-field safety conditions.
[0033] In a third aspect, the present invention provides an electronic device comprising: a memory and a processor, wherein the memory and the processor are communicatively connected to each other, the memory stores computer instructions, and the processor executes the method described in the first aspect by executing the computer instructions.
[0034] In a fourth aspect, the present invention provides a computer-readable storage medium, wherein the computer-readable storage medium stores computer instructions, and the computer instructions are used to enable the computer to execute the method described in the first aspect.
[0035] The technical solution provided in the embodiments of the present invention has at least the following technical effects or advantages:
[0036] The embodiment of the present invention provides a method, device, equipment and medium for optimizing the attitude angle of the rocket's final stage off-orbit, which obtains the first position and first velocity vector of the rocket's final stage in the launch coordinate system when it is off-orbit, as well as the target off-orbit time of the rocket's final stage, and understands the initial conditions for off-orbit; predicts that the rocket's final stage will deorbit at the first position with the first velocity vector and different target attitude angles, and the second position and second velocity vector of the rocket's final stage in the earth-fixed coordinate system when the off-orbit time reaches the target off-orbit time, and obtains the position and velocity vector of the rocket's final stage after deorbiting at different target attitude angles; according to the second position and velocity vector corresponding to each target attitude angle The corresponding second velocity vector determines the height of the first perigee of the rocket's final stage after it leaves the orbit, and the height of the first perigee reflects the trajectory after leaving the orbit; the difference between the height of each first perigee and the preset height threshold is determined respectively, and the target attitude angle corresponding to the difference less than or equal to the preset difference threshold is used to determine the candidate attitude angle. The height of the first perigee under the candidate attitude angle will be near the required height threshold; the optimal attitude angle is determined from the candidate attitude angles that meet the preset measurement and control conditions and far-field safety conditions. The optimal attitude angle must at least meet the requirements of the first perigee height, measurement and control conditions, and far-field safety conditions. The optimal attitude angle determined by this method can enable the rocket's final stage to have a better deorbit effect.
[0037] The above description is only an overview of the technical solution of the present invention. In order to more clearly understand the technical means of the present invention, it can be implemented according to the contents of the specification. In order to make the above and other purposes, features and advantages of the present invention more obvious and easy to understand, the specific implementation methods of the present invention are listed below. BRIEF DESCRIPTION OF THE DRAWINGS
[0038] Various other advantages and benefits will become apparent to those of ordinary skill in the art by reading the detailed description of the preferred embodiments below. The accompanying drawings are only for the purpose of illustrating the preferred embodiments and are not to be considered as limiting the present invention. Moreover, the same reference symbols are used throughout the accompanying drawings to represent the same components. In the accompanying drawings:
[0039] Figure 1 It is a flow chart of a method for optimizing the attitude angle of the last stage of a rocket off-orbit provided by an embodiment of the present invention;
[0040] Figure 2 It is a schematic diagram of the orbit of a rocket final stage before and after leaving orbit provided by an embodiment of the present invention;
[0041] Figure 3 It is a structural block diagram of a device for optimizing the attitude angle of a rocket's final stage off-orbit provided by an embodiment of the present invention. DETAILED DESCRIPTION
[0042] In order to make the purpose, technical solutions and advantages of the present application clearer, the implementation mode of the present application will be further described in detail below in conjunction with the accompanying drawings. It should be understood that the embodiments of the present disclosure and the specific features in the embodiments are detailed descriptions of the technical solutions of the present application, rather than limitations on the technical solutions of the present application. In the absence of conflict, the embodiments of the present application and the technical features in the embodiments can be combined with each other.
[0043] Figure 1 The present invention provides a flowchart of a method for optimizing the attitude angle of the rocket's final stage off-orbit provided by an embodiment of the present invention. The method can be used when designing a rocket's final stage off-orbit attitude scheme. The rocket's final stage off-orbit attitude scheme is used to design the attitude angle of the rocket's final stage off-orbit in advance, such as Figure 1 As shown, the method includes:
[0044] Step S110, obtaining the first position and first velocity vector of the rocket's final stage in the launch coordinate system when it leaves orbit, and the target deorbit duration of the rocket's final stage.
[0045] Among them, the launch coordinate system is a coordinate system established with the launch point where the last stage of the rocket leaves the orbit as the origin and the launch direction as the horizontal axis direction; its vertical axis is upward along the plumb line of the launch point, and its vertical axis, its horizontal axis and its vertical axis form a right-handed rectangular coordinate system.
[0046] In the embodiment of the present application, the first position and the first velocity vector are designed in advance, and the previously designed first position and the first velocity vector can be directly used when designing the rocket's final stage deorbit attitude plan.
[0047] Optionally, before step S110, a method for calculating the target off-orbit duration is further included, specifically including:
[0048] Obtain the remaining amount of propellant when the rocket's final stage leaves orbit and the amount of propellant consumed per unit time during the deorbit process; determine the ratio of the remaining amount of propellant to the amount of propellant consumed as the target deorbit time.
[0049] In the embodiment of the present application, the remaining amount of medicine and the consumed amount of medicine are also pre-designed. In order to avoid the pollution caused by the residual fuel of the rocket engine, the remaining amount of medicine is consumed as much as possible during the deorbiting process, so the time required to consume the remaining amount of medicine is the target deorbiting time. That is, the remaining amount of medicine is recorded as m, the consumed amount of medicine is recorded as k, and the target deorbiting time t = m / k.
[0050] Step S120, predicting that the rocket's final stage will deorbit at a first position with a first velocity vector and different target attitude angles, and the second position and second velocity vector of the rocket's final stage in the Earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration.
[0051] Among them, the target attitude angle uses the orbital coordinate system as the reference coordinate system and includes the target yaw angle, target roll angle and target pitch angle; the orbital coordinate system is a coordinate system established with the center of mass of the rocket's final stage as the origin and the direction of the first velocity vector as the horizontal axis. Its vertical axis is outside the orbit pointing to the deorbit point and is perpendicular to its horizontal axis; its vertical axis is perpendicular to its horizontal and vertical axes respectively, and its establishment satisfies the right-hand rule. Instantaneous orbital plane
[0052] The target pitch angle is the angle between the projection of the X-axis (horizontal axis) of the rocket body in the instantaneous orbital plane and the X-axis of the orbital coordinate system; the target yaw angle is the angle between the X-axis of the rocket body and the instantaneous orbital plane; the target roll angle is the rotation angle of the rocket body around the X-axis of the orbital coordinate system. When the target roll angle = 0, the Y-axis (vertical axis) of the rocket body is in the instantaneous orbital plane. The instantaneous orbital plane is the plane formed by the horizontal and vertical axes of the orbital coordinate system.
[0053] The target yaw angle, target roll angle and target pitch angle are all in the range of -180° to 180°. The range between -180° and 180° can be divided into multiple groups of small ranges with the same interval, and different attitude angles are taken from different small ranges. The number of small ranges of target yaw angle values is recorded as m, and the target yaw angle psi = -180 + [360 / (m×i)], i = 0, 1, 2, 3..., 360 / (m-1); the number of small ranges of target roll angle values is recorded as n, and the target yaw angle gam = -180 + [360 / (n×j)], j = 0, 1, 2, 3..., 360 / (n-1).
[0054] It should be noted that the two different attitude angles may refer to at least one of a target yaw angle, a target roll angle, and a target pitch angle being different.
[0055] Among them, the Earth-fixed coordinate system is a coordinate system established with the Earth's center of mass as its origin.
[0056] In the embodiment of the present application, it is assumed that the rocket's final stage starts to deorbit from the first position, the velocity vector at the time of deorbit is the first velocity vector, and the attitude angle is the target attitude angle, and the second position and the second velocity vector of the rocket's final stage in the earth-fixed coordinate system are predicted when the deorbit duration reaches the target deorbit duration (at the end of deorbit). Then, while keeping the first position and the first velocity vector unchanged, the target attitude angle is changed, and the deorbit at the changed target attitude angle is predicted again, and the second position and the second velocity vector of the rocket's final stage in the earth-fixed coordinate system at the end of deorbit are obtained, thereby obtaining the same first position and the same first velocity vector, and the second position and the second velocity vector under different target attitude angles.
[0057] The calculation of the velocity and position of the rocket's final stage during the deorbiting process generally uses the Runge-Kutta integration method. By integrating the force on the rocket's final stage and combining the first position and first velocity vector, the final position and final velocity vector of the rocket's final stage in the launch coordinate system at the end of the deorbiting process can be calculated. Then the final position and final velocity vector of the rocket's final stage in the launch coordinate system The second position and second velocity vector of the rocket's final stage converted to the Earth-fixed coordinate system During the deorbiting process, the velocity increment ΔV of the rocket's last stage can be a fixed value, and the size of ΔV can be designed in advance.
[0058] Step S130, determining the perigee height of the first first circle after the rocket's final stage leaves orbit according to the corresponding second position and the corresponding second velocity vector under each target attitude angle.
[0059] In the embodiment of the present application, the rocket's final stage deorbits from the final stage's initial orbit before deorbit into an elliptical orbit after deorbit. There is a point in the elliptical orbit that is closest to the earth, which is called the perigee. The distance between the perigee and the earth is called the perigee height. The perigee height of the first elliptical orbit that the rocket's final stage enters after deorbit is called the first-circle perigee height.
[0060] In the embodiment of the present application, the perigee height of the first lap after leaving the orbit is calculated by numerical integration, and step S130 includes:
[0061] The first step is to calculate the semi-major axis of the elliptical orbit according to formula (1).
[0062]
[0063] Among them, a represents the semi-major axis of the elliptical orbit, μ represents the gravitational constant of the earth, V1 represents the magnitude of the second velocity vector, R represents the magnitude of the distance vector from the last stage of the rocket to the center of the earth when it is in the second position at the end of deorbit, and E represents the orbital energy.
[0064] The second step is to calculate the angular momentum of the elliptical orbit according to formula (2).
[0065]
[0066] in, represents angular momentum, It represents the distance vector from the center of the earth to the last stage of the rocket at the second position at the end of deorbit. represents the second velocity vector.
[0067] The third step is to calculate the eccentricity according to formula (3).
[0068]
[0069] Here, e represents the eccentricity.
[0070] The fourth step is to calculate the perigee height of the first lap according to formula (4).
[0071] h=a×(1-e)-R e ; Formula (4)
[0072] Where h is the height of the first perigee, R e Represents the radius of the Earth.
[0073] Therefore, according to the corresponding second position and the corresponding second velocity vector at each target attitude angle, the above four formulas can be used to determine the perigee height of the first circle after the rocket's last stage leaves orbit, and the perigee height of the first circle at all target attitude angles can be obtained.
[0074] Step S140, respectively determine the difference between the perigee height of each first lap and a preset height threshold, and determine the candidate attitude angles by taking the target attitude angle corresponding to the difference being less than or equal to the preset difference threshold.
[0075] In an embodiment of the present application, a candidate attitude angle is first selected from all target attitude angles. The candidate attitude angle is a target attitude angle whose first-circle perigee height is near a height threshold. Deorbiting with the candidate attitude angle can make the first-circle perigee height after deorbit meet the requirements.
[0076] Step S150: determine the optimal attitude angle from the candidate attitude angles that meet the preset measurement and control conditions and far-field safety conditions.
[0077] In the embodiment of the present application, the candidate attitude angles that meet the preset measurement and control conditions and far-field safety conditions are determined as the preferred attitude angles, and then the optimal attitude angle is determined from the preferred attitude angles. Finally, deorbiting with the optimal attitude angle can ensure that the first perigee height of the rocket's final stage after deorbiting meets the requirements, and meets the measurement and control conditions and far-field safety conditions, so that the deorbit effect is optimal.
[0078] Optionally, the measurement and control condition is that after the last stage of the rocket leaves orbit, the angle between the pointing direction of the space-based antenna on the last stage of the rocket and the target straight line is less than or equal to a preset space-based beam angle, and the target straight line is the line between the last stage of the rocket and the Tianlian satellite.
[0079] In an embodiment of the present application, the angle between the space-based antenna pointing and the target straight line at different times can be calculated according to the position of the rocket's final stage at different times after deorbiting, the installation angle of the space-based antenna and the position of the Tianlian satellite, and then it is determined whether the angle meets the constraint requirements of the space-based beam angle, and the candidate attitude angles that do not meet the constraint requirements are eliminated, leaving the candidate attitude angles that meet the constraint requirements. In other words, deorbiting with the candidate attitude angles that meet the constraint requirements of the space-based beam angle can make the rocket's final stage after deorbiting can normally perform data transmission. Among them, the line between the rocket's final stage and the Tianlian satellite, i.e. the target straight line, can be determined according to the position of the rocket's final stage and the position of the Tianlian satellite.
[0080] Among them, the space-based antenna is used to transmit data to the Tianlian satellite, which is China's tracking and data relay satellite.
[0081] Optionally, the far-field safety condition is that after the rocket's last stage leaves orbit, the minimum distance between the rocket's last stage and the target satellite is greater than or equal to a preset distance threshold.
[0082] Among them, the target satellite is the satellite that is sent into the predetermined orbit by the last stage of the rocket.
[0083] In an embodiment of the present application, the distance between the rocket's final stage and the target satellite is calculated based on the positions of the rocket's final stage and the target satellite at different times after leaving orbit, and the candidate attitude angles whose minimum distance is less than a preset distance threshold are eliminated, leaving the candidate attitude angles whose minimum distance is greater than or equal to the preset distance threshold, that is, leaving the candidate attitude angles between the rocket's final stage and the target satellite that meet the far-field safety requirements.
[0084] Among them, after the last stage of the rocket begins to leave orbit, the distance between the last stage of the rocket and the satellite is monotonically increasing or decreasing over a period of time. If the distance is monotonically increasing, the distance will become larger and larger, and at this time it can be considered that the far-field safety meets the requirements; if the distance shows a decreasing trend over a period of time, and the minimum distance after the decrease is less than the distance threshold L0, it is considered that the far-field safety does not meet the requirements; if the minimum distance after the decrease is greater than or equal to the distance threshold L0, it is considered that the far-field safety meets the requirements. Among them, the distance threshold L0 can be set according to the actual situation and the needs of the satellite.
[0085] Optionally, step S120 includes:
[0086] The first step is to assign an initial attitude angle to the rocket's final stage when it leaves orbit.
[0087] The initial attitude angle includes the initial yaw angle, the initial roll angle and the initial pitch angle, and the initial pitch angle is 180°. The initial attitude angle uses the orbital coordinate system as the reference coordinate system.
[0088] In the embodiment of the present application, it can be known from formula (1) that the semi-major axis decreases as the second velocity vector decreases. When the semi-major axis decreases, the corresponding first perigee height will decrease. Therefore, the smaller the second velocity vector is, the lower the first perigee height will be. Therefore, theoretically speaking, when the rocket's final stage is off-orbit, if it is ignited completely in the opposite direction of the first velocity vector of the rocket's final stage, the speed of the rocket's final stage when it is off-orbit can be minimized, and the first perigee height of the rocket's final stage after leaving the orbit can be reduced to the greatest extent. If it is ignited completely in the opposite direction of the first velocity vector of the rocket's final stage, the rocket's pitch angle needs to reach 180°, so first allocate an attitude angle with an initial pitch angle of 180°, and then assume that the rocket's final stage is off-orbit when the pitch angle is 180°.
[0089] The second step is to predict the deorbiting of the rocket's final stage at the first position with the initial attitude angle and the first velocity vector, and the third position and the third velocity vector of the rocket's final stage in the earth-fixed coordinate system when the deorbiting time reaches the target deorbiting time.
[0090] In an embodiment of the present application, it is assumed that the rocket's final stage is deorbited at a first position with a first velocity vector, and the pitch angle at the time of deorbiting is the initial pitch angle, that is, 180°. Then, the third position and the third velocity vector of the rocket's final stage in the earth-fixed coordinate system are determined at the end of deorbiting.
[0091] The third step is to determine the perigee height of the second first circle after the rocket's final stage leaves orbit based on the third position and the third velocity vector.
[0092] According to the method described above, the second first-circle perigee height at the third position and the third velocity vector after the rocket's final stage leaves orbit is calculated, and the minimum first-circle perigee height after the rocket's final stage leaves orbit is obtained.
[0093] Step 4. If the perigee height of the second first circle is less than the height threshold, it is predicted that the rocket's final stage will deorbit at the first position with the first velocity vector and different target attitude angles, and the rocket's final stage will be at the second position and second velocity vector in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration.
[0094] In an embodiment of the present application, if the height of the second first perigee is less than the height threshold, it means that no matter which angle the pitch angle is selected when leaving the orbit, the height of the first perigee after leaving the orbit will be less than the height threshold, and the height requirement will be met. In this case, the target attitude angle is continuously changed to make a deorbit assumption, and the second position and second velocity vector of the rocket's final stage in the ground-fixed coordinate system are obtained when the rocket's final stage leaves the orbit at different target attitude angles and ends the deorbit. Among them, the height threshold can be 100km.
[0095] Figure 2 FIG. 1 is a schematic diagram of a trajectory of a rocket before and after the final stage leaves the orbit provided by an embodiment of the present invention. Figure 2As shown, the orbit 1 before the rocket's final stage leaves the orbit is the final stage initial orbit, and the minimum height (the first first circle perigee height) of the first orbit after leaving the orbit is the orbit of the same height from the earth 2 includes the first elliptical orbit 3 and the second elliptical orbit 4, so, under the same target yaw angle and the same target roll angle, there are two target pitch angles when the first circle perigee height is the same. That is to say, the direction of the speed V0 of the rocket's final stage before leaving the orbit is forward, and the direction of the speed V when leaving the orbit is relative to the speed direction before leaving the orbit, which can be upward or downward, and the speed variable ΔV of the deorbit process also has two directions, so the deorbit mode includes the deorbit mode of ascending and the deorbit mode of descending, and the corresponding target pitch angle also has two groups of solutions. These two groups of solutions are respectively called "upward solution" and "downward solution". At the same time, when the angle between ΔV and the direction of the first velocity vector at the time of leaving the orbit is 180°, the perigee height will be the lowest, and as the angle approaches 0°, the perigee height is getting higher and higher.
[0096] Therefore, if the height of the second perigee is less than the height threshold, it means that there is a solution for the target pitch angle. When designing the attitude plan for the final stage of the rocket to deorbit, in order to improve the design efficiency, the candidate attitude angles can be tried out little by little without determining the height of the first perigee under all target attitude angles.
[0097] Therefore, the fourth step includes: iteratively calculating the target pitch angle according to the upward de-orbiting method and the downward de-orbiting method respectively, until the calculated perigee height of the first circle after de-orbiting reaches the vicinity of the required height threshold.
[0098] Specifically, from the previously divided small ranges of different values of the target yaw angle and the target roll angle, different target yaw angles and different target roll angles are determined according to different i and j, and the target pitch angle that meets the off-track requirement is solved through trajectory integral iteration. Among them, the target pitch angle is iterated starting from 180°.
[0099] First, select a suitable target yaw angle and target roll angle, first determine the first perigee height corresponding to the initial value of the target pitch angle of 180°, and determine whether the difference between the first perigee height and the height threshold is less than or equal to the difference threshold. If the difference is less than or equal to the difference threshold, the target yaw angle and target roll angle, as well as the target pitch angle of 180°, are determined as candidate attitude angles. If the target yaw angle and target roll angle remain unchanged, if the target pitch angle is changed to other angles (not 180°), the first perigee height will become larger and larger, resulting in a larger difference. Therefore, target pitch angles at other angles will not meet the requirements of the first perigee height, and will not be selected as candidate attitude angles, so there is no need to determine the first perigee height at other target pitch angles.
[0100] If the difference is greater than the difference threshold, it means that the perigee height of the first lap is not near the height threshold, and the target pitch angle of 180° does not meet the requirement. Then the target pitch angle needs to be changed, and the adjustment amount of the target pitch angle can be determined according to the difference. Therefore, the adjusted target pitch angle is the target pitch angle before adjustment ± the adjustment amount. The target pitch angle before adjustment + the adjustment amount is the "upward solution" of the adjusted target pitch angle, and the target pitch angle before adjustment - the adjustment amount is the "downward solution" of the adjusted target pitch angle. When the target yaw angle and the target roll angle are kept unchanged, the target pitch angle after the first adjustment becomes 180° ± the adjustment amount, and the perigee height of the first lap under the first adjusted target pitch angle is calculated, and then it is determined whether the difference between the perigee height of the first lap and the height threshold is less than or equal to the difference threshold. If so, the target yaw angle and target roll angle, as well as the target pitch angle after the first adjustment, are determined as candidate attitude angles; if not, the target pitch angle is adjusted using the same iterative method until the difference between the first perigee height of the adjusted target pitch angle and the height threshold is less than or equal to the difference threshold, the iteration stops, the uplink and downlink solutions of the target pitch angle are obtained, and the corresponding target pitch angle, the corresponding target yaw angle, and the corresponding target roll angle are determined as candidate attitude angles; or, until the adjusted target pitch angle is 0, the iteration stops, and no candidate attitude angle is iterated this time. Finally, under the same target yaw angle and the same target roll angle, at most one uplink target pitch angle and one downlink target pitch angle, i.e., two sets of candidate attitude angles, can be determined.
[0101] Then, the target yaw angle or target roll angle is adjusted, and a new round of iteration is performed under a new combination of target yaw angle and target roll angle to determine a new set of target pitch angles and obtain two new sets of candidate attitude angles. In this way, the target yaw angle, target roll angle and target pitch angle that meet the requirements under different target yaw angles, different target roll angles and different target pitch angles are selected to obtain multiple sets of candidate attitude angles. The above method is used to iteratively determine all candidate attitude angles that meet the requirements, which can reduce the design time.
[0102] Exemplarily, the target pitch angle adjustment calculation formula is:
[0103] Δα=(H1-H) / 10000; Formula (5)
[0104] Wherein, Δα represents the adjustment amount, H1 represents the perigee height of the first lap, and H represents the height threshold.
[0105] Optionally, the third step is followed by a fifth step:
[0106] If the perigee height of the second first circle is greater than or equal to the height threshold, the remaining amount of charge when the rocket's final stage leaves orbit will be increased.
[0107] In an embodiment of the present application, if the height of the second first perigee is greater than or equal to the height threshold, it means that no matter what attitude angle the rocket's final stage uses to leave orbit, the height of the first perigee after leaving orbit will not meet the requirements. Therefore, the remaining amount of charge when the rocket's final stage leaves orbit is increased, thereby increasing the target deorbit time and increasing the distance between the rocket's final stage and the earth after leaving orbit, so that the height of the first perigee after leaving orbit meets the requirements.
[0108] Optionally, step S150 includes:
[0109] Obtain the historical attitude angle of the rocket's last stage before it leaves orbit; calculate the angle difference between the historical attitude angle and the preferred attitude angle; and determine the preferred attitude angle corresponding to the smallest angle difference as the optimal attitude angle.
[0110] The historical attitude angle can be the attitude angle of the rocket's last stage just before leaving orbit, and the preferred attitude angle is a candidate attitude angle that meets the preset measurement and control conditions and far-field safety conditions. The historical attitude angle uses the orbital coordinate system as the reference coordinate system.
[0111] It can be understood that, combined with the de-orbit attitude adjustment angle minimization standard, the angle with the smallest angle change is selected from the preferred attitude angles as the optimal attitude angle, that is, the preferred attitude angle with the smallest change between the de-orbit moment and the moment before de-orbit is selected as the optimal attitude angle, because this optimal attitude angle requires the smallest amount of adjustment, which is more conducive to the de-orbit operation.
[0112] In an embodiment of the present application, when designing a deorbit attitude plan, the plans are automatically screened in combination with measurement and control requirements, far-field safety requirements, and attitude adjustment angle minimization standards to quickly obtain the optimal rocket deorbit attitude plan, thereby reducing the difficulty and workload of manual design and improving design efficiency.
[0113] Based on the same inventive concept, an embodiment of the present invention also provides a device for optimizing the attitude angle of the last stage of a rocket off-orbit. Figure 3 : is a structural block diagram of a device for optimizing the attitude angle of the last stage of a rocket off-orbit provided by an embodiment of the present invention, such as Figure 3 As shown, the device 300 includes an acquisition module 301 , a prediction module 302 , a first determination module 303 , a second determination module 304 and a third determination module 305 .
[0114] The acquisition module 301 is used to acquire the first position and the first velocity vector of the rocket's final stage in the launch coordinate system when the rocket's final stage leaves the orbit, as well as the target departure time of the rocket's final stage; the launch coordinate system is based on...
[0115] The established coordinate system;
[0116] The prediction module 302 is used to predict the deorbiting of the rocket's final stage at a first position with a first velocity vector and different target attitude angles, and the second position and second velocity vector of the rocket's final stage in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration; the target attitude angle uses the orbital coordinate system as a reference coordinate system and includes a target yaw angle, a target roll angle, and a target pitch angle; the orbital coordinate system is a coordinate system established with the center of mass of the rocket's final stage as the origin and the direction of the first velocity vector as the horizontal axis; the earth-fixed coordinate system is a coordinate system established with the center of mass of the earth as the origin;
[0117] The first determination module 303 is used to determine the perigee height of the first first circle after the rocket's final stage leaves the orbit according to the second position corresponding to each target attitude angle and the corresponding second velocity vector;
[0118] The second determination module 304 is used to determine the difference between the perigee height of each first lap and a preset height threshold, and determine the candidate attitude angles according to the target attitude angles corresponding to the difference being less than or equal to the preset difference threshold;
[0119] The third determination module 305 is used to determine the optimal attitude angle from the candidate attitude angles that meet the preset measurement and control conditions and far-field safety conditions.
[0120] Optionally, the apparatus 300 further includes a fourth determining module, configured to:
[0121] Obtain the remaining amount of propellant when the rocket's final stage leaves orbit and the amount of propellant consumed per unit time during the process of leaving orbit;
[0122] The ratio of the remaining amount of medicine to the consumed amount of medicine is determined as the target deorbit time.
[0123] Optionally, the prediction module 302 includes:
[0124] The allocation unit is used to allocate the initial attitude angle of the rocket's final stage when it leaves the orbit; the initial attitude angle includes the initial yaw angle, the initial roll angle and the initial pitch angle, and the initial pitch angle is 180°; the initial attitude angle uses the orbital coordinate system as the reference coordinate system;
[0125] The first prediction unit is used to predict that the rocket's final stage deorbits at a first position with an initial attitude angle and a first velocity vector, and a third position and a third velocity vector of the rocket's final stage in an earth-fixed coordinate system when the deorbit duration reaches a target deorbit duration;
[0126] A determination unit, used for determining the second first perigee height of the rocket final stage after leaving the orbit according to the third position and the third velocity vector;
[0127] The second prediction unit is used to predict that the rocket's final stage will deorbit at a first position with a first velocity vector and different target attitude angles if the second first-circle perigee height is less than a height threshold, and predict the second position and second velocity vector of the rocket's final stage in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration.
[0128] Optionally, the prediction module 302 further includes an adding unit, configured to:
[0129] If the perigee height of the second first circle is greater than or equal to the height threshold, the remaining amount of charge when the rocket's final stage leaves orbit will be increased.
[0130] Optionally, the measurement and control condition is that after the last stage of the rocket leaves orbit, the angle between the pointing direction of the space-based antenna on the last stage of the rocket and the target straight line is less than or equal to a preset space-based beam angle, and the target straight line is the line between the last stage of the rocket and the Tianlian satellite.
[0131] Optionally, the far-field safety condition is that after the rocket's last stage leaves orbit, the minimum distance between the rocket's last stage and the target satellite is greater than or equal to a preset distance threshold, and the target satellite is a satellite sent into a predetermined orbit by the rocket's last stage.
[0132] Optionally, the third determining module 305 is further configured to:
[0133] Get the historical attitude angle of the rocket's last stage before it leaves orbit; the historical attitude angle uses the orbital coordinate system as the reference coordinate system;
[0134] Calculate the angle difference between the historical attitude angle and the preferred attitude angle, the preferred attitude angle being a candidate attitude angle that meets the preset measurement and control conditions and far-field safety conditions;
[0135] The preferred attitude angle corresponding to the smallest angle difference is determined as the optimal attitude angle.
[0136] It can be understood that the device provided in the above embodiment is only illustrated by the division of the above functional modules. In actual applications, the above functions can be assigned to different functional modules as needed, that is, the internal structure of the device can be divided into different functional modules to complete all or part of the functions described above.
[0137] An embodiment of the present invention further provides an electronic device, which may include a processor and a memory, wherein the processor and the memory may be communicatively connected to each other via a bus or other means.
[0138] The processor may be a central processing unit (CPU), or an application specific integrated circuit (ASIC), or may be configured to implement one or more integrated circuits of the embodiments of the present application, or may also be other general-purpose processors, digital signal processors (DSP), application specific integrated circuits (ASIC), field programmable gate arrays (FPGA) or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components and other chips, or a combination of the above-mentioned chips.
[0139] The memory may include a large capacity memory for data or instructions. By way of example and not limitation, the memory may include a hard disk drive (HDD), a floppy disk drive, a flash memory, an optical disk, a magneto-optical disk, a magnetic tape, or a universal serial bus (USB) drive or a combination of two or more of these. Where appropriate, the memory may include a removable or non-removable (or fixed) medium. Where appropriate, the memory may be inside or outside the electronic device. In a particular embodiment, the memory may be a non-volatile solid-state memory.
[0140] In one example, the memory may be a read-only memory (ROM). In one example, the ROM may be a mask-programmed ROM, a programmable ROM (PROM), an erasable PROM (EPROM), an electrically erasable PROM (EEPROM), an electrically rewritable ROM (EAROM), or a flash memory, or a combination of two or more of these.
[0141] The processor implements any one of the methods for optimizing the attitude angle of the rocket's final stage off-orbit in the above-mentioned embodiments by reading and executing computer program instructions stored in the memory.
[0142] In one example, the electronic device may further include a communication interface and a bus. Among them, the processor, memory, and communication interface are connected through a bus and communicate with each other. The communication interface is mainly used to realize the communication between the modules, devices, units and / or devices in the embodiment of the present application. Where appropriate, the bus may include one or more buses.
[0143] In addition, in combination with the method for optimizing the attitude angle of the rocket's final stage off-orbit in the above embodiment, the embodiment of the present invention can provide a computer-readable storage medium for implementation. The computer-readable storage medium stores computer program instructions; when the computer program instructions are executed by the processor, any method for optimizing the attitude angle of the rocket's final stage off-orbit in the above embodiment is implemented.
[0144] Those skilled in the art can understand that all or part of the processes in the above-mentioned embodiments can be implemented by instructing the relevant hardware through a computer program, and the program can be stored in a computer-readable storage medium, and when the program is executed, it can include the processes of the embodiments of the above-mentioned methods. Among them, the storage medium can be a read-only memory (ROM), a random access memory (RAM), a flash memory, a hard disk drive (HDD) or a solid-state drive (SSD), etc.; the storage medium can also include a combination of the above-mentioned types of memory.
[0145] The technical solutions in the above embodiments of the present application have at least the following technical effects or advantages:
[0146] The embodiment of the present invention provides a method, device, equipment and medium for optimizing the attitude angle of the rocket's final stage off-orbit, which obtains the first position and first velocity vector of the rocket's final stage in the launch coordinate system when it is off-orbit, as well as the target off-orbit time of the rocket's final stage, and understands the initial conditions for off-orbit; predicts that the rocket's final stage will deorbit at the first position with the first velocity vector and different target attitude angles, and the second position and second velocity vector of the rocket's final stage in the earth-fixed coordinate system when the off-orbit time reaches the target off-orbit time, and obtains the position and velocity vector of the rocket's final stage after deorbiting at different target attitude angles; according to the second position and velocity vector corresponding to each target attitude angle The corresponding second velocity vector determines the height of the first perigee of the rocket's final stage after it leaves the orbit, and the height of the first perigee reflects the trajectory after leaving the orbit; the difference between the height of each first perigee and the preset height threshold is determined respectively, and the target attitude angle corresponding to the difference less than or equal to the preset difference threshold is used to determine the candidate attitude angle. The height of the first perigee under the candidate attitude angle will be near the required height threshold; the optimal attitude angle is determined from the candidate attitude angles that meet the preset measurement and control conditions and far-field safety conditions. The optimal attitude angle must at least meet the requirements of the first perigee height, measurement and control conditions, and far-field safety conditions. The optimal attitude angle determined by this method can enable the rocket's final stage to have a better deorbit effect.
[0147] In the description provided herein, a large number of specific details are described. However, it is understood that embodiments of the present invention can be practiced without these specific details. In some instances, well-known methods, structures and techniques are not shown in detail so as not to obscure the understanding of this description.
[0148] Similarly, it should be understood that in order to streamline the present disclosure and aid in understanding one or more of the various inventive aspects, in the above description of exemplary embodiments of the present invention, various features of the present invention are sometimes grouped together into a single embodiment, figure, or description thereof. However, this disclosed method should not be interpreted as reflecting the intention that the claimed invention requires more features than those explicitly recited in each claim. More specifically, as reflected in the claims below, inventive aspects lie in less than all the features of the individual embodiments disclosed above. Therefore, the claims that follow the specific embodiment are hereby expressly incorporated into the specific embodiment, with each claim itself serving as a separate embodiment of the present invention.
[0149] It should be noted that the above embodiments illustrate the present invention rather than limit it, and that those skilled in the art may devise alternative embodiments without departing from the scope of the appended claims. In the claims, any reference symbol between brackets shall not be construed as a limitation on the claims. The word "comprising" does not exclude the presence of components or steps not listed in the claims. The word "one" or "an" preceding a component does not exclude the presence of a plurality of such components. The present invention may be implemented by means of hardware comprising a number of different components and by means of a suitably programmed computer. In a unit claim enumerating a number of devices, several of these devices may be embodied by the same hardware item. The use of the words first, second, and third, etc., does not indicate any order. These words may be interpreted as names.
Claims
1. A method for optimizing the attitude angle of a rocket's final stage off-orbit, characterized in that: The method comprises: Obtaining a first position and a first velocity vector of the rocket's final stage when it leaves the orbit, and a target off-orbit duration of the rocket's final stage in a launch coordinate system; the launch coordinate system is a coordinate system established with the launch point of the rocket's final stage leaving the orbit as the origin and the launch direction as the horizontal axis direction; Predicting that the rocket final stage deorbits at the first position with the first velocity vector and different target attitude angles, and the second position and the second velocity vector of the rocket final stage in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration; the target attitude angle uses the orbital coordinate system as a reference coordinate system and includes a target yaw angle, a target roll angle and a target pitch angle; the orbital coordinate system is a coordinate system established with the center of mass of the rocket final stage as the origin and the direction of the first velocity vector as the horizontal axis; the earth-fixed coordinate system is a coordinate system established with the center of mass of the earth as the origin; Determine the perigee height of the first first circle after the rocket final stage leaves the orbit according to the second position corresponding to each target attitude angle and the corresponding second velocity vector; Determine the difference between the perigee height of each of the first first laps and a preset height threshold, and determine the candidate attitude angles by taking the target attitude angles corresponding to the difference being less than or equal to the preset difference threshold; The optimal attitude angle is determined from candidate attitude angles that meet preset measurement and control conditions and far-field safety conditions.
2. The method for optimizing the attitude angle of the rocket's final stage off-orbit according to claim 1, characterized in that: Before obtaining the first position and the first velocity vector of the rocket's final stage in the launch coordinate system when the final stage leaves the orbit, and the target departure time of the rocket's final stage, the method further comprises: Obtaining the remaining amount of explosives when the rocket's final stage leaves orbit and the amount of explosives consumed per unit time during the process of leaving orbit; The ratio of the remaining amount of medicine to the consumed amount of medicine is determined as the target off-orbit duration.
3. The method for optimizing the attitude angle of the rocket's final stage off-orbit according to claim 2, characterized in that: The method predicts that the rocket final stage deorbits at the first position with the first velocity vector and different target attitude angles, and predicts the second position and the second velocity vector of the rocket final stage in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration, including: Assigning an initial attitude angle for the rocket final stage when leaving orbit; the initial attitude angle includes an initial yaw angle, an initial roll angle and an initial pitch angle, and the initial pitch angle is 180°; the initial attitude angle uses the orbital coordinate system as a reference coordinate system; Predicting that the rocket final stage deorbits at the first position with the initial attitude angle and the first velocity vector, and the rocket final stage is at a third position and a third velocity vector in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration; Determining the second perigee height of the rocket final stage after leaving orbit according to the third position and the third velocity vector; If the second first-circle perigee height is less than the height threshold, it is predicted that the rocket's final stage will deorbit at the first position with the first velocity vector and different target attitude angles, and when the deorbit duration reaches the target deorbit duration, the rocket's final stage will be at the second position and second velocity vector in the Earth-fixed coordinate system.
4. The method for optimizing the attitude angle of the rocket's final stage off-orbit according to claim 3, characterized in that: After determining the second first perigee height of the rocket final stage after leaving orbit according to the third position and the third velocity vector, the method further comprises: If the second first-circle perigee height is greater than or equal to the height threshold, the remaining amount of charge when the rocket's final stage leaves orbit is increased.
5. The method for optimizing the attitude angle of the rocket's final stage off-orbit according to claim 1, characterized in that: The measurement and control condition is that after the last stage of the rocket leaves orbit, the angle between the pointing direction of the space-based antenna on the last stage of the rocket and the target straight line is less than or equal to a preset space-based beam angle, and the target straight line is the line between the last stage of the rocket and the Tianlian satellite.
6. The method for optimizing the attitude angle of the rocket's final stage off-orbit according to claim 1, characterized in that: The far-field safety condition is that after the rocket's final stage leaves orbit, the minimum distance between the rocket's final stage and the target satellite is greater than or equal to a preset distance threshold, and the target satellite is the satellite sent into a predetermined orbit by the rocket's final stage.
7. The method for optimizing the attitude angle of the rocket's final stage off-orbit according to claim 1, characterized in that: The step of determining the optimal attitude angle from candidate attitude angles that meet preset measurement and control conditions and far-field safety conditions comprises: Obtaining a historical attitude angle before the rocket's final stage leaves orbit; the historical attitude angle uses the orbital coordinate system as a reference coordinate system; Calculating an angle difference between the historical attitude angle and a preferred attitude angle, wherein the preferred attitude angle is a candidate attitude angle that satisfies a preset measurement and control condition and a far-field safety condition; The preferred attitude angle corresponding to the minimum angle difference is determined as the optimal attitude angle.
8. A device for optimizing the attitude angle of a rocket's final stage off-orbit, characterized in that: The device comprises: An acquisition module is used to acquire a first position and a first velocity vector in a launch coordinate system when the rocket's final stage leaves the orbit, as well as a target off-orbit duration of the rocket's final stage; the launch coordinate system is a coordinate system established with the launch point of the rocket's final stage leaving the orbit as the origin and the launch direction as the horizontal axis direction; A prediction module, used for predicting that the rocket final stage deorbits at the first position with the first velocity vector and different target attitude angles, and predicting the second position and the second velocity vector of the rocket final stage in the earth-fixed coordinate system when the deorbit duration reaches the target deorbit duration; the target attitude angle uses the orbital coordinate system as a reference coordinate system and includes a target yaw angle, a target roll angle and a target pitch angle; the orbital coordinate system is a coordinate system established with the center of mass of the rocket final stage as the origin and the direction of the first velocity vector as the horizontal axis; the earth-fixed coordinate system is a coordinate system established with the center of mass of the earth as the origin; A first determination module is used to determine the perigee height of the first first circle after the rocket's final stage leaves the orbit according to the second position corresponding to each target attitude angle and the corresponding second velocity vector; A second determination module is used to determine the difference between the perigee height of each of the first first laps and a preset height threshold, and determine the candidate attitude angle by taking the target attitude angle corresponding to the difference being less than or equal to the preset difference threshold; The third determination module is used to determine the optimal attitude angle from the candidate attitude angles that meet the preset measurement and control conditions and far-field safety conditions.
9. An electronic device, characterized in that: include: A memory and a processor, wherein the memory and the processor are communicatively connected to each other, the memory stores computer instructions, and the processor executes the method according to any one of claims 1 to 7 by executing the computer instructions.
10. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores computer instructions, and the computer instructions are used to cause the computer to execute the method according to any one of claims 1 to 7.
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