CN113777955A - Dual-spacecraft tracking pointing device and method - Google Patents

Dual-spacecraft tracking pointing device and method Download PDF

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CN113777955A
CN113777955A CN202111018412.4A CN202111018412A CN113777955A CN 113777955 A CN113777955 A CN 113777955A CN 202111018412 A CN202111018412 A CN 202111018412A CN 113777955 A CN113777955 A CN 113777955A
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guide rail
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CN113777955B (en
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马广程
单钰
夏红伟
李莉
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Harbin Institute of Technology
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Abstract

The invention provides a double-spacecraft tracking and pointing device and a method, and belongs to the technical field of aircraft ground simulation. The accompanying spacecraft simulation device is arranged on a three-axis air bearing table, a large-diameter arc-shaped guide rail is overlapped with the central axis of the three-axis air bearing table, a follow-up platform is arranged on the large-diameter arc-shaped guide rail and can do arc motion, an upright post is arranged on the follow-up platform and can move on the follow-up platform along the radial direction of the large-diameter arc-shaped guide rail, a reference spacecraft simulation device is arranged on a linear guide rail on the upright post, the reference spacecraft simulation device can slide up and down on the side surface of the upright post and rotate around the axis of the reference spacecraft simulation device, and a pointing evaluation device is fixedly connected to the bottom end of the upright post and used for receiving a tracking pointing signal pointing from the accompanying spacecraft simulation device. The full-physical simulation device disclosed by the invention is more suitable for a real space microgravity environment, is high in simulation precision, simple in structure and easy to maintain, and can visually display the relative motion relationship of the two spacecrafts.

Description

Dual-spacecraft tracking pointing device and method
Technical Field
The invention relates to a double-spacecraft tracking and pointing device and a method, belonging to the technical field of aircraft ground simulation.
Background
At present, large ground test equipment for various spacecraft in China is few, and a verifiable space test is single. At present, few test devices aiming at tracking and pointing of double spacecrafts in space are used in China. Therefore, the method for verifying the tracking and pointing of the dual spacecrafts in the space with high precision and high reliability has great practical significance for the sustainable development of the aerospace industry in China.
The utility model of Chinese patent No. CN202010826219.2 entitled "a semi-physical satellite simulation system and simulation method" discloses a device and a method, which are mainly used for solving the problem of poor applicability of the satellite simulation system designed in the prior art, and in the scheme provided by the application embodiment, a 1553B interface simulation module and a universal 1553 simulation device are arranged between a satellite simulation computer and a satellite single machine, so that the mutual replacement between at least one satellite single machine software simulation module in the satellite simulation module and the satellite single machine (working under the cooperation of the 1553B interface simulation module and the 1553 simulation device) can be conveniently completed; when the simulation system is accessed to different satellite single machines, the configuration file is generally only required to be modified, and software and hardware in the semi-physical simulation system are not required to be modified, so that the problems that the configuration of the simulation system is not flexible and the openness of the system is insufficient are avoided, and the applicability of the simulation system is further improved.
The utility model discloses a device and method of chinese patent No. CN202010589212.3 name "a universalization satellite simulation test system", provides a universalization satellite simulation test system, including main controller board, bus mother board, first intelligent interface board able to programme, second intelligent interface board able to programme, third intelligent interface board able to programme and upper supervisory control machine, main controller board pass through the ethernet mouth with upper supervisory control machine connects, main controller board pass through the bus mother board respectively with first intelligent interface board able to programme, second intelligent interface board able to programme, third intelligent interface board able to programme is connected. The beneficial effects of the invention are: the universality of the satellite simulation test system is improved.
The utility model discloses a device and method of chinese patent number CN201410497678.5 name "a satellite simulation system and method", provides a satellite simulation system and method, relates to the satellite simulation field, and this system flexible operation, be convenient for improve, easily optimize, and this system includes: the system comprises a simulation control subsystem, an environment simulation subsystem, a view orbit simulation subsystem and a satellite simulation subsystem, wherein the simulation control subsystem is respectively connected with the environment simulation subsystem, the view orbit simulation subsystem and the satellite simulation subsystem, and the environment simulation subsystem is connected with the view orbit simulation subsystem.
Although the method in patent No. CN202010826219.2 improves the applicability of the simulation system, and can change the load to achieve different task requirements, the semi-physical simulation method still has the problem of lower accuracy than the full-physical simulation method. Meanwhile, the method cannot realize the simulation of high-precision tracking pointing of the space dual-spacecraft.
The method in patent number CN202010589212.3 adopts pure digital simulation, and programs the programmable intelligent board card according to different models, and builds different simulation conditions, and the accuracy of digital simulation is based on the accuracy of the model, and is not as accurate as full physical simulation.
The method in patent No. CN201410497678.5 adopts distributed simulation, and each sub-module can be configured according to the function of the simulation system, thereby improving the resource utilization rate. The method of the device still belongs to the category of semi-physical simulation, the precision is lower than that of full physical simulation, and the tracking direction of the double spacecrafts cannot be simulated.
The invention provides a simulation device and a simulation method capable of realizing high-precision tracking and pointing of space double spacecrafts aiming at the ground simulation problem of space motion of the spacecrafts. The device method is based on a full physical simulation platform, can realize tracking pointing simulation of space double spacecrafts, and has the characteristics of high precision and flexible satellite working condition configuration.
Disclosure of Invention
The present invention is directed to solving the above problems in the prior art, and further to providing a dual-spacecraft tracking pointing device and method.
The purpose of the invention is realized by the following technical scheme:
a dual-spacecraft tracking pointing device, comprising: the device comprises a base, a large-diameter arc-shaped guide rail, a structural supporting device, a three-axis air bearing table, an accompanying spacecraft simulation device, a pointing evaluation device, a reference spacecraft simulation device, an upright post pulley and a follow-up platform;
the device comprises a following spacecraft simulation device, a structure supporting device, a base, a large-diameter arc-shaped guide rail, a follow-up platform, a stand column pulley, a straight guide rail, a reference spacecraft simulation device and a pointing evaluation device, wherein the following spacecraft simulation device is arranged on a three-axis air bearing table, the three-axis air bearing table is arranged on the structure supporting device, the base is arranged below the structure supporting device, the large-diameter arc-shaped guide rail is laid on the ground and is overlapped with the central axis of the three-axis air bearing table, the follow-up platform is arranged on the large-diameter arc-shaped guide rail and can do arc motion on the large-diameter arc-shaped guide rail, the stand column stands on the follow-up platform, the stand column pulley is arranged at the bottom of the stand column and can move on the follow-up platform along the radial direction of the large-diameter arc-shaped guide rail, the reference spacecraft simulation device is arranged on the straight guide rail, the reference spacecraft simulation device can slide up and down on the side surface of the stand column and rotate around the axis of the stand column, and the pointing evaluation device is fixedly connected to the bottom end of the stand column and is used for receiving a pointing tracking pointing direction signal from the following spacecraft simulation device.
The analysis method for motion simulation of the dual-spacecraft tracking pointing device comprises the steps of firstly determining the relationship between an orbit coordinate system and a geocentric inertial coordinate system, then establishing a relative motion equation, and solving the relative motion equation to obtain the relative motion of the accompanying spacecraft and the reference spacecraft, wherein the relative motion can be decomposed into two mutually independent motions in an orbit plane (xy plane) and in a direction perpendicular to the orbit plane (z direction).
The invention has the beneficial effects that:
1. the double-spacecraft tracking and pointing device is a full-physical simulation device, is more suitable for a real-space microgravity environment, and has higher simulation precision than a semi-physical and digital software simulation device.
2. Compared with a single-shaft air bearing table, the three-shaft air bearing table has more degrees of freedom, and can improve the simulation precision.
3. The double-spacecraft tracking and pointing device is characterized in that a large-diameter arc-shaped guide rail is laid around the triaxial air bearing table, the device is simple in structure and easy to maintain, and the relative motion relation of the double-spacecraft can be visually displayed.
4. According to the invention, the stand column of the double-spacecraft tracking and pointing device moves on the follow-up platform along the radial direction of the guide rail, so that the fine adjustment of the orbit radius of the reference spacecraft can be realized, and the simulation precision is improved.
Drawings
Fig. 1 is a schematic structural diagram of a dual-spacecraft tracking pointing device according to the present invention.
Fig. 2 is an exploded schematic view of the relative motion of the dual-spacecraft tracking pointing device of the present invention.
FIG. 3 is a schematic diagram of an elliptical trajectory simulation of the dual-spacecraft tracking pointing device of the present invention.
In the figure, reference numerals, 1 is a base, 2 is a large-diameter arc-shaped guide rail, 3 is a structural support device, 4 is a triaxial air bearing table, 5 is a companion spacecraft simulation device, 6 is a pointing evaluation device, 7 is a reference spacecraft simulation device, 8 is an upright, 9 is an upright pulley, and 10 is a follow-up platform.
Detailed Description
The invention will be described in further detail below with reference to the accompanying drawings: the present embodiment is implemented on the premise of the technical solution of the present invention, and a detailed implementation is given, but the scope of the present invention is not limited to the following embodiments.
As shown in fig. 1 to 3, the dual-spacecraft tracking pointing device and method according to the present embodiment includes:
example 1
As shown in fig. 1, the present embodiment is composed of a base 1, a large-diameter arc-shaped guide rail 2, a structural support device 3, a triaxial air bearing table 4, a spacecraft simulation device 5, a pointing evaluation device 6, a reference spacecraft simulation device 7, a column 8, a column pulley 9, and a follow-up platform 10. With the spacecraft simulation apparatus 5 mounted on the three-axis air bearing table 4, the three-axis air bearing table 4 is mounted on the structural support apparatus 3. A base 1 is mounted below the structural support means 3. The large-diameter arc-shaped guide rail 2 is laid on the ground and is superposed with the central axis of the triaxial air bearing table 4. The follow-up platform 10 is arranged on the large-diameter arc-shaped guide rail 2 and can do arc-shaped motion on the large-diameter arc-shaped guide rail 2. The upright post 8 is erected on a follow-up platform 10, an upright post pulley 9 is arranged at the bottom of the upright post, the upright post can move on the follow-up platform 10 along the radial direction of the large-diameter arc-shaped guide rail 2, a linear guide rail is arranged on the upright post, and the reference spacecraft simulation device 7 is arranged on the linear guide rail, can slide up and down on the side surface of the upright post 8 and rotate around the axis of the upright post to simulate the position of the reference spacecraft. The pointing direction evaluation device 6 is fixedly connected to the bottom end of the upright post 8 and is used for receiving a tracking pointing signal from the accompanying spacecraft simulation device 5, and the tracking pointing signal can rotate in two dimensions.
At the beginning of the tracking pointing process, the accompanying spacecraft simulation device 5 is placed on the three-axis air bearing table 4 with an air film between the three-axis air bearing table 4 and the structural support device 3 to simulate the microgravity environment of the space. The upright column 8 and the follow-up platform 10 do rotary motion around the central axis of the three-axis air bearing table 4 on the large-diameter arc-shaped guide rail 2, and the reference spacecraft simulation device 7 slides on a linear guide rail on the upright column 8 to respectively simulate the decomposition motion of the relative motion of the dual-spacecraft in two-dimensional rotation (xy plane) and one-dimensional translation (z direction), and meanwhile, the upright column 8 can move on the follow-up platform 10 along the guide rail in the radial direction according to simulation requirements.
The analysis method for motion simulation of the dual-spacecraft tracking pointing device comprises the following steps:
the method comprises the following steps: orbital coordinate system s-xyz and earth-centered inertial coordinate system OEXYZ the following relationship exists:
as shown in fig. 2, s denotes a reference spacecraft, c denotes an accompanying spacecraft, and the reference spacecraft makes a counterclockwise motion on a near-circular orbit. The relative motion coordinate system is selected as an orbit coordinate system s-xyz of the accompanying spacecraft c, and the origin of the relative motion coordinate system is fixedly connected with the mass center of the reference spacecraft s and moves anticlockwise on the near-circular orbit along with the reference spacecraft. X-axis of relative motion coordinate system and geocentric vector r of reference spacecraftsAnd (3) coinciding, namely pointing to s from the geocenter, pointing to the direction of motion, wherein the y axis is perpendicular to the x axis in the orbital plane of the reference spacecraft, and pointing to the direction of motion, and the z axis can be determined by a right-hand rule, namely the z axis is consistent with the direction of the moment vector of the orbital momentum of the reference spacecraft.
Step two: establishing a relative equation of motion
Assuming that the orbit radius of the reference spacecraft is b, r iss=[b 0 0]TThe radius of the track can be adjusted by moving the upright post on the follow-up platform along the radial direction, and tracks with different eccentricities can be simulated by matching with the arc motion of the follow-up platform on the guide rail, and the simulation schematic diagram is shown in fig. 3.
Let the geocentric position vector associated with the space as rcIts position vector p relative to the reference spacecraft is then
Figure BDA0003240770960000061
In the geocentric inertial frame, the kinetic equations for the reference spacecraft and the companion spacecraft are as follows
Figure BDA0003240770960000062
Figure BDA0003240770960000063
In the above formula fsAnd fcThe acceleration vectors of the reference spacecraft and the accompanying spacecraft, respectively, are formed by the resultant of forces other than the earth's central gravity, i.e. the acceleration vectors of thrust and perturbation forces.
Wherein t is time; mu is gravitational constant, mu is 3.986005 × 1014m3/s2
The difference between the absolute accelerations of the accompanying spacecraft and the reference spacecraft can be obtained from (1) to (2)
Figure BDA0003240770960000064
Is composed of
Figure BDA0003240770960000065
The above formula can be further expressed as the following equivalent relation
Figure BDA0003240770960000066
Establishing a relative motion equation of the accompanying spacecraft and the reference spacecraft in a moving coordinate system s-xyz as follows
Figure BDA0003240770960000067
In the above formula
Figure BDA0003240770960000068
And v is the relative acceleration vector and the relative velocity vector of the accompanying spacecraft and the reference spacecraft in the moving coordinate system respectively, then
Figure BDA0003240770960000069
Figure BDA0003240770960000071
In the above formula
Figure BDA0003240770960000072
n is the angular acceleration vector and angular velocity vector of the rotation of the moving coordinate system respectively, and the average moving angular velocity of the reference spacecraft is
Figure BDA0003240770960000073
The following approximate formula can be obtained
Figure BDA0003240770960000074
By substituting the formula (5) and the formulae (7) to (9) for the formula (6)
Figure BDA0003240770960000075
Wherein f isx,fy,fzIs the component of Δ f on three coordinate axes;
the above formula can be further simplified for the case of relative motion at close range (about several meters to several tens of kilometers) with the spacecraft and the reference spacecraft
Figure BDA0003240770960000076
Step three: solution of equations of relative motion
The second expression in the expression (11) is integrated to obtain
Figure BDA0003240770960000077
Wherein x is0,y0,z0Is the initial value of the corresponding variable,
the formula (12) is substituted for the first formula in the formula (11), and the integral results
Figure BDA0003240770960000078
The formula (13) is substituted for the formula (12), and the integral
Figure BDA0003240770960000081
Integration of the third expression of (11) can be obtained
Figure BDA0003240770960000082
When the force on the right side of equation (11) is 0, the equation becomes a homogeneous differential equation. The equation is integrated for the first and second times to obtain a solution of free motion
Figure BDA0003240770960000083
From equation (16), it can be seen that the relative motion of the accompanying spacecraft and the reference spacecraft can be decomposed into two mutually independent motions in the orbital plane (xy-plane) and perpendicular to the orbital plane (z-direction).
The device realizes the two independent motions through the following design. The stand column and the follow-up platform move in a near-circular manner on the large-diameter arc-shaped guide rail around the central axis of the three-axis air bearing platform, the center of mass of the accompanying spacecraft simulation device is on the central axis of the three-axis air bearing platform, namely the reference spacecraft simulation device moves in a near-circular manner around the accompanying spacecraft simulation device, and therefore the simulation of the relative motion in the orbital plane is completed. The reference spacecraft simulation device moves up and down on the upright post to realize the simulation of the partial motion of the relative motion in the direction vertical to the plane of the orbit. The radial motion of the upright post on the follow-up platform along the arc guide rail can realize the adjustment of the orbit radius of the reference spacecraft. The position simulation of the reference spacecraft in the space is realized by the radial movement of the upright post along the arc-shaped guide rail, the sliding and rotation of the reference spacecraft simulation device on the upright post and the rotation of the upright post around the three-axis air bearing table. The two-dimensional rotation of the efficiency evaluation device on the upright column realizes the simulation of the attitude of the reference spacecraft in space.
The accompanying spacecraft simulation device adjusts the self posture and sends out a tracking pointing laser signal after the reference spacecraft simulation device reaches a preset position. The evaluation device at the bottom end of the stand column is provided with a device for receiving the evaluation signal, the efficiency evaluation device shoots the laser target position through a CCD camera on the efficiency evaluation device, and the efficiency evaluation device is connected with an upper computer in a bus mode to extract the laser position of the shot original image to obtain evaluation information.
The measurement method of the double-spacecraft tracking pointing device comprises the following steps:
1. and opening the air valve for ventilation, so that an air film is formed between the triaxial air bearing table and the structure supporting device.
2. And (5) operating a leveling program, and adjusting the level of the three-axis air bearing table.
3. And the upper computer imports track parameters and tracking pointing parameter files and converts the parameter files into data for driving the motion simulator.
4. The servo platform, the upright post and the reference spacecraft simulation device move according to the driving data calculated by the upper computer. The stand column moves along the radial direction of the arc-shaped guide rail, the reference spacecraft simulation device slides and rotates on the stand column, the stand column and the follow-up platform rotate around the central axis of the triaxial air bearing table, and the reference spacecraft reaches a preset position after the movement is finished.
5. The driving data traction efficiency evaluation device calculated by the upper computer rotates in two dimensions, and the posture of the driving data traction efficiency evaluation device is adjusted to simulate the posture of a reference spacecraft in space, so that the relative position relation between the driving data traction efficiency evaluation device and a satellite spacecraft simulation device is ensured.
6. And adjusting the posture of the spacecraft along with the spacecraft to enable the tracking pointing signal emitting device to point to the efficiency evaluation device and send out a tracking pointing signal.
7. The CCD camera arranged on the efficiency evaluation device scans the laser signal to obtain the tracking pointing position data and transmits the data back to the upper computer for the subsequent efficiency evaluation.
8. And after receiving the tracking pointing signal, the efficiency evaluation device records the speed and attitude information of the air bearing table, the position of the reference spacecraft simulation device and the attitude information of the efficiency evaluation device, compares the information with the relative position information calculated by the upper computer, and evaluates the tracking pointing performance.
The above description is only a preferred embodiment of the present invention, and these embodiments are based on different implementations of the present invention, and the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention. Therefore, the protection scope of the present invention shall be subject to the protection scope of the claims.

Claims (5)

1. A dual-spacecraft tracking pointing device, comprising: the device comprises a base (1), a large-diameter arc-shaped guide rail (2), a structural support device (3), a three-axis air bearing platform (4), an accompanying spacecraft simulation device (5), a pointing evaluation device (6), a reference spacecraft simulation device (7), a stand column (8), a stand column pulley (9) and a follow-up platform (10);
wherein, the accompanying spacecraft simulation device (5) is arranged on a three-axis air bearing platform (4), the three-axis air bearing platform (4) is arranged on a structure supporting device (3), a base (1) is arranged below the structure supporting device (3), a large-diameter arc-shaped guide rail (2) is laid on the ground and is superposed with the central axis of the three-axis air bearing platform (4), a follow-up platform (10) is arranged on the large-diameter arc-shaped guide rail (2) and can do arc-shaped motion on the large-diameter arc-shaped guide rail (2), an upright post (8) is arranged on the follow-up platform (10), an upright post pulley (9) is arranged at the bottom of the upright post and can move on the follow-up platform (10) along the radial direction of the large-diameter arc-shaped guide rail (2), a linear guide rail is arranged on the upright post (8), a spacecraft reference simulation device (7) is arranged on the linear guide rail, and the reference spacecraft simulation device (7) can slide up and down on the side surface of the upright post (8) and rotate around the axis of the aircraft, the pointing direction evaluation device (6) is fixedly connected to the bottom end of the upright post (8) and is used for receiving a tracking pointing signal from the accompanying spacecraft simulation device (5).
2. Dual spacecraft tracking pointing device according to claim 1, wherein there is a gas film between the three axis air bearing table (4) and the structural support means (3).
3. The dual spacecraft tracking pointing device according to claim 1, characterized in that the mast (8) and the follower platform (10) are in a revolving motion around the central axis of the three-axis air bearing table (4) on a large diameter arc-shaped rail (2), while the reference spacecraft simulation device (7) slides on a linear rail on the mast (8).
4. Dual spacecraft tracking pointing device according to claim 1, characterized in that the uprights (8) can be moved radially along rails on the follower platforms (10) according to simulation needs.
5. Method of dual-spacecraft tracking pointing device according to claim 1, 2, 3 or 4, characterized in that it comprises the following steps:
the method comprises the following steps: relationship between orbit coordinate system and earth center inertial coordinate system
Obtaining an orbit coordinate system s-xyz and a geocentric inertial coordinate system OE-XYZ existence relationship: s denotes a reference spacecraft, c denotes an accompanying spacecraft, and the reference spacecraft makes counterclockwise motion on a near-circular orbit; the relative motion coordinate system is selected as an orbit coordinate system s-xyz of the accompanying spacecraft c, the origin of the relative motion coordinate system is fixedly connected with the mass center of the reference spacecraft s and moves anticlockwise on the near-circular orbit along with the reference spacecraft; x-axis of relative motion coordinate system and geocentric vector r of reference spacecraftsThe earth center points to s, the y axis is perpendicular to the x axis in the orbit plane of the reference spacecraft and points to the motion direction, and the z axis can be determined by a right hand rule, namely the z axis is consistent with the direction of the orbit moment vector of the reference spacecraft;
step two: establishing a relative equation of motion
Assuming that the orbit radius of the reference spacecraft is b, r iss=[b 0 0]TThe radius of the track can be adjusted by moving the upright post on the follow-up platform along the radial direction, and tracks with different eccentricities can be simulated by matching the arc motion of the follow-up platform on the guide rail;
let the geocentric position vector associated with the space as rcIts position vector p relative to the reference spacecraft is then
Figure FDA0003240770950000021
In the geocentric inertial frame, the kinetic equations for the reference spacecraft and the companion spacecraft are as follows
Figure FDA0003240770950000022
Figure FDA0003240770950000023
In the above formula fsAnd fcAcceleration vectors of the reference spacecraft and the accompanying spacecraft, respectively, which are formed by resultant forces of other acting forces than the earth's central gravity, i.e., acceleration vectors of thrust and thrust;
wherein t is time; mu is gravitational constant, mu is 3.986005 × 1014m3/s2
The difference between the absolute accelerations of the accompanying spacecraft and the reference spacecraft can be obtained from (1) to (2)
Figure FDA0003240770950000024
Is composed of
Figure FDA0003240770950000031
The above formula can be further expressed as the following equivalent relation
Figure FDA0003240770950000032
Wherein Δ f ═ fc-fsEstablishing a relative motion equation of the accompanying spacecraft and the reference spacecraft in a moving coordinate system s-xyz as follows
Figure FDA0003240770950000033
In the above formula
Figure FDA0003240770950000034
And v is the relative acceleration vector and the relative velocity vector of the accompanying spacecraft and the reference spacecraft in the moving coordinate system respectively, then
Figure FDA0003240770950000035
Figure FDA0003240770950000036
In the above formula
Figure FDA00032407709500000310
n is the angular acceleration vector and angular velocity vector of the rotation of the moving coordinate system respectively, and the average moving angular velocity of the reference spacecraft is
Figure FDA0003240770950000037
The following approximate formula can be obtained
Figure FDA0003240770950000038
By substituting the formula (5) and the formulae (7) to (9) for the formula (6)
Figure FDA0003240770950000039
Wherein f isx,fy,fzIs the component of Δ f on three coordinate axes;
for the case of relative movements accompanying the close range of the spacecraft and the reference spacecraft, the above formula can be further simplified
Figure FDA0003240770950000041
Step three: solution of equations of relative motion
The second expression in the expression (11) is integrated to obtain
Figure FDA0003240770950000042
Wherein x is0,y0,z0Substituting the formula (12) into the first formula in the formula (11) for the initial value of the corresponding variable, and obtaining the integral
Figure FDA0003240770950000043
The formula (13) is substituted for the formula (12), and the integral
Figure FDA0003240770950000044
Integration of the third expression of (11) can be obtained
Figure FDA0003240770950000045
When the force on the right side of equation (11) is 0, the equation becomes a homogeneous differential equation; the equation is integrated for the first and second times to obtain a solution of free motion
Figure FDA0003240770950000051
As can be seen from equation (16), the relative motion of the accompanying spacecraft and the reference spacecraft can be decomposed into two mutually independent motions in the orbital plane, i.e., the xy-plane, and in the direction perpendicular to the orbital plane, i.e., the z-direction.
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